In this paper we give necessary and sufficient conditions that a continuous transformation f: A→A of a metric space A with the metric r should be a contraction with respect to an equivalent metric s. This is the solution of a problem stated by J. S. W. Wong [2].

J. A. Erdös proved recently [1] that every singular matrix over a field F is a product of idempotent matrices. He gave two proofs, one valid for matrices which are similar to triangular matrices and the other valid in general. We shall give a simple geometric proof of the above result. Instead of matrices we use linear operators. Moreover we get an explicit factorization in terms of projectors (idempotent operators).

In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.

In a recent paper Box and Cox (1964) considered the problem of transforming dependent variables in regression and analysis of variance problems, in order to achieve the usual assumptions of Normality, constant variance and additivity of effects. Here we adopt the same approach to investigate transformations of data which allow the transformed observations to follow a Gamma distribution. A special case of this is the exponential distribution, valuable in life-testing, for which examples are given.

Let A be an m×n (0, l )-matrix. Let C1, C2, …, Cn denote its columns. A sequence of distinct columns is said to form a chain if the inner product of and (for 1 ≤ t ≤ k-l) is at least one. k-1 is called the length of the chain and this chain is said to connect are said to be connected. As can be easily seen, connectedness is an equivalence relation on the set of columns. A matrix is called connected if all its columns belong to the same equivalence class. If Ci and Cj belong to the same equivalence class, then s(Ci, Cj) will denote the length of the shortest chain between Ci and Cj We define the distance between any two columns Ci and Cj to be denoted by d(Ci, Cj), in the following manner.

J.H. van Lint has recently shown [1] that if A(n, m) denotes the n number of representations of m in the form , where εk = 0 or 1 for -n ≤ k ≤ n then

(1)

Using this result, the fact that A(n, m) is a non-increasing function of |m|, and a simple recurrence relation for A(n, m) we derive the following extension of (1):

(2)

where [0 (n)] is any integral valued function m(n) = 0(n).

This is a continuation of the author's article [3], and it contains further problems connected with the theory of convex sets in En. To the list of general references in [3] may be added the recent book [2] on convex polyhedra.

1) Let A and B be two convex bodies in E2 and let a packing P = {B1, B2, …} be an infinite sequence of homothetic images of B sucn that:

1. a) each Bn is a subset of A,

2. b) no two of them share interior points,

3. c) Area (A) =

The existence of such packings is guaranteed by Vitali's Theorem. Let D(X) be the diameter of the set X and put .

The nth power Gn of a connected graph G is the graph with the same point set as G and where two points u and v are adjacent in Gn if and only if the distance between u and v in G is at most n. The graph G2 is called the square of G while G3 is referred to as the cube of G.

It is known [3], [5] that, the complex-valued solutions of

(B)

(apart from the trivial solution f(x)≡0) are of the form

(C)

(D)

In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],

(E)

In this paper, the functional equation

(P)

where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.

The distance d(x, y) between vertices x, y of a graph G is the length of the shortest path from x to y in G. The diameter δ(G) of G is the maximum distance between any pair of vertices in G. i.e. δ(G) = max max d(x, y). In this note we obtain an upper bound

x ε G y ε G

for δ(G) in terms of the numbers of vertices and edges in G. Using this bound it is then shown that for any complement-connected graph G with N vertices

where is the complement of G.

Let C denote the complex algebra of continuous functions of a non-negative real variable under addition, scalar multiplication and convolution. C has no divisors of zero and its quotient field F is called the field of Mikusiński operators [1]. It is well known that Mikusiński has defined a sequential convergence in F which is not topological [2]. Using a recent result due to T.K. Boehem [3] we shall provide F with a sequential convergence which is topological.