bluuebunny/crossref_metadata_2025 huggingface dataset:
Local theorems and group extensions
Authors: Hickin Kenneth K. , Phillips Richard E.
Published: November 2008

The concept of a local system of a set W is defined in ((8), p. 166) and ((12), p. 126). Recall that a set ℒ of subsets of W is a local system if Uℒ = W and ℒ is directed in the following sense: for every finite set H1, …, Hn of elements of ℒ, there is an M∈ℒ such that Hi⊂M for 1≤i≤n. If Σ is a class of groups, L(Σ) is the class of all groups G that possess a local system of Σ subgroups. Σ satisfies the local theorem, or is L-closed, if L(Σ)⊂Σ. Many classes of groups which satisfy the local theorem are discussed in ((12), pp. 126–144).
Grothendieck groups of quadratic forms andG-equivalence of fields
Authors: Szymiczek K.
Published: November 2008

Letkbe a field of characteristic other than 2 and letg(k)denote the multiplicative groupk* of the fieldkmodulo squares, i.e.g(k)=k*/k*2. This is an abelian group of exponent 2 and its order, if finite, is a power of 2. We denote byG(k)the Grothendieck group of quadratic forms overk.
On a conjectured analogue to Clifford's chain in [4]
Authors: Longuet-Higgins M. S.
Published: November 2008

Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):

(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.

(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.

(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).

(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).

And so on (Clifford(3)).

Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:

(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.

(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.

(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.

This leads naturally to the conjecture

(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.
The Kahn–Priddy theorem
Authors: Adams J. F.
Published: November 2008

Let ΦSr(X) be the stable homotopy group

where SnX means the n-fold suspension of X. For example, the groups ΦSr(S0) are the stable homotopy groups of spheres. Let

be the ‘infinite-dimensional’ orthogonal group. Then topologists are familiar with the ‘stable J-homomorphism’

G. W. Whitehead observed that J factors through an ‘even more stable’ J-homomorphism

he conjectured that J′ is epi (for r > 0).
Internal-inertia waves in a fluid of variable depth
Authors: McKee W. D.
Published: November 2008

Waves in a rotating, stratified fluid of variable depth are considered. The perturbation pressure is used throughout as the dependent variable. This proves to have some advantages over the use of the vertical velocity. Some previous three-dimensional solutions for internal waves in a wedge are shown to be incorrect and the correct solutions presented. A WKB analysis is then performed for the general problem and the results compared with the exact solutions for a wedge. The WKB solution is also applied to long surface waves on a rotating ocean.
Open decompositions on ordered convex spaces
Authors: Wong Yau-Chuen
Published: November 2008

Let (E, ) be a topological vector space with a positive cone C. Jameson (3) says that C given an open decomposition on E if V ∩ C − V ∩ C is a -neighbourhood of 0 whenever V is a -neighbourhood of 0. The concept of open decompositions plays an important rôle in the theory of ordered topological vector spaces; see (3). It is clear that C is generating if C gives an open decomposition on E; the converse is true for Banach spaces with a closed cone, by Andô's theorem (cf. (1) or (9)). Therefore the following question arises naturally:

(Q 1) Let (E, ) be a locally convex space with a positive cone C. What condition on is necessary and sufficient for the cone C to give an open decomposition on E?
On Moore graphs
Authors: Damerell R. M.
Published: November 2008

In this paper, we shall first describe the theory of distance-regular graphs and then apply it to the classification of Moore graphs. The object of the paper is to prove that there are no Moore graphs (other than polygons) of diameter ≥ 3. An independent proof of this result has been given by Barmai and Ito(1). Taken with the result of (4), this shows that the only possible Moore graphs are the following:
Some generalizations of the Borsuk–Ulam theorem and applications to realizing homotopy classes by embedded spheres
Authors: Fenn Roger
Published: November 2008

In this paper, some theorems of the Borsuk-Ulam type (1) are given. One of these can be applied to show that certain homotopy classes in manifolds cannot be realized by embedded spheres. The n-dimensional sphere Sn is the subset of the euclidean space

Rn+l consisting of all points (x1, …,xn+1) satisfying . Let be a piecewise linear (PL) involution on Sn without fixed points.
The reflexion of an acoustic pulse by a plane vortex sheet
Authors: Jones D. S.
Published: November 2008

This paper deals with the influence of a vortex sheet separating two fluids in relative motion on the radiation from a point source of sound. Both the harmonic and impulsive sources are considered and it is found that waves due to Helmholtz instability must be included in order to ensure that there is no field before the source is excited. The instability wave is confined to a finite region and dominates other disturbances in that region. It is suggested that the instabifity wave is initiated by the unrestricted growth of the specularly reflected wave.
On groups satisfying the converse of Lagrange's theorem
Authors: Humphreys J. F.
Published: November 2008

