bluuebunny/crossref_metadata_2025 huggingface dataset:
The scale effect in compressible turbulence
Authors: Crighton D. G.
Published: November 2008

The pressure fluctuations in a compressible fluid in statistically steady isotropic motion are studied. These depend crucially on the scale of the turbulent region. If the region is small enough, the mean square acoustic pressure varies as the linear scale of the region, and as the fourth power of the turbulence Mach number. In the limit of infinite scale, however, diffusive effects limit the otherwise infinite pressure fluctuations as first shown by Lighthill(6). The mean square acoustic pressure then varies as the Mach number and as the turbulence Reynolds number. The pressure fluctuations above a sheet composed of statistically uniformly fluctuating sources are also examined. Here the mean square pressure diverges, initially, in proportion to the logarithm of the scale of the sheet, until viscous effects again become significant in limiting the pressure.
The flow around a semi-infinite oscillating plate and the skin friction on arbitrarily cross-sectioned infinite cylinders oscillating parallel to their length
Authors: Wilks Graham
Published: November 2008

In Part I of the work that follows solutions for the flow around a semi-infinite oscifiating plate are obtained, one by an operational method using parabolic coordinates and one in the form of a series solution. The latter is used to obtain an estimate of the skin friction distribution about the edge of the plate and hence the excess skin friction over the infinite oscilliating plate value. A valuable check on this result is provided by an equivalent consideration of the operational solution. Having demonstrated the usefulness of the series solution we proceed to extend its application to a wedge of arbitrary angle in Part II and eventually use the results to derive a general expression for the skin friction on an infinite cylinder of arbitrary cross-section oscifiating rapidly parallel to its length.
A note on B- and Br- incomplete topological Abelian groups
Authors: sulley L. J.
Published: November 2008

From results of Baker (2) it appeared to be unlikely that, in Hausdorff topological Abelian groups, completeness would be implied by Br- or B-completeness (for definitions, see below). We show here (Corollary 1) that the group of all complex roots of unity, though not complete, is B-complete. Another example, for the suggestion of which we are indebted to Dr J. W. Baker, is used to show (Corollary 2) that B-completeness is not a consequence of Br-completeness. It is proved, however, that B- and Br-complete Hausdorff topological Abelian groups are embedded in their completions in special ways in relation to the closed subgroups of the completions. Also, the completions of B- and Br-complete Hausdorff topological Abelian groups are shown to be, respectively, B- and Br-complete.
On the axiomatic foundations of the theory of Hermitian forms
Authors: Wall C. T. C.
Published: November 2008

In recent work on some topological problems (7), I was forced to adopt a complicated definition of ‘Hermitian form’ which differed from any in the literature. A recent paper by Tits(5) on quadratic forms over division rings contains a new and simple definition of these. A major objective of this paper is to formulate both these definitions in somewhat more general terms, and to show that they are equivalent.
Random Fourier–Stieltjes transforms on partially ordered ǵroups
Authors: Fournier John
Published: November 2008

If one changes at random the signs of the Fourier coefficients of an L2 function the result is still the sequence of Fourier coefficients of an L2 function. L2 is the only LP space with this property; in fact if a sequence remains a Fourier transform for every insertion of ± signs then it is really the transform of an L2 function ((6), p. 215, Theorem 8·14). Our main theorem is a version of this principle, for sequences defined on suitable subsets of a group; it is phrased in terms of large sets of choices of ± signs rather than all sequences of ± signs.
Exact solutions for vertically polarized electromaǵnetic waves in a horizontally stratified maǵneto-plasma
Authors: Westcott B. S.
Published: November 2008

In a previous paper (11) refractive index profiles capable of yielding exact solutions for vertically polarized electromagnetic waves propagating in horizontally stratified isotropic media were derived systematically. The present work extends the method to deal with anisotropic media in which propagation is transverse to a horizontally applied static magnetic field. The relevance to ELF radio wave propagation in the terrestrial ionosphere is noted.
The propagation of waves in an elastic half-space containing a cylindrical cavity
Authors: Gregory R. D.
Published: November 2008

The problem of the propagation of time harmonic waves in an isotropic elastic half-space containing a submerged cylindrical cavity is solved analytically. Linear plane strain conditions are assumed. Using an expansion theorem proved in a previous paper (Gregory (3)), the elastic potentials are expanded in a series form which automatically satisfies the governing equations, the conditions of zero stress on the flat surface, and the radiation conditions at infinity. The conditions of prescribed normal and tangential stresses on the cavity walls are shown to lead to an infinite system of equations for the expansion coefficients. This system of equations is shown to be a regular L2-system of the second kind and from its unique l2-solution, the solution to the problem is constructed. The fundamental questions of existence and uniqueness are fully treated and methods are described for constructing the solution.

