bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on Smale's formulation of electrical network theory
Authors: Arrowsmith D. K. , Smith R.
Published: November 2008

We consider in this note two problems which arise naturally in Smale's paper (3).

First of all we show that it is easy to extend Smale's results on the dynamics of LCR circuits with non-linear elements to circuits containing triode valves. It should be noted that transistors and other n-ports (1) can also be easily accommodated within this theory with suitable differentiability assumptions. We then further extend the theory to include elements whose voltage depends explicitly on time.
Complex potential equations II. An application to general relativity
Authors: Collins C. B.
Published: November 2008

The results of a previous paper are applied to a study of the class of Kerr–Schild metrics in general relativity. These metrics have the form

where η is the flat (Minkowski) space-time metric, m is an arbitrary real number, and 1 is a null covector. It is already known that for a certain restricted subclass of these metrics, the vacuum Einstein field equations, viz.

can be written in the form

where γ is a complex potential. Using the methods developed in a previous paper, such space-times are characterized by means of a special family of complex surfaces in three-dimensional Euclidean space, and the exact solutions for the metric g are consequently recovered. It is also shown that the field equations for a much wider class of Kerr–Schild metrics can be expressed in terms of a potential formalism, not only in the vacuum case, but also for many electrovacuum solutions.
Azumaya algebras and rings with polynomial identity (Addendum)
Authors: Goldie Alfred
Published: November 2008

In the paper published in Mathematical Proceedings 79, 393–399, there is a gap in the proof of Theorem 2 which may present a difficulty to the reader. It occurs in the proof of non-degeneracy of the map g in the ring of matrices Zn, where Z is a finite field. Matrices are chosen, the first is diagonal with distinct entries and the others are

When n is larger than the number of elements in Z, then such an x0 does not exist in Zn. The argument can, however, be sustained in the following way.
Abstract length functions in groups
Authors: Chiswell I. M.
Published: November 2008

If F is a free group on some fixed basis X, there is a mapping from F to the non-negative integers, given by sending an element of F to the length of the normal word in X±1 representing it. A similar mapping is obtained in the case of a free product of groups. Lyndon (3) considered mappings from an arbitrary group to the non-negative integers having certain properties in common with these mappings on free groups and free products.
Weakly prime compact convex sets and uniform algebras
Authors: Ellis A. J.
Published: November 2008

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.
Exact sequences of conjugacy classes and rationalization
Authors: Steiner R. J.
Published: November 2008

The purpose of this paper is to exhibit two exact sequences involving the sets of conjugacy classes in various groups. One of them is applied to prove the following result.
Fully admissible binary relations in topology
Authors: McSherry D. M. G.
Published: November 2008

A fully admissible binary relation (3) is an operator , other than the equality operator and universal operator , which assigns to each space |S, τ|, a reflexive, symmetric, binary relation , and which is such that for any continuous mapping implies . With each such relation , we associate a ‘separation axiom’ , as well as ‘-regularity’ and ‘-connectedness’, where ≡ -regularity + T0, and -regularity + -connectedness ≡ indiscreteness.
On non-unital Jordan–Banach algebras
Authors: Smith R. R.
Published: November 2008

A unital JB-algebra is a Jordan algebra A with identity together with a complete norm satisfying, for all a, b ∈ A,

(i) (a2b) a = a2(ba),

(ii) ∥a2∥ = ∥a∥2,

(iii) ∥ab∥ ≤ ∥a∥ ∥b∥,

(iv) ∥a2 + b2∥ ≥ ∥a2∥, ∥b2∥.

(It should be noted that axiom (iii) is a consequence of (ii) and (iv).) Such spaces have been studied by several authors (3, 6, 11), and as a consequence their structure is now quite well understood. Many of the results of these papers, while relying on the existence of an identity for their proofs, can be formulated for algebras which lack this property. C*-algebra theory and operator theory abound in examples of spaces which fail to be unital JB-algebras only in this one respect, and this motivates the study of the general case undertaken in this note.
On stratifying 3-jets of functions
Authors: Bedford J. O. , Gibson C. G.
Published: November 2008

The crucial step in the proof of the Topological Stability Theorem is the construction of a certain contact-invariant Whitney stratification of the sufficient jets in the jet space Jk(n, p). There is therefore a certain intrinsic interest in the construction of natural contact-invariant Whitney stratifications of the jet space. It has long been known that if there are only finitely many contact orbits then these provide a Whitney stratification, but in general there is the double problem of finding a candidate for a Whitney stratification, and then establishing that it has the desired properties. Virtually nothing is known about this problem, so it seems sensible to start by attempting to understand the first non-trivial examples which arise. Since functions are rather better understood than mappings it seems natural to try the case p = 1 first. The complexity of the problem now increases with k. When k = 1, 2 there are only finitely many contact orbits, so we start with k = 3. Having fixed p = 1, k = 3 we turn our attention to n. When n = 1, 2 we still obtain only finitely many contact orbits, as easy computations verify, so we start with n = 3. Thus our first example is J3(3, 1). It is a relatively easy matter to list the contact orbits in this case, and in so doing one is presented with an obvious candidate for a Whitney stratification. Our objective is to show that this candidate is indeed a Whitney stratification.
Dirichlet's theorem on diophantine approximation
Authors: Baker R. C.
Published: November 2008

