bluuebunny/crossref_metadata_2025 huggingface dataset:
On the method of matched asymptotic expansions
Authors: Fraenkel L. E.
Published: November 2008

The method of matched (or of ‘inner and outer’) asymptotic expansions is reviewed, with particular reference to two general techniques which have been proposed for ‘matching’; that is, for establishing a relationship between the inner and outer expansions, to finite numbers of terms, of an unknown function. It is shown that the first technique, which uses the idea of overlapping of the two expansions, can be difficult and laborious in some applications; while the second, which is the ‘asymptotic matching principle’ in the form stated by Van Dyke(13) can be incorrect. Two different sets of conditions sufficient for the validity of the asymptotic matching principle are then established, on the basis of assumptions about the structure of expressions which approximate to the desired function f(x,∈) for all relevant values of x. Finally, it is noted that in four classes of singular-perturbation problems for which complete and rigorous asymptotic theories exist, uniform approximations to the solutions have a structure which is a particular form of the general one assumed in this paper.
Some infinite integrals involving Whittaker functions and generalized hypergeometric polynomials, with their applications
Authors: Shah Manilal
Published: November 2008

We have defined the generalized hypergeometric polynomial ((6), eqn. (2·1), p. 79) by means of

where δ and n are positive integers and the symbol Δ(δ, − n) represents the set of δ-parameters

The polynomial is in a generalized form which yields many known polynomials with proper choice of parameters and therefore the results are of general character.
An area theorem for bounded univalent functions
Authors: Chichra Pran Nath
Published: November 2008

Let f(z) be regular and univalent in |z| < 1 with a power series expansion

Then it is proved in (2) that

where z1 and z2 are points in |z| < 1.
Cubic forms over algebraic number fields
Authors: Ryavec C.
Published: November 2008

In 1935 Tartakowski (7) proved that, in general, a cubic form in sufficiently many variables with coefficients in an algebraic number field K has a non-trivial zero in that field; and in the case when K is the rational field 57 variables suffice. Here, ‘in general’ means that the coefficients of the form do not lie in a proper subvariety of the coefficient space. Hence, Tartakowski's result holds for almost all cubic forms. Later, Lewis (5) proved that if K is any algebraic number field such that [K: Q] = n, then there exists a function ψ(n) such that every cubic form over K in m ≥ ψ(n) variables has a non-trivial zero in K. His bound, ψ(n), is extremely large; e.g. when K is the rational field, ψ(1) > 500.
On irreducible polynomials of certain types in finite fields
Authors: Cohen Stephen D.
Published: November 2008

LetGF(q) be the finite field containingq=plelements, wherepis a prime andla positive integer. LetP(x)be a monic polynomial inGF[q, x] of degreem. In this paper we investigate the nature and distribution of monic irreducible polynomials of the following types:

(I)P(xr), whereris a positive integer (r-polynomials).

(II)xmP(x+x−1). (Reciprocal polynomials.) These have the form

(III)xrmP(xr+x−r). (r-reciprocal polynomials.) These have the formQ(xr), whereq(x) satisfies (1·1).
Congruence properties of the binary partition function
Authors: Churchhouse R. F.
Published: November 2008

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.
On semi-Markov processes on arbitrary spaces
Authors: Çinlar Erhan
Published: November 2008

Let E be an arbitrary set, a σ-algebra of subsets of

the Borel sets of F. We write A × B for the product of the sets A and B, and

for the product σ-algebra of and (i.e. the σ-algebra generated by the rectangles A × B with A ∈ and B ∈ ).
The space group corepresentations of antiferromagnetic NiO
Authors: Cracknell A. P. , Joshua S. J.
Published: November 2008

The geometry of the Brillouin zone of antiferromagnetic NiO (Cc2/c) is studied. The corepresentations of the magnetic little group are deduced for the points and lines of symmetry. The magnon symmetries are investigated and the selection ruies for two magnon electric dipole and magnetic dipole absorption are deduced.
A note on magnetohydrodynamic duct flow
Authors: Butler G. F.
Published: November 2008

This paper is concerned with the problem of the flow of an incompressible electrically conducting fluid along a rectangular duet under a transverse magnetic field. The case in which the walls perpendicular to the field are perfectly conducting and those parallel to the field are non-conducting has been considered by Hunt (1), who derives the solution in two ways; as the limiting cases of the flows with (a) non-conducting walls parallel and thin walls of arbitrary conductivity perpendicular to the field, and (b) thin walls of arbitrary conductivity parallel and perfectly conducting walls perpendicular to the field. We show that these two limiting solutions derived by Hunt are in fact equivalent. In addition, we extend the solution of case (b) above by removing the thin wall restriction.
A Tauberian theorem for absolute summability
Authors: Das G.
Published: November 2008

