Spaces homeomorphic to $(2\sp{a})\sb{a}$. II
H. H.
Hung;
S.
Negrepontis
1-30
Abstract: Topological characterizations and properties of the spaces ${({2^\alpha })_\alpha }$, where $\alpha$ is an infinite regular cardinal, are studied; the principal interest lying in the significance that these spaces have in questions of existence of ultrafilters (or of elements of the Stone-Čech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces ${({2^\alpha })_\alpha }$ in terms of a simple set of conditions, and (b) the $\alpha$-Baire category property of ${({2^\alpha })_\alpha }$ and the stability of the class of spaces homeomorphic to ${({2^\alpha })_\alpha }$ (or to ${({\alpha ^\alpha })_\alpha }$) when taking intersections of at most $\alpha$ open and dense subsets of ${({2^\alpha })_\alpha }$. Among the applications of these results are the following. Assuming ${\alpha ^ + } = {2^\alpha }$, the class of spaces homeomorphic to ${({2^{({\alpha ^ + })}})_{{\alpha ^ + }}}$ includes the space of uniform ultrafilters on $ \alpha$ with the ${P_{{\alpha ^ + }}}$-topology ${(U(\alpha ))_{{\alpha ^ + }}}$, its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming ${\omega ^ + } = {2^\omega }$ (or in some cases only Martin's axiom), the class of spaces homeomorphic to $ {({2^{({\omega ^ + })}})_{{\omega ^ + }}}$ includes the following: The space $ {(\beta X\backslash X)_{{\omega ^ + }}}$ where X is a noncompact locally compact realcompact space such that $\vert C(X)\vert \leq {2^\omega }$ and its subspaces of $ {P_{{\omega ^ + }}}$-points of $ \beta X\backslash X$ and/or (if X is in addition a metric space without isolated elements) the remote points. In particular the existence of good and/or Rudin-Keisler minimal ultrafilters and the existence of P-points and/or remote points follows always from a Baire category type of argument.
Moufang loops of small order. I
Orin
Chein
31-51
Abstract: The main result of this paper is the determination of all nonassociative Moufang loops of orders $\leq 31$. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Sylow's main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of orders p, ${p^2}$, ${p^3}$ or pq (for p and q prime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2.
$p$-absolutely summing operators and the representation of operators on function spaces
John William
Rice
53-75
Abstract: We introduce a class of p-absolutely summing operators which we call p-extending. We show that for a logmodular function space $A(K)$, an operator $ T:A(K) \to X$ is p-extending if and only if there exists a probability measure $\mu$ on K such that T extends to an isometry $ T:{A^p}(K,\mu ) \to X$. We use this result to give necessary and sufficient conditions under which a bounded linear operator is isometrically equivalent to multiplication by z on a space $ {L^p}(K,\mu )$ and certain Hardy spaces $ {H^p}(K,\mu )$.
On existence and uniqueness for a new class of nonlinear partial differential equations using compactness methods and differential difference schemes
Theodore E.
Dushane
77-96
Abstract: We prove existence and uniqueness results for the following Cauchy problem in the half plane $t \geq 0:{u_t} + {(f(u))_x} + {u_{xxx}} = {g_1}(u){u_{xx}} + {g_2}(u){({u_x})^2} + p(u),u(x,0) = {u_0}(x)$, where $u = u(x,t)$ and the subscripts indicate partial derivatives. We require that f, ${g_1}$, ${g_2}$, and p be sufficiently smooth and satisfy $f(u) = {u^{2n + 1}},{g_1}(u) = {u^{2m}},{g_2}(u) = - {u^{2r + 1}}$, and $p(u) = - {u^{2s + 1}}$, for n, m, r, and s nonnegative integers. To obtain a global solution in time, we perturb the equation by $- \epsilon ({u_{xxxx}} - {(f(u))_{xx}})$. The perturbed equation is solved locally (in time) and this solution is extended to a global solution by means of a priori estimates on the ${H^s}$ (of space) norms of the local solution. These estimates require the use of new nonlinear functionals. We then obtain the solution to the original equation as a limit of solutions to the perturbed equation as $\epsilon$ tends to zero using the standard techniques. For the related periodic problem, for which we require $ u(x + 2\pi ,t) = u(x,t)$ for all $t \geq 0$, we also obtain existence and uniqueness results. We prove existence for this problem via similar techniques to the nonperiodic case. We then consider differential difference schemes for the periodic initial value problem and show that we may obtain the solution as the limit of solutions to an appropriate scheme.
