bluuebunny/crossref_metadata_2025 huggingface dataset:
Diffraction by a perfectly conducting wedge in an anisotropic plasma
Authors: Faulkner T. R.
Published: November 2008

The problem considered is the diffraction of an electromagnetic wave by a perfectly conducting wedge embedded in a plasma on which a uniform magnetic field is impressed. The plasma is assumed to behave as an anisotropic dielectric and the problem is reduced, by employing a contour integral representation for the solution, to solving a difference equation. Surface waves are found to be excited on the wedge and expressions are given for their amplitudes.
The scattering cross-section of an obstacle in an elastic solid for plane harmonic waves
Authors: Barratt P. J. , Collins W. D.
Published: November 2008

It is shown that, when a plane harmonic P or S wave is incident upon a two-or three-dimensional obstacle in an infinite elastic solid, the scattering cross-section of the obstacle can be calculated from an appropriate far-field scattering amplitude.
An inversion formula for the Varma transform
Authors: Saxena R. K.
Published: November 2008

Recently Fox ((5)) has given an inversion formula for the transform whose kernel is xνKν(x), where Kν(x) is the modified Bessel function of the second kind, by the application of fractional integration theory. In the present paper it has been shown that the integral equation

can be thrown into the form of a Laplace integral, with the help of fractional integration, which can be solved by known methods.
Certain product theorems for bilateral hypergeometric series
Authors: Denis R. Y.
Published: November 2008

Introduction. Agarwalin 1953 ((1)) gave the most general transformations involving bilateral cognate trigonometrical series of the hypergeometric type. Later, Shukla in 1957 ((2)) gave certain relations involving the products of two bilateral hypergeometric series. In this paper an attempt has been made to establish certain new transformations involving the products of two and more bilateral hypergeometric type of series. (For notations and definitions see Agarwal ((1)).)
Two examples in homotopy theory
Authors: Brown R.
Published: November 2008

The examples are unconnected.

According to some published footnotes, examples are known of Serre fibrations which are not Hurewicz fibrations. However, as far as I know no such example has been published. The following example, which was stimulated by a more complicated example of G. Horrocks, seems worth mentioning, since it is extremely simple and does suggest some further questions.
Analogue of a theorem of Khintchine in fields of formal power series
Authors: Cusick T. W.
Published: November 2008

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and let

be any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):

For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such that

for all integers x1, x2, …, xm not all zero.
Some plane curve pathologies
Authors: Croft H. T.
Published: November 2008

We construct various pathological curves and sequences of curves to solve three problems of Steinhaus.
Absolute and Unconditional Convergence in Normed Linear Spaces
Authors: Rutovitz D.
Published: November 2008

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).
The Inverse Laplace Transform of the Product of Two Whittaker Functions
Authors: Ragab F. M.
Published: November 2008

The object of this paper is to obtain the original function of which the Laplace transform (l) is the product

where, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equation

In § 2 it will be shown that this original function is

where the symbol Δ(n; α) represents the set of parameters
A generalization of the Laplace transform
Authors: Dunn H. S.
Published: November 2008

An integral transformation is denned over a finite interval of the time domain. When the Laplace transform exists, the finite transform yields identical results. However, the finite transform is found to be considerably more general than the Laplace transform. It permits consideration of functions which are not of exponential order, leads to a simple scheme to determine system response, and is applicable to boundary-value problems.
A formal solution of certain dual integral equations involving H-functions
Authors: Saxena R. K.
Published: November 2008

The problem discussed is the formal solution of certain dual integral equations involving H-functions. The method followed is that of fractional integration. The given dual integral equations have been transformed, by the application of fractional integration operators, into two others with a common kernel and the problem is then reduced to solving one integral equation. In the first case the common kernel comes out to be a symmetrical Fourier kernel given earlier by Fox and the formal solution is then immediate. In the second case the common kernel is a generalized Fourier kernel and dual integral equations of this type have recently been studied by Fox.
Locally nilpotent ideals of a Lie algebra
Authors: Hartley B.
Published: November 2008

The purpose of this paper is to investigate the locally nilpotent radical of a Lie algebra L over a field of characteristic zero, its behaviour under derivations of L, and its behaviour with regard to finite-dimensional nilpotent subinvariant and ascendant subalgebras of L.
On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method
Authors: Mohanty R. , Ray B. K.
Published: November 2008

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us set

The sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformation

The series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, if

and is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).
Complex potentials in shallow shells of revolution
Authors: Yacoub A. R.
Published: November 2008

By introducing complex variable techniques in the theory of shallow shells, the problem is reduced to the solution of one fourth-order differential equation, and the displacement components and stress resultants and couples are expressed in terms of four complex potentials. Two of these complex potentials are those of thin plate theory, thus admitting of deducing the solution of a shallow shell problem from the limiting case of the circular flat plate. These results are applied to a case of non-rotational symmetry, a special case of which has practical significance in the design of tanks and boilers.
Fluctuating boundary layer on a magnetized plate
Authors: Chawla Sahib Singh
Published: November 2008

