bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on the unsteady motion of a compressible fluid
Authors: Pack D. C.
Published: November 2008

A solution is found in closed form for the one-dimensional motion of an ideal, inviscid and isentropic gas with ratio of specific heats (2m + 3)/(2m + 1), m being any positive integer, when the gas is supposed to start from rest with an inhomogeneous temperature distribution.

An example is given which is of interest in the study of interstellar gas clouds. A cloud is assumed to be initially at rest in contact with a vacuum; the Riemann invariant r is assumed to increase linearly for a finite distance into the cloud (from zero at the surface), beyond which it remains constant. The r-characteristics from the inhomogeneous strip meet simultaneously in the distance-time plane at a point of the gas surface; the surface does not move until this happens, when it acquires instantaneously a finite non-zero velocity. A numerical estimate is made of the time during which the surface remains at rest for an actual gas cloud, under the above conditions. In the subsequent development the gas motion takes the form of a growing simple wave.
On the circle polynomials of Zernike and related orthogonal sets
Authors: Bhatia A. B. , Wolf E.
Published: November 2008

The paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.

The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.
Laminar flow in an axially symmetrical jet of compressible fluid, far from the orifice
Authors: Pack D. C.
Published: November 2008

When the boundary-layer approximations are applied to the laminar flow in an axially symmetrical jet of gas with Prandtl number unity, a particular integral of the equations of motion satisfying the required boundary conditions is found to be , where i1, is the (constant) enthalpy on the jet boundary. By a suitable transformation an equation is obtained which, to a first approximation, may be replaced by the equation for flowin an axially symmetrical jet of incompressible fluid at a sufficiently large distance from the orifice. For an ideal gas, the known solution of this equ`ation may be used as the starting point of an iterative method to build up the stream function as a power series. The next approximation is found explicitly. It shows that for subsonic Mach numbers the density at a point on the axis is greater and the temperature less than on the boundary by amounts which may be very closely represented by terms inversely proportional to the square of the distance of the point from the orifice.
Eddington's theory in terms of the concept of order
Authors: Bastin E. W. , Kilmister C. W. , Sciama D. W.
Published: November 2008

The theory of a previous paper is used to discuss Eddington's work in fundamental physics.

1. Introduction. A previous paper (2), referred to here as I, described a theory depending on an unorthodox view of the relation of mathematics to physics. This theory was begun from the study of the work of Eddington which we take to be summarized in Fundamental Theory (5). In the present paper is shown the extent to which, and the way in which, the two theories are connected.
On multiple returns in the random-walk problem
Authors: Domb C.
Published: November 2008

Consider a random-walk problem on a simple lattice, the probabilities of the walker taking any direction in the lattice at each lattice point being equal. Then Polya (6) has shown that if a walker starts at the origin and continues to walk indefinitely, the probability of his passing through his starting point is unity in one and two dimensions, but less than unity in three or more dimensions. Recently, a generalization of this problem has been considered (1),(3) in which the walker is allowed to jump several lattice points with assigned probabilities. F. G. Foster and I. J. Good (3) have shown that if the assigned probabilities satisfy certain conditions, Polya's result still holds, and K. L. Chung and W. H. J. Fuchs (1) have shown that the result is valid under far less restrictive conditions. The above authors were primarily concerned with the question whether return is almost certain or not, and did not consider a detailed calculation of the probability at any stage. It is the purpose of the present paper to show that the use of contour integrals allied with the method of steepest descents (4) enables one to perform this calculation very simply.
Some inequalities concerning positive-definite Hermitian matrices
Authors: Fan Ky
Published: November 2008

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatM

and more generally,
The interpolation of atomic wave functions
Authors: Hartree D. R.
Published: November 2008

If r̄nl is the mean radius for the radial wave function of a complete (nl) group in an atom of atomic number N, the variation of 1/r̄nl with N is nearly linear. Further the variation of a given (nl) radial wave function with N is such that for a given value of (r/r̄nl), the variation of the quantity (r̄nl)½P(nl; r) with r̄nl is nearly linear. These relations between the radial wave functions for different atoms are examined from the point of view of using them as a means of interpolating, with respect to atomic number, between results for atoms for which solutions of Fock's equations have been carried out.
Temporally ordered graphs and bound state equations
Authors: Polkinghorne J. C.
Published: November 2008

