In this note we show that if we have an exact sequence of AH algebras (AH stands for “approximately homogeneous”) 0 → I → A → B → 0, then A has the ideal property (i.e., any ideal is generated by its projections) if and only if I and B have the ideal property. Also, we prove that an extension of two AT algebras (AT stands for “approximately circle”) with the ideal property is an AT algebra with the ideal property if and only if the extension is quasidiagonal.

Let σk(x) be the number of integers n≤x which are the product of just k prime factors, so that

and let πk(x) be the number of such n for which all the pi are different.

In comparison with the general plane quartic on the one hand, and the curves having either two or three nodes on the other, the uninodal curve has been neglected. Many of its properties may of course be deduced from those of the general quartic in the limiting case when an oval shrinks to a point or when two branches approach and ultimately unite. The modifications of properties of the bitangents are shewn more clearly by Geiser's method, in which these lines are obtained by projecting the lines of a cubic surface from a point on the surface. As the point moves up to and crosses a line on the surface, the quartic acquires a node and certain pairs of bitangents obviously coincide, viz. those obtained by projecting two lines coplanar with that on which the point lies. A nodal quartic curve and its double tangents may also be obtained by projecting a cubic surface which has a conical point from an arbitrary point on the surface. Each of these three methods leads us to the conclusion that, when a quartic acquires a node, twelve of the double tangents coincide two and two and become six tangents from the node, and the other sixteen remain as genuine bitangents: the twelve which coincide are six pairs of a Steiner complex.

The following proof of this theorem assumes only Euclid, I. 43, and its converse, with the well-known deductions, “the line joining the mid points of two sides of a triangle is parallel to the third side,” and “the mid point of one diagonal of a parallelogram is also the mid point of the other.” The proof given by Dr Taylor in his Conies which suggested the method, makes use of ratios.

Semigroup presentations have been studied over a long period, usually as a means of providing examples of semigroups. In 1967 B. H. Neumann introduced an enumeration method for finitely presented semigroups analogous to the Todd–Coxeter coset enumeration process for groups. A proof of Neumann's enumeration method was given by Jura in 1978.

In Section 3 of this paper we describe a machine implementation of a semigroup enumeration algorithm based on that of Neumann. In Section 2 we examine certain semigroup presentations, motivated by the fact that the corresponding group presentation has yielded interesting groups. The theorems, although proved algebraically, were suggested by the semigroup enumeration program.

In a previous paper (q.v.) by Mr W. L. Marr and the present writer, it was shown that, in accordance with Morley's Theorem, the angles A+2pπ, B+2pπ, C+2rπ of the triangles ABC be trisected, the three groups of six lines at the vertices give rise to 27 triangles DEF, the biangular coordinates of D with respect to BC being (B/3+2qπ/3, C/3+2rπ/3) or (βq, γr), and similarly for and F with respect to CA and AB.

The following examples illustrate a somewhat obvious extension of the method of factoring given in a former paper (Proceedings, Vol. XII., p. 32, q.v.). By means of it, if we are given any function of n variables, no one of which is of higher degree than the second, we can either find the factors of it, or prove that it has no factors with rational coefficients.

In a recent paper Segel (1) points out that the diverse techniques (which “comprise the core of the applied mathematicians art (or craft)”) of the applied mathematician, although in general reliably proven, are “rarely explicitly delineated but rather are transmitted indirectly and informally”. In his article Segel aims to clarify two such techniques, namely:

(i) Scaling—or how to choose dimensionless variables in such a way that the relative size of the various terms in an equation is explicitly indicated by the magnitudes of the dimensionless parameters which precede them,

(ii) Simplification—a procedure in which a term is neglected under the assumption that it is small, and the consistency of the assumption checked later.

For the measurement of a plane area, bounded by an irregular curve, various methods are adopted. Besides the well-known methods of approximation in use among land measurers, the following may be mentioned—

The area is divided by parallel ordinates. These are measured, and the results are then treated in several ways to produce more or less accurate results. See Williamson's Integral Calculus, third edition, p. 211.

I. If the Jacobian sextic of a pencil of binary quartics is known, the pencil itself is determined in five ways. The explicit determination of the pencil in terms of the irrational invariants of has been effected by Stephanos.

There are two known cases in which a pencil of quartics admits of rational determination from its Jacobian. In these, the rationally determinable pencil differentiates itself algebraically from its remaining four co-Jacobian pencils.

We consider interpolation at 2n equidistant nodes in [0,π) from the space ℱN spanned by sines and cosines of odd multiples of x. This interpolation problem is shown to be correct for an arbitrary sequence of derivatives specified at all the nodes. Explicit expressions for the fundamental polynomials are obtained and it is shown that under mild smoothness assumptions on the function f interpolant from ℱN converges uniformly to f as the node spacing goes to zero.