Let l be a prime and let v ≥ 1 be an integer (when l = 2 we assume v ≥ 2). Any ring, A, with unit, possesses mod lv algebraic K-groups [B] denoted by Ki(A; Z/v) (i ≥ 0). For i ≥ 2, Ki(A; Z/lv) = [Pi(lv), BGLA +], the group of based homotopy classes of maps from the Moore space , to BGLA+, the classifying space of algebraic K-theory [G–Q ; W].

We consider the second order linear differential equation

where p and q are real-valued and p(t) > 0 for all t ≥ T. Our interest here is the oscillatory nature of solutions of (1.1). More particularly we consider the following questions, (I), (II) and (III).

Attempts to extend known two-dimensional results (Ursell, 1947) to the fully three-dimensional case can lead to unpredictable results. We show how the use of a variational approximation for a finite plane vertical barrier leads to apparently different results when different formulations are used. The reason for this is not so much that the method is wrong, but rather that several different limits are taken in the process, which are hard to control. We suggest an alternative matching scheme, based on Ayad and Leppington (1977), which holds for the case ka → ∞, l/a → ∞, kl → ∞, where / is the length of the barrier, a its depth and k the wavelength of the incident wave. The method is applied to a channel with impeding side walls, as a model of French's (1977) wave-energy device.

For x, y ≥ 1, let Ψ(x, y) denote the number of positive integers less than or equal to x and free of prime factors greater than y. The behaviour of the function Ψ(x, y) has been the object of numerous articles (see e.g. Norton's memoir [5] and the bibliography there). It turns out that a good approximation to ψ(x, y)/x is given by ρ(log x/log y), where the function ρ(t) is defined for t ≥ 0 as the continuous solution of the equations

Cardinal functions of topologies have been extensively studied. Cardinal functions of measures have attracted less interest, perhaps because there are fewer straightforward results which are independent of special axioms. In this paper I consider the “additivity” and “cofinality” of a measure (Definition 1) and show that they can often be calculated in terms of certain fundamental cardinals (Corollary 11 and Theorem 16).

Let 1 ≤ α ≤ β ≤ γ be cardinals, and let denote the class of all graphs on γ vertices having no subgraph isomorphic to Kα,β. A graph is called universal if every can be embedded into Go as a subgraph. We prove that, if α < ω ≤ γ and the General Continuum Hypothesis is assumed, then has a universal element, if, and only if, (i) γ > ω or (ii) γ = ω, α = 1 and β ≤ 3. Using the Axiom of Constructibility, we also show that there does not exist a universal graph in .

Gale transforms are constructed for certain infinite dimensional α-polytopes. In a manner analogous to the finite dimensional case the Gale transform can be used to determine all closed faces and Radon partitions of the α-polytope. A by-product is a characterization of closed faces using nets of functionals.

A recent paper [1] indicates that the beginnings of dynamic stall, near an aerofoil's leading edge, for instance, can be regarded as the finite-time nonlinear breakdown of a boundary layer subjected to an angle of attack above the critical value for the existence of a steady solution. The present theoretical study shows that the same non-linear breakdown can occur even in the below-critical regime. This happens particularly when reversed flow is present since short wavelength disturbances are then unstable and accumulate, for certain confined initial conditions, to force the finite-time collapse. A number of marginal cases with forward or reversed, subsonic or supersonic, oncoming motion are also noted and shed extra light on the instability and subsequent breakdown.

It is proved here that, if G is a positive definite integral ternary quadratic lattice of discriminant d and c is a squarefree integer which is primitively represented by the genus of G, then G primitively represents all sufficiently large integers of the type ct2, with g.c.d. (t, 2d) = 1, which are primitively represented by the spinor genus of G.

The principal objective of this work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations will be found in response to a number of questions in a recent survey article by Rolf Schneider and Wolfgang Weil. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. We shall also show these classes are characterized by certain extremal geometric inequalities. The work concludes with a brief discussion of related results concerned with generalized zonoids.

We state some definitions belonging to the two halves of the title, going far enough to state our main results.

Fourier transforms. Let μ be a finite, complex-valued measure on R and its Fourier-Stieltjes transform. We define ℛ to be the set of μ with When μ ∈ ℛ and φ is of class (continuously differentiable of compact support), the identity shows that θ · μ ∈ ℛ.

Let K be a number field and E/K an elliptic curve. As is well known [3, 4,[ if K has class number 1, then there exists a global minimal Weierstrass equation for E. Our main goal in this paper is to prove the following converse to this statement.

In this paper the authors formulate a boundary-initial value problem for a linear elastic porous body saturated with an inviscid fluid and establish a continuous dependence theorem (Theorem 2) and two uniqueness theorems (Theorems 3, 4) for a particular class of such continua. Theorems 2, 3 are proved without hypotheses on the sign of the constants and, if the domain is unbounded, under mild assumptions on the spatial asymptotic behaviour of the field variables. Theorem 4 holds for body-forces not equal to zero and, if the domain is unbounded, without restrictions upon the behaviour of the unknown fields at infinity, but under suitable conditions on the sign of the constants.