Ordered graphs are used to obtain the kernels of the single-time, three dimensional, equations for bound states given by Lévy and Klein, and by Tamm and Dancoff.
On a conjecture of Nagata
Authors: Higman Graham , Hall P.
Published: November 2008

In a recent paper (1) Nagata proved that a (linear associative) algebra, not necessarily of finite dimension, over a field of characteristic 0 which satisfies the identical relation xn = 0 satisfies also the relation x1x2 … xN = 0, where N is an integer depending only on n. He remarked further that it is a corollary that the result remains true if the ground field is of prime characteristic p, provided that p is large enough compared with n; and he conjectured that the obviously necessary condition p > n is in fact sufficient. The object of this note is to prove Nagata's conjecture. To do this, we give a new proof of his theorem, and as a by-product we obtain a rather better bound for N than his, showing, namely, that we can take N = 2n − 1. The determination of the best possible value of N, or even of its order of magnitude, seems not to be easy; at any rate, the best I have been able to do in the opposite direction is to show that for large n we cannot take N as small as n2/e2, where e is the base of natural logarithms.
A measure of dispersion applied to cosmic-ray and other problems
Authors: Broadbent S. R.
Published: November 2008

The statistic is applied in several different situations to detect departure from uniform dispersion. The given sample points may lie on a square lattice (without repetition), a rectangular lattice (repetition being allowed), or within rectangular or circular plane boundaries. The problems considered include an analysis of cosmic ray shower structure, and the counting of dust particles and bacterial colonies.
Generalized Riemann spaces
Authors: Moffat John
Published: November 2008

The recent attempt at a physical interpretation of non-Riemannian spaces by Einstein (1, 2) has stimulated a study of these spaces (3–8). The usual definition of a non-Riemannian space is one of n dimensions with which is associated an asymmetric fundamental tensor, an asymmetric linear affine connexion and a generalized curvature tensor. We can also consider an n-dimensional space with which is associated a complex symmetric fundamental tensor, a complex symmetric affine connexion and a generalized curvature tensor based on these. Some aspects of this space can be compared with those of a Riemann space endowed with two metrics (9). In the following the fundamental properties of this non-Riemannian manifold will be developed, so that the relation between the geometry and physical theory may be studied.
Quantum electrodynamics and Hilbert space theory
Authors: Taylor J. G.
Published: November 2008

This paper describes an attempt to formulate quantum field theory, in particular quantum electrodynamics, in terms of Hilbert space theory. The work of Cook (1) is extended to give a precise description of non-interacting electrons and positrons. The hole interpretation is not required in this extension, and no subtraction formalism is required. It is shown that the formalism can never reduce to that of intuitive quantum field theory except by an abuse of language associated with the δ-function. Interaction cannot be introduced in a simple manner into the rigorous formalism, so it seems extremely difficult to develop the Hilbert space formalism for quantum field theory in any useful manner.

These difficulties indicate that an investigation of the Hilbert space basis of simple quantum theory is necessary before a rigorous mathematical formalism for intuitive quantum field theory can be developed.
An extension of the method of steepest descents
Authors: Chester C. , Friedman B. , Ursell F.
Published: November 2008

In the integral

the functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle points varies with α, and if for some a (say α = 0) two saddle points coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In the present paper we consider this case of two nearly coincident saddle points and construct uniform expansions as follows. A new complex variable u is introduced by the implicit relation

where the parameters ζ(α), A(α) are determined explicitly from the condition that the (u, z) transformation is uniformly regular near z = 0, α = 0 (see § 2 below). We show that with these values of the parameters there is one branch of the transformation which is uniformly regular. By taking u on this branch as a new variable of integration we obtain for the integral uniformly asymptotic expansions of the form

where Ai and Ai′ are the Airy function and its derivative respectively, and A(α), ζ(α) are the parameters in the transformation. The application to Bessel functions of large order is briefly described.
An approximate treatment of high-frequency scattering
Authors: Jones D. S. , Whitham G. B.
Published: November 2008

