A few years ago, in a short paper (4) Ryser introduced an interesting topic in number theory, viz. the connection between integer matrices (i.e., matrices having only integers as their elements) satisfying certain conditions and 0-1 matrices (i.e., matrices that have no element different from 0 and 1). In this series of papers we shall pursue this topic further.

To make the statements of our theorems short we introduce some terminology. We need the definitions of certain 0-1 matrices related to a few well-known combinatorial configurations. By an incidence matrix of a balanced incomplete block (b.i.b. for conciseness) design we mean a 0-1 matrix with v rows and b columns, such that the sum of the elements in each column of A is k, k < v, and the scalar product of any two row vectors of A is λ, λ ≠ 0.

In this paper we establish a connection between rectangular integer matrices and incidence matrices of resolvable balanced incomplete block designs. The definition of these terms has been given in paper I of this series.

Our theorem can be stated as follows:

THEOREM 2. Let A be a v X b matrix with integer elements such that

2.1

In this paper, we give a connection between incidence matrices of affine resolvable balanced incomplete block designs and rectangular integer matrices subject to certain arithmetical conditions. The definition of these terms can be found in paper II of this series or in (2). For some necessary conditions on the parameters of affine resolvable balanced incomplete block designs and their properties see (2).

Let n1 and n2 be two elements of a commutative field of characteristic different from 2 that satisfy: (i) n1 ≠ 0; (ii) n2 ≠ 0; (iii) n1 + n2 ≠ 0. We define the “weighted average” α * β of two arbitrary elements α and β of as

Recently Bullen (1) developed a J3-integral that gives a Fourier representation for trigonometric series when both the series and its conjugate are summable (C, 1) everywhere. Earlier Cross (4) had shown that the C2P-integral (3) was strong enough to integrate such series and, indeed, that any trigonometric series for which the series and its conjugate were summable (C, k) were Fourier series where the integral used in the formula for the coefficients was the Ck+iP-integral (3). In (1) Bullen defines a J4-integral and says, without proof, that a Fourier formula in terms of this integral may be obtained for series for which the series and its conjugate are summable (C, 2) and the coefficients o(n).

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.

Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

Let N be a finite set of n elements. A collection ﹛S1, S2, … , Sm﹜ of subsets of N is called a determining collection if an arbitrary subset T of N is uniquely determined by the cardinalities of the intersections Si ⋂ T, 1 ≤ i ≤ m. The purpose of this paper is to study the minimum value D(n) of m for which a determining collection of m subsets exists.

This problem can be expressed as a coin-weighing problem (1; 7).

In a recent paper Cantor (1) showed that D(n) = O(n/log log n), thus proving a conjecture of N. J. Fine (3) that D(n) = o(n). More recently Erdös and Rényi (2), Söderberg and Shapiro (7), Berlekamp, Mills, and Leo Moser have independently found proofs that D(n) = O(n/log n).

In Perron integration, majorants are usually functions of points. If the domain of definition is a Euclidean space of n dimensions, we can define a finitely additive n-dimensional majorant rectangle function by taking suitable differences of the majorant point function with respect to each of the n coordinates. The way is then open to a generalization, in that we need only suppose that the majorant rectangle function is finitely superadditive. Similarly, we need only suppose that a minorant rectangle function is finitely subadditive. These kinds of rectangle functions were used by J. Mařík (5) to prove the Fubini theorem for Perron integrals in Euclidean space of m + n dimensions. He also proved that for a function that is Perron, and absolutely Perron, integrable, the majorant and minorant rectangle functions can be taken to be finitely additive. As a result he posed the following problem.

Generalized direct products with amalgamated subgroups were introduced by B. H. Neumann and Hanna Neumann in their joint paper (4). In general, we call a given collection of groups with specified subgroups amalgamated an amalgam of groups; if all groups are abelian we speak of an abelian amalgam. The group freely generated by the amalgam is called the abelian free sum of the amalgam provided it contains the amalgam isomorphically. The free abelian sum need not exist. Hence one of the problems is to find necessary and sufficient conditions for its existence.

