bluuebunny/crossref_metadata_2025 huggingface dataset:
Ordinal recursion
Authors: Routledge N. A.
Published: November 2008

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.
Some probability limit theorems with statistical applications
Authors: Diananda P. H. , Bartlett M. S.
Published: November 2008

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 ofdn-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 form-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.
On the differential equations for the transition probabilities of Markov processes with enumerably many states
Authors: Reuter G. E. H. , Ledermann W. , Bartlett M. S.
Published: November 2008

Letpik(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, andpik(s, t) is the conditional probability that the system be in statekat timet,given that it was in stateiat an earlier times.If certain regularity conditions are imposed on thepik, they can be shown to satisfy the well-knownKolmogorov equations§
On the invariants of the canonical system of a Vd
Authors: Todd J. A.
Published: November 2008

In a note (3) published jointly with E. A. Maxwell in 1937, a set of relations was obtained connecting the arithmetic genera Ωi of the i-dimensional characteristic systems of the canonical system on an algebraic Vd. When d is odd these relations are

and when d is even they are
On the distribution of queueing times
Authors: Smith Walter L.
Published: November 2008

The hypothetical model that we shall be considering in this paper is referred to as the single-server queue, and the details of this model are given in a recent paper by Lindley(5). The present treatment involves exactly the same assumptions as Lindley has given already, and we refer to his paper for a rigorous statement of them. Briefly, we shall be assuming general independent service times and general independent input or arrival times. Theoretical studies of the single-server queue are capable of wide applications, many of which are described in a paper by Kendall (4) and in the discussion to that paper.
Stochastic processes associated with random divisions of a line
Authors: Ramakrishnan Alladi
Published: November 2008

A class of stochastic processes associated with points randomly distributed in a line of finite extensionL, is considered. A general integral equation for the function representing the probability distribution of the stochastic variable under consideration is derived and solved by using the Laplace transform technique. Examples of the above class of processes are cited. In particular, the problem of the fluctuations in brightness of the Milky Way is discussed in detail. The results of Chandrasekhar and Munch in regard to this astrophysical problem are derived in a simple and direct manner.
Lie rings satisfying the Engel condition
Authors: Higgins P. J.
Published: November 2008

1. Let be a Lie ring in which the product of elements x and y is denoted by xy. The inner derivations of , i.e. the mappings X:a→ax for fixed elements x of , form a Lie ring under the product [X, Y] = XY – YX, and the mapping x→ X is a homo-morphism of onto . We shall say that satisfies the nth Engel condition if Xn = 0 for all X in , i.e. if

for all a, x; in . If satisfies the maximum condition on subrings, it is known (1) that this condition implies the nilpotence of ; indeed, must then be nilpotent even if the integer n is allowed to depend on the element X of . We consider here the case in which n is independent of X but does not necessarily satisfy the maximum condition, and inquire whether is then nilpotent.
An extension of the Minkowski-Hlawka theorem
Authors: Churchhouse R. F.
Published: November 2008

If R is any n-dimensional convex region, symmetric about the origin, and V(R), Δ(R) denote the content and critical determinant of R respectively, then
The roots of Yn(λη)Jn(λ) – Jn(λη) Yn(λ) = 0
Authors: Chandrasekhar S. , Elbert Donna
Published: November 2008

Cylinder functions, (λx), of integral orders which vanish at x = 1 and x = η (where η is an assigned positive constant less than 1) occur in the solution of many problems in applied mathematics. Such functions can be expressed in terms of the Bessel functions Jn(x) and Yn(x) of the two kinds in the form

where λ is a root of the equation
The Monte Carlo solution of some integral equations
Authors: Page E. S. , Cox D. R.
Published: November 2008

Estimators are given for the solution by Monte Carlo methods of the Fredholm integral equation of the second kind and the variances of the estimators are compared. Two integral equations arising in sequential analysis are considered in detail; some numerical examples are given.
A characteristically simple group
Authors: McLain D. H.
Published: November 2008

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.
Physical properties of particles obeying generalized statistics
Authors: McCarthy I. E.
Published: November 2008

