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.

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.

Let G be any enumerable subset of the positive real numbers, with infinity as its only limit point. The purpose of this paper is to give a construction for a Lebesgue measurable set E ⊂ R+, with the following properties:

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.

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 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

A sequence {an} of integers is said to be primitive if ai × aj whenever i ≠ j. Let f be a polynomial with integer coefficients and A a sequence of positive integers. We discuss further a problem considered in [1] in which I. Anderson, W. W. Stothers and the author investigated primitive sequences of the form f(A) = {f(x), x ∈ A}. (Of course, we can assume f(x)→ ∞ as x → ∞.) We shall prove the following theorem in which A(n), as usual, denotes the number of memhers of A that are. less than or equal to n.

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.

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

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

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.

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.

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.)

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.

Hitherto, little use has been made of the well-known representation

of the Riemann zeta-function, ζ(s), within the critical strip l = {s:s = σ + it, 0 < σ < 1}. Certain variants of (1) have been used to deduce the functional equation of ζ(s), while a simple consequence of (1) itself is that ζ(s) does not vanish on the positive real axis.

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

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 .

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.