bluuebunny/crossref_metadata_2025 huggingface dataset:
On the spectral theorem for normal operators
Authors: Douglas R. G. , Pearcy Carl
Published: November 2008

It has been known for some time that one can construct a proof of the spectral theorem for a normal operator on a Hilbert space by applying the Gelfand representation theorem to the Abelian von Neumann algebra generated by the normal operator, and using the fact that the maximal ideal space of an Abelian von Neumann algebra is extremely disconnected. This, in fact, is the spirit of the monograph (8). On the other hand, it is difficult to find in print accounts of the spectral theorem from this viewpoint and, in particular, the treatment in (8) uses a considerable amount of measure theory and does not have the proof of the spectral theorem as its main objective.
A second anticommutant theorem for symmetric ternary algebras
Authors: Joseph A.
Published: November 2008

It is pointed out that many axiomatic systems include a condition of closure under binary composition. In the present development a study is made of the consequences of the weaker requirement of closure under ternary, or three-fold composition. As a preliminary step in the investigation, subsets of groups closed under n-fold multiplication are examined and characterized abstractly. Attention is then focused upon ternary algebras defined as linear subspaces of ℬ (ℋ) (bounded operators on some Hilbert space ℋ) closed under three-fold multiplication. These are shown to satisfy, under suitable conditions, a secondanticommutanttheorem, the analogue of the second commutant theorem which holds for (binary) algebras. It is a result which provides a useful technique for studying such ternary algebras, the structure of which are examined in some detail.
Summability methods which include the Riesz typical means. I
Authors: Russell Dennis C.
Published: November 2008

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.
Singularities of two-dimensional exterior solutions of the Helmholtz equation
Authors: Millar R. F.
Published: November 2008

A technique for locating possible singularities of two-dimensional ex-terior harmonic functions was discussed in a previous paper. In the present work, the method is generalized to exterior solutions of the Helmholtz equation. Although the procedure deviates in some of its details from the earlier exposition, the conclusions are similar. In particular, it is verified that solutions of the Laplace and Helmholtz equations that satisfy the same Dirichlet boundary condition on the same boundary, possess the same convex hull of singularities. The possibility of extending the method to more general equations is raised.
A note on residually finite groups
Authors: Knight J. T.
Published: November 2008

We shall prove the following Theorem. There are exactlyisomorphism classes of residually finite groups with every proper quotient finite.
A synthetic proof of de Longchamps' chain
Authors: Longuet-Higgins M. S.
Published: November 2008

A famous chain of theorems, due originally to de Longchamps (l) and afterwards rediscovered by Pesci(2), Morley(3) and Grace(4), goes as follows:

(1) Given four lines in a plane, the four circumcentres O3 of the triangles formed by omitting each one of the lines in turn lie all on the same circle C4 with centre O4 say.

(2)Given five lines in a plane, the centres O4 of the five circles C4 obtained by omitting each of the five lines in turn lie all on the same circle C6 with centre O5 say.

And in general

(3) Given (n+1) lines in a plane, the (n+1) centres On of the circles Cn+1 formed by omitting each of the lines in turn lie all on the same circle Cn+1 with centre On+1.
Stratified laminar flow of two immiscible fluids
Authors: Packham B. A. , Shall R.
Published: November 2008

We consider the steady co-current flow of two immiscible viscous liquids in a horizontal pipe, the fluid interface being ripple-free and plane. It is shown that if the cross-section of the duct is symmetric with respect to the interface, the velocity distribution may be expressed in terms of two separate pipe-flow solutions. One corresponds to the flow of a single fluid occupying the whole of the pipe, and the second to a similar flow in a pipe whose cross-section coincides with that of the region occupied by one fluid in the two-phase motion. Flow rates and other quantities of engineering interest are evaluated, and several particular flows are discussed.
The product of the distributions x−r and δ(r−1)(x)
Authors: Fisher B.
Published: November 2008

The product of two distributions f and g on the open interval (a, b), where −∞ ≤ a < b ∞, was defined in (1) as the limit of the sequence {fn·gn} provided this sequence is regular in (a, b), where

for n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:
The fractional parts of an additive form
Authors: Cook R. J.
Published: November 2008

Heilbronn (6) proved that for every ε ≥ 0 and N ≥ 1 and every real θ there is an integer x such that

,

where C(ε) depends only on ε and ∥α∥ is the difference between α and the nearest integer, taken positively. Danicic(1) obtained an analogous result for the fractional parts of nkθ, the proof of this is more readily accessible in Davenport(4). Danicic(2) also obtained an estimate for the fractional parts of a real quadratic form in n variables, and in order to extend this result to forms of higher degree it is desirable to first obtain results for additive forms.
Some extremum principles for magnetohydrodynamic flow in conducting pipes
Authors: Smith P.
Published: November 2008

