bluuebunny/crossref_metadata_2025 huggingface dataset:
Partition theory and thermodynamics of multi-dimensional oscillator assemblies
Authors: Nanda V. S.
Published: November 2008

The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the ‘Zustandsumme’ of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck (1) where the Hardy Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.
Random motion on a finite Abelian group
Authors: Good I. J.
Published: November 2008

1. The present paper is concerned with a class of finite Markoff chains. It is thought that there are some new points of interest here, although the literature dealing with finite Markoff chains is large. (For bibliographies see (2) and (7).)
The theory of queues with a single server
Authors: Lindley D. V.
Published: November 2008

1. The study of queues has been of interest to mathematicians and engineers for the past forty years, and a very extensive literature on the subject exists. The applications of the theory are many and varied, from Erlang's original work on telephone engineering, to present-day studies in the design of airports. The number of papers which deal with the theoretical side of the subject is, however, small, and it therefore seems desirable to attempt to develop a general theory which will cover the diverse practical requirements, so that the unity in the applications will become apparent. This paper gives such a development in the case where there is a single queue and a single server attending to it; the theory of multiple queues or many servers seems, except under simplifying assumptions which do not always correspond to reality, to be a problem of considerable difficulty. Previous work on the subject has mainly been confined to a special case where the customers arrive at random, as, for example, in a recent paper by Kendall(5);* the present theory makes no such assumption and allows the customers to join the queue in other ways, though the theory simplifies when the more restrictive assumption is made. The centre of interest in the present theory is the waiting times of the customers.
The spectra of turbulence in a compressible fluid; eddy turbulence and random noise
Authors: Moyal J. E.
Published: November 2008

The state of a real fluid is completely specified by its velocity, density, pressure and temperature fields. When the fluid is in turbulent flow, all these quantities fluctuate in a disordered manner. The method of space Fourier spectra is used to show that these field variables separate into two physically distinct groups, one corresponding to fluctuating acoustical waves, orrandom noise, and the other to fluctuating vorticity, oreddy turbulence. The corresponding decomposition of the spectral and correlation tensors in a homogeneous field of turbulence is given. The noise Fourier components are shown to be coupled to the eddy Fourier components only through the non-linear inertia terms in the dynamical equations of the fluid; whereas the former propagate as acoustical waves, the wave character of the latter is due entirely to the mean motion of the fluid. The measurement of the noise component, its attenuation through absorption by walls and its effects on the eddy component are discussed. Finally, the dynamical equations for the eddy component of the velocity spectral tensor in a homogeneous field of turbulence are compared with the corresponding equations for an incompressible fluid.
On an extension of Riemann's method of integration, with applications to one-dimensional gas dynamics
Authors: Ludford Geoffrey S. S.
Published: November 2008

The object of this paper is to investigate the influence of the initial values on the behaviour of the one-dimensional unsteady isentropic motion of a perfect gas, and in particular the occurrence of singularities. For this purpose it is expedient to develop an unfolding procedure in the theory of partial differential equations of hyperbolic type. This not only resolves a difficulty in the classical treatment of the initial value problem associated with this type of equation, but also simplifies the presentation.

The geometrical properties of the singularities (branch and limit lines, edges of regression, etc.) occurring in two-dimensional steady flow have been well discussed already, and their counterparts in one-dimensional unsteady motion have been indicated by Stocker and Meyer(5). However, here a method for finding the explicit analytic connexion between the occurrence of such singularities and the prescribed initial conditions is developed.

Assuming quite general initial conditions, two examples are considered. In the first, a conjecture of Riemann is verified; in the second a special type of periodic solution is discussed, and it is shown that this solution cannot be continued to all values of t > 0. Finally, this latter result is shown to be true for arbitrary periodic initial conditions.
Mass transport in gravity waves
Authors: Ursell F.
Published: November 2008

A proof is given of the existence of a slow drift associated with the passage of gravity waves over the surface of a frictionless fluid when the wave motion is irrotational. The drift is in the direction of propagation and is greatest near the surface, decreasing steadily towards the bottom, where it may be negative if the fluid is of finite depth. The particle displacement due to a solitary wave has similar properties. The proof confirms the approximate calculations of Stokes (6) and extends the proof of Rayleigh applicable only to deep water.
The number of non-homogeneous lattice points in n-dimensional point sets
Authors: Sawyer D. B.
Published: November 2008

