bluuebunny/crossref_metadata_2025 huggingface dataset:
Flexure with shear centres: a general treatment with complex variable
Authors: Capildeo R. , Davenport H.
Published: November 2008

This work is consequent on a forthcoming joint paper by A. C. Stevenson and the present writer on the use of the torsion function in the solution of flexure problems. The treatment is general, but special attention is paid to doubly-connected cross-sections which can be mapped conformally on a circular annulus. The eccentric circles cross-section (4) is solved in order to illustrate the power of the method, and results are obtained for the cross-section bounded by confocal limacons. This is of interest, since it has been customary to regard these problems as largely determined by the cross-section of the cylinder; but these results show that the terms involving the elastic constants are now no longer negligible. The method is capable of giving useful information to aeronautical engineers, and the writer hopes to present in a later communication a treatment of an aerofoil-like profile.
The Green's function of an elastic plate
Authors: Dean W. R.
Published: November 2008

In this paper a simple expression in finite terms is found for the small transverse displacement of a thin plane elastic plate due to a transverse force applied at an arbitrary point of the plate. The plate is clamped along the semi-infinite straight lines represented by AB, CD in Fig. 1, these lines being the only boundaries of the plate. The transverse displacement w at any point (x, y) of the plate is a biharmonic function of the variables (x, y) which vanishes together with its normal derivative at all points of the boundary. Clearly w is also a function of the coordinates (x0, y0) of the point of application of the force, and it is known ((5), p. 173) that it is a symmetrical function of the coordinate pairs (z, y) and (x0, y0); it is the Green's function associated with the differential equation and the boundary conditions.
Yet another proof of Minkowski's theorem on the product of two inhomogeneous linear forms
Authors: Cassels J. W. S.
Published: November 2008

Theorem 1. Let ξ = αx + βy, η = γx + δy be two homogeneous linear forms in x, y with real coefficients and determinant αδ − βγ = Δ ≠ 0. Then for any real constants p, q there are integers x, y such that
Hyperelliptic threefolds
Authors: Roth L.
Published: November 2008

It is a classical result (5) that the algebraic surfaces which admit a continuous group of birational self-transformations, or automorphisms, and which are neither rational nor scrollar, belong to one or other of two families: (i) the elliptic surfaces; these possess an elliptic group of ∞1 automorphisms; (ii) the hyperelliptic surfaces of rank 1 (which, for brevity, we shall call hyperelliptic); these possess a completely transitive permutable group of ∞2 automorphisms. The chief properties of the two classes of surface, in so far as they are required in the present work, are described in § 1.
A certain triple Whitehead product
Authors: Hilton P. J.
Published: November 2008

The triple Whitehead product we consider in this note is [[ɩn, ɩn], ɩn] ∈ π3n−2(Sn), where ɩn generates πn(Sn). It follows from the Jacobi identity for Whitehead products

α ∈ πp(X), β ∈ πq(X), γ ∈ πr(X), that 3[[ɩn, ɩn], ɩn] = 0. Now, if n is odd, 2[ɩn, ɩn] = 0, so that 2[[ɩn, ɩn], ɩn] = 0, whence
The coefficients of certain integral modular forms
Authors: Rankin R. A. , Rushforth J. M.
Published: November 2008

The notation which we use is that of a recent paper by one of us, and we quote results from that paper as they are required. It is known (see R, Theorem 1, for example) that the vector space k of all cusp-forms f(z) of even negative dimension – k (k ≥ 12), belonging to the full modular group Γ(1), possesses a finite basis of forms

where k is defined by (2·10) of R and the coefficients possess the following properties:

for a prime p, where p is a positive integer.
The product of non-homogeneous linear forms. I
Authors: Samet P. A.
Published: November 2008

1. Statement of problem and the method of Barnes and Swinnnerton-Dyer. 1·1. Let ξ1, ξ2, … ξn De n real, homogeneous linear forms in the real variables x1, x2,…, xn and of determinant Δ ǂ 0. Let y1y2, …, yn be any real numbers. We identify the n-tuplet (x1x2, …, xn) with the point P, coordinates (x1, x2, …, xn), of n-dimensional space.We write P≡P0 (mod 1) if x1≡y1, x2≡y2, …, xn≡yn (mod 1), and P0 is the point (y1, …, yn).
Some new results on equivalent products
Authors: Slater L. J.
Published: November 2008

In this paper, I prove some new results which are identities connecting groups of general infinite products, and I give two tables, calculated using EDSAC, of the function 1/II(1 − aqn−1) over the range a = − 0·90(0·05) + 0·95, q = 0·0(0·05) + 1·00, and of the function 1/n(1 − qn) over the range q = 0(0·005)+ 0·890.
On Kepler's equation
Authors: Cherry T. M.
Published: November 2008

1. Kepler's equation

has, when θ, x are real with 0 < x < 1, just one real root ξ = ξ*(θ, x). For this root there are the well-known formulae, dating from Lagrange and Bessel,

The objects of this paper are (i) to obtain analogous formulae for the unreal roots ξ of the equation, and (ii) to sum the conjugates (as Fourier series in θ) of the series on the right of (2) and (3).
On the infinity in the second Born approximation for the Coulomb field
Authors: Elton L. R. B.
Published: November 2008

