This note has its origin in the following problem: do there exist non-trivial increasing continuous functions on [0, l] to [0, l], which map the following sets in [0, l] onto themselves: the rational, the algebraic and the transcendental numbers? One such function is obviously f(x) = x; more generally, f(x) = (c + l)x/(cx + 1), with c rational and non-negative, satisfies the conditions. Let G denote the space of order-preserving homeomorphisms of [0, l} onto [0, l], in the uniform metric. It follows from Theorem 1 below that the set S of all such functions i s dense in G. S is clearly a subgroup of G and one may ask what a r e its group-theoretic properties. We shall not consider these questions.

Repeated application of the unitary reflections

to the point (1,0) and line x + iy = 1 yields 24 points and 24 lines. These are the vertices and edges of the regular complex polygon 4 {3} 4 whose group has the abstract definition R4 = I, RSR = SRS [l]. The purpose of this note is to introduce the notion of regular quaternion polygon and give an example analogous to the preceding one.

Let A = (aij), B = (bjj), be two n-square matrices over the complex numbers. Then the n-square matrix H = (hjj) = ij(aijb) is called the Hadamard product of A and B, H = AoB, [l; p. 174]. Let the n2 - square matrix K = A⊗B denote the Kronecker product of A and B.

In this note we use the method of generating functions to show that there is a unique way of splitting the non-negative integers into two classes in such a way that the sums of pairs of distinct integers will be the same (with same multiplicities) for both classes. We prove a similar theorem for products of positive integers and consider some related problems.

Let 1,θ1, θ2, …,θn be real numbers linearly independent over the rational field and let α1, α2,…, αn be arbitrary real numbers. Then, to each N > 0 and ε > 0, there correspond integers

which satisfy the set of inequalities

A

Let f(x) be a real analytic function of the real variable x and α a simple root of the equation f(x) = 0. It is well known that a function ϕ(x) can be associated with the equation in many different ways such that

1. (i) α is a root of the equation ϕ(x), i.e. α is a fixed point (invariant point) of the function ϕ(x);

2. (ii)

If f(x) is real-valued and continuous, it has the property that it takes on all intermediate values when it passes from one value to another. This means that whenever f(x1) and f(x2) are different and u is any number between them, then f(x) = u for at least one x between x1 and x2. We shall call this the Darboux property.

One of the most elegant results of the elementary theory of the distribution of primes is that

1

where the product runs over primes. A very simple proof of (1) has recently been given by Erdös and Kalmar [1], [2].

Students in Mathematics and Physics courses are commonly exposed to a method of treating data known as the "half-table method". The method is presented with encomiums by R. C. Dearle [1] and by Lucius Tuttle and John Satterly [2]. It can best be described by an illustration; since the determination of the period of a pendulum is one of the experiments most commonly used (and occurs in both [1] and [2]), let us borrow a set of data from [2] and some explanatory text from [1].