Let R be a set of points in n-dimensional Euclidean space, and let Δ′(R) denote the lower bound of the determinants of non-homogeneous lattices which have no point in R. For Δ′(R) to be non-zero it is necessary, as Macbeath has shown (2), that R should have infinite volume.
Stability of plane parallel flows of electrically conducting fluids
Authors: Michael D. H.
Published: November 2008

The ordinary theory of stability of plane parallel flows is considerably simplified by a result due to Squire (2) which says that if a velocity profile becomes unstable to a small three-dimensional disturbance at a given Reynolds number, then it will become unstable to a small two-dimensional disturbance at a lower Reynolds number. This result enables us to restrict investigation of the stability to the cases of two-dimensional disturbances.
A note on Stokes's stream function for motion with a spherical boundary
Authors: Butler S. F. J.
Published: November 2008

The circle theorem of Milne-Thomson(1) connecting the complex potential in a two-dimensional irrotational flow about a circular cylinder with that of the flow when the cylinder is absent has a three-dimensional counterpart in the result due to Weiss (3) for the perturbed velocity potential in an unlimited irrotational flow when the rigid spherical boundary r = a is inserted.
The serial test for sampling numbers and other tests for randomness
Authors: Good I. J.
Published: November 2008

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).
Note on hydrodynamics
Authors: Darwin Charles
Published: November 2008

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.
A theorem of Minkowski on the product of two linear forms
Authors: Chalk J. H. H.
Published: November 2008

1. Let Λ denote any plane lattice of determinant Δ ╪ 0. Then it is well known that the region

contains a pair of generating points of Λ; and that, unless the lattice Λ is of the special form

where both α│β, γ│δ are rational, there are an infinity of such pairs in the interior of the region. Minkowski (4) obtained this result by considering the ‘tangent’ parallelogram Пλ,

inscribed in the region (1). With simple geometrical arguments, he showed that

(i) Пλ always contains a point of Λ, other than 0,

(ii) if Пλ contains two primitive points of Λ then these generate Λ, except possibly when Λ has points, other than 0, on both coordinate axes.
On the oscillations of a bowed string
Authors: Friedlander F. G.
Published: November 2008

The small transverse oscillations of a string excited by bowing are treated by neglecting all external forces except that due to the bow, taking the ends of the string as fixed, and replacing the bow by a concentrated force acting transversely at the ‘bowing point’, whose magnitude depends on the velocity, v, of the string at this point. Non-linear recurrence equations for v are obtained, from which some of the principal features of the motion can be inferred, and some qualitative conditions about the dependence of the transverse force of the bow on v can be drawn. A detailed examination of the case when the string is bowed at the mid-point shows that, in order to account for the stability of the periodic motion excited, it appears to be necessary to allow for the effect of dissipative forces, such as air friction; to a first approximation this can be done by postulating a small concentrated dissipative force at the bowing point hi addition to the frictional force exerted by the bow. The theory predicts that for small bowing speeds a ‘noise’ is produced instead of a note; this phenomenon does not appear to have been noted in the literature, but can easily be verified in practice.
Commutators in associative rings
Authors: Drazin M. P. , Gruenberg K. W.
Published: November 2008

Let R be an arbitrary associative ring, and X a set of generators of R. The elements of X generate a Lie ring, [X], say, with respect to the addition and subtraction in R, and the multiplication [a, b] = ab − ba. In this note we shall be concerned with the following question: if [X] is given to be nilpotent as a Lie ring, what does this imply about R?
On counters with random dead time. I
Authors: Hammersley J. M.
Published: November 2008

The following work was undertaken in connexion with a device for counting blood cells electronically, which has been developed in the Clinical Pathology Department of the Radcliffe Infirmary with financial assistance from the Medical Research Council and the Nuffield Foundation. Although the results will apply to other types of counter, it will help to fix the ideas if we consider the problem for this specific device. A large number of blood cells, contained in a shallow chamber, are scanned by a photoelectric cell. The depth of the chamber and the concentration of blood cells in solution therein allow blood cells (supposed distributed at random throughout the chamber) to overlap when viewed from above by the scanner. The field of view of the scanner at any instant is somewhat larger than the size of a blood cell, but is, nevertheless, of much the same order of magnitude. With passage of time the chamber moves underneath the photocell so that the field of view traces out a long narrow path not crossing or overlapping itself and only embracing a portion of. the whole chamber. The blood cells have no motion relative to the chamber. As each blood cell comes under the photocell it produces an electrical impulse, whose duration depends upon the size and shape and orientation of the blood cell. These impulses go to a counter, which counts them except that it will not count any impulse which is overlapped by a previous impulse. The problem is to determine the number of blood cells in the chamber from a knowledge of the recorded count and the distribution of the lengths of individual impulses. A further complication arises because it is inadvisable to count two impulses separated only by a very short interval of time, and therefore the counter itself generates some self-paralysing impulses additional to the blood-cell impulses and in a manner which is correlated with them. Until § 3, however, we shall neglect paralysis, because a proper understanding of its effects cannot be reached unless we first know how the system behaves in the absence of paralysis.
Convergence in the intermediate representation
Authors: Hamilton J.
Published: November 2008

