bluuebunny/crossref_metadata_2025 huggingface dataset:
Bott Periodicity and the Parallelizability of the spheres
Authors: Atiyah M. F. , Hirzebruch F.
Published: November 2008

The theorems of Bott (4), (5) on the stable homotopy of the classical groups imply that the sphere Sn is not parallelizable for n ≠ 1, 3, 7. This was shown independently by Kervaire(8) and Milnor(7), (9). Another proof can be found in (3), §26·11. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that Sn with any (perhaps extraordinary) differentiable structure is not parallelizable if n ≠ 1, 3, 7. Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in (4), (5). The purpose of this note is to show how the refined form of Bott's results given in (6) leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the following theorem due to Milnor (9) which implies the non-parallelizability.
A Homological characterization of certain Abelian groups
Authors: Douglas A. J.
Published: November 2008

Let G be a monoid; that is to say, G is a set such that with each pair σ, τ of elements of G there is associated a further element of G called the ‘product’ of σ and τ and written as στ. In addition it is required that multiplication be associative and that G shall have a unit element. The so-called ‘Homology Theory’† associates with each left G-module A and each integer n (n ≥ 0) an additive Abelian group Hn (G, A), called the nth homology group of G with coefficients in A. It is natural to ask what can be said about G if all the homology groups of G after the pth vanish identically in A. In this paper we give a complete answer to this question in the case when G is an Abelian group. Before describing the main result, however, it will be convenient to define what we shall call the homology type of G. We write Hn(G, A) ≡ 0 if Hn(G, A) = 0 for all left G-modules A.
Construction of orders in Abelian groups
Authors: Teh H.-H.
Published: November 2008

Let G be an Abelian group. A binary relation ≥ denned in G is called an order of G if for each x, y, z ε G,

(i) x ≥ y or y ≥ x (and hence x ≥ x);

(ii) x ≥ y and y ≥ x ⇒ x = y,

(if x ≥ y and x ≠ y, we write x > y);

(iii) x ≥ y and y ≥ z ⇒ x = z;

(iv) z ≥ y ⇒ x + z ≥ y + z.
A problem on tangents to a sphere
Authors: Croft H. T.
Published: November 2008

The problem is no. 226, Colloq. Math. 5 (1958), a problem of Steinhaus. If T is a closed set of tangent lines to a sphere S, are conditions (1) and (2) compatible:

(1) Every great circle of S has at least one tangent belonging to T.

(2) No great circle of S has two parallel tangents belonging to T?
The number of real zeros of the solution of a linear homogeneous differential equation
Authors: Brearley M. N.
Published: November 2008

The following theorem will be established:

Provided the roots of the associated characteristic equation are all real, any solution of a linear homogeneous differential equation with constant coefficients has at most n − 1 zeros for real finite values of the independent variable, where n is the order of the equation.

The theorem applies to equations with a complex independent variable, but since the conclusion concerns only real values of the variable there is no loss of generality in considering an equation of order n in the form

with y = y(x), where x is real. The constant coefficients ar may be complex.
Note on the Hopf-James invariants
Authors: Hardie K. A.
Published: November 2008

The Hopf-James invariants are a series of homomorphisms

(see (3) or (13)). I. M. James (7) and H. Toda (13), have shown effectively that in the case n even and r a prime p the restrictions to the p-primary components may be embedded in exact sequences

In this note we show that there are similar exact sequences if n is even and r = pq or if n is odd and r = 2pq, and we establish connexions in certain cases between the homomorphisms Hmn and the composite homomorphisms HmHn. We are able to compute the action of the invariants on the groups of the r-stem for r ≤ 9 with the partial exception of π6(S2) and π7(S2). The exact sequences yield useful criteria for determining the reduced product filtrations (see (9)) of elements.
A method for the numerical integration of the one-dimensional heat equation using chebyshev series
Authors: Elliott David
Published: November 2008

A numerical solution of

with general linear boundary conditions alongx= ±1, is described where at any timetthe Chebyshev expansion of θ(x,t) in –1 ≤x≤ 1 is computed directly. Compared with the more usual finite difference methods, this method requires much less computation and there are no stability problems. Two cases are considered in detail.
The Marginal stability in porous inhomogeneous media
Authors: Gheorghitza St. I.
Published: November 2008

The breakdown of stability in a porous homogeneous medium of a fluid subject to a vertical temperature gradient was examined by Lapwood (4). However, in natural conditions the fluids move generally in inhomogeneous media and then it is important to deal with the problem of stability in these media; the purpose of this paper is to consider just such a problem.
Relativistic magnetohydrodynamics and irreversible thermodynamics
Authors: Hughes W. F.
Published: November 2008

The basic equations of non-relativistic magnetohydrodynamics are briefly reviewed. Beginning with the energy-momentum tensor for the coupled electromagnetic and fluid fields relativistic energy and momentum equations are formulated. The effect of heat conduction is included and relativistic constitutive equations are discussed. A relativistic first law of thermodynamics is developed in terms of internal energy and entropy. Finally, the second law and entropy production rate are derived in a covariant fashion.
A family of exact solutions of one-dimensional anisentropic flow
Authors: Weir D. G.
Published: November 2008

