bluuebunny/crossref_metadata_2025 huggingface dataset:
The nuclei of non-associative division algebras
Authors: Rees D.
Published: November 2008

The results of this paper have as their background two concepts fundamental in the theory of non-associative algebras (i.e. algebras in which the associative law of multiplication is not assumed). These concepts are:

(1) Isotopy,

(2)The three nuclei λ, μ, ρ of an algebra .
Surface waves on deep water in the presence of a submerged circular cylinder. II
Authors: Ursell F.
Published: November 2008

In the first part of this paper a method was given for constructing a wave potential when the normal velocity is a prescribed function of the angular variable on a submerged circular cylinder. It was shown that the method breaks down for values of the parameters Ka and Kf for which a certain infinite determinant vanishes. The vanishing of this determinant implies the existence of a non-trivial velocity potential, such that the normal velocity vanishes on the cylinder and both velocity components vanish at infinity. In this part of the paper it is shown that there can be no non-trivial solution of this kind; in other words the infinite determinant does not vanish. In the absence of a general uniqueness theory for surface waves it seems worth while to establish this particular result.
On Euclid's Algorithm in cubic self-conjugate fields
Authors: Heilbronn H.
Published: November 2008

In a paper published in these Proceedings I proved that there are only a finite number of quadratic fields in which Euclid's Algorithm (E.A.) holds. Recently Davenport has found a new proof of this theorem based on the theory of the minima of the product of linear inhomogeneous forms.
The translation of two bodies under the free surface of a heavy fluid
Authors: Coombs A.
Published: November 2008

1. Many investigations have been made to determine the wave resistance acting on a body moving horizontally and uniformly in a heavy, perfect fluid. Lamb obtained a first approximation for the wave resistance on a long circular cylinder, and this was later confirmed to be quite sufficient over a large range. In 1926 and 1928, Havelock (4, 5) obtained a second approximation for the wave resistance and a first approximation for the vertical force or lift. Later, in 1936(6), he gave a complete analytical solution to this problem, in which the forces were expressed in the form of infinite series in powers of the ratio of the radius of the cylinder to the depth of the centre below the free surface of the fluid. General expressions for the wave resistance and lift of a cylinder of arbitrary cross-section were found by Kotchin (7) using integral equations, and the special case of a flat plate was evaluated. He continued with a discussion of the motion of a three-dimensional body. More recently, Haskind (3) has examined the same problem when the stream has a finite depth.
The paradox of separated systems in quantum theory
Authors: Cooper J. L. B.
Published: November 2008

The aim of this article is to discuss a paradox of importance in the quantum theory of measurement. The paradox was first propounded by Einstein, Podolsky and Rosen in 1935(1), and was discussed subsequently by Bohr(2), Furry(3), Schrödinger(4, 5), as well as in texts (6, 7). We begin by formulating the paradox.
An extension of the law of the iterated logarithm
Authors: Cassels J. W. S.
Published: November 2008

1. Let x1, x2, …, xn, … be a set of independent variables each with a uniform probability distribution in 0 ≤ x ≤ 1. If 0 ≤ α < β ≤ 1 we denote by FN (α, β) the number of x1, …, xN which satisfy α < x ≤ β,

and put RN(α, β) = FN(α, β) − N(β − α).
Triple velocity correlations in isotropic turbulence
Authors: Stewart R. W.
Published: November 2008

The triple velocity correlation, in turbulence produced by inserting a square-mesh grid near the beginning of the working section of a wind tunnel, has been measured for mesh Reynolds numbers of RM = 5300, 21,200 and 42,400 (RM = UM/ν, where U is the mean wind speed in the working section of the tunnel and M is the centre to centre spacing of the rods making up the grid; ν is the kinematic viscosity of air). At the lowest Reynolds number the correlation has been measured at distances downstream of the grid varying from 20 to 120M. This range covers practically all of the initial period of the decay of turbulence, where the turbulent intensity varies as t−1.
The theoretical behaviour of a magnetic monopole in a wilson cloud chamber
Authors: Cole H. J. D.
Published: November 2008