In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).
Sub-Hopf-algebras of the Steenrod algebra
Authors: Adams J. F. , Margolis H. R.
Published: November 2008

In (3) the second author has shown that all sub-Hopf-algebras of the mod 2 Steenrod algebra have a certain form; this resulthas been used in (1) and (2). The analogous result for the mod p Steenrod algebra, where p is an odd prime, is contained in (6). The object of this paper is to give a shorter proof of a slightly sharper result; we construct all the subalgebras in question. In the mod 2 case this sharper result has also been obtained by Anderson and Davis (2).
Order and commutativity in C*-algebras
Authors: Archbold R. J.
Published: November 2008

In this paper, we use the results of (3) to show that a condition which is formally much weaker than that given by Sherman in (11) is necessary and sufficient for the commutativity of a C*-algebra. The basic idea behind the proof is then used again to obtain a characterization in order-theoretic terms of the self-adjoint elements in the centre of a C*-algebra. We shall use the notation and terminology of (3).
On isotropic tensors
Authors: Jeffreys Sir Harold
Published: November 2008

(1) Proc. Cambridge Philos. Soc. 73 (1973), page 173

Dr E. Leibowitz has called my attention to a mistake in my paper on isotropic tensors. (1) On p. 173, if 12 is annihilated as stated and then we annihilate l13 by the same method, a new contribution to l12 appears. The correct method is to choose α so as to annihilate l21. Then a suitable rotation on the 13 plane removes l31 without re-introducing l21. The rest of the argument stands.
A conjecture for polytopes
Authors: Lillington J. N.
Published: November 2008

All sets considered in this paper will be subsets of n-dimensional Eucidean space En. In this paper, we shall consider the ‘total edge-lengths’ of polytopes which are inscribed in a given sphere and which contain its centre. We first, however, mention some related problems considered by various authors which may be of interest to the reader.
De Sitter symplectic spaces and their quantizations
Authors: Rawnsley J. H.
Published: November 2008

The de Sitter group, Spin (4, 1), is a simply connected, semi-simple, ten-dimensional Lie group which can be contracted to the inhomogeneous Lorentz group. Physical systems with the de Sitter group as asymmetry group should resemble those of the inhomogeneous Lorentz group and may provide an alternative to these systems of special-relativistic physics. Details of physics in de Sitter space from the group theoretical view-point are given in (3). The de Sitter group is also known to be a symmetry group for the bound states of the hydrogen atom, and recent work has shown how this group acts on the corresponding classical system, the Kepler problem. See (8).
Fine topologies for Minkowski space
Authors: Williams Gareth
Published: November 2008

Fine topologies are discussed for Minkowski space. The conformal group and projective group have representations as C1 groups of homeomorphisms of various topologies.
The Kontorovich–Lebedev transformation on distributions of compact support and its inversion
Authors: Zemanian A. H.
Published: November 2008

1. During the past decade a variety of integral transformations have been extended to various classes of distributions. However, one important integral transformation, which has defied such generalization, is the Kontorovich–Lebedev transformation. The objective of this note is to alter that situation.
Geometrical aspects of the theory of non-homogeneous Markov chains
Authors: Kingman J. F. C.
Published: November 2008

A geometrical representation of the transition matrices of a non-homogeneous chain with N states, in terms of certain convex subsets of , is used to describe aspects of the chain. For example, an important theorem of Cohn on the structure of the tail σ-field is a simple corollary. The embedding problem is shown to be entirely geometrical in character. The representation extends to Markov processes on quite general state spaces, and the tail is then represented by the projective limit of these convex sets.
Proof of the Churchhouse conjecture concerning binary partitions
Authors: Gupta Hansraj
Published: November 2008

In a recent paper, R. F. Churchhouse has studied the function b(n) giving the number of partitions of n into powers of 2. The generating function of b(n) is

so that

Using this relation, Churchhouse has shown that

and

where

with

Moreover

and b(n) is even for n ≥ 2, while b(2n) ≡ (mod 4), for n = 22m−l(2k + 1); and, it is not ≡ (mod 8) for any n. He has conjectured that for k ≥ 1 and n odd,

and

The object of this note is to prove these conjectures.
Stochastic monotonicity and queues subject to tidal interruptions
Authors: Gittins J. C.
Published: November 2008

A generalization is given of the notion of stochastic monotonicity. This is used in setting up, and showing how to obtain approximate solutions for, an integral equation formulation of GI/G/l queues, when access to the waiting-line is subject to periodic interruptions.
Two-dimensional anisotropic elastic waves emanating from a point source
Authors: Payton Robert G.
Published: November 2008

A two (spatial) dimensional initially strained elastic body is excited by a point impulse. Expressions are found for the various displacement components in a form which is readily evaluated by residues. The solid itself is characterized by three parameters which depend on the material properties and the initial deformation. For the case when two of these parameters are equated, explicit expressions for the displacements are given along the Cartesian axes passing through the origin of the point impulse. Wave front singularities and lacunas are identified and discussed. Some typical numerical results are given.
Kernels with only a finite number of characteristic values
Authors: Swann Dale W.
Published: November 2008

