In this paper the values of m(6)2,9, m(5)2,8 and m(5)2,9 have been determined as well as improved bounds on other arcs, and in particular a general construction for certain large arcs in planes of square order has been given. Table 5 summarizes the known values of m(n)2, q for 2 ≤ n ≤ q and q ≤ 9.

Let F: n × r → be a sufficiently small representative, defined near (0,0), of a versal unfolding of a germ F0: (n, 0) → (, 0) with isolated singularity at 0∈n. Then a result of Teissier ((4), p. 338) says that, for u∈r sufficiently close to 0, F acts as a versal unfolding of all the various singularities, close to 0, on the fibre Fu = 0 (where Fu(x) = F(x, u)). Let us fix a small neighbourhood W of 0 in n and restrict u to be so close to 0 that all singularities of Fu lie in W. Suppose that the fibre Fu = 0 has singularities of (contact) type χ1, …, χm, all isolated. Suppose that, for each i, the collection {χij}(1 ≤ j ≤ k(i)) of singularities specializes to χi, that is it occurs on a single fibre arbitrarily close to χi in a versal unfolding of χi. Then Teissier's result shows that there will exist fibres Fv = 0 of F, with v arbitrarily close to u, such that the fibre Fv = 0 carries singularities, all in W, of all the types χij for 1 ≤ i ≤ m, 1 ≤ j ≤ k(i). In other words, if ‘despecializations’ {χij} of the separate singularities χi are possible, then they can all occur together on a single fibre of F, provided F is versal.

A quadratic complex Q is the set of lines in 3-dimensional projective space 3 given by a single non-trivial quadratic equation in the Plücker coordinates. The lines of the complex which pass through a fixed point of 3 are, in general, the generators of a quadric cone; this cone degenerates for the points of a sub variety K of 3. Thus one can associate with Q a fibration with base 3 - K and fibre isomorphic to 1, and ask whether this fibration is associated with an algebraic vector bundle of rank 2. When the base field is and Q is non-singular, the answer is negative; this was proved some years ago by Narasimhan and Ramanan ((8), proposition 8·1), and has the consequence that there is no universal family of stable vector bundles of rank 2 and degree 0 over a curve of genus 2.

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).

In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

Since their introduction by Fischer(12) and Fischer, Gaschütz and Hartley (13) Fitting classes of soluble groups have attracted attention on two fronts (all groups considered in this paper will be finite and soluble). On the one hand is their important role in the structure of finite soluble groups, a good account of which can be found in Gaschütz (14), and on the other is their intrinsic interest as classes of groups. This paper falls into the second category, and is a continuation and completion of (8). There we proved that a subgroup closed Fitting class is a formation if it consists of groups of nilpotent length at most three. Happily, at last, we can remove this qualification.

An ordinary differential equation for Hermite-Bell polynomials is derived. Bell's error concerning orthogonality of his polynomials is corrected.

1. Introduction. Suppose that K is a continuous function on the square Q = [ – 1, 1] x [– 1,1] satisfying , for – 1 ≤ s, t ≤ 1; then the Fredholm operator T on L2(-1,1)

is compact and symmetric. Suppose also that T is a positive operator, i.e.

then there is an eigenfunction expansion

where (λn) is a sequence of non-negative real numbers which decreases to 0 and (φn) is an orthonormal sequence in L2( – 1,1). In this paper we shall find asymptotic estimates for λn when K takes certain specific analytic forms. In all cases K will be real-valued on Q and analytic in a neighbourhood of Q in complex 2-space; for example

Let A and E be C*-algebras, let A ⊗ B denote the minimal C*-tensor product, and let ε A *. The right slice map R: A ⊗ B → B is the unique bounded linear mapping with the property that R (a ⊗ b) = (a)b (a ε A, b ε B)(10). A triple (A, B, D), where D is a C*-subalgebra of B, is said to have the slice map property if whenever x ε A ⊗ B and R(x) D for all ε A* then x ε A ⊗ D). It is known that (A, B, D) has the slice map property whenever A is nuclear (11,13), but it appears to be still unknown whether the nuclearity of B will suffice (unless some extra condition is placed on D (l)).

