The Nottingham group can be described as the group of normalized automorphisms of the ring Fp[[t]], namely, those automorphisms acting trivially on tFp[[t]]/t2Fp[[t]]. In this paper we consider certain proper subgroups of the Nottingham group. We prove that these subgroups are identical to their normalizers and that some of them are isomorphic to the Nottingham group.

For each non-empty subset Λ of the complex plane, let (Λ) be the set of all those operators (on a fixed Hilbert space H) whose spectrum is included in Λ. The problem of spectral approximation is to determine how closely each operator on H can be approximated (in the norm) by operators in (Λ). The problem appears to be connected with the stability theory of certain differential equations. (Consider the case in which Λ is the right half plane.) In its general form the problem is extraordinarily difficult. Thus, for instance, even when Λ is the singleton {0}, so that (Λ) is the set of quasinilpotent operators, the determination of the closure of (Λ) has been an open problem for several years (3, Problem 7).

Let A and B be function algebras. We generalise the Nagasawa theorem by proving that the Banach–Mazur distance between the underlying Banach spaces of A and B, is close to one if and only if they are almost isomorphic, that is if and only if there is a linear map T from A onto B such that ∥T−1(Tf · Tg)−fg∥≦ε∥f∥∥g∥.

In the answer to the book-work question, set in a recent examination to investigate the volume of a pyramid, one candidate stated that the three tetrahedra into which a triangular prism can be divided are congruent, instead of only equal in volume. It was an interesting question to determine the conditions in order that the three tetrahedra should be congruent, and this led to the wider problem – to determine what tetrahedra can fill up space by repetitions. An exhaustive examination of this required one to keep an open mind as regards whether space is euclidean, elliptic, or hyperbolic, and then to pick out the forms which exist in euclidean space.

This paper concerns criteria for assuring that every solution of a real fourth order nonselfadjoint differential equation

is oscillatory at x = ∞. Our technique is a generalisation of that used by Whyburn (1) for the study of the selfadjoint equation,

combined with the theory of H-oscillation of vector equations as introduced by Domšlak (2) and studied by Noussair and Swanson (3). Whyburn's technique consists of representing (1.2) as a dynamical system of the form

and then studying (1.3) in terms of polar coordinates in the y, z-plane. In Section 2 below we show how to represent (1.1) as a dynamical system of the form

The fundamental equation in this paper,

,

can be derived, easily and without using trilinears, from the following property of conics.

Sir T. L. Heath's translation of The Thirteen Books of Euclid's Elements, with its Introduction and Commentary, is not merely a worthy tribute to the lasting merits of Euclid's work, but is at the same time a most valuable history of elementary geometry; the language in which he describes the character of Camerer's edition of Euclid's first six books is even more applicable to his own: “No words of praise would be too warm for this veritable encyclopaedia of information.” There may, however, be room for difference of opinion on matters of detail, and I propose in this note to call attention to one or two passages in which I think he is in error in his criticism of Simson, whose edition of Euclid formed the basis of so many English text-books and kept alive the traditions of Greek geometry in this country long after Euclid's Elements had disappeared as a text-book on the Continent.

The problem of the incidence of a train of plane waves on a semi-infinite plane, when the edge of the plane is perpendicular to the direction of the waves, is a two-dimensional one, and was first fully treated in a paper by Sommerfeld. It was also discussed by me in a paper published in the Proceedings of the London Mathematical Society.

An asymptotic tract of a real function u harmonic and non-constant in ℂ is a component of the set {z:u(z)≠c}, for some real number c; a quasi-tractT(≠ℂ) is an unbounded simply-connected domain in ℂ such that there exists a function u that is positive, unbounded and harmonic in T such that, for each point ζ∈∂T∩ℂ,

and a ℱ-tract is an unbounded simply-connected domain T in ℂ whose every prime end that contains ∞ in its impression is of the first kind.

The authors study the growth of a harmonic function in one of its asymptotic tracts, and the question of whether a quasi-tract is an asymptotic tract. The branching of either type of tract is also taken into consideration.

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.