Given a set E of v elements and given positive integers k (k ≤ v) and λ, we understand by balanced incomplete block design (BIBD) B [k, λ, v] a system of blocks (subsets of E) having k elements each such that every pair of elements of E is contained in exactly λ blocks.

A necessary condition for the existence of a design B [k, λ, v] is known to be (4)

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.

In 1945, L. I. Kopejkina (4), at the suggestion of A. G. Kurosh, began a programme of studying the properties of free projective planes and the analogies between free planes and free groups. In this paper, this study will be extended by proving a tool theorem and several of its consequences. The theorem deals with the existence of "minimal free generators" for subplanes of free planes.

A set of points and lines and an incidence relation are said to form a projective plane if the following three axioms are satisfied.

In Bernstein's proof of the Weierstrass Approximation Theorem, the polynomials

are constructed in correspondence with a function f ∊ C [0, 1] and are shown to converge uniformly to f. These Bernstein polynomials have been the starting point of many investigations, and a number of generalizations of them have appeared. It is our purpose here to consider several generalizations in the form of infinite series and to establish some of their properties.

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,

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)

The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation

The letter k denotes an infinite cardinal. A space is a compact Hausdorff space unless otherwise indicated. A space is called extremally disconnected (k-extremally disconnected) if it is the Stone space for a complete (k-complete) Boolean algebra. A map is a continuous function from one space into another. A map f:X —> Y is called minimal if f is onto, but f(M) is properly contained in Y for each closed proper subset M of X. A space F is called free if F has a dense subset X such that every space-valued function on X extends to a map on all of F or, equivalently, if F is the Stone-Cech compactification of some discrete topological space X.

The distance between two vertices of a connected finite graph is the smallest number of edges forming a path that joins the two vertices, and the diameter of the graph is the largest integer which is realized as the distance between two vertices of the graph. We are concerned here with the diameters of two graphs associated with a d-dimensional convex polytope P (called henceforth a d-polytope). The graph Γ(P) of P is the 1- complex formed by the vertices and edges of P , and the polar graph II (P) of P is the 1-complex whose vertices correspond to the (d — 1)-faces of P, with two vertices joined by an edge in II (P) if and only if the corresponding (d — 1)- faces intersect in a (d — 2)-face of P. The diameters of Γ(P) and II (P) will be denoted respectively by δ(P) (called the diameter of P) and ϕ(Ps) (called the face-diameter of P ).

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .

Let π be the Desarguesian affine plane of order n = pr, for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags, of π. Further, G is doubly transitive on π if G is doubly transitive on the points of π. Clearly, G is flag transitive if G is doubly transitive on π.

The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).

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.

The set of all adjacency-preserving automorphisms of the vertex set of a graph form a group which is called the (automorphism) group of the graph. In 1938 Frucht (2) showed that every finite group is isomorphic to the group of some graph. Since then Frucht, Izbicki, and Sabidussi have considered various other properties that a graph having a given group may possess. (For pertinent references and definitions not given here see Ore (4).) The object in this paper is to treat by similar methods a corresponding problem for a class of oriented graphs. It will be shown that a finite group is isomorphic to the group of some complete oriented graph if and only if it has an odd number of elements.

Let K denote a connected finite 1-dimensional cell complex (1, p. 95), G(K) its group of homeomorphisms, and D(K) the group of homeomorphisms of K which are isotopic to the identity. The group (K) = G(K)/D(K) is a topological invariant of K and is called the homeotopy group ofK (4). K may be thought of as a linear graph (connected finite 1- dimensional simplicial complex) extended to admit loops and multiple edges and (K) as the topological analogue of the automorphism group A(L), (the permutations of vertices which preserve edge incidence relations) of a linear graph L.

A well-known consequence of the König theorem on maximum matchings and minimum covers in bipartite graphs (5) or of the P. Hall theorem on systems of distinct representatives for sets (4) asserts that an n by n (0, 1)-matrix A having precisely p ones in each row and column can be written as a sum of p permutation matrices:

Our main objective is a generalization of (1.1) along the following lines. Let A be an arbitrary m by n (0, 1)-matrix. Call an m by n (0, 1)-matrix P a permutation matrix if PPT = I, where PT is the transpose of P and I is the identity matrix of order m.

Let F1(x) and F2(x) be two distribution functions, that is, non-decreasing, right-continuous functions such that Fj(— ∞) = 0 and Fj(+ ∞) = 1 (j = 1, 2). We denote their convolution by F(x) so that

the above integrals being defined as the Lebesgue-Stieltjes integrals. Then it is easy to verify (2, p. 189) that F(x) is a distribution function. Let f1(t), f2(t), and f(t) be the corresponding characteristic functions, that is,

The purpose of this paper is to prove the following theorem:

Theorem. Given a convex curve C, of perimeter length I and three points M, N, and P which divide the perimeter of C into three parts of equal length, the perimeter length of the triangle MNP is never less than ½l. Equality holds only in the case where C is an equilateral triangle and M, N, and P are the mid-points of the three sides.

Let < be a non-reflexive partial ordering defined on a set P. Let G(P, <) be the undirected graph whose vertices are the elements of P, and whose edges (a, b) connect vertices for which either a < b or b < a. A graph G with vertices P for which there exists a partial ordering < such that G = G(P, <) is called a comparability graph.

In §2 we state and prove a characterization of those graphs, finite or infinite, which are comparability graphs. Another proof of the same characterization has been given in (2), and a related question examined in (6). Our proof of the sufficiency of the characterization yields a very simple algorithm for directing all the edges of a comparability graph in such a way that the resulting graph partially orders its vertices.

It is well known that for finite connected graphs the following are equivalent:

1. (i) X is Euler (i.e., every vertex of X has positive even degree);

2. (ii) X is traceable (i.e., the edges of X can be arranged in a sequence e1, . . . ,en such that ei ≠ ej if i ≠ j, and ei, ei+1 are adjacent, i = 1, . . . , n, subscripts considered mod n) ;

3. (iii) X is cyclically coverable (i.e., X contains a family of non-overlapping circuits whose union is X).

This paper is concerned with some of the properties of arcs in the real affine plane which are met by every parabola at not more than four points. Many of the properties of arcs of parabolic order four which we consider here are analogous to the corresponding properties of arcs of cyclic order three in the conformai plane which are described in (1). The paper (2), on parabolic differentiation, provides the background for the present discussion.

In Section 2, general tangent, osculating, and superosculating parabolas are introduced. The concept of strong differentiability is introduced in Section 3; cf. Theorem 1. Section 4 deals with arcs of finite parabolic order, and it is proved (Theorem 2) that an end point p of an arc A of finite parabolic order is twice parabolically differentiable.

Lower case italics will denote rational integers, while lower case Greek letters will denote algebraic integers. The Law of Quadratic Reciprocity can be formulated as follows:

If p and a are distinct odd primes, then a is a quadratic residue (mod p) if and only if ( — 1)(p-1)/2p is a quadratic residue (mod q).

Let Xp denote the non-principal quadratic character (mod p) and let ζp be a primitive pth. root of unity. Then