Dirac has suggested that the quantization of electric charge could be explained by the existence of magnetic monopoles. In view of this hypothesis, this paper investigates what theoretically would be the behaviour of such monopoles in a Wilson cloud chamber. The treatment, which for simplicity is basically classical, closely follows Bohr's work on the decrease of velocity and ionization properties of α- and β-particles, and expressions are derived for the rate of decrease of energy and the number of ion-pairs produced per centimetre by a monopole passing through a gas. These expressions are then discussed with particular reference to the case of heavy particles, and the main differences between them and the corresponding expressions for α-particles both as to range and ionization are indicated; these differences can be summarized by saying that monopoles have much shorter paths, but create many more ion-pairs per centimetre than α-particles. Also, the very sharp increase in the ionization at the end of the path of an electric particle is missing, the ionization for the monopole decreasing to a small amount near the end of the path.
Self-consistent field, with exchange, for Ne III
Authors: Garstang R. H.
Published: November 2008

Two of the strongest emission lines in the spectra of the planetary nebulae were identified by Boyce, Menzel and Payne (1) as due to forbidden transitions in the ground configuration of doubly ionized neon. The transitions are of wave-length 3967 and of wave-length 3869 in the configuration Ne III 1s22s22p4. These lines are excited by inelastic collisions of electrons with the neon ions. In order to investigate these transitions in detail a knowledge of the collision cross-sections is required. As a first step in this direction, wave functions have been calculated for the 3P state by the method of the self-consistent field, including the effect of exchange. In later papers these wave functions will be used in the calculation of collision cross-sections, in a study of the departures from the Landé rule in the triplets of the p2 and p4 configurations and, it is hoped, in the calculation of line strengths for certain permitted lines of astrophysical importance in the spectrum of Ne II.
The function and associated polynomials
Authors: Lawden D. F.
Published: November 2008

A transform method for the solution of linear difference equations, analogous to the method of the Laplace transform in the field of linear differential equations, has been described by Stone (1). The transform u(z) of a sequence un is defined by the equation
A note on the correspondence between the x, t-plane and the characteristic plane in a problem of interaction of plane waves of finite amplitude
Authors: Stocker P. M. , Meyer R. E.
Published: November 2008

The head-on interaction, in the one-dimensional, unsteady isentropic flow of a perfect gas, of a simple compression wave and a simple expansion wave is studied by considering typical examples. The physical aspect of the problem is discussed in (1); in this note the possibility of shock formation is ignored, and the correspondence defined by the complete mathematical solution of the equations of isentropic flow between the x, t-plane and the plane of the characteristic variables is elucidated.

The solution is distinguished by the appearance of two limit lines and a second-order limit point where they meet. It is found that the image of the characteristic plane in the x, t-plane is four-sheeted; all sheets overlap each other, but each covers only part of the plane, and the only point common to all sheets is the second-order limit point, where both limit lines are cusped (§ 3·1).

The solution also contains an edge of regression, and a discussion of the properties of this type of singularity will be found in §§ 2 and 2·1.
A trigonometrical inequality
Authors: Neville E. H.
Published: November 2008

The inequality is

which is established by an elementary argument and is shown to lead directly to the evaluation of the integral

and to the expression of sin x as an infinite product.
Some properties of analytically irreducible geometric quotient rings
Authors: Northcott D. G.
Published: November 2008

The purpose of this note is to show how a considerable part of the local theory of the prime divisors of a field of algebraic functions can be extended to analytically irreducible geometric quotient rings. In doing this we shall make frequent use of Chevalley's paper on local and semi-local rings (2). For brevity this paper will be referred to as ‘L.R.’. We begin the discussion by recalling for the reader's convenience the definition of the Kronecker product of two fields over a common subfield, since this concept will play an important role later.
A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order
Authors: Olver F. W. J.
Published: November 2008

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder function

as a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.
Statistics of the simple cubic lattice. II
Authors: Wakefield A. J.
Published: November 2008

Experimentally it is found for a ferromagnetic substance that the spontaneous magnetization decreases as the temperature increases. At a certain temperature called the Curie point, the magnetization disappears (or substantially disappears) and remains zero for higher temperatures. Associated with these magnetic properties is an anomaly in the specific heat. This quantity is greater than that which would be calculated theoretically for the material were it non-magnetic and becomes large when the temperature approaches the Curie point from below. Just above the Curie point there is a sharp but continuous decrease in the specific heat and the system is said to show critical behaviour. We shall examine the Ising model of the ferromagnet with a simple cubic lattice structure, determine the specific heat anomaly and the corresponding energy excess due to magnetism, and also see how the critical behaviour of the model compares with that actually observed.
An inequality for real positive functions
Authors: Hayman W. K.
Published: November 2008

In the theory of functions of a complex variable a positive quantity g(r), denned in terms of a circle |z| = r, can often be estimated by means of another similar quantity f(R) denned in terms of a circle |z| = R with R > r. Inequalities of the form

arise in this connexion. Putting R = r + h we have for every h > 0

.
Calculation of S-matrix elements
Authors: Chisholm J. S. R.
Published: November 2008