Let be the given infinite series with {sn} as the sequence of partial sums and let be the binomial coefficient of zn in the power series expansion of the function (l-z)-σ-1 |z| < 1. Now let, for β > – 1,

converge for 0 ≤ x < 1. If fβ(x) → s as x → 1–, then we say that ∑an is summable (Aβ) to s. If, further, f(x) is a function of bounded variation in (0, 1), then ∑an is summable |Aβ| or absolutely summable (Aβ). We write this symbolically as {sn} ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, (Aα) ⊂ (Aβ). Note that for β = 0, (Aβ) is the same as Abel method (A). Borwein (2) also introduced the (C, α, β) method as follows: Let α + β ╪ −1, −2, … Then the (C, α, β) mean is defined by
A ‘stick-slip’ problem related to the motion of a free jet at low Reynolds numbers
Authors: Richardson S.
Published: November 2008

A problem in fluid mechanics which has received some attention recently concerns the emergence of an incompressible Newtonian fluid jet from a uniform tube into an inviscid atmosphere. Both the axisymmetric case of a circular tube and the two-dimensional case of flow from between parallel planes are of interest. When the jet falls vertically under gravity, the motion far downstream is dominated by gravity and the expansion procedures of Clarke (3), and Kaye and Vale (10) give details of the flow in this region. When the flow near the exit is at a high Reynolds number, it is reasonable to expect the flow appropriate to that in an infinite tube to prevail right up to the exit (except, perhaps, near the point of discontinuity of the boundary conditions). With this assumption, Duda and Vrentas(5) use a numerical technique to solve for the flow in the axisymmetric jet beyond the exit, both with and without gravity acting in the axial direction. In the absence of gravity, the jet can be expected to attain a constant width some distance downstream, and at high Reynolds numbers the above assumption is sufficient to allow a mass and momentum balance to determine the contraction ratio of the jet as for the axisymmetric case, and for the two-dimensional case (see Harmon (8)). By treating the dynamics of the jet as those of a boundary layer growing on the free surface, Goren and Wronski (6) and Tillett (18) are able to examine the flow in greater detail.
A weak 4-dimensionals-cobordism theorem
Authors: Thomas C. B.
Published: November 2008

In this note we prove a gloss on the usuals-cobordism theorem, which may be of interest, since it holds one dimension lower than the theorem itself. Our result is that everys-cobordismis its own left and right inverse in the groupoid of allh-cobordisms, studied by Stallings in (3).
On the limiting normality of posterior distributions
Authors: Dawid A. P.
Published: November 2008

1.Introduction. Suppose that we wish to make inferences about an unknown real parameter θ, and that we have a random sample of observations (Xl,X2, …,Xn) from a distribution which depends on θ.
Mean field approximations to Couette flow and its stability
Authors: Bradley Roy
Published: November 2008

The preferred-mode method of Roberts (11) is applied to the stability of liquid contained between coaxial cylinders, the inner of which is rotating. Accordingly the velocity field of a steady Taylor vortex is approximated by a truncated Fourier series and the steady state corresponding to a chosen wave number is obtained. The stability of a steady state so obtained is investigated:

(a) With respect to an axisymmetric perturbation associated with a Taylor vortex of a different axial wave number, so that a stable or ‘ preferred ’ steady state is determined. This is done for a narrow gap, η (the ratio of the radii of the cylinders) = 0·95, for Taylor numbers up to 40%, and for a wide gap, η = 0·5, up to 30% beyond the critical values.

(b) For a narrow gap, η = 0·95, with respect to a non-axisymmetric perturbation of a different axial wave number. The solutions obtained for the steady states are found to be stable to such perturbations for a range of Taylor numbers 40% beyond the critical value.
Some rings of interest in the study of places
Authors: Knight J. T.
Published: November 2008

The rings studied in this paper were first constructed in connexion with the problem (Proposition 1) of extending a place from an integral domain to its field of fractions. They provide (Proposition 3) a series of examples of places not so extendable, generalizing an example implicit in the work of Samuel(3); and by the use of ultraproducts they show that such extendability is (in certain precise sense) not an elementary property (Proposition 4). On the other hand they have some unexpected properties, especially in connexion with the relation 1/y0 + … + 1/yr = 1, and they have very small automorphism groups (Proposition 2 and Corollaries); and they are not unique factorization rings (Proposition 5), though they are integrally closed (Proposition 6).
On the continuity of the P.S.-function
Authors: McCrudden M.
Published: November 2008

Let G be a locally compact, unimodular, topological group with μ Haar measure on G, and μ* the corresponding inner measure. If (G) denotes the Borel subsets of G of finite measure, and V(G) = {μ(E):E∈(G)}, then the Product Set function, or P.S.-function, of the group G, written ΦG, is defined by
On hearing the shape of a drum: further results
Authors: Stewartson K. , Waechter R. T.
Published: November 2008

