bluuebunny/crossref_metadata_2025 huggingface dataset:
On the axiomatics of nearness and compression
Authors: Ward A. J.
Published: November 2008

H. Herrlich ((2), p. 193) defines, for a given nearness ξ, a property ξ̃ (which by analogy with the case of filters on a proximity space may be called compression) and remarks that ξ is determined by ξ̃. He continues: ‘Consequently, each of the axioms (Ni) can be translated into a condition concerning ξ̃, thus providing an axiomatization of the concept of collections of sets containing arbitrary small members.’ This last remark seems slightly misleading; it might reasonably be taken to mean that the reciprocity between ξ and ξ̃ is a mere set-theoretic tautology, independent of the nearness axioms themselves. (Compare, for example, the relation between the ideas of ‘closure-point of a set’ and ‘neighbourhood of a point’ in the axiomatics of topology.) This is not in fact the case; however, one cannot select from Herrlich's axioms a subset which is necessary and sufficient for ξ to be determined by ξ̃. Moreover, the relation between ξ and ξ̃ appears to be asymmetrical. We shall exhibit, in terms of stacks ((4), p. 36), an elegant set of mutually independent axioms, first for the more general ‘Čech nearness’ discussed by Naimpally in (3) and then for a Herrlich nearness (called in (3) a LO-nearness, with a compatible topology as there defined); these axioms make the reciprocity between nearness and compression, and its relationship with the axioms, explicit and obvious.
Discreteness of the singular spectrum for Schrödinger operators
Authors: Schechter Martin
Published: November 2008

Under assumptions (1·1)–(1·3) given below, we prove discreteness of the positive singular spectrum of the forms realization of the operator – Δ + V. The limiting absorption principle is also proved.
On the elasticity of monatomic crystals
Authors: Parry Gareth P.
Published: November 2008

This is an exploration of the mechanical consequences of the symmetry inherent in a monatomic elastic crystal lattice. The investigation is based on a theorem by means of which the square of the right stretch tensor is uniquely decomposed. Generality is assured by the affine equivalence of all monatomic lattices. Response functions appropriate to perfect monatomic crystals are thereby constructed, and some simple deformation paths are considered.
Families of sets whose pairwise intersections have prescribed cardinals or order types
Authors: Erdös P. , Milner E. C. , Rado R.
Published: November 2008

For a given index set I, let us consider a family (Aν,: ν ∈ I) of subsets of a set E. In this note we deal with some aspects of the following question: to what extent is it possible to prescribe the cardinalities, or the order types in case E is ordered, of the sets Aν and of their pairwise intersections? In (1) the authors have shown that, given any regular cardinal a, there is a family of a+ sets of cardinal a whose pairwise intersections are arbitrarily prescribed to be either less than or equal to a. In Theorem 1 below we prove a stronger result which states that if a is regular, say a = ℵα, and if E is well-ordered and of order type , then one can find a+ subsets Aν, of E, each of type , whose pairwise intersections are arbitrarily prescribed to be either of type ωα or of a type less than ωα. By way of contrast, Theorem 2 below implies – this is its special case m = ℵω; n = ℵ2; p = ℵ0 – that, assuming the Generalized Continuum Hypothesis (GCH), there do not exist ℵω+1 sets Aν, each of cardinal at most ℵω such that ℵ2 of them have pairwise finite intersections, whereas all other pairs of sets Aν have a denumerable intersection. Theorem 3 gives another case in which some type of prescription of the sizes of the intersections cannot be satisfied. Finally, Theorem 4 asserts that in Theorem 3 the condition cfp ≠ cfm cannot be omitted. The paper concludes with some remarks on open questions.
Valentine convexity in n dimensions
Authors: Eggleston H. G.
Published: November 2008

A subset X of Euclidean space such that if a, b, c are points of X then at least one of the segments joining two of them lies in X, is said to be V-convex. Valentine (4) showed that in two dimensions a compact V-convex set is the union of at most three convex sets. We show here that if the set of star centres of X is of lower dimension than X and X is a compact V-convex set then it is the union of at most two convex sets.
Global stability of genetic systems governed by mutation and selection
Authors: Moran P. A. P.
Published: November 2008

This paper considers the behaviour of infinite haploid genetic populations under the influence of mutation and selection depending on a single locus. Under wide conditions the Perron–Frobenius theory of non-negative matrices and its generalization by Vere-Jones are used to prove that there is a single globally stable state of the population when there is a finite or, under more restrictive conditions, an infinite set of possible alleles.
New exact results for the defective square lattice
Authors: Ronca G.
Published: November 2008

