bluuebunny/crossref_metadata_2025 huggingface dataset:
Closed graph theorems for generalized inductive limit topologies
Authors: Ruess W.
Published: November 2008

The object of this and a subsequent paper is to investigate the locally convex structure of several strict topologies that are generalizations of R. C. Buck's strict topology β on C(S), S locally compact Hausdorff. If the topology τ of a locally convex space (lcs) (X, τ) is any of these strict topologies, then it is localizable on every absorbing disc T in X, i.e. it is the finest locally convex topology on X agreeing with τ on T. Topologies of this kind are said to be (L)-topologies. As our main tools for the analysis of the structure of strict topologies, we deduce in this paper several closed graph theorems for spaces of type (L). In particular, it is shown that every semi-Montel lcs with a fundamental sequence of bounded sets and every Bτ-complete Schwartz space belongs to the class Bτ(L) of all lcs Y with the property that every closed linear map from any (L)-space X into Y is continuous. Further closed graph theorems are established and many of the known closed graph theorems are deduced as special cases of our results. Moreover, the problem of Bτ-completeness of locally convex spaces belonging to Bτ(L) is considered.
A commutative Noetherian ring which possesses a dualizing complex is acceptable
Authors: Sharp Rodney Y.
Published: November 2008

Throughout this paper the word ‘ring’ will mean ‘commutative Noetherian ring with non-zero identity’, and A will always denote such a ring. It will only be assumed that A is local when this is explicitly stated, but when this is the case m (resp. k) will always be used to denote the maximal ideal (resp. residue field) of A. It is to be understood that ring homomorphisms respect identity elements.
The number of non-homogeneous lattice points in subsets of Rn
Authors: Barnes E. S. , Mather Michael
Published: November 2008

Let Zn denote the integer lattice in Rn, let A be a non-singular n × n matrix and ʗ ∈ Rn. Then G = AZn + ʗ is called a grid (non-homogeneous lattice) and its determinant det G is defined to be |det A|.
The transfer and stable homotopy theory
Authors: Kahn Daniel S. , Priddy Stewart B.
Published: November 2008

The purpose of this paper is to give a proof of the following splitting theorem in stable homotopy theory. We assume all spaces are localized at a fixed prime p. Let k be the symmetric group on {1, …, k}, Q(.) = lim ΩnΣn(.), and QkS0, k ∈ , denote the components of QS0.
Subgroup closed Fitting classes
Authors: Bryce R. A. , Cossey John
Published: November 2008

In (1) we showed that a subgroup closed Fitting formation is a primitive saturated formation, and in (2) we showed that a subgroup closed and metanilpotent Fitting class is a formation. Whether or not a subgroup closed Fitting class is always a formation is a question that has plagued us ever since. The purpose of this paper is to prove
Isotone measures on groups
Authors: Moran W. , Williamson J. H.
Published: November 2008

If E and F are ordered spaces, a map T: E → F is positive if

The map T is bi-positive if f ≥ 0 ⇔ Tf ≥ 0. It is a natural problem to seek information about the positive and bi-positive linear maps between pairs of ordered vector spaces. The present study is an attempt to elucidate one facet of this general problem (see, for example, (2), vol. II, § 16·7·1, p. 278).
Stochastic bounds for the queue GI/G/1 in heavy traffic
Authors: Köllerström Julian
Published: November 2008

It is often of greater practical value to have results about queueing theory which involve probabilities rather than characteristic functions. To quote Kendall (4), section 5, ‘These results of Prabhu, further exploited by himself and Takács, are rapidly raising the Laplacian curtain which has hitherto obscured much of the details of the queue-theoretic scene’. In this paper we derive the exponential limit formula for the equilibrium waiting time distribution functionG, for the queue GI/G/1 in heavy traffic, using stochastic bounds which are asymptotically sharp as the traffic intensity (defined below) increases to unity (which has not been done before to the author's knowledge). This formula was derived by Kingman (5), (6) and (9), using characteristic functions, who, in section 9 of the latter paper, stressed the need for improving the precision of the approximation ‘by giving inequalities, bounds for errors, and generally by setting the theory on a more elegant and rigorous basis’. Kingman (6) and (9) also sketched a proof of the same result using a Brownian approximation, which was done in detail by Viskov (18); but here again the same difficulties are present in practical interpretation, error bounds etc.
The structure of certain subgroups of the Picard group
Authors: Wielenberg Norbert
Published: November 2008

A torsion-free discrete subgroup G of PSL(2, C) acts as a group of isometries of hyperbolic 3-space H3. The resulting quotient manifold M has H3 as its universal covering space with G as the group of cover transformations. We shall give examples where M has finite hyperbolic volume and is a link complement in S3. In these examples, G is a subgroup of the Picard group and in most cases is given as an HNN extension or a free product with amalgamation of kleinian groups with fuchsian groups as amalgamated or conjugated subgroups.
An application of p-factorization methods to symmetric graphs
Authors: Weiss Richard
Published: November 2008