A method of calculating S-matrix elements is described. All factors corresponding to lines in Feynman-Dyson graphs are put in the form of scalar boson line factors acted on by certain operators and involving only the energy-momenta of external lines and particle masses. A general formula is given for the integral of a product of scalar boson factors, and the effect of a product of operators and on this result is described. The infinities appear in a different form, and it is shown how subtraction procedure can be applied to eliminate them.
Conformal transformations and kinematical relativity
Authors: Littlewood D. E.
Published: November 2008

The Einstein relative-velocity formula. Milne (3) has remarked that in kinematical relativity, whereas the Lorentz transformation is valid in t-time but fails in τ-time, the Einstein relative-velocity formula is valid both in t-time and τ-time, thus implying the validity of the latter in a more general system. It was thought that some interest might be attracted to a description of the most general system under which the Einstein formula holds.
An irreducible non-convex region
Authors: Ollerenshaw Kathleen , Mordell L. J.
Published: November 2008

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

has the minimum determinant Δ(K) = 2 and has an infinity of singular critical lattices. In this note I show that there is a unique irreducible star domain . That ís to say, there is just one star domain H contained in but different from K for which Δ(H) = Δ(K) = 2, and such that Δ(H′) < 2 for every star domain H′ contained in but different from H.
The Parseval formulae for monotonic functions. IV
Authors: Edmonds Sheila M.
Published: November 2008

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

hold in a number of cases when two of the functions concerned are monotonic, and that there are corresponding results for Fourier series. In some of these theorems, the existence of one side of the relevant equation (in the ordinary finite sense) is postulated; others contain no such existence condition, and show that either both sides are finite and equal, or both sides are ‘ + ∞’. In the case of sine transforms, we always obtain theorems of the second type (which is clearly the stronger); but cosine transforms are more troublesome. The first of the examples which we shall give is designed to show that the cosine formula is not always true in the ‘finite or infinite’ sense when two of the functions are monotonic.
A formalization of an ℵ0-valued propositional calculus
Authors: Rose Alan
Published: November 2008

In 1930 Łukasiewicz (3) developed an ℵ0-valued prepositional calculus with two primitives called implication and negation. The truth-values were all rational numbers satisfying 0 ≤ x ≤ 1, 1 being the designated truth-value. If the truth-values of P, Q, NP, CPQ are x, y, n(x), c(x, y) respectively, then
On the partial sums of series of hypergeometric type
Authors: Agarwal R. P.
Published: November 2008

About twenty years ago a number of results were given expressing the sum of n terms of an ordinary hypergeometric series with unit argument in terms of an infinite series of the type 3F2. The interest in the subject was aroused by a theorem due to Ramanujan, who stated that
Conformally Kähler manifolds
Authors: Westlake W. J.
Published: November 2008

Introduction. The present paper is concerned with the conformal geometry of Hermitian spaces. In the first part we find a necessary and sufficient condition for a Hermitian space to be conformally Kähler, that is, conformal to some Kähler space. The condition is that a certain conformal tensor, , vanishes identically. Then, defining a Hermitian manifold as in Hodge (3), we consider such a manifold where the restriction is made that at every point the tensor is zero. This will be called a conformally Kähler manifold, and conditions under which it may be given a Kähler metric are obtained. It is found that any conformally Kähler manifold may be given a Kähler metric provided it is simply-connected or that its fundamental group is of finite order.
Hydrodynamic stability of the flow in the laminar boundary layer between parallel streams
Authors: Lock R. C.
Published: November 2008

A method is given for determining the stability of small sinusoidal oscillations in the steady laminar flow of a horizontal wind over the surface of a liquid at rest (with particular reference to the flow of air over water), taking into account viscosity, gravity and surface tension. It is shown that there are two fundamental types of oscillation of the system, which may be called ‘water’ waves and ‘air’ waves, and curves showing the conditions for neutral stability of these two types of wave are given for a range of wind speeds from 100 to 300 cm./sec.
The consistency of cardinal series
Authors: Noble M. E.
Published: November 2008

1. Whittaker (7), generalizing a result of Ferrar(3), showed that the cardinal series based on the positive and negative integers is consistent in the sense that, if

and an integral function f (x) is denned by

then provided 0 < λ > 1

In this note I show that results of Paley- Wiener, Levinson and others on biorthogonal series can be made to yield a consistency theorem for cardinal series based on sequences λn, where

that is, series

where

We use Fourier transform technique and need hypotheses, a little more restrictive than Whittaker's, which would reduce in the case .
An invariant characterization of pseudo-valuations on a field
Authors: Cohn P. M.
Published: November 2008

