An optimal control for a nonlinear system is considered. The existence of an optimal-control pair, the characterization of the optimal control in terms of the optimal system and the uniqueness of solutions for the control problem are established. The uniqueness requires smallness assumptions on parameters in the functional.

(i) Let R be a semiprime right Goldie ring with dim R = n. Then a maximal chain of right annihilators in R has exactly n terms, (ii) A semiprime locally right Goldie ring with ACC and DCC on right annihilators is a right Goldie ring.

Let V and W be finite dimensional real vector spaces, k≧0 an integer. We write L(V, W) for the space of all linear maps V→W and Lk(V, W) for the subspace of maps with kernel of dimension k; in particular, L0(V, W) is the open subspace of injective linear maps. Thus Lk(ℝn, ℝn) is the space of n × n-matrices of rank n – k in the title. We also need the notation Gk(V) for the Grassmann manifold of K-dimensional subspaces of V.

A natural topology on the set of germs of holomorphic functions on a compact subset $K$ of a Fréchet space is the locally convex inductive limit topology of the spaces $\mathcal{O}(\sOm)$ endowed with the compact open topology; here $\sOm$ is any open subset containing $K$. Mujica gave a description of this space as the inductive limit of a suitable sequence of compact subsets. He used a set of intricate semi-norms for this. We give a projective characterization of this space, using simpler semi-norms, whose form is similar to the one used in the Whitney Extension Theorem for $C_\infty$ functions. They are quite natural in a framework where extensions are involved. We also give a simple proof that this topology is strictly stronger than the topology of the projective limit of the non-quasi-analytic spaces.

AMS 2000 Mathematics subject classification: Primary 46A13; 46F15

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

A (2, 3, 7)-group is a group generated by two elements, one an involution and the other of order 3, whose product has order 7. Known finite simple examples of such groups are PSL(2, 7), PSL(2, p) where p is prime and p ≡ ±1 (mod 7), PSL(2, p3) where p is prime and p≢0, ±1 (mod 7), groups of Ree type of order q3(q3 + 1)(q − 1) where q = 32n+1 and n > 0, the sporadic group of order 23 · 3 · 5 · 7 · 11 · 19 discovered by Janko, and the Hall–Janko–Wales group of order 27 · 33 · 52 · 7 [4, 2]. G. Higman in an unpublished paper has shown that every sufficiently large alternating group is a (2, 3, 7)-group. Here we show that the sporadic group Co3 discovered by Conway [1] is a (2, 3, 7)-group.

Crossed modules occur in the theory of group presentations, in group cohomology and in providing algebraic models for certain homotopy types. There are profinite analogues of each of these contexts. In this paper, we examine the problem of extending the profinite completion functor on groups to one on crossed modules thus providing a method for comparing the information contained in profinite and abstract crossed modules in each of these situations.

Mr D. E. Littlewood has recently discussed the properties of the quadratic equation over the real quaternions and shown that the solutions correspond to the common intersections of four quadrics in four-space. Although complex quaternion solutions may arise, the system of real quaternions to which the coefficients belong is a division algebra. It is of interest, therefore, to discuss the solution of the quadratic when the coefficients are drawn from a system containing divisors of zero.

To divide the straight line AB (containing a units) at C so that

By algebra, taking the positive root,

The number p may therefore have any positive value, integral or fractional, and when negative cannot exceed ¼. Secondly, AC and AB are incommensurable except when 4p +1 is a square:– e.g., if 4p = (q – l)(q +1) or if p = q(q+l), q being any positive integer or fraction.

The problem is

Between two sides of a triangle to inflect a straight line which shall be equal to each of the segments of the sides between it and the base.

This problem was brought before the Society at the January meeting in 1884, and a solution of it by Mr James Edward will be found in our Proceedings, Vol. II, pp. 5–6, a second by myself in Vol. II., p. 27 (10th April 1884), a third by Mr R. J. Dallas in Vol. III., pp. 41–2 (9th January 1885). Solutions of a slightly more general problem were also given by myself in Vol. III., pp. 40–1, and reference made to the Educational Times, Vol. 37, p. 328 (1st October 1884), and to Vuibert's Journal de Mathématiques Élémentaires, 9e année, p. 45 (15th December 1884).