bluuebunny/crossref_metadata_2025 huggingface dataset:
On the short-wave asymptotic motion due to a cylinder heaving on water of finite depth. II
Authors: Rhodes-Robinson P. F.
Published: November 2008

This paper continues the investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we now present a rigorous method for the special case of a half-immersed circular cylinder. The aim is to justify, in one particular case, those assumptions made in a previous paper giving a non-rigorous method for an arbitrary smooth cylinder which intersects the free surface normally, concerning the asymptotic values on the cylinder of the two related auxiliary potentials which were introduced to enable an evaluation of the coefficients describing wave-making and virtual mass. This is done by constructing an integral equation of the second kind whose kernel is small and deducing the leading term in the iteration solution. The method is an extension of the method for infinite depth and it is found that there is a simple relationship between the kernels. In conclusion, computations for these asymptotic results are presented and compared with the trend of known non-asymptotic results.
Sets of synthesis for general algebras
Authors: Körner T. W.
Published: November 2008

Following a program of Reiter, sets of synthesis in Banach algebras with certain Ditkin conditions are discussed. It is shown that the Ditkin Shilov lemma is in a general context best possible. Various methods for the construction of algebras are discussed. These are applied to show that very strong homogeneity conditions are needed to ensure that star sets are of synthesis for regular Ditkin algebras with Rn as carrier space.
Schauder decompositions in locally convex spaces
Authors: Kalton N. J.
Published: November 2008

A decomposition of a topological vector space E is a sequence of non-trivial subspaces of E such that each x in E can be expressed uniquely in the form , where yi∈Ei for each i. It follows at once that a basis of E corresponds to the decomposition consisting of the one-dimensional subspaces En = lin{xn}; the theory of bases can therefore be regarded as a special case of the general theory of decompositions, and every property of a decomposition may be naturally denned for a basis.
New types of factorizable equations
Authors: Humi Mayer
Published: November 2008

By a generalization of the Infeld-Hull Factorization Kernels we find it possible to factorize a wide family of second-order ordinary differential equations at specified points of their spectrum.
Bounded and invariant elements in 2-firs
Authors: Bowtell A. J. , Cohn P. M.
Published: November 2008

1.Introduction. In a principal ideal domainR, any two-sided ideal is of the formRc=cR, i.e. it has an invariant element as generator, and the customary development of ideal theory in a principal ideal domain (cf. e.g. (10), ch. III) takes on a more transparent form when expressed in terms of invariant elements. Likewise, the one-sided bounded ideals may be studied in terms of their generators.
Factorizations of completely positive matrices
Authors: Markham Thomas L.
Published: November 2008

1. Introduction. DEFINITION. If with aij = aji, is a real quadratic form inx1 …,xn, andwhere Lk = cklx1 + … + cknxn (ckj ≥ 0 for k = 1, …, t), then Q is called a completely positive form, and A = (aij) is called a completely positive matrix.
Homology, Hilbert functions, and the conservation of number
Authors: Northcott D. G.
Published: November 2008

1. Introduction. The principle of the Conservation of Number is concerned with the following situation. One starts with a system of algebraic equations having only a finite number of solutions and then applies a homomorphism whose domain contains the coefficients of those equations. This produces a new system. Let us suppose that the new system of equations also has only a finite number of solutions. The question then arises as to how the number of solutions before specialization compares with the number present afterwards. In a typical geometrical situation, one usually wishes to assert that the two systems have equally many solutions. However, it is easy to construct algebraic situations where the number changes,† and where the change is not to be explained away through the confluence of solutions or by their slipping off to infinity. At first sight this represents a breakdown of the conservation principle, but this principle has proved so useful in the past that one has a natural reluctance to discard it. The alternative is to attempt a reformulation and in (l) the present author gave such a reformulation for the case in which the specialization consisted in mapping a regular local ring on to its residue field. The modified theory requires that we take account of systems of equations which arise in connexion with the homology modules of a certain complex. The system associated with the homology module of degree zero is found to be the same as the one that arises in the naive theory, and usually this is the only one that makes a contribution. However, in cases where the number of solutions appears to change, the other systems become active and act in such a way that the balance is restored. For an amplification of these remarks we must refer the reader to (l). They are made here to indicate how the relevance of homological concepts first became clear in any detail. In the present paper these ideas are taken further, the principal gain being that it is no longer necessary to restrict the type of specialization to that which consists in mapping a regular local ring on to its residue field. Indeed one can use very general specializations provided that one transfers the homological requirement from the ring to the system of equations under consideration. In this way, one obtains a theory which is more general and, in some of its aspects, simpler as well.
A note on sufficient statistics and the exponential family
Authors: Tallis G. M.
Published: November 2008

