A positive integer n is called square-full if p2|n for every prime factor p of n. Let Q(x) denote the number of square-full integers up to x. It was shown by Bateman and Grosswald [1] that

Bateman and Grosswald also remarked that any improvement in the exponent would imply a ‘quasi-Riemann Hypothesis’ of the type for . Thus (1) is essentially as sharp as one can hope for at present. From (1) it follows that, for the number of square-full integers in a short interval, we have

when and y = o (x½). (It seems more suggestive) to write the interval as (x, x + x½y]) than (x, x + y], since only intervals of length x½ or more can be of relevance here.)

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K by

L can be viewed as a module over the ring , and a fortiori over the group ring KG.

In this paper we consider the conjecture of Kervaire, that a nonsingular system of equations in one or more variables over an arbitrary group G has a solution in some overgroup of G.

We determine the integral cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested independently by P. H. Kropholler and J. Huebschmann. This construction has also been used by the author to calculate the mod-p cohomology of the same groups and by B. Moselle to obtain partial results concerning the mod-p cohomology of the extra special p-groups [7], [9].

It is well-known that limits in 2-categories are more complex than limits in ordinary categories. Most readers will at least be familiar with terms such as ‘lax limit’ and ‘pseudo-limit’. In the more modern treatments, these become special cases of a more general class of ‘weighted’ or ‘indexed’ limits (see Kelly [7] and Section 1 of this paper).

The aim of this paper is to give a new description of Giraud's nonabelian cohomology set H2(L), [7], defined for a band L of a Grothendieck topos E as the set of L-equivalence classes of L-gerbes over E. Our description is similar to that for H1 by Čech cohomology, but with cocycles taken from the stack of bitorsors over E. We first continue to study the construction of gerbes by cocycle bitorsors or bouquets [13], originally given in [7] and [4]; Duskin [4, 5] showed that bouquets B and B′ of E give rise to equivalent gerbes if and only if there exist essential equivalences B″ → B and B″ → B′ for another bouquet B″.

It is an important result in frame theory that the coproduct of a family of regular Lindelöf frames is Lindelöf [3]. We show that this ‘Lindelöf Tychonoff Theorem’ or ‘LTT’ is independent of ZF and indeed lies close in logical strength to the Axiom of Countable Choice, quite unlike the case with the usual (frame) Tychonoff Theorem. Along the way we construct the regular Lindelöf coreflection and obtain a simple proof of the LTT as a corollary.

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.

How must a sphere be covered by n equal circles so that the angular radius of the circles will be as small as possible? In this paper, conjectured solutions of this problem for n = 15 to 20 are given and some sporadic results for n > 20 (n = 22, 26, 38, 42, 50) are presented. The local optima are obtained by using a ‘cooling technique’ based on the theory of bar-and-joint structures. Thus the graph of the coverings by circles is considered as a spherical cable net in which the edge lengths are uniformly decreased, e.g. due to a uniform decrease in the temperature, until the graph becomes rigid and tensile stresses appear in the cables.

Throughout this paper q = 2h with h ≥ 4, and PG(r, q) stands for the r-dimensional projective space over the finite field GF(q) with q elements.

The aim of this paper is to show one more facet of the role of crossed complexes as generalisations of both groups (or groupoids) and of chain complexes. We do this by defining and establishing the main properties of a classifying space functor B: from the category of crossed complexes to the category of spaces.