Extremum principles are obtained for the laminar flow of a conducting liquid in a straight pipe of arbitrary cross-section with conducting walls of arbitrary thickness, which is subject to a uniform applied magnetic field. These extrema provide upper and lower bounds for the mass-flow rate in the pipe. These bounds are particularly useful in the construction of asymptotic estimates for the flow rate at high Hartmann number. The method is illustrated by its application to the pipe of circular cross-section and to pipes with thin conducting walls.
Extremum principle for a nonlinear problem in magneto-elasticity
Authors: Anderson N. , Arthurs A. M. , Hall R. R.
Published: November 2008

A minimum principle associated with a class of magneto-elastic boundary-value problems is presented. The principle depends on a result on integral inequalities which appears to be new. An accurate variational solution is obtained for an illustrative one-dimensional problem.
Peeling and conservation laws in the Born-Infeld theory of electromagnetism
Authors: Porter John R.
Published: November 2008

1. Introduction Born and Infeld originally proposed their non-linear theory of electromagnetism (1) as one derivable from a Lorentz invariant Lagrangian and, in contradistinction to the usual Maxwell theory, possessing solutions of the ‘point charge’ type without singularities at the position of the charge. Subsequently, examination of the theory by means of iterative methods (2) has demonstrated the self-scattering of outgoing radiation and the existence of wave tails. In the present paper it is shown that the theory, under suitable asymptotic conditions, gives rise to peeling of the type exhibited by Maxwell's theory as well as by the asymptotically flat space-times of general relativity (3, 4, 5). It is also shown that certain conserved quantities exist paralleling those exhibited in the above theories. The Maxwell theory exhibits a sequence of such conserved quantities, the truncation of which is examined in detail for the Born-Infeld theory.
Uniform summability and Töplitz bases
Authors: Kurtz J. C.
Published: November 2008

If the series Σαn is convergent, it follows that Σαnzn is uniformly convergent in a Stolz angle at z = 1 ((7), p. 229). It was shown in (3) that for the Ces methods (C, α) with α > – 1, and for a certain class of general Nörlund means (containing (C, α) for 0 < α ≤ 1), summability of Σαn implies uniform summability in a Stolz angle at z = 1. In section 2 we prove this theorem for a class of methods which satisfy a mean value property ((6), p. 31), and for a wider class of Nörlund methods which includes (C, α) for all α > 0. An analog is also proved for absolute summability.
Positive definite functions on spheres
Authors: Bingham N. H.
Published: November 2008

Positive definite functions on metric spaces were considered by Schoenberg (26). We write σkfor the unit hypersphere in (k+ 1)-space; then σkis a metric space under geodesic distance. The functions which are positive definite (p.d.) on σkwere characterized by Schoenberg (27), who also obtained a necessary condition for a function to be p.d. on the it sphere σ∞ in Hilbert space. We extend this result by showing that Schoenberg's necessary condition for a function to be p.d. on σ∞ is also sufficient.
The variational derivative of degenerate Lagrange densities
Authors: Pavelle Richard
Published: November 2008

The variational derivative of Lagrange densities which are functions of the metric tensor and its first and second derivatives is considered. This tensor is generally of fourth order in the derivatives of the metric tensor. If derivatives of any order are not present in the variational derivative then the Lagrange density is said to be degenerate in these derivatives. An explicit expression for the variational derivative of Lagrange densities which are degenerate in third and/or fourth derivatives is displayed in tensor form.
Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods
Authors: Dumir Vishwa Chander , Khassa Dharam Singh
Published: November 2008

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined by

where V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured that

where ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.
The (11, 3)-arcs of the Galois plane of order 5
Authors: Bramwell D. L. , Wilson B. J.
Published: November 2008

1. It was shown by Barlotti(1) that the number, k, of points on a (k, n)-arc in a Galois plane S2, q, of order q, where n and q are coprime, satisfies

Regular arcs, in which all the points are of the same type have been studied by Basile and Brutti(2) and, for n = 3 by d'Orgeval(4). By means of an electronic computer Lunelli and Sce(5) have enumerated many arcs in Galois planes of low order. The object of this note is to show how the (11, 3)-arcs of S2, 5, none of which is regular, may be described using only geometrical properties.
On operator equations and variational principles
Authors: Arthurs A. M.
Published: November 2008

A necessary and sufficient condition for operator equations Tϕ = M(u, ϕ), T*u = N(u, ϕ) to arise from a variational principle is presented.
The asymptotic analysis of canonical problems in high-frequency scattering theory
Authors: Boyd W. G. C.
Published: November 2008

This is the second part of a two-part paper in which a systematic method is advanced for treating separation of variables problems asymptotically as the frequency becomes large. The method assumes an integral representation in which the contour of integration is the real axis. The contour is then deformed in the neighbourhood of the real axis to derive rigorous asymptotic expansions of the field. In this paper the method is applied to scattering in homogeneous media by the circular and parabolic cylinders.
Completely stretched sets
Authors: Lillington J. N.
Published: November 2008

Introduction. All the sets X considered in this paper will be compact convex subsets of the Euclidean plane R. We shall require the following definitions. For x, y, z ∈ R let L(x, y, x) denote the minimum of all the lengths of all connected sets containing x, y and z. Define the stretch of x with respect to X to be

and the stretch of X to be
On a separation theorem of Rogers
Authors: Davies Roy O.
Published: November 2008

In (1) (where references and terminology may be found), C. A. Rogers has proved the following surprising theorem.