It is shown that the infinity arising in the second Born approximation to the scattering amplitude of a particle scattered by a Coulomb field is due to the distortion at infinity of Coulomb waves. Ways of subtracting out this infinity, which does not lead to any observable effects, are indicated.
On the stretching of material lines and surfaces in isotropic turbulence with zero fourth cumulants
Authors: Reid W. H.
Published: November 2008

The convective effects of isotropic turbulence on arbitrary vector and scalar fields satisfying certain conservation conditions are studied under the assumption that all fourth-order cumulant tensors are zero. In the absence of molecular conduction, the behaviours of such vector and scalar fields are related to those of material lines and surfaces respectively, and the diffusive action of the turbulence results in a stretching of these lines and surfaces. The present analysis leads to a definite prediction for the corresponding strain rates which suggests that an initially spherical material volume element is drawn out into a long thin ribbon of constant width.
The interpolation of atomic fields
Authors: Ridley E. Cicely
Published: November 2008

Z(nl; r) is the contribution to Z(r) from an electron in the (nl) wave function. The Z(nl; r) vary systematically with atomic number and, as N becomes large, tend to the corresponding hydrogen-like functions, ZH(nl; r). A two-parameter method of fitting the Z(nl; r) to the ZH(nl; r) is described. This involves a ‘screening constant’ and a ‘slope constant’, both of which are defined. From published data, the two parameters have been obtained as functions of atomic number. The parameters for an unsolved atom can then be found by interpolation and approximate Z(nl; r) derived by appropriate adjustment of the functions for the nearest atom in the periodic table for which they are known. The method has been tested by interpolating for the (3d) function between Cu+ and Rb+ and by preparing estimates of the Z(nl; r) for the unknown structure Mo+. The results were good for all but Z(4d; r) for Mo+, where the number of values of the screening and slope constants already known was insufficient for reliable interpolation.
A remark on translated sets
Authors: Sawyer D. B. , Smithies F.
Published: November 2008

Let Λ denote the integral lattice in n-dimensional Euclidean space. A classical theorem of Minkowski's states that any bounded closed convex region K symmetrical in the origin 0 and with volume 2n contains a point of Λ other than 0. There will be a lattice point other than 0 in the interior of K except when K has certain forms, of which we will denote an arbitrary one by K*. An example of a K* is the cube |xi| ≤ 1 (i = 1, 2,..., n), and more generally a famous theorem of Hajós (3) states that if K* is a parallelepiped it is defined (except for integral unimodular transformations of the x's) by inequalities of the form
|x1| ≤ 1, |a21x1 + x2| ≤ 1, …, |an1x1 + … + an, n-1xn-1+xn| ≤ 1.
The grade of an ideal or module
Authors: Rees D.
Published: November 2008

In (6), the author introduced a numerical character of an idealof a Noether ringA, called the grade of. This can be defined as the least integerksuch that(the definition given in (6) differed from this, but, as will be seen below, is equivalent). The purpose of the present paper is to study this character in more detail.
On the notion of a first neighbourhood ring with an application to theAF+BΦ theorem
Authors: Northcott D. G.
Published: November 2008

In an earlier paper (3) the author developed a theory of the neighbourhoods of a local domain, in which the concept of thefirst neighbourhood ringplayed a central role. This notion has since proved useful in connexion with certain one-dimensional problems, but it has emerged, in the process, that a considerable advantage would be gained if the theory could be freed from the assumption that the basic ring was to be without zero-divisors. This parallels the situation in the geometry of plane curves, where it is desirable, so far as is possible, that results and methods should apply with equal facility toreducibleas well as to irreducible curves. Accordingly Part I of the present paper is devoted to a fresh account of the first neighbourhood ring from a more general standpoint than that used previously. Besides the greater generality thus obtained, the theory is extended by the addition of some new results. Furthermore, the proof of one of the main results of the original paper ((3), Theorem 10) has been considerably simplified. Of course, the ideas of (3) reappear here in a modified form, but, to spare the reader the irritating fragmentation of the subject which would otherwise be necessary, the revised account has been made independent of the earlier one. To this extent, Part I is self-contained.
On uniform acceleration in special and general relativity
Authors: Marder L.
Published: November 2008

Various definitions of uniform acceleration are possible in relativity theory. Two of these are considered with particular reference to their physical realizations.
The Stokes phenomenon and certain nth-order differential equations II. The Stokes phenomenon
Authors: Heading J.
Published: November 2008

1. Introduction. In paper I(2), solutions have been found for the nth-order differential equations (I, 3), (I, 9), (I, 18), namely,

where ϑ = wd/dw, m is any rational fraction and a any constant. Power-series solutions have been found for these equations, together with integral representations and their asymptotic expressions valid for restricted ranges of arg z. The object of this second paper is to consider these asymptotic solutions in more detail, and to extend these expressions to all values of arg z. The Stokes phenomenon will be manifest throughout, and this will be treated in a manner suitable for further application.
The concept of order III. General relativity as a technique for extrapolating over great distances
Authors: Bastin E. W. , Kilmister C. W.
Published: November 2008