The Feynman probability amplitudes for interactions between spin ½ fields and fields of integral spin quantized according to the generalized method of Green are investigated. It is shown that the scattering amplitudes are independent of the method of quantization though the amplitudes of the vacuum fluctuations are not. The question of distinguishing particles satisfying the generalized statistics, if they exist in nature, from Fermi or Bose particles is considered. It is shown that the statistical thermodynamics of the particles depends on the statistics, but as this requires a large number of particles in thermodynamical equilibrium, it offers no means of distinction for particles with short lifetimes. In practice, an examination of particles in bound states would identify particles obeying generalized statistics.
A differential equation for undamped forced non-linear oscillations. I
Authors: Morris G. R.
Published: November 2008

In this paper and its sequels I consider the differential equation

in which e(t) is an even periodic function of t and dots denote differentiation with respect to t. We can regard (1·1) as a special case of the equation

(the ‘differential equation of non-linear oscillations’), which describes the motion on the x–axis of a particle of unit mass subject to a restoring force g(x), a variable damping force f(x, ẋ)ẋ and an external force p(t); the dynamical description used in the title appeals to this interpretation. The most interesting problems on (1·2) and its specializations concern the behaviour of solutions as t → ∞ for example, we may seek conditions on f, g and p which allow us to assert that some or all solutions are bounded or, having assumed that p(t) is periodic, seek conditions on f and g which allow us to assert the existence of periodic solutions.
The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables
Authors: Cox D. R.
Published: November 2008

Certain stochastic processes with discrete states in continuous time can be converted into Markov processes by the well-known method of including supplementary variables. It is shown that the resulting integro-differential equations simplify considerably when some distributions associated with the process have rational Laplace transforms. The results justify the formal use of complex transition probabilities. Conditions under which it is likely to be possible to obtain a solution for arbitrary distributions are examined, and the results are related briefly to other methods of investigating these processes.
A notation for vectors and tensors
Authors: Powell F. C.
Published: November 2008

The vector notation commonly employed in elementary physics cannot be applied in its usual form to spaces of other than three dimensions. In plane dynamics, for instance, it cannot be used to represent the velocity (– ωx2, ωx1) at the point (x1, x2) due to a rotation ω about the origin, or the (scalar) moment about the origin of the force (F1, F2) acting at (x1, x2). In relativity physics the symbol ⋅ is often used to denote the scalar product of two vectors, it is true, and the tensor aαbβ – aβbα is sometimes denoted by a × b, but there exists no body of rules for the manipulation of these symbols that enables one to dispense with the suffix notation as in the case of vectors in three-dimensional space.
Molecules with tetrahedral and octahedral symmetry
Authors: Ballinger R. A. , March N. H.
Published: November 2008

Detailed energy calculations for CH4, SiH4 and GeH4 have been carried out using the Thomas-Fermi approximation together with a nuclear field which is averaged over all orientations about the central nucleus. The energy, including exchange, has been calculated using the unmodified Thomas-Fermi electron densities as variation functions, and in this way bond lengths and force constants for the three molecules have been obtained. The effect of exchange is very marked, agreement with experiment being quite reasonable when it is included in this way.

In order to test the accuracy of this procedure, a scale factor has been introduced and used as a variational parameter. In this refined approximation, calculations have been made for SiH4, thus enabling improved charge distributions to be obtained. Energy curves are also presented for this molecule, showing separately the variations of kinetic, potential and exchange energies with internuclear distance. It is shown that whilst the scale factor has an appreciable effect on the separate energy curves, the total energy is virtually unchanged.

Finally, the reason why the exchange energy is of considerable importance in the theory is briefly discussed, and some observations are made concerning the rather abnormal bonding in GeH4.
Molecules with tetrahedral and octahedral symmetry. III. Theoretical basis of the ‘smoothing approximation’
Authors: Ballinger R. A. , March N. H.
Published: November 2008

In Parts I and II of this series (March(6); Ballinger and March(1)), we considered in detail the application of the Thomas-Fermi (T.F.) approximation to molecules with tetrahedral and octahedral symmetry. In these papers, following the work of Buckingham, Massey and Tibbs(2), who obtained results for CH4, we averaged the nuclear field over angles (the ‘smoothing approximation’) and considered the electrons as though they moved in the resulting central field. In this way, it was possible in (6) to carry through self-consistent (T.F.) calculations giving the electron distributions and potential fields in tetrahedral and octahedral molecules.
Uniformly almost periodic solutions of non-linear differential equations of the second order I. General exposition
Authors: Obi Chike
Published: November 2008