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we mean

and let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfying

For such kernels, the iterated kernels,

are well-defined (1), as are the higher order traces

Carleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinant

the latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).
The additive and multiplicative orders of a complex line-bundle
Authors: Pittie Harsh
Published: November 2008

Let X be a finite CW-complex, and ξ: Eξ → X a complex line-bundle on X. This bundle determines an element c1(ξ) in H2(X:Z) and a class ; let m(ξ), a(ξ) be the orders (possibly infinite) of these elements in their respective groups. How are m(ξ) and a(ξ) related? The answer is contained in the following three propositions.
On a singular integro-differential equation in aerofoil theory
Authors: Porter D.
Published: November 2008

In this note a brief description is given of a method for reducing a type of singular integro-differential equation to both an integral equation with a quasi-regular kernel and an integral equation with a regular kernel. A particular integro-differential arising in aerofoil theory is used as illustration.
Uniform progressing wave solutions of the wave equation
Authors: Zauderer Erich
Published: November 2008

The solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions is discussed in the book by Courant(l). He also refers to the work of Hadamard, Friedlander, Ludwig and others on this subject. More recently, Ludwig (2), Lewis(3) and others have considered 'uniform' progressing wave expansions for various problems. These expansions are valid in regions where the standard expansions are not suitable and they can be re-expanded in the standard form outside these regions. Examples of such regions are given by envelopes of bicharacteristic curves or, equivalently, caustics and by shadow boundaries such as occur in diffraction problems. In each of these regions, which we term 'transition regions' different types of uniform expansions are required.
On the Stieltjes transform of generalized functions
Authors: Pandey J. N.
Published: November 2008

If f(t) belongs to L(0, R) for every positive R and is such that the integral

converges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis and

for any positive ξ at which f(ξ+) and f(ξ−) both exist.

We define an operator Lk, t[F(x)]by

Under the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),

Let Ln, x denote the differentiation operator

Suppose that

converges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,
On piecewise linear embeddings
Authors: Smith Anthony
Published: November 2008

Notation: Let M and Q be manifolds. The embedding f: M → Q is said to be proper if f−1∂Q = ∂M.
Uniform summability of power series
Authors: Kurtz J. C. , Sledd W. T.
Published: November 2008

It is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.

If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.

Applications are made to theorems on summability factors.
Integrals with a large parameter. Several nearly coincident saddle points
Authors: Ursell F.
Published: November 2008

In the contour integral

the functions g( ) and f( ) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with the l parameters, denoted by α = (α1,α2,…,αl), and it may happen that two or more saddle-points coincide at a certain value of α. The asymptotic expansions given by the ordinary method of steepest descents are then non-uniform. The case of a single parameter and two nearly coincident saddle- points was studied earlier and leads to Airy functions. Here we are concerned with the case of l parameters and of m saddle-points which are nearly coincident when α is near 0. It is then known that f(z,α) can be transformed locally into a polynomial of degree m + 1, and accordingly we here restrict f(z,α) to be such a polynomial. The form of the resulting asymptotic expansion had already been conjectured (correctly as we now see), and involves certain special functions of m–1 variables. To establish the validity of the expansion we must show that the remainder after any number of terms is of smaller magnitude than the last term retained in the expansion. In other words, we must establish an inequality between integrals which is to hold for sufficiently small α and sufficiently large N, and which involves 1 parameters and m nearly coincident saddle-points.
On the existence of transverse elastic acceleration waves
Authors: Chadwick P. , Currie P. K.
Published: November 2008

It is shown that, at an arbitrary point in a general elastic material and regardless of the prevailing state of deformation, there exists in every plane at least one direction in which a transverse acceleration wave may propagate. The possibility of a transverse wave actually being able to progress in such a specific direction depends upon the associated local wave speed being real and conditions are obtained under which this requirement is satisfied.
Partitions of a finite set into structured subsets
Authors: Pólya G.
Published: November 2008

The aim of this paper is to present a common generalization of and , the Stirling numbers of the first and the second kind ((1), ex. 186–210, pp. 42–45).
Large cycles in large labelled graphs
Authors: Wright E. M.
Published: November 2008

An (n, q) graph consists of n labelled nodes and q edges, i.e. unordered pairs of different nodes. The maximum possible value of q is N = ½n(n − 1) and the (n, N) graph is the complete graph. We write B(h, k) = h!/{k!(h − k)!}. The number of (n, q) graphs is clearly B(N, q). A k-cycle in a graph consists of a sequence

of k edges, where D1D2,…, Dk are all different nodes. We are concerned here with the probability P = P(n, q, k) that a k-cycle will be present in an (n, q) graph, that is, the proportion of all (n, q) graphs which contain at least one k-cycle. We investigate the limit of P as n → ∞ and in particular try to determine when P → 1 (when we say that almost all (n, q) graphs contain a k-cycle) and when P → 0 (when we say that almost no (n, q) graphs contain a k-cycle).
Topological measure spaces: two counter-examples
Authors: Fremlin D. H.
Published: November 2008