Let Γ be an undirected graph and G a subgroup of aut (Γ) acting transitively on the vertex set V(Γ) of Γ. Let x be an arbitrary vertex of Γ. We denote by T(x) the set of vertices adjacent to x and by G(x)Γ(x) the permutation group induced by the stabilizer G(x) of x in G on Γ(x); G(x)Γ(x) is called the subconstituent of G (with respect to Γ). Let G1(x) = {a ∈ G(x)|a ∈ G(y) for each y ∈ Γ(x)}. For each y ∈ Γ(x), let G(x, y) = G(x) ∩ G(y) and G1(x, y) = G1(x) ∩ G1(y). An s-path is an (s+ l)-tuple (x0, x1, …, xs) of vertices such that xi−1 ∈ Γ(xi) if 1 ≤ i ≤ s and xi−2 ≠ xi if 2 ≤ i ≤ s. Γ is called (G, s)-transitive if G acts transitively on the set of all s-paths but intransitively on the set of all (S+1)-paths in Γ.
Homotopy colimits in the category of small categories
Authors: Thomason R. W.
Published: November 2008

In (13), Quillen defines a higher algebraic K-theory by taking homotopy groups of the classifying spaces of certain categories. Certain questions in K-theory then become questions such as when do functors induce a homotopy equivalence of classifying spaces, or when is a square of categories homotopy cartesian? Quillen has given some techniques for answering such questions. F. Waldhausen has extended these ideas in (19), and broadened the range of applications to include geometric topology (20).
Theoretical plasticity of textured aggregates
Authors: Hill R.
Published: November 2008

The plasticity of metal polycrystals with preferred orientations is considered from a phenomenological standpoint. Some new general theorems are proved, in particular the existence of a work-equivalent function of the tensor strain-rate over any yield surface. The status of the classical theory of plastic anisotropy is re-appraised in the light of recent experiments, which are themselves critically reviewed. A new type of yield function is proposed to account for the so-called anomalous behaviour of some materials.
Augmentation quotients of group rings and symmetric powers
Authors: Sandling Robert , Tahara Ken-Ichi
Published: November 2008

LetGbe a group with the lower central series

Let

where Σ runs over all non-negative integersa1,a2,…,ansuch thatandis theaith symmetric power of the abelian groupGi/Gi+1where

LetI(G) be the augmentation ideal ofGin, the group ring ofGover. Define the additive groupQn(G) =In(G) /In+1(G) for anyn≥ 1. Then it is well known thatQ1(G) ≅W1(G) for any groupG. Losey (4,5) proved thatQ2(G) ≅W2(G) for any finitely generated groupG. Furthermore recently Tahara(12) proved thatQ3(G) is a certain precisely defined quotient ofW3(G) for any finite groupG.
Closure operations for Schunck classes and formations of finite solvable groups
Authors: Forster Peter
Published: November 2008

The purpose of the present note is to investigate the closure operations common to both the operations of forming classes and formations. Furthermore, new descriptions of formations and saturated formations in terms of closure operations are given.
Note on the invariant of plane cubics
Authors: Wall C. T. C.
Published: November 2008

It was noted by Klein(4), part I, ch. II, §9, that if a polyhedral group G acts on the 2-sphere identified with P1(ℂ), then the polynomials cΠg∈G(λ − ag μ) form, as a∈ℂ varies, a vector space of dimension 2 over ℂ. G is generated by elements T1, T2 fixing respectively a vertex and a face; T1T2 fixes an edge and these elements have orders p, q, 2 where 1/p + 1/q > ½ Taking a to be a vertex or the mid-point of a face or an edge yields polynomials of the form and which are thus linearly dependent. The ring of all (relatively) invariant polynomials is generated by F1F2 and F3 subject to this single relation.
Embedding theorems for groups with an integer-valued length function
Authors: Chiswell I. M.
Published: November 2008

(1) In this paper we are concerned with normalized integer-valued length functions on a group G, that is, mappings l:G→ℤ satisfying three axioms:

(A1′)l(1) = 0

(A2) l(x) = l(x−1)for all x∈G

(A4) d(x, y) > d(x, z) implies that d(x, z) = d(y, z) for all x, y and z in G, where d(x, y) = ½(l(x) + l(y)−l(xy−1)).

These axioms were first considered by Lyndon(6), where their significance is discussed. Lyndon's axiom A1, which we shall not use, stated that l(x) = 0 if and only if x = 1. His axiom A 3 was that d(x, y) ≥ 0 for all x and y in G, but it was noted in (2) that this follows from A1′, A2 and A4. In particular, taking x = y, we find that l(x) ≥ 0 for all x in G. The other major axioms used in (6) were:

(A0) l(x2) > l(x), provided that x ≠ 1, and (A5) d(x, y) + d(xl, y1) > l(x) = l(y) implies that x = y.
The edge-cracked circular disc under symmetric pin-loading
Authors: Gregory R. D.
Published: November 2008

A circular disc of radius a, made of homogeneous, isotropic, linearly elastic material, contains a radial edge crack of length b(0 < b < 2a). The disc is in equilibrium in a state of generalized plane stress under various loadings, which are motivated by the fact that this geometry is to become a standard test specimen configuration in the fracture testing of materials.