Martin(1) has shown that if the pressure p and the Lagrange variable ψ (ψ = constant on a trajectory) are taken as independent variables, then problems in one-dimensional unsteady flow of an ideal gas reduce to the question of finding solutions of the Monge-Ampère equation

where τ = 1/ρ, ρ being the density, and S = S(ψ) is the specific entropy. Thus the right-hand term is a function of p and ψ depending upon the equation of state and the function S(ψ).
Estimates of Zeros of a Polynomial
Authors: Mirsky L.
Published: November 2008

Throughout this note we shall consider a fixed polynomial with complex coefficients and of degree n ≥ 2. Its zeros will be denoted by ξ1, ξ2, …, ξn where the numbering is such that Making use of Jensen's integral formula, Mahler (4) showed that, for l ≥ k < n, A slightly weaker result had been established by Feldman in an earlier publication (2). Mahler's inequality (1) is of importance in the study of transcendental numbers, and our first object is to sharpen his bound by proving the following result.
A note on time-varying gravitational potentials
Authors: Surdin M.
Published: November 2008

Using Newtonian theory of gravitation and postulating the existence of informational waves of gravitation, a time-varying gravitational potential is obtained. When a point mass is submitted to such a potential a precession of its orbit, of the right order of magnitude, is obtained. The correct value of the angle of deflexion of light by massive bodies can also be calculated.
On some inequalities of H. Kober
Authors: Diananda P. H.
Published: November 2008

Throughout this paper q1,q2,… are positive and x1,x2,… non-negative real numbers. When q1+…+ qn = 1, we define Δn, Sn and Σn as follows:

and

We use the term ‘comparable’ in Kober's(2) sense. Two functions f(x1, …, xn) and g(x1, …, xn) are comparable if and only if αg ≤ f ≤ βg for some positive constants α and β.
On approximations to the densities and moments of a class of stochastic systems
Authors: Kozin† F. , Bahtlett M. S.
Published: November 2008

This paper is concerned with a study of a class of stochastic processes that we shall call weakly non-deterministic.
The propagation of magneto-thermo-elastic plane waves
Authors: Willson A. J.
Published: November 2008

In a recent paper (1), Paria has discussed the propagation through a solid of a certain type of magneto-thermo-elastic plane wave. The analysis is essentially a reconciliation of the equations governing three fields: the electromagnetic field, the thermal field and the elastic field, which interact one with another. The principal result which was obtained was the dispersion equation connecting the frequency and the wavelength of waves of this type.
Some two-dimensional punch and crack problems in classical elasticity
Authors: England A. H. , Green A. E.
Published: November 2008

General methods for solving two-dimensional punch and crack problems in classical elasticity have been developed by Muskhelishvili and others. An alternative method for solving such problems is presented here. Use is made of suitable integral representations of the complex potentials and each problem is reduced to the solution of an Abel or Fredholm type of integral equation.
The motion of a non-conducting sphere through a conducting fluid in a magnetic cross-field
Authors: Ludford G. S. S. , Singh M. P.
Published: November 2008

The title problem for a perfectly conducting sphere was considered by Stewartson(6). Here we use a different method and a more realistic assumption concerning the sphere. Although the flow is still ultimately cylindrical, there is no longer a column of fluid which moves with the sphere as if solid. There is no reason to expect that in practice the flow will break away into eddies. In fact the same is true in Stewartson's case, once his results are corrected.
h2- and h4-extrapolation in eigenvalue problems
Authors: Dennis S. C. R.
Published: November 2008

Two recent papers have discussed eigenvalue problems relating to second-order, self-adjoint differential equations from the point of view of the deferred approach to the limit in the finite-difference treatment of the problem. In both cases the problem is made definite by considering the differential equation

primes denoting differentiation with respect to x, with two-point boundary conditions

and given at the ends of the interval (0, 1). The usual finite-difference approach is to divide the range (0, 1) into N equal strips of length h = 1/N, giving a set of N + 1 pivotal values φn as the analogue of a solution of (1), φn denoting the pivotal value at x = nh. In terms of central differences we then have

and retaining only second differences yields a finite-difference approximation φn = Un to (1), where the pivotal U-values satisfy the equations

defined at all internal points, together with two equations holding at the end-points and approximately satisfying the end conditions (2). Here Λ is the corresponding approximation to the eigenvalue λ. A possible finite-difference treatment of the end conditions (2) would be to replace (1) at x = 0 by the central-difference formula

and use the corresponding result for the first derivative of φ, i.e.

whereq(x) = λρ(x) – σ(x). Eliminating the external value φ–1 between these two and making use of (1) and (2) we obtain the equation

where for convenience we write k0 = B0/A0. Similarly at x = 1 we obtain

withkN = B1/A1. If we neglect terms in h3 in these two they become what are usually taken to be the first approximation to the end conditions (2) to be used in conjunction with the set (4) (with the appropriate change φ = U, λ = Λ). This, however, results in a loss of accuracy at the end-points over the general level of accuracy of the set (4), which is O(h4), so there is some justification for retaining the terms in h3, e.g. if a difference correction method were being used they would subsequently be added as a correction term.
A Monte Carlo method applied to the Heisenberg ferromagnet
Authors: Handscomb D. C.
Published: November 2008

