Let ℝn → ℝ be a weighted homogeneous polynomial such that df(0) = 0, L = {x ∈ Sn−1|f(x) = 0}, and let χ(L) be the Euler characteristic of L. The problem is how to calculate χ(L) in terms of f.

A general theory of Hankel forms over domains in one or several variables has been set forth in [6]. In [7] the study of Hankel forms over an annulus in the complex plane ℂ was begun. (An extension of the results of [7] to multiply connected domains was given in [4].) The present paper amplifies the results of [7] in various respects. First of all we define and study more general Hankel forms associated with a one parameter family of projective structures on the annulus. This displays several new features. For instance, we are now dealing with quadratic integral metrics which do not correspond to integration of the square of the function with respect to a weight. Furthermore, whereas in [7] essentially only the issue of the boundedness of Hankel forms was studied, we obtain here rather satisfactory Sp-results, even for 0 < p < 1. The question which remains is, of course, to which extent all this extends to multiply connected domains (or more general (open) Riemann surfaces).

Let be the classical Cayley algebra defined over the reals with basis where gives a quaternion algebra ℋ4 with i0 = 1, i1i2i3 = −1, i1i4 = i5, i2i4 = i6 and i3i4 = i7. The multiplication table of the imaginary basic units follows:

In their paper [1], Campbell and Jamison attempted to give necessary and sufficient conditions for a weighted composition operator on an L2 space to be normal, and to be quasinormal. Those conditions, specifically Theorems I and II of that paper, are not valid (see [2] for precise comments on the other results in that paper). In this paper we present a counterexample to those theorems and state and prove characterizations of quasinormality (Theorem 1 below) and normality (Theorem 2 and Corollary 3 below). We also discuss additional examples and information concerning normal weighted composition operators which contribute to the further understanding of this class.

The oscillatory and asymptotic behaviour of the positive solutions of the autonomous neutral delay logistic equation

with r, c, T, K ∈ (0, ∞) has been recently investigated in [2]. More recently the dynamics of the periodic delay logistic equation

in which r, K are periodic functions of period τ and m is a positive integer is considered in [6]. The purpose of the following analysis is to obtain sufficient conditions for the existence and linear asymptotic stability of a positive periodic solution of a periodic neutral delay logistic equation

in which Ṅ denotes and r, K, c are positive continuous periodic functions of period τ at and m is a positive integer. For the origin and biological relevance of (1.3) we refer to [2].

An elementary derivation of the asymptotic formula for the number of cube-full numbers up to x is given. This derivation is used, together with an estimation of a three dimensional exponential sum, to establish the asymptotic formula for the number of cube-full numbers in the short interval x < n < x⅔+θ where 140/1123 < θ < 1/3.

We principally consider rings R of the form R = S[G], generated as a ring by the subring S of R and the subgroup G of the group of units of R normalizing S. (All our rings have identities except the nilrings.) We wish to deduce that certain semiprime images of R are Goldie rings from ring theoretic information about S and group theoretic information about G. Usually the latter is given in the form that G/N has some solubility or finiteness property, where N is some specified normal subgroup of G contained in S. Note we do not assume that N = G∩S; in particular N = 〈1〉 is always an option.

All graphs considered in this article are finite connected, without loops and multiple edges. Let G be a graph and x be a vertex. The vertex neighbourhood graph (or υ-neighbourhood) of x in G (denoted by is the subgraph of G induced by the set of all vertices of G adjacent to x Analogously if f = xy is any edge of G, the edge neighbourhood graph (or e-neighbourhood) of f in G is the subgraph of G (denoted or induced by the set of all vertices of G which are adjacent to at least one vertex of the pair x, y and are different from x, y.

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.

Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

In [5] we exhibited the construction of faithful irreducible matrix representations of p-groups E and constructed their extensions to a semidirect product E. H, in case E and H satisfied suitable conditions. One of the major conditions was that the prime p had to be odd.

In this paper we assume the same conditions as in [5], but now with p = 2, in order to see if similar results can be obtained. Henceforth we will work with the following hypothesis.

In the sequel, given k, n ∈ ℕ, p ∈ [1, ∞] and a compact real interval I, we denote by Wk, p(I, ℝn) (simply by Wk,p(I if n = 1) the space of all functions u ∈ Ck−1(I, ℝn) such that uk−1 is absolutely continuous in I and u(k) ∈ Lp (I, ℝn).

Very recently, in [11], J. R. L. Webb and S. C. Welsh obtained the following existence result.

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient if

Here φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) that

and infinitely often

If A is a semiprime Banach algebra, soc A, rad A the socle and radical of A, then Soc A ∩ rad A = (0). This elementary result enables us to prove some results concerning algebraic ideal and algebraic elements modulo the socle of A. We also deduce several conditions for A equivalent to the condition dim A <+∞.

A longstanding open question in low dimensional topology was raised by J. H. C. Whitehead in 1941 [9]: “Is any subcomplex of an aspherical, two-dimensional complex itself aspherical?” The asphericity of classical knot complements [7] provides evidence that the answer to Whitehead's question might be “yes”. Indeed, each classical knot complement has the homotopy type of a two-complex which can be embedded in a finite contractible two-complex. This property is shared by a large class of four-manifolds; these are the ribbon disc complements, whose asphericity has been conjectured, and even claimed, but never proven. (See [4] for a discussion.) It is reasonable and convenient to formulate the following.

Let S and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product S sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] of Jones and previous partial results [3], [5], [6] take this approach.