Three applications of the general theory are presented dealing respectively with the production, amplification and reflexion of Rayleigh waves.
A general compactness theorem
Authors: Collins P. J.
Published: November 2008

It is the purpose of this paper to prove a general compactness theorem and deduce from it, as corollaries, the Tychonoff theorem on the product of compact spaces and Ascoli-type theorems giving sufficient conditions for function spaces with the topology of uniform convergence on compact sets to be compact.
Automorphisms of metastably connected PL-manifolds
Authors: Hodgson J. P. E.
Published: November 2008

Let Mm be a compact connected PL-manifold, we say M is metastably connected if πi(M) = 0 for i ≤ [⅓m] + 1, and m ≥ 6. Our notation will follow (3).
Montel's theorem for subharmonic functions and solutions of partial differential equations
Authors: Beardon A. F.
Published: November 2008

1. A theorem of Montel (14), states that if f(z) is analytic and bounded in the half-strip

and if there exists an x0 in (a, b) such that

as y → ∞, then

as y → ∞ l. u. on (a, b)(locally uniformly on (a, b), i.e. uniformly on every compact subset of (a, b)). Bohr (3), has proved a version of this result applicable to functions analytic, but not necessarily bounded, in S(a, b) and Hardy, Ingham and Pólya (10), have considered whether or not f(z) can be replaced by |f(z)| in (1·1) and (1·2). They show that this is not so but prove that if f(z) is analytic and bounded in S(a, b) and if

as y → ∞ for three distinct values x1, x2 and z3 in (a, b) then there exist constants A and B such that

as y → ∞, l.u. on (a, b). Cartwright (Theorem 5, (6)) has proved that if f(z) is analytic and satisfies |f(z)| < 1 in, S(a, b) and if for some x0 in (a, b)

asy → ∞, then

as y → ∞ co, l.u. on (a, b) and Bowen (5), has shown that if (1·4) is replaced by the weaker condition

as n → ∞ for some suitable sequence of points Zn in S(a, b) then (1·5) is still valid. In sections 2–5 of this paper we shall consider whether or not these results are valid |f(z)| is replaced by a subharmonic function.
Two-dimensional flow past a flat plate in the presence of boundary walls
Authors: Todd L.
Published: November 2008

1. Introduction. Flow of viscous fluid at low Reynolds number has attracted a great deal of attention over the last hundred years or so and is still an area of active research at the present time. For a general background on this field, we refer the reader to Langois(4). The present paper is concerned with steady flow past a finite flat plate which spans a long rectangular channel, of large aspect-ratio, at the centre ‘line’ (see Fig. 1). The flow is motivated by a constant pressure difference which is applied

between the ends of the channel. We examine such flows when the Reynolds number based on the semi-channel height, a, is much less than 1. Our main task is to show that Stokes approximation, i.e. neglection of the inertia terms, is uniformly valid † over the whole flow field. (†Such a result contrasts with the non-uniformity of the approximation when the channel walls are removed and the external flow is a uniform stream.) Harper & Chang (1) who treat the corresponding problem for a circular cylinder, give a full account of recent theoretical and experimental work on this class of fluid flows. We shall therefore defer comments on related papers until the appropriate part of the text.
Extension of the Curie principle and constitutive relations for fluids with antisymmetric stress
Authors: Morris Rosa M. , Price W. G.
Published: November 2008

In recent years several papers [Grad(9), Condiff and Dahler(5), Baronowski and Romatowski(2), Eringen(8), Allen and de Silva(i)] have been written dealing with the possibility of antisymmetric stress in a fluid and its relationship with the internal micro-structure of the fluid. In the classical fluid dynamics of the Navier–Stokes equations, the molecules of the fluid element are treated as material points devoid of any internal spin motion and the only type of angular motion that the macroscopic elements of the fluid possess is the usual vorticity ½ curl v, the velocity of the fluid being v. In the case of fluids whose molecules are not regarded as material points, but are treated as micro-structures having internal spin, the total angular velocity of a macroscopic volume element will be equal to the vector sum of spins of the micro-structures and the part due to the vorticity.
Solubility theorems for finite groups
Authors: Laffey Thomas J.
Published: November 2008

In this paper we obtain various sufficient conditions for the solubility of a finite group. In particular, we show that if G is a finite group and p≥5 is a prime such that all p′-subgroups of G are nilpotent, then G is soluble. We show also that if G is a finite group which has a cyclic Sylow p-subgroup Pand such that for all p′-subgroups H of G, H is nilpotent and H′ is cyclic, then, if p≠3, either P◃G or G has a normal p-complement.
Some cyclic and other inequalities. III
Authors: Diananda P. H.
Published: November 2008

1. Introduction. In a recent paper (2) Daykin has made the Conjecture. If t is real, then

where

He has proved (1, 2) that (i) Φ(t) is a convex function of t, (ii) Φ(t)≥ n if t ≥ 0 or 2, and (iii) Φ(t) takes values arbitrarily close to ½n if n is even, x2 = x4 = … = xn = 1, and xl = x3 = … = xn-1 and t have suitable small positive values. It is known (3) that (1) is true if t = 1.
On particle-like representations of the complete homogeneous Lorentz group
Authors: de Wet J. A.
Published: November 2008