It is shown that a combination of the methods developed in the two previous papers in the series allows the reformulation of general relativity as a technique for extrapolating physical theory to distant regions, whose accessibility to experiment is limited. General relativity is thus shown to be one of a number of extrapolation theories allowed by the spread theory of Paper II.
On the existence of freely generated algebras
Authors: Lowig H. F. J.
Published: November 2008

Throughout this paper, the concept of a ‘freely generated algebra’ will be meant in the sense of (3), Definition 2·1. It is understood that all conventions concerning terminology and notation introduced in (3) hold in this paper. In particular, it is understood that the signdenotes a freely generated algebra, and that the definition-set and the free basis ofare denoted byCandDrespectively. The letterqwill denote a cardinal number.
Bounded representations of the positive values of an indefinite quadratic form
Authors: Lawton B.
Published: November 2008

Let

be a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfying

where F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.
Further computations of quadrupole line strengths
Authors: Garstang R. H.
Published: November 2008

The methods introduced into the theory of atomic spectra by Racah are applied to the calculation of electric quadrupole line strengths for transitions involving d and s electrons. The transitions considered are of the types dn−dn−1s, dn−1s−dn−1s and dn−1s−dn−2s2. Sum rules and rules for almost closed shells are obtained. Detailed numerical results are given for various transitions in and between the configurations d9 and d8s, and d7, d6s and d5s2.
The Saddle-Points of the Confluent Hypergeometric Function M(a, b; x)
Authors: Fieldhouse M.
Published: November 2008

This paper demonstrates the existence of an infinity of saddle-points of the function and gives a method of evaluating them suitable for automatic computation. A table of about 100 points is given to 5 decimal places.
The simple group of order 6048
Authors: Edge W. L.
Published: November 2008

The group of the title is one of the classical groups and can be represented—will indeed be represented here—as the group of those projectivities in a certain finite plane that leave invariant the set of zeros of a Hermitian ternary form. Its properties in the large, so to speak, are common knowledge, being particular instances of those of unitary finite groups; the conjugate sets, and sufficient information to determine, or perhaps nearly to determine, the group characters are given in (7). The methods to be used here are elementary; they take advantage of the base field, and consequently the numbers of points in the geometry, being relatively small. Much detailed information emerges that seems worth putting on record; it does not occupy a disproportionate amount of room and might elude the coarser net of an investigation based on generalities.
On two-dimensional waves in an elastic half-space
Authors: Craggs J. W.
Published: November 2008

Two-dimensional elastic waves in a half-space 0 ≤ r < ∞, 0 ≤ θ ≤ π are examined under the assumption of dynamic similarity, so that the stresses depend only on r/t, θ. Analytic solutions are given for constant surface traction on θ = 0, 0 < r/t < V, where V is constant, the rest of the surface being unloaded, and for a concentrated load at r = 0.

Numerical results are quoted for the particular case V → ∞, corresponding to a load on half the bounding plane.
On the convergence of iterative processes for the solution of simultaneous equations in several variables
Authors: Bodmer W. F.
Published: November 2008

The notion of the order of convergence of iterative processes may well have been known to Gauss or even Newton, but to the author's knowledge it was first considered in detail in 1870 by Schröder(5). The subject was later investigated independently by Hartree(4) and Bodewig(1). Hartree alone mentions, briefly, the extension from one to several variables, but gives no detailed analysis. Domb and Fisher(2) consider a particular type of iterative process in several variables, which they call ‘degenerate’, in which all the variables tend to the same value. We shall follow essentially Hartree's treatment, in considering the solution of r simultaneous equations in r variables and applications to a generalization of the Newton-Raphson process.
Some Additive Cohomology Operations which are not Suspensions
Authors: Browder William
Published: November 2008

If X is a space with base point, let ΩX denote the space of loops of X based at its base point. We will denote the homology and cohomology suspensions by σ* and σ*, respectively, , . Let K(π, n) be the Eilenberg-MacLane complex with only one non-vanishing homotopy group πn(K(π, n))= π, πi(K(π, n)) = 0 for i ≠ n. If π is abelian K(π, n) = ΩK(π, n+1). We will assume that π is a finitely generated abelian group throughout this note. Let K(π, n) × K(π, n) → K(π, n) denote the multiplication in K(π, n).
On the Adem relations
Authors: Cohen D. E.
Published: November 2008

Let p be a prime and let Pi denote the Steenrod p-power if p ≠ 2 and the Steenrod square Sqi if p = 2. If p ≠ 2 let β denote the Bockstein co-boundary (defined so as to anticommute with suspension).
Matrices and norms
Authors: Petersen G. M.
Published: November 2008

We shall define the norm h(A) of a regular summability matrix A = (amn) by Two matrices are said to be b-equivalent if every bounded sequence summable by on matrix is summable by the other. If A sums all bounded sequences that are summable by B, A is said to be b-stronger than B. The norm of a method is defined as , where the inf is taken over all the matrices equivalent to for bounded sequences. These norms have been investigated by Brudno(1). One of his main results is that, if sums all bounded sequences that are summable, then In paper we shall prove the following.
On the construction of irreducible representations of the symmetric group
Authors: Gabriel J. R.
Published: November 2008