1·1. A general problem in the theory of non-linear differential equations of the second order is: Given a non-linear differential equation of the second order uniformly almost periodic (u.a.p.) in the independent variable and with certain disposable constants (parameters), to find: (i) the non-trivial relations between these parameters such that the given differential equation has a non-periodic u.a.p. solution; (ii) the number of periodic and non-periodic u.a.p. solutions which correspond to each such relation; and (iii) explicit analytical expressions for the u.a.p. solutions when they exist.
Stochastic differential equations
Authors: Edwards D. A. , Moyal J. E.
Published: November 2008

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) asmean-squareconvergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.
The stability of a homopolar dynamo
Authors: Bullard Edward
Published: November 2008

The stability of the self-exciting disk dynamo is considered. If there is no electrical load in parallel with the field coil and no friction at the axle, the dynamo can perform oscillations of constant amplitude about its state of steady motion. Viscous forces on the axle cause the dynamo to settle to a steady motion. A parallel load may have an effect similar to that of viscous forces or may cause the oscillations to grow without limit. With both a parallel load and viscous forces the amplitude of the oscillations is bounded.

Possible analogies between these results and the motion of a body of electrically conducting fluid in a magnetic field are discussed. The main applications are astronomical and geophysical.
A property of Hausdorff measure
Authors: Davies Roy O.
Published: November 2008

From the fact that Hausdorffs-dimensional measure is a regular Carathéodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result:TheoremA.If {En} is any increasing sequence of sets, then∧sEn→ ∧s(ΣEn)as n→ ∞.

Since ∧sXis denned (for every setX) as, the problem arises whether for every positive δ and every increasing sequence of sets one can prove

Now there are four possible ways of defining; it is the lower bound oftaken over coverings ΣUrofXby convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be (i) open and of diameter less than δ, (ii) closed and of diameter less than δ, (iii) open and of diameter not exceeding δ, or (iv) closed and of diameter not exceeding δ.
Molecules with tetrahedral and octahedral symmetry
Authors: Banyard K. E. , March N. H. , Coulson C. A.
Published: November 2008

The modified Thomas-Fermi method reported in (2) has been applied numerically to calculate the distribution of valence electrons in CC14. Slater wave functions have been used to describe the Cl inner-shell electrons, whilst a description of the C ls electrons based on the Thomas-Fermi model for atoms has been adopted. The electron distribution thus found is compared with the two approximations discussed in part I of this series (3). The first of these, in which valence and inner-shell electrons are treated in the same way, is shown to give a distribution of valence electrons in good agreement with the present results. The second approximation, that of effective nuclear charges, is less satisfactory and could only lead to a distribution of valence electrons in reasonable agreement with the present work if the effective nuclear charges were considerably reduced. This seems to throw some light on the results of Bowers for the force constants of tetrachlorides and tetrabromides.

The overall distribution of electrons obtained here seems to be the inevitable result of assuming the valence electrons to move in any plausible central field. It is, of course, by no means certain that such an approximation is near to the truth. In fact, the results of (3) and the present paper leave little doubt that the electron cloud obtained by such an approximate method will be too diffuse. This seems at least partly responsible for our finding that rather more than 20 of the 28 Cl valence electrons lie outside the sphere drawn through the Cl nuclei.
Statistical dynamics of multiply-periodic systems
Authors: Born M. , Hooton D. J. , Mott N. F.
Published: November 2008

