The pair (S, M) is a matroid if S is a finite set and M a collection of subsets of S such that (1) every subset of a set of M is in M, and (2) all maximal sets in M have a common cardinality. The span of a set A ⊂ S is Γ(A) where y ∈ Γ (A) if and only if y ∈ A or there is A' ⊂ A, A' ∈ M and {y} ∪ A' ∉ M. A maximal set in M is called a base.

All homology and cohomology groups are reduced. is the class of finite abelian groups whose orders are prime to p. is the class of abelian groups whose orders are products of primes less than a. If G is a finitely generated abelian group, Gp is the quotient of G by the subgroup of G made up of all elements whose orders are prime to p.

For a finite abelian group G let λ(G) be the least positive integer such that λ(G)G = 0. Let be the least integer such that λ(G) | (λ(G) divides ) and if 2 | λ(G) then 4 | . For a finitely generated abelian group G let GT be the subgroup of G made up of all elements of G of finite order, and let GF = G/GT. For a simply-connected C-W complex X, let be the smallest class of abelian groups containing the groups .

The densest lattice packings of equal spheres in Euclidean spaces En of n dimensions are known for n ⩽ 8. However, it is not known for any n ⩾ 3 whether there can be any non-lattice sphere packing with density exceeding that of the densest lattice packing. W. Barlow described [1] a non-lattice packing in E3 with the same density as the densest lattice packing, and I described [6] three non-lattice packings in E5 which also have this property.

We consider ordinary graphs, that is, finite, undirected graphs with no loops or multiple lines. The product (also called cartesian product [4]) G1 × G2 of two graphs G1 and G2 with point sets V1 and V2, respectively, has the cartesian product V1 × V2 as its set of points. Two points (u1, u2) and (v1, v2) are adjacent if u1 = v1 and u2 is adjacent with v2 or u2 = v2 and u1 is adjacent with v1.

Let us call an equational class (variety) K of algebras permutable if and only if every pair of congruences on each K-algebra is permutable. Similarly, we will call K modular (distributive) if the congruence lattice of each K-algebra is modular (distributive).

We assume F is a purely inseparable field extension of the field K. The characteristic of K is p ≠ 0, and we assume F and K are not perfect. For x ∈ F, the exponent of x over K is the smallest

non-negative integer e such that and will be denoted by

e (x); ⨱ will denote . For any subset S of F, e(x; S) will denote the exponent of x over K(S); in case S = {y} we will write e(x; y) for e(x; S).

Let an open, quasi-compact subscheme of an affine scheme be called subaffine. This note will centre on an elementary characterization of such schemes in terms of their topology and global sections. Thence one can obtain simplifications and generalizations of some well-known theorems, such as Serre's Criterion [2, Thm. 1].

T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.

I is an interval of R, the set of real numbers, n is a positive integer and F ⊂ Cj (I) for j ≥ 0 large enough so that the following definitions are possible:

(i) Let λ(n) = (λ1, λ2,…,λk) where k, λ1, λ2,…, λk, are positive integers and λ1 + λ2 +… +λk = n. Then λ(n) is an ordered partition of n. The set of all such partitions of n is denoted by P(n).

In an earlier work [12] we considered the case of (0, 2, 3) interpolation by trigonometric polynomials at the points , i = 0, 1,…, n-1. By (0, 2, 3) interpolation we mean the problem of finding a trigonometric polynomial of suitable order whose values, second and third derivatives are prescribed at some given points. An interesting distinction between the (0, 2) interpolation studied by the Hungarian mathematician O.

In this note we give a brief exposition of the mathematical foundations of the theory of spin for both classical and quantum mechanical systems on oriented Riemannian manifolds. We shall use freely the notations and theory developed in Abraham [1] and Marsden [2, 3], From the physical point of view nothing new appears. The whole purpose of the note is to explain how the theory fits in the spirit of [1].

As is well known, there is an intimate connection between geodesic flows and Hamiltonian systems. In fact, if g is a Riemannian, or pseudo-Riemannian metric on a manifold M (we think of M as q-space or the configuration space), we may define a smooth function Tg on the cotangent bundle T*M (q-p-space, or the phase space). This function is the kinetic energy of q, and locally is given by

Let Sn, r be the semigroup generated by two elements a and b with two defining relations

(1)

By symmetry, we may assume that n ≤ r. In [l] and [2] it was shown that the semigroups

and

Let be the lower central series of a group G, where K2(G) = [G, G] and inductively Kn+1(G) = [Kn (G), G]. A theorem of Blackburn ([1], Hilfssatz) states that

Theorem 1. The exponent of Kn+1 (G)/Kn+2 (G) divides the exponent of Kn (G)/Kn+1 (G).

Let d be a positive integer and let p be a prime > d. Set q = pm, where m ≥ 1, and let I (q, d) denote the number of distinct primary irreducible polynomials of degree d over GF(q). It is a simple deduction from the well-known expression for I(q, d) that

(1)

where d* is the largest positive integer < d which divides d if d > 1, and d* is 0 if d = 1.

In [1] Cutler proved the following theorem.

Theorem. If G and K are abelian groups such that nG ≅ nK for some positive integer n, then there are abelian groups U and V such that U ⊕ G ≅ V ⊕ K and nU = 0 = nV.

Cutler's proof is long and fairly involved. Walker [3] obtains the theorem rather elegantly as a corollary of his results on n-extensions. We give here a proof that is extremely simple both in conception and execution. Our proof relies on the notion of p-basic subgroups introduced by Fuchs in [2]. Therefore we shall first recall certain pertinent facts from [2].

In a recent note [3] in this department C. Clark has shown that Rellich's theorem on the compactness of the imbedding is valid if G is the "spiny urchin" domain obtained by removing from the plane the union of the sets Sk (k = 1, 2,…) defined in polar coordinates by

Problem 7 of Section VI of H. T. Croft's "Research Problems" (August, 1967 edition) is by J. H. Conway:

A is a finite set of integers {ai}. A + A denotes {ai + aj}, A - A denotes {ai - aj}. Prove that A - A always has more members than A + A, unless A is symmetrical about 0.

First note that 'about 0' is redundant and erroneous.

The following question has been asked by J. Dénes [2]: If n - 1 elements of the diagonal of an n × n array are prescribed, is it possible to complete the array to form an n × n latin square?" It is known that if n diagonal elements are given such a completion is not always possible.

That the answer to Dénes' question is yes follows directly from a theorem of M. Hall Jr. [1].