Since the publication of the fundamental papers by Lifshitz (1, 2) and Montroll and Potts (3, 4) many authors have investigated the effect of an isotopic impurity on the lattice vibrations of a harmonic crystal at zero temperature. A fairly broad knowledge is now available on scattering amplitudes, localized modes and resonance modes (6, 7). Nevertheless, as pointed out by Maradudin and Montroll (see (7), p. 430), a closed form solution to the problem has been found only for the one-dimensional crystal, the work done on two and three-dimensional crystals being predominantly numerical. Unfortunately the one-dimensional crystal, as an approximation for a real crystal is an oversimplified model, incapable as it is of exhibiting resonance modes. To the author's knowledge the most significant exact result concerning the classical behaviour at zero temperature of crystals having a dimensionality higher than one is the connexion, calculated by Mahanty et al. (5) between localized mode frequency and impurity mass for the case of a square lattice undergoing planar vibrations.
Some generalizations of the Borsuk–Ulam theorem
Authors: Edmunds D. E. , Webb J. R. L.
Published: November 2008

In a recent paper, Fenn (3) has established the following theorem: if ø is a piecewise linear involution on Sn without fixed points, if f: Sn → Sn and g: are continuous and the degree of f is odd, then there are points x and y on Sn such that f(x) = ø(f(y)) and g(x) = g(y). The Borsuk–Ulam theorem is the special case of this in which f is the identity and ø corresponds to reflexion in the origin. Since an infinite-dimensional version of the Borsuk–Ulam theorem is known, involving compact maps (see, for example, page 72 of (2)), it is natural to ask whether Fenn's result can also be extended to general Banach spaces, and in this paper we give such an extension when ø is reflexion in the origin. More precisely, we prove that if B is the closed unit ball in a Banach space X and f, g: B → X are compact, with deg (I − f, B, 0) odd (this is the Leray–Schauder degree and I is the identity map) and (I − g) (B) contained in a proper, closed subspace of X, then there exist x, y on the boundary ∂B of B and apositive numberα such that α(I − f) (x) = − (I − f) (y) and (I − g) (x) = (I – g) (y).
The Hopf bifurcation theorem in three dimensions
Authors: Swinnerton-Dyer Peter
Published: November 2008

The Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.
On a conjecture of Carter concerning irreducible Specht modules
Authors: James G. D.
Published: November 2008

We study the question: Which ordinary irreducible representations of the symmetric group remain irreducible modulo a prime p?

Let Sλ be the Specht module corresponding to the partition λ of n. The definition of Sλ is ‘independent of the field we are working over’. When the field has characteristic zero, Sλ is irreducible, and gives the ordinary irreducible representation of corresponding to the partition λ. Thus we are interested in the problem of whether or not Sλ is irreducible over a field of characteristic p.
Derivatives of the spectral radius as a function of non-negative matrix elements
Authors: Cohen Joel E.
Published: November 2008

Let A = (aij) be a non-negative n × n matrix, that is, aij ≥ 0, i, j = 1, …, n; n > 1. We write A ≥ 0. Let r = r(A) be the spectral radius of A; assume r > 0 throughout to avoid trivial cases. Let be the mth derivative of r with respect to the element aij, all other elements of A being held constant.
On non-local continuum mechanics
Authors: Green A. E. , Naghdi P. M.
Published: November 2008

It is shown in this paper that the conservation laws of non-local continuum mechanics are equivalent to, and can be derived from, the conservation of energy and the invariance conditions under superposed rigid body motions. Also, the theory of non-local thermoelasticity is reconsidered in the light of recent developments in thermodynamics, taking invariance conditions fully into account.
Cocircuit coverings and packings for binary matroids
Authors: Oxley James G.
Published: November 2008

If M is an arbitrary loopless matroid with ground set E(M) and rank function ρ, then let α(M) be the minimum size of a set of cocircuits of M, whose union is E(M), and let β(M) be the maximum size of a set of pairwise disjoint cocircuits of M. The following conjecture is based on Gallai's theorem that the vertex-stability and vertex-covering numbers of a graph G sum to the number of vertices of G (1).
Finite amplitude effects in free and forced edge waves
Authors: Rockliff Nicole
Published: November 2008

The effect of non-linearity on standing edge waves is studied on the basis of shallow water theory. Four problems are considered: the decay of free edge waves and the forcing of edge waves by an incident wave of double the frequency, a synchronous incident wave and by a side-wall wavemaker. Hysteresis effects are predicted for all types of forcing.
On the iteration of rational functions
Authors: Garber V.
Published: November 2008

In the theory of the iteration of a rational function or transcendental entire function R(z) of the complex variable z we study the sequence of natural iterates, {Rn(z):n = 0, 1,…}, of R, where

The domain of definition of the iterates is , the extended complex plane (if R is rational), and (if R is entire transcendental) with the topology of the chordal metric and euclidean metric respectively. Fatou(5) and Julia(9) developed a global theory of the iteration of a rational function. In (6) Fatou extended the theory of (5) to transcendental entire functions. A central role is played in the theory by the F-set, F(R), of R, R rational or entire, which is defined to be the set of points at which the family of iterates do not form a normal family in the sense of Montel.
Algebraic solutions of partizan games with cycles
Authors: Shaki Ahiezer S.
Published: November 2008