Let N take the values 1, 2, … A theorem of Dirichlet asserts that for any (x, y) in the Euclidean plane R2 the inequality

is soluble in integers q1, q2, p with 0 < max (|q1|, |q2|) ≤ N.
Finiteness and decidability: II
Authors: Johnstone P. T. , Linton F. E. J.
Published: November 2008

It has been known for some time ((6), p. 270; (4), theorem 9·19) that if is a Boolean topos, then the full subcategory Kf of Kuratowski-finite objects in is again a topos. For a non-Boolean topos , however, Kf need not be a topos, as can be seen when is the Sierpinski topos ((1), example 7·1); on the other hand, two other full subcategories of , coinciding with Kf when is Boolean, suggest themselves as candidates for a subtopos of finite objects. Of one of these, the category dKf of decidable K-finite objects in , the Main Theorem of (1) asserts that it is always a (Boolean) topos. The other is the category sKf of -subobjects of K-finite objects. The inclusions

dKf ⊆ Kf ⊆ sKf are clear.

are clear.
Intersection multiplicities and tangent cones
Authors: Tennison B. R.
Published: November 2008

The following result has at least the appeal of intuitive plausibility. Let U and V be subvarieties of an algebraic variety X; let x ∈ X be an isolated point of the intersection of U and V, and let I(X, U. V, x) denote the intersection multiplicity (in some sense to be made precise) of U and V at x. Now let Tx be the tangent space to X at x, within which lie the tangent cones Cu, Cv to U and V at x. Then
Augmentation quotients of some nonabelian finite groups
Authors: Losey Gerald , Losey Nora
Published: November 2008

1. LetGbe a group,ZGits integral group ring and Δ = ΔGthe augmentation idealZGBy anaugmentation quotientofGwe mean any one of theZG-modules

wheren, r≥ 1. In recent years there has been a great deal of interest in determining the abelian group structure of the augmentation quotientsQn(G) =Qn,1(G) and

(see (1, 2, 7, 8, 9, 12, 13, 14, 15)). Passi(8) has shown that in order to determineQn(G) andPn(G) for finiteGit is sufficient to assume thatGis ap-group. Passi(8, 9) and Singer(13, 14) have obtained information on the structure of these quotients for certain classes of abelianp-groups. However little seems to be known of a quantitative nature for nonabelian groups. In (2) Bachmann and Grünenfelder have proved the following qualitative result: ifGis a finite group then there exist natural numbersn0and π such thatQn(G) ≅Qn+π(G) for alln≥n0; ifGωis the nilpotent residual ofGandG/Gωhas classcthen π divides l.c.m. {1, 2, …,c}. There do not appear to be any examples in the literature of this periodic behaviour forc> 1. One of goals here is to present such examples. These examples will be from the class of finitep-groups in which the lower central series is anNp-series.
On Grothendieck's local duality theorem
Authors: Bijan-Zadeh M. H. , Sharp R. Y.
Published: November 2008

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.
Stratifying jet spaces – some special cases
Authors: Bruce J. W.
Published: November 2008

Here we consider the problem of Whitney stratifying some very simple jet spaces. This type of problem has also been considered by Lander, Bedford and Gibson, (see (7), (1), (2)). Following Bedford and Gibson, we shall consider complex jet spaces.
On disjointness of dynamical systems
Authors: Auslander J. , Dowker Y. N.
Published: November 2008

By a dynamical system we mean one of several related objects: measure preserving transformations on probability spaces (processes), self homeomorphisms of compact metric spaces (compact systems), or a combination of these, namely compact systems provided with invariant Borel probability measures. It is the latter, which we call compact processes, which will be of most interest in this paper. In particular, we will study the dynamical properties of the product of two processes with respect to compatible measures – those measures which project to the given measures on the component spaces. This leads to the notion of disjointness of two processes – the only compatible measure is the product measure. As an application we obtain a theorem, a special case of which gives rise to a class of transformations which preserve normal sequences. Finally, we study a topological analog (topological disjointness) and briefly consider the relation between the two notions of disjointness.
Functionals of diffusion processes as stochastic integrals
Authors: Davis M. H. A.
Published: November 2008

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Then

is a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representation

For Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e:
Some expansion formulae for Fox's H-function involving exponential functions
Authors: Bajpai S. D.
Published: November 2008