It is shown that for the quantized φ3 field the series expansions for intermediate representation field operators are convergent for summation over all graphs except those containing self-energy and vertex parts. The method used does not permit an investigation of the self-energy and vertex parts.
Further solutions of the Falkner-Skan equation
Authors: Stewartson K.
Published: November 2008

In an attempt to understand the mechanism governing separation the possibility of there being further solutions to the Falkner-Skan equation f′′′ + ff″ + β(1 − f′2) = 0, in addition to those found by Hartree, is investigated.

It is found that when β < − 0·1988 there are no solutions satisfying |f′| ⩽ 1 for all η, and that if 0>β> − 0·1988 there are two acceptable solutions, one with f″(0) < 0. The new ones are computed to three places of decimals for various values of β and tabulated. In addition, it is shown that if − 0·5 < β < 0 there is a family of solutions corresponding to boundary layers bounded on one side by free streamlines. These are also computed and graphs of f′(0) and δ1 are displayed. The impact of these solutions on the theory of separation is discussed.
An alternative method for linear programming
Authors: Beale E. M. L.
Published: November 2008

The method of leading variables is presented as a new method of solving Linear Programming, and is compared with the Simplex and the Dual Simplex methods.
Bounds for the least solutions of homogeneous quadratic equations
Authors: Cassels J. W. S.
Published: November 2008

In this note I obtain bounds for the least integral solutions of the equation

in terms of

For ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1), Mordell (2) and Skolem (3), but these lead only to bad estimates for general ternaries. So far as I know there have not been given estimates for n≽4. Here I generalize Thue's method to prove:

Theorem. Suppose that n ≥ 2 and that f(ɛ) represents zero. Then there is an integral solution of f(a) = 0 with
A use of complex probabilities in the theory of stochastic processes
Authors: Cox D. R.
Published: November 2008

The exponential distribution is very important in the theory of stochastic processes with discrete states in continuous time. A. K. Erlang suggested a method of extending to other distributions methods that apply in the first instance only to exponential distributions. His idea is generalized to cover all distributions with rational Laplace transforms; this involves the formal use of complex transition probabilities. Properties of the method are considered.
The derivation of the equations of motion of an ideal fluid by Hamilton's principle
Authors: Herivel J. W.
Published: November 2008

A new formulation of Hamilton's principle for the case of an ideal fluid is proposed which is claimed to be the uniquely proper form for such a system. The resulting derivation of the equations of motion on varying with respect to the position of the fluid particles is free from the difficulties encountered in previous treatments based on incorrect forms of Hamilton's principle. The treatment also differs from previous treatments in transforming to the fixed (a, b, c) space before carrying out the variation, and in allowing for the equations of mass and entropy conservation by means of undetermined multipliers. The necessity for conservation of entropy, if Hamilton's principle is to be applied, is emphasized. Finally, the method, first used by Eckart, of deriving the equations of motion for an ideal fluid by means of a variational principle of the same form as Hamilton's, but varying with respect to the velocities of the fluid particles, is extended to the general case of rotational motion.
The concept of order
Authors: Bastin E. W. , Kilmister C. W.
Published: November 2008

From the idea of the continuous development of an investigation, as expressed in the use of indefinitely continuing sequences of procedures, the concept of a measurement together with its result is formulated. The resulting theory is initially very general, but it is possible to restrict it in such a way as to make the construction of a consistent world possible. The physical condition providing the restrictions that make this construction possible is found in the use of the general idea of a test-particle in all fundamental investigations, which is shown to be a case of the use of the theory-languages of the previous paper (1). Finally, the theory is applied to the solution of the problem of preferred inertial frames.
Deductions from the perfect cosmological principle
Authors: Dungey J. W.
Published: November 2008

To obtain certain results of their perfect cosmological principle, Bondi and Gold made use of a part of the theory of kinematic relativity. The same results are here deduced directly from the perfect cosmological principle.
Sampling from an isotropic Gaussian process
Authors: Hammersley J. M. , Nelder J. A.
Published: November 2008