1. Introduction. The relationship between sufficient statistics and the exponential family was first investigated by Pitman (8) and Koopman (7). The treatments assumed a density function f(x; θ) which was at least differentiable with respect to both arguments.
On the geodesic completeness of causal space-times
Authors: Clarke C. J. S.
Published: November 2008

A geodesic in space-time is complete if it can be extended to infinite values of its affine parameter: it is shown that all strongly causal spaces are conformal to space-times in which all null geodesies are complete, and that a wide class of space-times are conformal to ones in which almost no null geodesies are complete.
Matchings in Arbitrary Graphs
Authors: Brualdi R. A.
Published: November 2008

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.
Electromagnetic wave propagation in an inhomogeneous gyrational medium
Authors: Westcott B. S.
Published: November 2008

The investigation of a recently devised model in which the permittivity profile takes the form of an asymmetrical Epstein layer is adapted for use in a gyrational medium. Reflexion and transmission coefficients for normally incident electromagnetic waves are obtained and resonance properties are discussed.
An axisymmetric contact problem: the constriction of elastic cylinders under axial compression
Authors: Vaughan Henry , Allwood Derek
Published: November 2008

The compression of fairly short solid cylinders under axial load is considered. Radial expansion is prevented over a central region of the outer surface by a rigid constraint concentric with the cylinder. Physically the situation arises when a fairly soft rivet is expanded into a relatively rigid plate. Two types of elastic material are considered; firstly, rubber-like materials governed by a strain energy function of the Mooney form, and secondly, metals which have a quadratic strain energy function. In the former case a finite axial compression is permitted prior to contact between the cylinder and the constraint. In both cases the irregularities introduced by the constraint are sufficiently small that they can be described by infinitesimal elasticity theory. The analysis utilizes displacement potential functions and is reduced to solving a set of dual cosine series. The particular case in which the cylinder height and diameter are equal and the contact height is equal to the radius is examined in detail and the contact stresses are given graphically.
Axial radiation reception
Authors: Chee-Seng Lim
Published: November 2008

A radiating point source is abruptly switched on and, thereupon, constantly maintained while immersed within a multi-dimensional, non-dispersive, originally uniform, unbounded, propagative medium with transverse isotropy, equivalent to an intrinsic axisymmetry. The number of spatial dimensions concerned is odd. The pertinent radiation problem is posed as a partial differential equation coupled with a preactivation zero condition. This problem is, first, resolved by a Fourier integral, which is then ‘frozen’ onto (i.e. evaluated at points on) the axis of symmetry. Postulating appropriate hyperbolic-type as well as elliptic-type conditions, a systematic reduction of the ‘frozen’ integral is carried out, enabling an analytic exact solution to be constructed with reference to certain phase curves in a phase plane. Away from the source, this solution is valid throughout the symmetry axis except at certain discrete, moving singularities. These fall into three different categories. The behaviour near each typical singularity is estimated. Three significant physical features are demonstrated (as corollaries); namely, the implicit compliance with a radiation principle, the linear interception of the symmetry axis by a Petrow sky's lacuna (associated with the instantaneous point impulse), and, the eventual attainment of either a steady or quiet state after an infinite period. Interpretations are provided in terms of the group velocity. To assist in probing into these various aspects, two appendices are separately included. An application is made to consider magnetically aligned, radiation reception within a field permeated, electrically conducting gas.
On Riemann summability of functions
Authors: Kuttner B.
Published: November 2008

1. Let k be a positive integer. With the usual terminology, the series

will be said to be Riemann summable (R, k) to l if

converges for all sufficiently small x, and tends to l as x → 0. Here we take (sin nx)/(nx) as meaning 1 when n = 0. A more general summability method which has been considered by various authors ((1), (2), (5), (6), (8), (9), (10)), and is usually denoted by (ℜ, λ,k) is obtained by replacing (2) by

where λ = {λn} is a sequence of non-negative numbers increasing to ∞.
Manifolds of homotopy type K(π, 1). II
Authors: Johnson F. E. A.
Published: November 2008

The following Conjecture was made in (6).

CONJECTURE A. Let Xn be a closed manifold of type K(π, 1). Then X˜n is homeornorphic to.

In section 4, we prove the conjecture when n ≥ 5 and π is a non-trivial direct product. More general than Conjecture A would be

CONJECTURE B. Let Xn be a manifold of type K(π, 1) with ∂X = Ø. Then X˜n is homeomorphic to.

