In [3], D. H. Lehmer has analysed the incomplete Gaussian sum

where N and q are positive integers with N < q and e(x) is an abbreviation for e2πix. The crucial observation is that, for almost all values of N, Gq(N) is in the vicinity of the point ¼(1 + i)q1/2. This leads to sharp estimates of the shape Gq(N) = O(q½).

Throughout the paper, let m be a natural number and let F(x1,…, xn) be a form of degree k ≥ 2 with integer coefficients, n ≥ 3. We are concerned with finding solutions of the congruence

for which x is a small non-zero integer vector. For example, in the case k = 2 it was shown by Schinzel, Schlickewei and Schmidt [11] that is a solution of (1) satisfying

provided that n is odd. This is best possible for n = 3, as we shall see later. Of course we can get an exponent (1/2) + (1/(2n – 2)) trivially for even n. I do not know how to improve on this. D. R. Heath-Brown (private communication) can improve the exponent in (2) to (l/2) + ε for n ≥ 4 and prime m > C1(ε).

The classical theory of analytic sets works well in metric spaces, but the analytic sets themselves are automatically separable. The theory of K-analytic sets, developed by Choquet, Sion, Frolik and others, works well in Hausdorff spaces, but the K-analytic sets themselves remain Lindelof. The theory of k-analytic sets developed by A. H. Stone and R. W. Hansell works well in non-separable metric spaces, especially in the special case, when k is ℵ0, with which we shall be concerned, see [9, 10 and 16–20]. Of course the k-analytic sets are metrizable. For accounts of these theories, see, for example, [15].

The purpose of this paper is two-fold, (i) to establish the existence of a unique local solution (in time) of an initial boundary-value problem for the tidal equations in bay areas and inlets, and (ii) to show the existence of a time-periodic solution of the equations when the tide raising force satisfies a condition involving the amplitude of the force, the depth of the sea and the domain considered.

1.1. In an earlier paper [6] the author proved

Theorem A. If α ≥ 0, β ≥ 0 (α, β integers), p > –1, p–r > –1 then necessary and sufficient conditions for a sequence (εn) to be such that

are

and

If α ≥ 1 (α an integer) and (ii) holds, then (i)bis equivalent to

In 1891 Victor Eberhard proved the following theorem concerning the number pk(P) of k-gonal facets of simple polytopes P, [4].

Eberhard's Theorem. For each k ≥ 3, k ≠ 6, let pk be a non-negative integer. Then there exists a simple 3-polytope P such that pk(P) = pk (k ≠ 6), if, and only if,

Kan and Thurston, in their paper [5], asked whether each smooth closed manifold other than S2 or RP2 has the same integral homology as a closed aspherical manifold. F. E. A. Johnson in [3], [4] is concerned with the answer to this question when the smooth closed manifold is an n-dimensional sphere Sn. He asked whether there exist aspherical manifolds Xг which have the homology of Sn.