Topologically defined classes of going-down domains
Ira J.
Papick
1-37
Abstract: Let R be an integral domain. Our purpose is to study GD (going-down) domains which arise topologically; that is, we investigate how certain going-down assumptions on R and its overrings relate to the topological space $ {\text{Spec}}(R)$. Many classes of GD domains are introduced topologically, and a systematic study of their behavior under homomorphic images, localization and globalization, integral change of rings, and the ``$D + M$ construction'' is undertaken. Also studied, is the algebraic and topological relationships between these newly defined classes of GD domains.
Differential games with Lipschitz control functions and applications to games with partial differential equations
Emmanuel Nicholas
Barron
39-76
Abstract: In §1 we formulate a differential game when the dynamics is the inhomogeneous heat equation. In §2 we state the basic theory of differential games when the controls must choose uniformly Lipschitz control functions. We then prove some general theorems for the case when the controls may choose any measurable control functions. These theorems hold for games with any dynamics. In §3 we apply our theory developed to our particular example and in §4 we prove the existence of value for games with partial differential equations.
On analytic independence
S. S.
Abhyankar;
T. T.
Moh
77-87
Abstract: This article examines the concept of ``analytic independence". Several illustrative examples have been included. The main results are Theorems 1-4 which state the relations between analytic independence and the degree of field extensions, transcendence degree, order of poles and ``gap'' respectively.
Symmetrizable and related spaces
Peter W.
Harley;
R. M.
Stephenson
89-111
Abstract: A study is made of a family of spaces which contains the symmetrizable spaces as well as many of the well-known examples of perfectly normal spaces.
Limit properties of Poisson kernels of Siegel domains of type II
Lawrence J.
Dickson
113-131
Abstract: The results of [1] concerning tight $C_0^\ast$ limit of the Poisson kernel of a tube domain, as its parameter converges to a point on the cone boundary, are extended under certain hypotheses to Siegel domains of type II. In the case where the domain is polytopic, almost everywhere convergence of the ${L^p}$ Poisson integral to its boundary values is obtained. Examples and further conjectures conclude the paper.
On the existence of compact metric subspaces with applications to the complementation of $c\sb{0}$
William H.
Chapman;
Daniel J.
Randtke
133-148
Abstract: A topological space X has property $\sigma - {\text{CM}}$ if for every countable family F of continuous scalar valued functions on X there is a compact metrizable subspace M of X such that $ f(X) = f(M)$ for every f in F. Every compact metric space, every weakly compact subset of a Banach space and every closed ordinal space has property $\sigma - {\text{CM}}$. Every continuous image of an arbitrary product of spaces having property $\sigma - {\text{CM}}$ also has property $\sigma - {\text{CM}}$. If X has property $ \sigma - {\text{CM}}$, then every copy of ${c_0}$ in $C(X)$ is complemented in $C(X)$. If a locally convex space E belongs to the variety of locally convex spaces generated by the weakly compactly generated Banach spaces, then every copy of ${c_0}$ in E is complemented in E.
The \v Cech cohomology of movable and $n$-movable spaces
James
Keesling
149-167
Abstract: In this paper the Čech cohomology of movable and n-movable spaces is studied. Let X be a space and let $ {H^k}(X)$ denote the k-dimensional Čech cohomology of X with integer coefficients based on the numerable covers of X. Then if X is movable, there is a subgroup E of ${H^k}(X)$ which is the union of all the algebraically compact subgroups of ${H^k}(X)$. Furthermore, $ {H^k}(X)/E$ is an ${\aleph _1}$-free abelian group. If X is an n-movable space, then it is shown that this structure holds for ${H^k}(X)$ for $0 \leqslant k \leqslant n$ and may be false for $k \geqslant n + 1$. If X is an ${\text{LC}^{n - 1}}$ paracompactum, then X is known to be n-movable. However, in this case and in the case that X is an ${\text{LC}^{n - 1}}$ compactum a stronger structure theorem is proved for ${H^k}(X)$ for $0 \leqslant k \leqslant n - 1$ than that stated above. Using these results examples are given of n-movable continua that are not shape equivalent to any ${\text{LC}^{n - 1}}$ paracompactum.
Some examples in shape theory using the theory of compact connected abelian topological groups
James
Keesling
169-188
Abstract: In previous papers the author has studied the shape of compact connected abelian topological groups. This study has led to a number of theorems and examples in shape theory. In this paper a theorem is proved concerning the Čech homology of compact connected abelian topological groups. This theorem together with the author's previous results are then used to study the movability of general compact Hausdorff spaces. In the theory of shape for compact metric spaces, a number of significant theorems have been proved for movable compact metric spaces. Among these are a theorem of Hurewicz type due to K. Kuperberg, a Whitehead type theorem due to Moszyńska, and a theorem concerning the exactness of the Čech homology sequence for movable compact metric pairs due to Overton. In this paper examples are constructed which show that these theorems do not generalize to arbitrary movable compact Hausdorff spaces without additional assumptions.