In section 4 we give an example to show that Conjecture B is false in each dimension n ≥ 4.
On the rate of convergence in a local limit theorem
Authors: Basu Sujit K.
Published: November 2008

Let Zn = n−½(X1 + X2 + … + Xn), where {Xn} is a sequence of independent and identically distributed random variables with EX1 = 0, and a common distribution function F and characteristic function ω. Suppose |ω|r is integrable for some integer r ≥ 1. For all n ≥ r, then Zn has a probability density function fn obtained by using the inversion formula.
On pro-reductive groups
Authors: Moskowitz Martin
Published: November 2008

In the proof of the Freudenthal–Weil theorem in, for example (5), essential use is made of the fact that if G and H are compact analytic groups and ø: G → H is a continuous epimorphism then ø(Z(G)0) = Z(H)0 where the subscript 0 denotes the identity component of a topological group G and Z(G) its centre. Although this is sufficient for the proof of the Freudenthal–Weil theorem it raises the interesting question as to whether actually ø(Z(G)) = Z(H) (from which the above would follow) and, if so, in what generality this can be expected. The present paper deals with this question, in more general form, as well as certain of its structural consequences.
On Borchers class of Markoff fields
Authors: Karwowski W.
Published: November 2008

The possibility of constructing a quantum field theory by means of fields on Euclidean space is based on works by Schwinger and Symanzik (1). Probabilistic methods were used and Nelson has shown (2) that from so-called ‘Markoff fields’ one can construct Wightman fields. This idea turned out to be unusually fruitful as it made available the statistical mechanic's techniques for consideration of quantumfield theory problems. See for example (7). However as in the Minkowski space approach, the only two-dimensional space-time nontrivial models have been proved to fulfil all Wightman axioms. Since the problem for higher dimension theories is still open and extremely difficult, it is useful to have at least some criterion which allows us to eliminate certain procedures as leading to trivial theories. One such criterion existing in the Minkowski space approach is described by a notion of Borchers class (5).
The generation of surface waves over a sloping beach by an oscillating line-source. II. The existence of wave-free source positions
Authors: Morris Clare A. N.
Published: November 2008

An expression is obtained for the amplitude of the waves radiated to infinity by a line source of sinusoidally-varying strength situated in water over a beach of angle α (0 < α ≤ π). It is shown that, for certain positions of the source, this amplitude is zero. Equations for the loci of these positions, and approximations to their solutions in particular cases, are derived.
A complete group of odd order
Authors: Dark R. S.
Published: November 2008

A group G is said to be complete if the centre of G is trivial and every automorphism of G is inner; this means that G is naturally isomorphic to Aut G, the group of automorphisms of G. In response to a question of Rose (10) we shall describe the construction of an example demonstrating the following result. (Rose has pointed out that the problem was mentioned earlier by Miller (8).)
A note on the Fourier transform of generalized functions
Authors: Fisher B.
Published: November 2008

If F(f) denotes the Fourier transform of a generalized function f and f * g denotes the convolution product of two generalized functions f and g then it is known that under certain conditions

Jones (2) states that this is not true in general and gives as a counter-example the case when f = g = H, H denoting Heaviside's function. In this case

and the product (x−1 – iπδ)2 is not defined in his development of the product of generalized functions.
On the asymptotic behaviour of solutions of x″ + a(t)f(x) = 0
Authors: Burton T. , Grimmer R.
Published: November 2008

We consider the equation:

where a: [0, ∞) → R1, a(t) > 0, a'(t) is continuous,

f:( −∞, +∞) → R1, f is continuous, and xf(x) > 0 for x ≠ 0. The problem is to give conditions on a(t) and f(x) to ensure that all solutions of (1) tend to zero as t → ∞. First, however, we give some sufficient conditions and some necessary and sufficient conditions to ensure that all solutions of (1) are oscillatory or bounded.
Separating functions for V-Sidon sets
Authors: Galanis E.
Published: November 2008

We show that if E is a V-Sidon set and K a disjoint compact set then there exists a function of low tensor norm separating them.
Bounds on the Engel class of some group extensions
Authors: Shield David
Published: November 2008

In this paper upper bounds on the Engel length of a group G of prime-power exponent are obtained, under varying conditions on a normal subgroup H and the quotient G/H. Examples are constructed to show that the bounds are best possible when the nilpotency class of H is less than the relevant prime.
Serial subgroups of locally finite groups
Authors: Hartley B.
Published: November 2008

Serial subgroups seem as yet to have received little attention in the literature. In this short note we draw attention to some interesting properties which these subgroups possess in locally finite groups.
The symplectic bordism ring
Authors: Ray Nigel
Published: November 2008

This paper is concerned with the symplectic bordism ring MSp*, whose structure is still largely unknown despite the partial results of Liulevicius(3), Novikov(4) and Stong(lO). It is the first of two elaborating(8), and it relies completely on (7) for notation and prerequisites. In particular we assume the reader to be familiar with (7), section 8, which establishes the relation between the Sp Hattori-Stong conjecture (that KO decides MSp) and the spectral sequence

We are here concerned entirely with computations in this spectral sequence.
The order-bound topology
Authors: Wong Yau-Chuen
Published: November 2008

