bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalized splitting theorems
Authors: May J. P. , Taylor L. R.
Published: November 2008

In (5), we and Fred Cohen gave some quite general splitting theorems. These described how to decompose the suspension spectra of certain filtered spaces CX as wedges of the suspension spectra of their successive filtration quotients Dq X. The spaces CX were of the form Cr × Xr/(˜) for suitable sequences of spaces {Cr} and {Xr}, and the construction CX was intended to be a reworking in ‘proper generality’ of the constructions introduced in (9).
Transfer and ramified coverings
Authors: Smith Larry
Published: November 2008

In this note we introduce a general class of finite ramified coverings π X˜ ↓ X. Examples of ramified covers in our sense include: finite covering spaces, branched covering spaces and the orbit map Y ↓ Y/G where G is a finite group and Y an arbitrary G-space. For any d-fold ramified covering π: X˜ ↓ X we construct a transfer homomorphism

with the expected property that

is multiplication by d. As a consequence we obtain a simple proof of the Conner conjecture; viz. the orbit space of an arbitrary finite group action on a ℚ-acyclic space is again ℚ acyclic.
Smith equivalence of representations
Authors: Petrie Ted
Published: November 2008

An old question of P. A. Smith asks: If a finite group G acts smoothly on a closed homotopy sphere Σ with fixed set ΣG consisting of two points p and q, are the tangential representations Tp Σ and Tq Σ of G at p and q equal? Put another way: Describe the representations (V, W) of G which occur as (Tp ΣTq Σ) for Σ a sphere with smooth action of G and ΣG = p ∪ q. Under these conditions we say V and W are Smith equivalent (21) and write V ~ W. A stronger equivalence relation is also interesting. We say representations V and W are s-Smith equivalent if (V, W) = (Tp Σ, Tq Σ) and Σ is a semi-linear G sphere (23), i.e. ΣK is a homotopy sphere for all K and ΣG = p ∪ q. In this case we write V ≈ W.
Almost periodic generalized functions
Authors: Burkill H. , Rennie B. C.
Published: November 2008

In (4) a space C of generalized functions was defined which is rather larger than the simple space used to such effect by Lighthill in (3). At the core of C is the space C0 = T of test functions. These are entire (complex) functions f such that all derivatives of f and its Fourier transform F have order of magnitude not exceeding as x → ± ∞, where c is a positive number depending on the individual derivative concerned. If f, g∈ T, the inner product 〈f | g〉 is defined to be
Categorical characterization of uniform hyperspaces
Authors: Harvey J. M.
Published: November 2008

In this paper, we use categorical methods to arrive at a characterization of Hausdorff hyperspaces of separated uniform spaces within a category of ordered uniform spaces. We begin by showing that the hyperspace construction gives rise to a monad H in the category Uni of non-empty separated uniform spaces and uniformly continuous maps. We then identify the category of H-algebras and characterize hyperspaces in this context.
Further aspects of the evaluation of σ(m. n≠0, 0) (am2 + bnm + cn2)−s
Authors: Zucker I. J. , Robertson M. M.
Published: November 2008

In some previous publications, Zucker and Robertson [13, 14, 15] for certain cases evaluated exactly the double sum

with a, b, c integers. In (1·1) the summation is over all integer values of m and n, both positive and negative, but excluding the case where m and n are simultaneously zero. The term ‘exact’ used here is in the sense introduced by Glasser [5], and means that S may be expressed as a linear sum of products of pairs of Dirichlet L-series. S is then said to be solvable. Whether S may be solved or not depends on the properties of the related binary quadratic form (am2 + bmn + cn2) = (a, b, c). The cases considered in [15] were when a > 0 and d = b2 − 4ac < 0, i.e. (a, b, c) was positive definite. When this is so, Glasser [5] conjectured that S was solvable if and only if (a, b, c) had one reduced form per genus, i.e. the reduced forms of (a, b, c) were disjoint. Zucker and Robertson [15] were in fact able to solve S for all (a, b, c) for which d < 0 and whose reduced forms were disjoint. A complete list of solutions may be found in [6].
Some renewal theorems for random walks in multidimensional time
Authors: Maejima Makoto , Mori Toshio
Published: November 2008

Let Kr denote the set of r-tuples n = (n1n2, …, nr) with positive integers as coordinates (r ≥ 1) and {X, Xn, n ε Kr} be a family of independent, identically distributed random variables with positive mean 0 < EX ≡ μ < ∞ and finite positive variance 0 < var X ≡ σ2 ∞. The notation m ≤ n, where m = (mi) and n = (ni), means that mi ≤ ni (i = 1, 2,…, r) and |n| = n1n2 … nr. Denote Sn =Σj ≤ nXj (j, n ε Kr). When r = 1, {Xn, n ε Kr) reduces to the sequence {Xj, j ε 1} of independent random variables each distributed as X, and Sn becomes the ordinary partial sum .
On nearly uniformly convex and k-uniformly convex spaces
Authors: Istrăt‚escu V. I. , Partington J. R.
Published: November 2008

In this note we prove that every nearly uniformly convex space has normal structure and that K-uniformly convex spaces are super-reflexive.