A simple method for constructing irreducible representations of the symmetric group is given. It is particularly suitable for use in quantum mechanical calculation of interactions in many particle systems, using electronic digital computers.
The interaction of nuclear multipole moments with an external charge distribution in elliptic coordinates
Authors: Stephen M. J.
Published: November 2008

The calculation of the interaction between a point multipole moment and an external charge distribution is considered in elliptic coordinates. It is complicated by the singularity at the point moment. The results obtained are useful in calculating hyperfine interactions in diatomic molecules where the elliptic coordinates are the most natural system of coordinates. The results lead to an extension of Heine's expansion of (λ + μ)−1.
On a problem of Steinhaus about polygons
Authors: Croft H. T. , Fowler M.
Published: November 2008

We solve a problem of Steinhaus (New Scottish Book, Problem 328; also Colloq. Math. 5 (1958), Problem 225); if p is a prime, does there exist a regular p-gon such that three of its diagonals cut in a point? The answer is in the negative, except in the case where the said diagonals are all parallel, which can clearly occur for any p ≥ 7.
On the Sum of Differentiable Functions
Authors: Croft H. T.
Published: November 2008

Ryll-Nardzewski has proposed the following problem (New Scottish Book, no. 119). If fn(x) are continuous, differentiable* functions in a closed finite interval, do there always exist constants cn (no cn = 0) (depending on the ), such that converges and is also a continuous differentiable function?
On a Class of Tchebysheffian Approximation Problems Solvable Algebraically
Authors: Talbot A.
Published: November 2008

The general Tchebysheffian approximation problem is the following: Given a real continuous function g(x) in the interval a ≤ x ≤ b, and a real function f(x; Pj) of prescribed form, continuous in the variable x and the parameters Pj, determine the values of the parameters so that the measure of absolute error or ‘deviation’ of f, shall be as small as possible.
Bounds for the Torsional Rigidity of Isotropic Beams
Authors: Milne-Thomson L. M.
Published: November 2008

Consider a cylindrical or prismatic beam of isotropic material. Let the cross-section of the beam be a simply-connected region S bounded by the closed contour C.
Definite integrals involving G-functions
Authors: Saxena R. K.
Published: November 2008

The main aim of the present note is to evaluate certain definite integrals involving Meijer's G-function. The results proved are of general character and include as particular cases some of the results given earlier by MacRobert ((3), p. 81). For the definitions and properties of the G-functions see (1), §§5·3 and 5·3·1.
A note on the convergence of multi-point Taylor's series
Authors: Fine M. , Head J. W.
Published: November 2008

Series which behave rather like Taylor's series at more than, one point are described and their convergence is discussed briefly.

Let

where z, zs are complex numbers and zs are the points, (the base points)about which a function is to be expanded. They need not be distinct. p{z) is a polynomial of degree n. in z. Assume first that z, zs are real and the zs are distinct. Then polynomials pn-1,t(z) may be chosen so that the series

has contact of order N at the points zs with any specified real function which has N derivatives at these points. This is proved merely by rearranging the simplest polynomial in z which has contact of order N with/(z) at z = zs. The term given by t = 0 in (2) allows the series to have the values f(zs) at z = zs; adding the term for t = 1 does not alter the values of the series at z = zs; it allows the series to have a derivative with values f′(zs) at z = z; similarly adding the term for t = 2 does not alter the value of the series or its first derivative at z = zs, it allows the series to have a second derivative with values f″{zs) at z = zs; and so on. If m of the zs coincide then f(z) has mN + m− 1 derivatives there and contact is of order mN + m− 1. If z, zs are complex the result is seen to be valid subject to the obvious generalization of ‘ contact of the ith order’ to complex variables and functions.
On the director surface of a rectilinear congruence
Authors: Jha P. , Eliezer C. J.
Published: November 2008

In this paper, it is shown that certain quantities defined with reference to the rays of a rectilinear congruence, remain unchanged for a change in the director surface, provided that the spherical representation of the congruence is kept unchanged. These quantities may be called semi-invariants. Some known invariants have been expressed in terms of these semi-invariants. An attempt has been made to find how these semi-invariants alter when the parametric curves are changed in any manner.
The minimization of a general function subject to a set of non-linear constraints
Authors: Bigg M. D.
Published: November 2008

A general method is described in this paper for the minimization of a function subject to a set of non-linear constraints. This method has been used to solve some problems in the geometry of numbers. Critical lattices of several regions have been computed and some results are given.
An abstract theory of topological subspaces
Authors: Papert Seymour
Published: November 2008

1. Introduction. The open subsets of a topological space X form a complete Brouwerian, i.e. distributive and pseudo-complemented lattice L(X). Many topological properties of X can be formulated as properties of L(X), although, in general, X is not determined by L(X). Obvious examples are quasi-compactness and connectivity: X is quasi-compact if any set of elements of L(X) whose join is I has a finite subset whose join is I; X is connected if no element of L(X) has a complement. These properties which we shall call ‘paratopological’, can be defined for lattices that are not of the form L(X), and many topological theorems can be proved in this more general context. The purpose of this paper is to develop sections of general topology as part of the theory of complete Brouwerian lattices (CBL).
Quantum mechanics of the isotropic three-dimensional anharmonic oscillator
Authors: Zucker I. J.
Published: November 2008

