Let Fn be the free group on {ai: i ∈ Zn}, where the set of congruence classes mod n is used as an index set for the generators. Let φ be the permutation (1, 2, 3, …, n) of Zn and denote by θ the automorphism of Fn induced by φ, namely

The equation of the propagation of electric signals along cables, generally known as the equation of telegraphy, may be written

Particular solutions of this equation, adapted to various purposes have been found by Heaviside, Poincaré, A. G. Webster, T. W. Chaundy, § and others. The object of the present paper is to unify the theory of the equation by exhibiting the relations which these solutions bear to each other, and by obtaining them as particular cases of a general solution. The derivation of new particular solutions by the solution of integral equations is also discussed.

The infinite continued fraction

in which

is periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

In this paper we first of all solve the dual series equations

where ƒ(ρ) and g(ρ) are prescribed functions,

is the Jacobi polynomial (2).

It is well known that every monic polynomial of degree n with coefficients in a field Φ is the characteristic polynomial of some n × n matrix A with elements in in Φ . However, it is clear that this result is an extremely weak one, and that it should be possible to impose considerable restrictions upon the matrix A. In this note we prove two results in this direction. In section 2, we show that it is possible to prescribe all but one of the diagonal elements of A. This result was first proved by Mirsky (2) when the ground field Φ is the field of complex numbers. In section 3, we see that we can require A to have any prescribed non-derogatory n–l × n–1 matrix in the top left-hand corner.

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.