In this paper we have evaluated some integrals involving Fox's Η-function and employed them to establish some expansion formulae for the Η-function involving exponential functions. Some results recently given by MacRobert (7) and Kesarwani (5) have been shown as particular cases.
On cylindrical waves in stratified media: high frequency refraction and diffraction at a plane interface
Authors: Roebuck I.
Published: November 2008

Introduction. The problem of the scattering of high-frequency waves, which emanate from a line source in a homogeneous isotropic dielectric medium and impinge upon a cylindrical obstacle, has been attacked in a variety of ways. In certain cases, where both the shape of the obstacle and the conditions to be satisfied on its boundary are particularly convenient, an exact solution may be found by separation of the wave equation (see, for example, Marcuvitz (l)), but in general some form of approximation is necessary to obtain an explicit answer.
One-relator quotients of free products
Authors: Pride Stephen J.
Published: November 2008

LetIbe a set with |I| > 1, letHi(i∈I) be nontrivial groups and letH= *i∈IHibe their free product. LetRbe a cyclically reduced element ofHiand write

ifGis (isomorphic to)H/{R}H.
Canonical stratifications of functions: the simple singularities
Authors: Bruce J. W.
Published: November 2008

In this paper we shall describe that part of the canonical stratification of a versal unfolding of a function germ which deals with simple singularities. These results are doubtless known to many workers in this field; see especially the papers of E. J. N. Looijenga (e.g. (12)). We fill in the details however since we have in mind some geometrical applications (given in § 2) concerning natural partitions of the spaces of binary forms, cubic surfaces and quartic curves. The first two stratifications were considered in the author's Liverpool Ph.D. thesis; the quartic curves were considered in the Liverpool thesis of C. M. Hui, where the results were obtained by specific calculation. See (4) for an illustration of the techniques employed in these theses.
The distance to upper triangular operators
Authors: Power S. C.
Published: November 2008

We show how a generalization of a distance formula of W. Arveson(1) may be derived from an elementary but ingenious special case, due to S. Parrot(2), involving 2 × 2 matrices.
A constructive approach to the duality theorem for certain Orlicz spaces
Authors: Johns D. L. , Gibson C. G.
Published: November 2008

One of the intriguing complications of constructive analysis is that it does not seem possible to obtain examples of inseparable Banach spaces; more precisely, when one comes to study explicit examples one finds that one cannot assign a constructively denned norm to every vector in the space. One of the objectives of (2) was to make a detailed study of an explicit class of Banach spaces, containing both separable and inseparable examples, in the hope that one might begin better to understand these matters. We chose for this study the class of Orlicz spaces, primarily because the simplest examples (namely the Lebesgue spaces LP) had already been the objective of detailed constructive work in (1).
Invariants of pencils of binary cubics
Authors: Newstead P. E.
Published: November 2008

Let U denote the variety of pencils of binary cubics without base point defined over an algebraically closed field k whose characteristic is not equal to 2 or 3. (For full details of the terminology, see § 2.) There is a natural action of SL(2) on U; our main object is to show that U possesses a good quotient for this action in the sense of ((7), Definition 1·5) or ((4), p. 70), and to identify this quotient with the affine line over k. In fact, we prove

Theorem 1. There exists a morphism φ: such that (, φ) is a good quotient of U by SL(2). Moreover, all but one of the fibres of φ are orbits; the exceptional fibre consists of two orbits corresponding to pencils with one or two triple points.
Self-maps of BU
Authors: Clarke Francis
Published: November 2008

One of the results proved by Lance in (7) is the following.

Theorem 1. Let ø: BU(p) → BU(sp) be an H-map, where BU(p) denotes the localization of BU with respect to an odd prime p. Let ø*: π2k(BU(p))→ π2k(BU(p)) ≅ (p)) be multiplication by λk then

if n1 ≡ n2 mod(p − 1).
Infinite loop structures on the algebraicK-theory of spaces
Authors: Steiner Richard J.
Published: November 2008

LetXbe a topological space with base-point. Thealgebraic K-theory AXofXis a space invented by Waldhausen in (15) for use in geometric topology. It can be defined in two ways, which I shall callgeometricandring-theoretic; Steinberger ((12)) has shown them to be equivalent.