The laminar boundary layer on a magnetized plate, when the magnetic field oscillates in magnitude about a constant non-zero mean, is analysed. For low-frequency fluctuations the solution is obtained by a series expansion in terms of a frequency parameter, while for high frequencies the flow pattern is of the ‘skin-wave’ type unaffected by the mean flow. In the low-frequency range, the phase lead and the amplitude of the skin-friction oscillations increase at first and then decrease to their respective ‘skin-wave’ values. On the other hand the phase angle of the surface current decreases from 90° to 45° and its amplitude increases with frequency.
Clifford algebras in general relativity
Authors: Lord E. A.
Published: November 2008

The generalization of the Dirac and quaternion algebras to Riemannian spaces is outlined. The components of various elements of the algebras are interpreted as physical quantities (tensors) and their symmetries and algebraic properties are linked with the properties of the algebra. The generalization of the quaternion algebra is of particular interest in that it resolves the anomalies that arise in the usual identification of quaternions with rank two spinors. Algebraic expressions for the electromagnetic energy momentum tensor, the Ricci tensor and Einstein tensor are obtained in both E-number and quaternion form. Extension of the principles to EF-numbers yields a proof of the symmetry properties of Bel's tensor and a simple expression for its divergence.
On a conformal theory of gravitation
Authors: Islam Jamal N.
Published: November 2008

Recently Hoyle and Narlikar(2) have put forward a conformal theory of gravitation in which they obtain an explicit solution of the field equations using certain approximations and assumptions of symmetry. To obtain the solution, the metric is assumed to have a special form. In the present paper a more general form of the metric is considered, and it is shown that many of the features of the special solution are shared by the solution in the more general case. In particular, the implications of the special solution as concerns the sign of the gravitational constant remain valid in the more general case under certain assumptions.
An intersection theorem of Erdős and Rado
Authors: Davies Roy O.
Published: November 2008

Theorem. if a ≥ 2, b ≥ 1, and a + b ≥ ℵ0, then every collection of more than ab sets each of cardinal not exceeding b contains a subcollection of more than ab sets every two of which have the same intersection.
Plane stress theories
Authors: Sayer F. P. , Calder C. C.
Published: November 2008

Tiffen and Lowe in two recent papers (4,5) have developed exact theories for the bending and stretching of generally loaded elastic plates using the moments of the fundamental equations of infinitesimal elasticity. The two theories depend on the choice of independent variables so that a pair of functions, denoted by and are independent. The choice is correct in (4) but not in (5). This paper pro-poses a modification to the forms of the functions and in (5) and then shows that the modified forms are independent. The subsequent alteration to (5), as will be seen, involves only the biharmonic function χ Apart from a minor algebraic adjustment the rest of the theory, containing the real and complex eigenfunctions and particular integral, is preserved. It should be noted that the modified theory like the original is obtained entirely from the moment equations.
Abstract Köthe spaces. IV
Authors: Fremlin D. H.
Published: November 2008

In this paper I investigate the completed projective tensor product of two perfect Riesz spaces, and show how a natural order structure on this renders it also a perfect Riesz space. Sections 7–14 contain interesting order-topological properties of this tensor product. Finally, section 15 describes how the tensor product of function spaces may be represented as a function space, in the manner of (1 b).
Creeping modes in a shadow
Authors: Ursell F.
Published: November 2008

In the two-dimensional theory of diffraction by smooth curves there are certain canonical problems that can be solved explicitly in series form. The series converge slowly at short wavelengths but they can be transformed by Watson's transformation into another form (the residue series or creeping-mode expansion) which has been much used in shadow regions. It is found that throughout the shadow region the first few terms of the residue series are exponentially small and decrease rapidly, and these have often been used as an estimate of the wave potential without further justification. Leppington's recent work on the shadow of an ellipse has shown, however, that in part of the shadow some of the later terms of the residue series are exponentially large. In other words, the complete residue series in part of the shadow is even more slowly convergent than the original series.