Letbe a class of sets, all Souslin-in a topological space X. If A is a Souslin-set and B is an analytic set that does not meet A, then there is a Borelian-set C with A ⊂ C and C ∩ B = Ø
Correction to a paper on wreath products
Authors: Meldrum J. D. P.
Published: November 2008

Some time ago, Dr I. B. S. Passi pointed out to me the connexion between the structure of the terms of the α-central series of a wreath product and that of polynomial groups which he had defined and studied. In discussing this problem with him, we looked closely at a paper (1) which I had written on nilpotent wreath products. In the process of this, I found a mistake in Theorem 3·7, the main result of (1). The results in (2) depend onthis theorem. Fortunately, it is possible to obtain a correct version of Theorem 3·7 and to modify the proof in (2) which depends on this theorem in such a way that all the results of (2) remain valid.
Crystallography in spaces of arbitrary dimension
Authors: Schwarzenberger R. L. E.
Published: November 2008

The detailed study of n-dimensional crystallography appears to have been initiated by Hermann but unfortunately only a small fragment of his work was ever published(4). Recently, interest in the subject has been revived by the work of Bülow, Neubüser and Wondratschek(2), who not only obtained definitive results on 4-dimensional lattices and space groups but also proposed general definitions and procedures for the n-dimensional case. In particular they introduced the concept of crystal family and so resolved satisfactorily the controversy on whether there are six or seven 3-dimensional crystal systems (see(1)). Their point of view is arithmetic, that is, they use a coordinate system adapted to the particular lattice under consideration and represent elements of its symmetry group by integer matrices.
Some cyclic and other inequalities. IV
Authors: Diananda P. H.
Published: November 2008

For t > 0, let

where xn+r = xr ≥ 0 and xr+1 + xr+2 > 0 for each r.
Repeated Riemann integrals
Authors: Rennie B. C.
Published: November 2008

Let f(x, y) be a bounded real function on the unit square. Consider the Riemann integrals (if they exist)
Differential games with optional stopping
Authors: Kalton N. J.
Published: November 2008

Consider a differential game of survival governed by the differential equation

in , with pay-off

where tF is the entry time of the trajectory (t, x(t)) into a given terminal set F. Under suitable conditions on f, g, h and the terminal set F, it was shown in (3) that the question of existence of value of such a game can be approached by considering a certain pair of partial differential equations called the Isaacs-Bellman equations.
Bounds for surface solitary waves
Authors: Keady G. , Pritchard W. G.
Published: November 2008

In these notes we give proofs of some properties of surface solitary waves. Assuming the existence of solitary-wave solutions to the nonlinear boundary-value problem (P) defined below, it is shown (i) that the wave is a wave of elevation alone, and (ii) that at large distances it is asymptotic to a uniformsupercriticalstream (i.e. the Froude number, wherecis the speed of the stream,his its depth, andgis the gravity constant).

We also deduce a number of inequalities relatingF2to a/h, whereais the maximum displacement of the free surface from its value at infinity. In particular, it is shown for the wave of greatest height that 1·480 <F2< 2.
Cesàro iterations of Hausdorff matrices
Authors: Kuttner B.
Published: November 2008

1. Jurkat and Peyerimhoff(3) (or see (4), pages 50–56) have given a definition of a function, f(A), of an infinite matrix A = (ank). In particular, they considered Aα, where a is any real constant. This is defined wherever A is normal; that is to say, when ank = 0 for k > n, and when, for all n, ann ±0. They also defined the ‘Cesàro iteration’ Aα. This is obtained by defining a matrix B = (bnn) by

(so that every diagonal element of B is 1), forming Bα, and then dividing each row by the row sum; that is to say, if Bα = (b(α)nk), we take

where

The definition requires, of course, that ≠ 0.
On the relation of complex cobordism to connective K-theory
Authors: Johnson David Copeland , Smith Larry
Published: November 2008

The objective of this note is to complete in one essential way the study undertaken in (2) of the relation between complex bordism and the connective k-homology theory. Specifically, let us denote by MU*( ) the generalized homology theory associated to the Thorn spectrum MU(6), and by k*( ) the generalized homology theory associated to the connective bu spectrum (2, 4). Recall that
A theorem on bounded synthesis
Authors: Galanis E.
Published: November 2008

LetGbe a locally compact Abelian group.