In two previous papers (1, 2) representations of the unitary groups U4, U2 were found which described some of the properties of nucleons and electrons. In particular, the many electron wave functions were constructed from the irreducible representations of U2 restricted to the proper orthochronous Lorentz group Lp. In this paper the irreducible representations of U4 found in (1) will be shown to be also irreducible representations of the complete homogeneous Lorentz group L0 and the techniques of matrix contraction employed in (2) will be used to find the precise form of the matrices of the infinitesimal ring.
A lemma on finite p-groups and some consequences
Authors: Laffey Thomas J.
Published: November 2008

Let S be a finite p-group and let σ be a p'-automorphism of S. A well-known result of Huppert ((2), IV, (5·12), Satz) states that if σ acts trivially onΩ1(S) (ω2(S) if p = 2) then σ is the identity. In this paper, we give a short proof of this result and also give some applications of it. Our proof depends on a somewhat surprising Lemma (Lemma 1) which is of some interest in its own right.
A distributional Hardy transformation
Authors: Pathak R. S. , Pandey J. N.
Published: November 2008

In this paper the classical Hardy transformation (5) has been extended to distributions. The celebrated Hankel transformation, Y-transformation and H-transformation are particular cases of the Hardy transformation. The Hankel transformation has been extended to distributions by Zemanian(12). The distributional theory has been explored by Zemanian(13), Koh and Zemanian(6), and Dubey and Pandey(2). The famous Y- and its reciprocal H-transformations have not been extended to distributions. The inversion and some of the properties of these distributional transformations can be deduced as particular cases of the results contained in this paper.
Operators with large trace, and a characterization of
Authors: Garling D. J. H.
Published: November 2008

Let E be a real or complex n-dimensional normed space. Deschaseaux (1) has shown that if the absolutely summing constant π1(E) = n then E is isometrically isomorphic to . (For the definition of π1(E) and related quantities, see (2).) It is well-known ((3), (4) and (6)) that if E is a P1-space then E is isometrically isomorphic to . We shall show that Deschaseaux' result follows easily from this, and from the following elementary result:
Proposition. If S is a linear operator on a real or complex n-dimensional normed space E, then |trace S| ≤ ‖S‖, with equality if and only if S is a scalar multiple of the identity I.
A topology on the space of PL homeomorphisms of a PL manifold
Authors: Smith Anthony
Published: November 2008

0. Introduction. It has long been known that the space of all smooth homeomorphisms of a smooth manifold with the Cl topology can be given the structure of a smooth manifold. A path in this space corresponds to a diffeotopy. It has been shown more recently (1) that the space of all homeomorphisms of a topological manifold is locally contractible when given the C–0 topology. A path in this space corresponds to an isotopy.
The generation of surface waves over a sloping beach by an oscillating line-source. I. The general solution
Authors: Morris Clare A. N.
Published: November 2008

The problem of wave generation by a line source of sinusoidally varying strength situated in water above a beach of arbitrary angle α(0 < α ≤ π) is solved by the use of a Laplace-integral representation of the solution. It is shown that a solution can be constructed which is regular at the shoreline and gives an outgoing wave-train at infinity.
Absolute total-effectiveness of (N, pn) means. I
Authors: Dikshit H. P. , Kumar A.
Published: November 2008

1. Introduction. Using throughout the definitions and notations contained in (4), we state the following recent results by one of the present authors ((4), Theorem I and Theorem 2) which include as a special case the corresponding |C| – result due to Bosanquet (1), Bosanquet and Hyslop (2) and Hyslop(5).
Almost-conservative second-order differential equations
Authors: Swinnerton-Dyer H. P. F.
Published: November 2008

During the last thirty years an immense amount of research has been done on differential equations of the form

where ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the form

where the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assume

and for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)
Weighted tensors and Lie groups
Authors: Held A.
Published: November 2008

A class of objects, sometimes known as weighted tensors, are investigated. Their relation to a Lie group acting as a transform group on a manifold is analysed. An equivalence relation between weighted tensors is established along with a method for their generation. Several examples are given.
Another proof that the inclusion I: ι1→ι2 is 0-radonifying
Authors: Garling D. J. H.
Published: November 2008

Kwapień(2) and Schwartz (3) have shown that the inclusion mapping I: ι1→ι2 is 0-radonifying. We shall give another proof of this, using an inequality due to Khint-chine(1).
Some results in generalized homology, K-theory and bordism
Authors: Ray Nigel
Published: November 2008

This paper is designed to pave the way for the author's work on the symplectic bordism ring MSp*(12), (13). We here discuss all the notation used, and collect together all the theorems quoted therein. At the same time, I hope that some of the results presented here might be of interest in their own right. Our central theme is the study of the hurewicz map e: S*(X) → E*(X), both in general and certain specific cases.
On a summation method of S. Verblunsky
Authors: Kwee B.
Published: November 2008

Let 0 ≤ λ0 < λ1 < … < λn → ∞, and let

If as ω → ∞, we write

Suppose that

converges for σ > 0,

where τp(h) is finite for |h| < δ. If

we write
Some characterizations for the Poisson distribution starting with a power-series distribution
Authors: Shanbhag D. N. , Clark R. M.
Published: November 2008