In the present paper the Watson transformation is re-examined in the light of this result. The original series is expressed as the sum of a finite number of terms of the residue series and of a remainder. It is shown that throughout the shadow the remainder is at short wavelengths asymptotically of smaller order than the last term retained in the finite residue series. It follows that the residue series (which is a convergent infinite series) is also an asymptotic series, and this fact is sufficient to justify most of the usual applications. The proof is given in detail for the circle and in outline for the ellipse; it makes use of the theory of conformal mapping.
Diffraction of elastic waves by a rigid circular disc
Authors: Mal A. K. , Ang D. D. , Knopoff L.
Published: November 2008

The problem of the diffraction of axisymmetric harmonic elastic waves by a rigid circular disc is reduced to finding the solution of a pair of integral equations of the second kind suitable for iteration at low frequencies. Using the principle of contraction mapping, the rate of convergence of the iteration procedure is discussed and the error caused by stopping at any particular stage estimated. Detailed calculations are given for plane compressional waves at normal incidence.
The generalization of Lagrange's expansion and the enumeration of trees: a correction
Authors: Good I. J.
Published: November 2008

The determinantal factor on p. 512 of Proc. Cambridge Philos. Soc. 61 (1965), should have the off-diagonal element in position (s, t) of the array multiplied by bst. This slip was pointed out to me by Dr Donald Knuth. When the rooted-labelled tree is not necessarily ‘chromatic,’ so that the bss's can be 1, the diagonal terms of the matrix become
A note on a source in a rotating liquid
Authors: Stewartson K.
Published: November 2008

Consider an inviscid incompressible fluid confined within an infinite tube of radius b and assume that its motion initially consists of a uniform translation with velocity U and a uniform rotation, with angular velocity ω, about the tube axis. Suppose now that at time t = 0 a unit source of fluid situated at a point O of the axis begins to emit fluid at a uniform rate and that ultimately the flow pattern so modified becomes steady (at a fixed point as t → ∞). Under the further assumption that far upstream flow conditions are undisturbed by the source the stream function ψ of the disturbance satisfies

where z denotes distance along the axis from O, r distance from the axis and k = 2ω/U.
On a class of functions regular in the unit disc
Authors: Hornblower R.
Published: November 2008

MacLane(1) has introduced the class of functions with a dense set of asymptotic values on {|z| = 1} in the following sense:

Definition. Let f(z) be regular and not constant in the unit disc and let ξ be a point of {|z| = 1}. Suppose that there exists a curve Γξ in {|z| < 1} and a in the closed complex plane such that f(z) → a as z → ξ along Γξ. Then we say ξ ∈ Aa.
Bilinear and trilinear generating functions for Jacobi polynomials
Authors: Manocha H. L.
Published: November 2008

The writer in his paper (4) has shown that

where the Jacobi polynomial is defined as ((5), p. 255)

.
On the complexes of Hirsch and Eilenberǵ–Moore
Authors: Barcus W. D.
Published: November 2008

Both Hirsch (cf. (5), (3)) and Eilenberg and Moore (cf. (7)) have described filtered complexes for the cohomology of a fibre space, under different hypotheses. Provided both complexes are defined, we shall give an explicit filtration-preserving map of the first into the second which induces homology isomorphisms. In section 3 we shall use this to transfer the action of the group in a principal bundle from the Hirsch complex (as computed by the author in (1)) to that of Eilenberg and Moore. The latter complex has certain advantages, for example, it is natural, and admits a cup-product structure.
Exact potential functions in spherically stratified media
Authors: Westcott B. S.
Published: November 2008

A systematic analysis is made of the equation ∇. [k(r) ∇ψ] = 0 for useful profiles k(r) in spherically stratified media which enable the radial variation of the potential function ψ to be expressed exactly in terms of hypergeometric functions or confluent hypergeometric functions. Previously known profiles are shown to emerge as special cases of the present analysis.
A class of compressible embeddings
Authors: Carter Sheila
Published: November 2008

By a closed m-manifold we mean a C∞ compact connected m-dimensional manifold, without boundary. M will denote a closed m-manifold (m > 0), and En will denote Euclidean n-space. All embeddings will be C∞.
On the convergence factor of a Fourier series and a differentiated Fourier series
Authors: Mohanty R. , Ray B. K.
Published: November 2008

Definition A. The serieswith partial sumUn (or the sequence {Un}) is said to be summable by logarithmic means to the sum U, if
Jacobi series which converge to zero, with applications to a class of singular partial differential equations
Authors: Wimp Jet , Colton David
Published: November 2008

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose that

the series being boundedly convergent for 0 ≤ x ≤ 1.
On the method of matched asymptotic expansions
Authors: Fraenkel L. E.
Published: November 2008

The prescription for forming composite series, and a modified form of that procedure, are used systematically and extensively to derive uniform asymptotic approximations to various given functions which cannot be approximated uniformly by expansions of classical (Poincaré) type. The formal method of matched expansions is then applied to a boundary-value problem for an ordinary differential equation with a turning point. It is proved with the help of the uniform approximations found earlier that in this problem a restricted form of the asymptotic matching principle is valid, even when it is applied to truncated inner and outer expansions which do not overlap to the order of the terms being matched.
Doubling perturbation sizes and preservation of operator indices in normed linear spaces
Authors: Gustafson Karl
Published: November 2008

