In an earlier paper [3] we considered an inversion formula for the Gauss transformation G defined by

1.1

We noted there that formally G is inverted by,

1.2

and we showed that if e-D2 is interpreted via the power series for the exponential function, that is if

1.3

then under certain conditions on φ,

1.4

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

In this paper an attempt is made to generalize the rather special role played by certain "Young subgroups" of Sn. In Theorem 1, the substitutional equation involving Young's raising operators for Sn is generalized to apply to an arbitrary subgroup H of a finite group G. Applying this to Sn, it is shown how the multiplicities mλH can be calculated in terms of the raising operators in Theorem 2, which calculations are illustrated by a simple example.

The concept of a Boolean ring, as a ring A in which every element is idempotent (i. e., a2 = a for all a in A), was first introduced by Stone [4]. Boolean algebras and Boolean rings, though historically and conceptually different, were shown by Stone to be equationally interdefinable. Indeed, let (A, +, x) be a Boolean ring with unit 1, and let (A, ∪, ∩, ') be a Boolean algebra, where ∩, ∪, ', denote "union", " intersection", and "complement". The equations which convert the Boolean ring into a Boolean algebra are:

1. I

   Conversely, the equations which convert the Boolean algebra into a Boolean ring are:

2. II

When two uniform functions , are given, each with a finite radius of absolute convergence R1 R2 respectively, and {λn}, {μν} are real positive increasing sequences tending to infinity, a theorem due to Eggleston [1], which is a generalisation of Hurwitz1 s composition theorem, gives information about the position of the singularities of a composition function h(z), which is assumed to be uniform, in terms of the position of the singularities of f(z) and g(z). This result can be extended to Dirichlet series with real exponents by use of the transformation z = es.

Consider the plane symmetric random walk on a square lattice: a particle is initially at the origin in the xy-plane, it makes n consecutive steps of unit length, and each step is made with the probability 1/4 in each one of the four directions parallel to the axes. We call the path of the particle a spiral if the following conditions are met: a) the particle never occupies the same position twice, b) the path of the particle, whenever it turns, either turns always clockwise or always counter-clockwise throughout the path, and c) for every m > n the given n-step path can be continued in atleast one way to give an m-step path meeting the conditions.

The orthogonal latin squares displayed in [1] and [2] have the property that their row permutations are transformed amongst themselves by a permutation of order 7. In this note I present three examples of orthogonal latin squares of order 10 whose row permutations are transformed amongst themselves by a permutation of order 9.

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).

A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.

Throughout this note let be a polynomial of degree n. The following results are immediate.

For R ≧ 1

1

If p(z) has no zeros in |z| < 1, then

2

Given an analytic function f, the successive iterates of f are defined by

f[0](z) = z, f[n+1](z) = f{f[n](z)} for every z.

In particular f[1](z) = {f[0](z)} f(z). Extensive study has been given [1] to the problem of generalizing the iterates f[n], for integer n, to f[t] for arbitrary t, where the iterative character of f[t] is to be preserved by the conditions,

f[0](z) = z and f[s]{f[t](z)} = f[s+t](z) for arbitrary s and t.

Linear differential equations both ordinary and partial are often studied by means of Green's functions. One reason for this is that linearity permits superposition of solutions. A Green's function describes the "effect" of a point source, and the description of line, surface, or volume sources is achieved by superposing, that is to say, integrating, this function over the source distribution.

For equations with constant coefficients the use of integral transforms permits the calculation of such source functions in the form of integrals. Only in the simplest cases is explicit evaluation by elementary functions possible, and this has perforce led to the use of asymptotic estimates, which so thoroughly pervade the domain of applied mathematics.

It is well known that sets of constant width share several properties with spheres. In this note we consider a simple property of the circle and we show that it is shared by every plane set of constant width. As an application we derive a stronger form of the following theorem of D. Gale, [1]: every plane set of diameter 1 is a union of three sets of diameters not exceeding √3/2, and this constant is best possible. We shall make free use of the more elementary properties of convex sets and of sets of constant width; for these properties and for the terminology see the standard reference [2], or [3].

A note by N. Kimura gives the sums

explicitly linearly interms of the sums for positive integers p.

We note here that by a similar simple method, the Bernoulli numbers and Euler numbers may be related similarly to these sums.

Let a1 < a2 < … be any sequence of integers. Assume that the infinite sequence of numbers un satisfies the following condition: To every ɛ > 0 there is an no = no (ɛ) such that for all n > no and all k

1

Obreanu asked (Problem P. 35 Can. Math. Bull.) under what conditions on the sequence a1 < a2 < … does (1) imply that the sequence u is convergent. N. G. de Bruijn and P. Erdos proved that a necessary and sufficient condition for (1) to imply the convergence of un is that the sequence {an} be infinite and that the greatest common divisor of the a1 should be 1.

In Problem P 60 L. Moser has proposed the formula

1

This can be expressed in the more elegant form

2

The formula has been given in the literature a number of times. For example, it was proved by Bouniakovsky [1].