Baer (1) introduced the term "(p,L)-collineation" to denote a central collineation with centre p and axis L. We shall find it convenient to use a modification of the related notion of "(p, L)-transitivity."

Definition. Let π0 be a subplane of the projective plane π. Let L be a fixed line of π0, and let p be a fixed point of π0. Let r and s be any two points of π0 that are collinear with p, distinct from p, and not on L. If, for each such choice of r and s, there is a (p, L)-collineation of π that (1) carries π0 into itself and (2) carries r into s, we shall say that π is (p, L, π0)-transitive.

It is well known that for any real number θ there are infinitely many positive integers n such that

Here ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

The concept of a mosaic was recently introduced by A. A. Mullin (1). By the fundamental theorem of arithmetic, every integer n > 1 can be uniquely expressed in the form

where the pi are primes satisfying p1 < p2 < . . . < pr. We then express any exponents aj which are greater than unity in the same manner, and continue in this way until the process terminates. The resulting planar configuration of primes is called the mosaic of n.

Let Σ represent the class of functions

(1)

which are schlicht and regular, except for the pole at infinity, in |z| > 1. Further let Σ-1 be the class of inverse functions of Σ which at w = ∞ have the expansion

(2)

Sumner (3) discussed for arbitrary real λ and h, where the averaging operator ∇h is defined by

(1.1)

when f(z) is an entire function of exponential type <2π/|h|. Boas (2) gave an alternative definition of ∇h which gave Sumner's results quickly and showed that his definition is equivalent to that of Sumner.

Previously, the problem of expressing rational numbers as finite sums of rational numbers of a given type has been concerned with the Egyptian, or unit, fractions. It has long been known that any rational number is the sum of distinct unit fractions. In response to a problem proposed by E. P. Starke (4), R. Breusch (1) and B. M. Stewart (5) showed that every rational number with an odd denominator is a sum of distinct odd unit fractions. P. J. Van Albada and J. H. Van Lint (6) extended this result to show that any integer is a sum of unit fractions with denominators from an arithmetic progression.

Multipliers are useful in constructing difference sets as well as in impossibility proofs. The existence of a fixed set facilitates their application, so that the results of this paper should prove to be useful in numerical work concerning difference sets.

Let G be a group which we shall write multiplicatively. For any set A of elements of G, we define

(1)

The purpose of this paper is to point out a number of curious phenomena in the category of (real) vector lattices and linear lattice homomorphisms. Birkhoff (3, p. 221, Ex. 2 and Problem 96) called attention to the question of constructing models of the free objects with more than one generator in this category, a problem recently solved by E. C. Weinberg (9). In §6 we construct a more manageable class of (non-free) projective vector lattices. Here, however, there is a countability restriction which suggests strong connections with free and projective Boolean algebras (in the category of Boolean algebras and their homomorphisms, such algebras must satisfy the countable chain condition (6)).

Let Γn be the representation group or spin group (9; 4) of the symmetric group Sn. Then the irreducible representations of Γn can be allocated into two classes which we shall call (i) ordinary representations, which are the irreducible representations of the symmetric group, and (ii) spin or projective representations.

As is well known (3; 5), there is an ordinary irreducible representation [λ] corresponding to every partition (λ) = (λ1, λ2, . . . , λm) of n with

λ1 ≥ λ2 ≥ . . . ≥ λm > 0.

This paper deals with a problem raised in a paper by J. de Groot (1): Do there exist fields Ω whose full automorphism group is isomorphic to the additive group of integers Z?

The answer to this question is yes. In this paper we construct, given any subfield k of the complex numbers, extension fields Ω of k such that the automorphism group G(Ω/k) of Ω with respect to k is infinite cyclic. Fields having the infinite cyclic group as a full group of automorphisms are obtained by choosing the base field k in such a way that it does not contain any subfield k0 so that k possesses non-trivial automorphisms leaving k0 pointwise fixed.