Isotropic Gaussian processes, of which we shall give a formal definition presently, arise in various practical problems. The present inquiry arose from the consideration of the variability found in the yields of plots in agricultural field experiments. Samples of such patterns of variability can be obtained from uniformity trials, whereby a piece of land is treated uniformly throughout and the crop is harvested in small units or plots. The results from such trials have been widely used to determine optimum plot sizes for future experiments with the crop concerned, but it has never been clear how valid is the generalization from a uniformity trial to future experiments on other sites. One difficulty arises from the lack of a suitable model to express the variability; another difficulty arises from the formidable analytical problems besetting any attempts to apply deductive reasoning to even simple models. We may mention two approaches that have been made to the uniformity trial problem: Fairfield Smith (3) has supplied an empirical law connecting the variance of contiguous groups of plots with the size of the group, and Quenouille (6) and Whittle (8) have considered the fitting of two-dimensional isotropic Gaussian processes to uniformity trial data. Whittle has pointed out that a satisfactory model may have to incorporate an additional random element at each point, and has outlined the difficulties of estimation which arise when such an element is introduced. The connexion between Fairfield Smith's law and the approach via two-dimensional stochastic processes, if any, seems to be quite unknown.
The electron-phonon interaction, according to the adiabatic approximation
Authors: Ziman J. M.
Published: November 2008

The behaviour of the conduction electrons in a vibrating metallic lattice is treated by the Born-Oppenheimer method. Non-adiabatic processes, due to certain terms usually neglected, are shown to correspond to scattering of electrons by the lattice vibrations, as in Bloch's theory of electrical resistivity, but the matrix elements are not the same as Bloch's except when energy is conserved in the process. This may have important consequences in Fröhlich's theory of superconductivity.
Simultaneous linear differential equations with constant coefficients
Authors: Afriat S. N. , Eggleston H. G.
Published: November 2008

The equations to be considered are of the form

where Xk is to be determined as a function of t, is its ith derivative with respect to t, Fj is a given analytic function of t, a given constant, and i = 0, 1, …, n; j, k = 1,…, m, and they represent a system of simultaneous linear differential equations of order n in m functions Xk (k = 1,…, m).
On clumps formed when convex laminae or bodies are placed at random in two or three dimensions
Authors: Mack C. , Bartlett M. S.
Published: November 2008

A correction is made to a previous paper(2) on clumps formed by random placing of laminae, a useful property of such placings is deduced and formulae are found for clumps formed when convex bodies are randomly placed in space and for clumps formed by the orthogonal projections of such placings.
The image system of a vortex element in a rigid sphere
Authors: Lighthill M. J.
Published: November 2008

It will be shown that a vortex element has the following image system in a rigid sphere immersed in incompressible fluid:

(i) A transverse vortex element at the inverse point, equal in strength to minus the transverse resultant of the given vortex element, divided by the ratio of its distance from the centre to the radius of the sphere.

(ii) A radial vortex element at the inverse point, equal in strength to plus the radial resultant of the given vortex element, divided by the same ratio.

(iii) A uniform line vortex stretching from the inverse point to the centre of the sphere, equal and opposite in total strength to the radial vortex element at the inverse point.
Diffraction by a cylinder of finite length
Authors: Williams W. E. , Lighthill M. J.
Published: November 2008

The diffraction of a plane harmonic sound wave by a hollow circular cylinder of finite length is considered. The problem is treated by using Laplace transforms and is reduced to the solution of two complex integral equations. An approximate solution is obtained for these equations when the product of the wave number (k) and the cylinder length (l) is large. The resonance of the system is considered and an equation derived for the resonant lengths which is then solved approximately for kl large. An explicit expression is obtained for the end correction of an. r–resonant system (i.e. kl ≈ rπ), and also comparison is made with experimental results.
FC-nilpotent and FC-soluble groups
Authors: Duguid A. M. , McLain D. H.
Published: November 2008

Let an element of a group be called an FC element if it has only a finite number of conjugates in the group. Baer(1) and Neumann (8) have discussed groups in which every element is FC, and called them FC-groups. Both Abelian and finite groups are trivially FC-groups; Neumann has studied the properties common to FC-groups and Abelian groups, and Baer the properties common to FC-groups and finite groups. Baer has also shown that, for an arbitrary group G, the set H1 of all FC elements is a characteristic subgroup. Haimo (3) has defined the FC-chain of a group G by