This note is concerned with pseudo-valuations on a (commutative) fieldF. Apseudo-valuation on F† is defined as a real-valued functionWonFsuch that

(1)W(x) ≥ 0for all x∈FandW(0) = 0,but W does not vanish identically on F,

(2)W(xy) ≤W(x)W(y) for allx, y∈F,

(3)W(x−y) ≤W(x) +W(y)for all x, y∈F.

Wisnon-Archimedean, if it satisfies the ultrametric inequality

(3′)W(x−y) ≤ max {W(x),W(y)}.
Coupled forms of the differential equations governing radio propagation in the ionosphere
Authors: Clemmow P. C. , Heading J.
Published: November 2008

It is shown that the equations governing oblique propagation in a horizontally stratified ionosphere with an oblique magnetic field can be cast into a form suitable for solution by successive approximations. In the general case this ‘coupled’ form consists of four first order differential equations, each being associated with one characteristic wave. In the special cases (a) horizontal magnetic field, (b) plane of incidence perpendicular to the magnetic meridian, two second order coupled equations of a particular type can be derived, each of which is associated with a pair of corresponding upgoing and downgoing characteristic waves. These latter equations are similar to, and include, those already given by Försterling(7) for vertical incidence.

The cases in which there is no coupling are briefly considered from the point of view of the first order equations.

The general formulation provides a basis for assessing the validity of the standard ‘ray’ approximation, alternative to that developed by Booker (2), and brings out the nature of its breakdown in coupling and reflexion regions.

Specific applications and extensions of the theory are left for later consideration.
An investigation of the propositional calculus used in a particular system of logic
Authors: Harrop R.
Published: November 2008

This paper contains a summary of some of the results obtained by the author during a study of the propositional part (denoted by A) of a system of free-from-contradiction, type-free logic set up in 1950 by Ackermann (1). It is shown that in its original form the calculus does not possess the desired properties with respect to equivalence of formulae. A calculus A′, which it is shown may be considered as a ‘minimal’ satisfactory extension of A, is constructed. A′ is compared and contrasted with an alternative form A″ of A given by Ackermann in a paper published in 1952 (2). It is proved that A″ is a proper extension of A′. Among the properties of A′ and A″ which are obtained is the resuit that neither calculus possesses a finite complete model. Reference is made to the solution of the decision problem for A, and it is indicated that it is thought that the corresponding problems can probably be solved for A′ and A″. The proofs of many of the results mentioned in the paper are, if given in detail, rather long. In such cases, from space considerations, only outline proofs are given. Complete proofs are contained in (5), which reference also contains several additional properties of the calculi considered.
The statistics of scaled random events
Authors: Davies R. O. , Leech J. W.
Published: November 2008

When independent random events are scaled down by a factor n, the number of counts in a given time interval no longer obeys the Poisson Law. The correct distribution function is derived together with its moment-generating function and explicit formulae (n = 2, 4) for the first four moments about the mean. The theory is in agreement with experiments carried out with a scale-of-four counter. A more general treatment is given and is applied to find the variance for all even values of n.
A reciprocity theorem on the propagation of radio waves via the ionosphere
Authors: Budden K. G.
Published: November 2008

The reciprocity theorem for electrical systems which include a radiation link was believed to be true only when the media within the system have dielectric constants which are symmetric tensors. This condition is not fulfilled by the ionosphere, so that the reciprocity theorem is not generally applicable when the radiation link includes one or more reflexions from the ionosphere. It is here proved that, when the ionosphere is horizontally stratified, and when the path from transmitter to receiver is in the magnetic meridian (north-south and south-north transmission), the reciprocity theorem applies (a) when the transmitting and receiving aerials both radiate or receive waves whose electric vector is in the plane of incidence, and (b) when both aerials radiate or receive waves whose electric vector is horizontal. Further, (c) if the electric vector radiated or received is horizontal for one aerial and in the plane of incidence for the other, then there is reciprocity in signal amplitude, but the phase changes for transmission in the two directions differ by 180°. These results are valid for any law of variation of electron density and collision frequency with height. They are based on a ‘full-wave’ theory, and therefore apply to all frequencies. They are unaffected if the path includes multiple reflexions, and if allowance is made for the curvature of the earth.
Correspondences with unequal valencies
Authors: Scott D. B.
Published: November 2008