We recall [1] that a Banach space is said to be Kadec–Klee if whenever xn → x weakly and ∥n∥ = ∥x∥ = 1 for all n then ∥xn −x∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in [1]. For the reader's convenience we recall them here.
Finding fundamental units in totally real fields
Authors: Cusick T. W.
Published: November 2008

Let F a totally real cubic field. For any number α in F, let α, α′, α″ denote the conjugates of α. Define the function T(α) by

.
Various representations of a generalized hypergeometric function through statistical techniques
Authors: Anderson W. J. , Mathai A. M.
Published: November 2008

This paper deals with a statistical technique of deriving new results on Meijer's G-function. Multiple integral as well as computable series representations are given. The method used is to derive the density of a product of independent beta-distributed random variables by using different statistical techniques, and then compare the expressions to get the desired results. The results obtained in this article do not seem to be available through standard mathematical techniques.
An existence theorem for the discrete coagulation-fragmentation equations
Authors: Spouge John L.
Published: November 2008

This paper gives an existence result for the discrete coagulation-fragmentation equations:

(If k = 1, the first and last sums are 0.)
Boundary conditions and reducibility of differential operators
Authors: Fisherl B. , Zahreddine Z.
Published: November 2008

Examples were exhibited in [4] of both reducible and irreducible symmetric operators (of deficiency index (1:1)) associated with − d2/dt2 in the Hilbert space L2(I) (I = [0,1). Such symmetric operators are determined by three linearly independent boundary conditions which define their domains as restrictions of the domain of the maximal operator associated with — d2/dt2.
Ideals in group rings of soluble groups of finite rank
Authors: Brookes Christopher J. B.
Published: November 2008

The original motivation for this paper was the question of primitivity for group rings of soluble groups of finite rank. At the end this is touched upon as an application of a theorem about prime ideals in such rings. If a group Γ acts on a set S we say an element is (Γ)-orbital if its orbit is finite and write ΔΓ(S) for the subset of such elements. The FC-radical of a group G, denoted by Δ(G), is just ΔG(G) where the action of G on itself is by conjugation.
On Toda's fibrations
Authors: Gray Brayton
Published: November 2008

In 1956 Toda [5] introduced two fibrations localized at p > 2:

where is a subcomplex of the James construction . The construction of H′ was somewhat difficult and was discussed by Moore (see [3]). Moore's definition however is not natural (in the sense of Theorem 1 (b) below). It is our purpose to give another definition, closer to Toda's original definition, which is natural, is an H map* and behaves well with respect to the Dyer-Lashof map λ: BΣp → Q(So). We use this to settle an unanswered question in [2].
Asymptotic shapes of inflated noncircular elastic rings
Authors: Antman Stuart S. , Calderer M. Carme
Published: November 2008

In this paper we study the asymptotic behaviour of large deformations of nonlinearly elastic, noncircular rings under internal hydrostatic pressure. These rings can undergo flexure, extension, and shear. Their governing equations are the same as those for the inflation of cylindrical shells.
The maximal subgroups of the Lyons group
Authors: Wilson Robert A.
Published: November 2008

In this paper we complete the work begun in [2] on the subgroup structure of the Lyons simple group Ly of order
The Cauchy integral and analytic continuation
Authors: Brennan James. E.
Published: November 2008

One of the most important concepts in the theory of approximation by analytic functions is that of analytic continuation. In a typical problem, for example, there is generally a region Ω, a Banach space B of functions analytic in Ω and a subfamily ℱ ⊂ B, each member of which is analytic in some larger open set, and one might be asked to decide whether or not ℱ is dense in B. It often happens, however, that either ℱ is dense or the only functions which can be so approximated have a natural analytic continuation across ∂Ω. A similar phenomenon is also known to occur even for approximation on sets without interior. In this article we shall describe a method for proving such theorems which can be applied in a variety of settings and, in particular, to: (1)  the Bernštein problem for weighted polynomial approximation on the real line; (2)  the completeness problem for weighted polynomial approximation on bounded simply connected regions; (3) the Shapiro overconvergence problem for sequences of rational functions with sparse poles; (4) the Akutowicz-Carleson minimum problem for interpolating functions. Although we shall present no new results, the method of proof, which is based on an argument of the author [6], seems sufficiently versatile to warrant exposition.
On automorphisms of unique factorization domains
Authors: Brown M. L.
Published: November 2008

Chatters has asked whether a unique factorization domain (UFD) R, equipped with an automorphism σ transitive on the height 1 prime ideals of R, is necessarily Dedekind. If R contains an uncountable field, then Chatters observed that the answer is affirm ative. In this note was show:

Theorem. Let R be a local UFD equipped with an automorphism σ which is transitive on the height 1 prime ideals of R. Suppose that σ induces an automorphism of finite order on the residue field κ of R (for example, if κ is a global or finite field), Then R is Dedekind.
Generalized fractions, Buchsbaum modules and generalized Cohen-Macaulay modules
Authors: Sharp R. Y. , Zakeri H.
Published: November 2008