The first cases considered are those in which the disc is loaded by either (i) opposing point forces P normal to the crack, or (ii) opposing point couples M, in each case acting at the crack mouth. The problem of determining the resulting stress field throughout the disc is solved analytically in closed form in each case, and the respective stress intensity factors are given exactly by

where K+(0), K+(i) are constants whose values are

K+(0) = 0.966528…,

K+(i) = 0.355715…,

correct to 6 decimal places.
Not all H′-algebras are operator algebras
Authors: Carne T. K.
Published: November 2008

An H′-algebra which is not an operator algebra is constructed. This is used to prove that the class of operator algebras is not equal to the class of α-algebras for any tensor norm α.
Rational projective representations of finite groups
Authors: Humphreys J. F.
Published: November 2008

Let G bea finite group, let C1, …, Cn be the cyclic subgroups of G and let 1i be the identity linear character of Ci (1 ≤ i ≤ n). It is a well-known result of Artin ((l), 39·1) that a character x of a linear representation of G over the rational numbers may be written

where a1, …, an are integers. In this note, we establish an analogue of this result for characters of projective representations over the rational numbers.
On the relative stability of large order statistics
Authors: Hall Peter
Published: November 2008

Let X1, X2, … be independent and identically distributed (i.i.d.) random variables and let Xn, r denote the rth largest of X1, X2, …, Xn (so that Xn, r is the (n − r + l)th-order statistic of X1, X2, …, Xn). It is well known that if Xn, l/cn→1 in probability or with probability 1, for some sequence of constants cn, then Xn, r/cn→1 for each r ≥ 1. Therefore if r(n) → ∞ sufficiently slowly, Xn, r(n)/cn→1 for the same sequence of constants cn. In this paper we study behaviour of this type in considerable detail. We find necessary and sufficient conditions on the rate of increase of r(n), n ≤ 1, for the limit theorem Xn, r(n)/cn→1 to hold, and we investigate the rate of convergence in terms of a central limit theorem and a law of the iterated logarithm (LIL). The LIL takes a particularly interesting form, and there are five distinctly different modes of behaviour.
On the comparison of stability criteria in anisotropic materials
Authors: Parry Gareth. P.
Published: November 2008

The convexity of the internal energy, regarded as a function of any tensor strain measure, determines the stability of a material in a particular loading environment. Such convexity criteria reflect properties of the material alone and therefore have the alternative interpretation of constitutive inequalities. An assessment of the relative strengths of these criteria is therefore of interest because it provides an ordering of the constitutive inequalities. Existing comparison theorems, allowing such an assessment, suppose that work-conjugate measures of stress and strain are coaxial. The main content of this paper is a theorem which makes no such supposition and so enables a comparison of criteria of convexity, and an ordering of constitutive inequalities, for arbitrarily anisotropic materials.
A vector identity for the Dirichlet tessellation
Authors: Sibson Robin
Published: November 2008

A vector identity associated with the Dirichlet tessellation is proved as a corollary of a more general result. The identity has applications in interpolation and smoothing problems in data analysis, and may be of interest in other areas.
A local limit theorem for attractions under a stable law
Authors: Basu Sujit K. , Maejima Makoto
Published: November 2008

Let {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such that

as n → ∞, where v0 is the p.d.f. of Vo.
Analytic functions having distributional boundary values in W′-spaces
Authors: Pathak R. S.
Published: November 2008

It is shown that the functions which are analytic in tubular radial domains and satisfy certain growth conditions have distributional boundary values in the weak topology of (WΩ)′-space. Representation of analytic functions in terms of distributional boundary values are given. Converse results are also obtained. An analytic decomposition theorem is proved. The main theorems are established by means of a number of lemmas concerning WM, WΩ spaces and their dual spaces. Several new lemmas are proved for K{Mp} spaces from which results for WM-spaces can be easily deduced.
A partition of non-synthesis for tensor algebras
Authors: Drury S. W.
Published: November 2008

1.Introduction. To every closed subalgebra S of a regular algebra R(D) of functions tending to zero at infinity on a discrete space D we define the Rudin spectrum of S to be the partition of D consisting of the equivalence classes of the equivalence relation .
On the absolute summability of some series related to a Fourier series
Authors: Ray B. K.
Published: November 2008

1.Introduction. 1.1. Let f(t) be a periodic function with period 2π and integrable in the Lebesgue sense over ( -π,π). We assume as we may without loss of generality, that the Fourier series of f(t) is .
The simultaneous diophantine approximation of certain kth roots
Authors: Osgood Charles F.
Published: November 2008

In a recent paper in this journal(1) Baker obtained a result giving effectively computable bounds on the simultaneous approximation of certain numbers of the form (ab-1)nj/n by rational numbers of the formpιq-1 where a, b, n, the nι and q are positive integers while the pι are non-negative integers. In this paper we shall obtain explicit bounds on the simultaneous approximation of certain numbers of the form rβ1, …, rβn where β and each rι, 1 ≤ j ≤ n, are positive rational numbers.
ε- and γ-formalisms in the calculus of 4-spinors
Authors: Buchdahl H. A.
Published: November 2008

