Let T be a linear operator on a complex normed space X. Its spatial numerical range V(T) is denned as

Let = (Ai;i ∈ I) be an indexed family of sets. The family is said to contain the family = (Bj;j ∈ J) if there is a one-to-one map f from J into I such that Bj = Af(i) for all j in J. The indexed family = (Ai;i ∈ I) is said to be a Δ (λ)-family if |I| = λ and Ai ∩ Aj = Ak ∩ Ai for all pairs i, j and k, l from I. The family is said to be a (λ, k)-family if |I| = λ and |Ai| = k for all Ai.

This paper concerns, in the first instance, a sequence {t(1), t(2), t(3), …} of positive numbers denned successively by

where k ∈ ℕ = {1,2,3,…} and

Thus, when k is odd, A(k) consists of those divisors of k that do not exceed k/3; when k is even, A (k) consists of those even divisors n of k that make k/n odd and do not exceed k/3. The sets A(k) are listed in Table 1 for k ≤ 105; the numbers t(1) to t(10) are 1, 1, 4/3, 1, 6/5, 4/3, 8/7, 1, 14/9, 6/5.

An element a of a Banach algebra A over ࠶ will be called quasi-central if, for some K ≥ 1,

The set Q(A) of all quasi-central elements of A will be called the quasi-centre of A and the set of elements a which satisfy (1) for a particular value of K will be denoted by Q(K, A). The motivation for these definitions is the result of Le Page ([1], proposition 3) that Q(1, A) coincides with the centre Z(A) of A. The reader is referred to [2] and [3] for a study of the properties of quasi-central elements.

The difference between the maximal C*-norm ‖ ‖max and the Haagerup norm ‖ ‖h on the tensor product space of C*-aIgebras is studied. Let A and B be C*-algebras. It is shown that ‖ ‖max is equivalent to ‖ ‖h on A ⊗ B if and only if A or B is finite-dimensional.

Let A be a (commutative, Noetherian) local ring. The big Cohen-Macaulay conjecture asserts that if a1,…,an is a system of parameters for A there exists an A-module M such that a1,…,an is an M-sequence. Then we say that M is a big Cohen-Macaulay module with respect to a1,…,an. This conjecture implies some important conjectures in commutative algebra and has been established affirmatively by M. Hochster for any ring containing a field as a subring (see [9] for further information).

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 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|.

In this paper we consider certain questions concerning the differential geometry of generic hypersurfaces in ℝn. Our results prove, for example, that the curve of rib points of a generic surface in ℝ3 has transverse self-intersections.

In (4) Porteous discussed (amongst other things) the generic geometry of curves and surfaces in ℝ3. Subsequently Looijenga ((3) and see also (5)) gave a more precise definition of the term generic and showed that an open dense subset of smooth embeddings of manifolds in Euclidean space were indeed generic.

The existence of solutions is considered for equations of the form

for x ∈ H (a Hilbert space), P a compact linear operator on H; Q(x) a bounded linear operator on H and continuous in x and uniformly bounded; g(x) a continuous uniformly bounded map with range in H. Two situations are considered: Q(x) lies in a weakly compact set of operators for which (a) (I − PQ(x)) is invertible (non-resonance case) or (b) (I − P(Q(x) + λI)) is invertible for 0 < λ ≤ α (resonance case).

The purpose of this paper is to exhibit two exact sequences involving the sets of conjugacy classes in various groups. One of them is applied to prove the following result.

Delsarte [2], Povzner [9], Levitan [8], Leblanc [7], Dunford and Schwartz [3] (p. 1626) and Hutson and Pym [5] have discussed generalized translation operators (GTO) ‘associating with a differential operator’. The latter authors have also considered the topic in an abstract setting-the GTO ‘associates’ with a compact operator in a normed space. GTO are to have properties generalizing those of the translation operators defined by members of a group on a vector space E of functions defined on the group:

(πεE, s and t are group elements). In the case of a locally compact topological group the integration spaces E = L1,L2,L∞, for a Haar measure of the group, are of especial interest.

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for β-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.

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.

The knots and links which can arise as the closure of 3-string braids, and their relations to the braids which give rise to them have been studied by Murasugi [5] and others, including Hartley [2] and more recently Przytycki[6]. Three-braids appear to form a rather special class among braids from some points of view [3]; they are also the only group of braids for which Burau's representation is known to be faithful [1]. They are, however, varied enough to provide an interesting range of knots and links on which to test a number of conjectures.

Sawon and Warzecha established in (2) the general form of a subalgebra of codimension 1 in a commutative Banach algebra with identity. A short and elegant proof of their result was later given by Detraz in(1). This note is intended to show that an even more elementary proof is possible and which has the advantage that no use is made of the existence of maximal ideals.

The paper continues earlier work on the generalization in n-dimensional space of the differential equations governing the propagation of electromagnetic waves in inhomogeneous anisotropic ionospheric plasma. Here there are investigated certain properties of the equations and their solutions that are independent of the dimension n of the space in which propagation takes place. The isotropic case is discussed, together with a special anisotropic model for which the various planes of circular polarization are geometrically related to each other.

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.

We give a method to construct non-trivial $p$-harmonic morphisms via conformal change of the metric on the domain and/or the target manifold (Theorems 2.1, 2.5 and 2.8). As applications, we show the existence of higher dimensional harmonic spheres in general manifolds (Theorem 2.10) generalizing Sacks and Uhlenbeck's result on harmonic 2-spheres, prove some existence theorems for non-trivial $p$-harmonic morphisms between conformally flat spaces (Theorems 2.12, 3.1 and 3.2), give a method to construct minimal foliations via $p$-harmonic morphisms and show that many $\mathbf{R}^{m}$ with a conformally flat metric admit minimal foliations of codimension greater than two (Theorems 3.3, 3.4).

We refer to a beautiful and important result of Tutte(1), in the theory of graphs; that a linear graph G is prime if and only if it contains a set ∑ of vertices, such that u(G∑) > n(∑); where n(∑) is the number of vertices in ∑, G∑ is the graph obtained from G by deleting the star of ∑ (all the vertices of G in ∑, together with all the edges of G meeting vertices of ∑), and u(G∑) is the number of connected components of G∑ having an odd number of vertices.