Let A be a (commutative Noetherian) local ring (with identity) having maximal ideal m and positive dimension. This note is concerned with, among other things, a complex of A -modules which was studied in [8] and which involves modules of generalized fractions derived from A and subsets of systems of parameters for A; in ([8], 3·5), the complex was shown to have connections with local cohomology. The complex is described as follows.
On the quantificational logic of intuitionistic set theory
Authors: Friedman Harvey M. , Ščedrov Andrej
Published: November 2008

Formal propositional logic describing the laws of constructive (intuitionistic) reasoning was first proposed in 1930 by Heyting. It is obtained from classical pro-positional calculus by deleting the Law of Excluded Middle, and it is usually referred to as Heyting's (intuitionistic) propositional calculus ([9], §§23, 19) (we write HPP in short). Formal logic involving predicates and quantifiers based on HPP is called Heyting's (intuitionistic) predicate calculus ([9], §§31, 19) (we write HPR in short).
Perfect powers in values of certain polynomials at integer points
Authors: Shorey T. N.
Published: November 2008

1. For an integer v > 1, we define P(v) to be the greatest prime factor of v and we write P(1) = 1. Let m ≥ 0 and k ≥ 2 be integers. Let d1, …, dt with t ≥ 2 be distinct integers in the interval [1, k]. For integers l ≥ 2, y > 0 and b > 0 with P(b) ≤ k, we consider the equation

Put

so that ½ < vt ≤ ¾. If α > 1 and kα < m ≤ kl, then equation (1) implies that

for 1 ≤ i ≤ t and hence
Uniqueness and continuous dependence in the linear elastodynamic exterior and half-space problems
Authors: Galdi G. P. , Knops R. J. , Rionero S.
Published: November 2008

A method based upon the Lagrange identity has been used by Brun [2] in the linear theories of thermoelasticity and viscoelasticity to establish uniqueness of the solution to the initial boundary value problem on bounded three-dimensional regions. A major feature of Brun's analysis is that it does not require any sign-definiteness assumptions on, for instance, the elasticities. The technique was extended by Knops and Payne [14] to derive certain continuous dependence results in linear elastodynamics, again for a bounded region. These authors had earlier recovered Brun's uniqueness result for linear elasticity [11] and derived other continuous dependence results based upon logarithmic convexity arguments [12, 14] (see also [13] for a similar treatment of thermoelasticity). Levine [18] later treated an abstract version of the Brun approach and applied it to a family of abstract linear operator equations. Among his results is a simplified proof that equipartition of the kinetic and potential energies occurs. Other applications of the Lagrange identity in proofs of uniqueness for bounded regions include those by Naghdi and Trapp [19] for a Cosserat surface, and by Green [9] for a theory of linear thermoelasticity that allows second sound.
A field theory approach to stability of radial equilibria in nonlinear elasticity
Authors: Sivaloganathan J.
Published: November 2008

In this paper we study the stability of a class of singular radial solutions to the equilibrium equations of nonlinear elasticity, in which a hole forms at the centre of a ball of isotropic material held in a state of tension under prescribed boundary displacements. The existence of such cavitating solutions has been shown by Ball[1], Stuart [11] and Sivaloganathan[10]. Our methods involve elements of the field theory of the calculus of variations and provide a new unified interpretation of the phenomenon of cavitation.
A localic approach to minimal prime spectra
Authors: Shu-Hao Sun
Published: November 2008

Throughout this paper, A will denote a distributive lattice with 0 and 1; we shall write spec A for the prime spectrum of A (i.e. the set of prime ideals of A, with the Stone–Zariski topology), and max A, min A for the subspaces of spec A consisting of maximal and minimal prime ideals respectively. These two subspaces have rather different topological properties: max A is always compact, but not always Hausdorff (indeed, any compact T1-space can occur as max A for some A), and min A is always Hausdorff (in fact zero-dimensional), but not always compact. (For more information on max A and min A, see Simmons[3].)
Stable mappings of discriminant varieties
Authors: Bruce J. W.
Published: November 2008

Smooth mappings defined on discriminant varieties of -versal unfoldings of isolated singularities arise in many interesting geometrical contexts, for example when classifying outlines of smooth surfaces in ℝ3 and their duals, or wave-front evolution [1, 2, 5]. In three previous papers we have classified various stable mappings on discriminants. When the isolated singularity is weighted homogeneous the discriminant is not a local smooth product, and this makes the classification of stable germs considerably easier than in general. Moreover, discriminants arising from weighted homogeneous singularities predominate in low dimensions, so such classifications are very useful for applications.
On shrinking bases in p-Banach spaces
Authors: Ariño M. A.
Published: November 2008

Singer [7] proved that in a Banach space all basic sequences are shrinking if and only if all of them are boundedly complete. Afterwards; Zippin [2] proved a similar result for Schauder bases, and it was extended [1] to Schauder decompositions.
Large correlated families of positive random variables
Authors: Fremlin D. H.
Published: November 2008

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.
Fourier-Laplace transforms decaying exponentially
Authors: Kaneta H.
Published: November 2008

Let ƒ(x) be a C-valued function which is integrable with respect to a positive measure m on the Borel σ-field of R = (– ∞, ∞). We shall give necessary and sufficient conditions under which the following Fourier-Laplace transform f(λ) of ƒ(x) decays exponentially as λ → ∞:

where θ is a constant with –π/2 ≤ θ ≤ π/2.
The quasi-centre of a Banach algebra
Authors: Rennison J. F.
Published: November 2008

