We study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if X is not a Hilbert space then one may find a subspace of X for which there is no Aron–Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given X forces such X to contain no uniform copies of ℓpn for p ∈ [1, 2). In particular, X must have type 2 − ϵ for every ϵ > 0. Also, we show that the bilinear version of the Lindenstrauss–Pełczyński and Johnson–Zippin theorems fail. We will then consider the notion of locally α-complemented subspace for a reasonable tensor norm α, and study the connections between α-local complementation and the extendability of α*-integral operators.

We investigate inclusion indices for general function spaces, not necessarily symmetric. Using them, we estimate the grade of proximity between two spaces $E\,{\hookrightarrow}\, F$ when we have certain information on the inclusion. The results are based on ideas from interpolation theory.

In this paper we extend to certain domains in m-dimensional Euclidean space Rm, m ≥ 3, some results about the boundary behaviour of harmonic functions which, in R2, are known to follow from distortion theorems for conformal mappings.

The hyperbolic boundary value problem

has, for δ = 0 and each positive integer k, t-periodic solutions of period τk(ε) = 2Φ/(k2 + ε)½. For a reasonably large class of functions, f, and, for each k, certain values of ε, one at least of these solutions persists for small δ

On page 196 in his lost notebook, S. Ramanujan offers evaluations of two particular Dirichlet series. In this paper, we establish Ramanujan's evaluations and more general results by various approaches. The different evaluations arising from different methods yield intriguing, unsuspecting identities.

This paper is an attempt to apply a simple series solution of the wave equation to the reflexion of a sound pulse. The reflector is a paraboloid of revolution, and the incident pulse is spherically symmetrical and comes from the focus of the reflector, so that the wave fronts of the reflected pulse are planes at right angles to the axis of symmetry of the reflector. If the incident pulse consists, at any point, of a discontinuous pressure rise followed by constant excess pressure, the series reduces to a power series in the time counted from the onset of the reflected pulse; for other forms of the incident pulse it can be interpreted as the result of the superposition of elementary pressure pulses which are constant in a small time interval and vanish outside it. No convergence proof is given, so that the interest of the investigation is physical rather than mathematical; but the numerical results indicate that the convergence of the series is unsatisfactory except near the vertex of the reflecting paraboloid. The coefficients of the series are obtained with the aid of recurrence formulae, and the first seven coefficients have been calculated. The calculations become more laborious at each successive stage. A detailed numerical investigation of the reflexion of the ‘simple rectangular pulse’ referred to already (for which the series reduces to a power series) reveals that initially the maximum pressure on any plane section at right angles to the axis of the paraboloid, due to the reflected pulse alone, occurs at the axis and has the same value everywhere on it; but after some time a secondary pressure maximum is established over a circular ring at some distance from the axis. A consideration of the initial pressure gradient of the reflected pulse suggests that a similar state of affairs exists at all distances from the vertex, but the actual calculations only extend to a plane section whose distance from the vertex is four times the focal length. The unsatisfactory convergence of the series precludes the investigation of subsequent changes in the distribution of pressure. It is finally pointed out that these results apply to a certain extent to a finite parabolic mirror.

Let d be a square-free positive integer and let O (= Od) be the ring of integers of the imaginary quadratic number field ℚ(-d). The groups PSL2(O) are called the Bianchi groups after Luigi Bianchi who made the first important contribution 1 to their study in 1892. Since then they have attracted considerable attention particularly during the last thirty years. Their importance stems primarily from their action as discrete groups of isometries on hyperbolic 3-space, H3. As a consequence they play an important role in hyperbolic geometry, low-dimensional topology together with the theory of discontinuous groups and automorphic forms. In addition they are of particular significance in the class of linear groups over Dedekind rings of arithmetic type. Serre9 has proved that in this class the Bianchi groups (along with, for example, the modular group, PSL2(z), where z is the ring of rational integers) have an exceptionally complicated (non-congruence) subgroup structure.

1. Introduction. In this paper we consider Q-groups; that is, Q is a group and we consider groups N endowed with a Q-action, meaning a homomorphism of Q into the group of automorphisms of N. In (4) a lower central Q-series was defined for such a Q-group, N, generalizing the lower central series of a group, and results were obtained relating to the localization of such a series. Since the ideas in that paper were inspired by the homotopical localization theory of nilpotent spaces (see (6)), the main body of results in (4) was concerned with the case in which N is nilpotent, and perhaps also the group Q and the action of Q on N (in the sense that the lower central Q-series terminates after a finite number of steps with the trivial group {1}). We now adopt a broader view-point and only restrict ourselves to the nilpotent case when our results appear to require us to do so; thus the spirit of this paper is much more that of general group theory as presented in [(8), especially ch. VI]. Thus, while there is some overlap of results, the methods used are not the same and many results (for example, Theorem 3·1) are far more general than any obtained in (4). Moreover, the methods also appear to us to be more appropriate in that essential appeal was made in (4) to a sophisticated theorem of Norman Blackburn on nilpotent groups, whereas here we merely use homological methods, the construction of the semidirect product and, in section 4, some very classical facts of the commutator calculus.

1. Good (4) has proved the formula

where Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).

Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put

A recurrent problem over many years in the study of linear groups has been the determination of the central height of a unipotent normal subgroup of some matrix group of specified type. In the theory of matrix groups over division rings, unipotent elements frequently present special difficulties and these have usually been by-passed by the addition of some suitable hypothesis. In this paper we make a start on the removal of these extraneous hypotheses. Our motivation for doing this now conies from [9], where by 3·7 of that paper the additional assumptions have finally reduced us to degree one, a situation where unipotent elements present few problems!