Let X be a non-negative discrete random variable with distribution {Px} and Y be a random variable denoting the undestroyed part of the random variable X when it is subjected to a destructive process such that
A general Bieberbach inequality
Authors: Lutwak Erwin
Published: November 2008

A proof is given of a general Bieberbach inequality which has the Bieberbach, Urysohn and harmonic Urysohn inequalities as special cases. This establishes a stronger as well as more general inequality than the previously mentioned inequalities. Generalizations (as well as strengthened versions) of other inequalities previously proven by the author are also obtained.
Anticritical graphs
Authors: Harary Frank , Thomassen Carsten
Published: November 2008

The term ‘critical graph’ has been used most frequently to mean a graph such that the removal of any line decreases the chromatic number. This can be generalized to define a graph G to be ‘critical (μ)’ for an arbitrary invariant, μ if, for each line e, μ(G – e) ≠ μ(G). Analogously we may consider any line e which joins two points not already adjacent in G and define G as ‘anticritical (μ)’ if μ(G + e) ≠ μ(G). Anticritical graphs are investigated here with respect to the following invariants: chromatic number, line-chromatic number, point-connectivity, line-connectivity, point-independence number, line-independence number, diameter, radius, and cyclability number.
Banach–Mazur games, comeager sets and degrees of unsolvability
Authors: Yates C. E. M.
Published: November 2008

This paper provides a framework for studying the general theory of the upper semi-lattice of degrees of unsolvability. A less obvious and more complicated extension of this framework has been initiated in our papers (17) and (18); these are concerned not with the general theory but with the Priority Method, for which a useful formal framework has long been sought. Our purpose here is partly to provide some background for these other papers, which should help explicate their less obvious aspects; but mainly we wish to illustrate the advantages of this framework from a purely expository point of view. There is an additional advantage in that a formal framework makes it possible to pose certain precise questions which were vague before, and even on occasion to answer them.
Groups of hyperbolic crystallography
Authors: Macbeath A. M. , Hoare A. H. M.
Published: November 2008

The aim of this paper is to describe the possible structures for NEC (non-euclidean crystallographic) groups of the hyperbolic plane with non-compact quotient space. The case of compact quotient space was settled by Wilkie (6), and it has been shown by Hoare, Karrass and Solitar (3) that all subgroups of infinite index in a Wilkie group have a certain structure pattern. Their proof depends on showing that a subgroup of infinite index in a Wilkie group has a presentation satisfying certain restrictions on the occurrences of each generator. Our method here is to study the presentation of an arbitrary NEC group which is given by the classical construction of the Dirichlet region, and to prove that it satisfies the same restrictions as were found for the above subgroups of Wilkie groups.
Automorphism sequences of free nilpotent groups of class two
Authors: Dyer Joan L. , Formanek Edward
Published: November 2008

In this paper we prove that the automorphism group A(N) of a free nilpotent group N of class 2 and finite rank n is complete, except when n is 1 or 3. Equivalently, the centre of A(N) is trivial and every automorphism of A(N) is inner, provided n ≠ 1 or 3. When n = 3, A(N) has an our automorphism of order 2, so A(A(N)) is a split extension of A(N) by . In this case, A(A(N)) is complete. These results provide some evidence supporting a conjecture of Gilbert Baumslag that the sequence

becomes periodic if N is a finitely generated nilpotent group.
Classes of unipotent elements in simple algebraic groups. II
Authors: Bala P. , Carter R. W.
Published: November 2008

This paper is the second part of the work begun in reference (BC I) and the terminology and notation will be carried over from this earlier part.
Prototurbulent motions in dissipative model systems
Authors: Thyagaraja A.
Published: November 2008

Two different models of open, dissipative systems are considered. The stability and the asymptotic manifold of solutions are studied with respect to the variation of a model viscosity ν. Bifurcation theorems result from the analysis which shows the existence of a class of ‘prototurbulent’ motions. These are generalisations of the quasi-periodic asymptotic motions characteristic of Hopf's well-known mathematical model of turbulence. Given the statistics of the driving external field, it is shown how to calculate the correlation functions of the asymptotic motions (as t → ∞) for arbitrary values of the model viscosity. The systems treated are non-linear and except in special cases, cannot be solved in closed form. The relation to real systems is discussed. The effects of dimensionality on the qualitative nature of the motion is explored. It is found that for the models treated, perturbations of the external field do not affect the qualitative properties. Certain ‘mean’ quantities vary smoothly with the variation of the external field, but the details of the motion may be sensitive to the external perturbations. It is also found that even if the motion is strictly not quasiperiodic in the sense of Hopf and Landau, almost all asymptotic motions have a recurrent character under certain conditions. Some of the models also show the reverse of this behaviour, i.e. a non-dissipative, quasi-recurrent approach to ‘equilibrium’. This phenomenon can occur if the number of active degrees of freedom at a non-zero model viscosity is infinite.
On Baker's inequality for linear forms in logarithms
Authors: van der Poorten Alfred J.
Published: November 2008

Let α1, …, αn an be non-zero algebraic numbers with degrees at most d and heights respectively Al, …, An (all Aj ≥ 4) and let b1, …, bn be rational integers with absolute values at most B (≥ 4). Denote by p a prime ideal of the field and suppose that p divides the rational prime p. Write

