In this note, we study the well-posedness of the exterior traction value problem for linear anisotropic non-homogeneous elastostatics. We prove existence and continuous dependence upon the data. In particular, in the isotropic homogeneous case, provided the body force is “simple”, we show that solutions tend to zero uniformly at large spatial distances.

We examine the case of plane, time-harmonic acoustic waves in two dimensions, scattered by an obstacle on the surface of which an impedance boundary condition is imposed. A stable method is developed for solving the inverse problem ofdetermining both the shape of the scatterer and the surface impedance from measurements of the asymptotic behaviour of the scattered waves at low frequencies. We accomplish this by minimizing an appropriate functional over a compact set of admissible boundary curves and admissible impedances.

In this paper, we study the local structure of the secant mapping of a pair of disjoint curves. We show that for generic curves, the secant map and unit secant maps are locally stable. If we allow our curves to coincide, we can define anew unit secant map to be the natural unit tangent map near the diagonal. This is, for a generic curve, a locally stablemap away from the diagonal. Along the diagonal, it is locally stable as a ℤ2 symmetric germ (the ℤ2 symmetry originating with reflection in the diagonal).

A one-phase Stefan problem can be reduced to an equivalent variational inequality by using the Baiocchi-Duvaut transformation. In this paper, we study the variational inequality by formulating it as a set-valued partial differential equation. The existence of solutions is proved by applying a generalized Schauder fixed fixed point theorem for set-valued mappings. Uniqueness and regularity of solutions are also obtained. In §3, we regard the boundary value Neumann data as boundary controls and combine both the variational inequality and the classical approaches to study the effects of controls on the free boundary and the state (i.e. temperature). In §4, we further use the theory to study an optimal “ice-melting” problem. Our results show that if the controls have fixed total input heat flux and are constrained in magnitude, then the optimal control is “bang-bang”. If the admissible controls are not constrained in magnitude, then the optimal control is a Dirac delta type distribution which is no longer admissible. In the last section, our existence theory is combined with the finite difference method and non-linear programming techniques to obtain numerical solutions.

Comparison functions are constructed for the problem of minimizing

over maps u: D(⊆ℝ2)→ℝ2 with det≥0, subject to the constraint u= f on ∂D, D the unit disk. This is accomplished for maps / which are reparameterizations of ∂D or which are “graph-like” maps. Estimates involving half derivative boundary norms are obtained.

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.