In some recent work Kodis* considers the scattering of a plane harmonic wave by a circular cylinder and uses variational methods to deduce an asymptotic formula for the scattering coefficient in the case of high-frequency waves. The scattering coefficient, σ, is denned as the total energy flux outward from the cylinder in the scattered wave divided by the energy flux in the beam of the incident wave which falls on the cylinder. In the limit, of geometrical optics, the scattered wave is made up of a wave reflected from the forward half of the cylinder with energy flux equal to that in the incident beam, and a wave behind the cylinder cancelling the incident wave there to form the shadow. Thus σ → 2 as ka → ∞, k being the wave number of the incident wave and a the radius of the cylinder. The next term in the asymptotic expansion for σ is proportional to . It has been determined already from the exact series expression for the field by White (6), Wu and Rubinow (7) and Kear (3). But Kodis gives an approximate method which also supplies this result and, since the coefficient is in good agreement with the values obtained from the exact solution, concludes that his approximate procedure could also be used for more general obstacles. In the present paper, we propose an alternative approximate method which is related to the one given by Kodis but is very much simpler. Also we go on to obtain the correction terms for general obstacles in detail. Various writers (see, for example, Keller (4) and van de Hulst(8)) have suggested previously that the correction is still proportional to , where a is some length defined by the body shape. However, the determination of a and of the numerical coefficient has been limited to two-dimensional obstacles.
The charge density near a sharp point on a conductor
Authors: Cade R.
Published: November 2008

A brief discussion is given of attempts that have been made to justify, in terms of electrostatic principles, the conjecture that surface charge on a conductor tends to infinity towards a convex sharp point and to zero towards a concave point, and it is concluded that the problem has not hitherto been solved. A new attempt is then made using potential-theoretic methods, and the problem is solved with a fair degree of generality. The limitations are of a kind familiar in this type of analysis and, roughly speaking, concern the degree of convexity of the conducting surface.
Stability criteria for linear systems and realizability criteria for RC networks
Authors: Fuller A. T.
Published: November 2008

A new set of stability criteria for linear systems is derived. This shows that about half of the Hurwitz criteria are redundant when certain of the coefficients of the characteristic equation are known to be positive. The theory is applied to obtain a very short derivation of the known aperiodicity criteria. The conditions for realizability of RC networks are shown to be closely related to the stability and aperiodicity criteria, and are stated as sets of criteria in terms of the polynomial coefficients. Two basic theorems are involved which give the necessary and sufficient conditions for the roots of two polynomial equations to be real and separated.
Classical electrodynamics as a distribution theory. II
Authors: Taylor J. G.
Published: November 2008

In a previous paper by the author (3) it was shown how the theory of distributions of L. Schwartz enables a mathematically consistent formalism to be given for a system composed of point charges interacting through their classical electromagnetic field. In the present paper a definition of the energy and momentum of the field lying in a space-like surface is given, and it is shown that from this four-vector it is possible to derive the usual equation of conservation of total energy and momentum.
A note on summation over Feynman histories
Authors: Kilmister C. W.
Published: November 2008

It is pointed out that the free particle, the harmonic oscillator and certain other systems are such that the Feynman summation over histories is independent of the class of histories summed over. These examples are therefore useless as tests of methods of defining and carrying out the summation.
Analytic functions of finite dimensional linear transformations
Authors: Afriat S. N.
Published: November 2008

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulae

for the sum, whereCis a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼r<mα; α) at each characteristic root α of a, of orderrat most the multiplicitymαof α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expression

for the characteristic projectors.
A generalization of a theorem on the content of polynomials
Authors: Northcott D. G.
Published: November 2008

Introduction. By the content of a polynomial with coefficients in a ring R, one understands either the R-ideal or (better) the additive group which the coefficients generate. There are important connexions between the contents of two polynomials and the content of their product, and results of this kind are to be found in the work of Gauss. Here we shall give a wide generalization of a theorem of this type not involving polynomials at all; however, as the reader will observe, the proof is essentially a reduction to the polynomial case.
A theorem on the cumulative product of independent random variables
Authors: Ruben Harold
Published: November 2008

1. Introductory discussion and summary. Consider a sequence {ui} of independent real or complex-valued random variables such that E(ui) = 1, and a sequence of mutually exclusive events S1, S2,…, such that Si depends only on u1, u2, …,ui, with ΣP(Sj) = 1. Define the random variable n = n(u1, u2,…) = m when Sm occurs. We shall obtain the necessary and sufficient conditions under which

referred to as the product theorem.
Hilbert transforms and almost periodic functions
Authors: Cossar J.
Published: November 2008

The Hilbert transform, Hf, of a function f is defined by Hf = g, where

P denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists and

In the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).
A random walk problem
Authors: Gillis J.
Published: November 2008