1. Introduction. The underlying problem is to deduce the shape of a drum or plane uniform membrane from the knowledge of its spectrum of eigenvalues ωn = i√λn. It has been shown by Kac(3) that some progress is possible on establishing the leading terms of the asymptotic expansion of the trace function for small positive t. In particular, for a simply connected membrane Ω bounded by a smooth convex curve Γ for which the displacement satisfies the wave equation ∇2ø = ∂2ø/∂t2 and Dirichlet conditions on Γ

where |Ω| = area of Ω, L = length of Γ, and the constant ⅙ is determined (apart from a factor) on integrating the curvature of the boundary; moreover, if Ω is permitted to have a finite number of smooth convex holes then the constant becomes ⅙(1–r), where r = number of holes.
Modular spin representations of the symmetric group
Authors: Jones R. H.
Published: November 2008

1. Let Γn be the representation group or spin group (4, 9) of Sn. Then the irreducible representations of Γn are of two distinct types. These are (a) ordinary representations, which are the irreducible representations of the symmetric group and (b) spin or projective representations. Corresponding to every partition (λ) = (λ1, λ2, …, λm) of n with λ1 > λ2 > … > λm > 0 there is an irreducible spin representation 〈λ〉 of Γn.
Harmonic analysis of scalar and vector fields inn
Authors: Smith J. Denmead
Published: November 2008

It is shown that a real scalar function innwhich is of classCnand either has zero mean on all spheres of unit radius, or has zero mean in all balls of unit radius admits a unique expansion in terms of eigenfunctions of the Laplacian operator. In a similar manner, a suitably smooth vector-valued function innwhich has zero flux through all spheres cf unit radius is shown to be decomposable as the sum of a solenoidal part and a series of conservative parts that are eigenfunctions of the Laplacian. Applications are given, including some in complex analysis.
An eigenvalue problem for elastic contact with finite friction
Authors: Spence D. A.
Published: November 2008

The two-dimensional indentation of an elastic half space by a rigid punch under a slowly applied normal load is considered, for the case in which there is a finite coefficient of friction μ between the surfaces. The contact area is then divided into an inner adhesive region −c<x<cin which the surface displacements are known, surrounded by regionsc< |x| < 1 in which the friction is limiting and the lateral displacement (which must increase in proportion to the overall load) is not known in advance. The problem is formulated in terms of a coupled pair of singular integral equations for the normal and shear stresses σ and τ at the surface; these are combined to give a single homogeneous Fredholm equation with positive kernel for a quantity π proportional to the difference τ – μσ in the adhesive region. The largest eigenvalue of this equation, for which π > 0, gives the adhesive boundarycin terms of μ and Poisson's ratio ν. A similarity transformation shows thatchas the same value for both flat-faced and power law punches.
Common solutions to a pair of linear matrix equations A1XB1 = C1 and A2XB2 = C2
Authors: Mitra Sujit Kumar
Published: November 2008

Penrose (4) gave a necessary and sufficient condition for the consistency of the linear matrix equation AXB = C and also its complete class of solutions. A necessary and sufficient condition for the equations AX = C, XB = D to have a common solution was given by Cecioni (3) and an expression for the general common solution by Rao and Mitra ((6), p. 25). In the present paper, we obtain a necessary and sufficient condition for the equations A1XB1 = C1 and A2XB2 = C2 to have a common solution and also an expression for the general common solution. This result isuseful in computing a constrained inverse of a matrix, a concept originallyintroduced by Bott and Duffin(2) and recently extended by Rao and Mitra(7) who consider more general constraints with the object of bringing together the various generalized inverses and pseudoinverses under a common classification scheme.
Locally convex topologies and the convex compactness property
Authors: Ostling E. G. , Wilansky A.
Published: November 2008

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.
Measures vanishing off the symmetric maximal ideals of M(G)
Authors: Graham Colin C.
Published: November 2008

We show that if μ is a measure on the LCA group G whose Gelfand transform vanishes off the set Σ of symmetric maximal ideals, then μ µ M0(G), that is, then the Fourier-Steiltjes transform of μ vanishes at infinity. This result is then used to show μ µ L1(G)½.
A condition for a compact plane set to be a union of finitely many convex sets
Authors: Eggleston H. G.
Published: November 2008

All the sets with which we are concerned are subsets of the real Euclidean plane E2. By Lm we denote those subsets X of E2 for which, if pl, p2, …, Pm are any m points of X, then at least one segment pipj, i ≠ j consists entirely of points of X. L2 is the class of convex subsets of E2. We shall show that if X is closed and X ∈ Lm. then X is the union of finitely many convex sets. This extends a result of Valentine (4). See also (1),(2),(3).
C-spectral operators
Authors: Dayanithy K.
Published: November 2008

The importance of the spectral theory of operators in Functional Analysis cannot be over emphasized. Spectral theory is in its best form when one considers normal operators in Hilbert spaces. However, for te dimensional spaces one has the reduction to Jordan's Canonical form. In an attempt to generalize this reduction for arbitrary Banach spaces, Dunford introduced the concept of spectral operators. Considerable work has been done in recent times in the study of spectral theory in Banach spaces, almost all of which stems from the pioneering work of Dunford.
Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz
Authors: Cohn P. M.
Published: November 2008

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.
Inextensible Riesz spaces
Authors: Fremlin D. H.
Published: November 2008

My aim in this paper is to give an abstract characterization of the C∞ spaces described in (6) or (9), and to develop some of the remarkable special properties of these spaces. Although the subject is in some ways highly specialized, inextensible and sequentially inextensible spaces seem common enough (they include all spaces of the forms Rx and L0) to be worth studying, and I have already employed them in the proof of more general results (1).