An element a of a Banach algebra A over ࠶ will be called quasi-central if, for some K ≥ 1,

The set Q(A) of all quasi-central elements of A will be called the quasi-centre of A and the set of elements a which satisfy (1) for a particular value of K will be denoted by Q(K, A). The motivation for these definitions is the result of Le Page ([1], proposition 3) that Q(1, A) coincides with the centre Z(A) of A. The reader is referred to [2] and [3] for a study of the properties of quasi-central elements.
Shape morphisms and components of movable compacta
Authors: Sanjurjo José M. R.
Published: November 2008

The relationship between components and movability for compacta (i.e. compact metric spaces) was described by Borsuk in [5]. Borsuk proved that if each component of a compactum X is movable, then so is X. More recently Segal and Spiez[19], motivated by results of Alonso Morón[1], have constructed a (non-compact) metric space X of small inductive dimension zero and such that X is non-movable. The construction of Segal and Spiez was based on the famous space of P. Roy [16]. On the other hand, K. Borsuk gave in [5] an example of a movable compactum with non-movable components. The structure of such compacta was studied by Oledzki in [15], where he obtained an interesting result stating that if X is a movable compactum then the set of movable components of X is dense in the space of components of X. Oledzki's result was later strengthened by Nowak[14], who proved that if all movable components of a movable compactum X are of deformation dimension at most n, then so are the non-movable components and the compactum X itself.
Random dissections and branching processes
Authors: Kingman J. F. C.
Published: November 2008

For a time in the mid-1970s probabilists were tantalized by a seemingly simple problem posed by Araki and Kakutani[3]. An interval is repeatedly divided by points chosen successively at random, the nth point being uniformly distributed over the largest of the n intervals formed by the first n − 1 points. Is this sequence of points asymptotically uniformly distributed?
Symmetry relations for connection matrices in the phase-integral method
Authors: Fröman Nanny , Fröman Per Olof , Lundborg Bengt
Published: November 2008

In many cases of practical interest there is some kind of symmetry associated with an ordinary second-order differential equation of the Schrödinger type. This symmetry is reflected in certain relations between different F-matrices or different elements of the same F-matrix of the phase-integral method. These symmetry relations imply in turn further relations between the Stokes constants, in addition to those which exist for a general cluster of transition points, and thereby reduce the number of independent Stokes constants. The present paper deals with two important kinds of symmetry relations.
Lifting colimits in various categories
Authors: Heath Philip R. , Parmenter M. M.
Published: November 2008

In a recent publication [2], R. Brown and the first author proved a Lifting Theorem for groups (and topological groups) showing that if β: B → H is an epimorphism of groups and H is a certain type of colimit of groups, then this colimit can be lifted (or pulled back) through β; that is B is a colimit of the lifted diagram (see Corollary 2·3 below).
Constructing locales from quantales
Authors: Niefield Susan B. , Rosenthal Kimmo I.
Published: November 2008

We recall that a locale is a complete lattice L satisfying a ∧ (∨bα) = ∨(a ∧ bα), for all a ∈ L, and {bα} ⊆ L. Examples of locales include the lattices (X) of open subsets of topological spaces X. Following Joyal and Tierney [7], a morphism f: L → M of locales is a ∨-, ∧-, and τ-preserving map. Such functions are sometimes called ‘frame homomorphisms’, in which case the right adjoint f*:M → L (which exists since f preserves ∨) is then called a ‘morphism of locales’.
On the analytic equivalence of curves
Authors: Elias Joan
Published: November 2008

Throughout this paper k will be an algebraically closed field, R the ring k[[X1, …, XN]] and (kN, 0) the k-scheme Spec(R).
Vanishing sums in function fields
Authors: Brownawell W. D. , Masser D. W.
Published: November 2008

Let k be a field of zero characteristic, and let F be a function field over k of genus g. We normalize each valuation v on F so that its order group consists of all rational integers, and for elements u1, …, un of F, not all zero, we define the (projective) height as

The sum formula on F shows that this is really a height on the projective space .
Fourier operators on weighted Hardy spaces
Authors: Heinig Hans P.
Published: November 2008

In this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.
Inclusions between FK spaces and Kuttner's theorem
Authors: Maddox I. J.
Published: November 2008

The result known as Kuttner's theorem [2] asserts that if 0 < p < 1 and A is a Toeplitz matrix then there is a sequence which is strongly Cesàro summable with index p but which is not A summable. This theorem was extended by Maddox[3] to coregular matrices, and Thorpe [9] gave a further extension by showing that if 0 < p < 1 and X is a locally convex FK space with X ⊃ w0(p) then X ⊃ l∞. Here, w0(p) denotes the space of sequences strongly summable to 0, i.e. x∈w0(p) if and only if

and l∞ denotes the space of bounded sequences. Other proofs of Thorpe's extension and related results appear in Maddox[4, 5].
The quasi-strict topology on the space of quasi-multipliers of a B*-algebra
Authors: Kassem M. S. , Rowlands K.
Published: November 2008