A method of determining numerically to any degree of accuracy the eigen-values of Hamiltonians in the form of power series is presented. The case of a spherically symmetric potential function of the form V = ar2 + br4 + cr6 is treated in detail.
On one-sided densities of arcs of positive two-dimensional measure
Authors: Besicovitch A. S.
Published: November 2008

1. Let AB be a plane simple arc, which, considered as a point-set, has a positive two-dimensional measure. If x be a point of AB, a point x′ ≠ x is said to be to the left or to the right from x according as it belongs to the arc Ax or xB. The upper and the lower limits of the ratio [m2{Ax C(x, r)}]/πr2, where C(x, r) denotes the closed disc with centre x and radius r, as r → 0 define the upper and the lower left densities at the point x. When the two densities are equal their common value defines the left density. The right densities are defined similarly.
Surface tension effect on waves in a liquid layer
Authors: Sells C. C. L.
Published: November 2008

1. Introduction. Little has been written on the effects of surface tension (or cohesion) on waves in liquid media; this is not surprising as in general its effects are small compared with those of gravity or compressibility, except in the case of ripple motion (Lamb (3), sections 265, 267). Bondi (1) has investigated the problem of a compressible fluid half space; he found that under cohesive action ‘faster-than-sound’ propagation along the surface is possible, and he studied it with the aid of an elegant image-function-space method; but he carefully points out the small size of the parameters involved.
Inclusion of sets of regular summability matrices
Authors: Baker J. W. , Petersen G. M.
Published: November 2008

1. In this paper we wish to discuss some problems which arise from a paper by Lorentz and Zeller; see (5). If {μn} is a fixed sequence monotonically increasing to infinity, and every sequence {sn} summed by both of the regular matrices A = (amn) and B = (bmn) and satisfying sn = O{μn) is summed to the same value by both matrices, the matrices are called (μn)-consistent. The two matrices are called consistent if they are (μn)-consistent for all {μn}, μn↗∞; they are b-consistent if the bounded sequences summed by both are summed to the same value by both. The matrix A is said to be (μn)-stronger than the matrix B, if every sequence {μn} that is B summable and satisfying sn = O(μn) is also A summable. The matrix A is stronger than B if every B summable sequence is A summable; A is b-stronger if every bounded B summable sequence is A summable. The symbol A -lim x denotes the value to which the sequence x = {xn} is summed by A; Am(x) is the transformation

and A(x) is the sequence {Am(x)}. Let {A(i)}i ∈ I be any family, infinite or finite, of regular summability matrices. This family is called simultaneously consistent if, given any finite subset of I, say F, and any set of sequences {x(i)i ∈ F such that A(i) sums x(i) for each i in F, and such that is the null sequence, then .
On a class of infinite products
Authors: Verblunsky S.
Published: November 2008

1.Introduction.In this note, we consider some estimates for the modulus of infinite products of the form

where thelvare near to an arithmetic progression of common difference unity. In a subsequent note, an application will be given of these estimates to the study of a class of Cauchy exponential series which includes, as a special case, the series of Korous(l).
On a property of verbal and marginal subgroups
Authors: Stroud P. W.
Published: November 2008

Let G be a group, and let φ (x1, …, xn) be any word in the variables x1, …, xn. We denote by φ(G) and φ*(G) respectively the corresponding verbal and marginal subgroups of G, which are defined as follows.

(i) φ(G)is the subgroup generated by all elements of the form φ(a1; …, an) with ai in G(for i = 1, 2, …, n);

(ii) an element b of G lies in φ*(G) if and only if

for all choices of a1, …, an in G and i= 1, 2, …,n.
A theorem on cardinal numbers associated with inductive limits of locally compact Abelian groups
Authors: Reade J. B.
Published: November 2008

Our motivation for this paper is to be found in (2) and (3). In (2) Varopoulos considers inductive limits of topological groups, in particular what he calls ‘ℒ∞’. (He calls a topology an ℒ∞-topology when it is the inductive limit of a decreasing sequence of locally compact Hausdorff topologies.) In (2) he proves that much of the classical theory of locally compact Abelian groups also goes through for Abelian ℒ∞-groups, in particular Pontrjagin duality.
On multilattice groups
Authors: McAlister D. B.
Published: November 2008

Multilattice groups have been introduced by Benado ((1)) and considered by Vaida ((9)), who has initiated the study of the structure of a class of these groups. The purpose of this paper is to further the investigation begun by Vaida.