If X is a topological space with base point, then a based map m: X × X → X is called a multiplication if m restricted to each factor of X × X is homotopic to the identity map of X. The pair (X, m) is then called an H-space. If A is a based topological space and (X, m) is an H-space, then the multiplication m induces a binary operation on the set [A, X] of based homotopy classes of maps of A into X. A classical result due to James [6, theorem 1·1 asserts that if A is a CW-complex and (X, m) is an H-space, then the binary operation gives [A, X] the structure of an algebraic loop. That is, [A, X] has a two-sided identity element and if a, b ∈[A, X], then the equations ax = b and ya = b have unique solutions x, y ∈ [A, X]. Thus it is meaningful to consider loop-theoretic properties of H-spaces. In this paper we make a detailed study of the following loop-theoretic notions applied to H-spaces: inversivity, power-associativity, quasi-commutativity and the Moufang property – see Section 2 for the definitions. If an H-space is Moufang, then it has the other three properties. Moreover, an associative loop is Moufang, and so a homotopy-associative H-space has all four properties. Since many of the standard H-spaces are homotopy-associative, we are particularly interested in determining when an H-space, in particular a finite CW H-space, does not have one of these properties.

Let G be a locally compact group, and let P(G) be the convex set of all continuous, positive definite functions ø on G normalized by ø(e) = 1, where e denotes the group unit of G. For ø∈P(G) the spectrum spø of ø is defined as the set of all indecomposable ψ∈P(G) which are limits, for the topology of uniform convergence on compact subsets of G, of functions of the form

(see [5], p. 43). Denoting by πø the cyclic unitary representation of G associated with ø, it is clear that sp ø consists of all ψ∈P(G) for which πψ is irreducible and weakly contained in πø (see [3], chapter 18).

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.

A necessary and sufficient condition is given for a unital C*-algebra A to admit a discontinuous homomorphism into a Banach algebra which is continuous on its centre. The condition is that A must have a Glimm ideal G such that the C*-algebra A/G admits a discontinuous homomorphism into a Banach algebra.

Let ƒ be a smooth function of compact support defined in the plane and consider the integral

The estimate

is well-known, see for instance the work of Littman[4]. The operator T amounts to convolution with the measure σ carried by the parabola t → (t, ½t2) and given by dσ = dt. Usually one proves (1) by embedding σ in an analytic family of distributions σz in ℝ2 given by

In this paper we consider the problem of characterizing the variation of the spectrum of a holomorphic family of compact operators ƒ:G → KB(X), where G is an open subset of ℂ and X is a Banach space. The natural conjecture, which the author first heard in a lecture by Professor B. Aupetit, is that these spectra are characterized as those analytic multivalued functions which have as values null sequences. This is obviously a necessary condition, and we prove that this is also sufficient. It will be convenient to use the notation K(ℂ) for the set of compact non-empty subsets of the plane and K0(ℂ) for the subset of K(ℂ) consisting of null sequences.

Under natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.

Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to l∞, l2, B(l2), or a direct sum of such spaces. A more natural project, given the matricial nature of the definitions involved, would be the classification of such spaces up to completely bounded isomorphism. This was done for injective von Neumann algebras in [6] and for injective operator systems (i.e. unital injective operator spaces) in [19]. It turns out that the spaces l∞ and B(l2) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of l2. For there are two injective operator spaces which are each isometrically isomorphic to l2 but not completely boundedly isomorphic to each other. We shall resolve this problem by showing that these are the only two possibilities, in the sense that any injective operator space which is isometric to l2 is completely isometric to one of them. (See Corollary 3 below.) The Hilbert spaces in von Neumann algebras investigated in [17], [13] turn out to be injective matricial operator spaces and are therefore completely isometric to one of our two examples. Another Hilbert space in B(l2) which has been much studied in operator theory, complex analysis and physics is the Cartan factor of type IV [10]. This is the complex linear span of a spin system and generates the Fermion C*-algebra ([3], §5·2). We show that a Cartan factor of type IV is not even completely boundedly isomorphic to an injective matricial operator space. One curious property of all the aforementioned Hilbert spaces is that every bounded operator on them is actually completely bounded, a fact that is crucial in our proofs.

Let be the self-dual CCR algebra over a triplet (K, r, Γ), where K is assumed to be a Hilbert space, and let {τt}t ∈ R be a Bogoliubov automorphism group of induced by a strongly continuous one-parameter unitary group on K. In this paper, we introduce some continuity for linear functionals on and, under this continuity, we study existence, uniqueness, and non-uniqueness of KMS states for .