Following on from a previous paper (5), we apply the new Monte Carlo method described there to the estimation of order parameters of a simple Heisenberg ferromagnet. By way of illustration, we include some results on the simple cubic lattice, comparing them with results obtained by conventional methods.
Coulomb wave functions for low energies
Authors: Meligy A. S.
Published: November 2008

The irregular radial Coulomb wave function is expanded in a convergent series of Bessel functions in which the coefficients are expressed in powers of the energy and the argument of the Bessel functions depends on the radius only and not on the energy. This formulation is suitable for low-energy particles. The corresponding expansion for the regular Coulomb function can be deduced from previous work by Tricomi.
Equilibrium analysis of a stochastic model of traffic flow
Authors: Gordon W. J. , Newell G. F.
Published: November 2008

A. J. Miller has proposed a stochastic model for the study of highway traffic in one direction along a two-lane road in which the desired speed of a vehicle is sampled from a continuous probability distribution and queues of vehicles are Poisson distributed on the highway. Miller has derived the equilibrium expression for the relation between the spacial densities of all vehicles and the spacial densities of those which are freely travelling. This relationship takes the form of a non-linear integral equation which is left unsolved. The present model differs from the original only in the assumption of a discrete distribution of desired speeds. Although the derivations of the equilibrium equations are based upon different arguments from those of Miller, these relations transform to the corresponding integral equations in the limit of a continuous speed distribution. The main body of the paper is devoted to a detailed investigation of the case of two speeds and asymptotic results are obtained for three speeds.
Some comments on single-server queuing methods and some new results
Authors: Keilson Julian
Published: November 2008

1.Introduction. Over the past few years, the theory of single-server queues has grown substantially. Much of the effort expended has been applied to the study of transient multivariate queuing processes in continuous time, i.e. to the study of joint time-dependent distributions of two or more system variables given some known initial system configuration. A variety of methods have been employed in these studies to accommodate general interarrival and service time distributions, among them Erlang's method of fictitious phases, the method of supplementary variables, the extended imbedded chain method, and the semi-Markov method. In the first part of this paper an informal review of these transient studies and of the methods employed is given.
The exact Hausdorff measure of the sample path for planar Brownian motion
Authors: Taylor S. J.
Published: November 2008

It has long been known (see Lévy (3), pp. 256, 260) that the sample paths of Brownian motion in the plane form an everywhere dense set of zero Lebesgue measure, with probability 1. In (7), a capacity argument was used to show that the Hausdorff measure with respect to tα is infinite for 0 < α < 2 with probability 1 so that the dimension of the path set is known to be 2. If one considers the initial part of the sample path Cω = C(1, ω) for 0 ≤ t ≤ 1, then it becomes interesting to ask if there is a measure function ψ(t) such that, with probability 1,

for suitable positive constants c1, c2. The corresponding problem for paths in k-space (k ≥ 3) has been solved. In this case, if φ1(t) = t2 log log t−1, Lévy (4) obtained the upper bound and Ciesielski and Taylor (1) obtained the lower bound. For the planar case, the path is recurrent, and the intricate fine structure makes the measure function φ1(t) inappropriate. In (2), Erdő and Taylor showed that the measure is finite with probability 1 with respect to φ2(t) = t2 log t−1, and at that time we thought that (1) might be true with ψ(t) = φ2(t). Recently Ray (5) has obtained the lower bound in (1)with

The purpose of the present note is to obtain the upper bound in (1) for the same measure function, thus showing that (2) defines the correct measure function for measuring planar Brownian motion.
Reducible vector bundles on a quadric surface
Authors: Newstead P. E. , Schwarzenberger R. L. E.
Published: November 2008

0. Introduction. In a paper (1) with the same title, one of us attempted to classify the reducible k2-bundles on a quadric surface Q = P1 × P1 defined over an algebraically closed field k. This attempt suffered from two defects: first, the classification was given by a one-one correspondence rather than by an algebraic parameter variety, and, secondly, there is a mistake in the proof of Proposition 6 of (1), as a result of which Proposition 6, Proposition 7 and the exceptional clause of the Theorem in (1) are false. To correct these defects we use the definitions, notations and numbering of (1) without comment. Finally, we show how the classification of reducible k2-bundles on P1 × … × P1 (n factors) is determined by that of reducible k2-bundles on P1 × P1. The net effect of these corrections is to underline the moral of (1): while the classification of reducible k2-bundles (but with a distinguished factor of the product variety P1 × … × P1) is as nice as it could possibly be, the classification of reducible k2-bundles alone is as nasty as it could possibly be. Since every decomposable k2-bundle is just a sum of two line bundles, we consider only indecomposable reducible k2-bundles.
Chern characters and higher-order cohomology operations
Authors: Maunder C. R. F.
Published: November 2008