The notion of a left (right, double) multiplier may be regarded as a generalization of the concept of a multiplier to a non-commutative Banach algebra. Each of these is a special case of a more general object, namely that of a quasi-multiplier. The idea of a quasi-multiplier was first introduced by Akemann and Pedersen in ([1], §4), where they consider the quasi-multipliers of a C*-algebra. One of the defects of quasi-multipliers is that, at least a priori, there does not appear to be a way of multiplying them together. The general theory of quasi-multipliers of a Banach algebra A with an approximate identity was developed by McKennon in [5], and in particular he showed that the quasi-multipliers of a considerable class of Banach algebras could be multiplied. McKennon also introduced a locally convex topology γ on the space QM(A) of quasi-multipliers of A and derived some of the elementary properties of (QM(A), γ).
The Fourier–Plancherel transform as a spectral representation of a rigged Hilbert space
Authors: Fishel B.
Published: November 2008

In ‘Generalized Translation Operators…’ [3] algebras associated with a self-adjoint operator were investigated. Examples were given in the cases, inter alia, of the operators Mt and i d/dt in the space L2(ℝ, m) (m = Lebesgue measure). This paper shows that by suitably rigging the space the examples can be seen as natural generalizations of certain familiar algebras. The rigging enables us to introduce, rigorously, into L2(ℝ,m) the improper elements used by Akhiezer and Glazman [1] as cyclic vectors for Mt and i d/dt in order to identify the Fourier–Plancherel transform with a spectral representation of the space.
On the automorphism groups of certain Lie algebras
Authors: Segal Dan
Published: November 2008

We fix a ground field k and a finite separable extension K of k. To a Lie algebra L over k is associated the Lie algebra KL = K ⊗kL over K. If we forget the action of K, we can think of KL as a larger Lie algebra over k; in particular we can ask what is the automorphism group Autk KL of KL as a k-algebra. There does not seem to be any simple answer to this question in general; the purpose of this note is to give a simple condition on L which makes Autk KL quite easy to determine. Examples of algebras which satisfy this condition include the free nilpotent Lie algebras and the algebras of all n × n triangular nilpotent matrices.
Units in real cyclic quartic fields
Authors: Cusick T. W.
Published: November 2008

Let F be a totally real quartic field. For any α in F, let α, α′, α″, α‴ or α(0) = α(1), α(2), α(3) denote the conjugates of α. Define the function T(α) by

We define a triple of units ε1, ε2, ε3 in F as follows. Let ε1 be a unit which gives the least value of T(ε) for any unit ε ≠ = ± 1 in F. Let ε2 be a unit which gives the least value of T(ε) for any unit ε ≠ = ± ε1m with m a rational integer. Let ε3 be a unit which gives the least value of T(ε) for ε ≠ = ± ε1m ε2n with m and n rational integers. We call ε1, ε2, ε3 the successive unit minima for T(ε).
The complement of a codimension-k immersion
Authors: Hirsch Michael D.
Published: November 2008

In [1] Marc Feighn proves the following result: a proper, C2, codimension-1 immersion in a manifold with no first homology separates the ambient space. In [4] Michelangelo Vaccaro proves a related result: the C1-immersed image of a compact n-manifold with image a (curved) polyhedron has non-zero Hn with ℤ2 coefficients. (In Vaccaro's terminology f(M) is a curved polyhedron if f is smooth and f(M) is the (non-PL) embedded image of a simplicial complex.) Using ideas similar to Feighn's we prove here the following result.
On asymptotic stability for sets of prime ideals connected with the powers of an ideal
Authors: Melkersson Leif
Published: November 2008

The notion of associated prime ideal and the related one of primary decomposition are classical. In a dual way one defines attached prime ideals and secondary representation. This theory is developed in the appendix to §6 in Matsumura[5] and in Macdonald[3].
Birational invariants of crystals and fields with a finite group of operators
Authors: Farkas Daniel R.
Published: November 2008

It is well known that an n-dimensional crystallographic group can be reconstructed from its point group, the integral representation of the point group which arises from its action on the translation lattice, and the 2-cocycle which glues the point group to the lattice ([2]). In practice, this constitutes a complicated list of invariants. When confronted with the classification of objects possessing a rich structure, the algebraic geometer first attempts to find more coarse birational invariants. We begin such a programme for torsion-free crystallographic groups. More precisely, if Γ is a torsion-free crystallographic group and k is a field then the group algebra k[Γ] is a non-commutative domain (see [6], chapter 13). It can be localized at its centre to yield a division algebra k(Γ) which is a crossed product; the Galois group is the point group and it acts on the rational function field generated by k and the lattice (regarded multiplicatively), which is a maximal subfield ([3]). What are thecommon invariants of Γ1 and Γ2 when k(Γ1) and k(Γ2) are isomorphic k-algebras?
A uniqueness theorem for the coagulation-fragmentation equation
Authors: Stewart I. W.
Published: November 2008

