The main object of this paper is to find the number of classes in a genus of indefinite quadratic forms, with integral coefficients, in k ≥ 4 variables, distinguishing for even k two cases, according as improper equivalence is or is not admitted.

If n is a non-negative integer, define pr(n) as the coefficient of xn in

;

otherwise define pr(n) as 0. In a recent paper (2) the author established the following congruence:

Let r = 4, 6, 8, 10, 14, 26. Let p be a prime greater than 3 such that r(p + l) / 24 is an integer, and set Δ = r(p2 − l)/24.

A matrix P with elements from an arbitrary field is called a cogredient automorph (c.a.) of a symmetric matrix A if P'AP = A, where P' is the transpose of P.

This paper is primarily a brief elaboration on the axioms for a bicategory introduced in (3). From this point of view, the main aim is the development of the structure of certain systems of topological and uniform spaces, and the present paper merely points out some very general properties which follow from axioms so weak that they are satisfied by any system likely to be considered.

Let (X, ≤) be a partially ordered set, that is, X is a set and ≤ is a reflexive, anti-symmetric, transitive, binary relation on X.

We write

,

for each x ∈ X. If, moreover,

exists for each x and y in X, then (X, ≤) is said to be a semi-lattice.

A well-known theorem of Jacobson (1) asserts that if every element a of a ring A satisfies a relation an(a) = a where n(a) > 1 is an integer, then A is a commutative ring. Thus the condition used in Jacobson's theorem is a sufficient condition for commutativity. However the condition is by no means a necessary one, as it is satisfied by a very restricted class of commutative rings.

Let G be a group which has a faithful representation as a transitive permutation group on m letters in which no permutation other than the identity leaves two letters unaltered, and there is at least one permutation leaving exactly one letter fixed. It is easily seen that if G has order mh, a necessary and sufficient condition for G to have such a representation is that G contains a subgroup H of order h which is its own normalizer in G and is disjoint from all its conjugates. Such a group G is called a Frobenius group of type (h, m).

Let o be an integral domain with as field of quotients. W. Krull has shown (3 ; 4) that the following three conditions on o are equivalent:

There is a set of rank 1, discrete valuations of such that for each non-null element for all except a finite number, and such that for all if and only if for all i ∊ I.