Positive Clifford semigroups on the plane
Reuben W.
Farley
353-369
Abstract: This work is devoted to a preliminary investigation of positive Clifford semigroups on the plane. A positive semigroup is a semigroup which has a copy of the nonnegative real numbers embedded as a closed subset in such a way that 0 is a zero and 1 is an identity. A positive Clifford semigroup is a positive semigroup which is the union of groups. In this work it is shown that if S is a positive Clifford semigroup on the plane, then each group in S is commutative. Also, a necessary and sufficient condition is given in order that S be commutative, and an example is given of such a semigroup which is, in fact, not commutative. In addition, both the number and the structure of the components of groups in S is determined. Finally, it is shown that S is the continuous isomorphic image of a semilattice of groups.
Infinite general linear groups over rings
George
Maxwell
371-375
Abstract: We are interested in the normal subgroups of the group G of automorphisms of a free module of infinite type over a commutative ring A. To this end, we introduce a certain ``elementary'' subgroup E of G and find that the subgroups of G normalised by E are exactly those which lie in congruence layers determined by the ideals of A. The normal subgroups are thus to be found in such layers.
Continuity of Gaussian processes
M. B.
Marcus;
L. A.
Shepp
377-391
Abstract: We give a proof of Fernique's theorem that if X is a stationary Gaussian process and ${\sigma ^2}(h) = E{(X(h) - X(0))^2}$ then X has continuous sample paths provided that, for some $\varepsilon > 0,\sigma (h) \leqq \psi (h),0 \leqq h \leqq \varepsilon $, where $\psi$ is any increasing function satisfying $\displaystyle \int_0^\varepsilon {\frac{{\psi (h)}}{{h{{(\log(1/h))}^{1/2}}}}} dh < \infty .$ ($ \ast$) We prove the partial converse that if $ \sigma (h) \geqq \psi (h),0 \leqq h \leqq \varepsilon $ and $\psi$ is any increasing function not satisfying $( ^\ast )$ then the paths are not continuous. In particular, if $\sigma$ is monotonic we may take $\psi = \sigma$ and $(^\ast)$ is then necessary and sufficient for sample path continuity. Our proof is based on an important lemma of Slepian. Finally we show that if $ \sigma$ is monotonie and convex in $ [0,\varepsilon ]$ then $ \sigma (h){(\log \,1/h)^{1/2}} \to 0$ as $h \to 0$ iff the paths are incrementally continuous, meaning that for each monotonic bounded sequence $t = {t_1},{t_2}, \ldots ,X({t_{n + 1}}) - X({t_n}) \to 0$, w.p.l.
The genus of repeated cartesian products of bipartite graphs
Arthur T.
White
393-404
Abstract: With the aid of techniques developed by Edmonds, Ringel, and Youngs, it is shown that the genus of the cartesian product of the complete bipartite graph $ {K_{2m,2m}}$ with itself is $1 + 8{m^2}(m - 1)$. Furthermore, let $ Q_1^{(s)}$ be the graph $ {K_{s,s}}$ and recursively define the cartesian product $ Q_n^{(s)} = Q_{n - 1}^{(s)} \times {K_{s,s}}$ for $n \geqq 2$. The genus of $Q_n^{(s)}$ is shown to be $1 + {2^{n - 3}}{s^n}(sn - 4)$, for all n, and s even; or for $n > 1$, and $s = 1 \;$ or$\; 3$. The graph $Q_n^{(1)}$ is the 1-skeleton of the n-cube, and the formula for this case gives a result familiar in the literature. Analogous results are developed for repeated cartesian products of paths and of even cycles.
Measures with bounded convolution powers
Bertram M.