We consider a random walk on a two-dimensional rectangular lattice in which steps are strictly between nearest neighbour points. The conditions of the walk are that the walker must, at each step, turn either to the right or to the left of his previous step with respective probabilities ½(1+α), ½(1−α), (≤ α ≤ 1). To fix the ideas it is assumed that he starts from the origin and the probability of each of the four possible starting directions is ¼. If Ar denotes the probability of return to the origin after r steps we shall show that

where β = ½(α + α−1) and Pn is the nth Legendre polynomial. It is clear that Ar is zero for r ≢ 0(mod 4).
Non-linear Lagrangians and Palatine's device
Authors: Buchdahl H. A.
Published: November 2008

Field equations in general relativity theory have sometimes been generated by subjecting, in an invariant action integral, the components of linear connexion and the components of a covariant tensor of valence 2 to independent variation. The conceptual objections to this process, and some of the manifold formal difficulties inherent in it, are discussed in some detail. At the same time certain results obtained elsewhere are strengthened and in part corrected.
The Motion of a Biased Bowl with Perturbing Projection Conditions
Authors: Brearley M. N.
Published: November 2008

The equations of motion of a biased bowl rolling on a plane horizontal grass green are derived by Lagrange's method, using an appropriate representation of the effect of the green. The equations are solved approximately under general initial conditions which include the presence of a perturbing wobble.

An expression is derived for the total angle of precession of the bowl, and the equation of the path is found in parametric form. The theory predicts that (contrary to universal belief) the effect of an initial wobble on the bowl path is negligible under normal playing conditions.

A detailed comparison is made between the predictions of the theory and experimental results obtained on several greens of different speeds.
Note on the Jacobi identity for Whitehead products
Authors: Hilton P. J.
Published: November 2008

The Jacobi identity for Whitehead products,

where α ɛ πl(X), β ɛ πm(M), γ ɛ πn (X), l, m, n ≥ 2 is now well known. In this note we extend the identity to the case l, m, n > 1; the extension is easy and we are merely filling a small gap in the literature. However, the extension does enable us to improve a result in (2).
A property of balanced functors
Authors: Northcott D. G.
Published: November 2008

Notation. We consider an additive, right exact functor T(B, C) where B is a variable Λ-module, C a variable Γ-module and the values of T are in the category of Δ-modules. Here Λ, Γ, Δ are rings (which need not be commutative) each possessing an identity element.
The correlations between relatives in a random mating diploid population
Authors: Trustrum G. B. , Williamson J. H.
Published: November 2008

Malecot(4) under certain conditions derived the formula for the covariance of the genotypic values of a quantitative character in two individuals AI and AII, which were related but not by direct descent. This generalized some results of Fisher (l). Kempthorne (2) extended the theory to multiple allelic systems with any degree of epistacy (i.e. interlocular genie interaction) but without linkage. He gave the formula Here is the item in the population variance which can be attributed to the interaction of additive gene effects at r loci and dominance gene effects at s loci. φ and φ′ are the coefficients of relation between the two individuals. The various assumptions normally included under random mating equilibrium were made, i.e. no selection, mutations, maternal effects or differential viability. Kempthorne (2), (3) gave two rather different proofs of this important result. His second proof was the more straightforward, but it was somewhat condensed.
On the cohomology of certain groups
Authors: Wall C. T. C.
Published: November 2008

Recent work of Atiyah (1) on the Grothendieck rings of classifying spaces of finite groups has yielded, among many other interesting results, a spectral sequence relating the simple integer cohomology of a finite group to its representation ring. He has determined the first differential operator of the spectral sequence which affects the p-part of the cohomology, and the question naturally arises, when is it the only non-zero one. My object in this note is to show that this is so for a large class of groups.
An alternative proof of a theorem of Iwasawa on free groups
Authors: Neuwirth Lee
Published: November 2008

The following well-known theorem has been proved by Iwasawa(1) and can also be deduced without much difficulty from earlier work of Levi(2), Magnus and Witt.
Inequalities for functions regular in the unit circle
Authors: Linden C. N.
Published: November 2008

Let f(z) = u(z) + iv(z) be a function regular for |z| < 1 with real and imaginary parts u(z) and u(z) respectively. The following theorems have been proved by Miss M. L. Cartwright(1) and (2).

