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].

1. If w(mod 2ω1, 2ω2) is an elliptic parameter for points of a normal elliptic curve C = 1Cn[n − 1], then it is well known that the sets of n points in which C is met by primes have a constant parameter sum k (mod 2ωl, 2ω2), and we may express this for convenience by saying that k is the prime parameter sum for the parametrisation of C by w. If we take the origin of w (the point for which w ≡ 0) to be one of the points of hyperosculation of C, then k ≡ 0, and we may say that w is a normal parameter for C. In the same way, if Γ is the Grassmannian image curve of the generators of a normal elliptic scroll 1R2n[n − 1], then a normal parametrisation of Γ defines a normal parameter w for the generators of 1R2n, such that n of the generators have parameter sum zero if and only if they belong to a linear line-complex not containing all the generators of 1R2n; or, in particular, if they all meet a space [n – 3] that is not met by every generator of the scroll. In this paper we are concerned in the first instance with the type of normal elliptic scroll 1R22m+1[2m] whose points can be represented by the unordered pairs (u1; u2) of values of an elliptic parameter u(mod 2ω1, 2ω2); and we establish a significant connection between any normal parametrisation of the generators of 1R2m+1 and an associated parametric representation (u1u2) of its points. We also add a brief note to indicate the lines along which this kind of connection can be extended to apply to a general normal elliptic scrollar variety 1Rkmk+1[mk] whose points can be represented by the unordered sets (u1, …, uk) of values of an elliptic parameter u.

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.

A general convex body in Euclidean space can be approximated by smooth convex bodies, and many results, arising first in the differential geometry of smooth convex bodies, have been extended to yield corresponding results for general convex bodies. Although convex bodies can be approximated by convex polyhedra, very little of the rich theory of convex polyhedra has been extended to general convex bodies. In this paper, we extend the concept of the one-skeleton of a convex polytope to yield the concept of the one-skeleton of a general convex body. We investigate the connectivity properties of this one-skeleton, and we extend a result of Balinski [1], on paths in the one-skeleton of a convex polytope to the class of convex bodies.

All of our work takes place in Ed, d-dimensional Euclidean space, with unit ball B. Unless specifically noted to the contrary, all sets will be presumed to be closed and convex. If X is a subset of a sphere S with centre p, we will say X is spherically convex if X is contained in some open hemisphere of S and if the cone generated by X with vertex p is convex. The distance between two points x, y ε Ed will be denoted |x − y|. If K1K2 are two convex sets, ρ(K1, K2) will mean the usual Hausdorff distance between them.

Various methods have been developed for solutions of boundary value problems involving discs of finite radius and spherical caps. A recent account of this work is described in the book by Sneddon [1]. In the present paper a simple method is presented for the solutions of potential problems for the electrified disc and spherical cap by reducing the axially symmetric boundary value problems to a corresponding problems for the two-dimensional Laplace equation. The essence of the method is to employ integral operators which map two-dimensional harmonic functions into axially symmetric potentials and are closely related to the integral transformations given in [3]. In particular it is shown how the mixed boundary value problems for the disc and spherical cap are mapped into Dirichlet problems for the two-dimensional Laplace equation in the half plane and interior of the unit circle respectively. In both cases a standard Green's function approach is applied to determine the solution of the two-dimensional problem. Williams [2] demonstrated how the potential problem for the lens can be found using a similar method. It is noted that Rostovtsev [5] Mossakovskii [4] and Heins [7] have used techniques similar to that presented in this paper.

The study of linear stability of a layer of stratified fluid in horizontal shearing motion leads, in the absence of diffusive effects, to a second order differential equation, often called the Taylor-Goldstein equation. This equation possesses a singularity at any critical point, i.e. at any point at which the flow speed, U, is equal to the wave speed, c. If c is complex, a similar singularity arises at any point at which the analytic extension of U into the complex plane is equal to c. Assuming the stratification is thermal in origin, the introduction of a small viscosity and heat conductivity removes this singularity, but leads to a governing equation of sixth order, four solutions being of rapidly varying WKBJ form. The circumstances in which the remaining two solutions can be uniformly represented in the limit of small viscosity and conductivity by the solutions of the Taylor-Goldstein equation are examined in this paper.

It is natural to say that a set S in a metric space has infinite generalized Hausdorff dimension if there is no Hausdorff measure Λh with Λh(S) = 0. In this note we study such sets. We first need some definitions.

We say that h(x) is a Hausdorff measure function if it satisfies the conditions:

Let p be an odd prime and denote by [p] the field of p elements. Let C = C(n, p) be the set of points x = (x1 …, xn) of Zn (n ≥ 1) satisfying

A classical problem in the theory of convex polytopes is the enumeration of the distinct combinatorial types of d-polytopes with υ vertices (υ ≥ d + 1). Following Grünbaum [1] c(υ, d) will denote the number of such types. Apart from the general results c(d + l, d)= 1 and c(d + 2, d) = [¼d2], which may be established by elementary arguments, the only other known values of c(υ, d) are for small values of υ and d. These have been determined empirically; details of the most recent results are contained in [2].

In the present paper we consider the uniqueness of the solutions of the equations governing the motion of an incompressible second-order fluid in a bounded region. For such a fluid model the stress tensor S in a rectangular Cartesian coordinate system xi at a point xi of the fluid is determined from

where I is the unit tensor, p the hydrostatic pressure and μ, β, γ are material constants. The tensors A1 and A2 are the first and second Rivlin-Ericksen tensors which are defined in terms of the velocity field vi by

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].

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 f(z) be an entire function. The definition of a Phragmén–Lindelöf indicator of f(z) requires the preliminary construction of a fairly regular comparison function V(r).

If f(z) is of order λ (0 < λ < + ∞), and of mean type, one takes

and associates with f(z) the indicator

The assertions about the congruence properties of the coefficients Ci, Di, in formula (3) of the paper (Mathematika, 16 (1969), 101–105) referred to above are patently false. They are, moreover, irrelevant, though the succeeding argument is so condensed and badly expressed that the true and the false in the paper cannot easily be distinguished. In short, the paper ought to be re-written. My thanks are due to the critic who first complained of the mistakes to the Editors and who thereby took upon himself the arduous task of refereeing this Corrigendum. His patient criticisms led to many improvements of the arguments alluded to in my paper, as well as of those which are presented here, and I am most grateful to him for his help.