The geometric method ((15), corollary to lemma 2·1) givesAXas the group-completion of the geometric realization of a permutative category. It follows from the machinery of May ((4), 4) or Segal (11) thatAXis an infinite loop space in a well-defined way ((6)).
On a matroid generalization of graph connectivity
Authors: Oxley James G.
Published: November 2008

This paper relates the concept of n-connection for graphs to Tutte's theory of n-connection for matroids (12). In particular, we show how Tutte's definition may be modified to give a matroid concept directly generalizing the graph-theoretic notion of n-connection.
Characters and contact transformations
Authors: Hannabuss K. C.
Published: November 2008

If V is a symplectic space, each affine contact transformation is shown to define an automorphism of a certain algebra of Schwartz functions on V. This automorphism turns out to be a generalized inner automorphism and allows one to associate to the contact transformation a tempered distribution which can be found explicitly. It is shown by means of various examples that these distributions contain much information about the corresponding quantized system. For example the energy levels and stationary states of the harmonic oscillator can be deduced from the general formulae. Generalizations to other locally compact abelian groups V are described. Some connections with the theory of Fourier integral operators and with the characters of certain projective representations are also outlined.
Conserved currents of the Klein–Gordon field
Authors: Gordon T. J.
Published: November 2008

A method is presented whereby all locally defined conserved currents of the Klein-Gordon field are found. The mathematical background to the method includes a generalization of the Poincaré lemma of the calculus of exterior differential forms. It is found that the only conserved currents are essentially a countably infinite set of functions, bilinear in the field, together with a single current in the case where the mass is zero. The usual energy-momentum tensor is included amongst these functions. The method does not depend on the use of any canonical formulation of the field theory.
A boundary estimate for harmonic functions
Authors: Rippon P. J.
Published: November 2008

In this paper we extend to certain domains in m-dimensional Euclidean space Rm, m ≥ 3, some results about the boundary behaviour of harmonic functions which, in R2, are known to follow from distortion theorems for conformal mappings.
Asymptotic properties of random subsets of projective spaces
Authors: Kelly Douglas G. , Oxley James G.
Published: November 2008

A random graph on n vertices is a random subgraph of the complete graph on n vertices. By analogy with this, the present paper studies the asymptotic properties of a random submatroid ωr of the projective geometry PG(r−l, q). The main result concerns Kr, the rank of the largest projective geometry occurring as a submatroid of ωr. We show that with probability one, for sufficiently large r, Kr takes one of at most two values depending on r. This theorem is analogous to a result of Bollobás and Erdös on the clique number of a random graph. However, whereas from the matroid theorem one can essentially determine the critical exponent of ωr, the graph theorem gives only a lower bound on the chromatic number of a random graph.
The asymmetry of M0(G)
Authors: Brown G. , Karanikas C. , Williamson J. H.
Published: November 2008

Let G be a non-discrete locally compact topological group. A construction is given for a self-adjoint measure μ on G that has independent powers, and is, moreover, in the sub-algebra M0(G) of M(G). From this, the asymmetry of M0(G) (and of M(G) itself) is immediate.
Series identities and reducibility of Kampé de Fériet functions
Authors: Buschman R. G. , Srivastava H. M.
Published: November 2008

The work of H. M. Srivastava (9) on generalizations of an interesting identity of Carlson is expanded upon in order to obtain series identities analogous to all four of the Appell functions and their generalizations. Reduction formulas for some Kampé de Fériet functions are derived as consequences of the various double-series identities considered in this paper.
On polynomial symmetries of the sine-Gordon equation
Authors: Tu G. Z.
Published: November 2008

The sine-Gordon (SG) equation uxt = sin u arises from many branches of physics, and now is one of the most important equations in soliton theory. There have been many works concerning its soliton solutions, Backhand transformations, symmetries and conservation laws and other properties. In this paper we prove that every polynomial symmetry of the SG equation is Hamiltonian, that is, takes the form of D-l δh/δu.
On almost-everywhere convergent eigenfunction expansions of the Laplace–Beltrami operator
Authors: Meaney Christopher
Published: November 2008

Let M be a compact Riemannian manifold equipped with Laplace–Beltrami operator Δ. We use the Rademacher-Menchoff theorem and the asymptotics of the eigenvalues of Δ to show that if a function belongs to an L2 Sobolev space of positive index then its expansion, in terms of eigenfunctions of Δ, converges almost everywhere on M.
Asymptotic normality of lightly trimmed means – a converse
Authors: Maller R. A.
Published: November 2008

Let Xi be iidrv's and Sn = X1 + X2 + … + Xn. Let (r)Sn be Sn minus the r terms of largest absolute value. If (Sn/Bn)− An converges to normality for some Bn, An, then so does ((r)Sn/Bn) − An. We show that the converse is true if Xi have a continuous symmetric distribution, and give some related results.
Absolutely superficial sequences
Authors: Trung Ngô Viêt
Published: November 2008

Let A be a local ring with maximal ideal m. Let M be a finitely generated module over A. Let a1 …, ar be a sequence of elements of m. Let qi denote the ideal (a1,…, ai), i = 1, …, r, and set q0 = 0A (the zero ideal of A), q = qr.
Note on analytic spread and asymptotic sequences
Authors: Ratliff L. J.
Published: November 2008