Classical dynamics is generally considered as the prototype of a deterministic theory; the equations of motion determine the coordinates q and the momenta p at any time provided they are given at an initial instant. It is pointed out that this is an unrealistic assumption, for it is inevitable that there should be small uncertainties Δq, Δp; a point of the mathematical continuum has no physical significance. The uncertainties can be taken into account without violating the deterministic equations by introducing a probability density P in phase space p, q. The function P satisfies the partial differential equation which expresses Liouville's theorem. Thus it can be shown that an initial uncertainty spreads out in phase space, so that finally all states of the system consistent with the mean initial values of the constants of the motion are equally probable. This property holds for one degree of freedom just as well as for many, and is not a consequence of our ignorance concerning large numbers of particles. The essential difference between classical mechanics and quantum mechanics consists not in the physical necessity of a statistical interpretation, but in the further introduction of a probability amplitude, the square of which is the probability density; thisimplies the restriction of the initial uncertainties as expressed by the uncertainty laws, and the phenomenon of interference of probabilities which makes necessary a revision of our concepts of physical reality. It is remarkable that the product of the uncertainties of a properly chosen pair of variables is constant also in classical mechanics, although of course its value is not a universal constant as in the quantum case.
Approximation to transients by means of Laguerre series
Authors: Head J. W.
Published: November 2008

Ward(1) has discussed a method, introduced by Tricomi (6), of calculating transients by means of series involving Laguerre functions which in some cases makes it unnecessary to determine poles of the relevant characteristic function. This method is here investigated with special reference to conditions for convergence and adjustments for improving convergence; some of the examples discussed by Ward are reconsidered.

Both in Ward's paper and here the location of poles of the characteristic function is assumed to be approximately known. In some cases the determination of poles of outstandingly small or large modulus and their separation from the remainder may be the most satisfactory procedure. Lin's method (2) for determining a quadratic factor of a polynomial is more widely applicable than has previously been supposed, and this is discussed in bare outline, without proof, here, but in detail, with adequate numerical examples, elsewhere (3).
Some arithmetic properties of the Legendre polynomials
Authors: Carlitz L.
Published: November 2008

1. Good (4) has proved the formula

where Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).

Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put
Siegel's mean value theorem in the geometry of numbers
Authors: Macbeath A. M. , Rogers C. A.
Published: November 2008

Our main object is to give an exposition of a simplified version of Siegel's proof of his mean value formula in the Geometry of Numbers, also proved by A. Weil in a more general context ((1), (2), (3), (4)). In deriving the particular case which is Siegel's formula, however, certain difficulties of convergence arise, which were resolved by Siegel through rather tedious calculations, and by Weil through the use of Fourier analysis in topological groups. A new method of dealing with these difficulties, depending on Lebesgue's bounded convergence theorem and the relatively simple Lemma 8, seems to be our major contribution. In other respects our proof is very similar to Siegel's. At some points we have simplified his treatment; at others, where his work seems rather condensed, we have expanded it. We are indebted to Mahler (9) for some of the simplifications and grateful to him for letting one of us see his unpublished work. By using Lemma 8 and giving a separate proof of the existence of a fundamental domain for the unimodular group, we give a proof which is independent of Minkowski's reduction theory.
Many-valued logical machines
Authors: Rose Alan
Published: November 2008

McCallum and Smith (1) have developed a logical computer which solves problems which are essentially of the following kind.
A note on the representation of general recursive functions and the μ quantifier
Authors: Rose Alan
Published: November 2008

It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schema

subject to the condition that, for eachn–tuple of natural numbersx1,…,xnthere exists a natural numberxn+1such that
Homology and ringoids. II
Authors: Hilton P. J. , Ledermann W.
Published: November 2008

1. Introduction and recapitulation. In a previous paper (4) we have defined a ringoid (with identities) as a set of elements α, β,… in which addition and multiplication are defined for certain pairs of elements in such a way that, whenever operations are defined, the usual ring axioms are satisfied. Precisely, is an abstract category ((3), p. 108) with an addition operation such that the following axioms hold:

(R 1) For any two identity elements ι, ι′ of, let ℋ(ι, ι′) be the set of elements ofwhose left identity is ι and right identity is ι′; then ℋ(ι, ι′) is empty or an Abelian group under addition.
Equivalence of two absorption problems with Markovian transitions and continuous or discrete time parameters
Authors: Sack R. A.
Published: November 2008

1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given by

where the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form as

where P(t) is the vector with components pi(t) and the constant matrix A has elements
Note on quadratic forms over the rational field
Authors: Cassels J. W. S.
Published: November 2008