DEFINITION 1. A compact subset K ⊂ G is called Kroneclcer set if for every continuous function f on K of modulus identically one (|f(x)| = 1, ∀x ∈ K) and for every ε 0 there exists x ∈ Ĝ such that
Some forms of the closed graph theorem
Authors: Kalton N. J.
Published: November 2008

In this paper we shall establish some forms of the closed graph theorem for locally convex spaces, using the approach of Pták(17). Our interest is in classifying pairs of locally convex spaces (E, F) which have the property that every closed graph linear mapping T: E → F is continuous; if (E, F) has this property then we shall say that (E, F) is in the class ℒ if is a particular class of locally convex spaces then ℒ() is the class of all E such that (E, F)∈ℒ for all F ∈ .
On the tensor system of a semisimple Lie algebra
Authors: Murphy Timothy
Published: November 2008

From the tensorial point of view, the essential characteristic of a semisimple lie algebra A is the non-singularity of the Killing form, since this enables one to construct a system of Cartesian tensors over A. That system is the subject of this paper.
Extensions of the Hájek–Rényi inequality to moments of higher order
Authors: Dunnage J. E. A.
Published: November 2008

1. Introduction. Bernoulli trials. Consider a sequence of Bernoulli trials. Let p, assumed to satisfy 0
 0,

as n→∞, and it is clearly of great interest to know quantitatively how this probability depends upon n and ∈.
A Frattini-like subgroup
Authors: Tomkinson M. J.
Published: November 2008

The Frattini subgroup φ(G) of a group G is the intersection of G and all its maximal subgroups. The following results for finite groups are well known:

THEOREM A0. If G is a finite group, then the following three conditions are equivalent:

(i) G is nilpotent,

(ii) G/φ(G) is nilpotent,

(iii) φ(G) ≥ G′.
A quadratic parabolic group
Authors: Riley Robert
Published: November 2008

When k is a 2-bridge knot with group πK, there are parabolic representations (p-reps) θ: πK → PSL(): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group πKθ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 41, which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p-reps θ for this knot are isomorphisms of πK on πKθ. Furthermore, the universal covering space of S3\k can be realized as Poincaré's upper half space 3, and πKθ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S3\k. The group πKθ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.
On twin ‘primes’ and gaps between successive ‘primes’ for the Hawkins random sieve
Authors: Neudecker Werner
Published: November 2008

1. In this paper we shall use the methods of (6) to give a short proof of Wunderlich's result (7, 8) on the distribution of twin ‘primes’ for the Hawkins random sieve. We shall also obtain a sharp estimate on the gaps between successive ‘primes’.
A proof of a theorem of Valentine
Authors: Eggleston H. G.
Published: November 2008

An interesting extension of the idea of convexity has been introduced by Valentine (2). He considered a plane set X such that for any three points a, b, c in X at least one of the segments [a, b], [b, c], [c, a] is contained in X, and showed that such a set can be regarded as the union of at most three convex sets. See also (3). The five pointed star (Fig. 1) is an example that shows that X may not be the union of fewer than three convex sets.
A proof of Hartman's theorem on compact Hankel operators
Authors: Bonsall F. F. , Power S. C.
Published: November 2008

Let U be a unilateral shift of arbitrary (perhaps uncountable) multiplicity on a Hilbert space. Following Rosenblum (5), an operator A is said to be a Hankel operator relative to U if

Hartman (2) has characterized the compact Hankel operators relative to the unilateral shift of multiplicity one as the Hankel operators with symbol in H∞ + C(T). Using the usual function space model for representing the unilateral shift, Page ((4), theorem 10) has extended Hartman's theorem to unilateral shifts of countable multiplicity. We give a model-free proof of Hartman's theorem which applies to shifts of arbitrary multiplicity. The proof turns on the observation that a compact operator acts compactly on a certain algebra of operators.
On the comparison of measure functions
Authors: Hawkes John
Published: November 2008

Suppose that f and g are measure functions and that µf and µg are the corresponding Hausdorff measures. We are interested in how relationships between f and g are reflected in relationships between µf and µg and vice versa. For example, if

Rogers ((8), p. 80) has shown that there exists a metric space Ω which has subsets A and B with µf(A) = µg(B) = 0 and, µf(B) = µg(A) = ∞. We are more interested in what happens when the metric space in question is the real line. To get the above result we need to assume that both f and g are starshaped. This is just about a best possible result since if we assume only that f is a power of t the conclusion fails. Further-more we show that there exist measure functions f and g satisfying (1) such that µf and, µg are identical. We also consider related questions: when are, µf and, µg equivalent measures and when are they identical?

The proofs depend on the construction in Theorem 3.1 of a Cantor set having pre-scribed measure properties. Although the construction is not difficult it turns out to be quite useful to appeal to the existence of such a set. We illustrate this remark by giving a short proof of a known theorem on cartesian product sets. We also make use of these ideas in section 5 where we discuss some properties of a class of net meastues and give a partial answer to a problem posed by Billingsley.
The reflection of short gravity waves on a non-uniform current
Authors: Smith Ronald
Published: November 2008

A uniform asymptotic solution is obtained which describes the propagation near an isolated caustic of water waves on a slow irrotational current. Unlike earlier work there is no restriction to straight caustics or to steady currents.
General asymptotic distributions for additive arithmetic functions
Authors: Elliott P. D. T. A.
Published: November 2008

1. Let f(n) be a real-valued additive arithmetic function. Let α(x) and β(x) > 0 be real valued functions, defined for x ≥ 2. Define the frequencies
Two compounds of antiprisms in R4
Authors: Smith Anthony
Published: November 2008

