After showing that each screenable Moore space is pointwise paracompact and that the converse is not true, Heath in [4] asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be screenable. In [12], Traylor asked for a necessary and sufficient condition for a pointwise paracompact Moore space to be metrizable. It is the purpose of this paper to provide such conditions, and to establish relationships between those conditions and metrization problems in Moore spaces.

A Moore space S is a space (all spaces are T1) in which there exists a sequence G = (G1, G2, …) of open coverings of S, called a development, which satisfies the first three parts of Axiom I in [7]. The statement that a collection H of subsets of the space S is point finite (point countable) means that no point of S belongs to infinitely (uncountably) many elements of H.

Commutative Bass rings, which form a special class of Gorenstein rings, have been thoroughly investigated by Bass [1]. The definitions do not carry over to non-commutative rings. However, in case one deals with orders in separable algebras over fields, Bass orders can be defined. Drozd, Kiricenko, and Roïter [3] and Roïter [6] have clarified the structure of Bass orders, and they have classified them. These Bass orders play a key role in the question of the finiteness of the non-isomorphic indecomposable lattices over orders (cf. [2; 8]). We shall use the results of Drozd, Kiricenko, and Roïter [3] to compute the Grothendieck groups of Bass orders locally. Locally, the Grothendieck group of a Bass order (with the exception of one class of Bass orders) is the epimorphic image of the direct sum of the Grothendieck groups of the maximal orders containing it.

Let A be a C*-algebra and a Hilbert space which is infinite dimensional and of Hilbert dimension ≧ dim π for all π ∈ Â. Suppose that the set Irr of all non-null *-representations π of A on , irreducible on the essential space , is given the relative strong topology as a subspace of Rep [2; 4; 6]. That is, the topology is that of simple convergence in with the strong topology. Finally, let ∼ denote equivalence of representations in Irr implemented by partial isometries in if and only if there exists a partial isometry with vv* ⊃ H(π1) and v*v ⊃ H(π2) satisfying π2(a) = v*π1(a)v for all a ∈ A.

It was proved almost forty years ago that every mapping of a tree into itself has at least one fixed point, but not much is known so far about the structure of the possible fixed point sets. One topic related to this question, the study of homeomorphisms and monotone mappings of trees which leave an end point fixed, was first considered by G. E. Schweigert [6] and continued by L. E. Ward, Jr. [8] and others. One result by Schweigert and Ward is the following: any monotone mapping of a tree onto itself which leaves an end point fixed, also leaves at least one other point fixed.

It is further known that not only single-valued mappings, but also upper semi-continuous (use) and connected-valued multifunctions of trees have a fixed point [7], and that two use and biconnected multifunctions from one tree onto another have a coincidence [5].

The point-arboricity ρ (G) of a graph G is defined as the minimum number of subsets into which the point set V(G) of G may be partitioned so that each subset induces an acyclic subgraph. Equivalently, the point-arboricity of G may be defined as the least number of colours needed to colour the points of G so that no cycle of G has all of its points coloured the same. This term was introduced by Chartrand, Geller, and Hedetniemi [1], although the concept was first considered by Motzkin [4].

As with the chromatic number of a graph G, which we denote by χ(G), there is no explicit formula for the point-arboricity of a graph. However, Nordhaus and Gaddum [5] have shown that if G is a graph with p points, then

On dit qu'un nombre entier A est hautement composé si tout nombre M plus petit que A a moins de diviseurs que A. Si l'on définit d(n) = nombre de diviseurs de n, on sait que, si la décomposition en facteurs premiers de n est n = Πi piai, on a:

La définition devient: A est hautement composé si et seulement si:

1

Ramanujan [8] a défini et étudié les nombres hautement composés, démontrant les propriétés suivantes:

Si A = 2a2 3a3 … pkapk est un nombre hautement composé, on a:

et à l'exception de A = 4 et A = 36, on a: apk = 1 [8, §8].

In a topological plane with strong enough topological properties one can use [6] open triangular regions to define a base for the topology. Similarly, one can use these regions to define boundedness of a set. In this setting we show that in the absolute plane geometry, the holding of the Heine-Borel theorem is equivalent to every four points being contained in some such region and that this second condition is equivalent to the parallel postulate. Thus we give two new conditions equivalent to the parallel postulate.