Theorem A. if v(0) = 0, 0 < α < ∞ and

for 0 ≤ r < 1, 0 ≤ θ < 2π then

for 0 ≤ r < 1, 0 ≤ θ < 2π
A cosmological model of light propagation
Authors: Prokhovnik S. J.
Published: November 2008

A theory of light propagation consistent with our present cosmological views is investigated in the context of a uniformly expanding universe. It leads to the relativistic equivalence of fundamental observers and also provides a natural substratum associated with an observable fundamental reference frame for every locality of the universe. Special Relativity emerges as a theory satisfying the observable universe.

The model provides a quantitative interpretation of the recent findings in radio and optical astronomy. It has the advantage of simplicity and its approach may assist in an understanding of what is happening in the mathematics of more sophisticated systems.
The hydromagnetics of an ellipsoid moving in a cross-field
Authors: Ludford G. S. S. , Singh M. P.
Published: November 2008

Our previous discussion ((4), (5)) is here extended to an ellipsoid of arbitrary conductivity. Again, the flow is ultimately cylindrical and the magnetic field undisturbed. An ellipsoidal insulator allows fluid to pass through its circumscribing cylinder, but also forces some to go around. An ellipsoidal conductor behaves like a very tall insulator, irrespective of its own height or conductivity: all the fluid goes round. If it is very wide an unexpectedly large amount of fluid is carried with it.
A certain sublattice of the lattice of congruences on a regular semigroup
Authors: Munn W. D.
Published: November 2008

It is well known that, with respect to the natural partial ordering, the set of all congruences on a group forms a modular lattice. In the present paper we develop an extension of this result to the case of a regular semigroup S (α ∈ αSα for all α in S). Let Σ(ℋ) denote the set of all congruences on S with the property that congruent elements generate the same principal left ideal and the same principal right ideal of S. We show (Theorem 1) that, under the natural partial ordering, Σ(ℋ) is a modular lattice with a greatest and a least element.
Short surface waves in the presence of a finite dock. II
Authors: Holford R. L.
Published: November 2008

The generalized wave-making problem for the forced high-frequency oscillations of a finite dock is solved through use of the Green's function obtained in the first part of this paper. The specific cases of heave and roll are considered with particular reference to the forces on the dock and the amplitude and phase of the radiated waves. These results are then utilized to solve the problem of transmission of short waves under a fixed dock.
Quadratic forms as Liapunov functions for linear differential equations with real constant coefficients
Authors: Lehnigk Siegfried H.
Published: November 2008

This paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.
A note on Milne-Thomson's general solution of Navier–Stokes equations
Authors: Nigam G. D.
Published: November 2008

The Navier–Stokes equations governing the motion of viscous compressible fluids are

where ρ is the density, q is the velocity vector, V is the scalar potential of the external force field, I is the idemfactor and Φ is the stress-tensor

Milne-Thomson ((l)) has given a general solution of (1) and (2) in the form
Alternative formulation of quantum mechanics
Authors: Knight Brian
Published: November 2008

The two formulations of quantum mechanics in q-space only are compared by the construction of Schrödinger's operator identity. The examination in detail of the examples of the harmonic oscillator and free particle shows clearly the role of the re-normalization term in both formulations.
A remark on matrix rings over free ideal rings
Authors: Cohn P. M.
Published: November 2008

If R is any ring then the set of isomorphism types of finitely generated projective right R-modules may be made into a semigroup by defining

.
Laminar boundary layers in oscillatory flow along an infinite flat plate with variable suction
Authors: Messiha S. A. S.
Published: November 2008

The two-dimensional incompressible fluid flow problem along an infinite flat plate with no heat transfer between the fluid and the plate is discussed when the suction velocity normal to the plate, as well as the external flow velocity, varies periodically with time. Expressions for the velocity and temperature fields in the boundary layer have been obtained in a non-dimensional form. It has been found that the effect of increasing the suction velocity fluctuations is that the amplitude of the skin-friction fluctuations is increased while the phase is decreased and it may be negative for very small values of the frequency. Also the plate temperature decreases with increasing the suction velocity fluctuations for small values of the frequency while there is no effect for very large frequencies.
Minimal verbal subgroups
Authors: Kovács L. G. , Newman M. F.
Published: November 2008

Introduction. In this note we obtain a description of the structure of the minimal verbal subgroups of a finite group which has, among others, the following consequences:

Theorem 1. A finite group which belongs to the variety generated by its proper subgroups and proper factor-groups belongs either to the variety generated by its proper subgroups or to the variety generated by its proper factor-groups.
Singular measures with absolutely continuous convolution squares
Authors: Hewitt Edwin , Zuckerman Herbert S.
Published: November 2008

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such that

for every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.
Binary integral quadratic forms over R(t)
Authors: Knight James T.
Published: November 2008

We use the notations and definitions of the introduction to the previous paper ((1)), and say that 2 lattices K and L on a quadratic space over R(t) with its Dedekind set C are in the same genus if, for all , the lattices and are in the same class. We mean to count the number of classes of lattices in a given genus over a binary space: we shall find that it is either 1 or .
An example of an automorphism which is a mixing of all degrees but which is not a Kolmogorov automorphism
Authors: Dugdale J. K.
Published: November 2008

Rohlin (7) has shown that Kolmogorov automorphisms are mixings of all degrees. Later Sucheston(9) introduced regular automorphisms and after showing that they are mixings of all degrees asks if the converse of either of the above theorems holds. Since the equivalence of Kolmogorov and regular automorphisms follows from (1) and (8) we have only one problem. We show that the converse of either of the above theorems is false by constructing a stationary Gaussian process which is a mixing of all degrees but which is not a Kolmogorov automorphism.
The secular variation of the earth's magnetic field
Authors: Allan D. W. , Bullard E. C.
Published: November 2008

The magnetic field observable outside a body of conducting fluid in which field is imbedded may be considerably altered by convection currents in the fluid. One possible explanation of the geomagnetic secular variation foci is that localized convection cells in the earth's core disturb the main field present. An analytic solution for such a process is readily obtained by assuming the form and dimensions for such a cell, and shows that the magnitude of the secular variation cannot easily be explained on these lines without the presence of a subsurface toroidal magnetic field of some hundreds of gauss which is ‘convected through’ the surface of the core.
Strong summability of infinite series on a scale of Abel type summability methods
Authors: Mishra Babban Prasad
Published: November 2008

Introduction. In a recent paper, Borwein(1) constructed a new method of summability which would read: Let

and let {sn} be any sequence of numbers. If, for λ > − 1,

is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we say that the sequence {sn} is Aλ convergent to s and write sn → s(Aλ). The A0 method is the ordinary Abel method.
The concept of a filter
Authors: Hannan E. J.
Published: November 2008

It is proved that for a second-order, homogeneous, random process on a globally symmetric space a filter, that is a closed linear operator which is invariant under a group of isometries of the space, may be fully described through a response function, that is that it has a direct integral decomposition into components which are scalar multiples of the identity.
On operators which depend on some parameter
Authors: Okikiolu G. O.
Published: November 2008

The purpose of this paper is to study the mapping properties of certain operators which depend on a variable parameter. We prove a general theorem which is applied to some special cases. Among results obtained are new inequalities involving the Poisson operator and its conjugate.
Summability by Dirichlet convolutions
Authors: Segal S. L.
Published: November 2008

Ingham (3) discusses the following summation method:

A series ∑an will be said to be summable to s if

where, as usual, [x] indicates the greatest integer ≤ x. (An equivalent method was introduced somewhat earlier by Wintner (8), but the notation (I) for the above method and the attachment to Ingham's name seem to have become usual following [(1), Appendix IV].) The method (I) is intimately connected with the prime number theorem and the fact that the Riemann zeta-function ζ(s) has no zeros on the line σ = 1. Ingham proved, among other results, that (I) is not comparable with convergence but, nevertheless, for every δ > 0, (I) ⇒ (C, δ) and for every δ, 0 < δ < 1, (C, −δ) ⇒ (I), where the (C, k) are Cesàro means of order k.
Ergodic theorems for sequences of infinite stochastic matrices
Authors: Paz A. , Reichaw M.
Published: November 2008

In the theory of finite state, discrete time, non-homogeneous Markov chains, different notions of ergodicity have been introduced in the literature. These notions are concerned with the long-run behaviour of chains and with their tendency to get some stability properties after a sufficiently long period of time. The aim of this paper is the study of non-homogeneous Markov chains with a denumerable number of states. It will be shown that some theorems which are valid in the finite case are also valid for chains with a denumerable number of states as well. Moreover, a new notion of stability is introduced and it is shown to be satisfied for some chains. Although the paper is self-contained some familiarity with the theory of finite state non-homogeneous Markov chains is desired. Without any attempt of completeness we list for the interested reader the papers: (1–3, 8, 9).
Finite amplitude convection in a self-gravitating fluid sphere containing heat sources
Authors: Baldwin P.
Published: November 2008