In this paper we shall show how a large number of higher-order cohomology operations (see (4)) may be calculated in suitable spaces by means of Chern characters: the method is an extension of that used by Adams in (1) to relate ‘mod p reductions’ of Chern characters and certain primary operations.
Differential inequalities and extension of Lyapunov's method
Authors: Lakshmikantham V.
Published: November 2008

One of the most important techniques in the theory of non-linear differential equations is the direct method of Lyapunov and its extensions. It depends basically on the fact that a function satisfying the inequality

is majorized by the maximal solution of the equation

Using this comparison principle and the concept of Lyapunov's function various properties of solutions of differential equations have been considered (1–11).
A dam with inverse Gaussian input
Authors: Hasofer A. M.
Published: November 2008

Gani and Prabhu (1) have presented a general theory of storage for dams subject to a steady release. At timet≥ 0, let a dam of infinite capacity have a continuous random contentZ(t) ≥ 0; the inputX(t) ≥ 0 which has entered it during timetfollows an infinitely divisible distribution, while the dam is subject to a steady release at constant unit rate except if it is empty, when the release ceases. A procedure is obtained whereby various functions describing the transient behaviour ofZ(t) can be calculated in principle.
On well-quasi-ordering transfinite sequences
Authors: Nash-Williams C. St. J. A.
Published: November 2008

. Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q1,q2,… of elements of Q, there exist i and j such that i < j and qi ≤ qi. A restricted transfinite sequence on Q is a function from a well-ordered set onto a finite subset of Q. If f, g are restricted transfinite sequences on Q with domains A, B respectively and there exists a one-to-one order-preserving mapping μ of A into B such that f(α) ≤ h(μ(α)) for every α ∈ A, we write f ≤ g. It is proved that this rule well-quasi-orders the set of restricted transfinite sequences on Q. The proof uses the following subsidiary theorem, which is a generalization of a classical theorem of Ramsey (4). Let P be the set of positive integers, and A(I) denote the set of ascending finite sequences of elements of a subset I of P. If s, t∈A(P), write s ≺ t if, for some m, the terms of s are the first m terms of t. Let T1,…,Tn be disjoint subsets of A(P) whose union T does not include two distinct sequences s, t such that s ≺ t. Then there exists an infinite subset I of P such that T ∩ A(I)is contained in a single Tj.
On the differentials in the Adams spectral sequence for the stable homotopy groups of spheres. I
Authors: Maunder C. R. F.
Published: November 2008

In (8), we have shown how, in general, the problem of identifying the differentials in the Adams spectral sequence (see(1),(4)) is equivalent to that of calculating certain higher-order cohomology operations (in the sense of (6)). However, we propose to investigate here a slightly different method, based on the naturality of the spectral sequence, which can be used to show that certain elements are never boundaries, for any differential.
Feynman's kernel and the classical path
Authors: Clutton-Brock M.
Published: November 2008

The quantum mechanical kernel K(q″,t″; q′,t′) in Feynman's form of a sum over paths is evaluated by means of the canonical transformation to the system Q, P for which the Hamiltonian vanishes identically. The kernel is found to be completely determined by the classical paths, and is given by

The sum is over all classical paths leading from q′ at t′ to q″ at t′. S(q″, P, t′; q′, P, t′) is the classical action along the classical path characterized by the constant momentum P.
On semi-normal lattice rings
Authors: Bernau S. J.
Published: November 2008

A lattice ring is a lattice group ((l), page 214) and a ring in which ab ≥ 0 whenever a ∧ b ≥ 0.

In any lattice group (commutative or not) we define a+ = a ∨ 0, a− = (−a) ∨ 0 and |a| = a+ + a−. Itisknown((1). pages 219,220) that a+ ∧ a− = 0, a = a+ − a−, |a| = a+ ∨ a−, and that a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), and a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). For a non-empty subset M of a lattice group we define
The distribution of the sequence {nξ}(n = 0, 1, 2, …)
Authors: Halton John H.
Published: November 2008

Introduction and statement of results. We shall describe how, for successive integers N, the points {nξ}, with n = 0, 1, …,N – 1, are distributed in the closed unit interval U = [0, 1]; by showing how successive points {Nξ,} modify the partition of U produced by the previous points. The simple generalization to the k-dimensional sequence {nξ} = ({nξ(1)},{nξ(2)}, …,{nξ(k)}), in the unit hypercube Uk, is also made.
On well-quasi-ordering infinite trees
Authors: Nash-Williams C. St. J. A.
Published: November 2008

Let A be the set of all ascending finite sequences (with at least one term) of positive integers. Let s, t ∈ A. Write s ⊲ t if there exist m, n, x1, …, xn such that m < n and x1 < … < xn and s is x1, …, xm and t is x2, x3, …, xn. Call a subset S of A a P-block if, for every infinite ascending sequence x1, x2, … of positive integers, there exists an m such that x1, …, xm belongs to S. A quasi-ordered set Q (i.e. a set on which a reflexive and transitive relation ≤ is defined) is better-quasi-ordered if, for every P-block S and every function f:S → Q, there exist s, t ∈ S such that s ⊲ t and f(s) ≤ f(t). It is proved that any set of (finite or infinite) trees is better-quasi-ordered if T1 ≤ T2 means that the tree T1 is homeomorphic to a subtree of the tree T2. This establishes a conjecture of J. B.Kruskal that, if T1, T2, … is an infinite sequence of trees, then there exist i, j such that i < j and Ti ≤ Tj.
Linear equations with elements (l/m)k(n/m) + δnmand infinitely many variables
Authors: Wood W. W.
Published: November 2008