We extend the index theory of Fredholm operators in Banach spaces to general operators in normed linear spaces. In so doing we make use of a doubling technique, which we also apply to obtain improved results for the perturbation of semigroups and sesquilinear forms.
A new generating function for Jacobi polynomials
Authors: Saxena R. K.
Published: November 2008

In an earner paper ((6), p. 209) the author has given a generating function for the Jacobi polynomials defined by ((5), p. 254)

In the present note we obtain another generating function for these polynomials. The results given earlier by Manocha and Sharma ((4), p. 432) in these proceedings and Brafman in (2) follow as a particular case of our result.
On the slow motion of two spheres in contact along their line of centres through a viscous fluid
Authors: Cooley M. B. A. , O'Neill M. E.
Published: November 2008

An exact solution in terms of analytic functions is given for the Stokes stream function and forces acting on two spheres in contact when the spheres move uniformly with the same velocity along their line of centres. For the case of equal sized spheres, the values of the forces agree with the Faxén limit of the Stimson–Jeffery forces for separated spheres when the minimum clearance between the spheres tends to zero. Numerical values of the forces for arbitrary sized spheres are tabulated and compared with corresponding values when the spheres are separated over a range of values of the minimum clearance. Comparison of the theoretical results with those obtained experimentally for the case of equal sized spheres is also given.
Multiple scattering of plane harmonic elastic waves in an infinite solid by an arbitrary configuration of obstacles
Authors: Barratt P. J.
Published: November 2008

The multiple scattering of plane harmonic P and S waves in an infinite elastic solid by arbitrary configurations of obstacles is considered. Integral equations relating the far-field multiple scattering amplitudes to the corresponding single scattering functions are obtained and asymptotic solutions are found by an iterative procedure. The scattering of a plane harmonic P wave by two identical rigid spheres is investigated.
Compressible flows with circular sector hodographs
Authors: McLaughlin Maureen D. , Pack D. C.
Published: November 2008

The general solution of Chaplygin's equation for the motion of an inviscid compressible fluid is obtained for ‘simple wedge flows’. Some particular examples are given and a discrepancy in earlier Russian papers explained. After a discussion of the properties of the velocity potential and the expression for the physical coordinates in terms of the hodograph variables, a general theorem on sonic jets is proved, namely, that the physical changes due to the presence of solid boundaries in the flow are completed within a finite distance in those directions in which the flow becomes finally a sonic jet.
A ǵeneralization of the Meijer transform
Authors: Pandey R. N.
Published: November 2008

In this paper a new generalization of the Meijer transform is given. It also generalizes the Mainra transform and the transform due to Varma. An inversion theorem is established and the result obtained has been illustrated by several examples.
Simultaneous confidence intervals for two linear functions of population means when population variances are not equal
Authors: Banerjee Saibal
Published: November 2008

It is shown that given k samples of nj units from it is possible to construct simultaneous confidence intervals for two given linear functions of population means, (where cij are known constants), when population variances are not equal.
Dimensionally dependent identities
Authors: Lovelock David
Published: November 2008

In the general theory of relativity a number of apparently unrelated identities peculiar to a 4-dimensional space are frequently used. However, the proofs usually presented appear to have no common ideas and, furthermore, it is not clear at what stage the dimensionality restriction plays a significant role. In this note a technique is presented which explicitly exhibits the dimensionality dependence and thereby enables the above identities to be generalized forn-dimensional spaces. It is also shown that the above identities are all special cases of a more general identity valid forn= 4.
On the remainder in the two-dimensional central limit theorem
Authors: Dunnage J. E. A.
Published: November 2008

Our object here is to refine the theorem proved in (3), and we use the notation of that paper. Let Z1, Z2, …, Zn, where Zr = (Xr, Yr), be independent random variables in two dimensions with zero first-order moments and finite third-order moments; and et the covariance matrix of Zr be

We write

Let (x, y) be the distribution function of the sum and let (x, y) be the normal distribution function having the same first- and second-order moments as (x, y).
Integral means of subharmonic functions
Authors: Beardon A. F.
Published: November 2008

If u is subharmonic in some open subset of Euclidean space, RN (N ≥ 1), which contains the ball |x| ≤ ρ and if S(r) and B(r) denote the mean values of u taken over the sphere |x| = r and the ball |x| ≤ r respectively (0 ≤ r ≤ ρ) then, ((2), p. 196),
The space-time metric inside a rotating cylinder
Authors: Davies H. , Caplan T. A.
Published: November 2008

It is shown that the space-time region inside an axially symmetric, infinite, rotating, cylindrical mass distribution is necessarily Minkowskian.
On the Kunneth formula spectral sequence in equivariant K- theory
Authors: Snaith V. P.
Published: November 2008