There is perhaps some methodological interest in developing the theory of quadratic forms over the rational field using only the methods of elementary arithmetic. Hitherto it has appeared necessary to use theorems of a fairly deep nature, most often Dirichlet's theorem about the existence of primes in arithmetic progressions (e.g. Minkowski(1), Hasse(2), Dickson(8), Skolem(9), Burton Jones(6)). Skolem(5) uses a weaker form of Dirichlet's theorem which is rather easier to prove and Siegel(4) uses instead the machinery of the Hardy-Littlewood circle method. In this note I indicate how it is possible to develop the theory of quadratic forms over the rationals without using extraneous resources. Pall(10) states that he has also found such a development of the theory but he does not appear to have published it.
Some embeddings of projective spaces
Authors: James I. M.
Published: November 2008

1. Introduction. We say that a smooth compact manifold is embedded in q–space when there is given a regular one-one map of the manifold into q–dimensional Euclidean space. Whitney (5) has shown that embeddings always exist when q is not less than twice the dimension of the manifold.
Diffusion in branching regions
Authors: Jaeger J. C.
Published: November 2008

The problem of diffusion from an instantaneous point source in a thin plane sheet which is intersected by one or more other thin plane sheets of the same material is considered. This problem arises in the study of the diffusion of packets of transmitter substance in synaptic clefts, and some numerical results for the amphibian neuro-muscular junction are presented. A similar discussion applies to branching rods and to cases where there is loss of diffusing substance at the surfaces of the sheets.
Flexural problems of circular ring plates and sectorial plates. I
Authors: Bassali W. A. , Gorgui M. A.
Published: November 2008

In this paper explicit expressions in closed forms are first obtained for the complex potentials and deflexion at any point of a circular annular plate under various edge conditions when the plate is acted upon by general line loadings distributed along the circumference of a concentric circle. These solutions are then used to discuss the bending of a circular plate with a central hole under a concentrated load or a concentrated couple acting at any point of the plate. Solutions for singularly loaded sectorial plates bounded by two arcs of concentric circles and two radii are also derived when the plate is simply supported along the straight edges. The boundary conditions along the circular edges include the cases of a free boundary as well as the elastically restrained boundary which covers the usual rigidly clamped and simply supported boundaries as special cases. The usual restrictions relating to the small deflexion theory of thin plates of constant thickness are assumed. Limiting forms of the resulting solutions are investigated.
Expansions of certain Whittaker functions
Authors: Meligy A. S.
Published: November 2008

An expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.
Permutation endomorphisms and refinement of a theorem of Birkhoff
Authors: Farahat H. K. , Mirsky L.
Published: November 2008

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equations

A permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .
On complex Stiefel manifolds
Authors: Atiyah M. F. , Todd J. A.
Published: November 2008

In a recent series of papers (10), (11), (12), I. M. James has made an illuminating study of Stiefel manifolds. We shall begin by describing his results (for the complex case). Let Wn, k, for k > 1, denote the complex Stiefel manifold U(n)/U(n − k), where U(n) is the unitary group in n variables. Then we have a natural fibre map Wn, k → Wn, 1 = S2n−1, where Sr denotes the r-dimensional sphere. Let Pn, k, for k ≥ 1, denote the ‘stunted complex projective space’ obtained from the (n − 1)-dimensional complex projective space† Pn by identifying to a point a subspace Pn−k. Then we have a natural ‘cofibre map’ Pn, k → Pn, 1 = S2n−2. The space Pn, k is said to be S-reducible if some suspension of the map Pn, k → S2n−2 has a right homotopy inverse. The results of James can then be summarized as follows.
Degree Functions in Local Rings
Authors: Rees D.
Published: November 2008

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identity

The main purpose of this paper is to obtain a formula

where vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).
The number of polygons on a lattice
Authors: Hammersley J. M.
Published: November 2008

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d-dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours, i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an (n + 1)-sided self-avoiding polygon. Clearly, under this definition a polygon must have an even number of sides. Let f(n) and g(n) denote the numbers of n-stepped self-avoiding walks and of n-sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such that

Here I shall prove the truth of the long-standing conjecture that

I shall also show that (2) is a particular case of an expression for the number of n-stepped self-avoiding walks with prescribed end-points, a distance o(n) apart, this being another old and popular conjecture.
Transport properties of thin metallic films
Authors: Baines M. J.
Published: November 2008

In using the Boltzmann equation to calculate the transport properties of thin metallic films, it is usually assumed that a time of relaxation exists for the scattering in the film which is the same as for the bulk metal, although this is only strictly justified in the ideal case of isotropic scattering. The correct Boltzmann equation in the case of elastic scattering, when the cross-section depends on the angle of scatter, is set up and possible methods of solution are discussed.

