bluuebunny/crossref_metadata_2025 huggingface dataset:
The behaviour of the universe
Authors: Littlewood D. E.
Published: November 2008

In a previous paper (3) the method of conformai transformations was employed to study the behaviour of a universe assuming the perfect cosmological principle, i.e. that apart from local irregularities the general aspect of the universe is the same at all points independent of place and time. To conform with Einstein's general theory an anisotropic cosmological term had to be postulated.
The effect of configuration interaction on forbidden line strengths
Authors: Garstang R. H.
Published: November 2008

The calculation of the strengths of forbidden lines in the 1s22s22p2 configuration is extended to include the effect of interaction with the 1s22p4 configuration. For this case it is shown that if observed energy levels are used instead of theoretical energies in line-strength formulae which neglect configuration interaction the major part of the effect of configuration interaction is nevertheless taken into account. The results are illustrated by detailed computations on the Ca xv spectrum. The 1s22s22p4 configuration is also discussed.
Logic on electronic computers: a practical method for reducing expressions to conjunctive normal form
Authors: Routledge N. A.
Published: November 2008

In § 1 we introduce our system and prove a theorem about its syntax. In § 2 we recall some stock results about the propositional calculus. In § 3 we consider a method of deriving an expression from a given expression and a real number. In § 4 we use this to derive a sequence of expressions from a given expression. In § 5 this sequence is shown to be just all the terms of a conjunctive normal form of the given expression. In § 6 we note that we may not need to produce all of these terms. In § 7 we describe a practical method (suitable for a binary digital electronic computer) which results from all this, and in § 8 we attempt to explain just why this is so.
The use of the Airy functions in a potential barrier problem
Authors: Jeffreys Bertha
Published: November 2008

The exact solution for a trapezoidal potential barrier is given in terms of Airy functions and trigonometrical functions. Numerical values are found for the ratio of the amplitudes of the transmitted and reflected waves. These are compared with those given by the WKBJ approximation.
Phase calculations for nuclear scattering on the Pilot ACE
Authors: Robertson H. H.
Published: November 2008

The Pilot ACE has been used to solve integro-differential equations appearing in the theory of nuclear scattering. The application of finite-difference approximations results in a set of simultaneous algebraic equations which are solved to obtain the wave function and from this the phase is evaluated.
Kinematic wave theory of bottlenecks of varying capacity
Authors: De S. C.
Published: November 2008

This paper discusses a problem in traffic flow by the method of kinematic waves developed by Lighthill and Whitham(1,2). The theory of kinematic waves introduced by Lighthill and Whitham has been extended here to include the case when the flow q varies with the time, and it is seen that the expression for the wave velocity is of the same form as before, namely, ∂q/∂k, where q is the flow (quantity passing a given point in unit time) and k is the concentration (quantity per unit distance). The theory is applied to the problem of estimating how a uniform oncoming flow behaves on entering a bottleneck, the capacity of which varies with time. This capacity has initially a higher value than the oncoming flow but falls at a uniform rate to a lower value, where it remains constant for a time, and again rises at a uniform rate to the original value. A shock wave is found to move back from the bottleneck, and later forward again and through it, much as in the case of a bottleneck of constant capacity with varying oncoming flow studied by Lighthill and Whitham.
Entry of conducting and non-conducting fluids in pipes
Authors: Shercliff J. A.
Published: November 2008

The problem of estimating the entry length over which the laminar motion of an electrically conducting, viscous fluid in a pipe under a transverse field adjusts to a steady state from some initial state is linearized by a Rayleigh approximation adapted for cases where there is a pressure gradient. Solutions are found which express the difference between the initial and ultimate velocity profiles as a series of orthogonal, exponentially decaying terms. Approximate entry lengths for the most persistent terms are calculated for flow between two infinite parallel planes and in rectangular tubes when magnetic effects are present. Entry lengths in circular and rectangular pipes and between infinite planes are calculated in the case where magnetic effects are negligible to permit a comparison of the results with those derived by other methods. Pressure losses at entry are considered in the simpler cases.
Applications of the theory of cluster sets to a class of meromorphic functions
Authors: Collingwood E. F. , Lohwater A. J.
Published: November 2008

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined by

where m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned by

from which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.
Eigenvalue problems treated by finite-difference methods. II. Two-dimensional Schrödinger equations
Authors: Bolton H. C. , Scoins H. I.
Published: November 2008

A discussion is given of the convergence of the eigenvalues Λ (N) of two-dimensional finite-difference equations towards the eigenvalues Λ of the corresponding second-order differential equation, and it is shown that

where h = N−1 and ν2 is a constant. As in our previous paper (4), this can be used to make an accurate estimate of λ by extrapolating to h = 0. After an account of the relaxation method used for computing Λ(N) and a discussion of the residual vector, results are presented for an approximation to the lowest spatially symmetric and antisymmetric states of two electrons in a sphere, interacting through their Coulomb potential.
On the continuous solution of integral equations by an electronic analogue. I
Authors: Fisher Michael E.
Published: November 2008