The ‘Radon measures’ of N. Bourbaki(1) enjoy many striking properties. Among the most important of these is the ‘strong Radon-Nikodým theorem’ that the dual of L1-(μ) can always be identified with L∞(μ) ((1), chap. 5, §5, no. 8, theorem 4). As this is certainly not true of non-σ-finite measures in general, it is natural to ask what are the special properties on which it relies.
A characterization of certain compact sets of measure zero
Authors: Connolly D. M. , Williamson J. H.
Published: November 2008

We show that the only compact subsets of [0, 1] of measure zero are the obvious ones, and extend this idea to show that the only compact subsets of [0, 1] that generate semigroups of ℝ of measure zero are the obvious ones. We apply this last result to prove that any proper Raikov system of subsets of ℝ, with a single generator, is contained in a strictly larger proper Raikov system.
Well-quasi-orderings and sets of finite sequences
Authors: Laver Richard
Published: November 2008

Section 1 contains a list of the facts about well-quasi-orderings (wqo's) which were established in (1), (4), (7), (10), and (13), with proofs given except for the last three theorems. The theory of well-quasi-orderings is a precursor to Nash–Williams' theory of better-quasi-orderings (10, 11, 12, 6, 7). Section 1 of this paper may be viewed as a beginning to Section 1 of a forthcoming paper, in which an explication of bqo theory is given.
On complete subgraphs of different orders
Authors: Bollobás Béla
Published: November 2008

LetSbe a set and let {X1, …,Xn} =be a family of distinct subsets ofS. Theintersection graphΩ() ofhas vertex set {X1, …,Xn} andXiXj(i≠j) is an edge of Ω() if and only ifXi∩Xi≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph hasn≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph withn≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graphGiscoveredwith the subgraphsG1, …,Gk, if everyedgeofGis in at least oneGi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).
The wave front set of the solution of a simple initial-boundary value problem with glancing rays
Authors: Friedlander F. G.
Published: November 2008

1. The singularities of solutions of linear partial differential equations are most appropriately described in terms of wave front sets of distributions. (For the definition and properties of wave front sets, see (4) and (5).) In particular, let Ω ⊂ RN be an open set and let P be a linear partial differential operator defined on Ω, with smooth coefficients, real principal symbol p(x, ξ), and simple characteristics. Then if u ∈ ′ (Ω) and Pu = 0, the wave front set of u, WF(u), is a set in the cotangent bundle T*Ω (minus the zero section) that is contained in p−1(0) and is invariant under the flow generated in p−1(0) by the Hamiltonian vector field (gradξp, − gradxp). In other words, WF(u) is made up of bicharacteristic strips. This is a special case of an important theorem due to Hörmander (5), and Duistermaat and Hörmander (1), which generalizes classical results on the relation between discontinuities of solutions of a partial differential equation and the characteristics of the equation.
The disintegration of invariant measures
Authors: Ripley B. D.
Published: November 2008

Krickeberg (in (5) and (6)) showed that disintegration applied to invariant measures sometimes yields an integral representation which is useful in analysing the moment measures of point processes. His results, based on Bourbaki's disintegration theory, raised several questions. We refine the theory, using a more general disintegration theorem, and answer his questions by several examples. Finally we consider how far the enlarged theory is applicable in stochastic geometry.
The equation A(t, u(t))′ + B(t, u(t)) = 0
Authors: Calvert Bruce
Published: November 2008

The initial value problem for the equation

has been studied recently as a model for long waves in nonlinear dispersive systems. Benjamin, Bona and Mahony (2) introduced this equation as an alternative to the KdV equation of Korteweg-de Vries. Hence, it is referred to as the BBM equation. They studied solutions u(x, t) of the BBM equation for t ≥ 0 and x∈(− ∞, ∞), satisfying u(x, 0) = g(x). Bona and Bryant(1) carried through the study of the BBM equation for t ≥ 0 and x ∈ [0, ∞), satisfying u(x, 0) = g(x) and u(0, t) = h(t). The aim of this paper is to study the equation

where At and Bt are mappings defined on subsets of Banach spaces, especially when At is a second order elliptic operator and B is a differential operator of lower order, defined on an unbounded subset Ω of .
A general theory of separation, regularity, and connectedness properties
Authors: McSherry D. M. G.
Published: November 2008