The equation is solved for a simple anisotropic scattering law and the electrical resistivity and thermo-electric power are found to depend upon two parameters which are measures respectively of the thickness of the film and the anisotropy of the scattering mechanism. The numerical results for the electrical resistivity, however, differ only slightly from those given by the one-parameter formula obtained when a time of relaxation is assumed to exist.
Solutions of certain finite difference equations connected with Laplace's equation
Authors: Sander K. F.
Published: November 2008

The familiar difference equations approximating to Laplace's equation are investigated for exact solutions. Solutions corresponding to the electrostatic potentials for line dipole, point and ring charges are obtained analytically.
Single-centre expansions for the hydrogen molecular ion. II
Authors: Cohen M.
Published: November 2008

In an earlier paper (Cohen and Coulson(3), referred to hereafter as I), it was shown that satisfactory energy eigenvalues and eigenfunctions for various even σ-states of may be obtained using a single-centre expansion, provided that the radial functions are properly determined. In particular, the ground-state energy at the equilibrium internuclear separation of 2 a.u. was found to be within 0·25% of the exact value (Bates, Ledsham and Stewart (2)), and the eigenfunction reproduced all the characteristics of the exact wave-function. The method has now been extended to the odd σ-states, as well as to the two lowest π-states (2pπu, 3dπg), and the results are in good agreement with the calculations of Bates et al.
Road Traffic Flow Considered as a Stochastic Process
Authors: Miller A. J.
Published: November 2008

Equations are derived for the process of catching-up and overtaking which occurs in one lane of traffic for a model in which vehicles travel in random bunches and overtake at random times. The model allows vehicle velocities to be distributed and allows for several vehicles at a time to overtake. At first only stationary states are considered, that is states in which the rate of catching-up equals, on average, the rate of overtaking.
On Viscous Flow in a Pipe with Constrictions
Authors: Belinfante D. C.
Published: November 2008

Two methods of solution are given for the flow of a viscous liquid in a circular pipe whose radius has a slight sinusoidal variation. Numerical examples are given, and some related problems are briefly discussed.
The Piecewise-Linear Structure of Euclidean Space
Authors: Stallings John
Published: November 2008

It is known that, for n ≤3, ‘ought to have’ can truthfully be replaced by ‘has’ (see (4), and (5), Cor. 6·6). In this paper, this conjecture will be proved for n ≥ 5. The only unsolved case then will be in dimension four.
A topological proof in group theory
Authors: Cohen D. E.
Published: November 2008

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 groupFof a, single elementris freely generated by conjugates ofr. 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.
Gradient configurations and quadratic functions
Authors: Afriat S. N.
Published: November 2008

A normal preference system for a combination of goods is represented by an increasing function φ with convex levels. From Gossen's law, that preference and price directions coincide in equilibrium (a special consequence of his Second Law), it follows that, on the data that xr is the vector of quantities purchased at prices given by a vector pr (r = 1,…, k), the gradient gr = g(xr) of the function φ at the point xr is given by

for some positive multiplier μr;. There may be considered the class of preference functions thus satisfying Gossen's law in respect to the data, and thus with gradients taking prescribed directions at k prescribed points. In particular, there may be considered the subclass of these which are quadratic in some convex region containing the points xr. By choosing any multipliers μr, there is obtained a set of gradients gr associated with the points xr. It is asked if there exists a quadratic function which is increasing and has convex levels in a convex neighbourhood of the points xr, and whose gradient at xr is gr; also it is required to characterize the class of such functions, which, if any exist, form an infinite variety. This is the background of the questions which are going to be investigated, and which are of importance in a general method of empirical preference analysis in economics.
The solution of Bessel function dual integral equations by a multiplying-factor method
Authors: Noble B.
Published: November 2008