Our aim is to prove that certain topologies have complements in the lattice of all the topologies on a given set. Lattices of topologies were studied in (1-8). In (7) Hartmanis points out that the lattice of all the topologies on a finite set is complemented and poses the question whether this is so if the set is infinite. A positive answer is given here for denumerable sets. This result was announced in (6). The case of higher powers remains unsettled, although quite a few topologies turn out to have complements. As far as the author knows, no one has proved the existence of a topology that has no complement.

The Bernstein polynomials

1

associated with a function f denned on [0, 1] have been the subject of much recent research and have been generalized in several directions (1 ; 2 ; 5). The generalized Lototsky or [F, dn] matrix (3) has also been the subject of extensive research.

The following problem was raised by M. Brown. Let K be a finite simplicial complex, of dimension n, with αi(K) simplexes of dimension i. Which of the linear combinations have the property that they are unaltered by all stellar subdivisions of K? The most obvious invariant is the Euler characteristic; there are also some identities that hold for manifolds (2), so, if K is a manifold, they remain true on subdivision. We shall see that no other expressions are ever invariant, but if K resembles a manifold in codimensions ⩽2r (in a sense defined below) that r of the relations continue to hold.

Ingham (2) has defined the following summation method. A series ∑ an will be called summable (I) to s if

where as usual [x] is the greatest integer ⩽ x. (An equivalent method was described somewhat earlier by Wintner (7), who called it “an Eratosthenian method“; however, the notation (I) and the name “Ingham summability” introduced by Hardy (1) seem to have become usual.)

Geometrically, a graph is a collection of points (or vertices) together with a set of edges (or curves) each of which joins two distinct vertices of the graph, and no two of which have points in common except possibly end points. Two given vertices of the graph may be joined by no edge or one edge, but may not be joined by more than one edge. From an abstract point of view, a graph G is a collection of elements ﹛x1, x2, …﹜ called points or vertices, together with a second collection of certain pairs (xα, Xβ) of distinct points of G. It is helpful to retain the geometric language, and refer to any pair in as an edge (or a curve) of G that joins the points xα and Xβ.

If x- (i = 1, 2, … , 6) are homogeneous co-ordinates in [5] (the complex projective space of five dimensions), then the equation

1.1

represents the well-known (11) Perazzo cubic primalP43 of order 3 and dimension 4. With it is associated the Segre cubic threefoldS33 (12); specifically, S33 is the section of P43 by a tangent [4].

If a quasigroup (L, .) has finite order n, then there are n2 principal loop-isotopes. Some of these n2 loops may be isomorphic, and the main purpose of this paper is to obtain theorems that describe the isomorphism classes. Using these results and a computer, we have determined all the loops of order 6. These are listed (using the Fisher and Yates (2) designations) at the end of the paper.

Let A = [aij] be a matrix of 0's and 1's or a (0, 1)-matrix of size m by m′. The term rank of A is denned as the maximal number of 1's of A with no two of the 1's on the same row or colunn. A theorem due to D. König (3, Theorem 5.1, p. 55) asserts that the term rank of A is also equal to the minimal number of rows and columns of A that collectively contain all the 1's. The term rank of A will be denoted by ρ(A). Obviously it is invariant under arbitrary permutations of the rows and columns of A. We assume without loss of generality that all matrices considered have no rows or columns consisting entirely of 0's.

Let be a finite-dimensional linear associative algebra over the real field R or the complex field C and let F be a function with domain and range in .

Several classes of functions on have been discussed in the literature, and it is the purpose of this paper to discuss the relationships between these classes and to present some interesting examples. First we shall list the definitions of the classes we wish to consider here.

Let

be a positive quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then f(x) assumes the value M for a finite number of integral vectors x = ±mk (k = 1 , … , s) called the minimal vectors.

Over a field of characteristic 0 certain of the simple Lie algebras have a root theory, namely those called “split” in Jacobson's book (3). We shall assume some familiarity with the subject matter of this book. Then the finite-dimensional representations of these Lie algebras have a weight theory. Our purpose here is to present a kind of weight theory for the representations of these Lie algebras when their ground field is the real numbers, and when the representation comes from a unitary group representation.