Since the foundational paper (10) by Northcott and Rees in 1954 there have been quite a few papers concerning reductions of ideals and the analytic spread of an ideal. One particular line of investigation concerning the analytic spread l(I) of an ideal I in a local ring (R, M) was begun in 1972 by Burch in (5), where it was shown that l(I) ≤ altitude R – min (grade R/In; n ≥ 1). This result was sharpened in 1980–81 by Brodmann in three papers, (2, 3, 4). Therein he showed that the sets {grade R/In; n ≥ 1} and {grade In−1/In; n ≥ 1} stabilize for all large n, and calling the stable values t and t*, respectively, it holds that t ≤ t* and l(I) ≤ altitude R – t* when I is not nilpotent. He then gave a case (involving R being quasi-unmixed) when equality holds. In 1981 in (20) Rees used two new approaches to Burch's inequality, and he proved two nice results which may both be stated as: l(I) ≤ altitude R – s(I) with equality holding when R is quasi-unmixed; here, s(I) = min {height P; P is a minimal prime divisor of (M, u) R[tI, u]}– 1 (in the first theorem), and s(I) is the length of a maximal asymptotic sequence over I (in the second theorem).
Topological Hopf bifurcation in the plane
Authors: Erle Dieter
Published: November 2008

Classical bifurcation theorems for a 1 -parameter family of plane dynamical systems

assert the presence of closed orbits clustering at some distinguished parameter value (∈ = 0, say). Here, for any ∈, the origin is the only stationary point. The topological content of the mostly analytic hypotheses imposed is some change in the stability behaviour of the origin at ∈ = 0, roughly the passing of a kind of stability to a kind of instability. Topologically speaking, e.g. some of the conditions demanded are asymptotic stability of the origin for the negative system at ∈ > 0 and asymptotic stability of the origin for at ∈ < 0 (Hopf (8), Ruelle and Takens(11)) or ∈ = 0 (Chafee(2)).
Integrals with a large parameter: Hilbert transforms
Authors: Ursell F.
Published: November 2008

A simple and elegant method of treating one-sided Hilbert transforms

has recently been described by Wong (7). This method immediately gives the form of the expansion but some of the coefficients in the expansion are in an inconvenient form for either analytical or numerical calculation. In the present paper it is shown that these coefficients can be readily determined when the Mellin transform of f(t) is known. The case is treated as an example.
Singularity of generalised Riesz products
Authors: Bott R. T.
Published: November 2008

Criteria for singularity and absolute continuity analogous to Zygmund's for Riesz products, are demonstrated for those generalized products introduced by Modica and Mortola.
Two-generator arithmetic Fuchsian groups
Authors: Maclachlan C. , Rosenberger G.
Published: November 2008

In (12), Takeuchi characterized those Fuchsian groups which are arithmetic in terms of the traces of the matrices involved in the group. Further, he used these criteria (13) to show that there are only finitely many arithmetic Fuchsian triangle groups and indeed to determine the signatures of all such groups. We note that, if two Fuchsian triangle groups have the same signature, then they are conjugate in PGL(2, ) and hence either all Fuchsian triangle groups of a given signature are arithmetic or none of them are. This will not, in general, be the case for non-triangle Fuchsian groups and in this paper, we examine Fuchsian groups with signature of one of the forms (1; n; 0), n ≽ 2 or (0; 2,2, 2, n; 0), n ≽ 3 and odd in which case the space of conjugacy classes has dimension two. Our principal results state that for each n, there are only finitely many conjugacy classes of arithmetic Fuchsian groups of given signature (1; n; 0) or (0; 2, 2, 2, n; 0) and for large enough n there are no arithmetic Fuchsian groups of that given signature (Theorems 3 and 4). Together with the results of (13) and known results for non-cocompact Fuchsian groups, these results show that there are only finitely many conjugacy classes of two-generator arithmetic Fuchsian groups (see §7).
Ringing the changes
Authors: White Arthur T.
Published: November 2008

The ancient and continuing art of change ringing, or campanology (how the British ring church bells), is studied from a mathematical viewpoint. An extent on n bells is regarded as a hamiltonian cycle in a Cayley colour graph for the symmetric group Sn, embedded on an appropriate surface. Two methods for variable n (Plain Bob for all n and Grandsire for n =; 3 (mod 4)) are discussed, and a new method for n odd is introduced. All minimus methods (n = 4) and five doubles methods (n = 5) are depicted, one of these being the new No Call Doubles.
A note on Cusick's theorem on units in totally real cubic fields
Authors: Godwin H. J.
Published: November 2008

Let ε = ε1, with conjugates ε2, ε3, be a unit in a totally real cubic field, and let . Let ε be a unit for which T (ε) is least and let η be a unit, not a power of ε, for which T(η) is least. It was shown by Cusick[l] that ε,η form a pair of fundamental units under certain conditions. The purpose of the present note is to show that these conditions are unnecessary and that ε, η form a pair of fundamental units in all cases.
Cofinality of the partial ordering of functions from Ω1 into Ω under eventual domination
Authors: Jech Thomas , Prikry Karel
Published: November 2008

