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Volume 37 - October 1994

Contents

Page 2 of 3

Research Article

  • Sturm–Liouville problems with eigenparameter dependent boundary conditions

  • P. A. Binding, P. J. Browne, K. Seddighi
  • Published online by Cambridge University Press:
    20 January 2009, pp. 57-72
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  • Sturm theory is extended to the equation

    for 1/p, q, r∈L1 [0, 1] with p, r > 0, subject to boundary conditions

    and

    Oscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are ajy(j) = cj(py′)(j), j=0, 1, forms a key tool.

  • Multiresolution and wavelets

  • Rong-Qing Jia, Zuowei Shen
  • Published online by Cambridge University Press:
    20 January 2009, pp. 271-300
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  • Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L2(ℝd), let Sk be the 2k-dilate of S (k∈ℤ). A necessary and sufficient condition is given for the sequence {Sk}k∈ℤ to fom a multiresolution of L2(ℝd). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.

  • Sn-normal semigroups

  • I. Levi, R. B. McFadden
  • Published online by Cambridge University Press:
    20 January 2009, pp. 471-476
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  • Certain subsemigroups of the full transformation semigroup Tn on a finite set of cardinality n are investigated, namely those subsemigroups S of Tn, which are normalised by the symmetric group on n elements, the group of units of Tn. The Sn-normal closure of an element of Tn is determined, and the structure of the Sn-normal ideals consisting of the members of Tn whose image contains at most r elements is studied.

  • Radial growth and exceptional sets for Cauchy–Stieltjes integrals

  • D. J. Hallenbeck, T. H. MacGregor
  • Published online by Cambridge University Press:
    20 January 2009, pp. 73-89
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  • This paper considers the radial and nontangential growth of a function f given by

    where α>0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

  • Chern characters, reduced ranks and -modules on the flag variety

  • T. J. Hodges, M. P. Holland
  • Published online by Cambridge University Press:
    20 January 2009, pp. 477-482
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  • Let D be the factor of the enveloping algebra of a semisimple Lie algebra by its minimal primitive ideal with trival central character. We give a geometric description of the Chern character ch: K0(D)→HC0(D) and the state (of the maximal ideal m) s: K0(D)→K0(D/m) = ℤ in terms of the Euler characteristic χ:K0()→ℤ, where is the associated flag variety.

  • On weakly multiplicative inverse transversals

  • T. S. Blyth, M. H. Almeida Santos
  • Published online by Cambridge University Press:
    20 January 2009, pp. 91-99
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  • We show that an inverse transversal of a regular semigroup is multiplicative if and only if it is both weakly multiplicative and a quasi-ideal. Examples of quasi-ideal inverse transversals that are not multiplicative are known. Here we give an example of a weakly multiplicative inverse transversal that is not multiplicative. An interesting feature of this example is that it also serves to show that, in an ordered regular semigroup in which every element x has a biggest inverse x0, the mapping x↦x00 is not in general a closure; nor is x↦x** in a principally ordered regular semigroup.

  • The asymptotic distribution of the eigenvalues of multiparameter Sturm–Liouville systems II

  • Bryan P. Rynne
  • Published online by Cambridge University Press:
    20 January 2009, pp. 301-316
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  • In a previous paper we studied the asymptotic distribution of the multiparameter eigenvalues of uniformly right definite multiparameter Sturm–Liouville eigenvalue problems. In this paper we extend the analysis to deal with multiparameter Sturm–Liouville problems satisfying uniform left definiteness, and non-uniform left and right definiteness.

  • Unions of Cockcroft two-complexes

  • W. A. Bogley
  • Published online by Cambridge University Press:
    20 January 2009, pp. 317-324
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  • A combinatorial hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This combinatorial hypothesis has a component that can be expressed in terms of the second homology of groups. The hypothesis is applied to the study of the third homology of groups.

  • Inequalities in l1 and lp and applications to group algebras

  • Geoffrey V. Wood
  • Published online by Cambridge University Press:
    20 January 2009, pp. 483-491
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  • In this note, we show that, if (an) in l1 with Σ|an| < 2 and Σ|an|2 = 1, then max {|ai| + |aj|:i ≠ j} ≧ 1, but that the corresponding theorem for sequences in lp(1<p<2) fails—but only just! Applications to group algebras are given, when it is shown that elements in l1(G) with powers bounded by ½(1+ ) are bounded away from the identity e of G, but that the corresponding result for lp (G) is false.

  • A diffusion equation with localized chemical reactions*

  • John M. Chadam, Hong-Ming Yin
  • Published online by Cambridge University Press:
    20 January 2009, pp. 101-118
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  • In some chemical reaction–diffusion processes, the reaction takes place only at some local sites, due to the presence of a catalyst. In this paper we study the well-posedness of a model problem of this type. Sufficient conditions are found to ensure global existence and finite time blowup. The blowup rate and the blowup set are also investigated in the case of special nonlinearity.