Hi/Hi−1 is the subgroup of all FC elements in G/Hi−1.
A new Monte Carlo technique: antithetic variates
Authors: Hammersley J. M. , Morton K. W.
Published: November 2008

As we have stressed in a previous paper (9), the main concern in Monte Carlo work is to achieve without inordinate labour a respectably small standard error in the final result. Mere replication of the Monte Carlo results is unrewarding; for, to reduce a standard error by a factork, the labour must be increasedk2-fold, and this will be beyond the resources of even electronic computers whenk= 1000, say. The remedy lies in a skilful choice of sampling technique and the substitution of analytical methods for random processes wherever possible. The efficiency of a Monte Carlo process may be taken as inversely proportional to the product of the sampling variance of the final estimate and the amount of labour expended in obtaining this estimate; and it is profitable to allow some increase in the labour if that produces an overwhelming decrease in the variance. For instance, in the last example quoted below (Table 2), we reduce the variance by a factor of four million at the expense of only multiplying the labour sixteenfold, thereby attaining a 250,000-fold gain of efficiency.
On a Procrustean technique for the numerical transformation of slowly convergent sequences and series
Authors: Wynn P.
Published: November 2008

1.The study of the numerical transformation of slowly convergent series and sequences permits of a certain unity of approach, since the partial sums of the former may be regarded as members of a slowly convergent sequence, while the differences of the latter may be treated as terms in a slowly convergent series. In the ensuing discussion, no essential distinction between the two problems is made, and methods devised for one relate with equal facility to the other.
The electron-phonon interaction, according to the adiabatic approximation
Authors: Taylor J. C.
Published: November 2008

The interaction of conduction electrons with a vibrating metallic lattice has been considered by Ziman in terms of non-adiabatic perturbations to the adiabatic approximation of the Born-Oppenheimer theory. This programme is continued to calculate the self-energy of the electrons. When account is taken of the dependence of the energy levels of the adiabatic Hamiltonian itself upon the electron configuration, the final result is the same as the standard one.
The characters of the symmetric inverse semigroup
Authors: Munn W. D.
Published: November 2008

There are two natural analogues of the symmetric group on n symbols in the theory of semigroups, namely, the set of all mappings of a set of n symbols into itself, and the set of all partial transformations of such a set, with the obvious definitions of multiplication. We are concerned here with the latter system. This is an inverse semigroup, and accordingly we call it the ‘symmetric inverse semigroup’. It gives rise to a semisimple algebra over a field of characteristic zero or a prime greater than n, and its matrix representations over such a field are thus completely reducible.
On the symmetries of spherical harmonics
Authors: Altmann S. L.
Published: November 2008

It is often necessary to obtain expansions in spherical harmonics that belong to a given irreducible representation of a symmetry group. This is the case, for instance, when the cellular method is applied to investigate the band structure of a metal, where expansions are required that reproduce the symmetry of the group of the k vector (see Bouckaert, Smoluchowski and Wigner(4); von der Lage and Bethe (9)). Another instance where such expansions are necessary appears when hybrid orbitals are obtained for a central atom in a molecule of given symmetry (Kimball (8)). In this case lower order spherical harmonics are considered and tables for them up to l = 2 (functions s, p and d in real form) are given in the literature (cf. for example Eyring, Walter and Kimball (5)). However, interest has recently arisen in hybrids that include f functions (Shirmazan and Dyatkina (12)) and an extension of these tables appears to be desirable.
A theorem on infinite groups
Authors: Baumslag Gilbert
Published: November 2008

It is well known ((1), pp. 120–1) that if G is a finite p–group then the factor group G/G′ is cyclic if and only if G is itself cyclic. A trivial consequence of this fact is that, in such a case, the factor group G′/G″ is the unit group. (As usual, G(r) denotes the rth derived group of the group G.)