Introduction.Infinite systems

are discussed in whichk(θ) andb(n) are differentiable. The aim is to determine the behaviour ofa(n) for largenand to reduce (1) to a finite system. This involves solving

for the high ordera(n),n≥N, in terms of the precedinga(n), with a large enoughNto facilitate the iteration procedure employed. The desiredN− 1,N− 1 system follows on substitution for the high ordera(n) in the firstN− 1 equations (1).
Notes on general topology
Authors: Ward A. J.
Published: November 2008

It is well known that a path-connected Hausdorff space is arc-connected. Indeed, given a path P, that is, a continuous map. f of the unit interval I0(0 ≤ t ≤ 1) into the Hausdorff space X, there is an arc joining the points f(0), f(1) (supposed distinct) which is obtained by ‘cutting out loops in P’. More precisely, there exist a continuous increasing map α of I0 onto itself and a homeomorphism φ of I0 onto φ(I0) ⊂ X, such that if [t0, t1] is any maximal interval of constancy of α (including the case t0 = t1) then f(t0) = f(t1) = φοα(t0). The function φ, α, can be defined by a constructive process.
Refractive index profiles based on the hypergeometric equation and the confluent hypergeometric equation
Authors: Heading John
Published: November 2008

Isotropic ionospheric propagation of electromagnetic waves yields refractive index profiles via the standard transcendental equations. Some transformations of these equations have long been regarded as standard. Here, every possible transformation is derived by a systematic analysis, yielding all possible refractive index profiles satisfying certain stated criteria regarding the wave frequency.
The isotropic scattering of radiation from a point source in a finite spherical atmosphere
Authors: Smith M. G.
Published: November 2008

The investigation of the isotropic scattering of radiation in an infinite spherically symmetric atmosphere, which was carried out in a previous paper, is extended to a finite atmosphere bounded by a totally absorbing shell.

It is shown that the solution may be expressed as a series which will certainly converge when the radius of the shell is large enough, and whose terms are found by iteration.

Numerical investigation shows that even for quite modest values of the radius, the convergence is so fast that for most practical purposes all but the first term can be neglected. Except at the very ends of the range, a very simple expression then gives an excellent approximation.
On the forced torsional oscillations of an elastic cylinder
Authors: Shail R.
Published: November 2008

The forced torsional motion of a semi-infinite isotropic right-circular elastic cylinder is studied in this paper, the vibrations being excited by a harmonically oscillating rigid disc, securely attached to the plane face of the cylinder. Two distinctsituations are considered, namely, when the curved surface of the cylinder is clamped and when it is stress free. In both cases the problem is reduced to the solution of a Fredholm integral equation of the second kind which can be solved iteratively when the frequency of oscillation is small and the cylinder radius large. Using the iterative solutions, approximate expressions are deduced for the distribution of shear stress under the disc and the driving couple required. In an appendix a discussion is given of a mathematically similar problem in the theory of the swirling flow of a perfect fluid.
Polyhedral immersions
Authors: Irwin M. C.
Published: November 2008

1. We work in the category of compact polyhedral spaces and polyhedral maps ((5)). Given spaces M and Q and a map f: M → Q, we define Ur(f) to be the set of points x ∈ M such that f−1f(x) contains at least r points. Sr(f) is the closure of Ur(f) in M. f is an embedding if it is a homeomorphism into, and an immersion if it is locally an embedding. We shall call f a simple immersion if S3(f) = ø and the connected components of S2(f) are individually embedded by f. Obviously a simple immersion is an immersion. If M and Q are manifolds (as they will be for the rest of the paper)f: M → Q is proper if it takes ∂M, the boundary of M, into ∂Q. In (2) the following result was proved:

Theorem 1. Let Mmbe(2m−q)-connected and Qq be(2m−q + 1)-connected, m ≤ q−3. Then any proper map f: M → Q which embeds ∂M is homotopic rel ∂M to an embedding.
The propagation of magneto-elastic plane waves
Authors: Willson A. J.
Published: November 2008

The propagation of magneto-elastic plane waves is a topic which has recently aroused much interest. Thus Paria ((l)), for example, and the present author ((2)), have studied in some detail the interaction between the magneto-elastic field and the thermal field. Analyses published so far, however, have neglected the effects due to the passage of the wave upon the magnetic permeability of the medium. It is the object of the present paper to consider these effects and it will be shown that although the elastic strains induced cause but a small change in the numerical values of the permeability, nevertheless the effects upon wave propagation may be very considerable. The principal result is the demonstration that the effects considered are equivalent so far as the dispersion equation is concerned, to an anisotropic rescaling of the primary magnetic field, the direction of propagation of the wave being a preferred direction. For simplicity, thermal effects are not considered here.
Real two-dimensional representations of the free product of two finite cyclic groups
Authors: Lehner J. , Newman M.
Published: November 2008