Then it is shown that

for some effectively computable constant C > 0 depending only on n, d and p. The argument suffices to prove similarly that in the complex case, if

for any fixed determination of the logarithms, then

for some effectively computable constant C′ > 0 depending only on n and d (and he determination of the logarithms).
The case of the jumping critical point
Authors: Griffiths H. B.
Published: November 2008

The work in this paper arises from observing a simple example of a potential in the plane, as the potential varies with changes in a ‘control’ parameter. To fix ideas, we first discuss the example in intuitive terms, and the rest of the paper is a detailed technical discussion. The example aids understanding of bifurcation theory and suggests interesting generalizations. We begin by describing the example, as follows.
Oblique slot blowing into a supersonic laminar boundary layer
Authors: Riley N.
Published: November 2008

The response of a supersonic laminar boundary layer is considered when fluid is blown obliquely into it from a slot in the boundary. The velocity and length scales associated with the slot blowing are chosen in such a way that the triple-deck theory of Stewartson (9) may be exploited. It is shown that, depending upon the blowing angle, there can be either a compressive or expansive interaction between the boundary layer and free stream upstream from the slot.
The non-existence of a certain finite projective plane
Authors: Erbach D. W.
Published: November 2008

It is not known if a projective plane of order 10 can contain ovals, nor if there are any structural restrictions on the ovals it might contain. In this note we take a small step toward understanding the possible embeddings by showing that no plane can contain ovals of a certain type.
Spectra of derivations
Authors: Kyle J.
Published: November 2008

This paper follows (4) in a quantitative study of derivations. The object is to investigate the properties of the spectrum of an inner derivation. In particular, we shall be interested in obtaining a general description of the spectrum of a derivation on a W*-algebra.
The positive-definiteness of the complete symmetric functions of even order
Authors: Hunter D. B.
Published: November 2008

An important role in the classical theory of symmetric functions of a real n-tuple x = (x1, x2, …, xn) is played by the complete symmetric functions or homogeneous product sums hr defined by the generating function

(see Littlewood (5), p. 82). In an earlier paper (4) I conjectured that h2r is positive definite. The main object of the present paper is to prove this conjecture in a rather sharper form.
The number of non-homogeneous lattice points in plane subsets
Authors: Mather Michael
Published: November 2008

Let Z2 denote the integer lattice in the plane, let A be a non-singular 2 x 2 matrix and let c ∈ R2. Then G = AZ2 + c is called a grid, and its determinant det G is defined to be det A. The grid G is called a subgrid if G ⊂ Z2.
On plane deformations of incompressible isotropic elastic solids
Authors: Ogden R. W.
Published: November 2008

In two recent papers (6, 10) a new formulation of the governing equations for finite plane-strain deformations of compressible isotropic elastic solids was presented. This has been used to obtaingeneral solutions to the field equations for a certain class of strain–energy functions, and then closed-form solutions to a large class of boundary-value problems.
Rational BP operations and the Chern character
Authors: Kane Richard
Published: November 2008

This paper will study certain rational BP operations. We will work in the category of spaces having the homotopy type of CW complexes. Recall that for − 1 ≤ n ≤ ∞ there is defined in this category the homology theory BP(n)*(X). (See (7).) In particular, BP(∞)*(X) = BP*(X) is the usual Brown Peterson homology of X, BP(1)*(X) = l*(X) is the canonical direct summand of connective K-theory with Qp coefficients (Qp are the integers localized at the prime p), BP(0)*(X) = Hℤ*(X) is ordinary homology with Qp coefficients, and BP(−1)*(X) = Hℤ/p*(X) is ordinary homology with ℤp coefficients (ℤp are the integers reduced mod p). For 1 ≤ n ≤ ∞, BP(n)*(X) is a module over BP(n)*(pt) = Qp[v1, …, vn] (deg vs = 2ps − 2). The BP(n) theories are all related via canonical maps p(m, n):BP(m)*(X) → BP(n)*(X) whose performance is described in (7).
On the null spaces of linear Fredholm operators depending on several parameters
Authors: Shearer M.
Published: November 2008

Let X, Y be real Banach spaces, and let = {f(λ):λ ∈ m} be an m-parameter family of bounded linear operators from X to Y, with f(λ) depending continuously on λ. The cases m = 1 and m = 2 are considered, and conditions on are found which determine the null space of f(λ) for all λ near a given λ0 such that f(λ0): X → Y is a Fredholm operator. The results obtained are shown to be of particular interest in perturbed bifurcation theory.
On the norm of an inner derivation of a C*-algebra
Authors: Archbold R. J.
Published: November 2008

Let A be a C*-algebra with centre Z. If a ∈ A, the bounded linear mapping x → + ax – xa (x ∈ A) is called the inner derivation of A induced by a, and we denote it by D(a, A). A simple application of the triangle inequality shows that

where d (a; Z) denotes the distance from a to Z in the normed space A.
A second grammar of associators
Authors: Smith J. D. H.
Published: November 2008