We prove the following theorem.
Percolation processes
Authors: Broadbent S. R. , Hammersley J. M.
Published: November 2008

The paper studies, in a general way, how the random properties of a ‘medium’ influence the percolation of a ‘fluid’ through it. The treatment diifers from conventional diffusion theory, in which it is the random properties of the fluid that matter. Fluid and medium bear general interpretations: for example, solute diffusing through solvent, electrons migrating over an atomic lattice, molecules penetrating a porous solid, disease infecting a community, etc.
Percolation processes
Authors: Hammersley J. M.
Published: November 2008

The previous paper (1) omitted the proof of the theorem on the connective constant of a crystal. This paper supplies the proof. For ease of reference, we first recall the relevant nomenclature and enunciate the theorem; to this extent the present paper is self-contained.
Wave functions for the normal states of Ne+3 and Ne+4
Authors: Froese Charlotte , Hartree D. R.
Published: November 2008

Solutions of Fock's equations for the self-consistent field with exchange have been carried out for Ne+4, using the EDSAC at the Mathematical Laboratory, Cambridge. The initial estimates for the calculation were made by a simplified version of a method previously suggested for interpolating atomic wave functions with respect to atomic number. This gave good estimates for Ne+4, and it is probable that estimates for Ne+3, obtained similarly, are already accurate enough for practical use. Such wave functions for Ne+3 are given. The results have application in astrophysical contexts.
A statistical method for analysing observations of faint radio stars
Authors: Scheuer P. A. G.
Published: November 2008

The observation of faint radio sources is often limited by the angular resolving power of the radio telescope, and not by its sensitivity. Continuous recording of the output from the receiving apparatus then produces a trace which always shows the sum of the effects of a number of sources in the reception pattern, instead of showing sources passing through the reception pattern one at a time. The apparent intensities of sources found on the records are affected by the presence of several other, fainter, sources in the reception pattern, and even the existence of many of the faint sources is doubtful. Counts of numbers of sources at given intensity levels are therefore unreliable.

Any assumed number-intensity law for radio sources can, however, be tested by a method free from the uncertainties due to confusion between adjacent sources, using a direct relation between the number-intensity law and the frequency distribution of deflexions of various sizes on the observed records. The present paper shows the derivation of the relation between M (I) dI, the mean number of radio sources per steradian in the intensity interval dI, and P(D)dD, the probability that the deflexion D recorded at an arbitrary instant lies in the interval dD. Expressions for P(D) are worked out for some specific models of M (I); a comparison with observation has been given in an earlier paper (6).
Triple points of Brownian paths in 3-space
Authors: Dvoretzky A. , Erdős P. , Kakutani S. , Taylor S. J.
Published: November 2008

In (3), some of us proved that Brownian paths in n-space have double points with probability 1 if n = 2 or 3; but, for n ≥ 4, there are no double points with probability 1. The question naturally arises as to whether or not Brownian paths in n-space (n = 2 or 3) have triple points. The case of paths in the plane is settled by (4), where it is shown that, with probability 1, Brownian paths in the plane have points of multiplicity k (k = 2, 3,4, …). The purpose of the present paper is to settle the remaining case, n = 3. We prove that, with probability 1, Brownian paths in 3-dimensional space have no triple points. The general idea behind our proof is to show that there are not too many double points. That is, we show that the set of double points has sigma-fmite linear measure, and therefore zero capacity, with probability 1.
Finite groups in which any two subgroups of the same order are isomorphic
Authors: Armstrong Ruth
Published: November 2008

1. Introduction. The class of finite groups in which any two subgroups of the same order are isomorphic will be denoted by (C), and ‘G ∈ (C)’ will mean ‘G belongs to the class (C)’.
On a non-symmetric theory of the pure gravitational field
Authors: Sciama D. W.
Published: November 2008

It is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.

A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.
Oscillations of a system of disk dynamos
Authors: Rikitake Tsuneji
Published: November 2008

The behaviour of two disk dynamos coupled to one another is examined in relation to the earth's magnetic field. It is found that reversals of electric current and magnetic field occur, unlike the case of a single disk dynamo.

On assuming a simple engine that drives a disk dynamo, the coupling between electro-magnetism, motion and heat is discussed. A possible range of temperature fluctuation associated with a dynamo action is also estimated very roughly.
Weak ergodicity in non-homogeneous Markov chains
Authors: Hajnal J. , Bartlett M. S.
Published: November 2008

The paper deals with weak ergodicity, i.e. the tendency for a chain to ‘forget’ the distant past. This may occur in non-homogeneous chains even if the probabilities of being in a given state do not tend to a limit as the number of trials increases.

Investigation of the general conditions of weak ergodicity leads to the definition of a special subclass of regular matrices. Conditions for chains involving regular matrices are also given (§§2,3).

Section 4 discusses similarities in asymptotic behaviour between chains whose transition matrices differ only by small amounts.
Single-space axioms for homology theory
Authors: Kelly G. M.
Published: November 2008

Eilenberg and Steenrod(1) give a set of axioms for the homology theory of pairs of spaces and their maps, and prove that these axioms are categorical on triangulable pairs.