A scheme is proposed for solving a class of integral equations by electronic analogue computing techniques in times as short as one-tenth of a second. The scheme utilizes a recently developed high-speed analogue function store for carrying out a special iterative procedure which is shown to be more efficient than the classical Neumann process. The problem of the kernel generation at high repetition rates is considered and a novel method based on pivotal function generators is described. Likely errors are analysed and an overall accuracy of the order of 1% is shown to be attainable with known techniques.
Properties of random functions
Authors: Palmer D. S.
Published: November 2008

On page 674, equation (5) and the expression for H in the line above it are in error and should be deleted. The correct expressions are more complicated and will not be given.

On page 679, line 5 from the bottom should read

and the equation in the bottom line should read
An application of Sturm-Liouville theory to a class of two-part boundary-value problems
Authors: Karp Samuel N.
Published: November 2008

A simple solution of a general problem involving a bifurcated wave guide is presented. The purpose of the work is to explain a new and simple method of solving such problems and to exhibit an organic connexion between Sturm–Liouville theory and the theory of two-part boundary-value problems.
On the Crank-Nicolson procedure for solving parabolic partial differential equations
Authors: Juncosa M. L. , Young David
Published: November 2008

Proof of convergence of the Crank-Nicolson procedure, an ‘implicit’ numerical method for solving parabolic partial differential equations, is given for the case of the classical ‘problem of limits’ for one-dimensional diffusion with zero boundary conditions. Orders of convergence are also given for different classes of initial functions. Results do not support the validity of so-called h2-extrapolation in some cases.
The foundations of a generalization of gravitation theory
Authors: Moffat John
Published: November 2008

1. Introduction. Among the more notable attempts to derive a generalization of Einstein's gravitational theory is the recent one of Einstein and Schrodinger ((1)–(8)). This was formulated by dropping the symmetry of the fundamental tensor gμν and the components of the affine connexion. The most serious defect of these non-symmetric theories is that the field equations, in their original form, do not determine the motion of electrically charged particles in an electromagnetic field, as has been proved by Infeld(9), Callaway (10) and Bonnor (n). Together with the lack of an energy-momentum tensor and a geometric description of the paths of charged particles, this seems to indicate that the concept of motion is missing in this type of theory. It is clear that one of the most important results which should follow from a generalization of Einstein's gravitational theory is the correct equations of motion of charged particles in an electromagnetic field.
Formation of an alpha-group in the shell-model of a heavy nucleus
Authors: Strachan Charles
Published: November 2008

By means of a variation principle applied to a nuclear wave function for four nucleons in the field of a closed core, equations are derived giving the relative proportions of a component describing an alpha-particle group and components describing the four nucleons in shell-model states with spin-orbit coupling wave functions, the two neutron states and the two proton states each paired to give zero resultant angular momentum. Numerical calculations, taking into account only 1h2/3 states for protons and the 2g3/2 states for neutrons, give the alpha-group wave function a relative weight (in the linear combination) of about 1%.
The second-order effect of free electrons on lattice conduction
Authors: Pyle I. C.
Published: November 2008

Second-order perturbation theory, in conjunction with the usual treatment of electron-phonon interaction, allows us to calculate the correction to the first-order result for the scattering of phonons by electrons. It is shown that the second-order term is much smaller, and therefore negligible. This justifies the use of the first-order theory in the treatment of the interaction in metals and semi-conductors.
A simplified method for integrating over Feynman histories
Authors: Brush S. G.
Published: November 2008

In Feynman's ‘space-time’ formulation of quantum mechanics the Green's function for the Schrödinger equation is defined by an integral over all histories of the system. By integrating over one-parameter sets of functions, one gets the same Green's function as by integrating over a Fourier series, in simple cases. The method may be useful for estimating the result in cases when the integration over all histories cannot be performed exactly.
Potential and stream function of a vortex disk in the presence of a rigid sphere
Authors: Martinek J. , Yeh G. C. K. , Zorn H.
Published: November 2008