Let G be a compact, connected Lie group such that π2(G) is torsion free. Throughout this paper a vector bundle (representation) will mean a complex vector bundle (representation) and KG will denote the equivariant K-theory functor associabed with the group, G.
On the h-enclosability of spheres
Authors: Antonelli P. L.
Published: November 2008

Let Bk, Mn, Np be manifolds in the category C = Top, Duff or PL. Define Mn and Np to be h-enclosable in Bk if (1) k = n + p + 1, (2) there are C-imbeddings i:Mn ⊆ Bk and j: Np ⊆ Bk with disjoint images and (3) there are de formation retractions of Bk − i(Mn) onto j(Np) and of Bk − j(NP) onto i(Mn). This is expressed as Bk = [Mn, NP/i, j] (mod C). The manifolds are trivially h-enclosable in Bk if, in addition, each manifold has a product tubular neighbourhood in the category C.
Complementary variational principles for a class of differential equations
Authors: Arthurs A. M. , Coles C. W.
Published: November 2008

Maximum and minimum principles for certain ordinary differential equations of order 2m are derived in a unified manner from the theory of complementary variational principles for multiple operator equations. The minimum principle is known in the literature, but the maximum principle appears to be new.
Difference-covers that are not k-sum-covers. I
Authors: Jackson T. H. , Williamson J. H. , Woodall D. R.
Published: November 2008

Various constructions are described for sets F = F(q, k, l) of residues modulo q such that F–F contains all residues while (k)F = F + F + … + F (k summands) omits l consecutive residues, in the cases k = 2, 3, 4 and 5. Let Q(k, l) denote the set of moduli q for which such a set F(q, k, l) exists. The following results are proved: q∈Q (2, 1) if and only if q ≥ 6; q∈Q(2,l) if q ≥ 4l + 2; 29, 31 and 33∈Q(3, 1) and if q∈Q(3, 1) if q ≥ 35; 24l + 21∈Q(3, l) if q ≥ 44l + 41. Upper bounds are also obtained for the least elements in Q(4, l) and Q(5, l).
A decomposition identity for a class of Riemannian invariants
Authors: Pavelle Richard
Published: November 2008

An identity of fundamental interest in General Relativity is the breakup or ‘decomposition’ of the scalar curvature density into an ordinary divergence in addition to a term constructed solely from the metric tensor and its first derivative. It is proven that this identity is the most elementary example of an identity satisfied by all members of a particularly important class of Riemannian invariants. For a subset of these invariants, the decomposition identity undergoes a simplification that connects this material with some problems of current research interest.
Generalized Cayley-Hamilton identities for matrices with entries in a non-associative ring
Authors: Ilamed Yehiel
Published: November 2008

In (1) A. H. Boers has studied the properties of n-PA rings, i.e. rings where any n-product (any product of n factors) is independent of its bracketing; a private communication is quoted in that paper to the effect that the n-PA rings were defined by Y. Ilamed starting from problems in mathematical physics.
Transverse regularity for maps of homology manifolds
Authors: Martin N.
Published: November 2008

Recall that in (2) we showed that it was possible to make transverse a homology manifold and PL-manifold inside a large dimensional homology manifold, subject to being able to do some general position inside the large manifold. In (1) we were able to relax the condition that one of the submanifolds be a PL-manifold to it being a homotopy manifold. The way in which the making transverse was achieved was via a system of h-cobordisms from the original situation to the transverse one. The problem we tackle here is that of making a map between homology manifolds transverse regular. Thus we ask: given a map f: M → N of homology manifolds with P a proper submanifold of N, is it possible to homotop f to a map g: M → N such that g−1(P) is a proper submanifold of M and g induces a map from the normal bundle of g−1(P) in M to the normal bundle of P in N?
Mackey duals and almost shrinking bases
Authors: Kalton N. J.
Published: November 2008

Suppose (en) is a basis of a Banach space E, and that (e′n) is the dual sequence in E′. Then if (e′n) is a basis of E′ in the norm topology (i.e. (en) is shrinking) it follows that E′ is norm separable: it is easy to give examples of spaces E for which this is not so. Therefore there are plenty of spaces which cannot have a shrinking basis. This leads one to consider whether it might not be reasonable to replace the norm topology on E′ by one which is always separable (provided E is separable). Of course, the weak*-topology σ(E′, E) is one possibility (Köthe (17), p. 259); then it is trivial that (e′n) is a weak*-basis of E′. However, if the weak*-topology is separable, then so is the Mackey topology τ(E′, E) on E′, and so we may ask whether (e′n) is a basis of (E′,τ(E′, E)).
Incomplete reflexive spaces without Schauder bases
Authors: Knowles R. J. , Cook T. A.
Published: November 2008