Here we give a set of axioms for the homology theory of single spaces and their maps, that is, for absolute rather than relative homology. This axiomatization is shown to be essentially equivalent to that of Eilenberg and Steenrod, the relative homology groups being suitably denned in terms of the absolute groups.
On methods of summability based on integral functions
Authors: Borwein D.
Published: November 2008

Suppose throughout that

and that

is an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series
Generalized intervals in partially ordered groups
Authors: Burgess D. C. J.
Published: November 2008

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.
Random walk and electric currents in networks
Authors: Nash-Williams C. St J. A.
Published: November 2008

Let G be a locally finite connected graph and c be a positive real-valued function defined on its edges. Let D(ξ) denote the sum of the values of c on the edges incident with a vertex ξ. A particle starts at some vertex α and performs an infinite random walk

in which (i) the ξj are vertices of G, (ii), λj. is an edge joining ξj–1 to ξj (j = 1, 2, 3, …), (iii) if λ is any edge incident with ξj, then

Let υ be a set of vertices of G such that the complementary set of vertices is finite and includes α. A geometrical characterization is given of the probability (τ, say) that the particle will visit some element of υ without first returning to α. An essentially equivalent problem is obtained by regarding G as an electrical network and c(λ) as the conductance of an edge λ; the current flowing through the network from α to υ when an external agency maintains α at potential I and all elements of υ at potential 0 is found to be τD(α).

A necessary and sufficient condition (of a geometrical character) for the particle to be certain to return to α. is obtained; and, as an application, a new proof is given of a conjecture of Gillis (3) regarding centrally biased random walk on an n–dimensional lattice.
Wreath products and p–groups
Authors: Baumslag Gilbert
Published: November 2008

The wreath product is a useful method for constructing new soluble groups from given ones (cf. P. Hall (3)). Now although the wreath product of one soluble group by another is (obviously) always soluble, the corresponding result is no longer true for nilpotent groups. It is the object of § 3 of this note to determine precisely when the wreath product W of a non-trivial nilpotent group A by a non-trivial nilpotent group B is nilpotent; in fact I prove that W is nilpotent if and only if both A and B are (nilpotent) p–groups with A of finite exponent and B finite.
On Lie groups and their homotopy groups
Authors: James I. M.
Published: November 2008

We prove a theorem which facilitates homotopy classification of maps into a topological group G. Some information about homotopy groups of G is obtained, including the following two results. Consider the Samelson product, as defined in (7), which constitutes a bilinear pairing of πp(G) with πq(G) to πp+q(G). The product of a α ∈ πp(G) with β ∈ πq(G) is written in the form 〈α, β〉. There exist groups having Samelson products of infinite order. Homotopy-commutative groups have zero Samelson products. We shall prove

Theorem (1·1). If G is a connected Lie group then there exists a positive integer n such that n〈α, β〉 = 0 for every pair α, β of elements in the homotopy groups of G.
The shortest path through many points
Authors: Beardwood Jillian , Halton J. H. , Hammersley J. M.
Published: November 2008

We prove that the length of the shortest closed path throughnpoints in a bounded plane region of areavis ‘almost always’ asymptotically proportional to √(nv) for largen; and we extend this result to bounded Lebesgue sets ink–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values ofk; and we estimate the constant in the particular casek= 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.
New (Regge) symmetry relations for the Wigner 6j-symbol
Authors: Jahn H. A. , Howell K. M. , Mott N. F.
Published: November 2008

The Wigner 6j–symbol, written in the form

is shown to be invariant under separate permutations of A, B, C alone, separate permutations of α, β, γ alone and separate change in sign of any pair of α, β, γ; results equivalent to the new symmetry relations of Regge. Alternatively, written in the form

with J0 + J1 + J2 + J3 = K1 + K2 + K3, Jr(r = 0, 1, 2, 3) and K3 (s = 1, 2, 3) integral, it is invariant for separate permutations of the Jr and of the Ks. If Jm = max (J0, J1, J2, J3), then each 6j-symbol with distinct value may be associated with an ordered partition of Jm into 6 integral parts: Jm = n1 + n2 + n3 + p1 + p2 + p3, n1 ≥ n2 ≥ n3; p1 ≥ p2 ≥ n3. The 6j-symbol is proportional to the Saalschützian

of unit argument.
Homology and Ringoids. III
Authors: Hilton P. J. , Ledermann W.
Published: November 2008