Basic in the work contained here is the setHconsisting of the differences between the minimal upper bounds of pairs of elements of a multilattice groupG. By means of conditions imposed onGand based onH, we determine the structure of the congruence relations on a multilattice group and study direct sums and lexicographic products and extensions of such groups (sections 2, 3).
On a Sturm-Liouville expansion in series of Bessel functions
Authors: Naylor D.
Published: November 2008

1. The determination of potentials and wave functions defined in regions bounded by the natural coordinate surfaces of a cylindrical polar coordinate system leads to eigenvalue problems connected with Bessel's differential equation

We shall consider the annulus 0 < a ≤ r ≤ b and take first the boundary condition to be that of vanishing on both surfaces. It is then convenient to introduce the function

This function will automatically satisfy the condition of vanishing for r = a. It will also vanish on r = b if u and k are related by the condition that ψ(u, k, b) = 0. If we regard u as being fixed and real then there will be an infinite number of zeros kn(u)(n = 1,2,…).
On the corrector formulas for the numerical solution of the Schrödinger equation for central fields
Authors: Lovitch L. , Rosati S.
Published: November 2008

The evaluation of the eigenvalues and corresponding solutions of a Schrödinger equation is an ever-present problem in atomic and nuclear physics. For the numerical evaluation of the Schrödinger equation for a particle in a central potential, a useful corrector formula was derived by Douglas and Ridley ((5)) some years ago. Given a first estimate of the eigenvalue, this formula enables one to obtain a better estimate using any standard integration procedure for the differential equation. In a particular example, it was found ((5)) that it was necessary to start with an initial value which was accurate to about 10% in order that the procedure should converge to the desired eigenvalue, and that few iterations were required to obtain the eigenvalue with good precision.
A note on the Besicovitch dimension of the closest packing of spheres in Rn
Authors: Larman D. G.
Published: November 2008

Introduction. Let sn denote the infimum of the set of real numbers X, where x belongs to X if, and only if, x is the Besicovitch dimension of the residual set of a packing of n-spheres into the unit n-cube In, of Rn. In recent work ((1)) I have shown that s2 is greater than one, and, quite naturally, I have since been asked whether or not the proof can be generalized to prove the analogous result, sn greater than n − 1, in Rn. Whilst it is clear, in theory, that this could be done, in practice the details might become rather complicated. However, such a generalization is unnecessary, for the result sn > n − 1 is a trivial consequence of combining the result s2 > 1 with the following theorem.
A counter-example in uniformity theory
Authors: Ward A. J.
Published: November 2008

In a recent paper, Smith ((l)), considering a statement or conjecture of Isbell ((2)), proved that the Hausdorff uniformities arising from two different uniformities on the same set must induce different topologies on the ‘hyperspace’ of subsets, except possibly in one case which remained open, namely, when the two given uniformities induce the same proximity and neither is precompact. I now give an example to show that different uniformities can (in this case) induce the same topology on the hyperspace (in the example one of the uniformities is metric). This disproves Isbell's conjecture.
A note on system normalizers of a finite soluble group
Authors: Hawkes T. O.
Published: November 2008

Introduction. Hall ((3), (4)) introduced the concept of a Sylow system and its normalizer into the theory of finite soluble groups. In (4) he showed that system normalizers may be characterized as those subgroups D of G minimal subject to the existence of a chain of subgroups from D up to G in which each subgroup is maximal and non-normal in the next; he also showed that a system normalizer covers all the central chief factors and avoids all the eccentric chief factors of G (for definitions of covering and avoidance, and an account of their elementary properties), the reader is referred to Taunt ((5)). This note arises out of an investigation into the question to what extent this covering/ avoidance property characterizes system normalizers; it provides a partial answer by means of two elementary counter-examples given in section 3 which seem to indicate that the property ceases to characterize system normalizers as soon as the ‘non-commutativity’ of the group is increased beyond a certain threshold. For the sake of completeness we include proofs in Theorems 1 and 2 of generalizations of two known results communicated to me by Dr Taunt and which as far as we know have not been published elsewhere. Theorem 1 shows the covering/avoidance property to be characteristic for the class of soluble groups with self-normalizing system normalizers introduced by Carter in (1), while Theorem 2 shows the same is true for A -groups (soluble groups with Abelian Sylow subgroups investigated by Taunt in (5)).
On the least non-residue of a quartic polynomial
Authors: Williams Kenneth S.
Published: November 2008

Let p be a prime and let f(x) be a quartic polynomial with integral coefficients. I consider the problem of estimating the least non-negative non-residue k of f(x) (modp) (I omit the mod p hereafter), for large primes p, so f(x) ≡ r has a solution for

but not for r = k. The same problem for cubics has been considered by Mordell ((1)), who showed that

as p → ∞, where the constant implied in the O-symbol is independent of the coefficients of the cubic. In fact a more detailed examination of Mordell's proof gives the better estimate

It is the purpose of this paper to show that this same estimate also holds for quartic polynomials.
On the Feynman integral for classical waves
Authors: Arthurs A. M.
Published: November 2008

The Feynman integral is evaluated for classical waves using a Hamiltonian approach to the summation over histories.
On the homeotopy group of a 2-manifold (corrigendum)
Authors: Lickorish W. B. R.
Published: November 2008

A gap in the proof of the main theorem of (1) has been pointed out by C. D. Papakyriakopoulos and R. H. Fox. That proof was incomplete in that the table of homeomorphisms on page 776 of (1) did not cover all the possible cases (of which there are an infinite number). The corrected proof, given below, is intended to be a re-written version of the first half of section 3 of (1), the notation and references being those of (1).
On the Uniqueness and Boundedness of Solutions of Hyperbolic Differential Equations
Authors: Lakshmikantham V.
Published: November 2008