In an earlier note (3) we have shown by examples that it is possible to have an irreducible non-degenerate point-point correspondence on an algebraic surface such that the correspondence and its inverse have arbitrary unequal negative valencies. We have also remarked that, by multiplying such a correspondence by one with equal positive valencies, we can find correspondences with unequal positive valencies. In (4) we have given reasons, falling short of complete proof, which suggest that, on any surface whose intersection group is cyclic, correspondences with unequal Albanese valencies exist. But the only proofs of the existence of such correspondences are the examples due to the author, Severi and Todd in (3) which all refer to ruled surfaces. Severi has rightly pointed out that these examples are not entirely satisfactory, since, as ruled surfaces have zero geometric genus, there is a confusion between valency correspondences in Severi's and in Albanese's senses, whereas the theoretical arguments of (4) apply only to Albanese valency correspondences. In this note we remark that on hyperelliptic surfaces there exist correspondences with valencies γ and γ3 in the two directions for any integer γ, and that unless |γ| = 1, in which case the two valencies are the same, these correspondences cannot have a Severi valency in either direction. This result is an immediate consequence of some classical results on Abelian varieties (c.f.(2)), but the application seems of sufficient interest to note explicitly. We remark that we are still not able to find examples of correspondences which are irreducible and non-degenerate and whose valencies in the two directions are of opposite sign; it seems a plausible conjecture that such correspondences do not exist.
Finite groups having unique proper characteristic subgroups. I
Authors: Taunt D. R.
Published: November 2008

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.
Note on nilpotent and solvable algebras
Authors: Patterson E. M.
Published: November 2008

In general, the class of a nilpotent linear algebra of dimensionnis at mostn+ 1, and the index, or derived length, of a solvable linear algebra of dimensionnis at mostn. In this note it is shown that, for a nilpotent linear algebra of dimensionnsatisfyingx2= 0 for allx, the class is at mostn; and bounds are obtained for the indices of solvable Lie algebras.
On conformal mappings of domains of infinite connectivity
Authors: Brown L. J. M.
Published: November 2008

1. In the theory of conformal mapping, there are many problems associated with domains of infinite connectivity which have not yet been fully explored. We consider, in this paper, some of these problems in relation to domains in the closed complex plane. In the following all domains are assumed to be plane domains and all mappings to be (1, 1) and conformal in the domains in which they are defined.
The central limit theorem for m-dependent variables
Authors: Diananda P. H.
Published: November 2008

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.
The retardation of alpha particles when capture and loss occur
Authors: Scott J. M. C.
Published: November 2008

The usual formula for the rate of loss of energy by a fast particle contains a factor z2, the square of the charge, which might suggest that He+ should lose energy at a quarter of the rate that He++ does. It is shown that when Born's approximation is applicable the energy-loss is reduced instead by a factor of about ⅝.
A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group
Authors: Cohn P. M.
Published: November 2008

Let L be a Lie ring and denote the product of x and y in L by [x, y]. The ring L is said to satisfy the Engel condition (cf. (1)), if for every pair of elements x, yεL there is an integer k = k(x, y)such that

If k(x, y) can be taken equal to a fixed integer n for all x, y ε L then L is said to satisfy the n-th Engel condition.
Hypergeometric Mellin transforms
Authors: Slater L. J.
Published: November 2008

The Mellin transforms of generalized hypergeometric functions are discussed in this paper, and it is shown how some of the most general integrals of the Mellin type can be deduced from them. Four general theorems are considered and a number of special cases are given in detail.
Extensions of a renewal theorem
Authors: Smith Walter L.
Published: November 2008

Let X1, X2, X3,… be a sequence of independent, identically distributed, absolutely continuous random variables whose first moment is μ1. Let Sk = X1 + X2 + … + Xk, and let fk(x) be the frequency function of Sk, defined as the k-fold convolution of f1(x). When f1(x) has been defined for all x, fk(x) is uniquely defined for all k, x. Write
The self-consistent field for Mo+
Authors: Ridley E. Cicely
Published: November 2008

The results of a self-consistent field calculation without exchange for Mo+ are presented.
Centre-of-mass state vectors
Authors: Hamilton J.
Published: November 2008