In this paper we first of all consider the dual integral equations

where f(ρ), g(ρ) are given, A(t) is unknown, and α is a given constant. This system, with g(ρ) = 0, was originally considered by Titchmarsh ((13), p. 337), and Busbridge (1), who obtained a solution by the use of Mellin transforms and analytic continuation in the complex plane. The method described in this paper involves the application of certain multiplying factors to the equations. In the present case it is relatively easy to guess the multiplying factors and then the method is essentially a real-variable technique. It is presented in this way in § 2 below.
The vibrations of three two-dimensional lattices
Authors: Dean P.
Published: November 2008

Dispersion relations are derived for the triangular, honeycomb and Kagomé lattices using theorems for the eigenvalues of block circulant matrices. Constant frequency contours are plotted for each of the spectral branches of the three lattices and the distribution functions for squared frequencies are sketched.
Observations on the Künneth theorem
Authors: Kelly G. M.
Published: November 2008

Let A be a right Λ-complex and C be a left Λ-complex where Λ is a ring which is both left and right hereditary; a principal ideal domain is a special case of such a ring. This paper is concerned with the Künneth theorem expressing the homology H(A⊗C) of the product complex A⊗C (= A⊗ΛC) in terms of H(A) and H(C). Any notations we do not explain are those of (1).
The rotating body problem
Authors: Levy H. , Robinson W. J.
Published: November 2008

It is shown that for an axisymmetric stationary system there exists, in vacuo, a canonical coordinate system, analogous to Weyl's canonical coordinates for the static case. Some approximate solutions of the vacuum field equations are then obtained in these canonical coordinates.

In the second part of this paper, the vacuum field equations for an isolated axisymmetric stationary system are set up and approximate solutions obtained by means of a multipole expansion. The physical components of the Riemann tensor are then examined and it is found that the ‘mass term’ predominates, the ‘rotation term’ being of the next order of magnitude.
Mrs Perkins's quilt
Authors: Conway J. H.
Published: November 2008

In this paper we consider the problem of dividing a square of side N into squares of integral side. When we demand that these be unequal, the problem is soluble only for certain N, and has been treated in some detail by various authors (1,3,4,5). If we relax this condition, every side is permissible, for a square of side N may plainly be divided into N2 squares of side 1. It is the object of this paper to reduce this number: in fact we shall show that a square of side N is divisible into some number less than 6 ∛N + 1 squares of integral side.
Integrals involving products of G-function and Gauss's hypergeometric function
Authors: Sharma K. C.
Published: November 2008

1. The main object of this note is to evaluate two definite integrals involving the product of Meijer's G-function with Gauss's hypergeometric function. The first result established in this paper is the extension of the results recently given by Saxena ((3), page 490) in these proceedings and includes both of his results given there as particular cases.
On normal mode vibrations
Authors: Rosenberg R. M.
Published: November 2008

1.Introduction. In linear systems, the concept of ‘free vibrations in normal modes’ is well defined and fully understood. The meaning of this phrase is far less clear when it is applied to non-linear systems. It is the purpose here to define and examine the free vibrations in normal modes (and their stability) in certain non-linear systems composed of masses and springs and having a finite number of degrees of freedom. Of necessity, such a paper is in some degree conceptual in nature.
A finite set of generators for the homeotopy group of a 2-manifold
Authors: Lickorish W. B. R.
Published: November 2008

The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: A → A by

H is then fixed on the boundary of A. If now e: A → X is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on X − eA to a homeomorphism h:X → X. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.
On doubly stochastic Poisson processes
Authors: Kingman J. F. C.
Published: November 2008

The class of stationary point processes known as ‘doubly stochastic Poisson processes’ was introduced by Cox (2) and has been studied in detail by Bartlett (1). It is not clear just how large this class is, and indeed it seems to be a problem of some difficulty to decide of a general stationary point process whether or not it can be represented as a doubly stochastic Poisson process. (A few simple necessary conditions are known. For instance, Cox pointed out in the discussion to (1) that a double stochastic Poisson process must show more ‘dispersion’ than the Poisson process. Such conditions are very far from being sufficient.) The main result of the present paper is a solution of the problem for the special case of a renewal process, justifying an assertion made in the discussion to (1).
On complex Stiefel manifolds
Authors: Adams J. F. , Walker G.
Published: November 2008

We shall study the following complex Stiefel fibring:

.

