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).

In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.

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.

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’.

The aim of this paper is to investigate the structure of fuzzes, introduced by Hutton. We shall not be concerned here with fuzzy topology, but we think that a deeper knowledge of the properties of fuzzes may be helpful in several fields of fuzzy set theory, and in the study of fuzzy topological spaces in particular.

Let a finite group G act topologically on a closed smooth manifold Mn. One of the most natural and basic questions is whether such an action can be smoothed. More precisely, let γ:G × Mn → Mn be a topological action of G on Mn. The action γ can be smoothed if there exists a smooth action and an equivariant homeomorphism It is well known that for n ≤ 2 every finite topological group action on Mn is smoothable. However already for n = 3 there are examples of topological actions on 3-manifolds which cannot be smoothed (see [1, 2] and references there). All these actions fail to be smoothable because of bad local behaviour.

Beardon [1] gave a procedure for constructing a polygon with prescribed angles. For each ordered set of angles Beardon's polygon is unique up to congruence. The polygon obtained this way has an inscribed circle. It is possible to obtain by means of these polygons a fundamental region for a non-Euclidean crystallographic (NEC) group with a given signature having equal angles in each of the cycles: see [5]. In [5] the minimal number of sides of a convex fundamental region of an NEC group is calculated, and regions are explicitly obtained achieving the bound.

In this paper we prove the following

Theorem. Each element of the group π3(S2) = ℤ can be rendered harmonic, i.e. admits a harmonic representative, provided that the domain S3is given a suitable riemannian metric.

The series is said to be summable (R, k) to s, where k is a positive integer, if

converges for all values of h in some neighbourhood of the origin and if its sum tends to s as h → 0.

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s if

that is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].

Let B be the open unit ball of c0. We give a geometric characterization of the sequences {xn} ⊂ bB with the property that, given any bounded sequence {αn} ⊂ ℂ, there is a continuous function , analytic in B, such that f(xn) = αn for all n and such that supi∈B|f(x)| = supn∈N|αn|.

We extend to a wider class of rearrangement invariant spaces a result of S. Guerre concerning the extraction of almost symmetric sequences from a given weakly null sequence in Lp.

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 Riesz space tensor product of Archimedean Riesz spaces was introduced by D. H. Fremlin[2, 3]. His construction as well as a subsequent simplified version by H. H. Schaefer[10] depended on representation techniques and it is our aim to find a more direct way to prove the existence of the tensor product and to derive its properties. This tensor product proved to be extremely useful in the theory of positive operators on Banach lattices (see [3] and [10]) and should be considered as one of the basic constructions in the theory of Riesz spaces. It is therefore of interest to construct it in an intrinsic way. The problem to do this was already posed by Fremlin in [2]. In this paper we shall present two different approaches, the first of which is analogous to the formation of a free lattice generated by a given partially ordered set. (See [5], p. 41.) In the second one we first assume the Riesz spaces involved to have the principal projection property. In this case a simple method of construction by step-elements is available and the tensor product of arbitrary Archimedean Riesz spaces can then be obtained by embedding the spaces into their Dedekind completions. To complete the latter step we need results on the extension of Riesz bimorphisms which will be proved in §1. Both our approaches hinge on results about the tensor product of ordered vector spaces. It turns out that a unique tensor product for ordered vector spaces exists and is contained in the Riesz space tensor product. This is investigated in §2.

The relationship between the local growth of a continuous function and the packing measure of its level sets and of its graph is studied. For the Weierstrass function with b an integer such that b ≥ 2 and with 0 < α < 1, and for x ∈ Range (W) outside a set of first category, the level set W−1(x) has packing dimension at least 1 − α. Furthermore, for almost all x ∈ Range (W), the packing dimension of f is at most 1 − α. Finer results on the occupation measure and the size of the graph of a continuous function satisfying the Zygmund Λ-condition are obtained.

Reade[11] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order α on a bounded region have eigenvalues which are asymptotically O(1/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) = ∂2rK(x, t)/∂xτ ∂tτ is in Lipα and establish λn(K) = O(1/n2r+1+α). If ∂Krr/∂t is in Lipα, the anticipated asymptotic estimate is also derived.

The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.

In this note we characterize those sequences kn such that the Hill estimator of the tail index based on the kn upper order statistics of a sample of size n from a Pareto-type distribution is strongly consistent.

It is shown that each vector in the Hilbert space of certain quasi-free representations of the CCR can be written uniquely in terms of quantum stochastic integrals. As a consequence, we obtain general vector-valued and operator-valued boson quantum martingale representation theorems.

In recent years it has become clear that there are several ways in which complex Banach spaces can differ quite markedly from their real counterparts, and many of these concern martingales. Thus, in [6] complex uniform convexity was related to martingale inequalities, in [3] and [7] the convergence of L1-bounded analytic martingales was considered and in [8] this property was related to the analytic Radon–Nikodym property.

It is well known that the trace X([0, ∞) of the 3-dimensional Brownian motion X has positive capacity and, therefore, the harmonic measure is well defined on this set. It is shown that a.s., this harmonic measure is singular with respect to the occupation measure Lebesgue ο X−1.