In 1968, Amemiya and Kōmura ((1), p. 275) gave an example of a separable and incomplete Montel space. Their example depended upon constructing a subspace of ω, the countable product of real lines, which had several properties and in particular had codimension in ω. There is an error on page 276, line 5 of the proof of the Hilfssatz concerning this construction. The constructed subspace has codimension one. A more delicate construction is needed here; in fact, an example given by Webb ((7), p. 360) is sufficient to make this correction.
On Blasius's equation governing flow in the boundary layer on a flat plate
Authors: Richardson S.
Published: November 2008

The original approach of Blasius to the solution of the differential equation now associated with his name was to develop the unknown function as a power series. Unfortunately, this series has a limited radius of convergence, so that such a representation is not valid over the whole range of interest. It is shown here that, if we work instead with a particular inverse function, this can be expanded as a power series which converges for all relevant values of the independent variable. Moreover, the number associated with the solution which is of principal physical interest can be expressed in terms of the asymptotic properties of the coefficients of this series. Exploiting this relationship, we find upper and lower bounds for this number in terms of the zeros of two particular families of polynomials.
The asymptotic analysis of canonical problems in high-frequency scattering theory
Authors: Boyd W. G. C.
Published: November 2008

The asymptotic treatment of high-frequency scalar wave problems has in the past been rather unsatisfactory. Typically, the integral representations which arose were evaluated by stationary phase, or as a series of residues. The justification of these methods was usually heuristic and formal. In this paper, a method is advanced which, it is claimed, may be applied to any one-parameter separation of variables problem. The method assumes an integral representation whose contour of integration is the real axis. It is then only necessary to deform this contour in the neighbour-hood of the real axis to derive rigorous asymptotic expansions of the field in both the illumination and shadow. The method is applied to the particular example of scattering by a plane boundary in a general stratified medium with monotonically increasing refractive index.
A note on a centrality property of finitely generated soluble groups
Authors: Lennox J. C.
Published: November 2008

We recall from (3) that a group G is (centrally) eremitic if there exists a positive integer e such that, whenever an element of G has some power in a centralizer, it has its eth power. The eccentricity of an eremitic group G is the least such positive integer e.

In ((4), Theorem A) we proved that if A is a torsion free Abelian normal subgroup of a finitely generated group G with G/A nilpotent, then G has a subgroup of finite index with eccentricity 1. In this note we use a much simpler method to prove a stronger result.
Some wave solutions in general relativity
Authors: Patel L. K.
Published: November 2008

A general scheme for the derivation of wave solutions in general relativity is developed. Some solutions describing the flow of gravitational waves are discussed. Singular electromagnetic fields corresponding to one particular solution are also discussed.
Osculatory properties of a certain curve in [n]
Authors: Edge W. L.
Published: November 2008

1. When, as will be presumed henceforward, no two of a0, a1, …, an are equal the n + 1 equations

are linearly independent; x0, x1, …, xn are homogeneous coordinates in [n] projective space of n dimensions—and the simplex of reference S is self-polar for all the quadrics.
The nuclearity of operators generated by Hölder continuous kernels
Authors: Cochran James Alan
Published: November 2008

Introduction. The space Cp is the class of operators on a Hilbert space for which the norm ∥K∥p = [trace (KK*)P/2]/p is finite. Equivalently, a compact operator is in Cp if

where the μn are the so-called ‘singular values’ of K (characteristic values of the non-negative compact operator [K] ≡ (KK*)½). The case p = 2 gives the familiar class of Hilbert–Schmidt operators, while C1 is the collection of trace-class or nuclear operators considered by Schatten(12), Lidskii(11), and Gohberg and Krein(7), among others.
Lifting convergent sequences with networks
Authors: Webb J. H.
Published: November 2008

Definition (Moukoko Priso(2)). A locally convex spaceE[T] is said to have a strict absorbent network of type Σ if there exists in E a familyof absolutely convex absorbent sets such that

(1) if {nk} is a sequence of positive integers andfor each k, then the seriesconverges in E[T]

(2) for each sequence {nk} there is a sequence {λk} of positive real numbers such that, ifand 0 ≤ μk ≤ λkfor each k, then

(i) converges in E[T], and

(ii) for each p.
On the short cylindrical waves due to a body heaving on water. I
Authors: Rhodes-Robinson P. F.
Published: November 2008

In this paper we obtain the short-wave asymptotic value of the force coefficient for a heaving body partially immersed in the free surface of water which has finite constant depth. The body is required to intersect the free surface normally and have a vertical axis of symmetry. The motion of the water is thus axisymmetric with short outgoing cylindrical waves at a distance, and these must be calculated. For this we use a non-rigorous argument. The potential problem discussed is analogous to the two-dimensional problem afready investigated involving a heaving surface cylinder. Numerical values are obtained in the case of a half-immersed sphere in conclusion.
The response of an anisotropic elastic half-space to a rolling cylinder
Authors: Clements D. L.
Published: November 2008