The calculus of 4-spinors, covariant with respect to arbitrary transformations in a 6-dimensional Riemann space, was presented earlier in a formulation analogous to the so-called ε-formalism in the 2-spinor calculus. Like the latter it is burdened by the appearance of a multiplicity of fractional spin weights and anti-weights as well as the arbitrariness of the trace of the linear spinor connexion. This paper deals in detail with a variant of the calculus which is more akin to the so-called γ-formalism in the 2-spirior calculus; and which is finally extended to be both gauge-invariant and phase-invariant.
Absolute curvatures in integral geometry
Authors: Baddeley Adrian J.
Published: November 2008

Surface integrals of curvature arise naturally in integral geometry and geometrical probability, most often in connection with the Quermassintegrale or cross-section integrals of convex bodies. They enjoy many desirable properties, such as the ability to be determined by summing or averaging over lower-dimensional sections or projections. In fact the Quermassintegrale are the only functionals of convex bodies to meet certain, quite reasonable, requirements. The conclusion has often been drawn, especially in practical applications, that the Quermassintegrale and their associated curvature integrals have a canonical status to the exclusion of all other quantities.
Overall properties of a cracked solid
Authors: Hudson J. A.
Published: November 2008

The differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.

The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.
An extension of the Freiheitssatz
Authors: Baumslag B. , Pride S. J.
Published: November 2008

LetIbe a set and letH(i)(i∈I) be non-trivial groups. IfJis a subset ofI, we denote the free product of theH(j)(j∈J) byH(J). We denoteH(I) simply byH.LetRbe a cyclically reduced element ofHof length at least two, and let

Let μ:H→Gbe the natural homomorphism. IfJis a subset ofIsuch thatR∉H(J), we callH(J)a Magnus subgroup, or occasionallythe J-Magnus subgroup(ofHwith respect toR). We will say that the Freiheitssatz holds if μ|Mis an injection for each Magnus subgroupM. Magnus (4) showed that the Freiheitssatz holds if theH(i)are free, and this was extended by Pride (5) to the case where theH(i)are locallyfullyresidually free. In this paper we prove Theorem 1.The Freiheitssatz holds if the H(i)are locally residually free.
Homology of groups of surfaces in the 4-sphere
Authors: Gordon C. McA.
Published: November 2008

1. Let F be a closed, connected, orientable surface of genus g ≥ 0 smoothly embedded in S4, and let π denote the fundamental group π1(S4 − F). Then H2(π) is a quotient of H2(S4 − F) ≅ H1(F) ≅ Z2g. If F is unknotted, that is, if there is an ambient isotopy taking F to the standardly embedded surface of genus g in S3 ⊂ S4, then π ≅ Z, so H2(π) = 0. More generally, if F is the connected sum of an unknotted surface and some 2-sphere S, then π ≅ π1 (S4 − S), so again H2(π) = 0. The question of whether H2(π) could ever be non-zero was raised in (5), Problem 4.29, and (10), Conjecture 4.13, and answered in (7) and (1). There, surfaces are constructed with H2(π)≅ Z/2, and hence, by forming connected sums, with H2(π) ≅ (Z/2)n for any positive integer n. In fact, (1) produces tori T in S4 with H2(π) ≅ Z/2, and hence surfaces of genus g with H2(π) ≅ (Z/2)g.
Bol loops of order pq
Authors: Niederreiter Harald , Robinson Karl H.
Published: November 2008

The main results of this paper show that for any odd primes p and q with q dividing p2 − 1 there exist non-associative Bol loops of order pq. The construction is sufficiently flexible so as to allow the resulting loops to be either Bruck loops or non-Bruck loops. It also permits the determination of isotopy classes among the loops constructed. All Bol loops of order 15 are determined.
Some best-approximation theorems in tensor-product spaces
Authors: Light W. A. , Cheney E. W.
Published: November 2008

We begin by describing a concrete example from the class of problems to be considered. A continuous bivariate function f defined on the square |t| ≤ 1, |s| ≤ 1 is to be approximated by a tensor-product form involving univariate functions. For example, the approximation may be prescribed to have the form

in which the Ti are the Tchebycheff polynomials, and the coefficient functions xi(t) and yi(s) are to be chosen to achieve a good or best approximation. Will a best approximation exist? If so, how can it be obtained?
Compensators and Cox convergence
Authors: Brown T. C.
Published: November 2008

In a previous paper (4), the author showed that the martingale approach to point processes may be used to derive Poisson distributional limit theorems. Here this approach is extended to Cox convergence, general increasing processes are considered, and a regularity condition needed in (4) is removed (that of calculability). In Section 3 we give examples to show the need for the various hypotheses used in the main theorem of Section 2. This theorem is applied to various (dependent) thinnings and compound-ings of point processes in Section 4.
Groups whose irreducible representations have finite degree
Authors: Wehrfritz B. A. F.
Published: November 2008