In the first section I set out those properties that can be described in simple Riesz space terms; much of this work has already been published in slightly different forms. In the second part I go on to questions that arise when we impose a topology on an inextensible Riesz space. Finally, in the third section, I discuss some problems, arising from the work before, which are related to the famous measurable cardinal problem.
Dissociated random variables
Authors: MoGinley W. G. , Sibson R.
Published: November 2008

We introduce the idea of dissociatedness for multiply-subscripted random variables: this is a weaker property than independence in general, coinciding with it in the singly-subscripted case; but it is still strong enough to allow a suitable version of the Strong Law of Large Numbers to hold. This result can be applied to prove a weak convergence result of interest in data analysis.
Physical interpretation of N.U.T. metric
Authors: Sackfield A.
Published: November 2008

A metric is constructed, which represents the gravitational field of two spherical static bodies of masses +b and −b, together with a massless source of angular momentum between them. When the separation of the two masses is allowed to tend to infinity the N.U.T. metric can be obtained from the limiting process. This supports the view that the N.U.T. solution represents a static Schwarzschild body together with a rotating semi-infinite spike of pure angular momentum.
The groups of the generalized Petersen graphs
Authors: Frucht Roberto , Graver Jack E. , Watkins Mark E.
Published: November 2008

1.Introduction. For integersnandkwith 2 ≤ 2k <n, thegeneralized Petersen graph G(n, k)has been defined in (8) to have vertex-set

and edge-setE(G(n, k))to consist of all edges of the form

whereiis an integer.All subscripts in this paper are to be read modulo n, where the particular value ofnwill be clear from the context. ThusG(n, k)is always a trivalent graph of order 2n, andG(5, 2) is the well known Petersen graph. (The subclass of these graphs withnandkrelatively prime was first considered by Coxeter ((2), p. 417ff.).)
Greatest common divisor of several polynomials
Authors: Barnett S.
Published: November 2008

Given a polynomial a(λ) with degree n, and polynomials b1(λ), …, bm(λ) of degree not greater than n – 1, then the degree k of the greatest common divisor of the polynomials is equal to the rank defect of the matrix R = [b1(A), b2(A), …, bm(A)], where A is a suitable companion matrix of a(λ). Furthermore, it is shown that if the first k rows of R are expressed as linear combinations of the remaining n – k rows (which are linearly independent) then the greatest common divisor is given by the coefficients of row k + 1 in these expressions. A simple expression is derived for R and a permutation of the columns of this matrix establishes a direct connexion with controllability of a constant linear control system. Finally, when m = 1 a relationship between the corresponding R and Sylvester's matrix is exhibited.
On the forced surface waves due to a vertical wave-maker in the presence of surface tension
Authors: Rhodes-Robinson P. F.
Published: November 2008

The classical wave-maker problem to determine the forced two-dimensional wave motion with outgoing surface waves at infinity generated by a harmonically oscillating vertical plane wave-maker immersed in water was solved long ago by Sir Thomas Havelock. In this paper we reinvestigate the problem, making allowance for the presence of surface tension which was excluded before, and obtain a solution of the boundary-value problem for the velocity potential which is made unique by prescribing the free surface slope at the wave-maker. The cases of both infinite and finite constant depth are treated, and it is essential to employ a method which is new to this problem since the theory of Havelock cannot be extended in the latter case of finite depth. The solution of the corresponding problem concerning the axisymmetric wave motion due to a vertical cylindrical wave-maker is deduced in conclusion.
The uniform consistency of maximum-likelihood estimators
Authors: Moran P. A. P.
Published: November 2008

Wald(4) proved that under rather weak conditions, maximum-likelihood estimators are consistent. In an earlier paper (3) he had promised to publish conditions under which they are uniformly consistent but he did not do so. As the uniformity of consistence of maximum-likelihood estimators is important in studying the asymptotic power of certain tests, particularly when the true value of the parameter lies on the boundary of the parameter space, the purpose of the present note is to fill this gap. The method of proof follows that of Wald(4) very closely.
Maximum-likelihood estimation in non-standard conditions
Authors: Moran P. A. P.
Published: November 2008

The origin of the present paper is the desire to study the asymptotic behaviour of certain tests of significance which can be based on maximum-likelihood estimators. The standard theory of such problems (e.g. Wald(4)) assumes, sometimes tacitly, that the parameter point corresponding to the null hypothesis lies inside an open set in the parameter space. Here we wish to study what happens when the true parameter point, in estimation problems, lies on the boundary of the parameter space.
A group theoretical approach to the many nucleon problem
Authors: de Wet J. A.
Published: November 2008