Let N > 1 and let AN be the polydisc algebra, i.e. the algebra of all continuous functions on the closed polydisc δ¯N ⊂ N, analytic on the open polydisc δN, with sup norm. Call a closed set F ⊂ δ¯N a peak interpolation set for AN if given any f ε C(F), f ≠ 0, there is an extension f ε AN of f such that ¦f˜(z)¦ < ‖ f ‖ (z ε δ¯N - F); call F a norm preserving interpolation set for AN if given any f ε C(F) there is an extension f˜ ε AN of f such that ‖f˜‖ = ‖f‖. The paper gives a complete description of norm preserving interpolation sets for AN in terms of peak interpolation sets for AM, M ≤ N.

Let M be a closed simply-connected 4-manifold. All manifolds will be assumed smooth and oriented. The purpose of this paper is to classify up to conjugacy the topological subgroups of Diff(M) isomorphic to the 2-dimensional torus T2 (Theorem 1), and to give an explicit formula for the number of such conjugacy classes (Theorem 2). Such a conjugacy class corresponds uniquely to a weak equivalence class of effective T2-actions on M. Thus the classification problem is trivial unless M supports an effective T2-action. Orlik and Raymond showed that this happens if and only if M is a connected sum of copies of ± ;P2 and S2 × S2 (2), and so this paper is really a study of the different T2-actions on these manifolds.

Let n points be drawn independently and uniformly from a compact convex set K. The distribution of the shape of the resulting n–ad is determined and studied in the region corresponding to near alignment. Special attention is given to the case n ≥ 4 and a table provided to help in the assessment of practical data (e.g. megalithic ‘alignments’). The Broadbent factor, representing the effect of stretching the parent distribution, is computed explicitly.

Problems dealing with the generation of internal waves at the surface separating two fluids involves the consideration of different types of singularities in one of the two fluids. In this paper the velocity potentials describing line and point multipoles are obtained for the case when each fluid is of finite constant depth, neglecting effects of surface tension at the surface of separation.

In the previous Paper I, the Variational Principle has been presented as a principle of natural transformation between conserved quantities and infinitesimal in variances, which preserves the geometrical character of the various standard objects under point transformations. So as to establish the theory as easily as possible, Paper I (2) confined itself to the simple case of autonomous dynamical systems, and their 2n dimensional space. The aim of the present paper is to extend the work to the 2n + 1 dimensional theory, in which time is included as an extra variable, and time dependent systems can be handled. It will be seen that within a suitable framework the extension is straightforward, and there are some rather more satisfactory features of the 2n + 1 dimensional theory - even for autonomous systems when included in the new formalism - than in the previous 2n dimensional theory. The exposition will now be rather concise; readers are requested to refer to Paper I for fuller discussion, particularly of the underlying ideas.

I. J. GOOD. ‘A new formula for cumulants.’

Henryk Minc has pointed out in correspondence that Equation (12), which was a corollary of Theorem 2, does not agree with the computed values of Schur's matrix ω(rs) given by Graham and Lehmer (1976). In the course of the ‘proof’ I made the error of assuming that , overlooking that s could be a multiple of m, so that the sum can also equal m. Since this was the only error, the statement of Theorem 2 is correct modulo m. In particular it shows that perm(ωrs) ≡ 0 (mod m) but this result is much weaker than the results obtained by Graham and Lehmer.

Theorem 1 is unaffected by this error, and moreover has since been proved again by Groeneveld and van Kampen (1977).

R. A. Bryce and John Cossey. ‘Subgroup closed Fitting classes.’

The proof of the main theorem (Theorem 1.1) in our paper (1) is incomplete. The entire page 203 does not establish what it purports to, the error lying in the assumption in the last paragraph that the group P(F0/φF0) R has nilpotent length three precisely. However, it could be of nilpotent length two, indeed in particular examples is so. The technique used there arose as a generalization of an ad hoc argument.

We discovered the error in attempting to extend the proof of Theorem 1.1, and in (2) we have indeed succeeded in removing the nilpotent length restriction. Page 203 of (1) must be replaced by a special case of Lemma 7.2 of (2).

Leonard E. Dor. ‘Some inequalities for martingales and applications to the study of L1.’

The author wishes to point out that his affiliation when he prepared the paper should have been included. It was ‘Wayne State University, Detroit, Michigan 48202, and The University of Illinois, Urbana, Illinois 61801.’