Let Gk, n, be the Grassmann manifold consisting in all non-oriented k-dimensional vector subspaces of the space Rk+n. In this paper we will show that any differentiable mapping f: Gk, n → Rm, has infinitely many critical points for suitable choices of the numbers m, n, k.

A tournamentTn consists of a finite set of nodes1, 2, …, n such that each pair of distinct nodes i and j is joined by exactly one of the arcsij or ji. If the arc ij is in Tn we say that i beats j or j loses to i and write i→j. If each node of a subtournament A beats each node of a subtournament B we write A→B and let A + B denote the tournament determined by the nodes of A and B.

The purpose of this paper is to study the consequences of an endomorphism near-ring of a finite group being a local near-ring and the existence of such near-rings. As we shall see in Section 2, an endomorphism near-ring of a finite group being local gives us some information about both the structure of the group (Theorem 2.2) and the automorphisms of the group lying in the near-ring (Theorem 2.3). Existence of local endomorphism near-rings of finite groups is considered in Section 3 where we obtain as our main result that any p-group of automorphisms of a p-group containing the inner automorphisms always generates a local endomorphism near-ring. In particular, we get as a corollary that the endomorphism near-ring of a finite group G generated by the inner automorphisms of G is local if and only if G is a p-group. The third section concludes with a discussion of endomorphism near-rings of dihedral 2-groups and generalized quaternion groups.

In this note, we show that, if (an) in l1 with Σ|an| < 2 and Σ|an|2 = 1, then max {|ai| + |aj|:i ≠ j} ≧ 1, but that the corresponding theorem for sequences in lp(1<p<2) fails—but only just! Applications to group algebras are given, when it is shown that elements in l1(G) with powers bounded by ½(1+ ) are bounded away from the identity e of G, but that the corresponding result for lp (G) is false.

Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equation

where α, β, γ are functions of x and y.

Fourier's Series were first applied to the solution of the Differential Equations which occur in the Theory of the Conduction of Heat and that of the Vibrations of Stretched Strings, namely,

In the first, when the flow of heat is linear and the initial temperature is given in the range 0 < x < π by f(x), the solution can be put in the form

In this case there is no difficulty with regard to the differentiation of the infinite series, term by term, as the factor causes the series which we thus obtain to remain uniformly convergent with respect to x and t in the intervals concerned.

The sampling theorem, often referred to as the Shannon or Whittaker-Kotel'nikov- Shannon sampling theorem, is of considerable importance in many fields, including communication engineering, electronics, control theory and data processing, and has appeared frequently in various forms in engineering literature (a comprehensive account of its numerous extensions and applications is given in [3]). The result states that a band-limited signal, i.e. a real function f of the form

where w>0, is under reasonable conditions on the even function F, determined by its values on the sampling set (l/2w)ℤ and can be reconstructed from the samples f(k/2w), k∈ℤ, by the series

A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with respect to any family of dilations on G, not just the natural dilations. We may also consider pseudodifferential operators as well as partial differential operators.

The appearance of this Review is a significant indication of the enormous development of mathematical studies in recent years. Nearly every one who has attempted to keep himself abreast of mathematical research has been obliged sooner or later to recognise the practical impossibility of mastering the literature of every branch, and has resigned himself to a comparatively elementary study of the general subject while devoting his main energies to special departments. Even then, so numerous are the Societies that publish Proceedings and Transactions, and so varied are the Journals that are chiefly mathematical in their content, that it is no easy matter for the mathematician to get a knowledge of what is being done in any special field by workers outside (sometimes in) his own country. The need for a publication that will, without undue delay, furnish a conspectus of the literature of the subject is thus a very real one. That the need has been felt is sufficiently shown by the synopses of the contents of other Journals, given in such publications as Darboux's Bulletin, and more especially by the excellent Jahrbuch über die Fortschritte der Mathematik. The last completed issue of the Jahrbuch, that for 1889, extends to upwards of 1300 pages, while that for 1890, two parts of which have appeared, will evidently be as large.

A ring R is said to be a P-ancestral ring if all proper non-zero sub-rings of R have property P. If p is the property that every proper non-zero sub-ring of R is a (two-sided) ideal then the ring Z of rational integers furnishes an example of a P-ancestral ring.

In this paper we are dealing with oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the form

Some sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) is considered also.

I am indebted to Dr. B. Kuttner for having kindly drawn my attention to certain corrections necessary in my paper with the above title, appearing in these Proceedings, Vol. 15 (Series II), Part 1, June 1966, pages 47–55.

The major corrections are as follows. On pages 48-9, Lemma 1 should appear without the factor log (1 – z)−1 of the denominator in the integrand at the end of the lemma; the formula in the proof of Lemma 1 is incorrect, its correct form being