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Volume 25 - October 1982

Contents

Page 2 of 3

Research Article

  • Small fractional parts of quadratic forms

  • R. C. Baker, G. Harman
  • Published online by Cambridge University Press:
    20 January 2009, pp. 269-277
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  • Let ‖x‖ denote the distance of x from the nearest integer. In 1948 H. Heilbronn proved [5] that for ε>0 and N>c1(ε) the inequality

    holds for any real α. This result has since been generalised in many different directions, and we consider here extensions of the type: For ε>0, N>c2{ε, s) and a quadratic formQ(x1,…, xs) there exist integersn1,…,nsnot all zero with |n1|,…,|ns≦N and with

Front matter

  • PEM volume 25 issue 2 Cover and Front matter

  • Published online by Cambridge University Press:
    20 January 2009, pp. f1-f2
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Research Article

  • Automatic discontinuity of intertwining operators

  • Sandy Grabiner
  • Published online by Cambridge University Press:
    20 January 2009, pp. 49-54
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  • Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

Book Reviews

Research Article

  • Zero sets—consequences for primitive near-rings

  • S. D. Scott
  • Published online by Cambridge University Press:
    20 January 2009, pp. 55-63
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  • Throughout this paper all near-rings will be left distributive. We shall denote the zero-symmetric part of a near-ring N by N0. The fact that the near-rings under consideration may not be zero-symmetric has important consequences for what follows, particularly the results of the last section.

Back matter

  • PEM volume 25 issue 2 Cover and Back matter

  • Published online by Cambridge University Press:
    20 January 2009, p. b1
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Book Reviews

Research Article

  • Generators for the sporadic group Co3 as a (2, 3, 7) group

  • M. F. Worboys
  • Published online by Cambridge University Press:
    20 January 2009, pp. 65-68
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  • A (2, 3, 7)-group is a group generated by two elements, one an involution and the other of order 3, whose product has order 7. Known finite simple examples of such groups are PSL(2, 7), PSL(2, p) where p is prime and p ≡ ±1 (mod 7), PSL(2, p3) where p is prime and p≢0, ±1 (mod 7), groups of Ree type of order q3(q3 + 1)(q − 1) where q = 32n+1 and n > 0, the sporadic group of order 23 · 3 · 5 · 7 · 11 · 19 discovered by Janko, and the Hall–Janko–Wales group of order 27 · 33 · 52 · 7 [4, 2]. G. Higman in an unpublished paper has shown that every sufficiently large alternating group is a (2, 3, 7)-group. Here we show that the sporadic group Co3 discovered by Conway [1] is a (2, 3, 7)-group.

  • A remark on the radical of a group algebra

  • Peter Brockhaus
  • Published online by Cambridge University Press:
    20 January 2009, pp. 69-71
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  • This paper contains a new proof of a theorem of D. A. R. Wallace [5] in case of a p-solvable group. An alternative proof has been given by K. Motose [4].

    Let G be a finite group, F a field with prime characteristic p, P a Sylow p-subgroup of G. JFG will denote the Jacobson radical of FG.

Book Reviews

  • V. Bryant and H. Perfect, Independence Theory in Combinatorics (Chapman and Hall, 1980), £5.50 (soft cover).

  • Michael J. Ganley
  • Published online by Cambridge University Press:
    20 January 2009, p. 282
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Books Received

  • Book Received

  • Published online by Cambridge University Press:
    20 January 2009, p. 282
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Research Article

  • Embedding near-rings into polynomial near-rings

  • John D. P. Meldrum, Günter Pilz, Yong-Sian So
  • Published online by Cambridge University Press:
    20 January 2009, pp. 73-79
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  • In this paper we will show that large classes of near-rings are embeddable into (or even isomorphic to) near-rings Gν[x] of polynomials over G in a suitably chosen variety ν of Ω-groups.