In Sadowsky and Sternberg(1), elliptic integral representations of axially symmetric flows suggested essentially by Weinstein's work (2) on axially symmetric flows of an ideal incompressible fluid have been considered. The physical and practical significance of their investigation was twofold. First, a more transparent analytic description was obtained than that afforded by the representation through discontinuous integrals of Bessel functions originally used by Weinstein. Secondly, by superposition of such axially symmetric flows and appropriately chosen uniform streams, a variety of technically significant flows around solids, annular bodies and half-bodies of revolution can be constructed. On the other hand, in an attempt to utilize the symmetry properties of the potential field in reference to a spherical boundary, Weiss(3) has derived an (exterior) ‘sphere theorem’ by applying Kelvin's transformation and using the potential function. Butler (4) later obtained, by means of the Stokes stream function, a sphere theorem applicable to axially symmetric flows only. Ludford, Martinek and Yeh(5) found then the ‘interior sphere theorem’ as well as a theorem satisfying the general radiation condition. A general sphere theorem was consequently conceived, valid for all linear boundary conditions, and was recently published by Yeh, Martinek and Ludford (6). The significance of these theorems again lies in their application to physical problems. They often give closed form expressions of the disturbance potential in terms of higher transcendental functions whenever the undisturbed potential is given by means of transcendental functions. Furthermore, when the singularities (discrete or distributed) he near the perturbing spherical boundary the usual treatment by expansion in spherical harmonics leads to solutions in the form of infinite series which are, because of slow convergency, unsuited for numerical computation. For this situation the sphere theorems provide a remedy in the form of neat formulae readily adaptable to numerical work.
Abelian groups, graphs and generalized knights
Authors: Nash-Williams C. St J. A.
Published: November 2008

If g is a set of generatore of an enumerably infinite Abelian group A, it is proved that the elements of A can be arranged in both a one-ended and an endless infinite sequence in which successive terms differ by ± an element of g, except that the one-ended arrangement is impossible if g is finite and the rank of A is 1. Let ν be a cardinal number. Consider an infinite ‘chessboard’ whose positions are those lattice points of ν-dimensional space which have only finitely many non-zero coordinates. A piece associated with this chessboard is a generalized knight if every vector obtainable from a move of the piece by permuting its components and changing the signs of a subset of them is itself a permitted move. It is ascertained which positions a given generalized knight can reach in a finite sequence of moves starting at the origin, and whether or not, if it can trace out the whole chessboard in (i) a one-ended, (ii) an endless infinite sequence of moves visiting each position exactly once.
Hilbert's function in a semi-lattice
Authors: Learner A.
Published: November 2008

Samuel (1) introduced a generalized Hilbert function, written Xq(r, a) and defined for arbitrary ideals a in a local ring Q with maximal ideai m. where q is m-primary.

Northcott(2) proved that for a homogeneous ideal ã in a polynomial ring A[X1, …, Xn], where A = Q/q, this is equal to the ordinary Hilbert function χ(r, ã).
Extensions of ideals in associative algebras
Authors: Smithies F.
Published: November 2008

1. Let A be an associative algebra over a commutative field K; A is then a vector space over K, and multiplication is defined in A in such a way that

for x, y, z in A and λ in K. We suppose in addition that A does not contain a unit, and we denote by Ae the algebra obtained by adjoining a unit e to A.
On Methods of Summability Based on Integral Functions. II
Authors: Borwein D.
Published: November 2008

In two recent papers ((l), (2)) I investigated inclusion relations between IF (integral function) methods of summability. In the present paper the investigation is continued with the aid of standard Mellin transform theory and results due to Rogosinski and to Kuttner.
Quadratic Lagrangians and gauge-invariance in covariant field theories
Authors: Stephenson G.
Published: November 2008

The idea of gauge-invariance in general relativity was first introduced by Weyl(1) who proposed that the field equations of gravitation should be invariant, not only under the general group of coordinate transformations, but also under the gauge-transformation

where is the symmetric metric tensor, is the symmetric affine connexion and λ(x8) is an arbitrary scalar function of the coordinates. In this way it was possible to introduce into the theory a four-vector Ak which in consequence of (1·1) transformed as

such that the six-vector remained an invariant quantity under the gauge-transformation. It was Weyl's hope that by widening the invariance properties gauge-transformation. It was Weyl's hope that by widening the invariance properties of general relativity in this way the vector Ak and its associated six-vector Fik could be interpreted as representing the electromagnetic field. However, no obvious or unique way of doing this was found. More recently (see Stephenson (2,3) and Higgs (4)) gaugeinvariant variational principles formed from Lagrangians quadratic in the Riemann—Christoffel curvature tensor and its contractions have been discussed by performing the variations with respect to the symetric and symetric independently (following the palatini method).
Some Theorems on the Relation between Riesz and Abel Typical Means
Authors: Kuttner B.
Published: November 2008

Let {λn} be a sequence of non-negative numbers increasing to infinity. Let be any series. Following the usual terminology, we say that the series is summable (R, λn, k) to s if as u → ∞, where . If, further, u−kA(k)(u) is of bounded variation in (0, ∞), we say that the series is summable s. We say that the series is summable (A, λn) to s if converges for all σ > 0, and tends to s as σ →0+. If A, B are two summability methods, we write A → B (‘A implies B’) if they have the property that any series summable A is necessarily summable B to the same sum.
Ramanujan's Continued Fraction and the Bauer-Muir Transformation
Authors: Singh V. N.
Published: November 2008

