Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).

Marston Morse and William Transue (6, 8) have introduced and studied function spaces, called MT-spaces, for which the elements of the topological dual are of integral type. Their theory does not admit certain classical Banach function spaces including spaces of bounded functions and spaces. The theory of function spaces determined by a length function (λ-spaces) (4, 5), which depends on a fixed measure, admits many of the maximal MT-spaces, the spaces and spaces of locally integrable functions but does not admit certain maximal MT-spaces including the space of complex continuous functions with compact supports.

Considerable interest has attached to the problem of determining the type of a Riemann surface obtained by performing an identification between the edges of a strip or a half-strip (1, 2, 4, 5, 8). A fairly thorough analysis was made in 1946 by Volkovyskii (6) who gave various sufficient conditions for parabolic and hyperbolic type. The object of the present paper is to show that his principal sufficient condition for hyperbolic type can be substantially improved.

The spaces , w real, 1 ≤ p < ∞, consist of those functions f(s), analytic for Re s > w, and such that μp(f;x) is bounded for x > w, where

1.1

Doetsch (1) has shown that if e-wtϕ(t) ∈ Lp (0, ∞), 1 < p ≤ 2, and f is the Laplace transform of ϕ, that is,

then f ∈ , where

1.2

and that conversely if f ∈ , 1 < p ≤ 2, then there is a function ϕ, with e-wtϕ(t) ∈ Lq (0, ∞), such that f is the Laplace transform of ϕ.

There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical of these are the supply-demand theorem due to Gale (4), which states a condition for the existence of a flow satisfying demands at certain nodes from supplies at other nodes, and the Hoffman circulation theorem (received by the present author in private communication), which states a condition for the existence of a circulatory flow in a network in which each arc has associated with it not only an upper bound for the arc flow, but a lower bound as well. If the constraints on flows are integral (for example, if the bounds on arc flows for the circulation theorem are integers), it is also true that integral flows meeting the requirements exist provided any flow does so.

A considerable volume of research on the theory of regular maps is now in existence. Systematic enumerations of regular maps on the surfaces of genus 1 and 2 were begun by Brahana (1; 2) and completed by Coxeter (6; 7, p. 141). In addition Coxeter enumerated the regular maps on the simplest non-orientable surfaces (7, pp. 116, 139), and constructed tables of some interesting families of regular maps (3; 7, p. 140).

Most of the regular maps on a surface of genus 3 have appeared in these papers, but no systematic enumeration of them seems to have been attempted. The ultimate goal of this paper is a complete list of these regular maps. However, the families of maps {j.p,q}and {j.p,j.q} which are defined in § 4 and listed in Tables I and II are of considerable interest in themselves. Also of some importance is the complete list of regular maps of type {p,3} with six or fewer faces (§ 5 and Table III).

1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is -complete for some infinite ,3 we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we assume orthocomplementation as Kaplansky does in his remarkable paper (1).

1.2. Our exposition is elementary in the sense that it can be read without reference to the literature. Our brief preliminary § 2 should enable the reader to read this paper independently.

1.3. Von Neumann's theory of independence (3, Part I, Chapter II) leans heavily on the assumption that the lattice is continuous, or at least upper continuous.

Dans une note récente (1), Szekeres a donné le théorème suivant:

Sif est une fonction réelle, définie, strictement croissante et ayant la deuxième dérivée continue sur (a, b) ( — ∞ ≤ a < b ≤ ≤) alors il existe une fonction réelle ϕ, définie et strictement croissante sur (a, b), et une fonction réelle ψ définie et strictement croissante sur (ϕ(a), ϕ(b)), telles que ϕ˝,(x) ≥ 0 pour a < x < b, ψ ˝(u) < 0 pour ϕ(a) < u < ϕ(b) et f(x) = ψ(ϕ(x)) pour a < x < b.

Dans la démonstration de ce théorème, Szekeres utilise une expression où la dérivée f‘(x) apparaît au dénominateur, ce qui exige qu'on ait f'(x) ≠ 0 pour a < x < b, condition qui ne figure pas dans l'énoncé du théorème. Il est naturel alors de poser le problème si le théorème ci-dessus reste en vigueur si cette condition n'est pas remplie. Il y a lieu aussi de se demander si les autres conditions, par exemple l'existence et la continuité def“(x) sont nécessaires pour qu'une fonction strictement croissante puisse être représentée comme une superposition de deux fonctions croissantes, l'une convexe, l'autre concave.

In the theory of ordinary differential equations, there is a strange relationship between uniqueness of solutions and convergence of the successive approxi mations. There are examples of differential equations with unique solutions for which the successive approximations do not converge (8) and of differential equations with non-unique solutions for which the successive approximations do converge (2). However, in spite of the known logical independence of these two properties, almost all conditions which assure uniqueness also imply the convergence of the successive approximations. For example, the hypotheses of Kamke's general uniqueness theorem (5), have been shown by Coddington and Levinson to suffice for the convergence of successive approximations, after the addition of one simple monotonicity condition (4). There is one counterexample to this “principle,” a generalization of Kamke's result, to which another condition in addition to a monotonicity assumption must be added before convergence of the successive approximations can be proved (2).