The non-linear equations governing the convective flow of heat, beyond marginal stability, in a fluid sphere containing a uniform distribution of heat sources, and cooled at its surface, are expressed in the Boussinesq approximation as a perturbation of the steady-state conduction solution. A steady-state axisymmetric poloidal flow is assumed, and the equations for an approximate solution, involving the first spherical harmonic only, are obtained using a special case of the general evolution criterion of Glansdorff and Prigogine. The solution is found numerically using an iterative procedure, the results showing good agreement with those obtained by Stuart's ‘shape assumption’ for values of the Rayleigh number just above the critical value for the onset of marginal stability.
On the infinitesimal generator of the Poisson operator II. The n-dimensional case
Authors: Okikiolu G. O.
Published: November 2008

In a recent paper (5) the author gave an alternative proof of the result due to Hille(2) for the representation of the infinitesimal generator of the Poisson operator. In this paper we obtain the n-dimensional analogue of the result by using arguments similar to those employed in (5).
Coupled wave equations for propagation in inhomogeneous compressible plasmas
Authors: Burman R.
Published: November 2008

This paper deals with wave propagation in inhomogeneous compressible electron plasmas. The waves are described by Maxwell's equations coupled to the linearized single-fluid equations of hydrodynamics. Coupled wave equations are derived which describe the propagation of coupled electromagnetic and electron acoustic waves. Results are obtained for generally inhomogeneous plasmas and are specialized to planar and cylindrically stratified media. Particular attention is given to the planar case and several approximate techniques for treating the equations are discussed. The fields in a region of coupling are investigated.
Excitation of multimode surface waves
Authors: Williams W. E.
Published: November 2008

1. In a recent paper Karp and Karal(1) have suggested a generalization of the normal impedance boundary condition which might be applicable to surfaces which can support more than one surface wave and have determined the total field produced by a magnetic line dipole placed above an infinite plane characterized by such a condition. The theory is at this stage purely tentative and arguments concerning its plausibility are given in (1). The validity of the generalized impedance condition has also not yet been experimentally verified. From the point of view of experimental verification it would seem useful to have available a theoretical solution valid for an arbitrary electromagnetic field incident on a plane characterized by a generalized impedance condition and such a solution is given in the present note. By means of a technique used by the author in related problems (2,3) an explicit solution is given for an arbitrary incident field and it is shown that the radiation field and the amplitudes of the surface waves may be expressed in terms of the radiation field of the incident wave.
The multiplicative process in semiconductor radiation detectors. A practical application of Good's theorem in the theory of branching processes
Authors: Ogawa Iwao , Furukawa Masao
Published: November 2008

The electron avalanche that takes place within an internally amplifying semiconductor radiation detector has been analysed statistically, treating the process as an example of multi-dimensional branching processes. By extending the result obtained from Good's theorem to the case of continuous variable, an integral equation is derived for the generating function of the probability that a single primary electron-hole pair, formed at a depth x, induces an avalanche consisting of n pairs. On the basis of this equation, expressions are deduced for the mean value and the standard deviation σ of the total number n of pairs in an avalanche induced by n0 primary pairs produced at x, as well as the probability w(x, n0, n) of getting an avalanche of ‘size’ n. In particular cases these expressions lead to various simple relations of practical importance.
Double Meijer transformations of certain hypergeometric functions
Authors: Srivastava H. M. , Singhal J. P.
Published: November 2008

Following the usual notation for generalized hypergeometric functions we let

(a) denotes the sequence of A parameters

that is, there are A of the a parameters and B of the b parameters. Thus ((a))m has the interpretation

with a similar interpretation for ((b))m; Δ(k; α) stands for the set of k parameters

and for the sake of brevity, the pair of parameters like α + β, α − β will be written as α ± β, the gamma product Γ(α + β) Γ(α − β) as Γ(α ± β), and so on.
A theorem on Kampé de Fériet function
Authors: Srivastava G. P. , Saran S.
Published: November 2008

Kampé de Fériet (l) has defined a generalized hypergeometric function of two variables as

where ∏(σp)s stands for the product (σ1)s (σ2)s … (σp)s.
Note on the isotropic scattering of radiation from a point source in a finite spherical atmosphere
Authors: Cassell J. S.
Published: November 2008