Consider a characteristic initial value problem of partial differential equations

where the functions E (x) and F (y) are real valued, uniformly Lipschitz continuous on 0 ≤ x ≤ a, 0 ≤ y ≤ b, respectively. Suppose f (x, y, u, p, q) is a real-valued and continuous function defined on 0 ≤ ≤ b. By a solution of (1), we mean a real-valued continuous function u (x, y), having partial derivatives ux (x, y), uy (x, y) and ux, y (x, y) in the domain 0 ≤ x ≤ a, 0 ≤ y ≤ b almost everywhere.
A simple random walk and an associated asymptotic behaviour of the Bessel functions
Authors: Keilson J.
Published: November 2008

We consider a random walk defined in the following way. We have a set of states indexed by n where n takes on all negative and positive integral values and zero. When we are at state n, there is a probability per unit time λ of going to n + 1, and a probability per unit time λ of going to n − l. Let us start out at n = 0, and study Wn(t), the probability of being at n at time t. Continuity of probability requires that whence since G(s, 0) = 1, we have It follows from the well-known result .
Mod p cohomology theories and the Bockstein spectral sequence
Authors: Maunder C. R. F.
Published: November 2008

Introduction. Let h*(X, A) be an extraordinary cohomology theory defined on the category of finite CW-pairs, that is, a cohomology theory satisfying all the axioms of Eilenberg and Steenrod (7) except the dimension axiom. This paper has three objects: to define and investigate the properties of the associated mod p cohomology theory h*(X, A; Zp) (see also (9)), to show that (with certain restrictions) if h*(X, A) is a multiplicative cohomology theory, so is h*(X, A. Zp), and lastly to generalize the notion of Bockstein coboundary homomorphisms and the apparatus of the Bockstein spectral sequence (see (4)) to an extraordinary cohomology theory.
Bessel function Jν(z) of complex order and its zeros
Authors: Hall Laurence S.
Published: November 2008

Methods are developed for the computation of the complex zeros of (½z)−νJν(z) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the Jv(z), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of Jν(z) for the rapid computation of Bessel functions of complex order.
The sluice gate
Authors: Conway William E.
Published: November 2008

Consider the two-dimensional irrotational steady flow of an incompressible inviscid fluid under no body forces between two infinite horizontal planes. When a vertical wall which intersects the upper plane is introduced in the flow, the fluid will flow under the wall and will be bounded above by a free streamline along which the pressure is constant. This configuration is called a sluice gate. At infinity the flow will be uniform and parallel with input velocity V and exit velocity U.
Inhomogeneous minimum of indefinite quaternary quadratic forms
Authors: Dumir Vishwa Chander
Published: November 2008

Let Q (x1, …, xn) be a real indefinite quadratic form in n-variables x1,…, xn with signature (r, s),r + s = n and determinant D ≠ 0. Then it is known (see Blaney (2)) that there exists constant Cr, s depending only on r, s such that given any real numbers c1, …,cn we can find integers x1, …, xn satisfying
On a theorem of Hardy and Littlewood
Authors: Das G.
Published: November 2008

1. Given an infinite series ∑an, we consider its Nörlund transformation

where

It is know that if

then the (N, p)-mean reduces to the familiar (C, α)-mean.
On the summability |R, log ω, 1| of Fourier series and an associated series
Authors: Mohanty R. , Ray B. K.
Published: November 2008

1. Definition. Let λ ≡ λ(ω) be continuous, differentiable and monotonic increasing in (0, ∞) and let it tend to infinity as ω → ∞. A series is summable |R, λ, r|, where r > 0, if

where A is a fixed positive number(3).
Generalized basic hypergeometric series with unconnected bases
Authors: Agarwal R. P. , Verma Arun
Published: November 2008

In a series of recent papers Verma and Upadhyay (7,8,9) developed the theory of basic hypergeometric series with two bases q and q½. These investigations were made in an attempt to discover a summation formula for a bilateral basic hypergeometric series 2Ψ2 analogous to that for a 2H2 (cf. Bailey (2,3)) and in finding relations between certain q-infinite products. In one of their papers they mentioned that it did not seem possible to develop the corresponding general theory for basic series with two unconnected bases q and q1. A recent paper by Andrews (1) indicates that transformations between basic hypergeometric series with two unconnected bases can be very interesting and useful in the study of ‘mock’ theta functions and their extensions. Besides this interest, such a theory also enables one to extend the entire existing transformation theory of the generalized basic hypergeometric series.
On the laminar compressible boundary layer with stationary origin on a moving flat wall
Authors: Ackroyd J. A. D.
Published: November 2008

The growth of the laminar compressible boundary layer on a moving flat wall is considered analytically for the case of zero velocity in the free stream outside the boundary layer. The results of this analysis are compared with other published results for the cases in which the free stream has some finite velocity. In all the cases considered in the present paper, the boundary layer is taken to originate at some stationary point on the moving wall. This type of boundary-layer flow occurs behind moving shock waves and it is argued that the case of particular interest in the present paper, that of the stationary gas outside the boundary layer, provides bounding values of such parameters as displacement and momentum thicknesses for shock-induced laminar boundary-layer flows.
Concentrated sets
Authors: Eggleston H. G.
Published: November 2008