Representations of the four-dimensional unitary group U4 were considered long ago by Wigner(1)as a model for nuclear isobaric multiplets. However, the Hamiltonian did not include the components of isospin which together with the spin coordinates are known to label the nuclear states. In this paper we shall find representations characterized by the eigenvalues of angular momentum J, isospin T and parity π, and will find mass relations which give good agreement with the experimental energy levels of Li6 and Be8 labelled by the same quantum numbers. The representations found by Wigner give good results for the ground state energies, or masses, of all the nuclei up to a mass number of A = 110(2), and we shall derive Wigner's representations as a special case. In fact, unless these are satisfied it is impossible for particle-like representations of U4 to exist!
Modular augmentation ideals
Authors: Sandling Robert
Published: November 2008

An ideal of an integral group ring is divisible by a given integer if all of its elements share this common factor; the ideals most often encountered are rarely divisible in this sense. Only in the case of finite p-groups are powers of the augmentation ideal of the integral group ring ever divisible. For every e, there is some n for which the nth power of the augmentation ideal is divisible by pe. The smallest such integer n arises in many contexts; this paper describes its properties and interpretations.
On dual summability methods
Authors: Kuttner B.
Published: November 2008

1. Let A be a summability method given by the sequence-to-sequence transformation

We suppose throughout that, for each n

converges; this is a much weaker assumption than the regularity of A. Then we define

We also suppose throughout that the sequence {sk} is formed by taking the partial sums of the series Σak; that is to say that

Let A' denote the summability method given by the series-to-sequence transformation

Following Lorent and Zeller (4), (5), we describe A, A' as dual summability methods. We recall that formally,
Symmetrical treatment of a case of normal correlation
Authors: Jeffreys Harold , Singh Kehar
Published: November 2008

1. It often happens that the greater parts of two variables x, y may be connected linearly but there may be other disturbances of both. A treatment given by one of us ((1), section 4·42, pp. 216–17) assumes that the other disturbances are independent, as for observational errors. This is not altogether satisfactory because unless the variations of xr, yr are much greater than the standard errors there is an asymmetry in the treatment corresponding to the well-known distinction between the two regression lines.
Subalgebras of the dual of the Fourier algebra of a compact group
Authors: Dunkl Charles F. , Ramirez Donald E.
Published: November 2008

We let G denote an infinite compact group and G its dual. We use the notation of our book ((l), Chapters 7 and 8). Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and ℒ∞(G) denotes its dual space under the pairing 〈ƒ,φ〉 (ƒ ∈ A(G), φ ∈ ℒ∞(G)). Further, note ℒ∞(G) is identified with the C*-algebra of bounded operators on L2(G) commuting with left translation. The module action of A(G) of ℒ∞(G) is defined by the following: for ƒ ∈ A(G), φ ℒ∞(G), ƒ. φ ∈ ℒ∞(G) by 〈g, ƒ . φ〉 = 〈 ƒg, φ〉, g ∈ A(G) Also ‖ƒ . φ‖∞ ≥ ‖ƒ‖A ‖φ‖∞.
On Teh's construction of orders in Abelian groups
Authors: Minassian Donald P.
Published: November 2008

A claim of H. H. Teh on Archimedean full orders ((2), ‘only if’ of Theorem 2, p. 480) fails. His statement (p. 478, line 5) that the direct product of n copies of the additive rational numbers admits no full order of Archimedean rank more than n is proved without Hahn's embedding theorem. A few other remarks are made.
Compactifications of ordered topological spaces
Authors: McCallion T.
Published: November 2008

An ordered topological space is a set X endowed with a topology τ and a partial order ≤. We shall denote such a space by (X, τ), it being understood that (unless otherwise stated) the symbol ≤ is used to denote the partial order on X. An account of some of the properties of these spaces can be found in (5), (6) and (7).
Embeddings of bounded manifolds
Authors: Hudson J. F. P.
Published: November 2008

Introduction. This paper gives some embedding and unknotting theorems for bounded manifolds in the PL and differential categories. The theorems are of a similar nature to the embedding and unknotting theorems of Irwin and Zeeman (9, 18), except that the boundaries are allowed to move and the appropriate connectivity conditions become conditions on the relative homotopy groups of the manifolds modulo the boundary, or some suitable part of the boundary.
Wave-front singularities for two-dimensional anisotropic elastic waves
Authors: Payton Robert G.
Published: November 2008

Wave-front singularities for the displacement functions, associated with the radiation of linear elastic waves from a point source embedded in a finitely strained two-dimensional elastic solid, are examined in detail. It is found that generally the singularities are of order d–½ where d measures distance away from the front. However, in certain exceptional cases singularities of order d–-n where n = ¼, ⅔, ¾, may be encountered.
Discrete spectra criteria for singular differential operators with middle terms
Authors: Hinton Don B. , Lewis Roger T.
Published: November 2008