It is unknown whether there can exist a family of functions from Ω1 into Ω of size less than that dominates all functions from Ω1 into Ω. We show that there is no such family if the continuum is real-valued measurable, and that the existence of such a family has consequences for cardinal arithmetic, and is related to large cardinal axioms.
The general vacuum Bianchi type-V solution in the Brans-Dicke theory
Authors: Lorenz-Petzold D.
Published: November 2008

Amongst the various modifications of the general theory of relativity (GRT), the scalar-tensor theory of Brans and Dicke (BDT) is treated most seriously ([2], [17], [13]). The BDT is consistent with observations as long as the coupling parameter ω between the scalar and tensor components of gravitation is about equal to or greater than 500 [18]. However, there are noa prioritheoretical reasons for excluding other values of ω. In the limit ω → ∞, the BDT reduces to the GRT for a constant BDT-scalar field Φ
Integrals with a large parameter: Legendre functions of large degree and fixed order
Authors: Ursell F.
Published: November 2008

Suppose that a function f(z, n) depends on a large parameter n. A proposed expression g(z, n) is an asymptotic approximation for f(z, n) if it can be shown that the error (i.e. the difference between f(z, n) and g(z, n)) is small of an appropriate order when n → ∞. Effective error bounds are particularly useful in numerical work with asymptotic expansions. Most of the existing derivations of error bounds involve complicated calculations.
A family of elliptic curves and cyclic cubic field extensions
Authors: Thomas E. , Vasquez A.T.
Published: November 2008

LetKbe a field with charK≡ 2,3. We consider the problem of finding rational points overKon the family of elliptic curvesFλ, given in homogeneous coordinates (over) by
A note on the algebra of Poisson brackets
Authors: Atkin C. J.
Published: November 2008

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].
On invariant knots
Authors: Kwasik Sławomir , Vogel Pierre
Published: November 2008

It was first proved by R. Lashof in [4], using the work of S. Cappell and J. Shaneson on four-dimensional surgeryu (see [1]), that there exist locally flat topological knots S3 ∪ S5 which are not smoothable. In [2] (compare also [6]) S. Cappell and J. Shaneson have constructed infinitely many non-smoothable locally fat topological knots as the fixed points of locally nice (= locally smoothable) Zp actions on S5, therefore giving non-trivial examples of locally smoothable but equivariantly non-smoothable actions of Zp on S5.
Novel properties of Fibonacci and Lucas polynomials
Authors: Galvez F. J. , Dehesa J. S.
Published: November 2008

Average or global spectral properties of the Fibonacci and Lucas polynomials recently introduced in the theory of one-dimensional Ising chains are investigated. In addition, several partial sums of these polynomials are shown to have a compact form.
Cohomology and finite subgroups of small cancellation quotients of free products
Authors: Collins Donald J. , Perraud Jean
Published: November 2008

In this paper we study small cancellation quotients of free products. In particular we calculate their cohomology and, via a theorem of Serre, classify their finite subgroups. The results obtained are the natural analogues of the corresponding results for small cancellation quotients of free groups obtained in Huebschmann [6] (but see also Collins-Huebschmann[3]). They also generalize results of McCool[9] on elements of finite order in quotients of free products and similar conclusions have been obtained by Howie [5] for one-relator quotients of free products of locally indicable groups.
On the approximation of analytic functions in a strip
Authors: Klusch Dieter
Published: November 2008

1. Let

and denote by Aδ the class of functions f analytic in the strip Sδ = {z = x + iy| |y| < δ}, real on the real axis, and satisfying |Ref(z)| ≤ 1,z∊Sδ. Then N.I. Achieser ([1], pp. 214–219; [8], pp. 137–8, 149) proved that each f∊Aδ can be uniformly approximated on the whole real axis by an entire function fc of exponential type at most c with an error

where ∥·∥∞ is the sup norm on ℝ. Furthermore ([7], pp. 196–201), if f∊Aδ is 2π-periodic, then the uniform approximation Ẽn (Aδ) of the class Aδ by trigonometric polynomials of degree at most n is given by
A uniqueness lemma for groups generated by 3-transpositions
Authors: Weiss Richard
Published: November 2008

Let G be a group. A subset D of G will be called a set of 3-transpositions if |x| =2 for all xεD and |xy| = 3 whenever x, yεD do not commute. We will call the set D closed if xDx = D for each xεD. For each xεD, let
A Choquet theorem for general subspaces of vector-valued functions
Authors: Saab Paulette , Talagrand Michel
Published: November 2008

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.
Trigonometric sums of Heilbronn's type
Authors: Odoni R. W. K.
Published: November 2008