The ringoids discussed in the preceding paper of this series(3) are characterized by two axioms which permit the operations of an exact category to be carried out within the structure of the ringoid. These operations do not in general include the formation of direct sums or products, for which new axioms are required. For a finite number of constituents these axioms are equivalent to the corresponding axiom in Buchsbaum's paper(1).
Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes
Authors: Good I. J.
Published: November 2008

A generalization to two independent variables of Lagrange's expansion of an inverse function was given by Stieltjes and proved rigorously by Poincaré. A new method of proof is given here that also provides a new and sometimes more convenient form of the generalization. The results are given for an arbitrary number of independent variables. Applications are pointed out to random branching processes, to queues with various types of customers, and to some enumeration problems.
a-Transforms of Local Rings and a Theorem on Multiplicities of Ideals
Authors: Rees D.
Published: November 2008

In two papers, (5) and (6), D. G. Northcott and the author considered the notion of the reductions of an ideal a of a Noether ring A. A reduction of a is an ideal b contained in a which satisfies ar+1 = arb for all sufficiently large r. This notion was inspired by the following elementary property of a reduction. Suppose that A is a local ring with maximal ideal m, and that a is m-primary. It is well known (Samuel (10)) that the length of the ideal an is, for large values of n equal to Pa(n) where Pa(n) is a polynomial in n whose degree d is equal to the dimension of A. If we write the coefficient of nd in Pa(n) in the form e(a)/d!, e(a) is a positive integer termed the multiplicity of a. If now b is a reduction of a, then b is also m-primary, and e(b) = e(a).
Some Extensions of the Multivariate t-Distribution and the Multivariate Generalization of the Distribution of the Regression Coefficient
Authors: Kshirsagar A. M.
Published: November 2008

If the components x1, x2,…, xk of a vector X have a non-singular multivariate normal distribution having a null vector of means and variance-covariance matrix Σ= σ2, the matrix R=[ρij] (where ρii = 1) is known in certain cases but σ2 is unknown. If s2 is an estimate of σ2 based on ƒ degrees of freedom and is distributed independently of X, the distribution of the vector t=x/s is known as the multivariate t-distribution. This distribution was first obtained by Dunnett and Sobel (6) and independently by Cornish (3). Dunnett, Sobel and Bechhofer(2) have discussed some practical applications of this distribution. Cornish (3) obtained this distribution while considering the pre-treatment to be given to certain types of replicated experiments. This distribution possesses some useful properties and makes it suitable as a basis for exact tests of significance in various problems, and Dunnett and Sobel (6), by providing tables of the probability integral, have taken the first step towards its use in practice. Cornish, in a later paper (4) considered the sampling distribution of statistics derived from the multivariate t-distribution and using this he obtained the well-known ((7), (8)) distribution of the sample regression coefficient of one variate with respect to another, when both have a bivariate normal distribution.
On a Non-Linear Differential Equation for the Zero-point Energies of the Rare Gas Solids
Authors: Fisher Michael E. , Zucker I. J.
Published: November 2008

Domb and Salter (3) have postulated that the Debye characteristic temperature for the rare gas solids should be given by where C involves atomic constants and the atomic mass of the rare gas, r is the mean interatomic distance and Φ(r) is the interatomic potential. This formula provides a good approximation for the heavier rare gas solids but breaks down for the lighter inert elements. On the basis of Born's argument (1) that the dynamic rather than the static equilibrium positions should be considered, Domb (2) suggested that (1) might be replaced by , where the additional term is effectively the zero-point energy contribution to the lattice equilibrium. (R is the gas constant.) From experimental measurements of the Gruneisen parameter γ Domb obtained numerical values which, when substituted in the right-hand side of (2), yielded values of vD that appeared to compare favourably with direct experiments. More recently Hooton (6) considered the problem of lattice dynamics with anharmonic forces (which are the dominant factor in the lighter rare gas solids) and developed a theoretical argument which also led to Domb's equation (2).
On the Hausdorff measure of Brownian paths in the plane
Authors: Erdős P. , Taylor S. J.
Published: November 2008

Ω will denote the space of all plane paths ω, so that ω is a short way of denoting the curve. we assume that there is a probability measure μ defined on a Borel fieldof (measurable) subsets of Ω, so that the system (Ω,, μ) forms a mathematical model for Brownian paths in the plane. [For details of the definition of μ, see for example (9).]letL(a,b; μ) be the plane set of pointsz(t, ω) fora≤t≤b. Then with probability 1,L(a,b; μ) is a continuous curve in the plane. The object of the present note is to consider the measure of this point setL(a,b; ω).