Let (X, d) be a metric space with metric d, and h be a homeomorphism of X onto itself. Any point y in X is called a regular point (2) under A if for any given ϵ > 0 there exists a δ > 0 such that d(x, y) < δ implies that d(hn(x), hn(y)) < ϵ for all integers n, where hn is the composition of h or h-1 with itself |n| times, depending upon whether n is positive or negative, and h0 is the identity on X. If y is not regular under h, then y is called irregular. We shall denote the set of regular points by R(h) and the set of irregular points by I(h).

One of the authors published in 1965 a paper with identical title (1), in which the following result was proved:

Theorem A. Let ϕ(r) be increasing and convex in log r, with

For any ring we shall denote by the ring of all n × n matrices with elements from and by the set of all polynomials in x with coefficients from .

will denote the non-commutative four-dimensional division algebra of real quaternions with 1, i1, i2, i3 as generators

In (8) and (13) it has been shown that certain bisimple inverse semigroups, called bisimple ω-semigroups and bisimple Z-semigroups, can be represented as semigroups of ordered triples. In these cases, two of the components of each triple are integers, and the third is drawn from a fixed group. This representation is analogous to that given by the theorem of Rees (1, p. 94) concerning completely simple semigroups, and shares the same advantages.

In this paper we use a generalized form of Polya's theorem (1) to obtain generating functions for the number of ordinary graphs with given partition and for the number of bicoloured graphs with given bipartition. Both the points and lines of the graphs are taken as unlabelled. These graph enumeration problems were proposed by Harary in his review article (4). Read (7, 8) solved the problem for unlabelled general graphs and labelled ordinary graphs.

The object of this paper is to establish the equivalence of four functionrelated dimension concepts in arbitrary topological spaces. These concepts involve stability of functions (3, p. 74), the modification of covering dimension involving basic covers (1, p. 243) (which is equivalent to Yu. M. Smirnov's definition using normal covers), the definition involving essential mappings (2, p. 496), and a modification of the closed set separation characterization of dimension in (3, p. 35).

1. Let

be an entire function of two complex variables z1 and z2, holomorphic in the closed polydisk . Let

Following S. K. Bose (1, pp. 214-215), μ(r1, r2; ƒ ) denotes the maximum term in the double series (1.1) for given values of r1 and r2 and v1{m2; r1, r2) or v1(r1, r2), r2 fixed, v2(m1, r1, r2) or v2(r1, r2), r1 fixed and v(r1r2) denote the ranks of the maximum term of the double series (1.1).

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL(ℵ) has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL(ℵ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.

Let

1

2

Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.

The introduction of a mathematics student to formal integration theory usually follows the lines laid down by Riemann and Darboux. Later a change of ideas is necessary if he tackles Lebesgue's more powerful theory, and connections between the two are laboriously constructed. On the other hand, the commonest method of evaluating an integral is through the operation inverse to differentiation (the indefinite integral taken between limits). We refer to this as the calculus integral; few realize that this can succeed in cases where even the Lebesgue integral does not exist, let alone the Riemann one. An example is given later.

Let R be a ring. We denote by o the so-called circle composition on R, denned by a o b = a + b — ab for a, b ∊ R. It is well known that this composition is associative and that R is a radical ring in the sense of Jacobson (see 6) if and only if the semigroup (R, o) is a group. We shall say that R is a generalized radical ring if (R, o) is a union of groups. Such rings might equally appropriately be called generalized strongly regular rings, since every strongly regular ring satisfies this property (see Theorem A below). This definition was in fact partially motivated by the observation of Jiang Luh (7) that a ring is strongly regular if and only if its multiplicative semigroup is a union of groups.

A graph G = G(n, e) consists of a set of n nodes e pairs of which are joined by a single edge; we assume that no edge joins a node to itself. A graph with modes is called a complete -graph if each pair of its nodes is joined by an edge. The graphs belonging to some collection of graphs are independent if no two of them have a node in common. The maximum number of independent complete -graphs contained in a given graph G will be denoted by Ik(G).

This paper concerns an algorithm, proposed by C. C. Gotlieb (4) and modified by J. Csima (1; 2), for a recent combinatorial problem whose application includes the construction of school time-tables. Theoretically, the problem is related to systems of distinct subset representatives, the construction of Latin arrays, the colouring of graphs, and flows in networks (1; 2; 3). I t was conjectured by Gotlieb and Csima that if solutions to a given time-table problem existed, i.e. if time-tables incorporating certain pre-assigned meetings existed, their algorithm would find one.

The ultraspherical function

1.1

for |1 — x| < 2 is a solution of the differential equation

1.2

This equation has two independent solutions; of the two, only Pn(λ)(x) is analytic at x = 1, aside for some special values of λ, which we shall not consider.

Let ƒ(x) be a non-differentiable function, i.e. a realvalued continuous function denned on a linear interval which has nowhere a finite or infinite derivative. We shall say that ƒ(x) has symmetrical derivates at a point x if the four Dini derivates of ƒ(x) at x satisfy the relations

and otherwise we shall say that ƒ(x) has asymmetrical dérivâtes at x.

In (8) M. V. Menon investigates the diagonal equivalence of a non-negative matrix A to one with prescribed row and column sums and shows that this equivalence holds provided there exists at least one non-negative matrix with these row and column sums and with zeros in exactly the same positions A has zeros. However, he leaves open the question of when such a matrix exists. W. B. Jurkat and H.J. Ryser in (7) study the convex set of all m × n non-negative matrices having given row and column sums.

The axioms of Euclidean geometry may be divided into four groups: the axioms of order, the axioms of congruence, the axiom of continuity, and the Euclidean axiom of parallelism (6). If we omit this last axiom, the remaining axioms give either Euclidean or hyperbolic geometry. Many important theorems can be proved if we assume only the axioms of order and congruence, and the name absolute geometry is given to geometry in which we assume only these axioms. In this paper we investigate what can be proved using congruence axioms that are weaker than those used previously.

Nous nous proposons d'étudier une question posée par L. Lesieur et R. Croisot (12). Un G-anneau possède un anneau de fractions à gauche semi-simple qui est son enveloppe injective. Un J-anneau possède-t-il un anneau de fractions qui soit un sous-anneau de son enveloppe injective ? Nous donnerons plusieurs exemples de J-anneaux qui ne possèdent pas d'anneau de fractions. Cependant nous donnerons une réponse positive dans un certain nombre de cas.

Let K be a commutative field, let GL(n, K) be the multiplicative group of all non-singular n × n matrices with elements from K, and let SL(n, K) be the subgroup of GL(n, K) consisting of all matrices in GL(n, K) with determinant one. We denote the determinant of matrix A by |A|, the identity matrix by In, the companion matrix of polynomial p(λ) by C(p(λ)), and the transpose of A by AT. The multiplicative group of nonzero elements in K is denoted by K*. We let GF(pn) denote the finite field having pn elements.

In (1) R. G. Douglas says: “For a finite abelian group there exists a unique invariant mean which must be inversion invariant. For an infinite torsion abelian group it is not clear what the situation is.” It is not hard to see that if every element of an abelian group G is of order 2, then every invariant mean on G is also inversion invariant (see 1, remark 4). In this note we prove the following theorem (Theorem 1 below): An abelian torsion group G has an invariant mean which is not inverse invariant if, and only if, 2G is infinite. This result, together with the theorems of Douglas, answers completely the question of the existence (on an arbitrary abelian group) of invariant means which are not inverse invariant.