Classification of simply connected four-dimensional $RR$-manifolds
Gr.
Tsagas;
A.
Ledger
189-210
Abstract: Let (M, g) be a Riemannian manifold. We assume that there is a mapping $s:M \to I(M)$, where $I(M)$ is the group of isometries of (M, g), such that ${s_x} = s(x),\forall x \in M$, has x as a fixed isolated point, then (M, g) is called a Riemannian s-manifold. If the tensor field S on M defined by the relation ${S_x} = {(d{s_x})_x},\forall x \in M$, is differentiable and invariant by each isometry $ {s_x}$, then the manifold (M, g) is called a regularly s-symmetric Riemannian manifold. The aim of the present paper is to classify simply connected four-dimensional regularly s-symmetric Riemannian manifolds.
A study of graph closed subsemigroups of a full transformation semigroup
R. G.
Biggs;
S. A.
Rankin;
C. M.
Reis
211-223
Abstract: Let ${T_X}$ be the full transformation semigroup on the set X and let S be a subsemigroup of ${T_X}$. We may associate with S a digraph $ g(S)$ with X as set of vertices as follows: $i \to j \in g(S)$ iff there exists $\alpha \in S$ such that $\alpha (i) = j$. Conversely, for a digraph G having certain properties we may assign a semigroup structure, $S(G)$, to the underlying set of G. We are thus able to establish a ``Galois correspondence'' between the subsemigroups of ${T_X}$ and a particular class of digraphs on X. In general, S is a proper subsemigroup of $S \cdot g(S)$.
Equivariant stable homotopy and framed bordism
Czes
Kosniowski
225-234
Abstract: This paper gives an elementary proof of the result that equivariant stable homotopy is the same as equivariant framed bordism.
Yoneda products in the Cartan-Eilenberg change of rings spectral sequence with applications to ${\rm BP}\sb\ast ({\rm BO}(n))$
Ronald
Ming
235-252
Abstract: Yoneda product structure is defined on a Cartan-Eilenberg change of rings spectral sequence. Application is made to a factorization theorem for the ${E_2}$-term of the Adams spectral sequence for Brown-Peterson homology of the classifying spaces $ BO(n)$.
Concordances of Hilbert cube manifolds
T. A.
Chapman
253-268
Abstract: The main result of this paper asserts that homotopy groups of concordances of compact Hilbert cube manifolds are isomorphic to homotopy groups of concordances of compact finite-dimensional piecewise-linear manifolds. This enables us to apply some finite-dimensional results to obtain some new information about homotopy groups of homeomorphism groups of compact Hilbert cube manifolds. Our approach also yields a much shorter proof of the local contractibility of the homeomorphism group of any compact Hilbert cube manifold.
Simplicial triangulation of noncombinatorial manifolds of dimension less than $9$
Martin
Scharlemann
269-287
Abstract: Necessary and sufficient conditions are given for the simplicial triangulation of all noncombinatorial manifolds in the dimension range $5 \leqslant n \leqslant 7$, for which the integral Bockstein of the combinatorial triangulation obstruction is trivial. A weaker theorem is proven in case $n = 8$. The appendix contains a proof that a map between PL manifolds which is a TOP fiber bundle can be made a PL fiber bundle.
Complex space forms immersed in complex space forms
H.
Nakagawa;
K.
Ogiue
289-297
Abstract: We determine all the isometric immersions of complex space forms into complex space forms. Our result can be considered as the local version of a well-known result of Calabi.
Partitions of large multipartites with congruence conditions. I
M. M.
Robertson;
D.
Spencer
299-322
Abstract: Let $ p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ be the number of partitions of $({n_1}, \ldots ,{n_j})$ where, for $1 \leqslant l \leqslant j$, the lth component of each part belongs to the set ${A_l} = \bigcup\nolimits_{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} }$ and $M,q(l)$ and the $ {a_{lh(l)}}$ are positive integers such that $0 < {a_{l1}} < \cdots < {a_{lq(l)}} \leqslant M$. Asymptotic expansions for $ p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ are derived, when the ${n_l} \to \infty$ subject to the restriction that ${n_1} \cdots {n_j} \leqslant n_l^{j + 1 - \in }$ for all l, where $\in$ is any fixed positive number. The case $ M = 1$ and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case $ j = 1$ for different values of M.