Let (E, C) be a partially ordered vector space with positive cone C. The order-bound topology Pb(6) (order topology in the terminology of Schaefer(9)) on E is the finest locally convex topology for which every order-bounded subset of E is topologically bounded.
On an integral transform associated with a condition of radiation
Authors: Naylor D.
Published: November 2008

1. There has been considerable debate on the conditions under which a given function f(r) can be expanded as a series of the orthogonal functions where υ1, υ2, … are the zeros of the Hankel function regarded as a function of its order. Here k, a are positive constants and 0 < a ≤ r < ∞. The formal expansion can be written as

where
Weakly non-linear, slowly varying waves and their instabilities
Authors: Grimshaw R.
Published: November 2008

A non-linear Klein–Gordon equation is used to discuss the theory of slowly varying, weakly non-linear wave trains. An averaged variational principle is used to obtain transport equations for the slow variations which incorporate the leading order modulation and non-linear terms. Linearized transport equations are used to discuss instabilities.
The many electron problem
Authors: De Wet J. A.
Published: November 2008

1. Introduction. In a paper on the many nucleon problem (l), which we shall henceforth call I, the determination of the irreducible representations of the four-dimensional unitary group were found from a decomposition of its infinitesimal ring U04 The method of decomposition made use of the four primitive four-component idempotents (projection operators) of the Dirac ring each of which, as was recognized long ago by Eddington (2), can be identified with a possible charge-spin state of a Dirac particle. Some experimental justification for the representations was also provided, and it is the purpose of this paper to apply the same tools to the many electron problem. In particular, matrices will be derived for the spin multiplets of a system of r electrons, and it will be shown how the model can account for the atomic shell structure and orbital angular momentum.
Groups of automorphisms of Abelian 1-groups
Authors: Hartley B.
Published: November 2008

Following Robinson(4), we shall use the term Abelian1-group to describe an Abelian group A of finite (Mal'cev special or Prüfer) rank, whose torsion subgroup T is Černikov. These groups were called Abelian groups of type A3 by Mal'cev (3). Let G = Aut A. It is not hard to see that G/CG(A/T) and G/CG(T) are linear groups, and CG(A/T) ∩ CG(T) is Abelian. We improve on this observation by proving

Theorem 1. The group G contains a normal Abelian Černikov subgroup G0, such that G/G0is linear over a field of characteristic zero.
On the characters of the Weyl group of type D
Authors: Mayer S. J.
Published: November 2008

This paper is a continuation of (2), (3) in the development of a unified theory of the characters of the Weyl groups of the simple Lie algebras using their common structure as reflection groups; compare Carter (1) for a similar development for the conjugacy classes. We look at the Weyl group of type D, which is a subgroup of index two in the Weyl group of type C. It was first studied by Young (4), but rather less is known about the characters of this group than those of types A and C. Indeed, the situation is rather more complicated, but we are able to give, as before, an algorithm to determine irreducible constituents of the principal character of a Weyl subgroup induced up to the whole group. We shall also study the case where the rank of the Weyl group is even, when extra irreducible characters may arise, and after constructing these, we shall state some results on their occurrence in the induced principal character.
Note on the generalized Wallace theorem and related topics
Authors: Babbage D. W.
Published: November 2008

1. Wallace's theorem, that the circumcircles of the four triangles formed by the sets of three from four coplanar lines meet in a point, has a generalization to Euclidean affine space of any even dimension n: namely, if we have n + 2 general primes (or hyperplanes) in such a space, the n + 2 circumhyperspheres of the simplexes formed by the sets of n + 1 of the primes meet in a point.
Extreme points of convex sets of doubly stochastic matrices. II
Authors: Mauldon J. G.
Published: November 2008

We prove a conjecture of (5), namely that the convex set of all infinite doubly stochastic matrices whose entries are all strictly less than θ(0 < θ ≤ 1) possesses extreme points if and only if θ is irrational.
Lp(G)-isotone measures
Authors: Peers Beryl J.
Published: November 2008

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.
Finite simple groups with Sylow 2-subgroups of typePSL(5,q),qodd
Authors: Mason David R.
Published: November 2008

The purpose of this paper is to study finite simple groups whose Sylow 2-subgroups are isomorphic to those ofPSL(5,q) for some oddq.This work originally appeared in the author's Ph.D. Thesis, (14), and at that time, the structure of the centralizers of involutions was worked out – see below for details. Subsequently, in view of the fact that ifq≡ – 1 (mod 4), the Sylow 2-subgroup ofPSL(5,q) has sectional 2-rank 4 (being, in fact, isomorphic to the wreath product of a semidihedral group by a cyclic group of order 2), the complete classification, at least for this case, was needed for Gorenstein and Harada's monumental work on sectional 2-rank at most 4, (8), and therefore Collins and Solomon completed the characterization ofPSL(5,q) andPSU(5,q) (for all oddq) in a neat paper, (5). Combining our result with theirs, we are able to state the following Theorem.
Läuchli's algebraic closure of Q
Authors: Hodges Wilfrid
Published: November 2008

