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”.

Let X be an infinite set and S be a transformation semigroup on X invariant under conjugations by permutations of X. Such S is termed x-normal. In the paper, we describe elements of a x-normal semigroup S of one-to-one transformations.

The effect of imposing a certain finiteness condition on the group of central automorphisms of a finite-by-nilpotent group is investigated. In particular it is shown that, if each central automorphism of a finite-by-nilpotent group G has finite order, then the factor group G/Z(G) has finite exponent.

Let A and B be C*-algebras and A ⊗ B denote the minimal C*-tensor product of A and B. T. Huruya [1] gave examples of C*-tensor products A ⊗ B with C*-subalgebras A1⊗B2 and A2⊗B2 such that strictly contains , answering a question of S. Wassermann [3, Remark 23]. In this short note, we show that the same situation can occur even if A1 = A2

The method of extracting any root may be understood from the subjoined example of extracting the fifth root of 2,073,071,593.

A proof of Dupin's theorem with some simple illustrations of the method employed.

Before plunging into Dupin's theorem, I think it well to speak of certain infinitesimal rotations which play a part in the proof. By an infinitesimal angle of the first order is meant an angle subtended at the centre of a circle of finite radius by an arc whose length is an infinitesimal of the first order. If we neglect infinitesimals of the second order, equal infinitesimal rotations of the first order about axes which meet and are separated by a small angle of the first order are identical. For instance, if AB and BC be elements of a curve of continuous curvature, an infinitesimal rotation about AB may, if we prefer it, be regarded as taking place about BC; and again, if OA, OB, OC be a set of rectangular axes, small rotations about OA, OB, OC may be regarded as taking place in any order. For if P be a point on a sphere of finite radius, and PQ, PR be the displacements of P due to equal infinitesimal rotations of the first order about two diameters separated by a small angle of the first order, the angle QPR is the angle of separation of the axes, and it follows that QR is an infinitesimal of the second order. Further, if the radius of the sphere is an infinitesimal of the first order, QR is of the third order of small quantities.

The principal problem which we have in view may be stated as follows:

To construct a triangle ABC (Fig. 1) in which are given in magnitude only, the height h = AO from the vertex A, the median m = BI from the vertex B, and the bisector f= CD from the vertex C.

Take first the Addition-Formula of the Tangent :

Assume that 1 – tanx tanO is not zero for all value of x II.