Let l be the differential operator of order 2n defined by

where the coefficients are real continuous functions and pn > 0. The formally self-adjoint operator l determines a minimal closed symmetric linear operator L0 in the Hilbert space L2 (0, ∞) with domain dense in L2 (0, ∞) ((4), § 17). The operator L0 has a self-adjoint extension L which is not unique, but all such L have the same continuous spectrum ((4), § 19·4). We are concerned here with conditions on the pi which will imply that the spectrum of such an L is bounded below and discrete.
Stationary gravity waves on non-uniform free streams: jet-like streams
Authors: Peregrine D. H. , Smith Ronald
Published: November 2008

The basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.
Almost ellipsoidal sections and projections of convex bodies
Authors: Larman D. G. , Mani P.
Published: November 2008

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex body.
Resolvents and trace form
Authors: Fröhlich A.
Published: November 2008

This paper is a continuation of (F3). In its first part we shall expand and extend the general theory of the earlier paper, while in the second part we specialize to number fields. The theory of resolvents and of the trace form, presented here, complements the more arithmetic theory of module conductors and module resolvents as described elsewhere (cf. (F4)). Both these papers will be applied in work on the connexion, for tame extensions, between Galois module structure of algebraic integers on the one hand, and Artin conductors and root numbers on the other hand (cf. (F5)). The results of the present paper are however not restricted to the tame case and, it is hoped, will subsequently be applied in a more general context.
Convex polytopes with convex nets
Authors: Shephard G. C.
Published: November 2008

The idea of anetwill be familiar to anyone who has made a model of a three-dimensional convex polytope (3-polytope) out of a flat sheet of card or similar material. To begin with, one cuts out a polygon, and then the model is formed by folding this and joining its edges in an appropriate manner. For example, Fig. 1 illustrates two nets for a regular tetrahedron. The folds are indicated by broken lines and lettering (either of the edges or of the vertices) indicates how the edges are to be joined.
On Toeplitz sections in sequence spaces
Authors: Buntinas Martin
Published: November 2008

The concept of sectional convergence (AK) in FK-spaces was investigated by Zeller in (20). In (5) and (6), Garling investigated convergent and bounded sections in more general topological sequence spaces. Many of the results hold for Toeplitz sections in sequence spaces. A topological sequence space has the property of Toeplitz sectional convergence (TK) if and only if the unit sequences form a Toeplitz basis. In section 3, we present characterizations of Toeplitz sectional boundedness (TB) and functional Toeplitz sectional convergence (FTK) in terms of βT- and γT-duality. In section 4, we apply our results to summability fields. These results are related to the Hardy-Bohr property of multipliers for Cesàro summable sequences of positive order. In section 5, we characterize the properties TK and TB in FK-spaces by factorization statements.
The Schwarzian derivative and estimates of functions analytic in the unit disc
Authors: Essén Matts , Keogh F. R.
Published: November 2008

Suppose that w is analytic in the open unit disc Δ in the z-plane. The Schwarzian derivative of w is defined by

(for basic properties of {w, z} we refer to Hille ((3), p. 375)). Let denote the class of functions w which are analytic in Δ and satisfy the conditions
On a class of Volterra and Fredholm non-linear integral equations
Authors: Bushell P. J.
Published: November 2008

This paper concerns the existence and uniqueness of non-negative solutions of non-linear Volterra integral equations of the type

and

where the kernel function k(.,.) is non-negative and sufficiently smooth, and either 0 < p < 1 or – 1 < p < 1. We will consider also the corresponding Fredholm equations

and
On group actions on groups and associated series
Authors: Hilton Peter , Stammbach Urs
Published: November 2008

1. Introduction. In this paper we consider Q-groups; that is, Q is a group and we consider groups N endowed with a Q-action, meaning a homomorphism of Q into the group of automorphisms of N. In (4) a lower central Q-series was defined for such a Q-group, N, generalizing the lower central series of a group, and results were obtained relating to the localization of such a series. Since the ideas in that paper were inspired by the homotopical localization theory of nilpotent spaces (see (6)), the main body of results in (4) was concerned with the case in which N is nilpotent, and perhaps also the group Q and the action of Q on N (in the sense that the lower central Q-series terminates after a finite number of steps with the trivial group {1}). We now adopt a broader view-point and only restrict ourselves to the nilpotent case when our results appear to require us to do so; thus the spirit of this paper is much more that of general group theory as presented in [(8), especially ch. VI]. Thus, while there is some overlap of results, the methods used are not the same and many results (for example, Theorem 3·1) are far more general than any obtained in (4). Moreover, the methods also appear to us to be more appropriate in that essential appeal was made in (4) to a sophisticated theorem of Norman Blackburn on nilpotent groups, whereas here we merely use homological methods, the construction of the semidirect product and, in section 4, some very classical facts of the commutator calculus.
Deterministic epidemic waves
Authors: Atkinson C. , Reuter G. E. H.
Published: November 2008

