1. Notation and definitions. In this paper necessary and sufficient conditions are found for the spectrum of a Fredholm operator in a locally convex space (always taken to be Hausdorff) to lie on the non-negative real axis of the complex plane. Some results of Grothendieck(2) allow us to obtain the results in this general form; an interesting special case is the trace-class of operators in a general Banach space. We also deal with the case of Hilbert–Schmidt operators in a Hilbert space.

We show that the minimal dimension of a faithful action of a metacyclic group , for primes p and q, on a homology sphere coincides with the minimal dimension of a faithful linear action on a sphere; as a consequence, we obtain the analogous result for various finite simple groups.

This paper is concerned with estimates of the coefficients Cn for normalized polynomials, inverse to univalent functions in the unit disc, i.e. univalent polynomials of the form

considered in the largest domain D such that 0 ∈ D and |pn(w)| < 1 for w ∈ D; this domain will be denoted by . Polynomials of the type (1) have been introduced by Charzyński in (1), where it was proved among other results that the domain is simply connected.

This article is concerned with homotopy properties of H-spaces X that are reflected in the module of indecomposables QH*(X; [ ]p). It is shown that mod p H-spaces X of finite type with finite transcendence degree mod p cohomology and locally finite QH*(X; [ ]p) are Bℤ/p-null spaces, Eilenberg–MacLane spaces K(ℤˆp, 2), K(ℤ/pr, 1), and extensions of those. If we restrict attention to H-spaces with noetherian mod p cohomology algebra, then we are left with finite mod p H-spaces and Eilenberg–MacLane spaces.

In this paper we introduced atomic Hardy spaces on product domains Sn−1×Sm−1 and given the L2-boundedness of the Marcinkiewicz integral operator with Hardy space function kernel on Rn×Rm.

We discuss the possible cofinalities of ⊆ *-chains in ℕ which are ⊆ *-unbounded in their topological closures.

1. The problem. Throughout this paper, the power set ℕ of the set ℕ of natural numbers will be given its usual compact metrizable topology corresponding to the product topology of {0,1}N. We say that a ⊆ * b if a/b is finite.

The moduli space ${\frak N}^{c}_{g,1}$ of non-orientable surfaces of genus $g \geq 0$ with $c \geq 0$ distinguished points and one boundary curve is described via a model ${\frak P}(h,c)$: the space of configurations of $h=g+c+1$ pairs of parallel slits in $\mathbb{C}$. Based on this model, we prove that ${\frak N}^{c}_{g,1}$ is a non-orientable manifold, and we compute its homology for $h \leq 3$ with ${\mathbb Z}_2$-coefficients.

Throughout this brief note V denotes a fixed, smooth, real-valued function on such that there exists a bounded, open, strictly convex set C with (the boundary of C), V > 0 in C, V = 0 on and V′(0) = 0.

It is often of greater practical value to have results about queueing theory which involve probabilities rather than characteristic functions. To quote Kendall (4), section 5, ‘These results of Prabhu, further exploited by himself and Takács, are rapidly raising the Laplacian curtain which has hitherto obscured much of the details of the queue-theoretic scene’. In this paper we derive the exponential limit formula for the equilibrium waiting time distribution function G, for the queue GI/G/1 in heavy traffic, using stochastic bounds which are asymptotically sharp as the traffic intensity (defined below) increases to unity (which has not been done before to the author's knowledge). This formula was derived by Kingman (5), (6) and (9), using characteristic functions, who, in section 9 of the latter paper, stressed the need for improving the precision of the approximation ‘by giving inequalities, bounds for errors, and generally by setting the theory on a more elegant and rigorous basis’. Kingman (6) and (9) also sketched a proof of the same result using a Brownian approximation, which was done in detail by Viskov (18); but here again the same difficulties are present in practical interpretation, error bounds etc.

Recently, Good discovered the following curious identity (1). Let θ be a real-valued random variable having cumulants Km ≡ Km(θ) (m = 1, 2, …). Take a set of m statistically independent variables θ(1), θ(2), …, θ(m), each with the same distribution as θ, and form the random variable

where ω is any primitive mth root of unity (e.g. ω = e2πi/m). Then

where 〈.〉 denotes expectation.

We transfer the results of Dyer, Formanek and Kassabov on the automorphism towers of finitely generated free nilpotent groups to infinitely generated free nilpotent groups. We prove that the automorphism groups of infinitely generated free nilpotent groups are complete. By combining the results of Dyer, Formanek and Kassabov with the results in this paper, one gets that the automorphism tower of any free nilpotent group terminates after finitely many steps.

This paper is a continuation of (3), in which we proved extensions, to generalized Stieltjes transforms, of some inversion theorems of Widder(l) for ordinary Stieltjes transforms. The numbering of formulae, lemmas and theorems continues on consecutively from that in (3). The last formula in (3) is numbered (32), and the last lemma or theorem is Theorem 18.

The irregular radial Coulomb wave function in repulsive fields is expressed, for any angular momentum value, in series of Bessel functions of arbitrary order. Previous expansions, obtained for angular momenta zero and one, can now be obtained as special cases. The expansion can be most conveniently used for high-energy particles.

In this article we extend the study of ‘plumbing’ initiated in [10] to the case of fibred even-dimensional knots. Plumbing is a geometric operation on the fibre-surfaces of two fibred knots of the same dimension that produces another such knot.

Let E be a Banach function lattice such that L∞[0, 1] [rarrhk ] E [rarrhk ] L1[0, 1]. We characterize the strict singularity of the embedding L∞[0, 1] [rarrhk ] E and the strict cosingularity of E [rarrhk ] L1[0, 1] in terms of functionals defined by using characteristic functions.

Conflict sets are loci of intersecting wave fronts emanating from $l$ different manifolds. We show that for generic manifolds their conflict sets are projections of smooth conic Lagrangian manifolds. Thus conflict sets locally admit the structure of wave fronts. Simple stable singularities for this problem in $\mathbb{R}^n$ occur when $0\leq n-l \leq 4$. Relations with centre sets, billiards, pedals and orthotomics are also discussed. Throughout canonical relations are used as an essential tool to carry out these geometrical constructions.

In ([2], p. 310) the problem of relating the Arf invariant of a link to other link invariants, and finding practical ways of computing it, was raised. Murasugi gave a partial solution to this in [9]: if L has two components, then Arf(L), when defined, is given by

where l1, l2 are the components of L, and ΔL(t1, t2) denotes the Alexander polynomial of L.

We prove that an unbounded operator Z on a Banach space , which commutes with a representation of a Lie group G, is the generator of a contraction semigroup, under conditions on Z and G which have not previously been investigated. The case of an unbounded derivation Z on a C*-algebra is considered in particular detail.

A method of constructing projective representations of separable locally compact groups in reproducing kernel Hilbert spaces is presented, based on the generalized inducing process of Rieffel and Fell. Examples show that the method can be used to construct some well-known holomorphically induced representations. Some representations on cohomology spaces are also described.

A. Avilés and C. Brech proved an intriguing result about the existence and uniqueness of certain injective Boolean algebras or Banach spaces. Their result refines the standard existence and uniqueness of saturated models. They express a wish to obtain a unified approach in the context of category theory. We provide this in the framework of weak factorisation systems. Our basic tool is the fat small object argument.