This paper presents an algebraic model for a general class of two-person perfect-information partizan games which may contain cycles, i.e. sequences of positions that may be assumed more than once. An algorithm is given for determining the winning, losing and tie-positions of disjunctive compounds of partizan games.
Generalized approximate methods for transmission through a barrier governed by a differential equation of order 2n
Authors: Heading John
Published: November 2008

The potential barrier governed by a second-order ordinary differential equation has been studied for decades, both exactly and approximately. These concepts are generalized so as to be applicable to differential equations of arbitrary even order. The barrier is defined, and the approximate solutions both inside and outside are derived. Transmission through this barrier is investigated by deriving connexion formulae through the two transition points of arbitrary odd order. The transmitted solution is linked through the barrier to the incident and reflected solutions in a unique manner, enabling an approximate transmission coefficient to be calculated and its structure to be ascertained. Every concept in the investigation is a suitable generalization of the elementary case, the generalization possible being a product of the approximate processes adopted and how they limit the nature of the generalization. An inspection of the elementary case and its transmission coefficient gives no hint as to the nature of the generalization, the limitations necessarily imposed, and the structure of the final transmission coefficient, since the elementary case has special features not present in the generalization.
Transforms which almost vanish at infinity
Authors: Pigno Louis
Published: November 2008

Let be the circle group, M() the set of bounded Borel measures on and ℤ the additive group of integers. If μ ∈ M() and n ∈ ℤ, define

A well-known result of Rajchman states that

The following quantitative generalization of this result has been given in (2) by K. de Leeuw and Y. Katznelson.
The geometric realization of Wall obstructions by nilpotent and simple spaces
Authors: Mislin Guido
Published: November 2008

Let π denote a finite group. It is well known that every element of the projective class group K0 ℤπ may be realized as Wall obstruction of a finitely dominated complex with fundamental group π (cf. (13)). We will study two subgroups N0ℤπ and Nℤπ of K0ℤπ, which are closely related to the Wall obstruction of nilpotent spaces. If the group π is nilpotent and if S denotes the set of elements x ∈ K0ℤπ which occur as Wall obstructions of nilpotent spaces, then

It turns out that in many instances one has N0,ℤπ = Nℤπ (cf. Section 3) and one obtains hence new information on S. The main theorem (2·4) provides a systematic way of constructing finitely dominated nilpotent (or even simple) spaces with non-vanishing Wall obstructions.
Martingale inequalities
Authors: Bollobás Béla
Published: November 2008

The first result of this paper was proved in January 1975 in order to engage the interest of Professor J. E. Littlewood, who was in hospital at the time. This theorem is of some independent interest and we present it here with some related results.
On conjectures of Mahowald, Segal and Sullivan
Authors: Lin Wen-Hsiung
Published: November 2008

In this paper we prove some results about the stable homotopy and cohomotopy of spaces related to the infinite real protective space RP∞. These include M. E. Mahowald's conjecture on the limit of stable homotopy of stunted real projective spaces RP2N+m/RP2N−m as N, m → ∞, G. Segal's Burnside ring conjecture for

and the stable analogue of a conjecture of D. Sullivan on RP∞.
On the summability of Fourier integrals
Authors: Nayak M. K.
Published: November 2008

We say a series is summable L if

tends to a finite limit s as x → 1 in the open interval (0, 1) where
A generalization of McDiarmid's theorem for mixed Bernoulli percolation
Authors: Hammersley J. M.
Published: November 2008

Let G be an infinite partially directed graph of finite outgoing degree. Thus G consists of an infinite set of vertices, together with a set of edges between certain prescribed pairs of vertices. Each edge may be directed or undirected, and the number of edges from (but not necessarily to) any given vertex is always finite (though possibly unbounded). A path on G from a vertex V1 to a vertex Vn (if such a path exists) is a finite sequence of alternate edges and vertices of the form E12, V2, E23, V3, …, En − 1, n, Vn such that Ei, i + 1 is an edge connecting Vi and Vi + 1 (and in the direction from Vi to Vi + 1 if that edge happens to be directed). In mixed Bernoulli percolation, each vertex Vi carries a random variable di, and each edge Eij carries a random variable dij. All these random variables di and dij are mutually independent, and take only the values 0 or 1; the di take the value 1 with probability p, while the dij take the value 1 with probability p. A path is said to be open if and only if all the random variables carried by all its edges and all its vertices assume the value 1. Let S be a given finite set of vertices, called the source set; and let T be the set of all vertices such that there exists at least one open path from some vertex of S to each vertex of T. (We imagine that fluid, supplied to all the source vertices, can flow along any open path; and thus T is the random set of vertices eventually wetted by the fluid). The percolation probability

is defined to be the probability that T is an infinite set.
Basic stress analysis of hyperbolic regimes in plastic media
Authors: Hill R.
Published: November 2008