This paper presents a uniqueness result for solutions to the general nonlinear coagulation-fragmentation equation

where

Equation (1·1) has many applications in the applied sciences (cf. [1, 3, 8, 13, 15]) and a brief physical interpretation can be found in Melzak [12] or the survey article by Drake[7]. c(x, t), for x ≥ 0, t ≥ 0, denotes the number of particles of size x at time t and the non-negative kernels K and F describe, respectively, the rates at which particles of size x coalesce with those of size y and particles of size (x + y) break-up into those of sizes x and y.
Probabilistic factorization of a quadratic matrix polynomial
Authors: Kennedy Joanne , Williams David
Published: November 2008

A purely algebraic result. We begin by stating the following theorem. Theorem. Let E be a finite set, and letdenote the set of real E × E matrices with non-negative off-diagonal elements and with non-positive row sums. Let A be a symmetric element of, and let V be a diagonal real E × E matrix. Then there exists a unique pair (H+, H−) of elements ofsuch that

I denoting the identity E × E matrix, and the superscript T signifying transpose. It is an immediate consequence that
Word problems related to periodic solutions of a non-autonomous system
Authors: Devlin James
Published: November 2008

In this paper, we develop an abstract formulation of a problem which arises in the investigation of the number of limit cycles of systems of the form

where p and q are homogeneous polynomials. This is part of the much wider study of Hilbert's sixteenth problem, in which information is sought about the number of limit cycles of systems of the form

where P and Q are polynomials, and their possible configurations.
Character-pure locally compact abelian groups and their dual groups
Authors: Takahashi Yuji
Published: November 2008

We give some characterizations of the class of character-pure locally compact abelian groups. As an application we supplement a result of Venkataraman characterizing the class of locally compact abelian groups which are topologically pure in their Bohr compactifications. We also study the class of locally compact abelian groups which are pure closed in their Bohr compactifications.
Representability of invariant positive sesquilinear forms on partial *-algebras
Authors: Antoine J.-P. , Inoue A.
Published: November 2008

We consider invariant positive sesquilinear forms on a (partial) *-algebra A without unit. First we investigate the relationship between extendability and representability for such a form ø; in particular we discuss under which conditions the two concepts are equivalent. Then we introduce the notions of weak representability and strict unrepresentability, and we show that every fully invariant positive sesquilinear form on A × A is uniquely decomposed into a weakly representable part and a strictly unrepresentable part.
A diophantine problem in harmonic analysis
Authors: Poorten A. J. Van Der , Schlickewei H. P.
Published: November 2008

In the problem session of the Journées Arithmétiques 1989 in Luminy, Bourgain made the following

Conjecture. Suppose α is an algebraic number of degree d. Assume that for each n∈ ℕ

Then there exists a constant c = c(α) with the following property: given any subspace W of dimension ≤ d − 1 of the d-dimensional ℚ-vector space ℚ(α), the set {n|n∈ℕ,αn∈W} contains fewer than c(α) elements.
Toeplitz operators associated with analytic crossed products
Authors: Saito Kichi-Suke
Published: November 2008

The class of Toeplitz operators has attracted the attention of several mathematicians and plays an important part in operator theory and related fields. Here we have a special interest in connection with the theory of shift operators, Toeplitz operators, and Hardy classes of vector and operator valued functions as in [12].
The flow near non-trivial minimal sets on 2-manifolds
Authors: Athanassopoulos Konstantin
Published: November 2008

In this paper we give a description of the qualitative behaviour of the orbits near a non-trivial compact minimal set of a continuous flow on a 2-manifold. The first results in this direction were obtained in [1] and the present paper can be regarded as a continuation of that work. The main result can be stated as follows:

Theorem 1·1. Let (ℝ, M, f) be a continuous flow on a 2-manifold M and A ⊂ M a non-trivial compact minimal set.
On parallelepipeds of minimal volume containing a convex symmetric body in ℝn
Authors: Pelczynski A. , Szarek S. J.
Published: November 2008

Given a convex symmetric body C ⊂ ℝn we put a(C) = |C| sup |P|−1 where the supremum extends over all parallelepipeds containing C and |A| denotes the volume of a set A ⊂ ℝn. Let an = inf {a(C): C ⊂ ℝn}. We show that

which slightly improves the estimate due to Dvoretzky and Rogers [6]. In every dimension n we construct a convex symmetric polytope Wn such that the unit Euclidean ball is the ellipsoid of maximal volume inscribed into Wn and the volume of every parallelepiped containing Wn is greater than

for large n which shows ‘the limit’ to the Dvoretzky Rogers method for bounding an below. We present an alternative proof of the result of I. K. Babenko [l] that . We show that , and that a local minimum of the function C→a(C) for C ⊂ ℝn is attained only at an equiframed convex body (that is, a body such that every point of its boundary belongs to a parallelepiped of minimal volume containing the body).
On the geodesic distance and group actions on trees
Authors: Pavone Marco
Published: November 2008

The natural distance function on the vertices of a tree is a kernel of negative type. As a corollary, for any group G acting on a tree X, the length function |g| = d(υ, gυ) is a negative definite function on G for any given vertex υ of X.
A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators
Authors: Wassermann Simon
Published: November 2008

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such that

as i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.
The structure of noetherian modules over hyperfinite groups
Authors: Duan Z. Y.
Published: November 2008