Let Λ be the group of all real non-singular 2 × 2 matrices and let Ω be the group of all real 2 × 2 matrices with determinant 1 in which a matrix is identified with its negative. Then Ω is isomorphic to the group of all real linear fractional transformations of determinant 1. In a previous paper ((3)) the authors determined all faithful representations of the modular group (more generally of the Hecke groups) by a discrete subgroup of Ω, in which the representations were partitioned into conjugacy classes over Λ. In this paper we consider the question for the more general situation of the free product of any two cyclic groups of finite order. Our results parallel the results of (3) quite closely, but some significant differences in the details of the proofs arise. In particular Theorem 3 of section 3 below, which is purely group-theoretic, is of independent interest and should prove useful in other investigations.
Spectral representations for families of self-adjoint operators
Authors: Broido Michael M.
Published: November 2008

Spectral theorems and a functional calculus for families of permutable self-adjoint operators in a separable Hilbert space are developed in a form convenient for physical applications. A recent paper of Jauch and Misra is discussed.
On the rigorous foundation of short-wave asymptotics
Authors: Ursell F.
Published: November 2008

Certain physical theories are short-wave limits of more general theories. Thus ray optics is the short-wave limit of wave optics, and classical mechanics is the short-wave limit of wave mechanics. In principle it must be possible to deduce the former from the latter theories by a rigorous mathematical limiting process; in fact the arguments found in the literature are formal, plausible and non-rigorous. (We are here concerned with linear wave equations and time-periodic phenomena.) For some wave equations there are, however, a few explicit rigorous canonical solutions relating to simple geometrical configurations, e.g. to conics in two dimensions for the equations of acoustics, and for these the asymptotics can be found rigorously. For more general configurations the solution of a typical boundary-value problem can be reduced to the solution of a Fredholm integral equation of the second kind.
Some expansions in products of hypergeometric functions
Authors: Srivastava H. M.
Published: November 2008

In the usual notation let

where .

Also, for the generalized hypergeometric function pFq(x) let us employ a contracted notation and write

Throughout the present paper i will run from 1 to p, I from 1 to P, and so on. Thus ((a) )m is to be interpreted as

,

and similar interpretations hold for ((A))m, etc.
On a non-linear integral equation occurring in diffraction theory
Authors: Millar R. F.
Published: November 2008

The problem of diffraction of a plane wave by a semi-infinite grating of iso-tropic scatterers leads to the consideration of a non-linear integral equation. This bears a resemblance to Chandrasekhar's integral equation which arises in the study of radiative transfer through a semi-infinite atmosphere. It is shown that methods which have been used with success to solve Chandrasekhar's equation are equally useful here. The solution to the non-linear equation satisfies a more simple functional equation which may be solved by factoring (in the Wiener-Hopf sense) a given function. Subject to certain additional conditions which are dictated by physical considerations, a solution is obtained which is the unique admissible solution of the non-linear integral equation. The factors and solution are found explicitly for the case which corresponds to closely spaced scatterers.
Solutions of unsteady compressible boundary-layer equations
Authors: Sarma G. N.
Published: November 2008

The unsteady two-dimensional compressible boundary layers have been studied by Sarma ((5)) assuming that the Prandtl number is unity and that the wall is in an arbitrary motion, the main stream being steady. In this paper the Prandtl number is taken to be an arbitrary parameter which need not be equal to unity, and it is assumed that the main stream velocity and temperature are perturbing about a steady mean, the wall being in an arbitrary motion. Solutions are obtained in two parts, one when the temperature gradient at the wall is perturbing about a zero mean and the other when the temperature of the wall is perturbing about a steady mean. Thus in this paper the theory given in Sarma ((5)) is made still more general. Following the work of Sarma ((4)–(6)) two types of solutions are developed in each part, one for large times and the other for small times.
Forced vertical vibration of a rigid circular disc on a semi-infinite elastic solid
Authors: Robertson Ian A.
Published: November 2008

Introduction. The effect of a vibrating point force on the surface of a semi-infinite elastic solid was investigated in a classic paper by Lamb ((8)). When the point force is replaced by a vibrating rigid disc the problem is more complicated, because the displacement is specified under the disc and the remainder of the surface is stress-free. This gives rise to a mixed boundary-value problem.
Crossed homomorphisms of Lie algebras
Authors: Lue Abraham S.-T.
Published: November 2008

In an earlier paper (3), a non-abelian cohomology theory (in the dimensions 0 and 1) for associative algebras was developed. One of the objectives was to obtain equivalence classes of crossed homomorphisms by considering inner automorphisms of the coefficient-algebra. This paper is an adaptation of the methods employed there to the case of Lie algebras. Throughout, all our Lie algebras will be over the field of real numbers, and finite-dimensional.
The zeta function of a cubic surface over a finite field
Authors: Swinnerton-Dyer H. P. F.
Published: November 2008