Nonlinear problems of two-dimensional deformation or stress in solid continua are considered where the in-plane components of stress are self-equilibrated and subject to a scalar constraint. In applications the latter is often a yield condition for plastic media. Such field equations are frequently hyperbolic, with a pair of characteristic curves through any point. The primary aim is to express the integrable differential relations along the curves in their simplest form, by an optimal choice of coordinates and variables. Special cases of this problem are now classical, but the general case has received little attention and the universal canonic structure of the relations has escaped notice.
On singularities, envelopes and elementary differential geometry
Authors: Bruce J. W.
Published: November 2008

The aim of this paper is to exhibit a connection between certain types of envelope and the discriminant sets of function singularities. We show how conditions that the envelope has a certain local structure (which arise from the singularity theory) often have pleasant geometric interpretations. Moreover in many cases one can show that these conditions are generically (nearly always) satisfied.
A characterization of two-dimensional unmixed local rings
Authors: Trung Ngô Viêt
Published: November 2008

This note deals with some relations between unmixed local rings and the notion of generalized analytic independence.
The projective extensions of a finite group
Authors: Webb P. J.
Published: November 2008

Let G be a finite group and let g be the augmentation ideal of the integral group ring G. Following Gruenberg(5) we let (g̱) denote the category whose objects are short exact sequences of zG-modules of the form and in which the morphisms are commutative diagrams

In this paper we describe the projective objects in this category. These are the objects which satisfy the usual categorical definition of projectivity, but they may also be characterized as the short exact sequences

in which P is a projective module.
The volume of certain convex sets
Authors: McMullen P.
Published: November 2008

Let Ll …, Lr be independent linear subspaces of Ed, with di = dim Li ≥ 1 (i = 1, …, r), and Ed = L1 + … + Lr. For each i, let K¯;i be a convex body in Li with 0 ∈ K¯i, Ki = K¯i + ti(ti ∈ Ed) any translate of K¯i, and define K = con v (K1 ∪ … ∪ Kr), and similarl K¯. Then vol K ≥ vol K¯. Necessary and sufficient conditions for equality are also obtained.
On the law of nullity
Authors: Cohn P. M. , Schofield A. H.
Published: November 2008

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.
A note on the magnetic scalar potential of an electric current-ring
Authors: Hulme A.
Published: November 2008

This paper considers the problem of expressing the magnetic scalar potential associated with the steady flow of an electric current around a thin circular wire, in terms of ‘local’ toroidal coordinates. This potential is known to be a multi-valued function of position and so cannot be expressed directly in terms of fundamental (i.e. single-valued) solutions of Laplace's equation. It is shown that the potential can however be expressed quite simply, as an infinite series of multi-valued toroidal harmonics and that this series is rapidly convergent in the neighbourhood of the current-ring.
Note on the reflexion of water waves at a wall in the presence of surface tension
Authors: Rhodes-Robinson P. F.
Published: November 2008

In this note we examine the influence of surface tension on surface waves incident against a fixed vertical plane wall. The motion is time harmonic and is determined by making the initial assumption that the free-surface slope at the wall is prescribed. From the unique solution obtained for the velocity potential, the parameter involved in this specification can be determined, for small laboratory-scale waves at least, using some longstanding experimental results on meniscus behaviour at a moving contact line. The effect of surface tension is to produce a motion wherein reflexion from the wall is not complete and there is a local disturbance, in contrast to the classical standing-wave motion in the absence of surface tension.
On the probable behaviour of some algorithms for finding the stability number of a graph
Authors: Pittel B.
Published: November 2008

A class of f-driven algorithms due to Chvátal for finding the stability number of a graph are studied. It is shown that, for almost all graphs, the computation time grows subexponentially with n, the number of vertices. If, however, each edge exists with probability δ/n independently on other edges then asymptotically almost certainly the computation time is exponential. Still, for all large enough δ's, these algorithms perform noticeably better than a naive algorithm. The results are extended to random graphs with a fixed number of edges.
Maximally almost disjoint families of representing sets
Authors: Balanda Kevin P.
Published: November 2008

A family of κ-sized sets is said to be almost disjoint if each pair of sets from the family intersect in a set of power less than κ. Such an almost disjoint family ℋ is defined to be κ-maximally almost disjoint (κ-MAD) if |∪ℋ| = κ and each κ-sized subset of ∪ ℋ intersects some member of ℋ in a set of cardinality κ. A set T is called a representing set of a family if T ⊆ ∪ and T has non-empty intersection with each member of .
Determining the thickness of graphs is NP-hard
Authors: Mansfield Anthony
Published: November 2008

The thickness of a graph is a measure of its nonplanarity and has applications in the theory of printed circuits. To determine the thickness of an arbitrary graph is a seemingly intractable problem. This is made precise in this paper where we answer an open problem of Garey and Johnson (2) by proving that it is NP-complete to decide whether a graph has thickness two.
The Coxeter–Todd lattice, the Mitchell group, and related sphere packings
Authors: Conway J. H. , Sloane N. J. A.
Published: November 2008