H. Läuchli (9) constructed, within a model of a weak form of set theory, an algebraic closure L of the field Q of rationals which had no real-closed subfield. Läuchli's construction is easily transferred to a model N of ZF (= Zermelo–Fraenkel set theory without the axiom of Choice), and it follows at once that neither of the two following statements is provable from ZF alone:

Every algebraic closure of Q has a real-closed subfield. (1)

There is, up to isomorphism, at most one algebraic closure of Q. (2)
Centraliser rings of multiplication groups on quasigroups
Authors: Smith J. D. H.
Published: November 2008

Let (Q,.) be a finite quasigroup, i.e. a finite set Q with a binary operation. called multiplication such that in the equation x.y = z any two elements determine the third uniquely. Then the mappings R(x): Q → Q; q ↦ q.x and L(x): Q → Q; q ↦ x.q are permutations of Q. The multiplication group G of Q is the subgroup of the symmetric group on Q generated by {R(x), L(x) | x ∈ Q}. If S is a field, G has a faithful representation Ḡ by permutation matrices acting on the S-vector space with Q as basis. The set of matrices commuting with Ḡ forms an S-algebra (under the usual operations) called the centraliser ring V(G, Q) of G on Q. The purpose of this note is to show how the permutation-theoretic object ‘centraliser ring’ may be expressed in terms of the quasigroup structure of Q, both to prepare one tool for the long-term programme of classifying finite quasigroups by means of their multiplication groups, and for comparison with the Schur ring method of group theory.
Characteristics of GWrSn
Authors: Williams A. G.
Published: November 2008

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).
On the fractional parts of certain additive forms
Authors: Baker R. C. , Gajraj J.
Published: November 2008

In this paper, k is a positive integer, k ≧ 2; K denotes 2k−1, and ε is an arbitrary positive number. X = (x1, …, xs) and U = (u1, …, ur) are integer vectors; is the distance from the real number y to the nearest integer.
A geometrical proof that causality implies the Lorentz group
Authors: Nanda S.
Published: November 2008

That causality implies the Lorentz group was first proved by Zeeman (1). Later, Barruchi (2) and Teppati(3) proved a generalised version of the theorem by using completely different arguments. In this paper, we give a very simple and elementary geometric proof of the same theorem.
A wild automorphism ofUsl(2)
Authors: Joseph A.
Published: November 2008

It is known that the automorphism groups of various torsion-free associative algebras over two generators take a particularly simple form. It has been suggested (10), though never proved, that this fails over three or more generators. Here it is shown that this is indeed the case for the enveloping algebra ofsl(2), a result which answers a question implicitly raised in (4). A weaker hypothesis is proposed for such automorphism groups and this is related to the structure of locally nilpotent derivations.
Harmonic functions on a Riemannian ball
Authors: Nakai Mitsuru , Sario Leo
Published: November 2008

Consider the classes OHX of noncompact Riemannian manifolds M of dimension N ≥ 2 carrying no non-constant harmonic functions with properties X = P (positive), B (bounded), D (Dirichlet finite), or BD (B and D). Denote by OG the class of parabolic manifolds of dimension N ≥ 2. The following complete string of strict inclusion relations is known:
Singular n-tuples and Hausdorff dimension
Authors: Baker R. C.
Published: November 2008

1. Introduction. Throughout the paper θ = (θ1, …, θn), φ = (φ1, …, φn), … denote points of Euclidean space Rn. We write Kn for the set of θ in Rn for which θl, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y, … We write

If α is a real number, ∥α∥ denotes the distance from α to the nearest integer.

Let θ ∈ Rn. By a theorem of Dirichlet ((2), chapter 1, theorem VI).

for all X ≥ 1. We say that θ is singular if

Singular points form a set of n-dimensional Lebesgue measure zero. In fact, H. Davenport and W. M. Schmidt (3) showed that

for almost all θ in Rn. Although there are no singular numbers in Kl ((2), p. 94) there are ‘highly singular’ n-tuples in Kn for n ≥ 2.
A generalization of the Königsberg bridges problem
Authors: Domb C.
Published: November 2008

A multi-edge graph is said to be a restricted Euler graph when it can be decomposed into cycles of length greater than 2. Necessary and sufficient conditions are obtained in relation to vertex structure for a graph to be restricted Euler.
On the crystallography of infinite Coxeter groups
Authors: Maxwell George
Published: November 2008