Introduction. The object of this paper is to obtain an explicit formula for the zeta function of an arbitrary non-singular cubic surface over a finite field. Let k denote the finite field of q elements, and kn the field of qn elements which is the unique algebraic extension of k of degree n. Let be a non-singular variety defined over k, and for each n > 0 let be the number of points defined over kn which lie on . The zeta function of is given by

Dwork has shown in (3) that for any this is a rational function of q−s; and in particular it follows from the results he proves in (4) that if is a non-singular cubic surface then

and hence also

Here the numbers w o depend only on q and on .
Summability fields which span the bounded sequences
Authors: Baker J. W. , Petersen G. M.
Published: November 2008

If A = (am, n) is a regular matrix, then for any sequence x = {xn}, Am(x) will denote the transform and A-lim x denotes if that limit exists. We shall denote by the set of bounded sequences which are summed by A. If B is another regular matrix with then we say that B is b-stronger than A. In that case B must be b-consistent with A (see (4) and (6)), i.e. if then

If {μn} is a sequence of positive real numbers with we say that A and B are (μn)-consistent if every sequencer x = {xn} satisfying xn = 0(μn) which is summed by both A and B is summed to the same value by both matrices. A finite set of matrices A1, A2, …, AN is said to be simultaneously (μn)-inconsistent (b-consistent) if whenever is summed by Ai with (i = l, 2, …, N) then implies that The set of sequences, is denoted by
The isotropic scattering of radiation in a finite homogeneous two-dimensional atmosphere
Authors: Smith M. G. , Hunt G. E.
Published: November 2008

The integral equation for the isotropic scattering of radiation in a medium bounded laterally by planes ±y0 is solved as a series in powers of exp (−y0). This series is certainly convergent for y0 large enough, and rapidly convergent for most practical purposes.
Generating functions of Jacobi polynomials
Authors: Manocha H. L. , Sharma B. L.
Published: November 2008

In this paper we obtain generating functions of Jacobi polynomials. The results obtained are of general character and include as particular cases of the results given earlier by Bailey ((1), p. 102, Ex. 19) and Brafman (2).
M-structure in Banach spaces
Authors: Cunningham F.
Published: November 2008

L-structure in a Banach space X was defined in (3) by L-projections, that is projections P satisfying

for all x ∈ X. The significance of L-structure is shown by the following facts: (1) All L-projections on X commute and together form a complete Boolean algebra. (2) X can be isometrically represented as a vector-valued L1 on a measure space constructed from the Boolean algebra of its L-projections (2). (3) L1-spaces in the ordinary sense are characterized among Banach spaces by properties equivalent to having so many L-projections that the representation in (2) is everywhere one-dimensional.
Simultaneous rational approximations to certain algebraic numbers
Authors: Baker A.
Published: November 2008

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the form

is satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such that

This result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡
Lacunas in two-dimensional wave propagation
Authors: Burridge R.
Published: November 2008

A simple sufficient condition is given for the existence of a lacuna in two-dimensional wave propagation governed by an equation of the second order. This work was inspired by Petrowsky's very general work (Petrowsky (3)) but for waves in two space dimensions we do not need the sophisticated machinery developed by him.
Correction to ‘Mod p cohomology theories and the Bockstein spectral sequence’
Authors: Maunder C. R. F.
Published: November 2008

In (1), Theorem 4·1·4, and hence Theorem 4·3·1, are incorrect as stated, unless h* denotes ordinary cohomology theory. However, if h* denotes complex K-theory, corresponding results still hold, provided everything is graded over Z2 instead of Z.
Some computations for m-dimensional partitions
Authors: Atkin A. O. L. , Bratley P. , Macdonald I. G. , McKay J. K. S.
Published: November 2008

1. It was known to Euler that p(n), the number of unrestricted partitions of n into non-increasing integral parts, is generated by

with the usual convention that p(0) = 1.
A class of infinitely divisible random variables
Authors: Goldie Charles
Published: November 2008

The question has arisen in the theory of queues whether the product of two independent exponentially distributed random variables is infinitely divisible. In this note it is proved that the product of any non-negative random variable with an independent exponentially distributed variable is infinitely divisible. Such random variables are members of a wider class of infinitely divisible random variables, which will be exhibited first.
Compatible uniformities and closed mappings
Authors: Roberts J. E.
Published: November 2008

Gel'fand and Šilov(1) have introduced the concept of compatibility of norms on a vector space, defining two norms to be compatible if each sequence which is Cauchy with respect to both norms and converges to zero with respect to one norm, also converges to zero with respect to the other. They have found it convenient to build up their exposition of spaces of generalized functions (2) by developing the theory of locally convex spaces whose topology can be defined by a countable set of pairwise compatible norms. The author (3) has examined the properties of locally convex spaces which have been topologized by making a set of linear mappings into locally convex spaces continuous, and has shown that if this method is applied to a set of closed operators on Hilbert space, then the resulting spaces have topologies generated by a set of pairwise compatible norms. The purpose of this paper is to generalize this result by putting it in its natural context as a theorem on uniformities which have been defined by making a closed set of mappings uniformly continuous. The closed mappings of the title are mappings with closed graph.
Finite extension inflation and torsion of a hollow cylindrical reinforced dielectric
Authors: Chaudhry H. R.
Published: November 2008