Ramanujan's Continued Fraction may be stated as follows: Let where there are eight gamma functions in each product and the ambiguous signs are so chosen that the argument of each gamma function contains one of the specified number of minus signs. Then where the products and the sums on the right range over the numbers α, β, γ, δ, ε: provided that one of the numbers β, γ, δ, ε is equal to ± ±n, where n is a positive integer. In 1935, Watson (3) proved the theorem by induction and also gave a basic analogue. In this paper we give a new proof of Ramanujan's Continued Fraction by using the transformation of Bauer and Muir in the theory of continued fractions (Perron (1), §7;(2), §2).
Lower-bound energies and the Virial theorem in wave mechanics
Authors: Caldow G. L. , Coulson C. A.
Published: November 2008

Several forms of the lower-bound variational method for the calculation of the eigenvalues in a wave-mechanical problem are considered, and compared; the particular case of the harmonic oscillator being chosen. All forms have certain unsatisfactory features, but some of them are considerably worse than others. One reason why calculations of lower bounds are in general less satisfactory than Ritz-type calculations of an upper bound is shown to be that whereas, in the presence of a scale factor, this latter wave-function satisfies the virial theorem, in none of the lower-bound wave-functions is this true. Similar calculations are reported for the ground state of the helium atom.
The low-frequency scattering by a perfectly conducting strip
Authors: Jones D. S. , Noble B.
Published: November 2008

Various authors, e.g. Sommerfeld (8), Bouwkamp(i), Groschwitz and Hönl(2), Müller and Westpfahl(7), have considered approximate solutions to the problem of the diffraction of a plane wave by a narrow perfectly conducting strip. If the electric vector is polarized parallel to the strip they find that the scattering coefficient, for normal incidence, is given by, where k is the wave number, 2b the width of the strip and γ(= 0·5572…) being Euler's constant. Any attempt to discuss the convergence of such a pseudo-power series requires a knowledge of the higher order terms and naturally leads to the question of what is the proper expansion parameter.
On some dual series equations and their application to electrostatic problems for spheroidal caps
Authors: Collins W. D.
Published: November 2008

This paper is concerned with dual series equations of the form where is the associated Legendre function of degree n + m and order −m of the first kind, Hn is a known coefficient of n, f(θ) and g(θ) are given functions of the variable θ and the equations are to be solved for the unknown coefficients An. In hydrodynamics, elasticity and electromagnetism such equations occur in many problems involving a spherical or spheroidal boundary when different conditions are prescribed on different parts of the boundary. These dual series equations are analogous to the dual integral equations which arise in the application of Hankel transforms to problems involving plane or paraboloidal boundaries, the theory and applications of these latter equations having been discussed by several writers amongst whom are Sneddon, Noble and Lebedev(1), (2), (3). The main result of this paper is that the solution of the dual series equations (1·1) and (1·2) for m a positive integer or zero can be reduced to the solution of a Fredholm integral equation of the second kind in one independent variable. The restriction that m be a positive integer or zero covers those dual series equations which arise in three-dimensional boundary-value problems but excludes those which arise in two-dimensional ones where m is . These latter equations have, however, been discussed in a recent paper by Tranter (13).
Hyperspaces of a CANR*
Authors: Smith D. Hammond
Published: November 2008

If X is a compact Hausdorff space we denote by S(X), and by C(X), the hyperspaces of X consisting of all non-empty closed sets, and all non-empty connected closed sets. The topology in each case is the finite topology of Michael ((6), Definition 1·7), in which a sub-base for the open sets is taken consisting of all sets of either of the forms {F|F ⊂ G} and {F|F ∩ G ≠ φ} (where G is any open set of X). Michael has shown that S(X) is also compact Hausdorff ((6), Theorem 4·9·6), and S(X) contains in an obvious way sets which are homeomorphic with C(X) and X itself. We recall that if Xis also a metric space, the topology induced on S(X) (and on C(X)) by Hausdorff's metric is the same as the finite topology ((6), Proposition 3·6).
The number of trees with nodes of alternate parity
Authors: Scoins H. I.
Published: November 2008

Interest in the theory of linear graphs has grown recently in several fields which at first sight seem rather far apart. Recent publications include an excellent review by Bryant(1) of electrical network theory, statistical mechanical work by Sykes(2) and in the field of linear programming a paper by Glicksman, Johnson and Eselson (3). It is in this latter field that the subject of this paper arose, but we shall confine attention here to the graph theoretical aspect alone.
A Note on Cubic Equations
Authors: Fowler James
Published: November 2008

It was proved by Davenport (2) that a cubic form in n variables with rational coefficients always represents zero if n > 32; and this condition was later (3) relaxed to n ≥ 29. The object of the present paper is to establish conditions under which a cubic form will represent every rational number other than zero, for rational values of the variables. The main result is as follows.
Additive Functionals on Groups
Authors: Hayes Allan
Published: November 2008

The kernel of a non-trivial linear functional φ on a linear space E is a maximal proper linear subspace of E which determines φ up to a non-zero multiple. Does a similar result hold for homomorphisms of a group G into the additive group R of the real numbers?
The Effect of a Magnetic Pole on the Energy Levels of a Hydrogen-Like Atom
Authors: Eliezer C. J. , Roy S. K.
Published: November 2008