In problems of additive number theory one frequently needs to obtain a non-trivial estimate for the absolute value of a trigonometric sum of the type

wheref(X) ε ℤ [X]and 1 ≤ m ε ℤ. The general procedure is first to reduce the estimation to the case wheremis a prime power, by means of the Chinese Remainder Theorem. The casem = pr(pprime) can often be reduced to that of a lower power ofp, by a substitution of the typex = u + vps(where 0 ≤u < psand 0 ≤v < pr-s), followed by the use of ap-adic Taylor expansionf(u+psv)=f(u) +psvf′(u)+…. Frequently this givesT(f, pr)= 0 whenr≥ 2, or at least allows one to reduce to the casem = p.In the latter case an appeal to Weil's estimate

usually gives a good estimate for (0.1), at least if degf= o(√p).
Duality in twistor theory without Minkowski space
Authors: Singer M. A.
Published: November 2008

A modified form of the generalized Penrose-Ward Transform [4, 6] is set up to investigate the correspondence between twistor space and dual twistor space. The ways in which it differs from the ‘usual’ transform are discussed and it is used to give an alternative proof of Eastwood's recent generalization [7] of the twistor transform [9, 5] which avoids all mention of Minkowski space.
On the radical of semigroup algebras satisfying polynomial identities
Authors: Okniński Jan
Published: November 2008

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebraK[S] of an arbitrary semigroupSover a fieldKin the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties ofSor, more precisely, by some groups derived fromS. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).
Effective resolution of cusps on Hilbert modular varieties
Authors: Sankaran G. K.
Published: November 2008

In this paper, we use the Shintani decomposition, known to number theorists, to describe an effective method of finding a resolution of the cusps of a Hilbert modular variety, in any dimension.
Connectivity of finite anisotropic random graphs and directed graphs
Authors: Cohen Joel E.
Published: November 2008

For graphs on a finite set of vertices with arbitrary probabilities of independently occurring edges, the reliability is defined as the probability that the graph is connected, and the redundancy as the expected number of spanning trees of the graph. Analogous measures of connectivity are defined for random finite directed graphs with arbitrary probabilities of independently occurring directed edges. Recursive formulas for computing the reliability are known. Determinantal formulas, based on matrix-tree theorems, for computing the redundancy are given here. Among random graphs with a given sum of edge probabilities, the more evenly the probabilities are distributed over potential edges, the larger the redundancy. This inequality, proved using the theory of majorization, in combination with examples shows unexpectedly that conflicts between reliability and redundancy can arise in the design of communication networks modelled by such random graphs. The significance of these calculations for the command and control of nuclear forces is sketched.
Corrigenda
Authors: Dawson R. J. M.
Published: November 2008

Volume 98 (1985), 403–412

‘On the mobility of bodies in ℝn’

The proof of the following theorem which I gave in the above paper is incorrect. A correct proof is given here.

Theorem 2·1. Let ℬ be an intersection regular, finite collection of bodies, and ℱ a manoeuvre on ℬ. ℱ is strongly separating if and only if every component of S(ℬ, ℱ) is positive; and, if any S(i, j, p) (ℬ, ℱ) is negative, ℱ is colliding.
Some theorems in the metric theory of diophantine approximation
Authors: Harman Glyn
Published: November 2008

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequality

for almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, andfis a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint onf, the number of solutions of (1) is asymptotically equal to

asktends to infinity.
Some one-relator semigroup presentations with solvable word problems
Authors: Jackson David A.
Published: November 2008

We provide, for N ≥ 1, a solution to the word problem for the semigroup S having the presentation {a, b|a = abaNb}. For N = 2, this is example 7 in Howie and Pride [1]. Since solving the word problem for {a, b|a = abaNb} is equivalent to solving the word problem for {a, b|a = baNba}, the result here should be regarded as a special case of a more general result of Oganesyan [2], who solves the word problem for {a, b|a = bA}, A arbitrary.
K- theory and finite loop spaces of rank one
Authors: McGibbon C. A.
Published: November 2008

Let p be a prime and let d be a positive integer which divides p−1. Assume that X is a space such that
A Magnus theorem for T(6) groups
Authors: Howie James
Published: November 2008

A theorem of Magnus [4] says that, if R and S are two cyclically reduced words in a free group F whose normal closures are equal, then R is a cyclic permutation of S or its inverse. This is useful for comparing presentations of one-relator groups.
Which sets contain multiple points of Brownian motion?
Authors: Tongring Nils
Published: November 2008

In 1950, Dvoretzky, Erdös and Kakutani [2] showed that in ℝ3almost all paths of Brownian motionXhave double points, or self-intersections of order 2 (there are no triple points [4]); later the same authors proved that almost all sample paths of Brownian motion in the plane have points of arbitrarily high multiplicity (a pointxin ℝ2is ak-tuple point for the path ω, or a self-intersection of orderk, if there are timestl<t2< … <tksuch thatx=X(t1, ω) =X(t2, ω) = …X(tk, ω)).
Morita context functors
Authors: Nicholson W. K. , Watters J. F.
Published: November 2008