The properties of centre-of-mass systems at high energies are discussed, and the single time form of field theory is used to set up state vectors that describe the simpler centre-of-mass states of fields. These states, which are labelled by the eigenvalues of total angular momentum, are expanded in terms of the spherical harmonic resolution of the fields about a fixed point. A discussion of the application of the method to pion-nucleon scattering is given.
Molecules With tetrahedral and octahedral symmetry. IV. A modified Thomas–Fermi scheme for molecules with heavy atoms in the outer positions
Authors: Coulson C. A. , March N. H.
Published: November 2008

In Part I of this series it was shown that for CCl4 the approximation of smoothing out the Cl nuclei over the surface of a sphere did not lead to reasonable results for the Thomas–Fermi distribution of electrons in the molecule. An alternative method in which the inner-shell Cl electrons were first compressed into the nuclei was not theoretically satisfactory because of the neglect of the exclusion principle. In this paper therefore a general scheme is presented in which we describe the inner-shell electrons of the outer atoms wave mechanically, assuming them to be undisturbed by molecular binding, but still use a Thomas–Fermi description for the valence electrons. A method by means of which the exclusion principle can be imposed is formulated. The following paper of the series deals with the application of the method described here to obtain an improved valence electron distribution in CCl4.
Analytical degeneration of complete twisted cubics
Authors: Alguneid A. R. , Hodge W. V. D.
Published: November 2008

In Schubert's book Abzählende Geometrie(3) there are listed, with no formal verifications, eleven first-order degenerations of the twisted cubic curve in S3. The curve, in this connexion, was seen by Schubert as a union C = (CL, CE, CT) of three simply infinite element systems, the locus CL of its points, the envelope CE of its osculating planes and the system CT of its tangent lines, the complete twisted cubic so envisaged being regarded as a single geometric variable of freedom 12, and any degeneration (projective specialization) of it, C¯ = (C¯L, C¯E, C¯T), being said to be of the first order if it has freedom 11. Schubert used the symbols λ, λ′, k, k′ w, w′, θ, θ′, δ, δ′, η to denote the eleven first-order degenerations in question, the dashed symbol denoting always the dual of that denoted by the undashed. In this paper we exhibit analytically the existence of the above degenerations and indicate further how degenerations of higher order can be systematically investigated. We use for this purpose the analytical theory of complete collineations which has just recently been developed (1,4), and also some of the ideas contained in van der Waerden's earlier development of his general method of degenerate collineations (5).
The numerical evaluation of a class of integrals. II
Authors: Das S. C.
Published: November 2008

Consider the integral

where x1, x2, …, xN are jointly distributed in a multivariate normal distribution f(x1, x2, …, xN) with (pij) as the correlation matrix. The integral has been expressed in an infinite series of tetrachoric functions for N≥2. The infinite series is not only complicated, but also is very slowly convergent and is consequently not of much practical use. Plackett (8) obtains a reduction formula for expressing normal integrals in four variates as a finite sum of single integrals of tabulated functions. These integrals have then to be evaluated by a rather awkward numerical quadrature.
On a probability problem arising in the theory of counters
Authors: Takács L.
Published: November 2008

The paper contains some remarks on the problems treated by Hammersley(9). With the aid of a new and simple method the exact solutions of Hammersley's problems and the asymptotic behaviour of the solutions are found. The problems in question are: A Poisson process is transformed into a new process by a Type II counter, and this new process is transformed again by a Type I counter. The distribution of the number of events occurring in each transformed sequence is to be determined.
The G-values of various games
Authors: Guy Richard K. , Smith Cedric A. B.
Published: November 2008

A disjunctive combination of a finite set of two-person games Γ1, Γ2, …, Γk may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual games. If, in addition, the conditions are imposed that

(i) a player loses if unable to move (in any game),

(ii) the games are impartial, i.e. the allowable moves from any position do not depend on which player is about to play (or on the previous moves, though these can be ‘built in’ to the position if necessary),

(iii) the games are of bounded play, i.e. for each game Γi corresponding to any initial position Pj there is an integer bij such that the game must terminate after at most bij moves, then Grundy (6) has shown that there is a function G(P) (called by him Ω(P)) of the positions P with the following properties:

(1a) G(P) = 0 for a terminal position, from which no move is possible; for other positions G(P) is the smallest non-negative integer different from all values of G(Qi), where there is a permissible move from P to Qi,

(1b) for a disjunctive combination of games, G for the combined position is the nim-sum of the G's the individual positions. By the nim-sum, we mean that each separate G is to be written in the scale of 2, as Σar 2r, and then in forming the sum, the ar's are to be added mod 2 for each value of r, as in the theory of Nim((1), (7), (8), pp. 36–8).
A theorem of homological algebra
Authors: Rees D.
Published: November 2008