The possible existence of a magnetic pole has been under discussion for many years. Dirac worked out in detail an electro-dynamics in which a single magnetic pole is assumed to exist (1,2). He has shown that the existence of such a pole leads naturally to the quantization of charge, the magnetic pole strength g and the electric charge e being related by the equation where p is an integer, h is Planck's constant divided by 2π, and c is the velocity of light.
A static spherically symmetric solution of the Einstein—Maxwell—Yukawa field equations
Authors: Stephenson G.
Published: November 2008

The general relativistic field equations for regions of space containing electromagnetic fields but no matter are

,

where

,

and κ = 8ΠG/c4 is the gravitational constant. Here giκ is the metric tensor and Fiκ is the electromagnetic field tensor which satisfies Maxwell's equations if

and

,

where Ai is the electromagnetic four-potential.
A note on the coefficients of Hilbert characteristic functions in semi-regular local rings
Authors: Narita Masao
Published: November 2008

Let Q be a semi-regular local ring of dimension d, m be its maximal ideal, and q be an m-primary ideal. Then LQ(Q/qn+1), the length of Q-module Q/qn+1, is equal to the characteristic polynomial PQ(q,n) in n for a sufficiently large value of n:

where ei = ei(q), i = 0,1,2,…, d are integers uniquely determined by q, called normalized Hilbert coefficients of q according to (1). It was shown in (1) that e1(q) is a non-negative integer, and is equal to zero if and only if q is generated by a system of parameters. We shall prove, in this paper, that e2(q) is also a non-negative integer, and that this non-negativity is not necessarily true for other coefficients. We shall give an example with negative e3(q).
A note on natural maps of higher extension functors
Authors: Oort F.
Published: November 2008

Hilton and Rees have proved (cf. (1), Theorem 1·3) that every natural map

is induced by a map from A to B (or, Hom (A, B) → Next1,1 (A, B) is surjective). It follows that Ext1 (B, −) and Ext1 (A, −) are naturally isomorphic if and only if A and B are quasi-isomorphic (loc. cit., Theorem 2·6), i.e. if there exist projective objects P, Q and an isomorphism . One can ask whether these theorems remain true for higher extension functors.
Non-immersion theorems for differentiable manifolds
Authors: Sanderson B. J. , Schwarzenberger R. L. E.
Published: November 2008

Let X be a compact differentiable n-manifold of class C∞ and let f be a C∞ map of X into euclidean (n + k)-space. Then f is an immersion if its jacobian has rank n at each point of X. It is an embedding if it is also one-one. We write X ⊆ n + k to denote the existence of an immersion, X ⊂ n + k to denote the existence of an embedding. By a theorem of Whitney, we have X ⊆ 2n − 1, X ⊂ 2n.
Embeddings in Euclidean space
Authors: Schwarzenberger R. L. E.
Published: November 2008

Let X be a compact differentiable manifold without boundary. A differentiable regular map of X into Euclidean N-space, dim X < N, is called an immersion (written X £ N). An immersion which is also one-one is called an embedding (written XN). A nonembedding theorem of the form: if X is a manifold with the property P then XN − 1, is best possible if there exists a manifold X with property P and an embedding XN. The purpose of this note is to show that two frequently used non-embedding theorems are best possible.
Simple proof of a formula due to Racah
Authors: Griffith J. S.
Published: November 2008

After some manipulation from the general formula, Racah (1,2) obtained a closed formula for his V coefficient in the special case of all three mj- components zero. He also showed that the integral of the product of three Legendre polynomials followed at once from these special V coefficients. In this case the general formula ((2), eq. 10–14) becomes

where

and a, b, c are all non-negative integers. By changing the parameter of summation to (a+ b−c−p) it follows that is zero unless 2g = a+b+c is even (2).
A theorem on cardinal numbers associated with a locally compact Abelian group
Authors: Varopoulos N. Th.
Published: November 2008

1. Notation. For an arbitrary set X, |X| will denote its cardinal number. Rn and Tn will denote the n-dimensional real vector space and the n-dimensional torus respectively with the usual topologies.
Complete functors in homology I. Chain maps and endomorphisms
Authors: Kelly G. M.
Published: November 2008

There is a sense in which the homology group HA of a free Abelian chain complex A may be said to be a ‘complete system of invariants’ of A, to within chain equivalence; certainly any graded Abelian group G is isomorphic to HA for a suitable A, and if HA and HB are isomorphic then A and B are chain equivalent. Such a result is useful in showing that it is fruitless to seek other homotopy invariants of A; whatever depends only on the homotopy class of A depends only on HA, so that we can, for instance, predict the existence of a formula giving H(A ⊗ G), to within isomorphism, in terms of HA and G. The theorem on the existence and uniqueness to within chain equivalence of projective resolutions of modules is a variant of the above theorem, more general in one direction and more special in another.
The numerical solution of the heat conduction equation in one dimension
Authors: Wadsworth M. , Wragg A.
Published: November 2008

The replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.
wave functions
Authors: Holt A. R.
Published: November 2008

Cohen and Coulson (4) and Cohen (5) have obtained wave-functions and energy levels for the hydrogen molecular ion in a spherical polar representation. They expanded the wave-function in terms of spherical harmonics, and their expansion may be written as

where S = 2k for even states, and S = 2k + 1 for odd states.
On the fundamental group of an orbit space
Authors: Armstrong M. A.
Published: November 2008

Introduction. Let K be a connected simplicial complex, finite or infinite, its polyhedron ((2), page 45) being the space X. Then X is connected. Suppose further that X is simply connected. For any group G of simplicial transformations of X, H will denote the normal subgroup generated by elements which have a non-empty fixed-point set.
Vector bundles that fill n-space
Authors: Robertson S. A. , Schwarzenberger R. L. E.
Published: November 2008

The idea of exact filling bundle may be described roughly as follows. Suppose that ξk is a vector bundle with fibre Rk, total space E(ξk) and base X. We say that ξk is a real k-plane bundle on X. Let in be the trivial n-plane bundle on X so that E(in) = X × Rn. A bundle monomorphism j: ξk → in defines a map : E(ξk)→Rn obtained by composition of the embedding E(ξk)→E(in) and the product projection E(in) → Rn. The map represents each fibre of ξk as a k-plane in Rn.
Translation invariant measures which are not Hausdorff measures
Authors: Hirst K. E.
Published: November 2008

An important and much-investigated class of measures is the class of Hausdorff measures, first defined by Hausdorff (1). These measures form a subclass of the class of translation invariant measures, but just how wide a class they form is not known.
On the minimum of a positive quadratic form in n ( ≤ 8) variables (verification of Blichfeldt's calculations)
Authors: Watson G. L.
Published: November 2008

Let f = f(x1, …, xn) be a positive definite quadratic form in n ( ≤ 8) variables; then we consider the old problem of estimating the minimum of f (its least value for integers xi not all 0) in terms of the determinant Δ(f). Normalizing by supposing the minimum to be 1, the known results may be stated as

each inequality being best possible. Further, for each n, all forms for which equality holds in (1) have integral coefficients and are equivalent to each other.
On the direct solution of certain half-plane mixed boundary-value problems
Authors: Naylor D.
Published: November 2008

In this paper a method is proposed for solving certain half-plane elliptic boundary-value problems involving mixed boundary conditions. The equation considered is a generalization of the Tricomi equation which contains the space form of the damped wave equation as a special case. Existing methods depend on the use of Fourier integrals and lead to the solution of integral equations. The methods employed here are direct and yield explicit solution formulas without the necessity of solving integral equations and as such avoid the arguments inherent in the use of the Wiener-Hopf technique.
Electromagnetic diffraction by a circular disk
Authors: Williams W. E.
Published: November 2008

It is shown that the problem of the diffraction of an arbitrary electromagnetic wave by a circular disk can be reduced to the solution of independent scalar problems. These latter problems are of a type which can be reduced to the solution of a Fredholm integral equation of the second kind and are particularly suitable for solution by approximate means at low frequencies.
On the behaviour of systems of coupled dynamos
Authors: Allan D. W.
Published: November 2008

The behaviour of a system of two coupled disk dynamos is studied in considerable detail. It is shown that in several cases the stability criteria and general behaviour are closely related to those for the single dynamo studied by Bullard, which are resumed briefly. Extended numerical investigations of the non-linear behaviour of the system are presented: these show that a true ‘reversal of field’ may occur.
The least extension of a closure operator on a sublattice
Authors: Davies Roy O.
Published: November 2008

Introduction. We recall (see Morgan Ward (3), Birkhoff (1), p. 49) that a closure operator on a complete lattice ℒ is a mapping f from ℒ into ℒ satisfying the conditions

for all X, Y ∈ ℒ. The set of all closure operators on ℒ is itself a complete lattice with respect to the partial ordering in which f1 ≤ f2 if and only if f1(X) ≥ f2(X) for all X ∈ ℒ for every subset ℱ of and every element X ∈ ℒ we have

.
Diffraction of a plane longitudinal wave by a penny-shaped crack
Authors: Robertson Ian A.
Published: November 2008

Introduction. The distribution of stress produced in the interior of an infinite elastic solid when a constant pressure is applied over the entire surface of a penny-shaped crack has been solved by Sneddon(6), (7). The problem considered here is the closely allied one of a plane longitudinal wave, harmonic in time, moving in the positive direction of the z-axis and impinging on the surface of a penny-shaped crack. The analysis follows the methods adopted for dealing with an axisymmetrical vibrating punch acting on a semi-infinite elastic solid, Robertson (5).
Bending of a simply-supported shallow spherical shell under a uniform line load on a part of a parallel circle
Authors: Mitra D. N.
Published: November 2008