In particular we shall study the problem: for what values of n and k does the fibring(1.1) admit a cross-section? A necessary condition for the existence of a cross-section has been found by Atiyah and Todd (8). We shall show (Theorem 1.1 below) that the condition of Atiyah and Todd is sufficient (as well as necessary) for the existence of a cross-section. The problem stated above is therefore completely solved.
The approximation of Gδ-sets, in measure, by Fσ-sets
Authors: Larman D. G.
Published: November 2008

Some time ago Besicovitch(l) gave an example of a linear Gδ-set E of Besicovitch dimension 1, with the property that E – A also has dimension 1, for each Fσ.-set A contained in E. This shows that an Fσ-set contained in E cannot be a very close approximation to E, but it leaves plenty of scope for Fσ-sets contained in E to be reasonably good approximations to E.
Diffraction of a high-frequency plane electromagnetic wave by a perfectly conducting circular disc
Authors: Jones D. S.
Published: November 2008

A method already developed for the diffraction of scalar waves by a circular disc is generalized so as to be applicable to the diffraction of an electromagnetic wave. The generalization leads, in a straightforward manner, to an asymptotic development of the current density on the disc at high frequencies. The principal complication which arises is in solving three simultaneous linear equations for three constants which occur in the method of solution. Detailed calculations are given for a normally incident plane wave.
A problem of a thin circular plate transversely loaded and rigidly held on its boundary
Authors: Sayer F. P.
Published: November 2008

The plate is of thickness 2h and occupies the region defined by 0 ≤ x2+y2 ≤ x2, −h ≤ Z ≤ h, x, y, Z being rectangular Cartesian coordinates. A normal load , where p is a constant, is applied to the surface Z = h and the boundary is held so that the displacement on it is zero. Using the results of an earlier paper by Tiffen and Sayer (3), expressions are obtained for the mean stresses and stress couples at any point of the plate.
Two notes on spectral synthesis for discrete Abelian groups
Authors: Elliott R. J.
Published: November 2008

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).
Fourier transforms and the operator Hα
Authors: Okikiolu G. O.
Published: November 2008

This is one of a number of papers in which the author studies the properties of the transform Hα(f) and allied operators. In this case, identities involving these operators and the Fourier transform are obtained, and these are applied in giving alternative proofs to certain inversion processes derived by the author elsewhere.
Two-dimensional elastic inclusion problems
Authors: List R. D. , Silberstein J. P. O.
Published: November 2008

A system of equations is derived for determining the elastic fields in an inclusion and its surrounding finite matrix when the inclusion suffers a physical change and, if not constrained by the matrix, would undergo a deformation . A method for obtaining the exact solution of these equations, when the matrix and inclusion have the same elastic constants, is described and the particular problem of the square inclusion in an infinite matrix solved.
On projective spaces over two skew fields
Authors: Denniston R. H. F.
Published: November 2008

Introduction. It is a classical problem to construct a complex projective space, using as ‘imaginary points’ objects of some kind from the geometry of real projective space. (The various solutions to this problem have been reviewed in a book by Coolidge ((1)).) The object of the present paper is to solve the corresponding problem for n-dimensional projective spaces over two skew fields, one being a second-rank extension of the other. The method used seems not to have been published before, even in connexion with the classical problem.
Formations and a class of locally soluble groups
Authors: Stonehewer S. E.
Published: November 2008

In (3) W. Gaschütz introduced the concept of a formation of finite soluble groups, generalizing the results of R. W. Carter on the existence and conjugacy of nilpotent self-normalizing subgroups in finite soluble groups (2). Carter's results have already been extended to two classes of infinite groups in (9), (10), and the object of the present work is to extend the results of Gaschütz to a class of infinite groups, thereby generalizing (9).