The notion of basic topological binary relations is introduced, and corresponding to each such relation we define -regularity, -connectedness, and a separation axiom . The family of separation axioms so obtained is shown to include all the standard axioms T0, T1, …, T5, while familiar properties such as hyperconnectedness and ultraconnectedness are among the class of connectedness conditions. The regularity conditions include R0, R1, and regularity itself.
Stochastic models with electrical analogues
Authors: Kelly F. P.
Published: November 2008

1. Introduction. It has long been known that a connexion exists between reversible random walks and electrical networks ((2), (4) pp. 303–310, (6)). In section 2 we outline this connexion. More recently Kingman (5) has exhibited an electrical analogy for a flow model. In section 3 we show that this analogy follows from the results on random walks and we extend the analogy to cover the transient behaviour of Kingman's model. In so far as this model may be regarded as a naive description of the mechanism by which electrons move in a conductor it provides a physical explanation of the connexion between random walks and electrical networks. In section 4 we describe an invasion model which was inspired by the work of Clifford and Sudbury (1) on models to represent competition between species. Adapting their method we display the relationship of this model to random walks and hence obtain an electrical analogy for another class of stochastic processes.
Complex potential equations I. A technique for solution
Authors: Collins C. B.
Published: November 2008

This work amalgamates and solves certain problems arising in differential equation theory and in classical differential geometry.

It describes a novel technique for solving systems of non-linear partial differential equations of the form

where f(γ) and g(γ) are arbitrarily assigned functions. The circumstances are determined under which compatible solutions exist, not only when γ is real, but also when γ is complex, and all of the corresponding solutions are found. This is done by using a geometric technique that incorporates the equipotential surfaces of constant γ. In general, these surfaces are imaginary, and a fairly extensive treatment of such surfaces in (complexified) 3-dimensional Euclidean space is included. A close association is discovered between the set of equipotential surfaces and the class of surfaces of constant radii of principal normal curvature.
Laguerre polynomials and derangements
Authors: Jackson D. M.
Published: November 2008

Consider a sequence of n = n1 + … + nk objects of k distinct types such that there are ni objects of type i, i = 1, …, k. A derangement of this sequence is a permutation such that no object is in a position which was occupied by an object of the same type. To fix ideas, we may suppose that the sequence is initially ordered so that objects of type i precede those of type i + 1, i = 1, …, k – 1. Let Pn. denote the number of derangements of this sequence, where n = (n1, …, nk).
Some more characterizations of Banach spaces containing l1
Authors: Haydon Richard
Published: November 2008

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:

a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.
On groups of automorphisms of operator algebras
Authors: Moffat J.
Published: November 2008

In section 3 we shall prove the following results: Let G be a separable locally compact abelian group, R a von Neumann algebra acting on a separable Hilbert space, and α a weakly continuous representation of G by inner *-automorphisms of R, say α(g) = ad Wg with Wg ∈ U(R). Then there is a weakly continuous unitary representation of G, by unitaries in R, implementing α if and only if the Wg's commute with each other. The result was motivated by the proof of (7), theorem 1. Suppose now Gis a discrete amenable group of *-automorphisms of a countably decomposable von Neumann algebra R. In section 3 we give a necessary and sufficient condition for the existence of a faithful normal G-invariant state on R. This generalizes a result of Hajian and Kakutani on invariant measures (2).
Perturbations by operators converging compactly to zero
Authors: Martens Eric
Published: November 2008

This paper considers perturbations in locally convex spaces of semi Fredholm operators T by a sequence of operators {Kn} converging compactly to zero in a sense extending that from (2). It is shown that, under suitable conditions, α(T + Kn) ≤ α(T), β(T + Kn) ≤ β;(T) and κ(T + Kn) = κ(T) for large n.
The eigenvalue problem for infinite systems of linear equations
Authors: Sayer F. P.
Published: November 2008

Given an infinite system of linear equations

where the aij depend on a parameter λ, the eigenvalue problem is to determine values of λ for which xj (j = 1, 2, …) are not all zero. This problem (Taylor (3) and Vaughan (4)) can arise in the vibration of rectangular plates. Little theoretical work, however, appears to have been done concerning the existence and determination of the eigenvalues. The usual procedure (see (3) and (4)) is to consider a truncated or reduced system of N equations and find the values of λ for which the determinant of the N × N matrix [aij] vanishes. If a particular λ tends to a constant value as N is increased then this value is assumed to be an eigenvalue. The question therefore arises as to what happens if no limit exists. Can we assert that there are no eigenvalues? By constructing an appropriate example we show that the non-existence of a limit does not imply the non-existence of eigenvalues. In order to construct our example we first establish a result concerning the Legendre polynomials.
An algorithm for the total, or partial, factorization of a polynomial
Authors: Farmer M. R. , Loizou G.
Published: November 2008

