This paper is devoted to the determining of the irreducible linear representations of the generalized symmetric group (elsewhere written as , Cm ≀ Sn or G(m, 1, n)) by considering the conjugacy classes of and then constructing the same number of inequivalent irreducible linear representations of . These have previously been determined by Kerber [2, Section 5] using Clifford's theory applied to wreath products.

Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.

Various authors have obtained an eight term exact sequence in homology

from a short exact sequence of groups

,

the term V varying from author to author (see [7] and [2]; see also [5] for the simpler case where N is central in G, and [6] for the case where N is central and N ⊂ [G, G]). The most satisfying version of the sequence is obtained by Brown and Loday [2] (full details of [2] are in [3]) as a corollary to their van Kampen type theorem for squares of spaces: they give the term V as the kernel of a map G ∧ N → N from a “non-abelian exterior product” of G and N to the group N (the definition of G ∧ N, first published in [2], is recalled below). The two short exact sequences

and

where F is free, together with the fact that H2(F) = 0 and H3(F) = 0, imply isomorphisms

.

.

The isomorphism (2) is essentially the description of H2(G) proved algebraically in [11]. As noted in [2], the isomorphism (3) is the analogue for H3(G) of the Hopf formula for H2(G).

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

If we seek solutions of the hyperbolic differential equation

which depend only on the variables i and , we see that these solutions must be even in r and satisfy the differential equation

The object of this paper is to show that some recent results in the fractional calculus can be used to prove the following theorem.

Throughout this paper S will denote a given monoid, that is, a semigroup with an identity. A set A is a right S-system if there is a map φ: A × S → A satisfying

for any element a of A and any elements s, t of S. For φ(a, s) we write as and we refer to right S-systems simply as S-systems. One has the obvious definitions of an S-subsystem and an S-homomorphism.

Several properties are known to be inherited from the derived factor group of a group G by other factors Γi = γi(G)/γi + 1(G) of the lower central series. Derek Robinson proved [1] that the denning property of any class of groups which is closed under the forming of homomorphic images of tensor products is so inherited. Possibilities for here include the following classes.

Two numerical characterizations of commutativity for C*-algebra (acting on the Hilbert space H) were given in [1]; one used the norms of self-adjoint operators in (Theorem 2), and the other the numerical index of (Theorem 3). In both cases the proofs were based on the result of Kaplansky which states that if the only nilpotent operator in is 0, then is commutative ([2] 2.12.21, p. 68). Of course the converse also holds.

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

Let n = be the factorization of an integer n(>1) into prime powers, and set Φ(n):= . In particular, for squarefree n, Φ(n) = phi;(n). Consider the set

.

It is known (from [5]) that A consists precisely of those integers n for which there is no non-abelian group of order n. It is also known (from [7]) that the set

consists solely of integers n with the property that every group of order n is cyclic. We set C′ = A – C.

If q(≥2) is a fixed integer it is well known that every positive integer k may be expressed uniquely in the form

We introduce the ‘sum of digits’ function

Both the above sums are of course finite. Although the behaviour of α(q, k) is somewhat erratic, its average behaviour is more regular and has been widely studied.

C. Widland [14] has defined Weierstrass points on integral, projective Gorenstein curves. We show here that the Weierstrass points on a generic integral rational nodal curve have the minimal possible weights or, equivalently, that such a curve has the maximum possible number of distinct nonsingular Weierstrass points. Rational curves with g nodes arise in degeneration arguments involving smooth curves of genus g and they have also recently arisen in connection with g-soliton solutions to certain nonlinear partial differential equations [11], [13].

Let {ak} be a sequence of non-negative real numbers satisfying a1 = l and

(1)

Brannan [1] proved that the function

(2)

is close-to-convex univalent in the unit disc D. The example

shows that the conclusion in Brannan's theorem is sharp in that sense that “close-to-convex” cannot be replaced by the stronger one: “starlike”. It is therefore of interest to see which additional condition can guarantee this stronger conclusion.