Let V be a vector space over the field K. A group G of K-linear transformations of V onto itself is primitive in case no proper nontrivial subspace of V is G-invariant and V cannot be written as a direct sum of proper subspaces permuted among themselves by G. Equivalently, G is primitive on V in case G is irreducible and is not induced from a proper subgroup.

Suprunenko showed [3, Theorem 12, p. 28] that the n-dimensional general linear group GL(n, K) has a solvable primitive subgroup only if

(1) there is a divisor, m, of n such that K has an extension field of degree m containing a primitive p-th root of 1 for each prime p dividing n/m.

The main result of this note is the converse fact.

An ordered set (u1, …, un) of positive Cn(a, b)-solutions of the linear differential equation

1.1

will be called fundamental principal system on [a, b) provided that

1.2

and

1.3

A system (u1, …, un) satisfying just (1.2) will be called a principal system on [a, b). In any principal system (u1, …, un) the solution u1 will be called a minimal solution.

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.

Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

In [2], J. Gross provides an infinite collection of topologically distinct irreducible homology 3-spheres. In this paper, we construct for any finitely generated abelian group A, an infinite collection {Mi} of topologically distinct irreducible closed 3-manifolds such that H1(Mi) = A for each i.

The proof consists of first constructing a closed irreducible 3-manifold MA with H(MA) = A, and then providing a method for producing more such manifolds with the same first homology group.

All maps and spaces in this paper are assumed to be in the piecewise linear category, and all subspaces are assumed to be piecewise linear subspaces.

A 3-manifold M is irreducible if each 2-sphere in M bounds a 3-cell in M. A compact 2-manifold (or surface) F in a compact 3-manifold M is properly embedded in M if F ∩ bdM = bdF.

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity

1.1

in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to

1.2

In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of

1.3

to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

The following (so-called unitary equivalence) problem is of paramount importance in the theory of operators: given two (bounded linear) operators A1, A2 on a (complex) Hilbert space , determine whether or not they are unitarily equivalent, i.e., whether or not there is a unitary operator U on such that U*A1U = A2. For normal operators this question is completely answered by the classical multiplicity theory [7; 11]. Many authors, in particular, Brown [3], Pearcy [9], Deckard [5], Radjavi [10], and Arveson [1; 2], considered the problem for non-normal operators and have obtained various significant results. However, most of their results (cf. [13]) deal only with operators which are of type I in the following sense [12]: an operator, A, is of type I (respectively, II1, II∞, III) if the von Neumann algebra generated by A is of type I (respectively, II1, II∞, III).

The projective planes discovered in 1957 by Hughes [3] were originally described by means of a nearfield F satisfying the following conditions:

(a) F is finite,

(b) the centre and kernel of F coincide,

(c) F is of rank 2 over its kernel.

(The definitions of these terms will be given in § 2; the terminology used throughout the paper is that of [1].)

Rosati [5] showed in 1960 that condition (a) is not necessary, thus constructing the first “infinite Hughes planes”. Condition (b), however, plays an essential part also in Rosati's work.

The aim in this paper is to show that condition (b) is superfluous as well. For the finite case, this has been remarked by Ostrom [4] without proof; here we shall show that a “generalized Hughes plane” can be constructed over any nearfield satisfying condition (c) only.

In § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).

In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, then

and note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

Let G be a complete normed abelian group with norm N1. Let S be an interval (bounded or otherwise) of real numbers. We propose to study the Stieltjes integral equation

where p is in G, a is in S, and each Fk is a function on S each value of which is a function from G to G. Our primary tools of investigation will be the works of J. S. MacNerney [6; 7] and their extensions by the author [4; 5]. Our main result, Theorem 4, will show that the equation above can be solved by a product integral of infinite products of solutions for the summands of the integrator. After obtaining our results, we shall specialize them to a linear situation and then show how this specialization allows us to obtain representations for analytic functions having only invertible values in a complex Banach algebra with identity.