A polygon is said to be uniform if and only if it is connected and regular. An n-dimensional polytope (n > 2) is called uniform if it has uniform cells and possesses a symmetry group transitive on the vertices. A polytope is either connected or a compound. If it is a compound we also require the components to be isomorphic and to have no cells of dimension ≥ n − 1 in common. The connected 3-dimensional uniform polytopes have been investigated by Coxeter, Longuet-Higgins and Miller (1). The convex 4-dimensional polytopes have been investigated by Conway and Guy (2). Compound polytopes have been less extensively described than connected polytopes, but see, for example, (3), (4). Based on 2- and 3-dimensional experience, one might navely conjecture: ‘given a symmetry group , a uniform polytope P and an integer n, if a uniform compound polytope with symmetry group and consisting of n copies of P exists, then it is either unique or else a member of a one-parameter family’.
Orthonormal systems in Banach spaces and their applications
Authors: Kalton N. J. , Wood G. V.
Published: November 2008

By an orthonormal system in a general complex Banach space, we mean a collection {eα: α ∈ } it vectors such that, for each α, there is an hermitian (in the numerical range sense, see (4)) projection Pα whose range is lin (eα) and such that PαPβ = 0, if α ≠ β. This paper is devoted to the study of orthonormal systems in general Banach spaces, and their applications to problems of characterizing isometries and hermitian operators.
Iterated torus knots
Authors: Hacon D.
Published: November 2008

Let k be a (smooth) knot in S3, → X the infinite cyclic covering of X = S3 – k corresponding to a generator of and let t generate the (infinite cyclic) group of covering transformations of . Various authors (1), (2), have shown that, for iterated torus knots, is of finite order (i.e. some power of t* is the identity).
A Fitting class construction
Authors: Hawkes Trevor
Published: November 2008

This note is devoted to the construction of a 〈Q, EΦ〉-closed Fischer class which is not a formation. It provides a negative answer to the Questions 3–7 raised by Cossey in (1). All groups considered are finite and soluble.
A compact space having the cardinality of the continuum with no convergent sequences
Authors: Fedorčuk V. V.
Published: November 2008

1. Introduction. All spaces in this paper are Hausdorff. We recall that a space X is sequentially compact, if every countable subset of X contains a convergent sequence. Let us consider the three statements:

(1) Every compact space of cardinality ≤ ʗ contains a point of countable character.

(2) Every compact space of cardinality ≤ ʗ is sequentially compact.

(3) Every infinite compact space of cardinality ≤ ʗ contains a convergent sequence.
Irreducible representations of finitely generated nilpotent groups
Authors: Segal Daniel
Published: November 2008

1. Introduction. It is well known that every finite-dimensional irreducible representation of a nilpotent group over an algebraically closed field is monomial, that is induced from a 1-dimensional representation of some subgroup. However, even a finitely generated nilpotent group in general has infinite-dimensional irreducible representations, and as a first step towards an understanding of these one wants to discover whether they too are necessarily monomial. The main point of this note is to show how far they can fail to be so.
On the spatial oscillations of a horizontally heated rotating fluid
Authors: Daniels P. G. , Stewartson K.
Published: November 2008

A laboratory model simulating aspects of oceanic or atmospheric circulationconsists of a rotating fluid annulus which is subject to a temperature gradient on the lower surface and is insulated on the inner and outer curved walls and on the top surface. A theoretical study of the resulting steady axi-symmetric fluid motion in the limit of small Ekman number, E, by Daniels (2) indicated that if an appropriate convective parameter is sufficiently large, oscillatory solutions are generated in the layer which is located on the hotter of the two side-walls, which do not decay into the geostrophic interior. The present paper resolves this difficulty by using the method of multiple scales to trace the development of the oscillatory solutions in the interior of the fluid. This leads to a consistent overall flow pattern throughout the annulus, except at an infinite set of discrete values of the convective parameter at which resonance is predicted.
Families of sets whose pairwise intersections have prescribed cardinals or order types Corrigenda
Authors: Erdös P. , Milner E. C. , Rado R.
Published: November 2008

(i) J. Baumgartner has kindly drawn our attention to the fact that Theorem 2 as stated in (1) is false. A counter example is the case in which m = ℵ2; n = ℵ1; p = ℵ0. For by reference (3) of the paper (1) there is an almost disjoint family (Aγ: γ < ω1) of infinite subsets of ω̲ Put Aν = ω̲ for ω1 ≤ ν < ω2. Then, contrary to the assertion of that theorem, all conditions of Theorem 2 are satisfied. However, Theorem 2 becomes correct if the hypothesis

is strengthened to

In fact, Baumgartner has proved the desired conclusion under the weaker hypothesis
The Riemann integrability of f′2 + g′2, f′, g′
Authors: Rennie B. C.
Published: November 2008

Suppose that, for all x in an interval [a, b], the functions f, g have finite derivatives. If, further, f′ and g′ are Riemann integrable, so are f′2 and g′2 and hence f′2 + g′2. Is it true that, conversely, the R-integrability of f′2 + g′2 implies that of f′ and g′? (Marcus, (2)). The answer is No, and a counter-example is given in this note. It is an elaboration of Volterra's classical construction of a derivative which is not R-integrable (3).
On representation of Poisson mixtures as Poisson sums and a characterization of the gamma distribution
Authors: Godambe A. V.
Published: November 2008

