Let $X$ be a Banach space and $\xi$ an ordinal number. We study some isomorphic classifications of the Banach spaces $X^\xi$ of the continuous $X$-valued functions defined in the interval of ordinals $[1,\xi]$ and equipped with the supremum norm. More precisely, first we use the continuum hypothesis to give an isomorphic classification of $C(I)^\xi$, $\xi\geq\omega_1$. Then we present a characterization of the separable Banach spaces $X$ that are isomorphic to $X^\xi$, $\forall\xi$, $\omega\leq\xi lt \omega_1$. Finally, we show that the isomorphic classifications of $(C(I)\oplus F^*)^\xi$ and $\ell_\infty(\N)^\xi$, where $F$ is the space of Figiel and $\omega\leq\xi lt \omega_1$ are similar to that of $\R^\xi$ given by Bessaga and Pelczynski.

AMS 2000 Mathematics subject classification: Primary 46B03; 46B20; 46E15

Without prior assumptions about growth, fundamental inequalities for the Taylor series of an entire function are obtained, valid outside a certain exceptional set. The results are vacuous or not depending on the estimate for the exceptional set. Only then does the growth of the function enter.

An old question of Brauer that asks how fast numbers of conjugacy classes grow is investigated by considering the least number cn of conjugacy classes in a group of order 2n. The numbers cn are computed for n ≤ 14 and a lower bound is given for c15. It is observed that cn grows very slowly except for occasional large jumps corresponding to an increase in coclass of the minimal groups Gn. Restricting to groups that are 2-generated or have coclass at most 3 allows us to extend these computations.

Let

The asymptotic behaviours of ƒ(x) and g(x), as x→+0, were first given by G. H. Hardy in (4), (5). In his papers {an}; is a monotone decreasing sequence. Further results on the asymptotic behaviours of ƒ(x) and g(x), as x→+0, for monotone coefficients have been given in (9) and (1). Recently, the results have been generalized to quasi-monotone coefficients.

This paper is concerned with asymptotic behaviours of ƒ(x) and g(x) for δ-quasi-monotone coefficients.

In what follows, we shall denote by L(x) a slowly varying function in the sense of Karamata (6), i.e.

An n × n matrix An = (aij) is tri-diagonal if aij = 0 for |i−j|≧2. The latent roots of such matrices may be conveniently studied by forming the sequence of polynomials Ψk(λ)=|λI−Ak|, where Ak is the principal submatrix of Ak+1 obtained by deleting the last row and column of Ak+1, and then observing that these polynomials satisfy the following recurrence relation:

The paper studies some classes of dense *-subalgebras B of C*-algebras A whose properties are close to the properties of the algebras of differentiable functions. In terms of a set of norms on B it defines -subalgebras of A and establishes that they are locally normal Q*-subalgebras. If x = x* ∈ B and f(t) is a function on Sp(x), some sufficient conditions are given for f(x) to belong to B. For p = 1, in particular, it is shown that -subalgebras are closed under C∞-calculus. If δ is a closed derivation of A, the algebras D(δp) are -subalgebras of A. In the case when δ is a generator of a one-parameter semigroup of automorphisms of A, it is proved that, in fact, D(δp) are -subalgebras. The paper also characterizes those Banach *-algebras which are isomorphic to subalgebras of C*-algebras.

The countability index, C(S), of a semigroup S is the smallest integer n, if it exists, such that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists, define C(S) = ∞. The density index, D(S), of a topological semigroup S is the smallest integer n, if it exists, such that S contains a dense subsemigroup with n generators. If no such integer exists, define D(S) = ∞. S(X) is the topological semigroup of all continuous selfmaps of the locally compact Hausdorff space X where S(X) is given the compact-open topology. Various results are obtained about C(S(X)) and D(S(X)). For example, if X consists of a finite number (< 1) of components, each of which is a compact N-dimensional subspace of Euclidean Nspace and has the internal extension property and X is not the two point discrete space. Then C(S(X)) exceeds two but is finite. There are additional results for C(S(X)) and similar results for D(S(X)).

§1. In a triangle ABC (fig. 37), BE is made equal to CF; to find the locus of the middle point of EF.

Take K the middle point of BC and P the middle point of EF, then PK is the locus required. For if E′ and F′ are the middle points of BE and CF, the middle point of E′F′ will lie in PK (namely, at the middle point of PK); and again if BE′ and CF′ are bisected in E″ and F″, the middle point of E″F″ will lie in PK (namely, at the middle point of KR); and so on. At any stage we may double the parts cut off from BA and CA instead of bisecting them. Hence the locus required is such that any part of it, however small, contains an infinite number of collinear points; and hence the locus is a straight line.

A formally self-adjoint operator L is said to be of limit circle type at infinity if its highest order coefficients are zero-free and all solutions x of L(x) = 0 are square-integrable on [c, ∞) for some c. (We will drop “at infinity” in what follows.)

The generalised zeta-function ζ(s, α) is defined by

where α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operators

where λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

It is a singular honour to be invited to deliver a lecture commemorating the work of Sir Edmund Whittaker, especially before the Edinburgh Mathematical Society, whose development owes so much to his initiative and co-operation. But when I reflect on the difficulties of the task I can only exclaim in the words of St Jerome's preface to his translation of the New Testament, “Pius labor, sed periculosa praesumptio”.