The problem of the steady motion of a heavy cylinder, of infinite length, over the surface of an elastic half-space has been examined by Craggs and Roberts(1), Clements(2) and Roberts (3). In papers (1) and (2) the surfaces of both the cylinder and the half-space are assumed to be smooth so that the shear stress over the region of contact is zero. With this assumption Craggs and Roberts analysed the motion for an isotropic half-space while Clements considered the case of an anisotropic half-space. In the paper by Roberts (3) various problems concerning the rolling of rigid and non-rigid cylinders over an isotropic elastic half-space are discussed. In this paper we consider a similar class of rolling cylinder problems for an anisotropic half-space. We begin, in section 2, by deriving the relevant basic equations for the stress and displacement while in section 3 some properties of certain constants occurring in the equations of section 2 are derived for use in later sections. The problem of the steady motion of an inflated tyre is discussed in section 4 under the assumptions that the tyre exerts a constant pressure over the contact region which is instantaneously at rest. The solution shows that, with these assumptions, the shear stress is infinite near the ends of the region of contact. This indicates that there will be slipping of the tyre over the half-space at both ends of the contact region. If these zones of slip are large compared with the total contact area then the assumption of a contact region which is instantaneously at rest is invalidated so that the solution may only be regarded as being applicable when the zones of slip are small. An expression to determine the width of the zones of slip is derived and this expression is used, together with the numerical results of section 5, to obtain information about the influence of anisotropy on the width of the zones of slip. In section 6 the problem of the inflated tyre is discussed with allowance made for a zone of slip at both ends of the region of contact. The results of this section could be expected, particularly when the zones of slipping are wide, to more accurately predict the response of the anisotropic half-space to the rolling tyre than the results of section 4. Unfortunately, in order to satisfactorily complete the analysis, it is necessary to restrict the class of anisotropic materials for which the results are applicable. As a consequence, the work of section 4 is in many ways preferable to that of section 6 in indicating the influence which elastic anisotropy has upon the results.
Extending embeddings of Rn−1 in Rn
Authors: Smith Anthony
Published: November 2008

Notation and results. The main results of this paper are (a) a new proof of the canonical Schoenflies theorem and (b) a non-compact version of the canonical Schoen-flies theorem.

The result (a) is already known(5); the existing proof is based on the proof by Brown of the ordinary Schoenflies theorem, whereas ours is based on that of Mazur and Morse.
Subgroups dual to dimension subgroups
Authors: Sandling Robert
Published: November 2008

Associated with, the powers of the augmentation ideal are the dimension subgroups. In the integral group ring case, they have long been conjectured to be the terms of the lower central series. This paper investigates the subgroups associated with the chain of ideals dual to the chain of powers of the augmentation ideal. The study is reduced to the case of the modular group rings of p-groups. The subgroups are calculated for Abelian p-groups, p odd. They appear in the upper central series of wreath products and provide a new criterion for the nilpotence of an arbitrary wreath product. The nilpotence class of wreath products is considered here as well; calculations and bounds are given; in particular, a new method of computing the class of the Sylow p-subgroups of the symmetric group arises.
Two disc dynamos with viscous friction and time delay
Authors: Cook A. E.
Published: November 2008

Two extensions of the Rikitake model are examined; the first includes viscous friction and the second includes time delay. By use of Liapounov's direct method the viscous system is shown to have an unstable limit cycle associated with two of the three singular points. By numerical integration true ‘reversals of field’ are shown to occur for both the viscous and time delay cases.
Mutual spaces and mutuality
Authors: Amson J. C.
Published: November 2008

The theory of dual spaces and duality is extended from a pair of vector spaces in duality to a list of more than two vector spaces in mutuality. The notions of a canonical bilinear functional and of compatible topologies for a dual pair are extended to those of a canonical multilinear functional and of compatible topologies for a mutual list. Compatible topologies are characterized by means of an extended version of the Mackey–Arens Theorem. Directions for further work are noted.
Chromatic-durable graphs
Authors: Lick Don R. , White Arthur T.
Published: November 2008

One of the most studied parameters in all of graph theory is the chromatic number. Undoubtedly, its popularity as a subject for research is due to its intimate relationship with the famous Four Colour Problem.
On Kneser's conjecture for bounded 3-manifolds
Authors: Heil Wolfgang
Published: November 2008

1. The classical Kneser conjecture states that if M is a closed 3-manifold and π1(M) ≈ A * B, a free product, then there exist 3-manifolds MA, MB such that

and π1(MA) ≈ A, π1(MB) ≈ B. This has been proved by John Stallings(6).
A simple proof of the Fisher–Fuller theorem
Authors: Fisher Franklin M.
Published: November 2008

