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.

It is well known that Cantor's ternary set C, constructed on [0, 1], has a difference set, D(C), equal to [0, 1]. If E ⊂ R we define Dk(E)(⊂ Rk-1), by Dk(E) = {(d1, d2, …, dk-1); di ≥ 0, and there is x ∈ E such that x + di ∈ E for all i, 1 ≤ i < k}. Thus D2(E) ≡ D(E) and Dk(E) tells us whether or not a particular set of k real numbers can be translated into E. We call Dk(E) the k-difference set of E. In this work we seek criteria for finding the Besicovitch dimension of Dk(E) (written dim Dk(E))) and in particular, conditions on certain classes of linear sets E that ensure that Dk(E) should contain an open interval in Rk-1.

Closed form solutions are obtained for a class of singular integral equations of the first kind with difference kernels. The kernel function is the sum of a polynomial and a second polynomial multiplied by a logarithm, with the possible addition of a strong singularity. A wider class of kernels have approximate representations in one of the above forms, where, for a specified accuracy, the polynomial orders will depend on the range of any parameter present. For example, the modified Bessel function kernels K0(|γx|), K1(|γx|)sgn x in the interval |x| ≤ 2 can be so expressed using 16th order polynomials with a maximum error 10-5 for real γ in the range 0 < γ ≤ 4.

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.

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].

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.

§1. Preliminaries. A Cauchy process in d-dimensional Euclidean space, Rd, is a stochastic process, Xt(ω), with stationary independent increments and with a continuous transition density, p(t, y − x) defined by

and

where m, the isotropic measure, is a probability measure on Sd, the unit sphere in Rd, such that when d > 1 the support of m is not contained in any d − 1 dimensional subspace. In (2) w is given by

where . It follows that for each t > 0 and y we have p(t, y) > 0 and that for each t > 0 p(t, y) is a bounded and continuous function of y. Xt(ω) can be considered as being a standard Markov process (for a full description of the definition of such a process see Chapter 1 of [1]) and in particular we can assume that the sample functions of Xt(ω) are right continuous and have left limits. We can also assume that Xt(ω) enjoys the strong Markov property. We write Px and Ex for probabilities and expectations conditional on X0(ω) = x, and we write P for P0.

Suppose K is a convex body in Euclidean n-space En and that all the orthogonal projections of K onto p-dimensional linear subspaces have the same p-dimensional volume, that is K has constant outer p-measure. If p = 1, this means that K is of constant width; if p = n − 1, this means that K has constant brightness. It is known that, when the boundary of K is smooth enough to admit principal radii of curvature R1, …, Rn-1 as functions of the outer normal u, and if we define Fp(u) to be the p-th elementary symmetric function of these radii, then, when K has constant width,

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

In [1] Fröhlich considers the kernel D(Z(Γ)) of the map of class-groups C(Z(Γ)) → C(), Γ a finite abelian group, the maximal order in the rational group ring Q(Γ). We obtain under mild hypotheses a non-trivial lower bound for the cardinality k(Γ) of the finite group D(Z(Γ)) when Γ is the cyclic group of order 2pn, p an odd prime. In fact, let f be the smallest positive integer such that 2f ≡ 1 mod.pn, If 2|f then k(Γ) > 1 for pn ≠ 3, 32, 5 and k(Γ) is divisible by primes ≠ 2, p except possibly when pn is a Fermat prime or when pn = 32. The latter result contrasts with the fact that D(Z(Γ)) is a p-group if Γ is a p-group [1; Theorem 5].

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.