A globally convergent algorithm is presented for the total, or partial, factorization of a polynomial. Firstly, a circle is found containing all the zeros. Secondly, a search procedure locates smaller circles, each containing a zero, and the multiplicities are then calculated. Thirdly, a simultaneous Iteration Function is used to accelerate convergence. The Iteration Function is chosen from a class of such functions derived herein to deal with the general case of multiple zeros; various properties of these functions are also discussed. Finally, sample numerical results are given which demon-strate the effectiveness of the algorithm.
A constrained isoperimetric problem
Authors: Singmaster David , Souppouris D. J.
Published: November 2008

Of all sets lying within a given convex plane set S, which one gives the greatest ratio of area A to perimeter P? This was posed to one of us by E. Bombieri for the case when S is a square. He asserted that T. Bang used an easy upper bound on A/P in an elementary proof of the Prime Number Theorem. After solving this problem from scratch, we learned from M. Silver that Besicovitch (1) had solved a closely related problem, which we call Besicovitch's problem and which will be discussed shortly. Later, we found that the problem of maximizing A/P had occurred to Garvin (4, 5) and that he solved the problem when S is a triangle. Garvin inspired DeMar (2) to consider and solve Besicovitch's problem for the triangle and for polygons. DeMar found that Steiner (8, p. 166) had solved Besicovitch's problem for triangles, but Steiner was only interested in subsets which touched all sides of a polygon (8, p. 168). More recently, J. Wills has referred us to Herz and Kaapke (6) who use Besicovitch's results to solve our problem of maximizing A/P for polygons circumscribed about a circle.
Weak convergence of randomly indexed sequences of random variables
Authors: Aldous D. J.
Published: November 2008

Let Yn ⇒ Y∞ be a sequence of random variables converging in distribution, or more generally a sequenceof random elements of a suitable metric space whose distributions are converging weakly. Let τn → ∞ be positive integer-valued random variables. If {τn} and {Yn} are independent, it is trivial that
Resolving zero-dimensional singularities in generalized manifolds
Authors: Bryant J. L. , Lacher R. C.
Published: November 2008

A generalized m-manifold is a euclidean neighbourhood retract X that is a homol-ogy m-manifold, i.e. for which H*(X, X–x;Z) = H*(Rm, Rm–0;Z) for each point x of X. In a generalized m-manifold X, a pointy is called singular, or a singularity of X, if p has no neighbourhood homeomorphic to Rm. We shall denote the set of singular points of X by S(X).
Dual modules
Authors: Mitchell W. J. R.
Published: November 2008

Introduction and statement of results. It is well known that if V is a vector space over a field k then its dual vector space Homk (V, k) has dimension not less than that of V, and that the dimensions are equal precisely when V is finite-dimensional. In this note we introduce a ‘dual module’ for a module over a principal ideal domain, and show that some though not all of the properties of the field case carry over.
Dualizing complexes and flat homomorphisms of commutative Noetherian rings
Authors: Hall Janet E. , Sharp Rodney Y.
Published: November 2008

All rings considered in this paper will be commutative and Noetherian, will have identities, and will be assumed to be non-trivial unless otherwise specified. The letter A will always denote such a ring. It is to be assumed that ring homomorphisms respect identity elements.
A localization theorem in homological algebra
Authors: Miller Haynes R.
Published: November 2008

In (1), J. F. Adams showed that for p odd, the Adams E2-term for a sphere, , is zero for s ≤ t ≤ (2p − 1)s − 1, while is the one-dimensional vector space generated by where corresponds to the Bockstein.
A Vidav theorem for Banach Jordan algebras
Authors: Youngson M. A.
Published: November 2008

In the present article we prove a result characterizing the Jordan analogues of B*-algebras among the complex Banach Jordan algebras in terms of the algebra and norm structures alone.
Transport theory in anisotropic media
Authors: Williams M. M. R.
Published: November 2008

We develop a theory of particle scattering in anisotropic media. That is, a medium in which the microstructure causes the mean free paths of the particles to become dependent on their direction of motion with respect to some fixed axis. The equation which results is similar to the normal, one-speed Boltzmann transport equation but has cross-sections which are functions of direction. This equation is solved for arbitrary cross-sectional dependence on direction in plane geometry.
A remark on the construction of random measures
Authors: Mecke J.
Published: November 2008

In his paper (2), Ripley has shown that some of the basic results of point process theory can be established without making any use of topological assumptions concerning the phase space. For the construction of distributions of point processes he needs pseudo-topological assumptions. In the present paper, another sufficient condition is given for the fact that a distribution of a random measure can be constructed by help of finite-dimensional distributions. This condition is expressed by purely measure-theoretic notions. Roughly speaking, it is supposed that concerning the existence of conditional probabilities the phase space behaves as if it were a Polish space. Measureable spaces with such a property are called ‘full measurable spaces’; they are identical with the measurable spaces of type (B) in (1).
Eberlein compacts and spaces of continuous functions
Authors: Hunter Richard J. , Lloyd J. W.
Published: November 2008