Throughout this paper F denotes a (commutative) field. Let XF denote the class of all groups G such that every irreducible FC-module has finite dimension over F. In (1) P. Hall showed that if F is not locally finite and if G is polycyclic, then G∈XF if and only if G is abelian-by-finite. Also in (1), if F is locally finite he proved that every finitely generated nilpotent group is in XF and he conjectured that XF should contain every polycyclic group. This turned out to be very difficult, but a positive solution was eventually found by Roseblade, see (8). Meanwhile Levič in (4) had started a systematic investigation of the classes XF. Although his paper contains a number of errors, obscurities and omissions, it remains an interesting work, and it, or more accurately its recent translation, stimulated this present paper.
On Banach modules I
Authors: Kaijser Sten
Published: November 2008

The purpose of this series of papers is to present a general theory of Banach modules and to give some applications of it. The applications of the theory arise from observations that certain important notions of functional analysis are very closely related to certain Banach algebras and thereby to a module structure. This holds in particular for Banach lattices (which are C(X)-modules), interpolation spaces, tensor products and operator ideals.
Codimension one immersions and the Kervaire invariant one problem
Authors: Eccles Peter John
Published: November 2008

Let i: M↬ℝn+1 be a self-transverse immersion of a compact closed smooth n-dimensional manifold in (n + 1)-dimensional Euclidean space. A point of ℝn+1 is an r-fold intersection point of the immersion if it is the image under i of (at least) r distinct points of the manifold. The self-transversality of i implies that the set of r-fold intersection points is the image of an immersion of a manifold of dimension n+1-r (the empty set if r > n + 1). In particular, the set of (n + l)-fold intersection points is finite of order, say, θ(i). In this paper we are concerned with the set of values of θ(i) for (self-transverse) immersions of all (compact closed smooth) manifolds of given dimension n.
Recognizing singularities of surfaces in ℂP3
Authors: Soares M. G. , Giblin P. J.
Published: November 2008

In this paper we consider complex projective surfaces V, defined by an equation of the form fn–1 (x, y, z) w + fn (x, y, z) = 0, where fi is homogeneous of degree i, and relate the geometry of the intersections of the piane projective curves fn–1 = 0 and fn = 0 with the singularities of V. The results we obtain clarify and generalize some of those presented by Bruce and Wall (3).
Wave solutions for the deterministic host-vector epidemic
Authors: Radcliffe J. , Rass L. , Stirling W. D.
Published: November 2008

Recent papers by Atkinson and Reuter(1), Brown and Carr(5), and Barbour(2) have proved several important results concerning wave solutions of the usual deterministic model for the spatial spread of an epidemic, such as measles or influenza. In particular Atkinson and Reuter showed that non-trivial wave solutions, in a population along a line, only exist provided the contact distribution function has an exponentially bounded tail, and that the speed of propagation c must be at least some critical value c0. They constructed a solution for c > c0, which Barbour later showed to be the unique solution, modulo translation, at that speed. Brown and Carr showed that a solution was also possible at speed c0, though it has not been possible to show uniqueness at this speed.
More 3-manifolds with multiple knot-surgery and branched-cover descriptions
Authors: Livingston Charles
Published: November 2008

For any integer N ≽ 3 we will construct a 3-manifold which can be described as + 1 surgery on N distinct knots in S3. We will also give examples of 3-manifolds which are N-fold cyclic branched covers of S3 over 2 distinct knots. Brakes (2) discovered the first examples of 3-manifolds with multiple knot surgery descriptions. Our construction is much different and follows directly from the construction used by Lickerish (6) to describe a manifold which has 2 distinct knot surgery descriptions. Giller (5) has given examples of 3-manifolds which arise as cyclic branched covers over distinct knots in S3. Qur construction is similar, but the knots are much easier to distinguish, being iterated torus knots.
Primary rings and tensor products of algebras
Authors: O'Carroll L. , Qureshi M. A.
Published: November 2008

All rings considered in this paper are non-trivial commutative algebras over a field k; unless indicated otherwise, all tensor products are understood to be taken over k. (Some of the results and concepts extend to general rings, but it is not worth while noting such generalizations.)
Lower bounds for the Perron root of a non-negative irreducible matrix
Authors: Deutsch Emeric
Published: November 2008

We derive a family of lower bounds for the Perron root of a non-negative irreducible matrix. These lower bounds are better than certain lower bounds of the Rayleigh quotient type, also derived in this paper. For the particular case of a symmetric non-negative irreducible matrix, our lower bound is always better than a corresponding Rayleigh quotient and, as shown in example 4, it can be infinitely better.
Infinitely divisible probabilities on SL(2,) are continuously embedded
Authors: McCrudden M.
Published: November 2008

For any locally compact Hausdorff topological group G, let M(G) denote the topo-logical semigroup of all probability measures on G, where M (G) is given the weak topology and the multiplication is convolution.
Infinite intensity mixtures of point processes
Authors: Daley D. J.
Published: November 2008

Let the stationary point processN(·) be the mixture of scaled versions of a stationary orderly point processN1(·) of unit intensity with mixing distributionG(·), so that

With

N(·) has finite or infinite intensity asis finite or infinite, and it is Khinchin orderly when the function γ(·) is slowly varying at infinity. Conditions forN(·) to be orderly involve bothG(·) and the Palm distribution ofN1(·).
Representing elements of π1M3 by fibred knots
Authors: Harer John
Published: November 2008