This paper studies the Coxeter–Todd lattice its automorphism group (which is Mitchell's reflection group 6·PSU(4, 3)·2), and the associated 12-dimensional real lattice K12. We give several constructions for , which is a Z[ω]-lattice where ω = e2πi/3; enumerate the congruence classes of and where θ = ω − ω¯; prove the lattice is unique; determine its covering radius and deep holes; and study its connections with the lattice E6 and the Leech lattice. A number of new dense lattices in dimensions up to about 107 are constructed. We also give an explicit basis for the invariants of the Mitchell group. The paper concludes with an extensive bibliography.
Localization in enveloping algebras
Authors: Lichtman A. I.
Published: November 2008

Let L be a finite-dimensional Lie algebra and U(L) its universal envelope. It is known that U(L) is a Noetherian domain (see (5), theorem v. 3·4) and therefore U(L) has a field of fractions. (Throughout the paper we use the term ‘field’ in the sense of skew field.) We prove in this article the following theorem.
The geometry of the slice-ribbon problem
Authors: Hass Joel
Published: November 2008

One of the principal problems in low-dimensional topology concerns the question of whether all slice knots are ribbon(3),(4). It will be shown in this paper that this problem is closely related to geometric properties of discs properly embedded in the unit 4-ball in Euclidean space. The main result, Theorem 1·13, states that a knot is ribbon if and only if a representative within its isotopy class bounds an embedded minimal disc, and that this in turn happens if and only if a representative bounds a disc of sufficiently small total curvature.
The S-unit equation over function fields
Authors: Silverman Joseph H.
Published: November 2008

In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equation

in S-units of the given ring. To accomplish this over number fields, the only known effective method is to use Baker's deep results on linear forms in logarithms, which yield relatively weak upper bounds. For function fields, R. C. Mason [2] has recently given a remarkably strong effective upper bound. In this note we give an independent proof of Mason's bound, relying only on elementary algebraic geometry, principally the Riemann-Hurwitz formula.
Properties of isolated singularities of some functions taking values in real Clifford algebras
Authors: Ryan John
Published: November 2008

The study of solutions to special examples of elliptic differential equations, using Clifford algebras, has been developed by a number of authors [4–12, 14–16, 18, 19, 21–30, 32]. This study has also been applied [3, 13, 17, 28] within a number of areas of theoretical physics, including Yang–Mills field theory, and the Kähler equation. Most of the function theories associated to the solutions of these elliptic differential equations, referred to as generalized Cauchy–Riemann equations [21,32], generalize in a natural manner many aspects of classical one-variable complex analysis [1]. For instance, each of these function theories contain analogues of the Cauchy theorem, Cauchy integral formula and Laurent expansion theorem. However, no information has yet been obtained on the local topological behaviour of a general solution to any of these equations.
Threshold distribution functions for some random representable matroids
Authors: Oxley James G.
Published: November 2008

A random submatroid ωrof the projective geometryPG(r− 1,q) is obtained fromPG(r− 1,q) by deleting elements so that each element has, independently of all other elements, probability 1 −pof being deleted and probabilitypof being retained. The properties of such random structures were studied in [5] and [6]. In the first of these papers,pwas kept fixed, while in the second, motivated by Erdös and Rényi's work ([3), [4]) on random graphs,pwas taken to be a function ofr. A recent paper of Bollobás[2] strengthens and extends a number of the results of Erdös and Rényi. In this paper we prove matroid analogues of several of Bollobàs's results thereby extending some of the results of [6].
Characters of nilpotent groups
Authors: Carey A. L , Moran W.
Published: November 2008

The characters (extremal positive definite central functions) of discrete nilpotent groups are studied. The relationship between the set of characters of G and the primitive ideals of the group C*-algebra C*(G) is investigated. It is shown that for a large class of nilpotent groups these objects are in 1–1 correspondence. One proof of this exploits the fact that faithful characters of certain nilpotent groups vanish off the finite conjugacy class subgroup. An example is given where the latter property fails.
Prime characteristic functions on compact separable groups
Authors: Wulfsohn Aubrey
Published: November 2008

Existence is established on some groups for Characteristic functions which are prime in the sence of being indecomposable for product factorization.
Alexander polynomials of closed 3-braids
Authors: Morton H. R.
Published: November 2008

The knots and links which can arise as the closure of 3-string braids, and their relations to the braids which give rise to them have been studied by Murasugi [5] and others, including Hartley [2] and more recently Przytycki[6]. Three-braids appear to form a rather special class among braids from some points of view [3]; they are also the only group of braids for which Burau's representation is known to be faithful [1]. They are, however, varied enough to provide an interesting range of knots and links on which to test a number of conjectures.
A note on unknotting number
Authors: Bleiler Steven A.
Published: November 2008