Let V be the vector space of translations of a finite dimensional real affine space. The principal aim of this paper is to study (generally non-Euclidean) space groups whose point groups K are ‘linear’ Coxeter groups in the sense of Vinberg (4). This involves the investigation of lattices Λ in V left invariant by K and the calculation of cohomology groups H1(K, V/Λ) (3). The first problem is solved by generalizing classical concepts of ‘bases’ of root systems and their ‘weights’, while the second is carried out completely in the case when the Coxeter graph Γ of K contains only edges marked by 3. An important part in the calculation of H1(K, V/Λ) is then played by certain subgraphs of Γ which are complete multipartite graphs. The only subgraphs of this kind which correspond to finite Coxeter groups are of type Al× … × A1, A2, A3 or D4. This may help to explain why, in our earlier work on space groups with finite Coxeter point groups (3), (2), components of r belonging to these types played a rather mysterious exceptional role.
The eighty-one types of isohedral tilings in the plane
Authors: Grünbaum Branko , Shephard G. C.
Published: November 2008

1. Atilingis a collection= {Ti|i= 1, 2, …} of closed topological discs which covers the Euclidean planeE2, and of which the individualtiles Tihave disjoint interiors. We shall assume throughout that the intersection of any two tiles is a connected set. If each tile iscongruent(directly or reflectively isometric) to a given setT, then the tilingis calledmonohedralandTis called theprototileof. Clearly every monohedral tiling is locally finite.
Some further results in infinite divisibility
Authors: Shanbhag D. N. , Pestana D. , Sreehari M.
Published: November 2008

Goldie (2), Steutel (8, 9), Kelker (4), Keilson and Steutel (3) and several others have studied the mixtures of certain distributions which are infinitely divisible. Recently Shanbhag and Sreehari (7) have proved that if Z is exponential with unit parameter and for 0 < α < 1, if Yx is a positive stable random variable with , t ≥ 0 and independent of Z, then for every 0 < α < 1

Using this result, they have obtained several interesting results concerning stable random variables including some extensions of the results of the above authors. More recently, Williams (11) has used the same approach to show that if , where n is a positive integer ≥ 2, then is distributed as the product of n − 1 independent gamma random variables with index parameters α, 2α, …, (n − 1) α. Prior to these investigations, Zolotarev (12) had studied the problems of M-divisibility of stable laws.
Non-singular bilinear maps and stable homotopy classes of spheres
Authors: Lam Kee Yuen
Published: November 2008

A bilinear map ø Ra x Rb → Rc is non-singular if ø (x, y) = 0 implies x = 0 or y = 0. For background information on such maps see (4, 5, 6, 14). If we apply the ‘Hopf construction’ to ø, we get a map

defined by 2ø(x, y)) for all x ∈ Ra, y ∈ Rb satisfying ∥x∥2 + ∥y∥2 = 1. Homotopically, Jø coincides with the map obtained by first restricting and normalizing ø to , and then applying the standard Hopf construction ((13), p. 112). In any case, one gets an element [Jø] in , which in turn determines a stable homotopy class of spheres {Jø} in the d-stem , where d = a + b − c −1. An element in which equals {Jø} for some non-singular bilinear map ø will be called bilinearly representable. The first purpose of this paper is to prove
On the transfer in the homology of symmetric groups
Authors: Kahn Daniel S. , Priddy Stewart B.
Published: November 2008

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphism

in group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.
On the closure of a class of subsets of the real line
Authors: Calvert I.
Published: November 2008

A subset, X, of R belongs to the family T(a) if |X| ≥ 2 and, for all x and y belonging to X, ax + (1 − a) y ∈ X, where a ∈ R.

I consider the problem of determining for which values of a > 1 all elements of T(a) are dense in R.
Refined positivity theorem for semigroups generated by perturbed differential operators of second order with an application to Markov branching processes
Authors: Hering H.
Published: November 2008

The search for a result such as the one presented in this note was motivated by an application in the theory of Markov branching processes. The limiting behaviour of a Markov branching process is determined mainly by properties of the set of its first moments, usually given as a semigroup of non-negative, linear-bounded operators on a Banach space. The principal case is that in which these operators are in some sense primitive. If the underlying space is finite-dimensional, the case of primitivity is described to complete satisfaction by Perron's theorem. For more general spaces we have the well known extension of Perron's result by Kreĭn and Rutman (8). Unfortunately, the use of this extension in the theory of general branching processes has so far not led to limit theorems as strong as the best results known in the finite-dimensional case. At least for this specific purpose the classical Kreĭn-Rutman theorem seems to be too crude. In fact, already the simplest branching diffusions on bounded domains (3) suggest a more refined, though necessarily less general extension of Perron's theorem.
Uniform primitivity of semigroups generated by perturbed elliptic differential operators
Authors: Hering H.
Published: November 2008

Our problem and the motive for attacking it are the same as in (1). Using a different method, we obtain a result admitting disconnected domains and higher dimensions without symmetry assumptions. For a detailed introduction we refer to (1).
On blowing up conductor ideals
Authors: Wilson P. M. H.
Published: November 2008