A necessary and sufficient condition for a Poisson mixture with an exponential type mixing distribution to be equivalently represented as a Poisson sum is obtained. The problem of deriving a similar condition under any mixing distribution on (0, ∞) is discussed. Finally, a characterization of the gamma distribution is obtained.
Littlewood–Paley theorems for sum and difference sets
Authors: Gaudry Garth I.
Published: November 2008

Let α be a positive integer, andEl, …,EαHadamard sets of positive integers. It is shown thatE = E1+ … +Eαdetermines a Littlewood–Paley decomposition ofZ.

Suppose thatis a Hadamard set of positive integers such thatnj+1/nj≥ 2 for allj. Let α be a positive integer, and

We show thatF(α) also determines a Littlewood-Paley decomposition of Z.
Multiple points for symmetric Lévy processes
Authors: Hawkes John
Published: November 2008

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies
On Adams' splitting of K-theory and complex cobordism
Authors: Hansen Idar
Published: November 2008

In (2) Adams gave a splitting of complex K-theory with coefficients in the ring R(d) of rationals a/b such that b contains no prime p with p ≡ 1 (mod d). The splitting comes from a complete set of projection operators on K(X; R(d)). One of the operators is then used to obtain a stable, multiplicative idempotent ε on complex cobordism with coefficients in the same ring R(d) and hence a splitting of the representing spectrum MUR(d). However, the idempotent is initially defined over the rational numbers and work is needed to show that it actually gives an operation on MUR(d). Since Novikov (6) has shown that multiplicative cobordism operations are distinguished by their values on the generator ω ∈ MU2CP∞, it is natural to seek an explicit formula for ε(ω) which wi11 show that ε gives an operation over the subring R(d).
Riesz spaces with the order-continuity property. II
Authors: Fremlin D. H.
Published: November 2008

I continue to investigate Riesz spaces E with the property that every positive linear map from E to an Archimedean Riesz space is sequentially order-continuous. In order to give a criterion for the product of such spaces to be another, we are forced to investigate their internal structure, and to develop an ordinal hierarchy of such spaces.
A note on the fundamental group of a manifold of negative curvature
Authors: Wood J. C.
Published: November 2008

Let Y be a compact connected C∞ Riemannian manifold with negative sectional curvatures. Let G be a non-trivial subgroup of the fundamental group π1(Y). G is known to be cyclic if it is abelian (Preissmann (6)) or contains a subnormal abelian (hence cyclic) subgroup (Yau(9)). These results may be generalized as follows: Say that a group G is of type (α) if ∃a ∈ G, a ≠ e, such that for all b belonging to a set of generators for G we have ambn = bqap for some integers m, n, p, q with either m = p or n = q.
Determinacy in a class of germs on a reflexive Banach space
Authors: Magnus R. J.
Published: November 2008

Given Banach spaces X and Y let (X, Y) be the space of germs at 0 of C∞ Y-valued mappings defined on neighbourhoods of 0 in X. Let j(X, Y) be those f ∈ (X, Y) such that the ith derivative f(i)(0) = 0 for i = 0,1,…, j – 1. Abbreviations: (X) for (X, ℝ) and j(X) for j(X, ℝ). For ease of exposition germs will often be replaced by mappings which ‘represent’ them.
On the nilpotence class of commutative Moufang loops
Authors: Smith J. D. H.
Published: November 2008

The nilpotence class of the free commutative Moufang loop on n generators (n > 3) is the maximum allowed by the Bruck-Slaby Theorem, namely n − 1. This is proved by setting up a presentation of an extension of the loop's multiplication group as a nilpotent group of class at most 2n − 2, and then using the Macdonald-Wamsley technique of nilpotent group theory to show that this class is exactly 2n − 2.
Convex fundamental regions for Fuchsian groups
Authors: Nicholls P. , Zarrow R.
Published: November 2008

1. Introduction. Let G be a Fuchsian group which acts on the unit disc Δ. A fundamental region D for G acting in Δ is a subset of Δ such that D is open and connected and each point of Δ is G-equivalent to exactly one point in D or at least one point in D̅ (the closure of D in Δ). Throughout this paper we consider only fundamental regions which are (hyperbolically) convex. Beardon has shown (2) that it is possible for a convex fundamental region to have certain undesirable properties. It can happen that a convex region is not locally finite, i.e. there exist points of Δ where infinitely many G images of D accumulate. For a domain D we denote by F the set of such points.
Universal matrices
Authors: Burkill H.
Published: November 2008

Mathematical literature contains a number of theorems which assert the existence of a structure ‘universal’ in relation to some given system of objects. Two examples will suffice to indicate the nature of these theorems.
On centralizers of involutions in 2-groups
Authors: Ronse Christian
Published: November 2008