Every smooth, closed, orientable manifold of dimension 3 contains a fibred knot, i.e. an imbedded circle K such that M – K is the total space of a fibre bundle over S1 whose fibre F is standard near K. This means the boundary of the closure of F is K so that K is null-homologous in M. A natural problem suggested by Rolfsen (see (2), problem 3·13) is to determine which elements of π1M3 may be represented by fibred knots. We deal with this by proving the

Theorem. The collection of all elements of π1 M3 which can be represented by fibred knots is exactly the commutator subgroup [π1 M3, π1 M3].
Strategies for compounds of partizan games
Authors: Fraenkel Aviezri S. , Tassa Uzi
Published: November 2008

In this paper we define two order relations between every two partizan games with possible draw positions. We use these relations to prove two ‘simplest form’ theorems, and to describe an algorithm for computing the simplest form of the ‘finite games’ which are an extension of Conway's short games.
On nearly uniformly convex Banach spaces
Authors: Partington J. R.
Published: November 2008

A real Banach space (X, ‖ · ‖) is said to be uniformly convex (UC) (or uniformly rotund) if for all ∈ > 0 there is a δ > 0 such that if ‖x| ≤ 1, ‖y‖ ≤ 1 and ‖x−y‖ ≥ ∈, then ‖(x + y)/2‖ ≤ 1− δ.
The hyperelliptic equation over function fields
Authors: Mason R. C.
Published: November 2008

Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients. Siegel's method reduces the hyperelliptic equation to a finite set of Thue equations f(x, y) = 1, where f denotes a binary form with algebraic coefficients and at least three distinct linear factors; x and y are integral in a fixed algebraic number field. Siegel had already proved that the Thue equations so obtained have only finitely many solutions. However, as is well known, the work of Siegel is ineffective in that it fails to provide bounds on the integer solutions of y2 = g(x). In 1969 Baker (1), using the theory of linear forms in logarithms, employed Siegel's technique to establish explicit bounds on x and y; Baker's result thus reduced the problem of determining all integer solutions of the hyperelliptic equation to a finite amount of computation.
Free subgroups of certain one-relator groups defined by positive words
Authors: Baumslag Gilbert
Published: November 2008

Let ℒ be the class of those groupsGwhich can be presented in the form

whereuandvare positive words in the given generators. Here a wordwis termed positive if only non-negative powers ofa,b,…,coccur inw. If each generator occurs with exponent sum zero inuv-1, we term the ℒ-groupGa-group. This classcontains, in particular, the classXof those groupsGwhich can be presented in the form

whereuandvare positive words, and where [u,v] is the commutatoruvu-1v-1.
An irreducible 4-string braid with unknotted closure
Authors: Morton H. R.
Published: November 2008

Every oriented knot or link in S3 can be represented in many ways as the closure of a braid β ∈ Bn, the braid group on n strings, for some n. Braids β ∈ Bn, γ ∈ Bm are called closure-equivalent if and are equivalent as oriented knots. It is a well-known result of Markov, see, for example, (l), that β and γ are closure equivalent if and only if there is a sequence of elementary (Markov) moves in which β ∈ Bn is replaced by

(a) a conjugate in Bn, or , or

(c) β1 ∈ Bn−1, where ,

and the process repeated until γ is reached.
Wavefronts and parallels in Euclidean space
Authors: Bruce J. W.
Published: November 2008

Given a smooth plane curve or surface in ℝ3 its parallels consist of those curves or surfaces a fixed distance d down the normals in a fixed direction. Generically they have Legendre singularities. We are concerned here with the way in which these parallels change as we alter the distance d. (Alternatively the manner in which wave-fronts change as they evolve from an initial smooth wavefront.)

This problem was considered in (1) by V. I. Arnold. In a very beautiful paper he describes the generic evolution of wavefronts but does not prove that for a generic initial wavefront in ℝ2 or ℝ3 the evolution is of the type described there. This we do here, using the tool of transversality. A more positive outcome of our investigation is that some of Arnold's generic forms do not occur (those corresponding to A2 singularities).
Finite particle systems and infection models
Authors: Donnelly Peter , Welsh Dominic
Published: November 2008

Infinite particle systems on lattices have been extensively studied in recent years. The main questions of interest concern the ergodic and limiting behaviour of these processes, and their relationship with the dimension of the underlying lattice. A comprehensive review is given by Durrett(6).

One of the more tractable of these processes is the voter model introduced by Clifford and Sudbury(3) and much studied since, see for example the monograph by Griffeath(8), or the papers by Harris(11), Holley and Liggett(13), Bramson and Griffeath(1) and (2) or, for a more general approach, Kelly(16).