Schreiber
405-431
Abstract: For an element x in a Banach algebra we study the condition $\displaystyle \mathop {\sup }\limits_{n \geqq 1} \left\Vert {{x^n}} \right\Vert < \infty .$ ($ 1$) Although our main results are obtained for the algebras $M(G)$ of finite complex measures on a locally compact abelian group, we begin by considering the question of bounded powers from the point of view of general Banach-algebra theory. We collect some results relating to (1) for an element whose spectrum lies in the unit disc D and has only isolated points on $\partial D$. There follows a localization theorem for commutative, regular, semisimple algebras A which says that whether or not (1) is satisfied for an element $x \in A$ with spectral radius 1 is determined by the behavior of its Gelfand transform $\hat x$ on any neighborhood of the points where $\vert\hat x\vert = 1$. We conclude the general theory with remarks on the growth rates of powers of elements not satisfying (1). After some applications of earlier results to the algebras $M(G)$, we prove our main theorem. Namely, we obtain strong necessary conditions on the Fourier transform for a measure to satisfy (1). Some consequences of this theorem and related results follow. Via the generalization of a result of G. Strang, sufficient conditions for (1) to hold are obtained for functions in ${L^1}(G)$ satisfying certain differentiability conditions. We conclude with the result that, for a certain class $ \mathcal{G}$ of locally compact groups containing all abelian and all compact groups, a group $ G \in \mathcal{G}$ has the property that every function in ${L^1}(G)$ with spectral radius one satisfies (1) if and only if G is compact and abelian.
Orbits of the automorphism group of the exceptional Jordan algebra.
John R.
Faulkner
433-441
Abstract: Necessary and sufficient conditions for two elements of a reduced exceptional simple Jordan algebra $\Im$ to be conjugate under the automorphism group $\mathrm{Aut} \Im $ of $\Im$ are obtained. It was known previously that if $\Im$ is split, then such elements are exactly those with the same minimum polynomial and same generic minimum polynomial. Also, it was known that two primitive idempotents are conjugate under $\mathrm{Aut} \Im$ if and only if they have the same norm class. In the present paper the notion of norm class is extended and combined with the above conditions on the minimum and generic minimum polynomials to obtain the desired conditions for arbitrary elements of $ \Im$.
On topologically invariant means on a locally compact group
Ching
Chou
443-456
Abstract: Let $\mathcal{M}$ be the set of all probability measures on $\beta N$. Let G be a locally compact, noncompact, amenable group. Then there is a one-one affine mapping of $ \mathcal{M}$ into the set of all left invariant means on ${L^\infty }(G)$. Note that $\mathcal{M}$ is a very big set. If we further assume G to be $\sigma$-compact, then we have a better result: The set $\mathcal{M}$ can be embedded affinely into the set of two-sided topologically invariant means on ${L^\infty }(G)$. We also give a structure theorem for the set of all topologically left invariant means when G is $\sigma$-compact.
A spectral sequence for the homotopy of nice spaces
A. K.
Bousfield;
E. B.
Curtis
457-479
The plane is not compactly generated by a free mapping
S. A.
Andrea
481-498
Abstract: Let X denote the plane, or the closed half-plane, and let $ T:X \to X$ be a self homeomorphism which preserves orientation and has no fixed points. It is proved that, if A is any compact subset of X, then there exists an unbounded connected subset B of X which does not meet $ {T^n}(A)$ for any integer n.
Homology of deleted products of one-dimensional spaces
Arthur H.
Copeland;
C. W.
Patty
499-510
Abstract: The object of this paper is to investigate the homology of deleted products of finitely triangulated one-dimensional spaces. By direct calculation, we obtain upper bounds for the two-dimensional Betti numbers, and, using a rather small system of topological types of spaces appearing as subspaces of the space under consideration, we obtain lower bounds for these Betti numbers. We demonstrate that, in general, the two-dimensional Betti numbers are larger than they were originally thought to be.
Isotopy invariants in quasigroups
Etta
Falconer
511-526
Abstract: The purpose of this paper is to investigate quasigroup and loop identities that are invariant under isotopy. The varieties of quasigroups and the varieties of loops that are closed under isotopy form isomorphic lattices. Some methods of generating isotopically closed varieties of loops are given.
On the symmetric cube of a sphere
Jack
Ucci
527-549
$k$-groups and duality
N.