In the well-known deterministic model for the spread of an epidemic, one considers a population of uniform density along a line and divides the population into three classes: susceptible but uninfected, infected and infectious, infected but removed. If we denote space and time variables by s, t and let x(s, t), y(s, t), z(s, t) be the proportions of the population at (s, t) in these three classes, then x + y + z = 1 and we suppose that

Here Ῡ(s, t) denotes a space average ∫ y(s + σ) p(σ) dσ, where p is a probability density function; b is the removal rate; the scale of t has been adjusted to remove a constant that would otherwise occur in (1).
Uniqueness of BSO
Authors: Adams J. F. , Priddy S. B.
Published: November 2008

This paper will show that after localization at any given prime p, the infinite loop space structure on the space BSO is essentially unique. If the word ‘localization’ is replaced by ‘completion’, the result continues to hold; and both results continue to hold if the space BSO is replaced by the space BSU.
A new derivation of the W.K.B. solution for the coupled equations of radio waves in the ionosphere
Authors: Bennett J. A.
Published: November 2008

The W.K.B. solution for radio waves in a plane stratified ionosphere was obtained by Budden ((4), §§ 18.10 and 18.11) by a process using approximate diagonalization of the governing equations. Alternatively the solution may be obtained by specializing to the one-dimensional case the result of Bennett (1) which employs frequency scaling. Formally similar results were obtained by Ludwig (8) without explicitly introducing frequency scaling.
A simple proof of an identity of Good
Authors: Groeneveld J. , Kampen N. G. Van
Published: November 2008

Recently, Good discovered the following curious identity (1). Let θ be a real-valued random variable having cumulants Km ≡ Km(θ) (m = 1, 2, …). Take a set of m statistically independent variables θ(1), θ(2), …, θ(m), each with the same distribution as θ, and form the random variable

where ω is any primitive mth root of unity (e.g. ω = e2πi/m). Then

where 〈.〉 denotes expectation.
On the differentiability of space-time
Authors: Clarke C. J. S.
Published: November 2008

It is shown that the differentiability of a space-time is implied by that of its Riemann tensor, assuming a priori only boundedness of the first derivations of the metric. Consequently all the results on space-time singularities proved in earlier papers by the author hold true in C2− space-times.
The determination of the elastic field of an ellipsoidal inclusion in an anisotropic medium
Authors: Walpole L. J.
Published: November 2008

1. Introduction. In studying the elastic behaviour of inhomogeneous systems certain inclusion and inhomogeneity problems are fundamental. In the ‘transformation problem’, a region (the ‘inclusion’) of an unbounded homogeneous anisotropic elastic medium would undergo some prescribed infinitesimal uniform strain (because of some spontaneous change in its shape) if it were not for the constraint imposed by the surrounding matrix. When the inclusion has an ellipsoidal shape, Eshelby (3, 4) was able to show that the stress and strain fields within the constrained inclusion are uniform and that calculations could be completed when the medium was isotropic. A generally anisotropic medium seemed to raise forbidding analyses, but Eshelby (3) did point the way to an evaluation of the uniform strain which several authors (referred to later) developed into an expression amenable to numerical computation. Here we offer an elementary and immediate route to this expression of the uniform strain, which has been accessible hitherto only by the circuitous procedures of Fourier transforms. It is available as soon as the uniform state of strain in the inclusion is perceived and before an alternative evaluation is commenced. First, we appeal to a theorem (not it seems previously known) which reveals (in particular) the vanishing of the mean strain in the infinitesimally thin ellipsoidal homoeoid lying just outside the inclusion. Secondly, we need only reflect that at each point of the interface there is an immediate algebraic expression of the strain just outside the inclusion in terms of the uniform strain just inside.
The forced two dimensional oscillations of a rigid strip in smooth contact with a semi-infinite elastic solid
Authors: Wickham G. R.
Published: November 2008

In this paper we investigate a time harmonic plane strain problem in a homogeneous isotropic semi-infinite elastic solid with mixed boundary conditions on its surface. The boundary is free from applied stress apart from on an infinite strip where the tangential components of stress are zero, and the normal component of displacement is a prescribed function. By reducing the mixed boundary value problem to the solution of a Fredholm integral equation of the second kind, an existence theorem is proved and a simple low frequency asymptotic formula relating the normal stress to the prescribed displacement is derived rigorously. The results are applied to the forced heaving and rocking motion of a rigid cylindrical body in smooth contact with an elastic half space. Some further applications to scattering problems are suggested.
Weakly compactly generated locally convex spaces
Authors: Hunter Richard J. , Lloyd John
Published: November 2008

Locally convex spaces which are generated by a weakly compact subset (or a sequence of weakly compact subsets) and their subspaces are studied. Various characterizations and the permanence properties of these spaces are obtained. Certain results valid for weakly compactly generated Banach spaces are extended. These spaces are shown to have sequential properties which extend well-known properties of separable locally convex spaces.
Scattering from infinite sheets
Authors: Davies E. B.
Published: November 2008

