A characterization is given of those unital, 2-subhomogeneous, Fell C*-algebras which have only inner derivations. This proves Sproston and Strauss's conjecture from 1992. Various examples are given of phenomena which cannot occur for separable C*-algebras. In particular, an example is given of a C*-algebra with only inner derivations which has a quotient algebra admitting outer derivations. This answers a question of Akemann, Elliott, Pedersen and Tomiyama from 1976.

A key step in establishing the validity of the linear sampling method of determining an unknown scattering obstacle $D$ from a knowledge of its far-field pattern is to prove that solutions of the Helmholtz equation in $D$ can be approximated in $H^1(D)$ by Herglotz wave functions.

To this end we establish the important property that the set of Herglotz wave functions is dense in the space of solutions of the Helmholtz equation with respect to the Sobolev space $H^1(D)$ norm.

AMS 2000 Mathematics subject classification: Primary 35R30. Secondary 35P25

Consider the Darboux problem

where φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:

(A1) f(.,.,z) is measurable for every z ∈ Rd;

(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;

(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.

Let υ0 be a valuation of a field K0 with value group G0 and υ be an extension of υ0 to a simple transcendental extension K0(x) having value group G such that G/G0 is not a torsion group. In this paper we investigate whether there exists t∈K0(x)/K0 with υ(t) non-torsion mod G0 such that υ is the unique extension to K0(x) of its restriction to the subfield K0(t). It is proved that the answer to this question is “yes” if υ0 is henselian or if υ0 is of rank 1 with G0 a cofinal subset of the value group of υ in the latter case, and that it is “no” in general. It is also shown that the affirmative answer to this problem is equivalent to a fundamental equality which relates some important numerical invariants of the extension (K, υ)/(K0, υ0).

In a paper (1) by Harding there is a tacit invitation to seek the connection between the following two problems:

(i) Find the number, ηk(N), of regions into which a k-dimensional space is partitioned by a set of N (k- l)-dimensional hyperplanes.

(ii) Find the number, vk(N), of distinct partitions of a given set of N points in a k-dimensional space E that can be induced by (k- 1)-dimensional hyperplanes.

The separating subspace of any Lie derivation on a semisimple Banach algebra A is contained in the centre of A.

The matrices considered in the following note are non-singular, and are related to a given matrix A having elements taken from the field of positive and negative integers and zero. The invariant properties of such a matrix A, under multiplications by matrices of determinant equal to unity, can be formulated, as is well known, in terms of the “elementary divisors” of the determinant |A|. Thus if A is of the nth order, and p is a prime occurring in |A| to the power hn, in the H.C.F. of the first minors of |A| to the power hn−1, in the H.C.F. of the second minors to the power hn−2, and so on, h0 by convention being zero, then the first differences of the h's,

are invariant under the transformations considered, and it is known that r ≥ er−1. The numbers

where the product includes all prime factors of |A|, are called the elementary divisors of |A| and

We construct an uncountable family of pairwise non-isomorphic rings Si, such that the corresponding full 2 by 2 matrix rings M2(Si) are all isomorphic to each other. The rings Si are Noetherian integral domains which are finitely-generated as modules over their centres.

Among the tricyclic Hamiltonian graphs with a prescribed number of vertices, the unique graph with maximal index is determined. Some subsidiary results are also included.

In this paper we consider the correspondence between tangential quadrics of [3] and points of [9]. Godeaux has considered this geometrically, with the object of obtaining a representation for a twisted cubic of three dimensions. We have considered it from a standpoint more algebraic than that of Godeaux, with particular reference to the types of pencils of quadrics that correspond to special lines of [9], and to the interpretation in [9] of the fact that the condition for a net of quadrics to be part of the polar system of a cubic surface is poristic.

We prove the following: Assume that , where p is an odd positive integer, g(ζ is a transcendental entire function with order of growth less than 1, and set A(z) = B(ezz). Then for every solution , the exponent of convergence of the zero-sequence is infinite, and, in fact, the stronger conclusion holds. We also give an example to show that if the order of growth of g(ζ) equals 1 (or, in fact, equals an arbitrary positive integer), this conclusion doesn't hold.

1. Introduction. In an earlier paper1 the function

was considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points eiav (v = 1, 2, .., k). It was there proved,2 in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points eiav are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients cn can be interpolated in the form

where the Gv(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.

In a paper entitled “On differentiating a Matrix” H. W. Turnbull deduced some interesting and elegant results by the use of a matrix operator Ω, a matrix whose elements were the partial differential operators with respect to the elements of a square matrix x. Throughout the present paper the differential operator Ω. is used, or rather a matrix operator, which is the product of Ω and another square matrix Y.

By means of this operator in §§ 1 and 2 Bazin's matrix and Reiss's matrix are considered from the standpoint of matrices as distinct from that of determinants. Reiss's matrix is shown to be a constant times a compound of Bazin's matrix; and the latent roots of Reiss's matrix are immediately determined in terms of the latent roots of Bazin's matrix. From this result a theorem, discovered by Deruyts, is deduced as well as a more general theorem.