For a classical knot K in the 3-shere, the unknotting number u(K) is defined to be the smallest number of crossing changes required to obtain the unknot, the minimum taken over all the regular projections. This dependence on projection makes the unknotting number a difficult knot invariant. While some algebriac methods exist to give a lower bound for u(K) the unknotting number for approximately one-sixth of the 84 knots with nine or fewer crossings remains unddetermined, see [9] or [7]. For an upper bound, one is usually forced to intellgently experiment knotting various projections of the knot under study. Usual practice is to work with a minimal crossingprojection; indeed, it has long been a ‘folk’ conjecture that the unknotting number is realized in such a projection ([5], p. 21). This note shows by example that this conjecture is false. This remarkable knot is rational, i.e. a 2-bridge knot, and hence alternating. Thus there is also the surprising result that the unknotting number of an alternating kniot is not necessarily realized in a minimal alternating projecting.
On ∧-Mackey convergence in locally convex spaces
Authors: Fourie Jan H.
Published: November 2008

In this note we introduce the concepts of Λ-Mackey sequence, Λ-Mackey convergence property, Λ-Schwartz family and associated Λ-Schwartz family and consider some applications of these to locally convex spaces. Hereby Λ denotes a Banach sequence space with the AK-property — the results of this paper generalize those in [4] where the case Λ = I1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the Λ-Mackey convergence property and characterize the dual Λ-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra-bornological is proved.
Incompressible surfaces in the exterior of a closed 3-braid
Authors: Lozano María Teresa , Przytycki Józef H.
Published: November 2008

Letbe a closed 3-braid inS3with axisLγ.The exterior of γU Lγ inS3is a fibre bundle overS1with fibre a disc with 3 holes. The monodromy is given by a matrix inPSL(2,) and the braid γis called hyperbolic if its matrix is hyperbolic.
Unbounded subharmonic functions bounded in one tract
Authors: CÁmera G. A.
Published: November 2008

In this paper we shall consider the class Hm of subharmonic functions u, unbounded in, which have a thick tract where u is bounded. We recall here that non-constant subharmonic functions in space (i.e. m ≥ 3) are not necessarily unbounded as in the two-dimensional case. For basic concepts in this area we refer the reader to [3].
Note on propagation of classical waves: II
Authors: Kanseta H.
Published: November 2008

1. Main results and notations. We consider Cauchy's problem for the classical wave equation in :

with initial conditions u(0, x) = uo(x) and ∂tu(0, x) = u1 (x). In (1·1) δ and m2 stand for the Laplacian in and a constant respectively. Note that a second order hyper-bolic differential equation with real constants can be reduced to (1·1) ([1], p. 183). Let uj (j = 0,1) be C∞ funotions on whose support is contained in the closed ball BR = {xεl; |x| ≥R} for some R > 0. In this note we shall show that the solution u(t, x) of (1·1) possesses the following properties.
On the micro-to-macro transition in constitutive analyses of elastoplastic response at finite strain
Authors: Hill R.
Published: November 2008

The theoretical transition between basic properties of elastoplastic media at two levels of description is examined rigorously. At the micro-level the material response is heterogeneous, whereas at the macro-level it appears homogeneous. A broad class of constitutive relations is envisaged, and no restriction is placed on the magnitude of deformations and rotations at the micro-level. The investigation is concerned with quadratic differential forms that feature prominently in constitutive analyses, and is complementary to a previous study of bilinear differential forms. A principal objective is to access the transmissibility of measure-invariant inequalities from one level to the other.
A note on random walks in multidimensional time
Authors: Galambos Janos , Kátai Imre
Published: November 2008

Let Kr denote the set of r-tuples n = (n1, n2, …, nr), r ≥ 1, where the components ni are positive integers. Let {X, Xn, n ∈ Kr} be a family of independent and identically distributed random variables with positive mean EX = μ < + ∞ and finite variance VX = σ2 < + ∞. In a recent work, M. Maejima and T. Mori [2] have shown that, if X is integer valued, aperiodic and E∣X∣3 < + ∞, then, for r = 2 or 3,

where

the summation being extended over all members j = (j1,j2, …, jr) of Kr that satisfy jt ≤ nt for all 1 ≤ t ≤ r.
Fronsdal *-quantization and Fell inducing
Authors: Hennings M. A.
Published: November 2008

The process of Fronsdal inducing is discussed as a method of quantizing physical systems, and is seen to reduce to a technique for obtaining representations of *-algebras.

It is asked whether this above process is at all successful, and whether it is at all related to the representation inducing methods of Fell. The two inducing techniques are shown to be equivalent for algebras satisfying three specified properties, and it is also shown that all the strictly irreducible representations of such algebras can be found by these methods.