In recent years many mathematicians have studied the semigroups S, the Stone-ech compactification of the discrete semigroup S, and more particularly ℕ, where ℕ is the usual semigroup of positive integers with addition (see the surveys by Hindman and Pym 6). An important role in this theory is played by sequences in S which have distinct finite sums, for these sequences are closely linked to idempotents in S (see the survey by Hindman 6). Pym 8 introduced the concept of an oid in order to show that the structure of ℕ in 5 could be obtained in a simple way for a large class of semigroups. Papazyan 7 pointed out that the theories of sequences with distinct finite sums and of oids are the same. Thus oids have a central position in the theory of semigroup compactifications.

The object of this note is to draw attention to a family of eight uniform compounds of regular polyhedra which arise out of an action of the group on the regular icosahedron.

In this paper we give a description of the qualitative behaviour of the orbits near a non-trivial compact minimal set of a continuous flow on a 2-manifold. The first results in this direction were obtained in [1] and the present paper can be regarded as a continuation of that work. The main result can be stated as follows:

Theorem 1·1. Let (ℝ, M, f) be a continuous flow on a 2-manifold M and A ⊂ M a non-trivial compact minimal set.

1. In a recent paper in these Proceedings by Mr H. Lob, and iu an earlier paper by Mr F. P. White, it has been shown how the well-known chains of theorems in plane geometry discovered by Morley and Clifford may be proved by projection from higher space. A curve of order n in space of n dimensions and certain derived loci are projected from one, or two, or three, …, or n − 2, out of n + 1 chosen points of the curve upon a plane which contains two further points of the curve. The n + 1 lines of the plane which form the starting-point of each chain of results as originally proved (and which are obtained in various ways in the course of the projections) are actually the lines in which the plane of projection is met by the primes containing n out of the n + 1 points. Now, with the standard equations of such a curve in [n], viz.

Orthogonality is defined for all elements in a JB-algebra and Topping's results on annihilators in JW-algebras are generalized to the context of JB- and JBW-algebras. A pair (a, b) of elements in a JB-algebra A is said to be orthogonal provided that a2 ∘ b equals zero. It is shown that this relation is symmetric. The annihilator S⊥ of a subset S of A is defined to be the set of elements a in A such that, for all elements s in S, the pair (s, a) is orthogonal. It is shown that the annihilators are closed quadratic ideals and, if A is a JBW-algebra, a subset I of A is a w*-closed quadratic ideal if and only if I coincides with its biannihilator I⊥⊥. Moreover, in a JBW-algebra A the formation of the annihilator of a w*-closed quadratic ideal is an orthocomplementation on the complete lattice of w*-closed quadratic ideals which makes it into a complete orthomodular lattice. Further results establish a connection between ideals, central idempotents and annihilators in JBW-algebras.

In the paper [3] we obtained a lower bound for the Hausdorff dimension of certain sets of singular n-tuples in ℝn. In this paper we will obtain an upper bound for the dimension of these sets. These two bounds are not sharp enough to yield the exact dimension of all the sets considered but they do yield the dimension of the set of ‘highly singular’ (in a sense to be made precise below) n-tuples.

Groups associated to a knot in S3 which classify the knot have been constructed by Conway and Gordon [4], Simon [12] and Whitten[16]; see also [1] and [19] for certain special classes of knots. In this note we show that for a knot K in S3 of the groups π1(S3/K)/〈malb〉 classify the knot; here m resp. l denote a meridian resp. longitude of the knot and 〈〉 denotes normal closure. This is based on Thurston's orbifold geometrization theorem [13–15] and the following result (see also [8, 19]):

Theorem 1. Let O1 and O2 be good closed irreducible 3-orbifolds which possess a decomposition into geometric pieces (along Euclidean 2-suborbifolds, see [2, 15]) and have infinite (orbifold-) fundamental group. Then O1 and O2 are diffeomorphic (as orbifolds) if and only if π1 O1 and π1 02 are isomorphic.

Let G be a locally compact group and let UG denote the spectrum of the C*-algebra LUC(G) of bounded left uniformly continuous complex-valued functions on G, with the Gelfand topology. Then there is a multiplication on UG extending the multiplication on G (when naturally embedded in UG) such that UG is a semigroup and for each x ∈ UG, the map y ↦ yx from UG into UG is continuous, i.e. UG is a compact right topological semigroup. Consequently UG has a unique minimal ideal K which is the union of minimal (closed) left ideals UG. Furthermore K is the union of the set of maximal subgroups of K (see [3], theorem 3·ll).

An ingenious construction due to Connelly and Henderson (2) has shown that there exists a rectilinearly triangulated convex polyhedron P in having the property that at least one vertex of the triangulation lies in the interior of a face of P, and yet there is no isomorphic triangulation of a convex polyhedron P′ all of whose vertices are vertices of P′. Thus the assertion beginning on the top line of p. 354 of (1) is false, which leaves a gap in the proof of essentially the main result of (1), namely that any rectilinearly triangulated convex polyhedron incan be simplicially collapsed onto its boundary minus a 2-simplex σ. The purpose of this note is to show that the theorem is nevertheless still true. In any case the Corollaries 2 and 3 in (1) are unaffected by the error.