Let G be a hyperfinite locally soluble group and let A be a noetherian ℤZ;G-module. In [2], we proved that A is the direct sum of a ℤZ;G-submodule Af each of whose irreducible ℤZ;G-module sections is finite and a ℤZ;G-submodule each of whose irreducible ℤZ;G-module sections is infinite. In this paper we study the structure of the ℤZ;G-submodules Af and . Our main result gives a complete description of Af.
Shape-preserving polynomial approximation in C[– 1, 1]
Authors: Ditzian Z. , Jiang D. , Leviatan D.
Published: November 2008

A recent result of the first two authors has bridged the gap between the Timantype pointwise estimate and the norm estimate for polynomial approximation in C[– 1, 1]. Here we show that, when considering the second order modulus of smoothness, the aforementioned result holds with polynomials that preserve monotonicity, convexity and the values of the function at the end points of the interval [–1, 1]. This yields a generalization of results by Telyakovskii, Gopengauz, DeVore, Yu and the third author. Moreover, the construction here does not depend on the construction for the non-constrained approximation by polynomials.
Localization methods in the study of the homology of virtually nilpotent groups
Authors: Casacuberta Carles , Castellet Manuel
Published: November 2008

In a series of papers [13, 14, 15], Hilton introduced the terminology relative group to denote a group epimorphism ∈:G↠Q, and relative space to denote a map ƒ:E→B between connected spaces inducing an epimorphism of fundamental groups. He pointed out the desirability of relativizing the theory of P-localization of nilpotent groups and spaces developed in [17], and carried out the algebraic part of this project in [14, 16]. The homotopy-theoretic part was settled by Llerena in [18, 19].
Uniform Kadec–Klee–Huff properties of vector-valued Hardy spaces
Authors: Dowling P. N. , Lennard C. J.
Published: November 2008

In [8] Partington showed that a Banach space X is uniformly convex if and only if Lp([0, 1], X) has the uniform Kadec–Klee–Huff property with respect to the weak topology (UKKH (weak)), where 1 < p < ∞. In this note we will characterize the Banach spaces X such that HP(D, X) has UKKH (weak), where 1 ≤ p < ∞. Similar results for UKKH (weak*) are also obtained. These results (and proofs) are quite different from Partington's result (and proof).
Automorphisms of free nilpotent-by-abelian groups
Authors: Bryant R. M. , Gupta C. K.
Published: November 2008

Let Fn be a free group of finite rank n with basis {x1,…, xn}. Let be a variety of groups and write for the verbal subgroup of Fn corresponding to . (See [11] for information on varieties and related concepts.) Every automorphism of Fn induces an automorphism of the relatively free group Fn/V, and those automorphisms of Fn/V arising in this way are called tame. If is the variety of all metabelian groups and n ╪ 3 then every automorphism of Fn/V is tame [2, 4, 12]. But this is an exceptional situation. For many (and probably most) other varieties , Fn/V has non-tame automorphisms for all sufficiently large n. This holds for the variety of all nilpotent groups of class at most c where c ≥ 3 [1, 3] and for nearly all product varieties including, in particular, the variety of all groups whose derived groups are nilpotent of class at most c, where c > 2 [10, 13].
Discrete least squares approximation and prewavelets from radial function spaces
Authors: Buhmann M. D.
Published: November 2008

In this article we study the convergence behaviour of least squares approximations of various types by radial basis functions, i.e. least squares approximations from spaces spanned by radially symmetric functions φ (‖· −xj‖). Here the xj are given ‘centres’ in ℝn which we assume to lie on a grid. The inner products with respect to which the least squares problem is considered are discrete and Sobolev, i.e. may involve derivative information. Favourable estimates for the least squares errors are found that are shown to decrease as powers of the gridspacing. The work on discrete least squares approximations also gives rise to the construction of prewavelets from radial function spaces with respect to a discrete Sobolev inner product, which are discussed in the paper as well.
Lefschetz numbers of periodic orbits of pseudo-Anosov homeomorphisms
Authors: Guaschi John
Published: November 2008

Given a surface homeomorphism isotopic to the identity which is pseudo-Anosov relative to a finite set, we show that the sum of the Lefschetz numbers of periodic points of any period greater than one is non-negative. If this period is odd and greater than a number which depends only on the surface, the sum is zero. If we consider sequences of periods such that each element is twice that of its predecessor, then this sum is increasing beyond a certain point also depending on the surface. As a corollary, for each periodic orbit contained within the boundary of the surface there exists one of the same period contained in the interior.
Embedding derivatives of weighted Hardy spaces into Lebesgue spaces
Authors: Girela Daniel , Lorente María , Sarrión María Dolores
Published: November 2008

Let 1 ≤ p < ∞ and let ω be a non-negative function defined on the unit circle T which satisfies the Ap condition of Muckenhoupt. The weighted Hardy space Hp(ω) consists of those functions f in the classical Hardy space H1 whose boundary values belong to Lp(ω). Recently McPhail (Studia Math. 96, 1990) has characterized those positive Borel measures μ on the unit disc Δ for which Hp(ω) is continuously contained in Lp(dμ). In this paper we study the question of finding necessary and sufficient conditions on a positive Borel measure μ on Δ for the differentiation operator D defined by Df = f′ to map Hp(ω) continuously into Lp(dμ). We prove that a necessary condition is that there exists a positive constant C such that

where for any interval I ⊂ T,

We prove that this condition is also sufficient in some cases, namely for 2 ≤ p < ∞ and ω(et0) = |θ|α, (|θ| ≤ π), – 1 < α < p – 1, but not in general. In the general case we prove the sufficiency of a condition which is slightly stronger than (A).
Existence of R-matrix for a quantized Kac–Moody algebra
Authors: Lyubashenko Volodimir
Published: November 2008