Examples are produced whereby it is shown that algebras of the type required by the theory are of physical interest.
Real forms of cusp singularities
Authors: Wall C. T. C.
Published: November 2008

Cusp singularities were introduced and described in detail in Hirzebruch's fundamental paper [3] (se recall some of the basic results in § 1 below). They form a natural and well-behaved class, included in Laufer's ‘minimally elliptic’ singularities [5]. Those which occur on hypersurfaces appear also as hyperbolic singularities in Arnold's [1] classification of 1-modal singularities.
Estimates for polynomial norms on Lp(μ) spaces
Authors: Sarantopoulos I.
Published: November 2008

If L is a symmetric m-linear form on a Banach space and L^ is the associated polynomial then

For special choices of Banach space this inequality can be improved. This has been done by Harris [5] in the case of the Lp(μ) spaces. In this paper we improve his estimates and disprove one of his conjectures.
Bernstein's theorem for compact, connected Lie groups
Authors: Gaudry Garth I. , Pini Rita
Published: November 2008

In what follows, G will denote a compact, connected Lie group.

Definition. A complex-valued function f ε L2(G) lies in the space A(G) if it can be written in the form

where

The A(G) norm of f is the infimum of all sums (2) with respect to all decompositions (1).
Stable decompositions of classifying spaces of finite abelianp-groups
Authors: Harris John C. , Kuhn Nicholas J.
Published: November 2008

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecomposition

intoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.
Geometry of the tensor product of C*-algebras
Authors: Blecher D. P.
Published: November 2008

When and ℬ are C*-algebras their algebraic tensor product ⊗ ℬ is a *-algebra in a natural way. Until recently, work on tensor products of C*-algebras has concentrated on norms α which make the completion ⊗α ℬ into a C*-algebra. The crucial role played by the Haagerup norm in the theory of operator spaces and completely bounded maps has produced some interest in more general norms (see [8; 12]). In this paper we investigate geometrical properties of algebra norms on ⊗ ℬ. By an ‘algebra norm’ we mean a norm which is sub-multiplicative: α(u.v) ≤ ≤ α(u).α(v).
Relative characters for H-projective RG-lattices
Authors: Symonds Peter
Published: November 2008

If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.
A generalized Bernstein approximation theorem
Authors: Prolla João B.
Published: November 2008

A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of f

one for each integer n ≥ 1. Bernstein's Theorem states that Bn(f) → f uniformly on [0, 1] and, since each Bn(f) is a polynomial of degree at most n, we have as a consequence Weierstrass' theorem. See for example Lorentz [9]. The operator Bn, defined on the space C([0, 1]; ℝ) with values in the vector subspace of all polynomials of degree at most n has the property that Bn(f) ≥ 0 whenever f ≥ 0. Thus Bernstein's Theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials. This raises the following natural question: consider a compact Hausdorff space X and the convex cone C+(X):= {f ∈ C(X; ℝ); f ≥ 0}. Now the analogue of Bernstein's Theorem would be a theorem stating when a convex cone contained in C+(X) is dense in it. More generally, one raises the question of describing the closure of a convex cone contained in C(X; ℝ), and, in particular, the closure of A+:= {f ∈ A; f ≥ 0}, where A is a subalgebra of C(X; ℝ).
A relationship between link polynomials
Authors: Lickorish W. B. R.
Published: November 2008

The discovery by V. F. R. Jones of a Laurent polynomial invariant VL(t)∈ℤ[t±½] for every oriented link L in the 3-sphere prompted the finding of a more general 2-variable Laurent polynomial invariant PL(l, m)∈ℤ[l±1,m±1], (see [4], [3] and [6]).
Sequence spaces defined by a modulus
Authors: Maddox I. J.
Published: November 2008

Ruckle[4] used the idea of a modulus function ƒ (see Definition 1 below) to construct the sequence space

This space is an FK space, and Ruckle proved that the intersection of all such L(f) spaces is ø, the space of finite sequences, thereby answering negatively a question of A. Wilansky: ‘Is there a smallest FK-space in which the set {e1, e2, …} of unit vectors is bounded?’
Locally finite approximation of Lie groups. II
Authors: Friedlander Eric M. , Mislin Guido
Published: November 2008

In an earlier paper [10], we constructed a ‘locally finite approximation away from a given prime p’ of the classifying space BG of a Lie group with finite component group. Such an approximation consists of a locally finite group g and a homotopy class of maps which in particular induces an isomorphism in cohomology with finite coefficients of order prime to p. The usefulness of such a construction is that it reduces various homotopy-theoretic questions concerning the space BG to the corresponding questions concerning Bπ for finite subgroups π. For example, we demonstrated in [10] how H. Miller's proof of the Sullivan conjecture concerning maps from , where π is a finite group and X is a finite-dimensional complex, can be extended to maps BG→X for G a Lie group with finite component group.
A class of non-Kählerian manifolds
Authors: Johnson F. E. A.
Published: November 2008

Let S+ (resp. S−) denote the class of fundamental groups of closed orientable (resp. non-orientable) 2-manifolds of genus ≥ 2, and let surface = S+ ∪ S−. In the list of problems raised at the 1977 Durham Conference on Homological Group Theory occurs the following([7], p. 391, (G. 3)).
The derived group of a 2-group
Authors: Blackburn Norman
Published: November 2008