Eringen's theory(1) of homogeneous, isotropic, hyperelastic dielectric is applied to consider the effects of polarization on the surface tractions when a hollow cylindrical reinforced dielectric carrying a uniform surface charge at its inner surface is subjected to simultaneous large extension, inflation and torsion. New effects due to reinforcement and polarization have been observed separately. For vanishing charge, the results reduce to those of classical finite strain theory (2).
The fundamental group of the orbit space of a discontinuous group
Authors: Armstrong M. A.
Published: November 2008

A group G of homeomorphisms of a topological space X will be called discontinuous if

(1) the stabilizer of each point of X is finite, and

(2) each point x ∈ X; has a neighbourhood U such that any element of G not in the stabilizer of x maps U outside itself (i.e. if gx ≠ x then U ∩ gU is empty). The purpose of this note is to prove the following result.
A new expansion formula for hypergeometric functions of two variables
Authors: Sharma B. L.
Published: November 2008

The main object of this paper is to derive an expansion formula for a generalized hypergeometric function of two variables in a series of products of generalized hypergeometric functions of two variables and a Meijer's G-function. The result established in this paper is the extension of the results recently given by Srivastava (5) and Verma (6). It is interesting to note that some interesting expansions can be derived from the result by specializing the parameters.
A problem of bounded expressibility in free products
Authors: Rhemtulla A. H.
Published: November 2008

1·1. The main result. Let ø = ø(x1, x2, …, xn) be a word in n variables and let G be a group. A ø-element of G is any element of the form ø(x1, x2, …, xn)±1 with xi ∈G(1 ≤ i ≤ n). The subgroup generated by all the ø-elements of G is the verbal subgroup ø(G) of G. If there is a positive integer l such that every element of ø(G) can be expressed as the product of l or fewer ø-elements of G we say that G is ø-elliptic. If there is no such integer we say that G is ø-parabolic.
A characterization of sets of n points which determine n hyperplanes
Authors: Basterfield J. G. , Kelly L. M.
Published: November 2008

Suppose N is a set of points of a d-dimensional incidence space S and {Ha}, a ∈ I, a set of hyperplanes of S such that Hi ∈ {Ha} if and only if Hi ∩ N spans Hi. N is then said to determine {Ha}. We are interested here in the case in which N is a finite set of n points in S and I = {1, 2,…, n}; that is to say when a set of n points determines precisely n hyperplanes. Such a situation occurs in E3, for example, when N spans E3 and is a subset of two (skew) lines, or in E2 if N spans the space and n − 1 of the points are on a line. On the other hand, the n points of a finite projective space determine precisely n hyperplanes so that the structure of a set of n points determining n hyperplanes is not at once transparent.
Steady electrically driven flows
Authors: Waechter R. T.
Published: November 2008

Certain simple steady electrically driven flows under transverse applied magnetic fields, for which exact solutions are available, are considered. Electric potentials are prescribed on fixed perfectly conducting electrodes and the resulting Lorentz force j × B0 gives rise to a velocity field in the fluid. It is assumed that the magnetic Reynolds number of the flow is small (Rm ≪ 1) so that the induced magnetic field may be neglected. In particular, we examine in detail the simplest flat plate configuration which is correctly posed as a two-part mixed boundary value problem of the Wiener–Hopf type. At large Hartmann number, the fluid vorticity and the current flow transverse to the applied magnetic field are suppressed. Thin current transition layers, which emanate from an electrode edge and are aligned with the applied magnetic field, provide a smooth transition from one region of almost constant current to another. The thickness of these layers is found to be of the same order as the thickness of certain velocity transition layers occurring in other situations in magnetohydrodynamics.
On ideals of finite homoloǵical dimension in local rings
Authors: Burch Lindsay
Published: November 2008

In this paper I shall demonstrate certain algebraic properties of ideals of finite homological dimension in local rings.

In the first section, I show that no non-zero ideal of finite homological dimension in a local ring can be of zero grade (this is stated by Auslander and Buchsbaum in the appendix to (l), but I cannot find a proof in the literature). Using this result, together with the complex defined by a matrix, which is described by Eagon and Northcott in (2), I prove that an ideal of homological dimension one in a local ring Q may always, for some integer n, be described as the set of determinants of matrices obtained by adjoining to a certain (n–l)× n matrix with elements in the maximal ideal of Q another row with elements arbitrarily chosen in Q. (This result was established in (3), under the additional condition that Q should be a domain.)
A note on the asymptotic expansion of the second Appell function
Authors: Minton B. M.
Published: November 2008

In a long series of papers published some years ago both here and elsewhere, Dr L. J. Slater has derived a large number of valuable results concerned with the theory of generalized hypergeometric functions.
On some results involving generalized hypergeometric polynomials
Authors: Shah Manilal
Published: November 2008

The generalized hypergeometric polynomial ((7), equation (2·1)) has been defined by

where the symbol Δ(δ, −n) represents the set of δ-parameters:

and δ, n are positive integers. The polynomial is in a generalized form which yields many known polynomials on specializing the parameters.