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.

The well-known theorem of Minkowski, [1], [2], states that;

(M) a plane convex region, symmetric about the origin O, includes a lattice point other than O if its area is greater than 4.

By a lattice point we shall understand a point in the plane, both of whose coordinates are rational integers. In connection with (M) a critical region is defined to be a convex symmetric region of area 4, which includes no lattice point other than O. One such region is the open square S={(x, y)| |x| < 1, |y|<1} an infinite set of critical regions is formed by the parallelograms bounded by the lines y = x+1, y = x-1, y-1 = k (x-l), y+1 = k(x+l), 0≤-k<∞. Finally, there is a critical hexagon H, bounded by the following six lines: y-1 = -(x-l), y+1 = -(x+1), y-1 = mx, y+1 = mx, y = (1/m)(x-1), y = (l/m)(x+l), where m = tan π/12. All the vertex angles of H are equal to 2π/3.

In general topology, one knows several standard extension spaces defined for one class of spaces or another and it is a natural question concerning any two such extensions which are defined for the same space whether they can ever be equal to each other. In the following, this problem will be considered for the Stone-Čech compactification βE of a completely regular non-compact Hausdorff space E[4] and Katětov's maximal Hausdorff extension κE of E[5]. It will be shown that βEκE always holds or, what amounts to the same, that κE can never be compact. As an application of this it will be proved that any completely regular Hausdorff space is dense in some non-compact space in which the Stone-Weierstrass approximation theorem holds.

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.

In a recent note in this Bulletin [3], W.A.J. Luxemburg has shown in two different ways that a condition of Krein and Krasnosel'skii [2] for the uniqueness of solutions of a differential equation also implies the convergence of the successive approximations. Here, a third proof of the uniqueness and the convergence of successive approximations, formulated for systems of differential equations, will be obtained. This third proof is modelled on the methods used in proving general uniqueness and convergence theorems [l]. The approach is suggested by Luxemburg's idea of breaking the argument into two stages and using one of the hypotheses in each stage. Since the proofs given here are hardly shorter than the earlier direct proofs, their main interest lies in the fact that they fit what appeared to be an isolated result into the framework of a general theory.

A bipartite graph is a system consisting of two sets of vertices S and T and a set of edges K, each edge joining a vertex of S to a vertex of T. A set U of edges of K is said to be independent if no two edges of U have a vertex in common. The largest possible number of independent edges has been variously called the exterior dimension [3], term rank [4, 5, 7], etc. This number is the same as the smallest number of vertices in a set W such that each edge of K has at least one of its vertices in W. The edges of a finite bipartite graph can be represented as a set of cells in a matrix as follows. If S = a1, a2, …, an T = b1, b2, … bm, the edges of K are represented by some of the cells of an n by m matrix as follows: if K contains the edge joining ai to bj then the (i, j)th cell of the matrix represents this edge. It is convenient sometimes to represent the set K by a matrix A with real entries aij where aij = 0 if ai is not joined to bj in K and aij > 0 if ai is joined to bj in K. Any non-null graph K will have infinitely many matrix representations.

A list of known finite simple groups has been given by Dickson [3, 4]. With but five exceptions, all of them fall into infinite families. The five exceptional groups, discovered by Mathieu [8,9], were further investigated by Jordan [7], Miller [10], de Séguier [11], Zassenhaus [13], and Witt [12]. In Witt's notation they are M11, M12, M22, M23, M24. Generators for them may be seen in the book of Carmichael [1, pp. 151, 263, 288]; but only for the smallest of them, M11 of order 7920, has a set of defining relations been given.

Let (XxY, SxT, μxν) denote the completion of the Cartesian product of the σ-finite and complete "measure spaces (X, S, μ.) and (Y, T, ν) [3]. Let λx and λy denote arbitrary length functions defined on (X, S, μ.) and (Y, T, ν) respectively, the conjugate length functions [2]. We suppose that

1

is defined for every f(x, y) measurable (SxT). The Fubini theorem implies that f(x, y) is measurable (T) for almost all x. Thus λ xy(f) will be defined when λy(f) Is measurable (S). If Lλ y = Lp, 1 ≤ p < ∞, this is implied by the Fubini theorem. General conditions ensuring that λ y (f) is measurable (S) are given in [l, Theorem 3.2] When λxy(f) is defined for every f(x, y) measurable (SxT), it is a length function and Lλ xy is a Banach space [l, Theorem 3.l].

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.

Historically, the first example of a ring of quotients was the quotient field of an integral domain. Later on, conditions were found under which a noncommutative integral domain has a quotient division ring. More recently, R.E. Johnson [4], Y. Utumi [5], and G.D. Findlay and J. Lambek [3] have discussed the existence and structure of a maximal ring of quotients of any ring.

The present paper uses the methods of Findlay and Lambek to recast the results of Johnson on the quotient ring of a ring with zero singular ideal. It is also shown that such a ring has a unique left-right maximal ring of quotients.

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 Un be an n-dimensional unitary space with inner product In Un let Sn-1 denote the unit sphere:

Let A be an arbitrary linear transformation of Un.

The more important properties of the class κ of all bounded convex bodies in E3 with non-empty interior include: uniform approximability by polyhedra, existence of volume and surface area, and Blaschke's selection principle, [l ], [2 ]. In this note we define and consider a class ℋ of star-shaped bodies in E3, which enjoys many properties of κ, among them the above-mentioned ones, and is considerably larger. Roughly speaking, ℋ consists of closed bounded sets in E3 with nonempty interior, whose boundary is completely visible from every point of a set with non-empty interior. It turns out that ℋ is identifiable with the class of all real-valued positive functions on the sphere S3 which satisfy a Lipschitz condition.

Let k be the kernel of a “general transformation”; that is, k(x) / x ϵ L2 (0,∞), and if x and y are positive

1

Then it is well known (see for example [8; Theorems 129 and 131]) that if the transform of fϵL2 (0,∞) is g, that is, if

2

then the inverse transform is given by

3

In practice, the inversion formula (3) is often hard to use. For example, the integral may be too difficult to evaluate; moreover, since (2) requires a differentiation, it is not well suited for numerical calculation. Hence it seems worthwhile to find other methods for inverting the transformation.

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

Two important mathematical constructions are: the construction of the rational s from the integers and the construction of the reals from the rationals. The first process can be carried out for any ring, producing its maximal ring of quotients [4, 5]. The second process can be carried out for any partially ordered set producing its Dedekind-MacNeille completion [2, p. 58]. We will show that for Boolean rings, which are both rings and partially ordered sets, the two constructions coincide.

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)

W.A. Wythoff [1] in 1907 defined a modification of the game of Nim by the following rules:

1. (i) there are two players who play alternately;

2. (ii) initially there are two piles of matches, an arbitrary number in each pile;

3. (iii) a player may take an arbitrary number of matches from one pile or an equal number from both piles but he must take at least one match;

4. (iv) the player who takes the last match wins the game.

If, after his move, a player leaves one match in one pile and two in the other he can force a win; for if his opponent takes one match from the pile containing two he can take both remaining matches; and similarly for the other possibilities.

Two sequences of natural numbers are said to be complementary if they contain all the positive integers without repetition or omission. S. Beatty [l] observed that the sequences

(1)

(2)

(where square brackets denote the integral part function) are complementary if and only if α > 0 and α is irrational. We call the pair (1),(2) Beatty sequences of argument α.

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.