In 1912, shortly before his death, Poincaré (8) conjectured the following theorem in his investigation of the restricted problem of three bodies.

Poincaré's Last Geometric Theorem. Given a ring 0 < a ⩽ r ⩽ b in the r, θ plane and a homeomorphic, area-preserving mapping T of the ring onto itself under which points on r = a advance and those on r = b regress, there will exist at least two points of the ring invariant under T.

Poincaré was able to prove this theorem in only a few special cases. Shortly thereafter, Birkhoff was able to give a complete proof in (2) and in, (3) he gave a generalization of the theorem, dropping the assumption that the transformation was area-preserving. Birkhoff's proofs were very ingenious; however, they did not use standard topological arguments.

The question of solvability of the differential equation

1

with x ranging over an interval (0, a], and with the boundary condition ƒ(0+) = 0, can be investigated as an initial-value problem at 0, which may be a singular point for the equation.

The aim of this paper is to characterize the finite simple group PSp4(3) by the structure of the centralizer of an involution contained in the centre of its Sylow 2-subgroup. More precisely, we shall prove the following result.

In this paper we shall give necessary and sufficient conditions for three lines, passing respectively through the vertices of a proper triangle PQR in the real Euclidean plane, to be concurrent. Of course, the theorem of Ceva deals with this problem, but it is useful to have a criterion which involves only vectors localized at a point O of the plane, and the exterior products of these vectors. Applications are made to theorems which are not easily proved by other methods.

The central problem in the present paper is the construction of symmetric and of skew-symmetric ( = skew) matrices C of order v, with diagonal elements 0 and other elements + 1 or — 1, satisfying

The following necessary conditions are known: v ≡ 2 (mod 4) and

a and b integers, for symmetric matrices C (Belevitch (1, 2), Raghavarao (14)), and v = 2 or v ≡ 0 (mod 4) for skew matrices C.

The purpose of this paper is to introduce a generalization of metric space that arises naturally out of the notion of signal function as it occurs, for example, in (5). In §§ 2-5, the basic definitions and motivation are given. In §§ 6 and 7 several elementary topological properties are proved, and in §§ 8 and 9 an important example from special relativity is developed.

In a recent paper (6), S. T. Hu investigated the initial projection from the mth enveloping space of a topological space X into X and proved that, under some local conditions on X, the initial projection is a fibering. In a subsequent paper (7), Hu showed that the terminal projection from the mth enveloping space is a fibering without assuming the local conditions on X and in (8) he used the terminal projection from the second enveloping space in his topological immersion theorem.

This paper is concerned with generalizations of the classical Hardy spaces (8, p. 39) and the question of boundary values for functions of these various spaces. The general setting is the “big disk” Δ discussed by Arens and Singer in (1, 2) and by Hoffman in (7). Analytic functions are defined in (1). Classes of such functions corresponding to the Hardy Hp spaces are considered and shown to possess boundary values in (2). Contrary to the classical case, such functions do not form a Banach space; hence they are not the functional analytic analogue of the classical spaces. In (3) quasi-analytic functions are defined while in (4) Hardy spaces of such functions are considered and are shown to have boundary values and to form a Banach space.

The principal theorem of this paper, a generalization of a theorem given by R. H. Cameron (2), provides a means of approximating certain Wiener integrals to any desired degree of accuracy by an (n + 1)-fold Riemann integral with sufficiently large n. The generalization is in the use of a general complete orthonormal set of functions, whereas Cameron's paper used only the odd harmonic set.

Let C′ be the class of real-valued functions x(t) defined on [0, 1] and such that x(0) = 0 and which are continuous except perhaps for one left continuous jump. Let C be the class of continuous members of C′.

It is well known that planes of characteristic 2 behave differently from other Pappian projective planes. For this reason their detailed properties are usually ignored in books on synthetic projective geometry, especially when conies are being discussed. This can give rise to the misleading impression that planes of characteristic 2 are more difficult to deal with, while a cursory introduction to conies in such planes (e.g. Theorem 4.3) may suggest that the notion of “pole and polar” no longer exists.

A compactum is a compact, metrizable space. A continuum is a connected compactum. All polyhedra will be finitely triangulable spaces. If a is an open cover of a compactum X, a map of X onto a compactum Y is called an α-map provided that the inverse image of each point in Y is contained in some member of α.

If is a class of polyhedra, then, following Mardešić and Segal (10), we say a compactum X is -like provided that for each open cover α of X there exists an a-map of X onto some member of .

A ring R is called a J-ring if there exists an integer n > 1 such that xn = x for every x ∈ R. The following beautiful theorem due to Jacobson (4, 5) is a generalization of Wedderburn's theorem, which asserts that every finite division ring must be a field.

We call a digraph “antisymmetrical” if there is an automorphism θ of its graph, of period 2, which reverses the direction of every edge and maps no edge or vertex onto itself. We construct a theory of flows invariant under θ for such a diagraph. This theory is analogous to the Max Flow Min Cut theory for ordinary flows in digraphs. It is found to include that part of the theory of undirected graphs which discusses the existence of spanning subgraphs with a specified valency at each vertex.

Many years ago, I defined (8) three types of exceptional unode on an algebraic surface, which I called U*8, U*9, U*10, corresponding, on a non-singular model of the surface, to sets of six, seven, and eight rational curves, each of grade — 2, with the intersection patterns represented by the Coxeter-Dynkin graphs now usually known as E6, E7, E8:

where each dot represents a curve, and linked dots intersecting curves. In each case we shall denote the curves in the horizontal sequence by S1, s2, … from left to right, and the extra curve meeting s3 by s*.

Let X1 X2, … , Xn be mutually independent random variables with a common continuous distribution function F(t). Let Fn(t) be the corresponding empirical distribution function, that is Fn(t) = (number of Xi ⩽ t, 1 ⩽ i ⩽ n)/n.

If a convex polytope P is centrally symmetric, and has the property that all its faces (of every dimension) are centrally symmetric, then P is called a zonotope. Zonotopes have many interesting properties which have been investigated by Coxeter and other authors (see (4, §2.8 and §13.8) and (5) which contains a useful bibliography). In particular, it is known (5, §3) that a zonotope is completely characterized by the fact that all its two-dimensional faces are centrally symmetric.

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set.

The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn, which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn, or (ii) in Tn, if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.

A coalgebra over the field F is a vector space A over F, with maps δ: A → A ⊗ A and ∊: A → F such that

1

and

2

The notion of coalgebra is dual to the notion of algebra with unit, with δ as coproduct (equation (1) says that δ is associative) and ∊ as the unit map (equation (2) is just the statement that ∊ is a unit for the coproduct δ). If A is also an algebra with unit and δ and ∊ are algebra homomorphisms, A is a Hopf algebra.

If 0 → A → C → B → 0 is an exact sequence of abelian groups, if ƒ is a 2-cocyle for this extension, if α ∈ End A, and if β ∈ End B, then a necessary and sufficient condition that α extend to an endomorphism γ of C which induces β is that (M) αƒ and ƒβ be cohomologous ; see Montgomery (2). We shall extend this result to the case where 1 → A → G → B → 1 is an exact sequence of groups and A is abelian.

This paper continues work begun in a previous paper of the same title (7), which will be called I; results from I will be referred to as Theorem 1.4, Axiom 1.1 etc. The notation used in the present paper will, except where noted, be that of I, to which reference should be made for further details.

In § 2, certain ideas presented in I are modified to give a neater and more general theory and then some new results of this theory are added. The remaining two sections develop some of the examples mentioned in I, § 5.