A non-enumerable set of real numbers,Z, isconcentratedprovided that there exists an enumerable setKsuch that ifGis open andG⊂KthenZ−Gis at most enumerable.Zis said to be universally concentrated if we can take forKany enumerable setHfor which
An extreme point theorem for inverses in a Banach algebra with identity
Authors: Stampfli J. G.
Published: November 2008

A version of Corollary 2 for Hilbert space was orally communicated to the author by C. Pearcy. His proof employed facts about the numerical range, inner products, and orthogonality, and thus made use of structure peculiar to Hilbert space. A Banach algebra with identity is a more comfortable setting for these results, and there it is much clearer just what is important in each proof.
Some finite integrals involving F4 and H-functions
Authors: Rathie P. N.
Published: November 2008

1. The object of the present paper is to obtain some finite integrals involving the Appell's function F4 and the H-function of Fox by utilizing the results recently given by Saxena, Sharma and Tranter respectively. The first three results proved in this paper are the extension of the results recently established by Saxena and Sharma in these proceedings. MacRobert's result follows as a very special case of one of our results. A few very interesting special cases of the main results have also been given.
Elastic wave fields generated by scalar wave functions
Authors: Chadwick P. , Trowbridge E. A.
Published: November 2008

In this paper we obtain a representation of elastic wave fields (i.e. solutions of the equations of motion of classical elastokinetics) in terms of three functions satisfying scalar wave equations. Although the form of the representation implies no restriction upon the choice of coordinate system or upon the shape of the elastic body, it is found that the result can only be applied advantageously to initial-boundary-value problems having spherical polar coordinates as a natural frame of reference. The representation is shown to be complete in the sense that every (sufficiently smooth) elastic wave field in a homogeneous, isotropic body bounded by two concentric spheres can be expressed in the given form.

Under conditions of axial symmetry the representation generates wave fields which can be decomposed into poloidal and toroidal constituents, the former arising from two scalar wave functions and comprising both dilatational and rotational waves, and the latter being associated with a single scalar wave function and a state of pure shear of the elastic solid. Finally, the representation is used to obtain a formal solution describing the elastic pulse generated in an infinite body by the application of time-dependent tractions to the surface of a spherical cavity.
The refraction of sound pulses
Authors: Grimshaw Roger
Published: November 2008

The problem considered is that of the diffraction of sound pulses by a plane interface separating two stratified media in each of which the refractive index increases with distance from the interface. Two types of diffraction phenomena occur, glancing ray diffraction and total reflection. The solution is obtained as a series of ‘pulse propagation modes’, using a double transform technique. The main results are some formulae for the pulse field near a diffracted wavefront.
Products in homotopy groups with certain prime coefficients
Authors: Green Paul
Published: November 2008

Let p be a prime and n ≥ 3. Then there is a simply connected CW-complex, , unique up to homotopy type, such that

Let X be a CW-complex. Write , the group of pointed homotopy classes of pointed maps from to X. The group structure derives from the fact that, under the restriction on n, is a suspension.
A representation theorem for partially ordered Banach algebras
Authors: Ng Kung-Fu
Published: November 2008

Let be a real algebra which is also a Banach space. Then is called a partially ordered Banach algebra if there is specified a non-empty subset of , called the positive cone, such that

and

(A 5) is 1-normal and closed. (We recall that is said to be 1-normal if
A mini-gap theorem for Fourier series
Authors: Hayman W. K.
Published: November 2008

Suppose that

belongs to L2( − π, π). If most of the coefficients vanish then f (x) cannot be too small in a certain interval without being small generally. More precisely Ingham ((2), Theorem 1) has proved the following

THEOREM A. Suppose that f (x) is given by (1·1) and that an = 0, except for a sequence n = nν, where nν+1 − nν ≥ C. Then given ∈ > 0 there exists a constant A (∈), such that we have for any real x1
Integral equations and relations for Lamé functions and ellipsoidal wave functions
Authors: Sleeman B. D.
Published: November 2008

Non-linear integral equations and relations, whose nuclei in all cases is the ‘potential’ Green's function, satisfied by Lamé polynomials and Lamé functions of the second kind are discussed. For these functions certain techniques of analysis are described and these find their natural generalization in ellipsoidal wave-function theory. Here similar integral equations are constructed for ellipsoidal wave functions of the first and third kinds, the nucleus in each case now being the ‘free space’ Green's function. The presence of ellipsoidal wave functions of the second kind is noted for the first time. Certain possible generalizations of the techniques and ideas involved in this paper are also discussed.
Exact solutions for electromagnetic wave propagation in spherically stratified isotropic media
Authors: Westcott B. S.
Published: November 2008

Following Heading (4) a systematic search is made for useful refractive index profiles in spherically stratified isotropic media which are capable of yielding wave functions expressed in terms of the standard transcendental functions. Previously known profiles are shown to emerge as special cases of the analysis. Subject to the criteria adopted all possible profiles are derived.