Showing papers in "Pacific Journal of Mathematics in 1957"

A theorem on flows in networks

Abstract: The theorem to be proved in this note is a generalization of a well-known combinatorial theorem of P. Hall, [4].

A congruence theorem for trees.

Abstract: Let A and JB be two trees with vertex sets au α2, , an and bi, b2, • ••, bn respectively. The trees are congurent, are isomorphic, or \"are the same type\", (Aζ^B), if there exists a one-to-one correspondence between their vertices which preserves the join-relationship between pairs of vertices. Let c(at) denote the (n-l)-point subgraph of A^obtained by deleting at and all joins (arcs, segments) at at from A. It is the purpose here to show that if there is a one-to-one correspondence in type, and frequency of type, between the sub-graphs of order n — lmA and J5, that is, if there exists a labeling such that φ ^ ) ^ cφi), i = l , 2, •••, n, then A ~ B. It is assumed throughout, therefore, that there is a labeling of the two trees A and B such that c(α4)^c(δ4), i = l , 2, •••, n9 where n^Z. Some lemmas to the main theorem are established first. Let T denote a certain type of graph of order j , where 2

Lie algebras of locally compact groups.

Abstract: l IntroductionWe call an LP-group, a group which is the projective limit of Lie groups. Yamabe [8] has proved that every connected locally compact group is an LP-group. This permits the extension to locally compact groups of the notion of a Lie algebra. In §§ 2 and 3 we prove the existence and uniqueness of the Lie algebra of an LP-group and show the connection of the Lie algebra with the group by means of the exponential mapping. In § 4, we extend the notion of a universal covering group for connected groups with the same Lie algebra. A covering group of a connected group g, in the extended sense used here, means a pair (g, w), where ~g is a connected LP-group and w is a continuous representation of g into g which induces an isomorphism of the Lie algebra of g onto the Lie algebra of g (see Definition 4.5). The universal covering group of a connected locally compact group is not necessarily locally compact and may not map onto the group. It turns out that the arc component of the identity in g is a covering space in the sense of Novosad [5] of the arc component of the identity of g (these components are dense subgroups, Lemma 3.7). Finally, in § 5, we establish a one-to-one correspondence between \" canonical LP-subgroups \" of a group and subalgebras of its Lie algebra.

On the least primitive root of a prime.

Abstract: (1.5) This last result, of course, is not directly comparable with the others, giving better results for some primes and worse results for others. In any event, all of the results are very weak (as is evidenced by a glance at tables of primitive roots [1]) in relationship to the conjecture that the true order of g(p) is about log p. In this connection, Pillai [5] has proved (1.6) g(p)>loglogp for infinitely many p. In this note we shall give a very simple way of handling character sums, which not only yields (1.3) and (1.4) but allows a small improvement of these results; for example (1.7) g(p)=0(mp) , (c a constant).

On spaces with a multiplication

Abstract: Introduction. This paper is divided into three parts, together with an appendix. In the first part we discuss the homotopy theory of mappings into a space with a multiplication, such as a topological group. These spaces are more general than the group-like spaces considered by G. W. Whitehead in [6], and our treatment, as far as it goes, is quite different from his. In the second and third parts we apply the theory to the reduced product spaces of [2] and the loop-spaces of [4]. We arrive at useful new definitions of the Hopf construction and the Whitehead product, such that the relations between them are plainly exhibited. In many respects this completes the theory of the suspension triad as developed in [3].

Some remarks on a paper of Aronszajn and Panitchpakdi

Abstract: In the paper of the title [1], a number of problems are posed. Ne­ gative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self­ contained. 1. A Hausdorff space X is said to satisfy (Qm), where m is an in finite cardinal, if, whenever U and V are disjoint open subsets of X such that each is a union of the closures of less than m open subsets of X, then U and V have disjoint closures. In particular, a normal (Hausdorff) space X satisfies (Q*,) if and only if disjoint open F.-SUbsets of X have disjoint closures. (For, an open set that is the union of less than ~, closed sets is a fortiori an F.. Conversely if U is the union of countably many closed subsets Fo> then since X is normal, for each n there is an open set Un containing Fn whose closure is contained in U. Thus U is the union of the closures of the open sets Un.) In Prob­ lem 3 of [1], it is asked if every compact (HaUSdorff) space satisfying (Qm) for some m>~o is necessarily totally disconnected, and it is re­ marked that this is the case if the first axiom of countability is also as­ sumed. If X is a completely regular space, let C(X) denote the ring of all continuous real-valued functions on X, and let Z(f)= {x EX: f(x)=O}, let P(f)={xEX:f(x»O}, and let N(f)=P(-f). As usual, let (IX denote the Stone-Cech compactification of X. If every finitely generated ideal of C(X) is a principal ideal, then X is called an F-space. The fol­ lowing are equivalent. ( i) X is an F-space. ( ii) If f E C(X), then P(f) and NU) are completely separated [2, Theorem 2.3]. (iii) If f E C(X), then every bounded g E C(X-ZU» has an ex­ tension g E C(X) [2, Theorem 2.6]. A good supply of compact F-spaces is provided by the fact that if X is locally compact and ,,-compact, then {lX-X is an F-space [2, Theo­ rem 2.7].