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

We shall use the notation Σnr= lr + 2r+…+ nr. Mr A. J. Gray suggested to me that n(n+ 1) is always a factor of Σnr, and that, in addition, 2n + 1 is a factor when r is even.

Much has been written on inequalities concerning positive definite matrices, but a new insight may be gained by examining inequalities from the standpoint of the inverse matrix. The standard inequality of Hölder can then be used in a more fruitful manner. This leads to some new results and a rediscovery of some known results.

Lazarsfeld proved a bound for the excess dimension of an intersection of irreducible and reduced schemes. Flenner and Vogel gave another approach for reduced, non-degenerate schemes which are connected in codimension one, using the intersection algorithm of Stückrad and Vogel and defining a new multiplicity k. Renschuch and Vogel considered a condition to ensure that there is no degeneration for more than two schemes. We define an integer which enables us to unify these methods. This allows us to generalize the result of Flenner and Vogel to non-reduced schemes by comparing the multiplicities j and k. Using this point of view we give applications to converses of Bézout's theorem; in particular we investigate the Cohen-Macaulay case.

1. The chief purpose of this paper is to demonstrate the existence of a plane quartic curve with eight undulations, an “undulation” being a point at which the tangent has four-point contact. It is shown that the curve

where (x, y, z) are homogeneous point-coordinates and f a constant, has undulations at the eight points

The curve has, in addition to these undulations, eight inflections which are; in general, distinct. But there are two geometrically different possibilities of their not being distinct, and in either instance they coincide in pairs at four further undulations. Thus two types of curve arise without any ordinary inflections at all, their 24 inflections coinciding in pairs at 12 undulations.

What I intend to put before you first is a graphical view of the solution of indeterminate equations of the third degree.

Not the least interesting portions of the wonderful “Mathematical Collections” of Pappus are those which reproduce parts of the νε⋯σεις, or the two lost books of Apollonius (247–205 B.C.). Pappus (c. 300 A.D.) writes:— “A line is said to verge (using Heath's translation) toward a point if, being produced, it reach the point,” and among other particular cases of the general problem he gives the following as treated by Apollonius:

Problem A: Between two lines, given in position, to place a straight line given in length and verging toward a given point.

We consider rational functions of the form fm(z) = zm/(z – p) which are analytic in |z|<p, p>1, and establish that the asymptotic distribution of the zeros of their Taylor sections and Lagrange interpolants at uniformly distributed nodes is similar. This notion is also illustrated computationally. We conjecture that a similar result can be expected for any function analytic in |z| < p.

[The present paper is a translation of the second part of the third book of Pappus's Mathematical Collection. Pappus's date is uncertain, but 300 a.d. may be taken as an approximation to it.

Throughout the translation I have used the word “progression” as a rendering of the Greek μεσότης, which has no English equivalent. The only other alternative was to employ the term mediety, from the Latin medietas.

The account of the various progressions given by Nicomachus, in his Arithmetical Introduction, differs somewhat from that of Pappus. I hope to have something to say about Nicomachus in a future paper.]

The theorems given in the present section are fundamental in the theory of triangles in multiple perspective. They are all perfectly well known, but are given here because without them the succeeding sections would be unintelligible.

Two triangles A1,A2A3, B1,B2B3, can be in perspective in six different ways, indicated by the following symbols, in which the A's are to be understood as connecting collinearly with the B's standing directly underneath.

The usual methods of proving the existence of regular polyhedra, as given in Wilson and in Todhunter, appear to most students somewhat difficult. It seemed worth while trying, therefore, whether a simpler or more direct proof could not be obtained. The following note shows how this may be done.