1. Introduction. It is well known (see (5) and (6)) that if T is a 2-group containing an involution t such that |CT(t)| = 4, then T is dihedral, semidihedral or cyclic of order 4. Fomin(1) has studied 2-groups T containing an involution t such that |CT(t)| ≤ 8. He showed in particular that such a group has sectional 2-rank at most 4. In this paper we will prove the following:

Theorem A. Let T be a 2-group containing an involution t such that |CT(t)| ≤ 16. Then T has sectional 2-rank at most 6.
The Adams–Mahowald conjecture on real projective spaces
Authors: Lin Wen-Hsiung
Published: November 2008

Let A denote the mod 2 Steenrod algebra. Let ℤ2[x, x−l] be the (graded) ring of finite Laurent series over ℤ2 in the variable x with dim (x) = 1. ℤ2[x, x−1] is a module over the Steenrod algebra A by

where are binomial coefficients modulo 2 and m > 0 is large compared with |k| and i. Let M be the A-submodule of ℤ2[x, x−1 ] generated by all powers xi with i ≠ −1. It is easy to see that ℤ2 [,x, x−1]/M ≅ σ−1ℤ2 (means ℤ2 on dimension − 1). Let ρ: ℤ2[x, x−1] → σ−1 ℤ2 be the projection map.
Equivalent norms on Banach Jordan algebras
Authors: Youngson M. A.
Published: November 2008

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.
Spectra of approximate identities in Banach algebras
Authors: Dixon P. G.
Published: November 2008

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 1].
Sums of stably trivial vector bundles
Authors: Allard Jacques
Published: November 2008

We say that a real vector bundle ξ over a finite C.W. complex X is stably trivial of type (n, k) or, simply, of type (n, k) if ξ ⊕ kε ≅ nε, where ε denotes a trivial line bundle. The following theorem is an immediate corollary (see (12)) of a theorem of T. Y. Lam ((9), theorem 2).
A stratification of the space of cubic surfaces
Authors: Bruce J. W.
Published: November 2008

In (4) the classification of (complex, projective) cubic surfaces by the number and nature of their singularities is carried out. This gives a natural partition of the vector space of cubic surfaces (which we denote by H3(4, 1)). In this paper we investigate the differential geometric properties of this partition; we show that it provides a finite constructible stratification of H3(4,1) which, in the notation of (10), is Whitney (A) regular. In fact Whitney (B) regularity holds over each stratum other than E6, but this stratum of cubic cones has an exceptional (equianharmonic) orbit at which (B) regularity fails. It remains to be seen whether or not this is the only exceptional orbit.
Unilateral quasi-regularizers of closed operators
Authors: Cross R. W.
Published: November 2008

Let X and Y be normed spaces and let L(X, Y) denote the set of linear transformations from X into Y, with domain D(T) and range R(T). For a given T ∈ L(X, Y) we investigate the existence and properties of a closed densely defined operator S ∈ L(Y, X) such that ST − I/D(T) (or TS − I/D(S)) is a bounded operator of finite dimensional range. These results were previously announced without proof in (2).
On certain Subgroups of the fundamental group of a closed surface
Authors: Jaco William
Published: November 2008

1. Introduction. Although Corollaries 4 and 5 to Theorem 1 below appear elsewhere in the literature (1, 2), the proofs given seem to use rather long and involved arguments and refer to other results in the literature for their completeness. The proofs given below are brief and follow quite naturally in sequence with the other corollaries from Theorem 1. The arguments presented are independent of references to the literature except for the reference in the proof of Theorem 1 to Lemma 2·1 of (3), which is well known to topologists. For our purposes this theorem may be stated as: For each open 2-manifold M (non-compact and without boundary), there is a subcomplex L made up of edges of some triangulation of M such that the open simplicial neighbourhood of L is piecewise linearly homeomorphic to M.
Diffusion and convection of heat in the potential flow past a circular cylinder
Authors: Norminton E. J.
Published: November 2008

The problem of the diffusion and convection of heat from an arbitrarily placed heat source in the potential convection field due to uniform streaming past an infinite circular cylinder is considered. Solutions are obtained for the regions inside and outside the cylinder and are given in terms of Mathieu Functions. An appendix giving the required properties of these functions is included.
The numerical range in the second dual of a Banach algebra
Authors: Smith R. R.
Published: November 2008

An elementary consequence of the Hahn-Banach theorem is that every Banach space Y is ω*-dense in its second dual Y**, so that every element y ∈ Y** is the w*-limit of a net {ya}α ∈ Λ from Y. There is, of course, a great deal of choice in the selection of such a net, and so one may impose extra conditions on the net related to some special property of the limit point, and then ask for existence. The object of this paper is to present such a result in the context of a unital Banach algebra and its second dual , and then to give two applications to Banach algebra theory. The theorem to be proved is this: if the numerical range W(a) of an element in has non-empty interior then a is the ω*-limit of a net {aa}α ∈ Λ from whose numerical ranges are contained in W(a), while if W(a) has empty interior then the numerical ranges W(aα) are contained in a shrinking set of neighbourhoods of W(a).
A note on a theorem of Maddox on strong almost convergence
Authors: Das G. , Mishra S. K.
Published: November 2008

