The position of a point in a plane may be determined by its distances (r, ρ) from two fixed points which may be termed foci. These distances are termed vectorial coordinates. The determination is unique if attention is confined to the half-plane bounded by the line of foci. In certain cases where the properties of a curve are defined with respect to two points, representation of the curve by an equation between its vectorial coordinates possesses certain advantages. For example, the equation of the ellipse takes the extremely simple form r + p = 2α.

In this note we present some rather loosely connected results on Banach algebras together with some illustrative examples. We consider various conditions on a Banach algebra which imply that it is finite dimensional. We also consider conditions which imply the existence of non-zero nilpotents, and hence the existence of finite dimensional subalgebras. In the setting of Banach algebras quasinilpotents figure more prominently than nilpotents. We give an example of a non-commutative Banach algebra in which 0 is the only quasinilpotent; this resolves a problem of Hirschfeld and Zelazko (4).

In this note we shall consider the problem of uniquely continuing solutions of the parabolic equation

across an analytic arc σ: x=s1(t) satisfies the boundary data

We assume that u(x,t) is a classical solution of (1) in the domain D ={(x,t): s1(t)< x < s 2(t), 0 < t < t0}, continuously differentiate in D ∪ σ and define the “reflection” of D across σ by

In the part just issued of the American Journal of Mathematics (vii, pp. 3, 4) Proffessor Cayley has occasion to deal with the function

where stands for i (i – 1)…(i – j + 1). He Points out that

expresses {i, 1}, {i, 2},…{i, 5} each in descending powers of i, and tabulates the function from {1, 1} to {5, 5}.

A new interpretation of the Baker–Pym theorem is given in terms of operators and applies to a characterization of multipliers on a Banach algebra.

On pages 338 and 339 in his first notebook, Ramanujan records eighteen values for a certain product of theta-functions depending on two integral parameters m and n. When (m, n) = 1, it can be seen that each of these values is a unit. The purpose of this paper is to establish each of these eighteen values and to prove that under certain general conditions this product is indeed a unit. Lastly, we prove that certain quotients of theta-functions are algebraic integers.

Let d(<0) denote a squarefree integer. The ideal class group of the imaginary quadratic field has a cyclic 2-Sylow subgroup of order ≦8 in precisely the following cases (see for example [5] and [6]):

where p and q denote primes and g, h, u and v are positive integers. The class number of is denoted by h(d) and in the above cases h(d) = 0(mod 8). For cases (i), (ii) and (iii) the authors [6] have given necessary and sufficient conditions for h(d) to be divisible by 16. In this paper we do the same for case (iv) extending the results of Brown [4].

1. Let [ast] (s, t=0, 1, … n) be a square matrix of order n+1 and determinant |ast| and suppose that by repeated “isolation” of the variables the corresponding bilinear form has been expressed as

where, for all r,

Then

Now (1) implies, and is implied by, the identities

Thus, from any known identity of the form (4), subject to the condition (2), we may at once infer, using (3). the value of the corresponding determinant |ars|.

In [3] Fuller introduced an index (now called the Fuller index) in order to study periodic solutions of ordinary differential equations. The objective of this paper is to give a simple generalisation of the Fuller index which can be used to study periodic points of flows in Banach spaces. We do not claim any significant breakthrough but merely suggest that the simplistic approach, presented here, might prove useful for the study of non-linear differential equations. We show our results can be used to study functional differential equations.

This well-known theorem, published in Newton's Arithmetica Universalis, may be formulized thus:—

where sm ≡ the sum of the mth powers of the roots of the equation

Of course, when m>n, certain coefficients pn+1, pn+2, … pm will be zero, and the last term that does not vanish will be .

In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.

Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k ∈ ℕ, showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i ∈ ℕ is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 ≇ G1, 1 for all i > 1 and if i, k are primes then Gi, 1 ≅ Gk, 1 if and only if i = k.

[The correspondence which is here printed was bought by me on the 28th of March 1887 at the sale of the Gibson-Craig collection of Scottish MSS.

Simson's letters, which are beautifully written, seem all to have passed through the post, but Stewart's letters are, I conjecture, merely the drafts of what he proposed to send. The handwriting of the latter, though legible, is not elegant, and there are frequent erasures. I have scrupulously respected, in all the letters, the spelling, the punctuation (or want of it), the use or disuse of capitals, and I have made no attempt to improve the style.

In this note, we determine fields K and groups G that are either nilpotent or FC and such that the set of torsion elements of the group ring KG forms a subgroup.