1. Introduction. This paper is concerned with proving identities in commutative Moufang loops. Many such identities were derived in chapter VIII of (1) in the course of demonstrating the local nilpotence of commutative Moufang loops. The results there are regarded as constituting the ‘first grammar of associators’: the reader is assumed to have a good knowledge of them. The current paper develops additional material required for the determination in (5) of the precise nilpotence class of the free commutative Moufang loop on any given finite number of generators. It is called a ‘grammar’ because it lists formal ways in which the language of associators works, and is merely meant to serve a reader of ‘literature’ in the language such as (5). However, it may be of interest for other purposes, such as answering Manin's question ((3), Vopros 10·3; (4), problem 10·2) on the 3-rank of the free commutative Moufang loop of exponent 3. There is also the problem raised below as to whether the Triple Argument Hypothesis is a consequence of the commutative Moufang loop laws. Finally, the Möbius function in Section 9 may tempt someone to look at lattice-theoretical aspects of associators.
Consistency conditions for a finite set of projections of a function
Authors: Falconer K. J.
Published: November 2008

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.
Convex fundamental regions for Fuchsian groups: II
Authors: Nicholls P. , Zarrow R.
Published: November 2008

1.Introduction. In this article we continue the work begun in (5). We will consider only finitely generated Fuchsian groups of the first kind. LetGbe such a group acting on the unit disc Δ. A fundamental domainDforGis a connected open set with the property that any point of Δ isG-equivalent to exactly one point inDor at least one point in(the closure ofDin Δ). A fundamental domain is said to belocally finiteif there are no points in Δ where infinitely manyG-images ofDaccumulate.
Multipliers of certain convolution algebras over locally compact semigroups
Authors: Todd D. G.
Published: November 2008

In this paper we extend a result of Johnson and Lahr(3), which characterizes the multiplier algebra of L1(a, b) (the algebra of Lebesgue integrable functions on the interval of real numbers from a to b, under order convolution) to the L1 algebra of a general totally ordered semigroup. Similar work has been done in (l), but under more restrictive conditions.
Existence of periodic orbits of autonomous retarded functional differential equations
Authors: Smith Russell A.
Published: November 2008

For proving the existence of periodic orbits of autonomous ordinary differential equations, three different methods are available, namely, the Hopf bifurcation theorem, the torus principle and the Poincaré–Bendixson theorem. Until recently, the Poincaré–Bendixson theorem was applicable only to equations of the second order. However, it was extended in (13, 14) to certain equations of higher order by means of a plane projection technique. In §§ 2, 3 of the present paper this technique is adapted to prove analogues of the Poincaré–Bendixson theorem for a class of autonomous retarded functional differential equations. For such equations the orbits lie in a Banach space studied by Hale ((7), p. 43). The main hypothesis of this theorem is that the equation has a bounded semi-orbit whose ω-limit set contains no critical points. The problem of finding such a semi-orbit is solved in § 4 for a class of dissipative equations. The practical application of these results requires the construction of certain functionals which are similar to the Liapunov functionals of stability theory. In § 5, these functionals are constructed for a large class of retarded feedback control equations and explicit conditions for the existence of periodic orbits are deduced. Some practical details are illustrated in § 6 by applying the theory to certain scalar delay-differential equations.
An extension theorem for holomorphic mappings
Authors: Wood J. C.
Published: November 2008

Let X, Y be complex manifolds with smooth (C∞) boundaries ∂X, ∂Y. We give conditions which ensure that a smooth map Φ: ∂X → ∂Y has an extension to a holomorphic map X → Y. Let J denote the complex structure on X. We say that Φ satisfies the ‘tangential Cauchy-Riemann equation ∂¯bΦ = 0’if the differential dΦ restricted to the complex subspace Tp(∂X) ∩ JTp(∂X) of the tangent space Tp(∂X) is complex linear at all points p ∈ ∂X. Clearly this is a necessary condition for the existence of a holomorphic extension. A further necessary condition is that there exists no topological obstruction to extension, hence we assume that a smooth extension φ: X → Y is given and we shall look for a holomorphic map f: X → Y with the same boundary values.
On real simple singularities
Authors: Bruce J. W. , Giblin P. J.
Published: November 2008

In (3), Cor. 2 to Th. 1, the first author proved that in the complex jet space Jk (n, 1) the orbits of simple singularities form canonical strata. By definition the canonical stratification is contact invariant so the proof consisted essentially of two steps: firstly, any two functions in the same canonical stratum are C0-equivalent by right-left changes of coordinates, and secondly, the right codimension (and hence the Milnor number) of an isolated complex singularity is a topological invariant.
Representation of an integer as a prime plus a product of two small factors
Authors: Heath-Brown D. R.
Published: November 2008