Let G(z) be a function of a complex variable, regular in the annulus 0 ≤ a ≤ |z| < b ≤ ∞. We shall assume there exists a curve within the annulus for which

provided z is restricted to be a point of this curve. Under these restrictions G (z) has a Laurent expansion of the form

1.1

where the Laurent coefficients an have the integral representation

1.2

and C can be any contour, within the domain of regularity, that encloses z = 0. We shall also assume that the an are all real numbers. Using the usual complex conjugate notation, we can, therefore, write

1.3

The problem of determining the asymptotic behaviour of an as n —> co is very old in mathematical literature and appears in many forms and disguises.

This paper continues the study of continuous selections begun in (13; 14; 15) and the expository paper (12). The purpose of these papers, which is described in detail in the introduction to (13), can be summarized here as follows. If X and Y are topological spaces, and ϕ a function (called a carrier) from X to the space 2Y of non-empty subsets of F, then a selection for ϕ is a continuous f: X → Y such that f(x) ∈ ϕ(x) for every x ∈ X. For reasons which are explained in (13), we restrict our attention to carriers which are lower semi-continuous (l.s.c), in the sense that, whenever U is open in Y, then is open in X. Our purpose in these papers is to find conditions for the existence and extendability of selections.

The principal purpose of this paper is to generalize the following result, which is half of the principal theorem (Theorem 3.2˝) of (13) (and is repeated as Theorem I of (12)).

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.

By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.

Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.

Let be the Fourier transform of a Borel measure of finite total variation. The formula

can be justified if the integral on the right converges absolutely. For

where

Now let h → 0 in both sides of this equation and use the Lebesgue dominated convergence theorem.

If f(z) is regular in |z| < 1, the expression

is called the characteristic of f(z). This is the notation of Nevanlinna (4) for the special case of regular functions; in this note it will not be necessary to discuss meromorphic functions. If m(r,f) is bounded for 0 < r < 1, then f(z) is called quasi-bounded in |z| < 1. In particular, every bounded function is quasibounded. The class Q of quasi-bounded functions is important because, for instance, a “Fatou theorem” holds for such functions (4, p. 134).

Measures may be obtained from suitable non-negative valued functions in a number of ways. It is the purpose of this paper to present an abstract formulation of certain principles which may be used to construct measures, and to show that the various methods most frequently encountered in the literature are in fact all special applications of these principles.

The basic requirements A1-A4 are set forth in § 2, and it is there shown that a measure can be denned when these are fulfilled. These requirements are satisfied in every case known to the writer. Usually, however, conditions stronger than A1-A4 hold, and it is these extra restrictions which yield information on the class of measurable sets and other matters. In §§ 3, 4, and 5 certain abstracts form of such restrictions are considered, and results are derived thereforom. The paper concludes with an analysis of how a number of measures occur as special cases of the theory.

A theorem due to von Staudt states that a null polarity in complex projective space of three dimensions is determined by a selfpolar skew pentagon. By allowing an element of the self-polar pentagon to vary in a suitable manner we can arrive at a family of ∞1 null polarities, which we term a pencil of null polarities. Each polarity of the pencil distinguishes a linear complex as the class of self-polar lines. Thus, associated with the pencil is a family of ∞1 linear complexes, which we term a pencil of linear complexes.

It is the purpose of this paper to continue an earlier investigation of pencils of polarities (2), by applying analogous techniques to the study of pencils of null polarities and pencils of linear complexes.

Since it develops that the lines common to all linear complexes of a pencil are the lines of a linear congruence, the central question has been: How many of the different types of linear congruences can be achieved in this manner? Happily, it can be reported that the classification of pencils of null polarities yields all of the three types of linear congruences (4, pp. 140-141).

Let G be a connected finite graph in which each edge has two distinct ends and no two distinct edges have the same pair of ends. We suppose further that G is cubic, that is, each vertex is incident with just three edges.

An s-path in G, where s is any positive integer, is a sequence S = (v0, v1… , vs) of s + 1 vertices of G, not necessarily all distinct, which satisfies the following two conditions:

(i) Any three consecutive terms of S are distinct.

(ii) Any two consecutive terms of S are the two ends of some edge of G.

About half the following pages are concerned with the “cubic surface group” G, of order 51840, and the geometry cognate to a certain representation thereof. The literature of this group, with its subgroup of order 25920, is already voluminous; the addition of these few pages to it will not, it is hoped, be regarded askance as a perverse and misdirected indulgence in archaism.

In the preceding paper, Edge represented the celebrated “cubic surface group” of order 72.6! = 51840 as the group of automorphisms of a senary quadratic form over the field of residue-classes mod 2. The object of this sequel is to compare Edge's finite space with a real space, thus identifying his non-ruled quadric in PG(5, 2) with a modular counterpart of the semi-regular polytope 221 which was discovered by Gosset in 1897.

Furstenberg (1) and the author (4), among others, have exhibited postulate systems for groups in terms of the operation x — y = x + ( — y).Furstenberg also investigated a system obtained by removing one of his postulates which defines what he called a half-group. A structure theorem for half-groups was given in (1). In the present paper, we prove another structure theorem for half-groups. A more restricted entity, called a pseudo-group, is introduced, and its structure, together with that of a half-group satisfying the left and right cancellation laws, is studied. Finally, some topological questions concerning their structure are also considered.