Noble
551-561
Tensor product bases and tensor diagonals
J. R.
Holub
563-579
Abstract: Let X and Y denote Banach spaces with bases $({x_i})$ and $({y_i})$, respectively, and let $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ denote the completion in the $\varepsilon$ and $\pi$ crossnorms of the algebraic tensor product $X \otimes Y$. The purpose of this paper is to study the structure of the tensor product spaces $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ through a consideration of the properties of the tensor product basis $({x_i} \otimes {y_j})$ for these spaces and the tensor diagonal $({x_i} \otimes {y_i})$ of such bases.
Continua for which the set function $T$ is continuous
David P.
Bellamy
581-587
Abstract: The set-valued set function T has been studied extensively as an aid to classifying metric and Hausdorff continua. It is a consequence of earlier work of the author with H. S. Davis that T, considered as a map from the hyperspace of closed subsets of a compact Hausdorff space to itself, is upper semicontinuous. We show that in a continuum for which T is actually continuous (in the exponential, or Vietoris finite, topology) semilocal connectedness implies local connectedness, and raise the question of whether any nonlocally connected continuum for which T is continuous must be indecomposable.
Metrizability of compact convex sets
H. H.
Corson
589-596
Abstract: It is proved that a compact convex set is metrizable if the set of extreme points is the continuous image of a complete separable metric space.
Close isotopies on piecewise-linear manifolds
Richard T.
Miller
597-628
Locally convex topological lattices
Albert R.
Stralka
629-640
Abstract: The main theorem of this paper is: Suppose that L is a topological lattice of finite breadth n. Then L can be embedded in a product of n compact chains if and only if L is locally convex and distributive. With this result it is then shown that the concepts of metrizability and separability are equivalent for locally convex, connected, distributive topological lattices of finite breadth.
Homological dimension and cardinality
B. L.
Osofsky
641-649
Abstract: Let $\{ e(i)\vert i \in \mathcal{I}\}$ be an infinite set of commuting idempotents in a ring R with 1 such that $\displaystyle \prod\limits_{\alpha = 0}^n {e({i_\alpha })\prod\limits_{\beta = n + 1}^m {(1 - e({i_\beta })) \ne 0} }$ for $\{ {i_\alpha }\vert \leqq \alpha \leqq n\} \cap \{ {i_\beta }\vert n + 1 \leqq \beta \leqq m\} = \emptyset$. Let I be the right ideal generated by these idempotents. We show that the projective dimension of I is $n < \infty $ if and only if the cardinality of $I = {\aleph _n}$. As a consequence, a countable direct product of fields has global dimension $k + 1$ if and only if ${2^{{\aleph _0}}} = {\aleph _k}$. The same is true for a full linear ring on a countable dimensional vector space over a field of cardinality at most ${2^{{\aleph _0}}}$. On the other hand, if $ {2^{{\aleph _0}}} > {\aleph _\omega }$, then any right and left self-injective ring which is not semi-perfect, any ring containing an infinite direct product of subrings, any ring containing the endomorphism ring of a countable direct sum of modules, and many quotient rings of such rings must all have infinite global dimension.
On the geometric means of entire functions of several complex variables
A. K.
Agarwal
651-657
Abstract: Let $f({z_1}, \ldots ,{z_n})$ be an entire function of the $n( \geqq 2)$ complex variables ${z_1}, \ldots ,{z_n}$, holomorphic for $ \vert{z_t}\vert \leqq {r_t},t = 1, \ldots ,n$. We have considered the case of only two complex variables for simplicity. Recently many authors have defined the arithmetic means of the function $ \vert f({z_1},{z_2})\vert$ and have investigated their properties. In the present paper, the geometric means of the function $\vert f({z_1},{z_2})\vert$ have been defined and the asymptotic behavior of certain growth indicators for entire functions of several complex variables have been studied and the results are given in the form of theorems.
Applications of the Tumura-Clunie theorem
Chung-chun
Yang
659-662
Abstract: Some applications of the Tumura-Clunie theorem are given. Most of these concerned fixed points of compositions of entire functions.