The generalized equations of bisymmetry, associativity and transitivity are, respectively,

1. (1) (x1y)2(z3u) = (x4z)5(y6u)

2. (2) (x1y)2z = x3(y4z)

3. (3) (x1z)2(y3z) = x4y.

The numbers 1, 2, 3,…, 6 represent binary operations and x, y, z and u are taken freely from certain sets.

We shall be concerned with the cases in which x, y, z, and u are from the same set and each operation is a quasigroup operation. Under these conditions the solution of all three equations is known [1], [2]; equations (1) and (3) having been reduced to the form of (2) and a solution of (2) being given. We wish to present a new approach to these equations which we feel has the advantages that the equations may be resolved independently, the motivation behind the proof is clear, and the method lends itself to application on algebraic structures weaker than quasigroups. (Details of these generalizations will be given elsewhere.)

The celebrated Fredholm theory of linear integral equations holds if the kernel K(x, y) or one of its iterates K(n) is bounded. Hilbert utilizing his theory of quadratic form was able to extend the theory to the kernels K(x, y) satisfying

a

b

where k is independent of u(x).

These theories were extended considerably by T. Carleman who deleted condition (b) above.

Equations involving this Carleman kernel have been found useful in connection with Hermitian forms, continued fractions, Schroedinger wave equations (see [1], [2]) and more recently in scattering theory in quantum physics, etc. [3]. See also [5] for a variety of applications and extensions.

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by

1

E0 = 1 and Er=0(r>m).

We define the rth symmetric mean by

2

where denote the binomial coefficient. If α1 α2,…,αm are positive reals then

we have two well-known inequalities

3

and

4

In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

The purpose of this note is to generalize a result of Gulliksen, Ribenboim and Viswanathan which characterized local group rings when both the ring and the group are commutative.

We assume throughout that all rings are associative with identity. If R is a ring we call R local if R/J(R) is a division ring where J(R) denotes the Jacobson radical of R. It is well known that R is local if and only if each element of R\J(R) is a unit. We need the following.

The following theorem is due to J. L. Walsh (see [2, Theorem 17, 2a]):

Theorem. If all the zeros of f1(z)=zn+a1zn-1+ … + an lie in or on the circle C1 with centre c1 and radius r1 and if all the zeros of f2(z)=zn+b1zn-1+ … + bn lie in or on the circle C2 with centre c2 and radius r2, then each zero of the polynomial

lies in at least one of the circles Γk with centre γk and radius ρk, where

and where the ωk (k= 1, 2,…, n) are the nth roots of λ.

The purpose of this note is to give a basis with respect to which the companion matrix of an equation (over a field of any characteristic) is in Jordan canonical form.

Let k be a field. Define a k-linear map Di:k[X]→k[X] by where the integer is the binomial coefficient n!;/i!;(n —i)!;. We adopt the usual convention that if i>9 or j<0. Then D=(D0, D1D2,…) is a higher derivation (see [1, p. 192]).

We give an alternative construction of a Hausdorff compactification due to Fell [2]. We say that a space is compact if it has the Heine-Borel property, locally compact if each point has a fundamental system of compact neighbourhoods. The interesting spaces from the point of view of this paper, are the non-Hausdorjf ones since for locally compact Hausdorff spaces Fell's compactification is the usual one-point compactification. The motivation for the compactification comes from the theory of continuous fields of C*-algebras: the primitive spectrum of a C*- algebra A is a locally compact T0 space X and Fell [3] realizes A as an algebra of fields of operators over the compactification of X. This note is based on a discussion of the author with Professor Fell.