Maximal orders and reflexive modules
J. H.
Cozzens
323-336
Abstract: If R is a maximal two-sided order in a semisimple ring and $ {M_R}$ is a finite dimensional torsionless faithful R-module, we show that $m = {\text{End}_R}\;{M^\ast}$ is a maximal order. As a consequence, we obtain the equivalence of the following when ${M_R}$ is a generator: 1. M is R-reflexive. 2. $k = {\text{End}}\;{M_R}$ is a maximal order. 3. $k = {\text{End}_R}\;{M^\ast}$ where $ {M^\ast} = {\hom _R}(M,R)$. When R is a prime maximal right order, we show that the endomorphism ring of any finite dimensional, reflexive module is a maximal order. We then show by example that R being a maximal order is not a property preserved by k. However, we show that $k = {\text{End}}\;{M_R}$ is a maximal order whenever ${M_R}$ is a maximal uniform right ideal of R, thereby sharpening Faith's representation theorem for maximal two-sided orders. In the final section, we show by example that even if $R = {\text{End}_k}V$ is a simple pli (pri)-domain, k can have any prescribed right global dimension $\geqslant 1$, can be right but not left Noetherian or neither right nor left Noetherian.
Breadth two topological lattices with connected sets of irreducibles
J. W.
Lea
337-345
Abstract: Breadth two topological lattices with connected sets of irreducible elements are characterized by these sets.
On the zeros of Dirichlet $L$-functions. III
Akio
Fujii
347-349
Abstract: It is shown that the ordinates of the zeros of the Riemann zeta function are uniformly distributed. Similar results pertain to zeros of L-functions.
Infinite-dimensional Whitehead and Vietoris theorems in shape and pro-homotopy
David A.
Edwards;
Ross
Geoghegan
351-360
Abstract: In Theorem 3.3 and Remark 3.4 conditions are given under which an infinite-dimensional Whitehead theorem holds in pro-homotopy. Applications to shape theory are given in Theorems 1.1, 1.2, 4.1 and 4.2.
A Hausdorff measure inequality
Lawrence R.
Ernst;
Gerald
Freilich
361-368
Abstract: We prove that the Hausdorff $(m + k)$-measure of a product set is no less than the product of the Hausdorff m-measure of the (measurable) first component set in ${{\mathbf{R}}^m}$ and the (finite) Hausdorff k-measure of the second component in ${{\mathbf{R}}^n}$.
On the trivial extension of equivalence relations on analytic spaces
Kunio
Takijima;
Tetsutaro
Suzuki
369-377
Abstract: In this paper, we shall consider the problem: let X be a (reduced) analytic space and A a nowhere dense analytic set in X. And let R be a proper equivalence relation on A such that the quotient space $ A/R$ is an analytic space, and $\tilde R$ the trivial extension of R to X. Then, is $ X/\tilde R$ an analytic space? To this, we have three sufficient conditions. Moreover, using this result we shall extend Satz 1 of H. Kerner [8].
Asymptotic relations for partitions
L. B.
Richmond
379-385
Abstract: Asymptotic relations are obtained for the number ${p_A}(n)$ of partitions of the integer n into summands from a set A of integers. The set A is subject to certain conditions; however the only arithmetic condition is that A have property ${P_k}$ of Bateman and Erdös. A conjecture of Bateman and Erdös concerning the kth differences of ${p_A}(n)$ may be verified using these asymptotic relations.
Extreme points of univalent functions with two fixed points
Herb
Silverman
387-395
Abstract: Univalent functions of the form $f(z) = {a_1}z - \Sigma _{n = 2}^\infty {a_n}{z^n}$, where $ {a_n} \geqslant 0$, are considered. We examine the subclasses for which $f({z_0}) = {z_0}$ or
Asymptotic equipartition of energy for differential equations in Hilbert space
Jerome A.
Goldstein;
James T.
Sandefur
397-406
Abstract: Of concern are second order differential equations of the form $ (d/dt - i{f_1}(A))(d/dt - i{f_2}(A))u = 0$. Here A is a selfadjoint operator and $ {f_1},{f_2}$ are real-valued Borel functions on the spectrum of A. The Cauchy problem for this equation is governed by a certain one parameter group of unitary operators. This group allows one to define the energy of a solution; this energy depends on the initial data but not on the time t. The energy is broken into two parts, kinetic energy $K(t)$ and potential energy $P(t)$, and conditions on A, ${f_1},{f_2}$ are given to insure asymptotic equipartition of energy: $ {\lim _{t \to \pm \infty }}K(t) = {\lim _{t \to \pm \infty }}P(t)$ for all choices of initial data. These results generalize the corresponding results of Goldstein for the abstract wave equation ${d^2}u/d{t^2} + {A^2}u = 0$. (In this case, ${f_1}(\lambda ) \equiv \lambda ,{f_2}(\lambda ) \equiv - \lambda$.)