In this paper we consider the following problem, which seems to have been brought to light fairly recently by M. Car. Can every sufficiently large integer n be expressed as n = p + ab with p prime and 1 ≤ a, b ≤ n½? Certainly one should expect this to be possible. Taking b = 1, for example, p will be restricted to the range n − n½ ≤ p < n, and this interval is conjectured to contain a prime, for large enough n. Alternatively, providing that n is not a square, we expect n = p + a2 to be solvable for sufficiently large n. However, although the statement that n = p + ab, with a, b ≤ n½, is far weaker than either of the aforementioned conjectures, it is nevertheless rather tricky to show that solutions must in fact exist.
On Moore geometries. II
Authors: Damerell R. M.
Published: November 2008

In a previous paper, M. Georgiacodis and the author proved that there are no non-trivial Moore geometries of diameter greater than 12. Here, it is proved that there are none of diameter greater than 4.
Convolution mixtures of infinitely divisible distributions
Authors: Kent John T.
Published: November 2008

A great deal is known about infinitely divisible distributions on [0, ∞). In this paper, a simple mapping which can be used to extend this information to more general convolution algebras is examined. Some examples are given where this approach has proved fruitful.
Stability of universal unfoldings: a correction
Authors: Magnus R. J.
Published: November 2008

The author is grateful to Les Lander for pointing out an error in the stability section of (1). In fact Theorems 5 and 7 are incorrect. Recently Arkeryd proved a stability theorem for the infinite-dimensional case in the context of the imperfect bifurcation theory of Golubitsky and Schaeffer(3). In his result finitely many derivatives are controlled, the number depending on the codimension of the singularity unfolded. In this note we shall present a stability theorem involving the determinacy of the singularity. The context is the parameter-free potential case, that is, catastrophe theory. The proof is without recourse to the finite-dimensional results, and the theorem concludes an account of a part of singularity theory in Banach spaces, in which the author has tried to use as little as possible of the finite-dimensional theory (1, 2).
Normal bundles in flag bundles
Authors: Ilori Samuel A.
Published: November 2008

Let i: Y ↪ X be an inclusion map of non-singular irreducible algebraic quasi-projective varieties defined over an algebraically closed field. Let E be an algebraic vector bundle over X and H be a sub-bundle of the induced bundle, i*E. If j:F(H) ↪ F(E) is the corresponding inclusion map of (incomplete) flag bundles, then we derive the normal bundle N(F(H), F(E)) in terms of the bundles H and E, the tangent bundles of Y and X as well as the tautological bundles over F(H).
The structure of type I JBW-algebras
Authors: Stagey P. J.
Published: November 2008

A complete normed real Jordan algebra A with 1 is said to be a JB-algebra if and only if, for all a, b ∈ A,

where o denotes the Jordan product on A. If A is a Banach dual space then A is known as a JBW-algebra. In this case, by Theorem 3·9 of (7),

where Aex is an exceptional Jordan algebra and Asp is Jordan isomorphic to a JW-algebra (i.e. a weakly closed Jordan algebra of self-adjoint operators on a complex Hilbert space). Theorem 3–9 of (7) gives a description of Aex and so in this paper attention will be restricted to JW-algebras. Such an algebra A is said to be of type I if there exists an idempotent e in A with central cover equal to 1 such that e is abelian i.e. such that eAe is commutative as a set of operators. (See § 7 of (11) for further details.) A type I algebra A is said to be of type In, where n is some cardinal number, if there exists an orthogonal family {eα: α ∈ Λ} of n abelian projections in A such that Σαeα = 1 and such that, for each α ≠ β, there exists a symmetry sαβ in A with eβ = sαβeαsαβ. (A symmetry is an element sαβ of A such that ). By theorems 15 and 16 of (11) any type I JW-algebra A can be uniquely decomposed into a direct sum of type In(α) JW-algebras for suitable cardinal numbers n(α). Thus in order to investigate the structure of type I JW-algebras it is sufficient to consider type In JW-algebras.
Quadratic complexes. II
Authors: Newstead P. E.
Published: November 2008

A quadratic complex Q is the set of lines in 3-dimensional projective space 3 given by a single non-trivial quadratic equation in the Plücker coordinates. The lines of the complex which pass through a fixed point of 3 are, in general, the generators of a quadric cone; this cone degenerates for the points of a sub variety K of 3. Thus one can associate with Q a fibration with base 3 - K and fibre isomorphic to 1, and ask whether this fibration is associated with an algebraic vector bundle of rank 2. When the base field is and Q is non-singular, the answer is negative; this was proved some years ago by Narasimhan and Ramanan ((8), proposition 8·1), and has the consequence that there is no universal family of stable vector bundles of rank 2 and degree 0 over a curve of genus 2.
Subrings of the first neighbourhood ring: II
Authors: Kirby D.
Published: November 2008

In (1) and (2) we studied a lattice of extension rings associated with a commutative ring R with identity. When R, M is a one-dimensional Cohen-Macaulay local ring the elements of are just those integral extensions of R contained in the total quotient ring T(R) and such that lengthR(S/R) is finite. Experiments with local rings of singular points on algebraic curves indicate that only the simplest singularities give rise to finite lattices. So the problem arises as to which local rings R give rise to which finite lattices. In later papers this problem will be investigated in detail, at least when R is of low embedding dimension. The purpose of the present note is to establish some general results which indicate the size of the problem.
Isomorphic embeddings of l1(Г) into subspaces of C(Ω)
Authors: Argyros Spiros A. , Tsarpalias Athanasios
Published: November 2008