Given a Morita context (R, V, W, S), there are functors W⊗() and hom (V, ) from R-mod to; S-mod and a natural transformation λ from the first to the second. This has an epi-mono factorization and the intermediate functor we denote by ()° with natural transformations and . The tensor functor is exact if and only if WR is flat, whilst the hom functor is exact if and only if RV is projective. We begin by determining conditions under which ()° is exact; this is Theorem 1.
An upper bound for the breadth of the Jones polynomial
Authors: Thistlethwaite Morwen B.
Published: November 2008

In the recent article [2], a kind of connected link diagram called adequate was investigated, and it was shown that the Jones polynomial is never trivial for such a diagram. Here, on the other hand, upper bounds are considered for the breadth of the Jones polynomial of an arbitrary connected diagram, thus extending some of the results of [1,4,5]. Also, in Theorem 2 below, a characterization is given of those connected, prime diagrams for which the breadth of the Jones polynomial is one less than the number of crossings; recall from [1,4,5] that the breadth equals the number of crossings if and only if that diagram is reduced alternating. The article is concluded with a simple proof, using the Jones polynomial, of W. Menasco's theorem [3] that a connected, alternating diagram cannot represent a split link. We shall work with the Kauffman bracket polynomial 〈D〉 ∈ Z[A, A−1 of a link diagram D.
Statistical convergence in a locally convex space
Authors: Maddox I. J.
Published: November 2008

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.
Tori in metaplectic covers of GL2 and applications to a formula of Loxton–Matthews
Authors: Patterson S. J.
Published: November 2008

In this paper we have two objectives. The first is to investigate the restriction of a metaplectic cover to an arbitrary torus in GL2. This will be explained at greater length below, and the main results are Theorems 1 and 2. The second is an application of the same ideas to introduce the arithmetic function P, which has already appeared in a special case in [9], and to prove the fundamental property given by Theorem 3. These theorems will be proved in §§ 2 and 3. In §§ 4 and 5 we remark on the appearence of the function P in the formula of Loxton and Matthews [5], [6] for the biquadratic Gauss sum and discuss the structure of this formula.
Quadratic forms between spheres and the non-existence of sums of squares formulae
Authors: Yiu Paul Y. H.
Published: November 2008

Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the type

where are bilinear forms with real coefficients in and . Such an [r, s, n] formula is equivalent to a normed bilinear map satisfying . We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.
Framed cobordisms in real algebraic geometry
Authors: Kucharz Wojciech
Published: November 2008

Akbulut and King [3, 4] have obtained several interesting results investigating the effects on homology of real algebraic varieties of the blowing-up construction. Here we apply this technique to study the behaviour of homotopy classes.
Integrable almost cotangent structures and Legendrian bundles
Authors: Thompson G.
Published: November 2008

Recently, the present author together with M. Crampin proved a structure theorem for a certain subclass of geometric objects known as almost tangent structures (Crampin and Thompson [8]). As the name suggests, an almost tangent structure is obtained by abstracting one of the tangent bundle's most important geometrical ingredients, namely its canonical 1–1 tensor, and using it to define a certain class of G-structures. Roughly speaking, the structure theorem referred to above may be paraphrased by saying that, if an almost tangent structure is integrable as a G-structure and satisfies some natural global hypotheses, then it is essentially the tangent bundle of some differentiable manifold. (I shall have a further remark to make about the conclusion of that theorem at the end of Section 2.)
An algebraic model of graded calculus of variations
Authors: Kupershmidt B. A.
Published: November 2008

The modern theory of integrable systems rests on two fundamental pillars: the classification of Lax [13] and zero-curvature equations [14, 1, 2]; and algebraic models of the classical calculus of variations [9,5] specialized to the residue calculus in modules of differential forms over rings of matrix pseudo-differential operators [9, 6]. Both these aspects of the theory are by now very well understood for integrable systems in one space dimension.
Lifting amalgamated sums and other colimits of groups and topological groups
Authors: Brown Ronald , Heath Philip R.
Published: November 2008

Suppose a group H is given as a free product with amalgamation

determined by groups A0, A1, A2 and homomorphisms α1: A0 → A1, α2: A0 → A2. Thus H may be described as the quotient of the free product A * A2 by the relations i1 α1 (α0) = i2α2 (α0) for all α0 ∈ A0, where i1, i2 are the two injections of A1, A2 into A1 * A2. We do not assume that α1, α2 are injective, so the canonical homomorphisms α′i: Ai → H, i = 0,1,2, also need not be injective.
The central limit problem for trimmed sums
Authors: Griffin Philip S. , Pruitt William E.
Published: November 2008

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.