In this paper we investigate the procedure of blowing up a non-normal variety V in its conductor ideal (denned in Section 1). If V is a hypersurface then corresponds to the subadjunction conditions of Italian algebraic geometry (see (4), section 15), and so we would expect the blow up of V in , denoted , to be closely connected with the normalization Ṽ.
Limit theorems for weakly exchangeable arrays
Authors: Eagleson G. K. , Weber N. C.
Published: November 2008

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).
A footnote to ‘On the coefficients of certain automorphic functions’
Authors: Patterson S. J.
Published: November 2008

This note is a sequel to the paper referred to in the title, (2). Its purpose is to sharpen theorem 1 of that paper. We first set up some notation.
Some numerical results on the characters of exceptional Weyl groups
Authors: Beynon W. M. , Lusztig G.
Published: November 2008

1. Let V be an l-dimensional real vector space and let W be a finite subgroup of GL(V) generated by reflexions such that the space of W-invariant vectors in V is zero. Then W acts naturally on the symmetric algebra S of V preserving the natural grading . LetI be the ideal in S generated by the w-invariant elements in . The quotient algebra inherits the W-action and also a grading

is the image of Sk under S → S̅. It is well known that the W-module S̅ is isomorphic to the regular representation of W (see (3), ch. v, 5·2); in particular, S̅k = 0 for large k. (More precisely, S̅k = 0 for k > ν, where ν is the number of reflexions in W.)If ρ is an irreducible character of W, we denote by nk(ρ) the multiplicity of ρ in the W- module S̅k. The sequence n(ρ) = (n0(ρ), n1(ρ), n2(ρ), …) is an interesting invariant of the character ρ For example, in the study of unipotent classes in semisimple groups, one encounters the following question: what is the smallest k for which nk(ρ) 4= 0 (with ρ as above) and, then, what is nk(ρ) Also, the polynomial in q

can sometimes be interpreted as the dimension of an irreducible representation of a Chevalley group over the field with q elements. For these reasons it seems desirable to describe explicitly the sequence n(ρ) (or, equivalently, the polynomial Pρ(q)) for the various irreducible characters ρ of W. When W is a Weyl group of type Al, this is contained in the work of Steinberg (8); in the case where W is a Weyl group of type Bl or Dl this is done in (7),§2.
The double cusp has five minima
Authors: Callahan J.
Published: November 2008

The double cusp is the real, compact, unimodal singularity

see (2), (4). Functions in a universal unfolding of the double cusp can have nine non-degenerate critical points near the origin, but no more. Index considerations show that precisely four of the nine are saddles, and it has long been part of the folklore of singularity theory that one of the other five must be a maximum. Indeed, a standard form of the unfolded double cusp (1), (3) is a function having a pair of intersecting ellipses as one of its level curves; see Fig. 1(a). There are saddles at the four intersection points, a maximum inside the central quadrilateral, and a minimum inside each of the other four finite regions bounded by the ellipses. The rest of Fig. 1 suggests, however, that a deformation of this function (in which one of the saddles drops below the level of the other three) might turn the maximum into a fifth minimum. The following proposition shows that a function similar to the one in Fig. 1(d) can be realized in an unfolding of the double cusp.
Singularities, incompleteness and the Lorentzian distance function
Authors: Beem John K. , Ehrlich Paul E.
Published: November 2008

A space–time (M, g) is singular if it is inextendible and contains an inex-tendible nonspacelike geodesic which is incomplete. In this paper nonspacelike incompleteness is studied using the Lorentzian distance d(p, q). A compact subset Kof M causally disconnects two divergent sequences {pn} and {qn} if 0 < d(pn,qn) < ∞ for all n and all nonspacelike curves from pn to qn meet K. It is shown that no space–time (M, g) can satisfy all of the following three conditions: (1) (M, g) is chronological, (2) each inextendible nonspacelike geodesic contains a pair of conjugate points and (3) there exist two divergent sequences {pn} and {qn} which are causally disconnected by a compact set K. This particular result extends a theorem of Hawking and Penrose. It also implies that if (M, g) satisfies conditions (1) and (3), then there is a Co-neigh-bourhood of g in the space of metrics conformal to g such that any metric in this neighbourhood which satisfies the generic condition and the strong energy condition is nonspacelike incomplete.
Some qualitative properties of solutions of a generalised diffusion equation
Authors: Atkinson C. , Bouillet J. E.
Published: November 2008

Atkinson and Peletier (2,3) have considered similarity solutions of the differential equation

where the function k(s) is defined, real and continuous for s ≥ 0 and k(s) > 0 if s > 0 (in (2) k(0) = 0 is also assumed). In particular they look for similarity solutions of the form u(x, t) = f(η) where η = x(t+l)−½ with boundary conditions f(0) = A and . They show that if k(s) satisfies the condition

then for any A > 0 there is a unique similarity solution which is non-negative and has compact support in [0, ∞). They also show in (2) that

is a necessary condition for the solution to have compact support. In (3) they prove existence of similarity solutions when

and show that in this case the similarity solution has the property that f(η) > 0 for all η > 0.
Affine cubic functions
Authors: Wall C. T. C.
Published: November 2008

