In (5) Hammer and Sobczyk defined an outwardly simple line family in the plane as a family of lines in the plane having the property that each point outside some given circle lies on exactly one line of the family, and they characterized planar outwardly simple line families as follows (5): the extended diameters of a convex body, whose boundary has no pair of parallel line segments in it, form an outwardly simple line family; moreover, each outwardly simple line family is the family of extended diameters of a convex body having constant width.

The following problem is as yet unsolved: Given a convex polytope with N vertices in n-space, what is the maximum number of (n — 1)-faces which it can have? Aside from its geometric interest this question arises in connection with solving systems of linear inequalities and linear equations in non-negative variables. The problem is equivalent to asking for the best bound on the number of basic solutions for such problems and hence a bound (though a weak one) for the number of iterations needed in the simplex method for solving linear programmes.

Differentiation of a set function μ with respect to another v at a point x involves taking the limit of the ratio μA/vA as A "converges" to x in some sense which requires that A belong to some family of sets N(x) (e.g. spheres with centre x). For the development of a reasonable theory of differentiation certain restrictions must be placed on the families N(x). The best-known restriction is that they form a Vitali system. However, other systems have been considered.

We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.

In a recent article on Moulton planes (8), I have generalized the non-Desarguesian planes introduced by F. R. Moulton (6) and Pickert (7, pp. 93-94). Let F = (0, 1; a, b, . . . , x, y, . . .} denote a given field, having P as a multiplicative subgroup of index 2. Define x > 0 or x < 0 according as x lies in P or in the other coset of non-zero elements. The function ϕ is order-preserving, and can be assumed to satisfy ϕ(0) = 0, ϕ(1) = 1. For any n < 0, the maps x —> ϕ(x) and x —> ϕ(x) — nx both carry F onto itself. The elements of F form a Cartesian group, {+,0} being defined so that

This paper discusses projective planes from the viewpoint of their classification into singly-generated and non-singly-generated planes. (Singly-generatedness, a property explicit in Hall (6) and Wagner (19), and implicit in Hughes (10), is defined in Section 5.) The elements (points and lines) of a singly-generated plane are expressible in four basic points called a quadrangle. A t'completion procedure" enables us to obtain expressions for the elements of a plane.

In § 1 the existence of two non-isotopic loops satisfying the same system of relations is proved: C1 is isomorphic to the well-known Cayley loop (see 1 or 6); C2 seems not to have been mentioned in the literature. In § 2 a characterization of the second type is given in terms of Tamari's generalized normal multiplication table (7; 8) as a complete symmetric quasiregular partition, i.e., a quasi-associative loop (4, 5), or a so-called near-group (9). In § 3 these distinctions are followed by comparing the corresponding algebras.

If A = (aij) is a matrix whose elements are 0's and l's, more briefly a (0, l)-matrix, the trace of A, defined as the sum of the diagonal elements aii and denoted by Tr A, is the number of l's on the main diagonal. Matrices which can be obtained from A by permutation of its rows or of its columns, or both, may be expressed as PA, AQ, or PAQ, respectively, where P and Q are permutation matrices.

Let n be a positive integer and put N = {1, 2, . . . , n}. A collection {S1, S2, . . . , St} of subsets of N is called determining if, for any T ⊂ N, the cardinalities of the t intersections T ∩ Sj determine T uniquely. Let €1, €2, . . . , €n be n variables with range {0, 1}. It is clear that a determining collection {Sj) has the property that the sums

Let α(dx) be a finite measure defined on the Borel subsets of [—1, 1], the spectrum of which is infinite. Let be the family of orthonormal polynomials associated with a, so that

The are uniquely determined by this and by the condition

All the matrices considered in this paper have their elements in the field of residues mod 2.

Two non-singular matrices are equivalent if each row of either matrix is a linear combination of rows of the other. The matrices then have equal numbers of rows and equal numbers of columns.