1. In (1) M. G. Smith solved the equation

where

for R > 0, ø( − R) = − ø(R) and 0 < ω ≤ 1. In order to determine a constant, A(R1), appearing in the solution, two expressions for q(R1) were obtained, namely

and
A proof of Newman's theorem
Authors: Cohen Marshall M.
Published: November 2008

If Σn is a piecewise linear n-sphere and Qn is a piecewise linear n-ball which is a subpolyhedron of Σn thenis a piecewise linear n-ball.
Rectangulations
Authors: Biggs Norman
Published: November 2008

The subject of squared rectangles and squared squares was developed by Brooks, Smith, Stone and Tutte(1) using techniques based on the theory of electrical networks. In this note we shall treat the (apparently) more general topic of rectangulations using purely algebraic techniques. The starting point is the observation that the incidence matrices of the pair of dual networks associated with a squared rectangle in (l) can be more naturally derived as incidence matrices of the squaring (or rectangulation) itself. Further, the topological information contained in the networks is a simple consequence of Euler's formula applied directly to the rectangulation.
Complementary variational principles for ∇2φ = f(φ) with applications to the Thomas–Fermi and Liouville equations
Authors: Arthurs A. M. , Robinson P. D.
Published: November 2008

Complementary variational principles associated with the boundary-value problem ∇2φ = f(φ) in ∇,φ = φ0 on the boundary of ∇, are presented. The theory is applied to the Thomas–Fermi equation in quantum mechanics and the Liouville equation in equilibrium statistical mechanics.
On ‘translated quasi-Cesàro’ summability. II
Authors: Kuttner B.
Published: November 2008

In a recent paper (1), I considered the summability method (D, α) defined, for α > 0, by the sequence-to-sequence transformation

We note that, as is easily verified (and as was pointed out in (1)) a necessary and sufficient condition for the convergence of (1), and thus for the applicability of (D, α), is that

should converge. It was proved in (1) that, provided that (2) converges, a sequence summable (C, r) for any r > − 1 is necessarily summable (D, α). We now show that we can strengthen this result by replacing Cesàro by Abel summability. Moreover, we can omit the hypothesis that (2) converges provided that we interpret (1) as an Abel sum.
Sums of three roots of unity. II
Authors: Jones A. J.
Published: November 2008

The present paper is a sequel to a previous paper (1) in which I proved a conjecture made by R. M. Robinson concerning sums of three roots of unity. For any algebraic integer α, we denote by the maximum absolute value of α and all its algebraic conjugates. We say that α and β are equivalent if α = εβ′, where ε is some root of unity and β′ is some conjugate of β; when this happens we obviously have
Some results on generalized hypergeometric polynomials
Authors: Shah Manilal
Published: November 2008

In this paper, using a generalized hypergeometric polynomial defined by

where Δ(m, − n) denotes the set of m-parameters:

and m, n are positive integers, we have established some infinite series, transformations, integrals and expansion formulae for generalized hypergeometric polynomials. The polynomial is in a generalized form which yields many known polynomials with proper choice of parameters. Special cases have also been given.
Polynomial functors
Authors: Passi I. B. S.
Published: November 2008

1. Introduction: If G is a group, Z(G) its integral group-ring and AG the augmentation ideal, then we can form the Abelian groups

In (5) we have studied the structure of these Abelian groups which we called polynomial grouups. If C denotes the category of Abelian groups, then Pn and Qn are functors from C into C. We call these functors polynomial functors. The object of this work is to study the nature of these funtors. Except for n = 1, these functors are non-additive. In fact, in the sense of Eilenberg–Maclane (4) these are functors of degree exactly n (Theorem 2·3). Because of their non-additive nature, their derived functors cannot be calculated in the traditional Cartan–Eilenberg(1) method. We have to make use of the more recent theory of Dold–Puppe (3).
Absolute convexity in certain topological linear spaces
Authors: Maddox I. J. , Roles J. W.
Published: November 2008

For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U and

It is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).
On Mill's Ratio
Authors: Gupta Baikunth N.
Published: November 2008

The inequalities related to Mill's Ratio were conjectured true by Birnbaum (2) and were proved by Murty (4) for sufficiently large x > 0. In the present paper, these are proved for all finite x and have been used to calculate an upper limit for Mill's Ratio over the range x > – 1.