Let f be a positive-definite ternary quadratic form with integer coefficients; by c(f), the class-number of f, is meant the number of classes in the genus of f. The object of this paper is to find all the f with c(f) = 1; these f are the ones for which , where f′ is an arbitrary ternary form and ∼, denote equivalence and semi-equivalence respectively. Trivially, it suffices to find the primitive f with c(f) = 1.

Let Λ be a lattice in Cn such that the field of Abelian functions on the quotient space Cn/Λ is of transcendence degree n. This implies that is an algebraic extension of a field o of pure transcendence degree n. Thus there exists a vector A = (A1 …, An) of algebraically independent functions of the variable z = (z1, …, zn) and a function B = B(z), algebraic over

such that

In a recent paper [1], R. A. Rankin proved that

where τ(n) is Ramanujan's arithmetical function,

Let K be an algebraic number field. By a. full module in K [l,p.83] we mean a finitely-generated (necessarily free) subgroup M of the additive group of K whose rank is equal to the degree [K : ℚ] of K over the rational field ℚ. The intersection of M with ℤK, the ring of integers of K, is also a full module I, and we shall concern ourselves chiefly with the latter, in that we wish to count the number of rational integers in a given interval which can be expressed as the norms of elements of I. More precisely, we shall adapt the methods of [2] to prove the following

Motivated by questions of computational complexity, Rabin [7] introduced the notion of a complete proof of a system of inequalities. His work and the related paper of Spira [8] should interest geometers as well as computer scientists, for both papers involve convexity in an essential way. Spira's results concern the possibility of covering the intersection of a convex set C and a convex polyhedron Q with a finite collection P of polyhedra subject to certain conditions, while in Rabin's work the members of P may be more general than polyhedra. Both papers are interesting and treat important questions, but only Rabin's paper is correct in all respects. The present note contains counterexamples to some of Spira's results and establishes a correct version of one of them.

In an article [2] in a volume of papers dedicated to the memory of Edmund Landau, Heilbronn investigates the following problem on continued fractions, posed by Dr. J. Gillis:

Let a and N be natural numbers such that 1 ≤ a < N and (a, N) = 1. Then there exist unique natural numbers ci such that

The following theorem appears in Weyl's famous memoir [3] of 1916.

THEOREM A. Let λ1 ≤ λ2 ≤ … ≤ λn ≤ … be an increasing sequence of positive integers. Suppose that, of the numbers λ1, … λn, the first h1 are equal to each other, then the following h2 and so on, and finally that the last hm coincide. Let hj. If

the sequenceis uniformly distributed (mod 1)for almost all x, in the Lebesgue sense.

At high frequency, the leading terms in the virtual mass and wavemaking coefficients for a heaving, axisymmetric body depend only on the limit potential (Rhodes-Robinson, 1971). Here this result is applied to closed and open tori and solutions found in closed form. Some numerical values of the coefficients are tabulated.

In this note we derive an implicit representation of the solution to the problem of plane, inviscid, irrotational flow from a symmetric nozzle of arbitrary wall shape. For the case in which the nozzle wall has a slope which is everywhere much less than unity, we are able to convert this implicit representation into an explicit one in an asymptotic sense (based upon the smallness of the wall slope). Particular attention is paid to the contraction ratio of the jet. This work is complementary to that of Lesser [2] and Larock [l].

The system of equations governing the stress in a shear-strained prismatical body are examined and it is shown that, provided the stress-strain law adopts certain multi-parameter forms, the system may be reduced to the Cauchy-Riemann equations. Integration of the system is then immediate and the analysis of the stress concentration round certain notches thereby facilitated.

In a recent article [2] D. G. Larman and C. A. Rogers proved the following two results in Descriptive Set Theory (where R = the space of real numbers): (1) There is no analytic set in the plane R2, which is universal for the countable closed subsets of R; (2) there is no Borel set in R2, which is universal for the countable Gδ subsets of R. Recall that, if b is a class of subsets of a space X, a set U ⊆ X × X is called universal for b if (a) for each x ∈ X, Ux = def {y : (x, y) ∈ U} ∈ b, and (b) for each A ∈ b there is an x such that A = Ux. (Larman and Rogers have also shown that in both cases coanalytic universal sets exist.)

Let k be a knot of type K and with group πK. Let θ: nK → PSL(ℂ) = PSL (2, ℂ) be a parabolic representation (p-rep) as defined in [14]. We shall call the representation discrete when its image πKθ is a discrete subgroup of PSL(ℂ). It is known that PSL(ℂ) can be identified with the group of orientation preserving isometries of hyperbolic 3-space , and that each discrete subgroup of PSL(ℂ) acts discontinuously on . Hence each discrete p-rep θ has an associated orbit space . The present paper is a study of the general relations between the algebraic properties of a discrete image πKθ and the geometric properties of its orbit space.

Stewartson and Rickard [1] have shown that Rossby waves in thin spherical shells of an inviscid fluid contain singularities at the critical circles. These may be removed by introducing another wave containing square-root singularities in the velocity on the characteristics which by reflection touch the inner boundary at the critical circles. Stewartson and Walton [2] continued this wave all round the shell and showed it to be an inertial wave. Here we consider the effect of a weak kinematic viscosity v on these waves.

Let q be a power of an odd prime, [q] denote the Galois field GF(q) and write X(x) = xq − x. Let f(x) be a polynomial, having no linear factors, over [q], of positive degree, and write . Consider the continued fraction expansions

and

where the Ai(x) and aj,(x) are polynomials over [q] of degree ≥ 1 (if i ≥ 1, j ≥ 1). Plainly A0(x) = ao(x). Suppose that n = nf is the integer denned uniquely as the largest m such that

The notion of stability at infinity for an infinite finitely presented group with one end was introduced in [14], and for groups stable at infinity, the end invariant e was defined and studied for some (non-trivial) direct products. In this paper we study the corresponding problem for extensions.

We shall prove the following

THEOREM 1. Let α1, …, αn be any positive algebraic numbers and let u1…, un, ν be positive integers, relatively prime in pairs, such that ν ≥ 2 and ui > v for at least one i (1 ≤ i ≤ n). Then for every ε > 0 there are only a finite number of positive integers v such that the inequality

is satisfied, where for real α we understand by ‖α‖ the distance of α from the nearest integer.

This paper arises from an attempt to generalise the following result of [1].

THEOREM (Armitage-Fröhlich). If K is a cyclic extension of Q of degree lr, where 2 has even order mod l, then

In [1], the author considered the distribution of rational integers which are norms of elements of a given algebraic number field K, and in particular obtained the result

where A(K), B(K) and C(K) are positive parameters depending only on K, as does the implied constant in the O-term. In fact 0 < B(K) < 1, and B(K) is a rational number related to the Galois groups Gal and Gal, where is the normal hull of .

In this paper we shall discuss the following problem. Let G be a Fuchsian group of the first kind acting on the upper half-plane H. For z1, z2 ∈ H we set

Davies and Rogers [5] constructed a compact metric space Ω which is singular for a certain Hausdorff measure μh, in the sense that all subsets of Ω have μh-measure zero or infinity and μh(Ω) = ∞. (For a further study of this example see Boardman [3]). The interest lies in its extremely good descriptive character, which was lacking in the earlier examples given by Besicovitch [2] (a plane set singular for linear measure) and Choquet [4] (a plane set singular for any Hausdorff measure for which a segment has positive measure).