Let X be a Hausdorff topological space. We consider various locally convex spaces of continuous real valued functions on X and give necessary and sufficient conditions in order that (i) they contain an absolutely convex weakly compact total subset and (ii) they contain an absolutely convex total subset which is an Eberlein compact, when given the weak topology.
Phase-integral treatment of wave reflexion by real potentials
Authors: Lundborg Bengt
Published: November 2008

Approximate analytical expressions for the wave reflexion amplitude for transmission above one-dimensional real smooth and slowly varying potentials in the absence of classical turning points are obtained in two different ways by means of the phase-integral method developed by N. Fröman and P. O. Fröman. The wave reflexion is shown to be governed by the complex transition points for the problem. The great accuracy of the formulae obtained, as well as their respective ranges of applicability, is illustrated by applying them, up to the eleventh order phase-integral approximation, to an exactly solvable model, for which the potential represents a monotonic step.
Multi-indexing and multiple clustering
Authors: Melzak Z. A.
Published: November 2008

1. Simple linear clustering. One of the simplest clustering problems is the following: if n points are taken independently and uniformly at random on the interval I = [0, L], what is the probability Q = Q(n, a, L) that no two points are closer than a? The well-known answer

if (n − 1) a ≤ L, and Q = 0 otherwise, can be obtained in a variety of ways. Perhaps the simplest one is the original method of E.C.Molina(5). If x1,…,xn are the abscissas of the n points, then there are n! equiprobable orderings of the xi's consistent with the condition that no two points are closer than a. The probability of each one is

Let yi = xi − (i − 1) a for i = 1,…,n, then the above becomes

which is
Proof of the antithetic-variates theorem for unbounded functions
Authors: Wilson James R.
Published: November 2008

The method of antithetic variates introduced by Hammersley and Morton (2) is one of the most widely used Monte Carlo techniques for estimating an unknown parameter θ. The basis for this method was established by Hammersley and Mauldon(l).in the case of unbiased estimators with the form

where each of the variates ξj is required to have a uniform marginal distribution over the unit interval [0,1]. By assuming that n = 2 and that the gj are bounded Borel functions, Hammersley and Mauldon showed that the greatest lower bound of var (t) over all admissible joint distributions for the variates ξj can be approached simply by arranging an appropriate strict functional dependence between the ξj. Handscomb(3) extended this result to the case of n > 2 bounded antithetic variates gj(ξj). In many experiments involving distribution sampling or the simulation of some stochastic process over time, the response functions gry are unbounded. This paper further extends the antithetic-variates theorem to include the case of n ≥ 2 unbounded antithetic variates gj(ξj) each with finite variance.
The modular curves X0(65) and X0(91) and rational isogeny
Authors: Kenku M. A.
Published: November 2008

Let N be an integer ≥ 1. The affine modular curve Y0(N) parameterizes isomorphism classes of pairs (E; F), where E is an elliptic curve defined over ℂ, the field of complex numbers, and F is a cyclic subgroup of order N. The compacti-fication X0(N) is an algebraic curve defined over ℚ.
JV-algebras
Authors: Moreno J. Martinez
Published: November 2008

Let J be a complex Banach space and a complex Jordan algebra equipped with an algebra involution *. Then J is a Jordan C*-algebra if the following conditions are satisfied:

(Ua is defined on page 3).
Hamiltonian and algebro-geometric integrals of stationary equations of KdV type
Authors: Wilson George
Published: November 2008

In this paper we shall generalize a theorem of Bogoyavl'enskii (2) showing the equivalence of two apparently different families of integrals of the ‘higher stationary Korteweg–de Vries (KdV)’ equations. We recall that the KdV equations form a hierarchy of evolution equations

for an unknown function u(x, t). The equations can be written in ‘Lax form’ (7)

where L is the Schrödinger operator (δ2/δx2) + u, and P+ runs through all the (ordinary) differential operators such that the commutator on the right of (1·1) has order zero.
A theorem on the roots of a certain equation involving Bessel functions
Authors: Steinitz R.
Published: November 2008

It is well known (Watson, A treatise on Bessel functions) that if Jν(z) is a Bessel function of the first kind, and A is a real constant, then all roots of the equation are either real or purely imaginary.
Einstein–Maxwell fields with plane symmetry. I
Authors: Patnaik S.
Published: November 2008

An exact static solution of the system of Einstein–Maxwell field equations in the absence of charge is obtained when both the metric tensor gαβ and the Maxwell field tensor Fαβ exhibits plane symmetry in the sense that both remain invariant under the group of motions that characterize plane symmetry. Our solution generalizes the empty space-time solution with plane symmetry previously obtained by Taub to the situation when static charge-free electromagnetic fields are present. We hope to discuss non-static solutions of Einstein–Maxwell fields with plane symmetry as well as some other considerations in a subsequent paper.
The attenuation of surface waves by scattering
Authors: Hudson J. A.
Published: November 2008

Small variations in the elastic parameters of a solid medium give rise to scattering of elastic waves and hence to a reduction in their amplitudes. The attenuation coefficients associated with this process have been calculated for surface waves.