There is a pairing between two Borel subalgebras of a quantized Kac–Moody algebra, which plays the rôle of R-matrix. Over the field ℚ(q) this pairing is non-degenerate. We show the existence of a braiding in some categories of representations of a quantized Kac-Moody algebra.
Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices
Authors: Mason A. W. , Odoni R. W. K.
Published: November 2008

Let d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.

The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.
Complexes acyclic up to integral closure
Authors: Katz D.
Published: November 2008

In [R] D. Rees introduced the notions of reduction and integral closure for modules over a commutative Noetherian ring and proved the following remarkable result. Let R be a locally quasi-unmixed Noetherian ring and I an ideal generated by n elements. Suppose that height (I) = h. Then the ith module of cycles in the Koszul complex on a set of n generators for I is contained in the integral closure of the ith module of boundaries for i > n − h. This result should be considered a dimension-theoretic analogue of the famous depth sensitivity property of the Koszul complex demonstrated by Serre and Auslander-Buschsbaum in the 1950s. At roughly the same time, Hoschster and Huneke introduced the notion of tight closure and thereafter gave a number of theorems in the same (though considerably broader) vein for tight closure. In particular, in [HH] they showed that if R is an equidimensional local ring of characteristic p > 0, which is a homomorphic image of a Gorenstein ring, then for all i > 0, the ith module of cycles is contained in the tight closure of the ith module of boundaries for any complex satisfying the so-called standard rank and height conditions (see the definitions below). Since the tight closure is contained in the integral closure for such rings, the result of Hochster and Huneke extends (in characteristic p) considerably the result of Rees. In fact, their result could be considered a dimension-theoretic analogue of the Buchsbaum-Eisenbud exactness theorem ([BE]), which in a certain sense is the ultimate depth sensitivity theorem. Moreover, using the technique of reduction to characteristic p, Hochster and Huneke have shown that their results hold in equicharacteristic zero as well, whenever the tight closure is defined.
A polynomial invariant of integral homology 3-spheres
Authors: Ohtsuki Tomotada
Published: November 2008

In 1988 Witten [W] proposed invariants Zk(M) ∈ ℂ (what we call, quantum G invariants) for a 3-manifold M and any integer k associated with a compact simple Lie group G. The invariant Zk(M) is formally expressed by an integral (Feynman path integral) over the (infinite dimensional) quotient space of the all connections in G-bundles on M modulo gauge transformations. If one believes in Feynman path integrals, one can expect the asymptotic formula of Zk(M) for large k predicted by perturbation theory. As in [W], the asymptotic formula (which is a power series in k−1) is given by a sum of contributions from flat connections, since the integral contains an integrand which is wildly oscillatory apart from flat connections for large k. More precise forms of the asymptotic formula are studied in [AS1], [AS2] and [Ko].
Higher-order differential equations with a slowly decaying coefficient containing a periodic factor
Authors: Eastham M. S. P.
Published: November 2008

Let p(x) be periodic with mean value c and let α > 0. Then the investigation of the asymptotic solution of the second-order equation

can be split into four cases: (i) α > 2, (ii) α > 1 and c = 0, (iii) 0 < α ≤ 1 and c = 0, (iv) 0 < α ≤ 2 and c ≠ 0. The first two cases are straightforward because then there are solutions y1 and y2 of (1·1) such that y1(x) ~ x and y2(x) → 1 as x → ∞ [5, theorems 2·7·2 and 2·7·5; 7, p. 375; 8]. Thus the coefficient of y in (1·1) represents a perturbation which has no effect on the dominant asymptotic behaviour of solutions. Case (iii) is less simple because here the nature of p(x) does influence the asymptotic form of solutions and, indeed, when α < 1 the solutions of (1·1) are oscillatory. This case has been fully covered by a number of authors: Atkinson, Eastham and McLeod[2, §3], Cassell[3, 4], Eastham[5, §4·13], Mahoney[10], Willett[13] and Wong[14, 15]. In addition, corresponding results for a higher-order self-adjoint analogue of (1·1) have been obtained recently by Al-Hammadi and Eastham [1, 6] as an application of general asymptotic methods for differential systems.
Studying the topology of Morin singularities from a global viewpoint
Authors: Saeki Osamu
Published: November 2008

Letf:M→Nbe a smooth map of a closedn-manifold into ap-manifold (n≥p) having only Morin singularities [17]. We study the topology of such a map and obtain a modulo 2 congruence formula involving the Euler characteristics ofM, N, the singular sets and the regular fibres off. We also consider applications of this formula to the existence problem of maps having only fold singular points. Stable maps into 3-manifolds are also studied and we obtain a modulo 2 congruence formula involving the swallow tails and the number of triple points of the discriminant set.