Burnside[1] considered possible restrictions on the derived group G′ of p-group G and showed that if G′ is non-Abelian, the centre Z(G′) of G′ is not cyclic. This implies that |G′: G″| ≥ p3. Many other restrictions on G′ are to be found in Hall's famous paper [2], but in 1954 Hall proved that if p is odd and |G′: G″| = p3, then |G′| ≤ p. So far as I know, no proof of this is to be found in the literature, but it follows from the lemma below. Our concern here is with the case p = 2, and we shall prove the following.
Corrigenda
Authors: Antman Stuart S. , Calderer M. Carme
Published: November 2008

‘Asymptotic shapes of inflated noncircular elastic rings’

‘Asymptotic shapes of inflated spheroidal nonlinearly elastic shells’

Formulae (2·12) of both papers, giving an expression for the curvature, k̅, are wrong. To correct them, let ο(s) be the angle between r̅′ = ν̅a+ηb and i. Then

whence formulae (3·2) of both papers imply that

Formula (C) corrects the erroneous expression for k by adding the second term on the right-hand side of (C). Since this term is of order ∈, the analysis in the remainder of both papers is unaffected by this correction. (The analyses of higher-order terms do not include a study of corrections to the curvature.)
On the dual and hessian mappings of projective hypersurfaces
Authors: Choudary A. D. R. , Dimca A.
Published: November 2008

We investigate the first-order Thom–Boardman singularity sets of the dual mapping for an arbitrary (and then for a generic) smooth hypersurface in the complex projective space ℙn. Our results focus on nonemptiness, connectedness, regular stratifications and numerical invariants for these sets.
Stability of rotating compressed rod with imperfections
Authors: Atanacković T. M.
Published: November 2008

The stability of a rotating compressed rod is studied as a two-parameter bifurcation problem. Imperfections in shape and loading are assumed to be present. A number of solutions and their local behaviour is analysed.
Ordered products of topological groups
Authors: Henriksen M. , Kopperman R. , Smith F. A.
Published: November 2008

The topology most often used on a totally ordered group (G, <) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology.
Geodesic completeness and stability
Authors: Beem John K. , Ehrlich Paul E.
Published: November 2008

A (connected) Riemannian manifold M is geodesically complete (i.e. each geodesic may be extended to a geodesic with domain ( −∞, + ∞)) iff, as a metric space under the induced Riemannian distance function, M is Cauchy complete ([12], p. 138). Furthermore, all compact Riemannian manifolds are complete. On the other hand, compact pseudo-Riemannian manifolds exist which are not geodesically complete. For example Fierz and Jost[8] have constructed incomplete metrics of the form 2dx dy + h dy2 on T2. Furthermore, Williams [16] has shown that geodesic completeness may fail to be stable for pseudo-Riemannian manifolds. Here a property is said to be stable if the set of metrics for M with this property is open. This failure of stability may occur for both compact and non-compact manifolds. In particular, Williams has found that M = S1 x S1 = ℝ/(2πZ x 2πZ), which may be given the complete Lorentzian metric g = 2dx dy, also admits Lorentzian metrics gn = 2dx dy+(sin(x)/n) dy2 which are geodesically incomplete yet which for large n are arbitrarily close to g in the Whitney fine Cr topologies. In this example the set S = {(0, y)|0 ≤ y ≤ 2π} represents the image of a closed null geodesic for (M, g) and also for all (M, gn). However, for the metrics (M, gn) this closed null geodesic has affine parametrizations which are incomplete because the velocity vector of this null geodesic returns to a scalar multiple of itself after each complete circuit of S. In general, a closed null geodesic will be complete iff its velocity vector returns exactly to itself after each trip around its image.
On the fundamental group of the complement of certain singular plane curves
Authors: Némethi András
Published: November 2008

Let C be a complex algebraic curve in the projective space ℙ2. The purpose of this paper is to calculate the fundamental group G of the complement of C in the case when C = X ∩ H1 ∩ … ∩ Hn−2, where

and Hi are generic hyperplanes (i = 1, … n − 2).
The group of almost automorphisms of the countable universal graph
Authors: Truss J. K.
Published: November 2008

The group Aut Γ of automorphisms of Rado's universal graph Γ (otherwise known as the ‘random’ graph: see [1]) and the corresponding groups Aut Γc for C a set of ‘colours’ with 2 ≤ |C| ≤ ℵ0, were studied in [4]. It was shown that Aut Γc is a simple group, and the possible cycle types of its members were classified. A natural extension of Aut Γc to a highly transitive permutation group on the same set is obtained by considering the ‘almost automorphisms’ of Γ. It is the purpose of the present paper to answer similar questions about the resulting group AAut Γc. Namely we shall classify its normal subgroups and the cycle types of its members. The main result on normal subgroups is summed up in Corollary 2·9, which says that the non-trivial normal subgroups of AAut Γc form a lattice isomorphic to the lattice of subgroups of the free Abelian group of rank n where n = |C| – 1, and for cycle types it will be shown that those occurring in AAut Γc are precisely the same as in Aut Γc except for those which are the product of finitely many cycles.