Consider an isospectral manifold formed by matrices M ∈ glr(ℂ)[x] with a fixed leading term. The description of such a manifold is well known in the case of a diagonal leading term with different eigenvalues. On the other hand, there are many important systems where this term has multiple eigenvalues. One approach is to impose conditions in the sub-leading term. The result is that the isospectral set is a smooth manifold, bi-holomorphic to a Zariski open subset of the generalized Jacobian of a singular curve.

I have not seen the following properties of the Polar Conic and the Polar Conic of the Hessian given in treatises on the Cubic Curve. The results can be extended to space of n dimensions.

The formulae given in Herman's Optics, pages 80, 82, 98, 111, and called Cotes's formulæ, are a little difficult to grasp, and do not lend themselves to manipulation. The notation explained below is useful as a mnemonic. I think it also renders the proofs simpler.

An algorithm is given for determining presence or absence of injectively (at the fundamental group level) immersed tori (and constructing them, if present) in a branched cover of S3, branched over the figure eight knot, with all branching indices greater than 2. Such tori are important for understanding the topology of 3-manifolds in light of (for example) the Jaco-Shalen–Johannson torus decomposition theorem and the fact that the figure eight knot is universal, i.e., that all 3-manifolds are representable as branched covers of S3, branched over the figure eight knot.

The algorithm is principally geometric in its derivation and graph-theoretic in its operation. It is applied to two examples, one of which has an incompressible torus and the other of which is atoroidal.

In [6] Sands proved that the semisimple classes of associative rings are exactly the coinductive and closed under ideals and extensions classes. This characterization was transferred to the alternative case by Van Leeuwen, Roos and Wiegandt in [3]. Answering a question of [9], Sands [7] has recently proved that in the associative case the condition of being closed under ideals can be replaced by the regularity of the class. The same result for alternative rings has been proved by Anderson and Wiegandt in [2]. Thus the following result holds.

We give a necessary and sufficient condition for a sequence {ak}k in the unit ball of ℂn to be interpolating for the class A–∞ of holomorphic functions with polynomial growth. The condition, which goes along the lines of the ones given by Berenstein and Li for some weighted spaces of entire functions and by Amar for H∞ functions in the ball, is given in terms of the derivatives of m ≥ n functions F1, …,Fm ∈ A–∞ vanishing on {ak}k.

Let μ ≡ refractive index of the prism.

Let α, α′, β′, β be the successive angles of incidence and refraction of a ray, in a plane perpendicular to the edge of the prism.

This account of our Society is based to some extent on my Presidential address, which was given on 19 October 1977 and was devoted to the first fifty years.

In the latter half of the nineteenth century there was an upsurge of interest in mathematics that resulted in the foundation of a number of mathematical societies in different countries. The London Mathematical Society (1865), the Moscow Mathematical Society (1867), the Société Mathématique de France (1873), the Edinburgh Mathematical Society (1883) and the New York (later American) Mathematical Society (1888) were all founded in this period. There had, of course, been earlier more local societies, such as the Spittalfields Mathematical Society, which flourished over a long period before becoming defunct, as well as one or two much older bodies, for example the Mathematische Gesellschaft in Hamburg (1690), which still survive.

The theory of Inversion presents one of the simplest examples of those Birational Transformations of plane figures, whose general theory is due to Cremona. It has a distinguishing feature to which it owes its name. If the point P “inverts” into Q, then Q inverts into P. It is therefore a simple case of these involutive point transformations much of the general theory of which was developed by the late Admiral de Jonquières in a paper printed as late as 1864 in the Nouvelles Annales, but which had originally been addressed to the Institute of France in 1859. This memoir is not only highly interesting, but is eminently readable and very ingenious.

In what follows, a triangle ABC is taken, from it another triangle A′B′C′ is formed so that

A is the mid point of B′C′,

B is the mid point of C′A′,

and C is the mid point of A′B′.

Erdős (1) has proved the following result:

Theorem A.Every integral polynomial g(n) of degree k ≧ 3, represents for infinitely many integers n a(k-1)th power-free integer provided, in the case where k is a power of 2, there exists an integer n such that g(n)≢0 (mod 2k-1).

If ρ be the vector of a corner of a square in one system, σ that in a system derived without inversion, we must obviously have

k being the unit-vector perpendicular to the common plane.

August Ferdinand Möbius was born at Schulpforta, in Saxony, in the year 1790. He studied in the Universities of Leipsic and Göttingen, and, at the age of 2G, was appointed extraordinary Professor of Astronomy and Superintendent of the Observatory at Leipsic. There he remained till his death, in 1868, being appointed ordinary professor in 1844. Between the years 1817 and 1868 Möbius wrote his Barycentric Calculus, a Treatise on Statics, another on the Mechanics of the Heavens, and a large number of papers on Mathematical, Dynamical, and Astronomical questions. Most of these papers were contributed to Crelle's Journal, which was founded in 1826. The works of Mobius have recently been collected under the direction of the Royal Scientific Society of Leipsic, and under the editorship of Klein, Scheibner, and Baltzer.

We answer a question raised in an earlier paper concerning the pure state space of a separable C*-algebra.