The two fourth-order partial differential equations giving the transverse displacement and the stress functions for shallow shells are reduced to a sixth-order differential equation in one variable which has been solved, and the values of displacements, stress resultants and couples are all expressed in terms of Bessel Functions of imaginary argument. Numerical values of the displacement presented for points on the shell on different meridian lines, showed that on the symmetrical line deflexion is maximum at the point of loading but the point of maximum deflexion shifts gradually towards the pole.
Janko's group as a collineation group in PG(6,11)
Authors: Whitelaw T. A.
Published: November 2008

In (2) Janko gives 7 × 7 matrices over GF (11) which generate a copy of his new simple group. This matrix group induces a representation of Janko's group as a collineation group in PG (6,11)—i.e. 6-dimensional protective space over GF (11)—and the present paper is devoted to the results of study of this geometrical representation.
A Banach space that cannot be made into a BIP space
Authors: Olagunju P. A.
Published: November 2008

1. A Banach space E over the complex field C is said to be a Banach inner-product (BIP) space if there exists a mapping 〈.,.〉 of E × E into C satisfying:

(i) 〈x, x〉 ≥ 0 (x ∈ E) with equality only if x = 0;

(ii) 〈x, y〉 = 〈y, x〉 (x, y ∈ E);

(iii) 〈x + λy, z〉 = 〈x, z〉 + λ〈y, z〉 (x, y, z ∈ E, λ ∈ C);

(iv) 〈x, x〉 ≤ k2 ‖x‖2 (x ∈ E),

where k is a fixed positive number. Thus 〈.,.〉 is an inner product on E, which induces a norm ‖·‖1 by the relation
The low frequency scalar diffraction by an elliptic disc
Authors: Sleeman B. D.
Published: November 2008

The problem of scalar Dirichlet diffraction of a plane wave by an elliptic disc is discussed. A scheme is given whereby the low frequency expansion of the scattered field may be readily obtained. Series expansions are obtained for the far-field amplitude up to and including the second order in the wave number. The first two terms of the scattering cross-section are also derived.
A quotient ordered space
Authors: McCartan S. D.
Published: November 2008

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the property

a partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.
Sequential convergence in locally convex spaces
Authors: Webb J. H.
Published: November 2008

Given a locally convex Hausdorff linear topological space, we construct and examine the following topologies in the space:

the finest locally convex topology with the same convergent sequences as the initial topology, andthe finest locally convex topology with the same precompact sets as the initial topology.
Sums associated with Farey series
Authors: Robertson M. M
Published: November 2008

We let h1/k1,h2/k2…hN(n)/kN(n)represent the Farey series of order n. It is well known that

Recently, Gupta (l) proved that

and Hans and Chander (2) proved that
Expansions in series of Gontcharoff polynomials
Authors: Read G. A.
Published: November 2008

The Gontcharoff polynomials Gn(z; α0, α1,…, αn − 1); n = 0, 1, 2,… are defined to be the polynomials orthogonal to the linear functionals Ln(f) = f(n)(αn), for a specified sequence of complex numbers {αn}. We will abbreviate Gn(z; α0, α1,…, αn−1) to Gn(z) when the sequence {αn} is not in doubt, so that
Control systems and differential inequalities
Authors: Lakshmikantham V. , Tsokos C. P.
Published: November 2008

LetJdenote the half line 0 ≤t< ∞,Rnthe Euclidiann-space andR+= [0, ∞). Let ║x║ denote the Euclidian norm ofx∈Rn, andSdenote the set [x: ║x║ < ρ].
Path representations of the quantum-mechanical energy-level density
Authors: Bezák Viktor
Published: November 2008

The quantum-mechanical energy-level density g(E) is given as a functional of the quantum-mechanical kernel K(q″, q′, t″ −t′). On taking the kernel K in the Feynman's form, one obtains the function g(E), without solving a Schrödinger equation. As an example, the embedding of a particle in the one-dimensional square well with infinitely high walls is analysed. The functions K(x″, x′, t−t′) and g(E) are represented as sums of terms corresponding to classical paths of different types. By an adequate choice of some terms due to the ‘most important’ paths, one may construct partial sums giving approximations of the function g(E). The utilization of such approximations for estimation of energy levels is demonstrated.
The effect of surface tension on the waves produced by a heaving circular cylinder
Authors: Evans D. V.
Published: November 2008

In this paper the effect of surface tension on water waves is considered. The usual assumptions of the linearized theory are made. A uniqueness theorem is derived for the waves at infinity for a general class of bounded two-dimensional obstacles in a free surface by means of an energy argument. It is shown how the wave amplitude at infinity depends on the prescribed angle at which the free surface meets the normal to the obstacle. The particular case of a heaving half-immersed circular cylinder is considered in detail, and an expression obtained for the velocity potential in terms of a convergent infinite series, the coefficients of which may be computed.
Large families of singular measures having absolutely continuous convolution squares
Authors: Stromberg Karl
Published: November 2008