The present note is concerned with the proof and applications of the following theorem: Let A be a commutative ring, N be an A-module, and g1, …, gk be elements of A such that (g1,…, gi−1) A:gi = (g1,…, gi−1) N:gi = (g1,…, gi−1) N (i = 1,…, k), so that 0:g1 = 0 in N. Let g denote the ideal (g1 …, gk), let B = A/g and let M be any A-module such that gM = 0. then

where, on the right-hand side of (ii), M, N/gN are considered as B-modules.
Approximate self-consistent fields for In3+ and Sb3+
Authors: Ridley E. Cicely
Published: November 2008

Preliminary self-consistent field calculations without exchange have been performed for the ground states of the structures In3+ and Sb3+. The results are presented.
A proof of the comparison theorem for spectral sequences
Authors: Zeeman E. C.
Published: November 2008

The comparison theorem is an algebraic theorem, which corresponds to the following topological situation. A map between two fibre spaces induces a homomorphism between the two corresponding spectral sequences relating the homologies of the base, fibre and fibre space; if the map induces an isomorphism on any two of these quantities, then it also does on the third. J. C. Moore (l) proved the theorem using a mapping cylinder, which requires that the spectral sequences arise from filtered differential groups. The present proof assumes only the existence of the spectral sequences from the E2 terms onwards. Moreover, we generalize the theorem, assuming that the homomorphisms on the two given quantities are isomorphic only up to given dimensions, and deducing the same for the third quantity. For algebraic translucence we use a simpler hypothesis (property (iii)) than Moore's concerning the E2 terms. However, if his hypothesis is assumed, together with the existence of E1 terms, then his theorem can be deduced as a corollary. The analogous result for cohomology is similar, and is stated at the end.
The computation of quadrupole line strengths
Authors: Garstang R. H.
Published: November 2008

The methods introduced into the theory of atomic spectra by Racah are applied to the problem of the computation of the strengths of electric quadrupole lines. Detailed numerical results are given for two configurations of equivalent d-electrons.
On the unitary irreducible representations of the universal covering group of the 3 + 2 deSitter group
Authors: Ehrman Joachim B.
Published: November 2008

The unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.

Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)

In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.

Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.
Linear measures of some sets of the Cantor type
Authors: Mcminn Trevor J.
Published: November 2008

1. Introduction. Let 0 < λ < 1 and remove from the closed unit interval the open interval of length λ concentric with the unit interval. From each of the two remaining closed intervals of length ½(1 − λ) remove the concentric open interval of length ½λ(1 − λ). From each of the four remaining closed intervals of length ¼λ(1 − λ)2 remove the concentric open interval of length ¼λ(l − λ)2, etc. The remaining set is a perfect non-dense set of Lebesgue measure zero and is the Cantor set for λ = ⅓. Let Tλr be the Cartesian product of this set with the set similar to it obtained by magnifying it by a factor r > 0. Letting L be Carathéodory linear measure (1) and letting G be Gillespie linear square(2), Randolph(3) has established the following relations:
The Stokes phenomenon and certain nth-order differential equations I. Preliminary investigation of the equations
Authors: Heading J.
Published: November 2008

1. Introduction. The object of this investigation is to obtain by means of contour integrals exact solutions of certain nth-order differential equations, together with their n independent power-series solutions, their asymptotic solutions and the relationship between these two types of solution. The Stokes phenomenon associated with the changing constants in these asymptotic solutions will then be investigated by various methods. Paper I is concerned with the properties of the various contour-integral solutions of the equations under consideration. The manner in which the arbitrary constants in the general asymptotic solution must change as the argument of the independent variable z varies is dealt with in paper II. This phenomenon is considered in a way that has proved profitable for the approximate solution of more general linear differential equations of the nth order that are approximately identical with those considered here hi certain regions of the complex z–plane. In later publications, these approximations will be described together with their application to explicit physical problems.
Coupled forms of the differential equations governing radio propagation in the ionosphere. II
Authors: Budden K. G. , Clemmow P. C.
Published: November 2008

The four first-order ‘coupled’ equations governing the propagation of electromagnetic waves in the ionosphere, previously obtained in symbolic matrix form (Clemmow and Heading (4)), are expressed explicitly in terms of the ionospheric parameters. The physical significance of the equations is illustrated by considering the energy flux in one characteristic wave when coupling and damping are neglected. Three special cases are then discussed for which second-order coupled equations are also given, namely, the cases of (a) vertical incidence with oblique magnetic field, (b) oblique incidence with vertical magnetic field, (c) horizontal magnetic field in the plane of incidence. For case (a) the second-order equations are those previously derived by Försterling(5).