Expressions have been derived for waves with wavelengths long compared with the mean wavelength of the variation of the elastic parameters in space. The attenuation coefficients were found to be the same as if the attenuating mechanism were a particular type of visco-elasticity.
A quadratic recurrence of Faltung type
Authors: Wright E. M.
Published: November 2008

We write and x1 = 1. In a recent paper (3), Stein and Everett consider the sequence defined by

and investigate whether

tends to a limit as n → ∞. The case b = 0 has a combinatorial interpretation (see (1)) and, in (2), they use this to prove that xn → e−1 in this case. Even for positive integral b, the number Sn has no known combinatorial interpretation, but they prove (3) that xn → x under the hypothesis that Sn+1Sn−1 ≥ Sn i.e. that Sn is convex. By computation and induction, they prove this hypothesis for b = k/5, where k = 1, 2,…, 50.
Some inequalities for martingales and applications to the study of L1
Authors: Dor Leonard E.
Published: November 2008

We obtain here estimates for the expectation of a restricted quadratic variation of L1 bounded (weak) martingales and positive submartingales, and for related functionals. These inequalities are then used to give a quantitative version of a recent result of D. J. Aldous and D. H. Fremlin concerning the L1 norms of expansions with respect to uniformly integrable L1-normalized martingale difference sequences. Another application relates precisely the degree of norm divergence of an L1-bounded martingale to the degree of non-uniform integrability of that martingale. A partial analogue of this result is proved for certain vector valued martingales with values in L1 which are related to the Radon–Nikodým property.
Existence of solutions of Hammerstein equations of compact type
Authors: Becker Ronald I.
Published: November 2008

The existence of solutions is considered for equations of the form

for x ∈ H (a Hilbert space), P a compact linear operator on H; Q(x) a bounded linear operator on H and continuous in x and uniformly bounded; g(x) a continuous uniformly bounded map with range in H. Two situations are considered: Q(x) lies in a weakly compact set of operators for which (a) (I − PQ(x)) is invertible (non-resonance case) or (b) (I − P(Q(x) + λI)) is invertible for 0 < λ ≤ α (resonance case).
Shearing transformation of a linear system at an irregular singular point
Authors: Gilbert Richard C.
Published: November 2008

Consider an nth-order linear ordinary differential equation

Suppose the αj(x) are holomorphic for x ∈ S, 0 < x0 ≤ |x| < ∞, where S is an open sector of the complex plane with vertex at the origin and positive central angle not exceeding π. Suppose as x → ∞ in each closed subsector of S. The problem of finding a basis for the solutions of (1) can be reduced by a sequence of transformations (see, for example, Gilbert (1)) to the problem of finding a fundamental matrix for a system of the form

where q is a non-negative integer, A(x) is an m by m matrix which is holomorphic for x ∈ S, x0 < ≤ |x| < ∞, and as x → ∞ in each closed subsector of S. If m ≥ 2, A0 is an m by m matrix with elements all zero except for l's on the upper off-diagonal, and the elements of Ar for r ≥ 1 are all zero except possibly in the last row. (There is one such problem (2) for each root, μ of the equation with multiplicity m ≥ 2. Recall that the coefficient αn−r, 0 the first term in the asymptotic expansion of the coefficint αn−r(x) of equation (1).) We denote the elements of the last row of Ar by 1 ≤ k ≤ m. The system (2) has an irregular singular point at infinity.
Localized stable homotopy of some classifying spaces
Authors: Snaith Victor
Published: November 2008

In this note I will give new, simplified proofs of some of the results announced in (18) and proved in ((19), I, § 4, II, §§ 2, 9).These results mostly date from 1975/6 at which time they were proved using my crude stable decomposition of ωn σnX (20) and differential-geometric techniques with the Becker-Gottlieb transfer. Since then more systematic approaches have been developed towards the stable decompositions ((4); (5); (6); (10); (12)) and towards the transfer ((7), I and II; (8)). Actually, the system-atization of the stable decompositions was already latent in (15) if only I had realized it!
On the convergence of certain singular integrals
Authors: Falconer K. J.
Published: November 2008

Hardy (3) provides one of the earliest rigorous discussions of the convergence of Riemann multiple integrals of real functions with a general type of algebraic singularity. More recently, and with the advent of Lebesgue integration, various authors such as Garsia(2) have investigated singular integrals involving more general measurable functions. In this paper we shall develop an indirect potential theoretic method to obtain convergence criteria for a class of multiple integral that seems to have been neglected to date.