Introduction. The embeddability of l1(Γ), for uncountable sets Γ, into subspaces of Banach spaces of the form C(Ω) was investigated first by Hagler in (6) and subsequently by Haydon in (7), (8) and Argyros and Negrepontis in (1). An important role in the development of the above subject is played by a lemma of Rosenthal (12) that translates the functional analytic problem of finding a family {fξ: ξ Γ} of elements of C(Ω) equivalent to the usual basis of l1(Γ) into the problem of the existence of an independent family {(Aξ, Bξ,): ξ є Γ} of closed subsets.
Finding fundamental units in cubic fields
Authors: Cusick T. W.
Published: November 2008

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.
Holomorphic sectional curvatures of indefinite complex Grassmann manifolds
Authors: Montiel Sebastian , Romero Alfonso
Published: November 2008

In (2), indefinite Kählerian manifolds have been examined from the point of view of holomorphic sectional curvature. Examples in (2) show that the analogue of Kulkarni's theorem (see (4), p. 173) for the holomorphic sectional curvature is false and the best possible result in the direction is:

Theorem 1 (known, (2)). Let M be a connected indefinite Kählerian manifold with complex dimension n ≥ 2. If the holomorphic sectional curvature of M is bounded above and bounded below, then M is an indefinite complex space form.
On intrinsic eigenstates in plasticity with generalized variables
Authors: Hill R.
Published: November 2008

The classical specification of elastic/plastic or rigid/plastic response in metals is reformulated in generalized variables. The allowed measures of strain are sets of any geometric magnitudes that jointly determine the shape of a material element; the allowed measures of stress are generated by work-conjugacy. The choice of variables affects the parameters and qualitative features of the constitutive framework; these dependences are made explicit by concise formulae of transformation.

Eigenstates intrinsic to the material are considered wherein the strain can change while the conjugate stress is either stationary or coupled differentially with the strain. Such configurations are associated with incipient branching of the strain response to a prescribed variation of the conjugate stress. Sensitivity to the loading environment is evaluated for both elastic/plastic and rigid/plastic materials. This synoptic approach to eigenproblems is illustrated in the context of materials testing.
Defective big Cohen-Macaulay modules
Authors: Brown M. L.
Published: November 2008

Let R be a noetherian local ring and x = x1, …, xn a system of parameters for R. If R is an equicharacteristic local ring then Hochster(3) proved there is a big Cohen-Macaulay module with respect to x, i.e. an R-module M, not necessarily noetherian, with x1, …, xn a regular sequence on M and M/(x) M ≠ 0. Such modules are important for the study of the homological conjectures in commutative algebra(3). Nevertheless, for mixed characteristic local rings virtually nothing is known about their existence.
Resolving homologies in BPL and PL characteristic numbers
Authors: Buoncristiano Sandro , Hacon Derek
Published: November 2008

In (1) it was shown that PL characteristic numbers determine (unoriented) PL bordism. In §§ 1 and 2 of this paper we use D. Sullivan's obstruction theory (9), (5) to provide some geometric insight into this result, as follows. We take an unoriented PL manifold M and let νM: M → BPL classify its stable normal bundle, and assume that all its PL characteristic numbers vanish, i.e. that (νM)* [M] = 0, where [M] is the fundamental class of M and . To show that M bounds, we first ‘resolve the homology to zero’ given by (νM)* [M] = 0. This is done by defining obstructions in whose vanishing ensures that the homology can be ‘resolved’, i.e. can be replaced by a bordism F: W → BPL between νM and a constant map, g: N → BPL, say. Then it is shown that these obstructions all vanish in this case.
Localization of stable homotopy rings
Authors: Arthan R. D.
Published: November 2008

V. P. Snaith has shown that a number of interesting homology theories (e.g. K-theory, complex cobordism) arise when one localizes the stable homotopy type of certain H-spaces (BU(1), BU, etc.). In this paper we sketch a construction which, given an H-space X and , produces a ring spectrum IΣ−nX which lies between Xand its localization with respect to b, and in many cases has a tractable structure. We apply this when X is an Eilenberg-MacLane space K(ℤ/, n) and is a generator (p = 0 or a prime), and get complete results on the localizations in this case. The results confirm one's suspicions that localization is a fairly drastic manoeuvre, cutting down the large and complicated ring to something more or less trivial, except when p = 0 and n = 2, in which case the rich geometry of K(ℤ, 2) = BU(1) gives complex K-theory.
Indecomposable knots and concordance
Authors: Bayer-Fluckiger Eva , Stoltzfus Neal W.
Published: November 2008

R. C. Kirby and W. B. R. Lickorish have proved (cf. (4)) that any classical knot is concordant to an indecomposable knot. In the present note we show that this statement is also true for higher dimensional knots: more precisely, for any higher-dimensional knot K there exist infinitely many non-isotopic indecomposable simple knots which are concordant to K. This, together with the result of Kirby and Lickorish, gives a complete solution of problem 13 of (1).