In (4) Malcev generalizes the notion of the Lie algebra of a Lie group to that of an anti-commutative "tangent algebra" of an analytic loop. In this paper we shall discuss these concepts briefly and modify them to the situation where the cancellation laws in the loop are replaced by a unique two-sided inverse. Thus we shall have a set H with a binary operation xy defined on it having the algebraic properties

(1.1) H contains a two-sided identity element e;

(1.2) for every x ∊ H, there exists a unique element x-1 ∊ H such that xx-1 = x-1x = e;

In this paper integrals are considered from the point of view of inverting differential operators. In order to do this it is necessary to introduce integrals more general than the Lebesgue integral and these integrals turn out to have other interesting properties (6, 7, 12). The integral introduced here is defined in the setting of axiomatic potential theory (2, 4). By defining it as generally as possible it not only includes the James P2-integral but inverts many of the standard second-order differential operators.

Let π be a finite projective plane of order n containing a finite projective subplane π* of order u < n. Bruck has shown (1, p. 398) that if π contains a point that does not lie on any line of π*, then n ≥ u2 + u, while if every point of π lies on a line of π* then n = u2.

Let π be a finite projective plane of order n containing a finite affine subplane π0 of order m < n.

This viewpoint of studying projective planes was given in my previous paper (12). It is discussed in other papers: Hall (4, 6, 7), Maisano (16), Lombardo-Radice (14, 15), Wagner (19). In particular, we consider how to make identifications in the free plane, or how identifications are forced when one begins with a non-degenerate quadrangle and makes free extensions of this quadrangle of a known plane. We shall continue to develop this topic using the notations and definitions of the previous paper (12). We consider the number of subplanes of certain planes, finding exact values in the cases of the known order-nine planes, and deriving a lower bound in a general case. We prove a theorem concerning the structure of all singly generated planes. We give an example to show that this structure is not definitive. Finally, we prove that a specific Walker-Knuth plane is singly generated.

Denote by {t(x)} some linear transform of the sequence

of the form

where x attains continuous or only integer values. The problem of estimating |t(x) — sm| as x and m tend to ∞ with some connection between them was considered first by H.

Let us consider the following problem. Let there be v elements x1 , . . . , xv and v sets S1, . . . , Sv such that every set contains exactly k distinct elements and every pair of sets has exactly λ distinct elements in common. To avoid trivial situations we shall in general assume that 0 < λ < k < v — 1. This is known as a v, k, λconfiguration or design. We can give an equivalent characterization of a configuration in terms of a matrix A = [aij], called its incidence matrix, by writing the elements x1 , . . . , xv row and the sets S1, . . . , Sv in a column and setting aij = 1 if xj is in Si and aij = 0 if xj is not.

Since Beurling (1 ), the study of closed invariant subspaces of Hp(X) and Lp(X) on the unit circle X = {z:|z| = 1} has been done extensively and culminated into a very fine theory of generalized analytic functions ; cf. Hoffman (4), Srinivasan (7). Here we say that a space E of complex-valued functions on X is invariant if zE ⊆ E. Little is known, however, about the structure of closed invariant subspaces E of the space C(X) of continouus functions on X.

In the first part of this paper we shall consider associative rings, pointing out where associativity is required. In the second part we shall consider not necessarily associative rings and in particular alternative rings.

A property S of rings is said to be a radical property, in the sense of Kurosh (4), if it satisfies the following three conditions:

(a) Every homomorphic image of an S-ring (i.e. a ring with property S) is again an S-ring.

In (1), the writer defined a process of integration that leads to a kind of Riemann integral under certain rather general conditions. The purpose of this paper is to show how it is possible to use the process of integration of (1) to obtain integrals in a product space that satisfy a Fubini theorem. In this connection, we define a class of integrands that are the analogues of continuous functions in the product space, establish some of their properties, and finally arrive at a Fubini theorem for this class.