Let m be the set of all real sequences x = (xn) with norm . A linear functional L on m is said to be a Banach limit (see Banach(1), p. 32) if it has the following properties:

(i) L(x) ≥ 0, if x ≥ 0 (i.e. xn ≥ 0, for all n ∈ N)

(ii) L{e) = 1, where e = (1, 1, 1,…),

(iii) L(Sx) = L(x), where (Sx)n = xn+1.
On -convergence of Markov chains
Authors: Aldous D. J.
Published: November 2008

Let I be a countable set, with discrete topology, and let X = (Xn), Y = (Yn) be stationary stochastic processes taking values in I. To a probabilist, the natural topology on processes (strictly speaking, on distributions of processes) is weak convergence:
Absolute summability functions for Hausdorff methods
Authors: Kuttner B.
Published: November 2008

1. Following Lorentz, we suppose throughout that Ω(n) is a non-negati ve non-decreasing function of the non-negative integer n such that Ω(n)→ ∞ as n → ∞. Consider the summability method given by the sequence-to-sequence transformation corresponding to the matrix A = (ank). We say that Ω(n) is a summability function for A (or absolute summability function for A) if the following holds: Any bounded sequence {sn} such that the number of values of ν with ν ≤ n, sν ≠ 0 does not exceed Ω(n) is summable A (or is absolutely summable A, respectively). These definitions are due to Lorentz (4), (6). We shall be concerned with the case in which A is a regular Hausdorff method, say A = H = (H, μn). Then H is given by the matrix (hnk) with

with

X(0) = X(0 + ) = 0, X(1) = 1;(see e.g.(1), chapter XI). We shall suppose throughout that these conditions are satisfied. It is known that H is then necessarily also absolutely regular.
Linear invariants of Riemannian almost product manifolds
Authors: Carreras Francisco J.
Published: November 2008

Using the decomposition of a certain vector space under the action of the structure group of Riemannian almost product manifolds, A. M. Naveira (9) has found thirty-six distinguished classes of these manifolds. In this article, we prove that this decomposition is irreducible by computing a basis of the space of invariant quadratic forms on such a space.
Imbedding four-manifolds and slicing links
Authors: Ruberman Daniel
Published: November 2008

At the 1977 Georgia Topology Conference, Cappell and Shaneson(6) announced the following theorem.

Let M be a closed oriented smooth four-manifold.
Multipole expansions for two superposed fluids, each of finite depth
Authors: Kassem S. E.
Published: November 2008

Problems dealing with the generation of internal waves at the surface separating two fluids involves the consideration of different types of singularities in one of the two fluids. In this paper the velocity potentials describing line and point multipoles are obtained for the case when each fluid is of finite constant depth, neglecting effects of surface tension at the surface of separation.
On Ext1 for semisimple groups and infinitesimal subgroups
Authors: Donkin Stephen
Published: November 2008

Let G be an affine, connected, semisimple, simply connected algebraic group over an algebraically closed field k of non-zero characteristic p and, for a positive integer n, let Gn be the nth infinitesimal subgroup (the kernel of the nth power of the Frobenius morphism). The purpose of this note is to point out that, for certain rational G-modules V, the natural homomorphism

is the zero map, for m ≽ 0. For such modules V, taking m = 0, we see from the 5-term exact sequence of Lyndon-Hochschild-Serre that the inflation-restriction sequence

is exact and thus the dimension of H1(G, V), when finite, may be calculated by adding the dimensions of the outer terms. The practical value of this result seems to lie in the special case n = 1 and V = L(λ) ⊗ L(µ), the tensor product of simple rational G-modules. In this case H1(G/G1) VG1 is H1(G, L(λ′) ⊗ L(¯′)) for λ′, ¯′ ‘smaller’ than λ µ and the above exact sequence gives an inductive procedure for computing Ext1G between simple rational (G-modules once the (G-module structure of Ext1G1 (equi valently Ext1u1 where% is the restricted enveloping algebra of the Lie algebra of G), for simple G1-modules, is known. Using the rather complicated structure of Weyl modules Cline, in (4), calculated Ext1G between simple modules for G = SL2(k) (this process was reversed in (5)). The present note has been used by S. EIB. Yehia, (17), in finding the corresponding result for SL3(k). We believe these Ext groups to be of increasing importance in view of their connection with Lusztig's conjecture, (14), on the characters of the simple rational G-modules and the related conjecture, 7–2 of (l), of Andersen.
An extremum variational principle and error estimate procedure for qIV = f(q, x)
Authors: Djukic Dj. S. , Atanackovic T. M.
Published: November 2008

An extremum variational principle for boundary value problems described by a non-linear fourth order differential equation qIV = f(q, x) is constructed. For approximate solutions an error estimate procedure, based on the value of the functional, is developed and applied to two concrete problems.
Conditions for subnormality of a join of subnormal subgroups
Authors: Williams J. P.
Published: November 2008

The object of this paper is to prove a necessary and sufficient condition on two groups H, K for their join always to be subnormal in a group G whenever they are embedded subnormally in G.