Several years ago, Fisher and Fuller(1) proved that if P is a real square matrix with all members of its ‘nested set’ of principal minors non-zero, then there exists a real diagonal matrix, D, such that the characteristic roots of DP are all real, negative and distinct. This interesting and powerful result, used by the authors to derive further results concerning convergence of linear iterative processes, has since also proved of interest to economists studying the stability of economic general equilibrium.
An inequality for the trace of the product of two symmetric matrices
Authors: Theobald C. M.
Published: November 2008

Let A, B be real symmetric n × n matrices having orthogonal reductions

where ΛA = diag (λ1(A), …, λn(A)) and ΛB = diag (λ1(B), …, λn(B)) and the sets of latent roots are each in descending order. (Thus we may say that the latent vectors of A, B are ordered with respect to respectively.) Let the multiplicities of the roots and in order of occurrence be respectively and , so that
On the measurability of functions of two variables
Authors: Grande Zbigniew
Published: November 2008

Let f be a real-valued function of two real variables. It is well known(5) that the Lebesgue measurability or even upper semicontinuity of all the sections fx and fy does not imply the measurability of f. Lipiński proved (3) that there exists a non-measurable function f such that each section fx and fy is Baire one, has the Darboux property, and fails to be approximately continuous at no more than one point. Davies (1) noted that the continuum hypothesis implies the existence of a non-measurable function f, such that each fy is measurable and each fx approximately continuous. Ursell (6) proved that if each fy is measurable and each fx continuous (or monotonic), then f is measurable.
One-transversal families
Authors: Rado Richard
Published: November 2008

Let I be a set and

be a family of sets Aν labelled by the elements of I. Throughout, parentheses ( ) denote families and curly brackets {} denote sets. A transversal, or system of distinct representatives, of F is, by definition, a family (xν: ν ∈ I) of objects xν, for ν ∈ I, such that †

Thus the family ({1, 2,} {1, 2}) has exactly two transversals, namely (1,2) and (2, 1). Let TF denote the set of all transversals of F. Much work has been done on the question of characterizing those F for which TF ≠ Ø. We are here going to characterize those F for which TF has exactly one element. In contrast to the more familiar case TF ≠ Ø, our argument is effective irrespective of the cardinalities of I and Aν.
Extension of normal functionals on W*-tensor products
Authors: Wassermann Simon
Published: November 2008

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.
Generators for II*(MU)
Authors: Kozma Ilan
Published: November 2008

The aim of this note is to construct generators for II*(MU). Generators of a similar type were constructed by various authors; Hazewinkel (2), Kozma (3) and Liulevicius (4). The construction in (4) is in fact for II*(BP) at the prime 2.
Representations of a semi-direct product by quantization
Authors: Rawnsley J. H.
Published: November 2008

1.Introduction. The purpose of this note is to apply the Kostant-Souriau quantization theory (2, 3, 4, 5, 7) to construct representations of a semi-direct product.
Spectral asymmetry and Riemannian geometry. III
Authors: Atiyah M. F. , Patodi V. K. , Singer I. M.
Published: November 2008

In Parts I and II of this paper ((4), (5)) we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we defined

where λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that ηA(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s = 0 was not a pole. The real number ηA(0), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.
Strictly singular and cosingular operators
Authors: Martens Eric
Published: November 2008

The results given here are a natural continuation of those given by G. P. Shannon (4). The extension of various operators to the completions of the respective spaces are considered and the adjoint operator is more closely examined. A perturbation result for strictly cosingular operators is also given.
Magnetohydrodynamic flow past sources in the presence of a wall
Authors: Chee-Seng Lim
Published: November 2008

An incompressible magnetohydrodynamic flow, bounded below by a semi-infinite solid wall insulator with a plane surface, is weakly perturbed by two-dimensional (i.e. cylindrical) sources immersed within the fluid. Certain results of Chee-Seng(1) are employed to establish an exact analytic solution for arbitrary source functions. Both direct emission and reflected disturbance fields are hyperliptic, i.e. partly hyperbolic and partly elliptic. The reflected hyperbolic mode is propagated as a Riemann invariant of a modified incident field and its Hilbert transform. This propagation occurs along reflected characteristics whose (acute) inclination with the interface is generally unequal to that of the incident characteristics, but which are, instead, parallel to those non-incident characteristic rays emanating away from the interface.
Gleason parts for measure algebras
Authors: Brown Gavin , Moran William
Published: November 2008

We shall verify Miller's conjectured characterization of parts in measure algebras and obtain also a simpler proof of the main result of (2). The basic pattern of argument follows Miller while the chief novelty resides in the approximation method using Wiener–Ditkin sets which is introduced in section 2.
Uniform compounds of uniform polyhedra
Authors: Skilling J.
Published: November 2008

The work of Coxeter, Longuet-Higgins and Miller (1953) and of Skilling (1975) is extended to give a complete list of uniform compounds of uniform polyhedra. Symmetry relationships between polyhedra and their compounds are discussed.