The form of the coupled equations is physically illuminating and, in principle, suitable for solution by successive approximations. Extensive numerical work has indeed been carried out on the second-order coupled equations in case (a) (e.g. Gibbons and Nertney(6)), and it is probable that the first-order coupled equations would prove more advantageous. The present authors, however, feel that better methods are available for purely numerical work (e.g. Budden(3)), and that the chief interest of the coupled form is that it shows the scope and limitations of the physical conception of characteristic waves.
The transverse flexure of thin elastic plates supported at several points
Authors: Bassali W. A.
Published: November 2008

This paper depends upon the method developed by Kolossoff and Muskhelishvili for problems of plane elasticity and later extended to plate problems by Lechnitzky. Exact solutions in closed forms are obtained for the problem of a thin circular plate supported at several interior or boundary points and normally loaded over the area of an eccentric circle, the load being symmetrical with respect to the centre of the circle and the boundary of the plate being free. Explicit formulae for the deflexion, the bending and twisting moments and shearing stresses are given at any point of the plate. As limiting cases plates in the form of the infinite plane and half plane are also considered.
Orthogonal and oblique projectors and the characteristics of pairs of vector spaces
Authors: Afriat S. N.
Published: November 2008

The statistical operation of multiple linear regression by least squares is equivalent to the orthogonal projection of vectors of observations on a space spanned by vectors of observations; and a partial regression can be similarly represented as an oblique projection. This connexion between a statistical and a formal algebraical operation gives the main source of interest for this investigation. Its object is to develop algebraical theory which supplies terms necessary for a unified algebraical and geometrical formulation of concepts in multivariate analysis. An exposition of the statistical theory is to be made in a separate paper.
An extension of the general variational principle of transport theory
Authors: Moliner F. Garcia , Simons S.
Published: November 2008

A variational principle is obtained for the Boltzmann transport equation in the presence of a magnetic field and modified by boundaries, when the standard formulation of Kohler breaks down because of the lack of microscopic reversibility. The connexion with entropy production is discussed.
On the existence of finite models and decision procedures for propositional calculi
Authors: Harrop R.
Published: November 2008

1. Introduction. In this paper we consider certain general properties of propositional calculi. Two forms of the definition of a finite model of such a calculus are discussed, these forms differing in the manner prescribed for the satisfaction of the rules of the calculus. The methods of definition are shown to be equivalent for the application of the finite model method for the demonstration of the unprovability of formulae in the calculus. It is further proved that, although the decidability of a calculus follows from the existence of a finite model counter-example for each unprovable formula of the calculus, the converse result is not true.
Random motion and analytic continued fractions
Authors: Good I. J.
Published: November 2008

In a previous note (Good (7)) it was shown that there is an intimate connexion between Legendre polynomials and ‘trinomial’ random walks, that is walks on a one-dimensional lattice with the same probabilities p–1p0, p1 at each point of taking a step of – 1,0,1. The purpose of the present note is to point out that if the probabilities depend on the current position of the particle, then there is scope for applications of the theory of analytic continued fractions.
Equivalent functions: hybrids and Wannier functions
Authors: Altmann S. L. , Coulson C. A.
Published: November 2008

In the last two decades a family of closely related functions have been introduced for the solution of quantum-mechanical problems. In dealing with atomic and molecular systems Pauling (12) introduced in 1928 linear combinations of the eigenfunctions of a single atom, which were called hybrids. These have specific symmetry properties which were utilized by Kimball (6) to construct a group-theoretical method for their determination. More recently Lennard-Jones (8) defined linear combinations of molecular eigenfunctions which he called equivalent orbitals. These are formally similar to the hybrids and Lennard-Jones took over the Kimball method for their determination.
A modified form of Siegel's mean value theorem. II
Authors: Macbeath A. M. , Rogers C. A.
Published: November 2008

The object of this note is to prove the following generalization (from Riemann to Lebesgue integrable functions) of a theorem already proved by the authors (1). The notation and terminology of the earlier paper are used.
Summability and bounded sequences
Authors: Petersen G. M.
Published: November 2008

It is our purpose in this paper to present concise proofs of four theorems which take up the greater part of a paper due to Brudno (1). The proofs of Theorems 1 and 2 have appeared already ((2), (3)), but we present them again so that comparisons can be made with the other two.