In this paper we consider the case where the underlying spatial structure is finite and examine the transient behaviour of the voter process and also the infection process introduced by Williams and Bjerknes(21).
Decomposition theorems for certain C*-crossed products
Authors: de Brabanter Marc
Published: November 2008

Let G be an abelian discrete group, A a unital C*-algebra and an action of G on A, i.e. (A, G,) is a C*-dynamical system. Let K denote the kernel ker of and put R = G/K. The main purpose of this article is to determine the roles of K and R in the crossed product G A. This goal is achieved in Section 2, where we prove that G A is *-isomorphic to a twisted crossed product of R with C*(K) A with respect to the action 1 and a 2-cocycle related to the 2-cocycle determined by the extension G of R by K. Here is the obvious action of R on A.
Best approximation in LP (I, X)
Authors: Khalil Roshdi
Published: November 2008

Let T be a measure space and m a finite measure on T. The space of p-Bochner integrable functions defined on T with values in a Banach space X is denoted by Lp(T, X). It is well known (1) that Lp (T, X) is a Banach space under the norm

A subspace E in a Banachh space F is said to be proximinal if for each xF there is at least one xE such that

The element y is called best approximant of x in E.
Locally cartesian closed categories and type theory
Authors: Seely R. A. G.
Published: November 2008

It is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/A are cartesian closed. In such a category, the notion of a ‘generalized set’, for example an ‘A-indexed set’, is represented by a morphism B → A of C, i.e. by an object of C/A. The point about such a category C is that C is a C-indexed category, and more, is a hyper-doctrine, so that it has a full first order logic associated with it. This logic has some peculiar aspects. For instance, the types are the objects of C and the terms are the morphisms of C. For a given type A, the predicates with a free variable of type A are morphisms into A, and ‘proofs’ are morphisms over A. We see here a certain ‘ambiguity’ between the notions of type, predicate, and term, of object and proof: a term of type A is a morphism into A, which is a predicate over A; a morphism 1 → A can be viewed either as an object of type A or as a proof of the proposition A.
Non-commutative unique factorization domains
Authors: Chatters A. W.
Published: November 2008

We introduce a concept of unique factorization for elements in the context of Noetherian rings which are not necessarily commutative. We will call an elementpof such a ringR prime if (i) pR=Rp, (ii)pRis a height-1 prime ideal ofR, and (iii)R/pRis an integral domain. We define a Noetherianu.f.d.to be a Noetherian integral domainRsuch that every height-1 primePofRis principal andR/Pis a domain, or equivalently every non-zero element of R is of the formcq, whereqis a product of prime elements ofRandchas no prime factors. Examples include the Noetherianu.f.d.'s of commutative algebra and also the universal enveloping algebras of solvable Lie algebras. The latter class provides a rich supply of genuinely non-commutative examples.
Norms and spectra for certain subalgebras of M(G)
Authors: Karanikas C. , Williamson J. H.
Published: November 2008

We are concerned in this paper with the same general problem as in [4], that is, the relation between the spectral and norm properties of a sub-algebra of M(G) and the corresponding properties of the algebra of its ‘transforms’ in some sense. The spirit of the investigation is akin to that of the earlier paper, and some of the results obtained here directly generalize those of [4]. There are, however, considerable differences in scope and method.
Some combinatorial series identities
Authors: Srivastava H. M. , Raina R. K.
Published: November 2008

While expanding upon the work of H. M. Srivastava [6] on generalizations of an interesting identity of Carlson, R. G. Buschman and H. M. Srivastava [2] proved a number of double-series identities and listed various cases of reducibility of certain hypergeometric series in two variables (cf. [1], p. 150, equation (29)). The object of the present paper is to derive three new classes of combinatorial series identities (contained in Theorems 1, 2 and 3 below) which unify and extend the results of these earlier papers ([2], [6]). A multiple-series analogue of one of the combinatorial series identities presented here is also recorded.
On the connectedness of a random graph
Authors: Grimmett G. R. , Keane M. , Marstrand J. M.
Published: November 2008

Let p = (p(i): i ≥ 0) be a sequence of numbers satisfying 0 ≤ p(i) < 1 for i = 0,1,2,…, and let G be a random graph with vertex set ℤ = {…, — 1, 0, 1,…} and with edge set defined as follows: for each pair i, j of vertices, where i ≤ j, there is an edge joining i and j with probability p(j — i), independently of the presence or absence of all other edges. We explore the connectedness of G, showing that G is almost surely connected if and only if Σip(i) = ∞ and the (positive) greatest common divisor of the set {i ≥ 1: p(i) < 0} equals 1; if one of these two conditions fails to hold then G is almost surely disconnected. Corresponding results hold in higher dimensions, for random graphs defined on the vertex sets ℤd where d ≥ 2.
Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumerations
Authors: Flajolet P. , Odlyzko A. M.
Published: November 2008

This paper studies coefficientsyh, nof sequences of polynomials

defined by non-linear recurrences. A typical example to which the results of this paperapply is that of the sequence

which arises in the study of binary trees. For a wide class of similar sequences a general distribution law for the coefficientsyh, nas functions ofnwithhfixed is established. It follows from this law that in many interesting cases the distribution is asymptotically Gaussian near the peak. The proof relies on the saddle point method applied in a region where the polynomials grow doubly exponentially ash→ ∞. Applications of these results include enumerations of binary trees and 2–3 trees. Other structures of interest in computer science and combinatorics can also be studied by this method or its extensions.
Graphs in the plane invariant under an area preserving linear map and general continuous solutions of certain quadratic functional equations
Authors: McCarthy P. J. , Crampin M. , Stephenson W.
Published: November 2008

The requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the form

for all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).
Characterizations of invariant distributions
Authors: Brown Timothy C. , Cartwright Donald I. , Eagleson G. K.
Published: November 2008

Let (S, ρ) be a separable metric space and G a group of transformations of S. Necessary and sufficient conditions for a distribution on S to be invariant under G are derived in terms of the behaviour of the convolution of a random transformation from G and a random element of S.
A simpler proof of Levinson's theorem
Authors: Conrey J. B. , Ghosh A.
Published: November 2008

In this paper we present a proof of the mean-value theorem required by Levinson to show that at least one-third of the zeros of ζ(s) are on the critical line. As in Levinson [3], let

where Χ(s)=ζ(s)/ζ(1−s) is the usual factor from the functional equation, and let

where

and

where
Conservation laws and null divergences
Authors: Olver Peter J.
Published: November 2008

It is shown that P, depending on x, u and derivatives of u, satisfies Div P ≥ 0 for all such x, u, if and only if Div p = Φ(x) ≥ 0 where Φ is independent of u. Applications to theories of continuum thermomechanics are discussed.
Distances in Gaussian point sets
Authors: Clifford Peter , Green N. J. B.
Published: November 2008

The joint distribution of the n(n− l)/2 distances between n normally distributed points in d dimensions is studied. Moment generating functions and probability density functions are obtained. It is shown that when n = d the squared distances are jointly exponentially distributed subject only to the constraint that a valid n point configuration is prescribed. In the case n = d = 3 the distributions of the ordered distance are obtained explicitly.
Projective actions, invariant sigma-curves and quadratic functional equations
Authors: McCarthy Patrick J.
Published: November 2008

The quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.
Periodicity of Clifford algebras and exact octagons of Witt groups
Authors: Lewis D. W.
Published: November 2008

Exact octagons, i.e. circular eight-term exact sequences, have cropped up recently in a few places in the literature. Papers of the author [11], and implicitly [10], the book of M. Warshauer[14], and the notes [5], all contain exact octagons. The first three references involve octagons of Witt groups of quadratic and other kinds of forms, the last reference extending the octagons to the setting of L-groups, i.e. surgery obstruction groups.
Every graph is contained in a sparsest possible balanced graph
Authors: Györi E. , Rothschild B. , Runciński A.
Published: November 2008

In this paper we prove that, for any graph G, there exists a graph F such that G is a subgraph of F and
On the mobility of bodies in ℝn
Authors: Dawson R. J. MacG.
Published: November 2008

We may define a mobility problem to be one of the type: ‘Given a (fully or partlys pecified) collection of bodies in ℝn, assumed not to interpenetrate, can they be moved in some (fully or partly specified) fashion?’ Early examples of mobility problems nclude ‘Chinese puzzles’ and Sam Loyd's ‘15 puzzle’ (to which could be added the more recent Rubik's Cube!) Less combinatorial examples include the ‘piano mover’ problem (see, e.g. [2, 3, 9]) and the several recent papers on collision avoidance, such as [1, 5, 6, 7, 8].
Nonlinear random equations with maximal monotone operators in Banach spaces
Authors: Kravvaritis Dimitrios
Published: November 2008

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such that

Random equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].
Central limit theorems for sums of extreme values
Authors: Csörgoő Sándor , Mason David M.
Published: November 2008

Given a sequence of non-negative independent and identically distributed random variables, we determine conditions on the common distribution such that the sum of appropriately normalized and centred upperknextreme values based on the firstnrandom variables converges in distribution to a normal random variable, wherekn→ ∞ andkn/n → 0 asn→ ∞. The probabilistic problem is motivated by recent statistical work on the estimation of the exponent of a regularly varying distribution function. Our main tool is a new Brownian bridge approximation to the uniform empirical and quantile processes in weighted supremum norms.
The word problem for one-relator semigroups
Authors: Howie James , Pride Stephen J.
Published: November 2008

Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relatorgroupshas been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.
Knot surgery and primeness
Authors: Acuña Francisco González , Short Hamish
Published: November 2008

The aim of this paper is to prove some new results towards answering the question: When does Dehn surgery on a knot give a non-prime manifold? This question has been raised on several occasions (see for instance [5] or [4]; concerning the latter see below). Recall that a 3-manifold is prime if, in any connected sum decomposition

one of M1, M2 is S3. (For standard definitions of low-dimensional topology see [2] or [16].)
Stopping non-commutative processes
Authors: Barnett Chris , Lyons Terry
Published: November 2008

Stopping times are a powerful tool in the theory of stochastic processes, so it is natural to ask whether they have a counterpart in the theory of non-commutative processes. This paper is a part answer to that question. We show that the ‘formalism’ of stopping times carries over to a non-commutative context and prove an Optional Stopping Theorem.
Asymptotically optimal sampling schemes for periodic functions
Authors: Dahmen W. , Micchelli C. A. , Smith P. W.
Published: November 2008

In this paper we are concerned with the following question: Given a function class , the space of L2-one periodic complex valued functions, when does sampling a function give optimal information for approximation? To make this question precise, we introduce the quantity

, where A is any mapping from ℝN and I is any continuous linear mapping from into ℝN.