  • Finitely-generated solutions of certain integral equations

  • D. Porter, D. S. G. Stirling
  • Published online by Cambridge University Press:
    20 January 2009, pp. 325-345
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  • Recent work has shown that the solutions of the second-kind integral equation arising from a difference kernel can be expressed in terms of two particular solutions of the equation. This paper establishes analogous results for a wider class of integral operators, which includes the special case of those arising from difference kernels, where the solution of the general case is generated by a finite number of particular cases. The generalisation is achieved by reducing the problem to one of finite rank. Certain non-compact operators, including those arising from Cauchy singular kernels, are amenable to this approach.

  • Topologically free actions and ideals in discrete C*-dynamical systems

  • R. J. Archbold, J. S. Spielberg
  • Published online by Cambridge University Press:
    20 January 2009, pp. 119-124
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  • A C*-dynamical system is called topologically free if the action satisfies a certain natural condition weaker than freeness. It is shown that if a discrete system is topologically free then the ideal structure of the crossed product algebra is related to that of the original algebra. One consequence is that a minimal topologically free discrete system has a simple reduced crossed product. Sharper results are obtained when the algebra is abelian.

  • Mellin multipliers and radially symmetric Riesz potentials

  • W. Lamb, S. E. Schiavone
  • Published online by Cambridge University Press:
    20 January 2009, pp. 493-507
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  • Riesz potentials with radially symmetric densities are examined from the standpoint of Mellin multipliers. Various results are deduced from the underlying multipliers, including a decomposition of the potential into a product of Erdélyi-Kober fractional integrals. Distributional versions of these results are also produced and shown to be valid under less severe restrictions on the parameters than those required in a weighted Lp setting.

  • Stability of Gelfand–Kirillov dimension for rings with the strong second layer condition

  • T. H. Lenagan
  • Published online by Cambridge University Press:
    20 January 2009, pp. 347-354
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  • We study the influence of the link structure of the prime spectrum of a Noetherian ring on the representation theory of the ring in the case that the ring satisfies the strong second layer condition and has exact integer Gelfand–Kirillov dimension. In particular, we show that Jategaonkar's density condition is satisfied and that the growth of an injective module is controlled by the growth of its first layer.

  • Nilpotency indices of the radicals of p-group algebras

  • Yasushi Ninomiya
  • Published online by Cambridge University Press:
    20 January 2009, pp. 509-517
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  • Let k be a field of characteristic p>0. We classify all finite p-groups G satisfying the inequality p−2|G|≦t(G) < p−1|G|, where t(G) is the nilpotency index of the Jacobson radical of k[G].

  • Invertible disjointness preserving operators

  • C. B. Huijsmans, B. De Pagter
  • Published online by Cambridge University Press:
    20 January 2009, pp. 125-132
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  • It is shown that an invertible disjointness preserving operator from a uniformly complete vector lattice onto a normed vector lattice has a disjointness preserving inverse and is necessarily order bounded.

  • On locally soluble periodic groups with Chernikov centralizer of a four-subgroup

  • Pavel Shumyatsky
  • Published online by Cambridge University Press:
    20 January 2009, pp. 133-138
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  • Let G be a locally soluble periodic group having a four-subgroup V. We show that if CG(V) is Chernikov then G is hyperabelian-by-Chernikov, if CG(V) is finite then G is hyperabelian.

  • Corrigendum et addendum: the Frattini subalgebra of a Bernstein algebra

  • Jesús Laliena
  • Published online by Cambridge University Press:
    20 January 2009, pp. 519-520
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  • In a previous paper it is supposed that if A is a Bernstein algebra, every maximal subalgebra, M, verifies that dim M = dim A − 1. This is not true in general. Therefore Proposition 2 in this paper is not correct. However other results there, where this assertion was used, are correct but their proofs need some modifications now.

  • Linear metric spaces and analytic sets

  • Robert Kaufman
  • Published online by Cambridge University Press:
    20 January 2009, pp. 355-358
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  • A problem in descriptive set theory, in which the objects of interest are compact convex sets in linear metric spaces, primarily those having extreme points.

  • A specific form of Grothendieck's inequality for the two-dimensional case, with applications to C*-algebras

  • G. J. O. Jameson
  • Published online by Cambridge University Press:
    20 January 2009, pp. 521-537
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  • We characterize bilinear forms V on such that V(e, e) = ‖V‖ = 1 in terms of their matrices. For such V we prove that |V(x, y)|2≦φ(|x|2)ψ(|y|2) for all x, y, where φ(x)= V(x, e), ψ(y) = V(e, y). Some other properties of such forms are given.

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