In 1966, Hewitt and Zuckerman(3,4) proved that if G is a non-discrete locally compact Abelian group with Haar measure λ, then there exists a non-negative, continuous regular measure μon G that is singular to λ(μ ┴ λ) such that μ(G)= 1, μ * μ is absolutely continuous with respect to λ(μ * μ ≪ λ), and the Lebesgue-Radon-Nikodym derivative of μ * μ with respect to λ is in (G, λ) for all real p > 1. They showed also that such a μ can be chosen so that the support of μ * μ contains any preassigned σ-compact subset of G. It is the purpose of the present paper to extend this result to obtain large independent sets of such measures. Among other things the present results show that, for such groups, the radical of the measure algebra modulo the -algebra has large dimension. This answers a question (6.4) left open in (3).
Computer-calculated explicit forms for representations of the three-dimensional pure rotation group
Authors: Albert R. H.
Published: November 2008

Explicit forms are given for the Wigner for values of J from J = 0 to J = 7/2. The tedious labour involved in evaluating available formulas is reduced by use of a computer.
Non-Newtonian effects in axially symmetric rotational flows of some elastico-viscous liquids
Authors: Jones J. R. , Lewis M. K.
Published: November 2008

In recent years new materials of industrial importance have become known whose rheological properties cannot be adequately characterized by the classical Newtonian working model, many of these materials exhibiting both viscous and elastic properties. The large-scale flow behaviour of some of these new materials can be strikingly different from that of purely viscous liquids. For example, when sheared between coaxial cylinders, some liquids are observed to ‘climb up’ the inner cylinder, i.e. the free surface rises spectacularly near the inner boundary. This phenomenon has been observed by many authors (see, for example (1–4)), and is now widely known as the Weissenberg effect. Again, Merrington (5) observed that a stream of rubber solution ‘swells’ radially immediately on emerging from a capillary tube. In order to attempt to explain behaviour such as this, we must examine the consequences of a working model which is more complex than that for Newtonian liquids.
A finite set of generators for the homeotopy group of a non-orientable surface
Authors: Chillingworth D. R. J.
Published: November 2008

Let X be a closed surface, i.e. a compact connected 2-manifold without boundary. If Gx denotes the group of all homeomorphisms of X to itself, and Nx is the normal subgroup consisting of homeomorphisms which are isotopic to the identity, then the quotient group Gx/Nx is called the homeotopy group of X and is denoted by ∧x.
Couette flow of dipolar fluids
Authors: Shirkot K.
Published: November 2008

Dipolar fluids have been discussed by Bleustein and Green in quite a recent paper (3). They have obtained explicit constitutive equations for homogeneous incompressible linear dipolar fluids. An application of this theory is made here to the problem of Couette flow. Monopolar and dipolar stress fields are determined. Numerical values for velocity and rate of shear for different values of radius in the annulus are given. Graphs are also drawn to compare our results with the classical ones. It is found that the rate of shear does not change sign within the annulus.
Absolutely faithful group actions
Authors: Rose John S.
Published: November 2008

The definition and main result. It has been shown ((1), § 3) that if G is any finite group and p any prime number not dividing |G|, then the number of conjugacy classes of maximal nilpotent subgroups in the regular wreath product of a cyclic group of order p by G is equal to the number of conjugacy classes of all nilpotent subgroups in G. This fact, together with various properties of the map by means of which it was established, proved helpful in dealing with questions of construction raised in (1). The present note isolates the key property of the wreath product on which the argument rests, and from this shows how the argument can be carried over to a more general context. The essential situation is that a group G acts on a group A in a way which will be called ‘absolutely faithful’.
Changes of signs of restrictions of Fourier–Stieltjes transforms
Authors: Gaudry G. I.
Published: November 2008

Our principal result deals with a function φ defined on the ‘non-negative half’ X+ of a discrete ordered group X, the dual of a compact Abelian connected group G. Suppose φ vanishes outside a countable set S = {χ1, χ2 …} and has the following property: for every t in a set 0 ⊂ (0, 1) of positive Lebesgue measure, the function on X+ defined to be 0 off S and to be rn(t)φ(χn) for χ = χn, where rn is the nth Rademacher function, is the restriction to X+ of a Fourier–Stieltjes transform. We conclude that φ ∈ ℓ2(X+).
The least limit point of the spectrum associated with sinǵular differential operators
Authors: Eastham M. S. P.
Published: November 2008

Let τ be the formally self-adjoint differential operator denned by

where the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.
Symmetry adapted functions for double point groups II. Cubic point groups
Authors: Cracknell A. P. , Joshua S. J.
Published: November 2008

A method of deriving the basis functions of the double-valued representations of a point group by the reduction of Kronecker products is described. The method has been used to derive expressions for these bases for cubic point groups for which the results are tabulated.