We prove the existence and completeness of the wave operators for quantum-mechanical scattering by a potential which does not decrease to zero at infinity in two of the three space directions. We also obtain a new abstract result concerning the continuous dependence of the wave operators on the potential.
Harmonic balance and the Hopf bifurcation
Authors: Allwright D. J.
Published: November 2008

A form of the Hopf bifurcation theorem specially suited to systems in block-diagram form has been developed, which allows one to deal with high or infinite order linear elements solely in terms of their transfer functions. The results are proved by the method of harmonic balance, and, for general non-linear systems, lead to criteria for the existence and stability of bifurcated orbits generalizing those derived by various authors for systems of ordinary differential equations. In the particular case of control loops with a single non-linearity, a simple addition to the Nyquist diagram of the loop determines the amplitude and frequency of bifurcated orbits, and whether they occur when the equilibrium is stable or unstable. The analysis is independent of the central manifold theorem, and of Floquet theory.
On the differentiation of tensor functions
Authors: Durban David
Published: November 2008

A new method for generating tensorial derivatives of tensor functions is proposed. The method is based on the use of tensors as absolute entities along with the advantages offered by their decomposition on an orthonormal base.
Overstable convection in a horizontally heated rotating fluid
Authors: Daniels P. G. , Stewartson K.
Published: November 2008

Fluid is contained in a rotating annulus of rectangular cross-section which is subject to a radial temperature gradient along the base and is insulated around its upper surfaces. If E is an Ekman number, which is assumed small, a finite amplitude cellular motion of wavelength ∼ E⅓ develops near the inner sidewall of the annulus by an exchange process if the local Rayleigh number, R, exceeds its critical value Rc and the Prandtl number of the fluid, σ, is greater than a critical value, σc (Daniels and Stewartson(5)). In the present study we consider the remaining range of values 0 < σ < σc for which overstability is preferred, occurring at a value of the Rayleigh number, Rc0, less than Rc. Disturbances to the basic flow near the sidewall now oscillate in time with frequency ∼ E⅓ and are amplified if R exceeds Rc0, the vertical cellular velocity attaining values ∼ E in a region of extent ∼ E near the sidewall. On the time-scale ∼ E disturbances take the form of two wave-like components which travel in opposite directions with the characteristic group velocity of the system. The first is simply convected away from the sidewall into the stably stratified interior and decays, while the second travels towards the sidewall where it is reflected. The reflexion coefficient is determined by the dynamics of an E⅓ boundary layer where an oscillatory motion is generated by the incident wave.
A counter-example in summability theory
Authors: Kuttner B.
Published: November 2008

Let Σ denote the set of all series

of complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.
Simple Moufang loops
Authors: Doro Stephen
Published: November 2008

If H is a Moufang loop, and x ∈ H, there are defined permutations of H, L(x):y ↦ xy and R(x): y ↦ yx. The group Gr (H), generated by these permutations for all choices of x, is called the multiplication group of H. It has a close connexion with the structure of H, as shown, for instance, in the papers of Albert(1). The purpose of this paper is to investigate the correspondence between groups and loops, so that group theoretic results may be applied to determine the structure of Moufang loops.
Solution of two problems of Arhangel'skiĭ and Wiegandt
Authors: Ward A. J.
Published: November 2008

In their paper (l) on the general theory of connectednesses and disconnectednesses, Arhangel'skiĭ and Wiegandt conclude by proposing several problems. We shall show in this note, by elementary constructions (assuming of course the axiom of choice), that the answer to their questions (d) and (e) is negative in both cases; we also consider a modified form of question (d) to which the answer is positive. Our notation is that of (l) and references, unless otherwise stated, are to that paper; thus ‘(1·3)’ means ‘Proposition 1·3 of (l)’, but ‘(2·1 above)’ means ‘Proposition 2·1 of the present note.’
Conformal deformations, Ricci curvature and energy conditions on globally hyperbolic spacetimes
Authors: Beem John K. , Ehrlich Paul E.
Published: November 2008

We consider globally hyperbolic spacetimes (M, g) of dimension ≥ 3 satisfying the curvature condition Ric (g) (v, v) ≥ 0 for all non-spacelike tangent vectors v in TM. This curvature condition arises naturally as an energy condition in cosmology. Suppose (M, g) admits a smooth globally hyperbolic time function h: M → such that for some t0, the Cauchy surface h−1(t0) satisfies the strict curvature condition Ric (g) (v, v) > 0 for all non-spacelike v attached to h−1(t0). Then M admits a metric g′ conformal to g satisfying the strict curvature condition Ric (g′) (v, v) > 0 for all non-spacelike v in TM. If the curvature and strict curvature conditions are restricted to null vectors, the analogous result may be obtained. Similar results may also be obtained for the scalar curvature in dimension ≥ 2 and for non-positive Ricci curvature.