The number of self-complementary graphs and the number of self-complementary digraphs were expressed by Read [4] in terms of cycle indexes of the appropriate pair groups. These formulas for and , together with a modification of the method employed by Oberschelp [3] for graphs, can be used to obtain estimates for and and a bound on the error. For graph theoretic definitions not given here, we refer to the book [2].

Let Γ be a finite group of order n and let K be a field whose characteristic does does not divide n. The group ring K(Γ) is then an involution algebra if we define for y є Γ and extend by linearity, so that ¯ is trivial on K. A subalgebra T of K(Γ) is said to be an S-ring on Γ (see [4]) if there exists a decomposition

of Γ into non-empty, pairwise disjoint subsets Fi with the properties that the elements of K(Γ) form a K-basis of T and that for each τi there exists a τj such that .

It is not difficult to construct an unbounded set E on the positive real line such that, if x1, x2 belong to E, then x1/x2 is never equal to an integer. Our object is to show that it is possible to find such a set E which is measurable and of infinite Lebesgue measure. We were led to consider this problem through a study of those sets E, which are of infinite measure, yet, for each x > 0, nx є E for only a finite number of integers n. Sets of this type were first discovered by C. G. Lekkerkerker [2]. The set that we consider has both these properties. For another result on lattice points in sets of infinite measure see [1].

Let P be a d-dimensional convex polytope (briefly, a d-polytope) in d-dimensional euclidean space Ed. Associated with P is a vector f(P), known as the f-vector of P, defined by

where fj(P) is the number of j-faces of P for 0 ≤ j ≤ d − 1 and the superscript T denotes transposition. (We regard f(P) as a column vector, and identify it with the vector

where (e1, …, ed) is some fixed basis of Ed.) Let d be the set of all d-polytopes in Ed, and d be any subset of d. Using tghe notation of [1; §8.1], we donate by f(d) the set of vectors {f(P): P ε d}, and write aff f(d) for the (unique) affine subspace of lowest dimension in Ed which contains all the vectors of f(d). Then it is well-known that

the equation of the hyperplane aff f(d) being that given by the Euler relation between the numbers fj(P) [1; Theorem 8.1.1].

§0. We begin by stating a theorem of Jarník [1]. Let m be a positive integer, 0 < β < m, and Ψ(q) a positive function of integers q > 0. Let E be the set of points (x1, …, xm) for which the inequality

admits infinitely many solutions q ≥ 1. Then, supposing

Jarník proves that E has infinite Hausdorff β-measure.

It is well-known and easy to prove that the maximal line segments on the boundary of a convex domain in the plane are countable. T. J. McMinn [1] has shown that the end-points of the unit vectors drawn from the origin in the directions of the line segments lying on the surface of a convex body in 3-dimensional Euclidean space E3 form a set of σ-finite linear Hausdorff measure on the 2-dimensional surface of the unit ball. A. S. Besicovitch [2] has given a simpler proof of McMinn's result. W. D. Pepe, in a paper to appear in the Proc. Amer. Math. Soc., has extended the result to E4. In this paper we generalize McMinn's result to En by use of Besicovitch's method, proving:

THEOREM 1. If K is a convex body in En, the set S, of end-points of the vectors drawn from origin in the directions of the line segments lying on the surface of K, is a set of σ-finite (n − 2)-dimensional Hausdorff measure on the (n − 1)-dimensional surface of the unit ball.

If D is an integral domain with quotient field K, then by an overring of D we shall mean a ring D′ such that D ⊂ D′ ⊂ K. Gilmer and Ohm in [GO], Davis in [D] and Pendleton in [P] have studied the class of integral domains D having the property that each overring D′ of D is of the form Ds for some multiplicative system S of D. In [D] and [P] a domain with this property is called a Q-domain and in [GO] such a domain is said to have the QR-property. Slightly altering the terminology of [D] and [P], we shall say “QR-domain” instead of “Q-domain”. Noetherian QR-domains are precisely the Dedekind domains having torsion class group [GO; p. 97] or [D; p. 200]. Pendleton in [P; p. 500] classifies QR-domains as Prüfer domains satisfying the additional condition that the radical of each finitely generated ideal is the radical of a principal ideal. It is pointed out in [P; p. 500] that this characterization of QR-domains still leaves unresolved the question of whether the class group of a QR-domain is necessarily a torsion group. We show in §1 that a QR-domain need not have torsion class group. Our construction is direct; however, the problem can be viewed in terms of the realization of certain ordered abelian groups as divisibility groups of Prüfer domains, and we conclude §1 with a brief discussion of this approach to the problem.

Let E be a real normed linear space. Let K be a closed convex set containing 0, the origin, as an extreme point. Let A be a linear operator with AK ⊆ K. Stated below are theorems concerning eigenvectors and spectral (partial spectral) radius of A which generalize the well-known theorems of Bonsall [3] and Krein and Rutman [7] on positive operators. Proofs are given in §2.

In a recent article [3] L. Mirsky proved a theorem which gives necessary and sufficient conditions for the existence of a finite integral matrix whose elements, row sums, and column sums all lie within prescribed bounds. Mirsky suggested to me the problem of extending his theorem to infinite matrices, and it is the solution of this problem that is presented in this note. To allow for extra generality, instead of prescribing upper and lower bounds for the row and column sums we shall prescribe upper and lower bounds for the row and column deficiencies (a term to be explained later). The theorem when upper and lower bounds for the row and column sums are prescribed is then a special case of the deficiency theorem. The solution of the problem depends on a construction of Mirsky [3] and a theorem of mine [1] concerning the existence of a partial transversal of a family of sets satisfying certain properties. As will be seen, we shall take a rather broad view of the notion of a matrix.

When the Helmholz equation

is separated in ellipsoidal coordinates [1; §1.6] and the technique of separation of variables applied, there results the ordinary differential equation

known as the ellipsoidal wave equation or Lamé wave equation. In this equation k is the modulus of the Jacobian elliptic function sn z, and is related to the eccentricity of the fundamental ellipse of the ellipsoidal coordinates; a, b are separation constants, and the parameter q is connected with the wave number χ by

l being a real constant, the dimensional parameter of the coordinate system.