Asymptotic approximations to solutions of perturbations of Meijer's differential equation are derived. These are used to determine deficiency indices for certain subspaces associated with related pairs of symmetric differential expressions.

Two-dimensional differential systems

are considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

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,

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 [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).

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.

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.

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].

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).

A prime p > 2 is called irregular, if it divides the numerator of at least one of the Bernoulli numbers B2, B4, …, Bp – 3 (in the even suffix notation). The study of irregular primes has its origin in the famous theorem of Kummer which states that p divides the class number of the p-th cyclotomic field, if, and only if, p is irregular. Carlitz [1] has given the simplest proof of the fact that the number of irregular primes is infinite.

In line with the Ritt–Seidenberg elimination theorem in differential algebra [RIT], [SEI], and with an “approximation theorem” by Denef and Lipshitz [DEL] for formal power series, and with an elimination theorem by the author [RUB1] for C∞ solutions of systems of algebraic differential equations (ADE's), one is led to consider the corresponding elimination question for Cn solutions. Somewhat in the spirit of [RUB2], though, we produce a negative result in this direction.

As usual, if π is a finite group and ℳ ⊆ ℚπ a maximal order containing , we set

This is well known to be a finite group, and independent of the choice of ℳ.

Analytical and numerical properties are described for the free interaction and separation arising when the induced pressure and local displacement are equal, in reduced terms, for large Reynolds number flow. The interaction, known to apply to hypersonic flow, is shown to have possible relevance also to the origins of supercritical (Froude number > 1) hydraulic jumps in liquid layers flowing along horizontal walls. The main theoretical task is to obtain the ultimate behaviour far beyond the separation. An unusual structure is found to emerge there, involving a backward–moving wall layer with algebraically growing velocity at its outer edge, detached shear layer moving forward and, in between, reversed inertial flow uninfluenced directly by the adverse pressure gradient. As a result the pressure then increases like (distance)m, with m = 2(√(7)–2)/3 ( = 0.43050 …), and does not approach a plateau. Some more general properties of (Falkner–Skan) boundary layers with algebraic growth are also described.

The purpose of this paper is to prove the inequality of Theorem 1. The problem is due to B. Korenblum, who asked about it in connection with the characterization of the zero sets of functions analytic in the unit disc satisfying a growth condition. The problem was communicated to us by W. K. Hayman.

§1. Let E = E(A) be the set of real numbers x ε (0, 1) whose regular continued fraction expansion

contains only partial denominators ai from a given set A of positive integers. For finite A the (Hausdorff-) dimensional numbers dim E have been studied by I. J. Good ‘2’ and T. W. Cusick ‘1’. C. A. Rogers ‘8’ introduced a natural probability measure on E. He showed that excluding sets with measure between 0 and 1 (in the strict sense) from E does not reduce the dimensionality more than excluding sets of measure zero, and that the minimal (“essential”) dimensions ess dim E arising in this way is smaller than dim E, at least for A = {1,…, r} when r = 2 or r is sufficiently large.

Let η(z) be the Dedekind eta function defined by

where q = exp(2πiz) and Im (z) > 0. Then it is famous that

Let F be a number field, l a prime, and K a normal extension of F for which Gal(K/F) is topologically isomorphic to the additive group Zl. Then, corresponding to the subgroups ln Zl, there is a chain of fields F = F0 ⊂ F1 ⊂ F2 ⊂ … ⊂ K so that Gal (Fn/F) is cyclic of order ln. Let An be the l-Sylow subgroup of the ideal class group of Fn. By the now well-known theorem of Iwasawa, there are constants μ, λ and ν so that, for all sufficiently large n, |An| = len, where en = μln + λn + ν. Much work has been done investigating these constants, especially showing μ to be 0 for cyclotomic Zl-extensions of abelian fields F, [2], and computing λ and ν in special cases. In many cases, it has been shown that μ = λ = 0, and I know of no cases where μ or λ are positive when F is totally real (cf. Greenberg [3]). Here we investigate the maps in,m: An → Am induced by the inclusion of Fn in Fm for m ≥ n, with a view to determining their kernels. When μ = λ = 0, we prove that in, n+s(An) = for large enough n, and that, for large enough n, Ker in, n+s coincides with the kernel of the ls power map on An. I thank A. Brumer for helpful conversations on this matter.