The temporal evolution of nonlinear, incompressible ensembles is examined first for the one-dimensional Burgers' equation and then for the incompressible, unsteady Navier-Stokes equations. It is shown that local closure of the averaged problem can be obtained for finite ensembles of Burgers' equation in the limit as the number of moments tends to infinity. This limit behaviour is verified via direct numerical computations for the onedimensional inviscid and viscous Burgers' equation. Closure is found to occur at reasonably low order. It is shown that this technique can be extended to obtain a local closure of the convective terms of the Navier-Stokes Reynoldsaveraged equations.

Abstract. We study the envelope of the family of planes which bisect the volume of a tetrahedron.

In this paper, the Opial's inequality, which has a wide range of applications in the study of differential and integral equations, is generalized to the case involving m functions of n variables, m, n ≥ 1.

A topological ordered space (or pospace) is a poset (X, <) with a topology on X for which the relation < is closed in the product X × X. The topology of X is then necessarily Hausdorff. The basic theory of pospaces was developed by Nachbin in his book [5]; and others have extended it, but the resulting body of knowledge is not very geometrical. There are few concrete examples, other than the unit interval I with its natural order, and Euclidean spaces (Rn, ≤), the Hilbert cube (H, ≤) (each with the vector order), and some function spaces.

The Shirshov-Cohn theorem asserts that in a Jordan algebra (with 1), any subalgebra generated by two elements (and 1) is special. Let J be a Jordan algebra with 1, a, b elements of J and let a1, a2, …, an be invertible elements of J such that

Where

are Jordan polynomials. In [2, p. 425] Jacobson conjectured that for any choice of the Pi the subalgebra of J generated by 1, a, b, a1…, an is special.

Abstract. Let Φ be in the disc algebra H∞ ∩ C(T) with its restriction to T in the Zygmund space of smooth functions λ*(T). If P(Φ') ⊂ T is the set of Plessner points of Φ' and if F = Φ + Ψ, where Ψ∈C1(T), it is shown that F(P(Φ')) ⊆ C is a set of zero linear Hausdorff measure. Applications are given to the spectral theory of multiplication operators.

Abstract. Sufficient conditions are derived for all bounded solutions of general classes of integrodifferential equations of arbitrary order with variable coefficients to be either oscillatory or convergent to zero asymptotically.

Let D be an atomic integral domain (i.e., a domain in which each nonzero nonunit of D can be written as a product of irreducible elements) and k any positive integer. D is known as a half factorial domain (HFD) if for any irreducible elements α1, …, αn, β1, …, βm of D the equality α1… αn = β1… βm implies that n = m. In [5] the present authors define D to be a k-half factorial domain (k-HFD) if the equality above along with the fact that n or m ≤ k implies that n = m. In this paper we consider the k-HFD property in Dedekind domains with small class group and prove the following Theorem: if D is a Dedekind domain with class group of order less than 16 then D is k-HFD for some integer k > 1, if, and only if, D is HFD.

Let f(x) be a monic polynomial of degree n with complex coefficients, which factors as f(x) = g(x)h(x), where g and h are monic. Let be the maximum of on the unit circle. We prove that , where β = M(P0) = 1 38135 …, where P0 is the polynomial P0(x, y) = 1 + x + y and δ = M(P1) = 1 79162…, where P1(x, y) = 1 + x + y - xy, and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.

§1. Introduction. Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X. The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ-diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each ε>0, it is possible to write

where each set Xi, i≥1, has the property that each non-empty subset of Xi, has a non-empty relatively open subset of ρ-diameter less than ε. If is any family of subsets of X, we say that X is σ-fragmented by the metric ρ, using sets from, if, for each ε>0, the sets Xi, i ≥ 1, in (1.1) can be taken from

Let K be an algebraic number field, [ K: ] = KΣ. Most of what we shall discuss is trivial when K = , so that we assume that K ≥ 2 from now onwards. To describe our results, we consider the classical device [2] of Minkowski, whereby K is embedded (diagonally-) into the direct product MK of its completions at its (inequivalent) infinite places. Thus MK is -algebra isomorphic to , and is to be regarded as a topological -algebra, dimRMK = K, in which K is everywhere dense, while the ring Zx of integers of K embeds as a discrete -submodule of rank K. Following the ideas implicit in Hecke's fundamental papers [6] we may measure the “spatial distribution” of points of MK (modulo units of κ) by means of a canonical projection onto a certain torus . The principal application of our main results (Theorems I–III described below) is to the study of the spatial distribution of the which have a fixed norm n = NK/Q(α). In §2 we shall show that, with suitable interpretations, for “typical” n (for which NK/Q(α) = n is soluble), these α have “almost uniform” spatial distribution under the canonical projection onto TK. Analogous questions have been considered by several authors (see, e.g., [5, 9, 14]), but in all cases, they have considered weighted averages over such n of a type which make it impossible to make useful statements for “typical” n.

§1. Introduction. The literature on solving a system of linear equations in primes is quite limited, although the multi-dimensional Hardy-Littlewood method certainly provides an approach to this problem. The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation. Recently Liu and Tsang [4] studied this case in full generality and obtained a result with excellent uniformity in the coefficients. Almost no other general result has appeared so far, due probably to the fact that such a theorem is clumsy to state.

The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equations

with 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writing

in which e(α) denotes e2πiα, we observe that

where Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).

Let k ≤ 2 be an integer, and set

Ek (X) = |{n ≤ X, n ≠ mk, n not a sum of a prime and a k-th power}|.

We prove that there exists δ = δ(k) > 0 such that Ek (X)Ek(X)≪kX1−δ.