A nodal matrix is a non-singular matrix in which no column has more than two l's. A graphic matrix is a non-singular matrix equivalent to a nodal matrix.

As in my earlier paper (2), a graph on n labelled nodes is a set of n objects called "nodes" distinguishable from each other and a set (possibly empty) of "edges," i.e. pairs of distinct nodes. Each edge is said to join its pair of nodes and no edge joins a node to itself. By a k-colouring of the nodes of such a graph we mean a mapping of the nodes onto a set of k distinct colours, such that no two nodes joined by an edge are mapped onto the same colour. By a colouring of the edges of such a graph we mean a mapping of the edges onto a set of colours.

A construction of the free lattice generated by a partially ordered set P and preserving every least upper bound (lub) and greatest lower bound (glb) of pairs of elements existing in P has been given by Dilworth (2, pp. 124-129) and, when P is finite, by Gluhov (5).

The results presented here construct the free lattice FL generated by the partially ordered set P and preserving

(1) the ordering of P

(2) those lub's of a family of finite subsets of P which possess lub's in P

Let A be a complex Banach algebra and Lr (Ll) be the lattice of all closed right (left) ideals in A. Following Tomiuk (5), we say that A is a right complemented algebra if there exists a mapping I —> IP of Lτ into Lr such that if I ∊ Lr, then I ∩ Ip = (0), (Ip)p = I, I ⴲ Ip = A and if I1, I2 ∊ Lr with I1 ⊆ I2 then .

If in a Banach algebra A every proper closed right ideal has a non-zero left annihilator, then A is called a left annihilator algebra. If, in addition, the corresponding statement holds for every proper closed left ideal and r(A) = (0) = l(A), A is called an annihilator algebra (1).

Shapiro and Warga (2) have proved in an elementary way that all large integers are expressible as the sum of at most 20 primes. In so doing, they proved that

as x —> ∞, where u is a positive square-free integer,

Metric spaces in which the distances are not real numbers have been studied by several people (2, 3, 4, 7, 9). Any ring R together with a mapping, X —> ϕ(X), of R into a lattice A with 0 and 1 satisfying

is called a "lattice-valued ring," where the operations union, ∪, and intersection, ∩, are the usual lattice operations. The mapping ϕ is called a "valuation" and A is a 'Valuation lattice." If R is a lattice-valued ring and a mapping d is defined by

In the following, V is a vector space over an arbitrary field F, dimFV = n. Let {e1, . . . , en} be a basis for V, and {f1, . . . , fn} be the dual basis for and , then the operators *(u) and i(g) (exterior and inner multiplication by u and g respectively) set up an equivalence between the ideal = range of ∊(u) and the sub-algebra = range of i(g) considered as vector spaces. That is, ∊(u)i(g) is the identity on , i(g) ∊(u) is the identity on .

This paper follows naturally a note on parabolic differentiation (2) in which the parabolically differentiable points in the real affine plane were discussed. In the parabolic case, the four-parameter family of parabolas in the affine plane led to three differentiability conditions. In the present paper, the five-parameter family of conies in the real projective plane gives rise to four differentiability conditions and a point of an arc in the projective plane is called conically differentiable if these four conditions are satisfied. The differentiable points are classified by the nature of their families of osculating conies, superosculating conies, and their ultraosculating conies.

The purpose of this paper is to clarify and sharpen the argument in the last two chapters of the author's Representation theory of the symmetric group(3). When these chapters were written the peculiar properties of the case p = 2 were not fully appreciated. No difficulty arises in the definition of the block in terms of the p-core, or in the application of the general modular theory based on the formula

The purpose of this paper is to develop characterizations of weakly compact subsets of a Banach space in terms of separation properties. The sets A and B are said to be separated by a hyperplane H if A is contained in one of the two closed half-spaces determined by H, and B is contained in the other; A and B are strictly separated by H if A is contained in one of the two open half-spaces determined by H, and B is contained in the other. The following are known to be true for locally convex topological linear spaces.