The classification of affine cubic functions in the real case is a fairly easy corollary of that in the complex case (9). However as the results can be easily interpreted by diagrams, one can obtain a much richer understanding. For example, the question of which types of cubic curve occur as level curves of which types of function is now much less trivial. This will lead us first to re-examine the classification of cubic curves going back to Newton (4). Next the ‘dynamic’ approach of considering these curves as members of families leads to the diagrams associated with the umbilic catastrophes of Thorn (8). However the consideration of functions rather than of curves gives a 1-dimensional foliation of these diagrams which we describe next. We conclude by placing the results back in a protective setting.
On lifting Fuchsian Groups
Authors: Hoare A. H. M.
Published: November 2008

Let Γ be a discrete subgroup of G = PSL(2,ℝ) and p be the canonical map from the universal covering group on to PSL(2,ℝ). By a continuity argument on a fundamental polygon for Γ acting on the hyperbolic plane 2, Milnor (4) obtained a presentation for p−1(Γ) whenever 2/Γ is compact and of genus zero. Using Teichmüller theory and a double Reidemeister-Schreier process, Macbeath(2) showed that the general compact case can be deduced from Milnor's result. We give here a method of obtaining a presentation which uses only the geometry of a Dirichlet region and which applies equally to the non-compact case.
Integrals with a large parameter: a double complex integral with four nearly coincident saddle-points
Authors: Ursell F.
Published: November 2008

The method of steepest descents for finding the asymptotic expansion of contour integrals of the form ∫ g(z) exp (Nf(z)) dz where N is a real parameter tending to + ∞ is familiar. As is well known, the principal contributions to the asymptotic expansion come from certain critical points; the most important are saddle-points where df/dz = 0. The original contour is deformed into an equivalent contour consisting of paths of steepest descent through certain saddle-points, the relevant saddle-points. The determination of these is a global problem which can be solved explicitly only in simple cases. The function f (z) may also depend on parameters. The position of the saddle-points depends on the parameters and at a certain set of values of the parameters it may happen that two or more saddle-points coincide. The ordinary expansion is then non-uniform, but appropriate uniform expansions have been shown to exist in earlier work.
Applications of multiparameter spectral theory to special functions
Authors: Browne Patrick J. , Sleeman B. D.
Published: November 2008

A recently discovered abstract relation arising in multiparameter spectral theory is used to unify some of the integral relations and equations associated with the ellipsoidal wave equation.
The fundamental solution for the axially symmetric wave equation II
Authors: Heins Albert E.
Published: November 2008

In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equation

could be written as

if b > r

or

if r > b.
On functional Nörlund methods
Authors: Choudhary B.
Published: November 2008

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a function

regular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.
Strongly equivalent invariant measures
Authors: Rosenblatt J.
Published: November 2008

Two measures are strongly equivalent if they have the same sets of zero measure and the same sets of infinite measure. Given a group G of strongly non-singular measurable transformations of a non-atomic positive measure space (X, β, p), if G is amenable, then a necessary and sufficient condition for there to be a G-invariant positive measure on (X, β) which is strongly equivalent to p is that p(E) > 0 implies inf p(gE) > 0 and also p(E) < ∞ implies
Mod p associative H-spaces of given rank
Authors: Hubbuck J. R.
Published: November 2008

1. Few mod 2 finite simply-connected spaces are known to support an H-structure. These examples arise from products of Lie groups and seven spheres. It has been shown in (12) and (14) that the space of a mod 2 finite H-space of given rank can belong to only a finite number of different homotopy types. The situation at odd primes is very different; many examples of mod p finite H-spaces are known, such as p-localized odd-dimensional spheres. Therefore there are infinitely many homotopy types of prescribed rank. If one insists that the mod pH-space has a strictly associative multiplication, or equivalently has the homotopy type of a topological group, the picture is less clear. Examples have been constructed which do not arise from Lie groups and spheres but there are strong restrictions on just what can occur by these techniques, see (5). There are only finitely many known examples in each rank. We establish that there can in fact only be a finite number of possible homotopy types for the spaces.
On waves at an interface between two liquids
Authors: Rhodes-Robinson P. F.
Published: November 2008

In this paper it is shown that a class of linearized interface-wave problems for two superposed inviscid liquids of unequal densities occupying regions which are symmetric about the interface can be reduced to a surface-wave problem in the lower region together with a classical hydrodynamical problem for potential flow in the lower region under a plane lid. The effect of interfacial tension is included. Examples of fundamental singularities in two semi-infinite liquids are given.