Let k and t be positive integers; let q1, q2, . . . , qt be distinct prime numbers; and let ζ1, ζ2, ... , ζt be kth roots of unity, not necessarily primitive. Recent investigations on consecutive kth power residues have led to the following question: Under what conditions do there exist primes p that have a kth power character X such that

If R is a sequentially continuous linear lattice, a function f(x), denned on R+ = {x:0 ≤ x ∊ R} with 0 ≤ f(x) ≤ + ∞, will be called a monotone function if it satisfies

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.

Let Fn(k) denote the total number of k-coloured graphs on n labelled nodes and let Mn(k) denote the number of graphs on n nodes that are coloured in at most kcolours ; also let fn(k) denote the number of connected k coloured graphs on n nodes. Read (3) has proved the following formulas:

Hall's algebra (3) is an algebra over the field V(p) of rational functions in the indeterminate p with coefficients in the field V of complex numbers. The basis of the algebra consists of elements Gλ which are in one-one correspondence with the set of all partitions (λ) and whose multiplication "constants" are the "Hall polynomials" , i-e.

Put

In a recent paper (4), Lohne showed that

For a given class of transformations on graphs there arises naturally the problem of deciding when one graph can be transformed into another by a finite sequence of such transformations. We consider, in this paper, the special case of this problem when the graphs are finite trees and the transformations consist of rearranging the order of the segments from a point and replacing subtrees by other trees according to a given set of pairs of interchangeable trees. This decision problem is, in fact, equivalent to the word problem for one-generator algebras in a certain variety of algebraic systems and we exhibit a procedure for solving the word problem for these algebras.

In what follows we prove that a finite graph of n nodes in which each node has degree ≥ 1 but ≤ possesses a set of at least n/(1 + ) pairwise disjoint edges. Our principal theorem states an analogue of this result for the case when each node has degree ≥2: we show that in this case the graph possesses a set of at least 2n/(2 + max(4, )) mutually disjoint edges.

In the series of "Census" papers, of which this is the fourth, we attempt to lay the groundwork of an enumerative theory of planar maps (12, 13, 14). The maps concerned are rooted in the sense that some edge is fixed as the root, and a positive sense of description and right and left sides are specified for it. This device simplifies the theory by ruling out the possibility of a map being symmetrical.

In his fine and stimulating book Introduction to geometry (1), Coxeter considers three mutually tangent circles C1 , C2, and C3 (Fig. 1) and calculates the radii of the two circles E1 and E2 that touch all these three. If the centres of Ci are the vertices of a triangle ABC their radii are s — a, s — b, and s — c; α, β, and γ are defined by

Let denote the set of points x = (x1, . . . , xn) with integral co-ordinates in Euclidean n-space. For any fixed integer m ≥ 2, let C = C(m) be the set of such points in the cube 0 ≤ xi < m (i = 1, 2, . . . , n) and let be any subset of C. Suppose that f(x) is any single-valued, integral-valued function, defined for all x ∊ . We consider solutions x ∊ of the congruence

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).

The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.

Given a set E of v elements, and given positive integers k, l (l ≤ k ≤ v), and λ, we understand by a tactical configurationC[k, l, λ, v] (briefly, configuration) a system of subsets of E, having k elements each, such that every subset of E having l elements is contained in exactly λ sets of the system.

A necessary condition for the existence of a configuration C[k, l, λ, v] is known (6) to be

Let A be a simple ring with minimum condition, and B1, B2, and C be regular subrings of A such that Bi > C, i = 1, 2. A pair of isomorphisms σi of Bi into A such that σi|C is the identity, and that Biσi are regular subrings of A (i = 1, 2), is called compatible if σ1|B1 ∩ B2 = σ2|B1 ∩ B2. Here σ|X means the restriction of σ to X. Bialynicki-Birula has proved some necessary and sufficient conditions that every compatible pair (σ1, σ2) has a common extention to an automorphism σ of A (1 ). When A is a division ring, he shows that the linear disjointness of the division subrings B1 and B2 is necessary and almost sufficient for the existence of a common extension of any compatible pair.

In an earlier paper (2) reflexive transitive binary relations were considered on a connected ordered space. These relations were topologically restricted and their minimal sets were either an end point of the space or empty. It was shown that these relations could be characterized as one of the two orders of the space. Viewing the situation somewhat differently as suggested by I. S. Krule, one could say that this class of relations was characterized in terms of the identity function on the space. In this case the relations are considered in their natural setting, the product of the space with itself.

In (2), Tutte has shown that the number, Bn, of rooted non-separable planar maps having n edges is [2(3n — 3)!]/[n! (2n — 1)!]. Rooting was accomplished by designating one edge as the root, orienting it, and distinguishing between its sides as left and right. We shall here compute the number, Bn,m, of rooted non-separable planar maps having n edges and such that the face to the left of the root is incident with exactly m edges, which maps will be said to be of type [n, m].

The following elementary facts about certain commutative diagrams, called "squares," are stated and proved in terms of abelian groups and their homomorphisms. However, they are valid for arbitrary abelian categories and can be proved also for them. This does not need to be shown, since every abelian category can be embedded into the category of abelian groups with preservation of exact sequences according to a result due to S. Lubkin (1). Proofs are often omitted or given only for one half of a theorem, the other half being dual to the first. Generalizations to a larger class of diagrams containing all finite commutative diagrams are possible.

Let Un be an n-dimensional unitary space with inner product (u, v). For vectors u1, . . . , ur ∊ Un, r ≤ n, let denote the Grassmann exterior product (4) of the ut; it is a vector in Um where m = nCr. If also , then is the determinant of the r X r matrix If A is a linear transformation of Un to itself, the rth compound of A is defined by

In a recent paper, Olver (2) obtains a set of formulae that completely determine the asymptotic behaviour of the Hermite polynomials, Hn(z), as n —> ∞ and z is unrestricted. His proof depends on a technique that he has developed for discussing the asymptotics of solutions of second-order, linear, homogeneous differential equations satisfying certain conditions. We believe it fair to say that Olver's work follows the tradition of most of the major theorems of classical asymptotics. The results contained in theorems such as Watson's lemma and Perron's proof of the Method of Laplace are based on an acceptance, on an a priori basis, of the Poincaré type expansion.

Let G be a finite group and R a Dedekind domain with quotient field K. We denote by RG the group ring of formal linear combinations of elements of G with coefficients in R. By an RG-module we understand a unital left RG-module which is finitely generated and torsion-free as R-module. In particular, if R is a principal ideal domain this is equivalent to considering representations of G by matrices with entries in R.