    For near-rings see [2], for polynomials [1]; for near-rings of polynomials over Ω-groups see [3] [4], [5].

  • The subnormal subgroup structure of the infinite general linear group

  • David G. Arrell
  • Published online by Cambridge University Press:
    20 January 2009, pp. 81-86
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  • Let R be a ring with identity, let Ω be an infinite set and let M be the free R-module R(Ω). In [1] we investigated the problem of locating and classifying the normal subgroups of GL(Ω, R), the group of units of the endomorphism ring EndRM, where R was an arbitrary ring with identity. (This extended the work of [3] and [8] where it was necessary for R to satisfy certain finiteness conditions.) When R is a division ring, the complete classification of the normal subgroups of GL(Ω, R) is given in [9] and the corresponding results for a Hilbert space are given in [6] and [7]. The object of this paper is to extend the methods of [1] to yield a classification of the subnormal subgroups of GL(Ω, R) along the lines of that given by Wilson in [10] in the finite dimensional case.

Front matter

  • PEM volume 25 issue 3 Cover and Front matter

  • Published online by Cambridge University Press:
    20 January 2009, pp. f1-f2
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Back matter

  • PEM volume 25 issue 3 Cover and Back matter

  • Published online by Cambridge University Press:
    20 January 2009, pp. b1-b5
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Research Article

  • The weak-star closure of the unit ball in a subspace

  • G. J. O. Jameson
  • Published online by Cambridge University Press:
    20 January 2009, pp. 87-95
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  • Let X be a (real or complex) Banach space, and let K be a linear subspace of its dual X*. Denote by K1 the unit ball in K. If K is not weak-star closed, then the Krein-Šmul'yan theorem says that K1 is not weak-star closed. What, however, is its weak-star closure? Inner and outer extimates were obtained in [3] for the special case where K is a hyperplane. In the present paper we generalise these estimates to arbitrary linear subspaces. For f to belong to w*(K1) it is sufficient to have |φ(f)| ≦ 1 −∥f∥ for all φ in K0 (the annihilator of K in X**) with dist (φ, X)≦1. It is necessary to have |φ(f)| ≦ 1 + ∥f∥ for all such φ. These estimates depend on the action of each φ in K0 separately, which will often make them hard to apply in practice; in both cases, we derive a second estimate, expressed only in terms of certain constants that describe the position of X and K0 in X**.

  • Rigid Artinian rings

  • K. R. McLean
  • Published online by Cambridge University Press:
    20 January 2009, pp. 97-99
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  • In [4], Maxson studied the properties of a ring R whose only ring endomorphisms φ: R → R are the trivial ones, namely the identity map, idR, and the map 0R given by φ(R) = 0. We shall say that any such ring is rigid, slightly extending the definition used in [4] by dropping the restriction that R2 ≠ 0. Maxson's most detailed results concerned the structure of rigid artinian rings, and our main aim is to complete this part of his investigation by establishing the following

    Theorem. Let R(≠0) be a left-artinian ring. Then R is rigid if and only if

    (i) , the ring of integers modulo a prime power pk,

    (ii) R ≅ N2, the null ring on a cyclic group of order 2, or

    (iii) R is a rigid field of characteristic zero.

  • The number of Cayley integers of given norm

  • P. J. C. Lamont
  • Published online by Cambridge University Press:
    20 January 2009, pp. 101-103
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  • Using results obtained by J. W. L. Glaisher [1, 2] for the number of representations Rr,s(n) of n as a sum of r odd and s even squares, formulae are derived for the number of Cayley integers of given norm n in certain orders ℴ. When computer generating order elements of given norm, the formulae can be used to verify that all the required elements have been obtained.

Book Reviews

  • A. W. Chatters and C. R. Hajarnavis, Rings with chain conditions (Pitman, 1980), £8·00.

  • P. F. Smith
  • Published online by Cambridge University Press:
    20 January 2009, p. 105
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Page 2 of 3