Recapturing $H\sp{2}$-functions on a polydisc
D. J.
Patil
97-103
Abstract: Let ${U^2}$ be the unit polydisc and $ {T^2}$ its distinguished boundary. If $E \subset {T^2}$ is a set of positive measure and the restriction to E of a function f in $ {H^2}({U^2})$ is given then an algorithm to recapture f is developed.
On proper homotopy theory for noncompact $3$-manifolds
E. M.
Brown;
T. W.
Tucker
105-126
Abstract: Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincaré conjecture, that a noncompact 3-manifold having the proper homotopy type of a closed product $F \times [0,1]$ or a half-open product $F \times [0,1)$ where F is a 2-manifold is actually homeomorphic to $F \times [0,1]$ or $F \times [0,1)$, respectively. By defining a concept for noncompact manifolds similar to boundary-irreducibility, a well-known result of Waldhausen concerning homotopy and homeomorphism type of compact 3-manifolds is extended to the noncompact case.
Wave equations with finite velocity of propagation
Stephen J.
Berman
127-148
Abstract: If B is a selfadjoint translation-invariant operator on the space ${L^2}$ of complex-valued functions on n-dimensional Euclidean space which are square-summable with respect to Lebesgue measure, then the wave equation ${d^2}F/d{t^2} + {B^2}F = 0$ has the solution $ F(t) = (\cos tB)f + ((\sin tB)/B)g$, for f and g in ${L^2}$. In the classical case in which $- {B^2}$ is the Laplacian, this solution has finite velocity of propagation in the sense that (letting supp denote support of a function) ${\text{supp}}\;F(t) \subset ({\text{supp}}\;f \cup {\text{supp}}\;g) + {K_t}$ for all f and g and some compact set ${K_t}$ independent of f and g. We show that a converse holds, namely, if $\cos \;tB$ has finite velocity of propagation (that is, if $ {\text{supp}}\,(\cos tB)f \subset {\text{supp}}\;f + {K_t}$ for all f and some compact ${K_t}$) for three values of t whose reciprocals are independent over the rationals, then $ {B^2}$ must be a second order differential operator. If Euclidean space is replaced by a locally compact abelian group which does not contain the real line as a subgroup, then $ \cos \;tB$ has finite velocity of propagation for all t if and only if it is convolution with a distribution ${T_t}$ such that all ${T_t}$ are supported on a compact open subgroup. Problems of a similar nature are discussed for compact connected abelian groups and for the nonabelian group $ {\text{SL}}(2,{\mathbf{R}})$.
Uniformly distributed sequences in locally compact groups. I
Leonora
Benzinger
149-165
Abstract: We investigate the notion of uniformly distributed sequences in locally compact groups. Our main result is the following: A locally compact group G possesses a uniformly distributed sequence if and only if it possesses a sequence whose homomorphic images are dense in each of the compact quotients of G.
Uniformly distributed sequences in locally compact groups. II
Leonora
Benzinger
167-178
Abstract: We consider the following question. When is there a compactification $ {G_0}$ of a locally compact group G (recall that a compact group $ {G_0}$ is a compactification of G if there is a continuous homomorphism $ \phi :G \to {G_0}$ so that $ \phi (G)$ is dense in G) with continuous homomorphism $\phi :G \to {G_0}$ with the property that $\{ {g_\nu }\}$ is uniformly distributed in G if and only if $\{ \phi ({g_\nu })\}$ is uniformly distributed in $ {G_0}$? Such a compactification ${G_0}$ is called a D-compactification of G. We obtain a solution to this problem and thereby generalize to locally compact groups some results of Berg, Rajagopalan, and Rubel concerning D-compactifications of locally compact abelian groups.
Representations and classifications of stochastic processes
Dudley Paul
Johnson
179-197
Abstract: We show that to every stochastic process X one can associate a unique collection $(\Phi ,{\Phi _ + },T(t),E(U),{p^\ast})$ consisting of a linear space $\Phi$, on which is defined a linear functional ${p^ \ast }$, together with a convex subset ${\Phi _ + }$ which is invariant under the semigroup of operators $T(t)$ and the resolution of the identity $ E(U)$. The joint distributions of X, there being one version for each $\phi \in {\Phi _ + }$, are then given by $\displaystyle {P_\phi }(X({t_1}) \in {U_1}, \cdots ,X({t_1} + \cdots + {t_n}) \in {U_n}) = {p^ \ast }E({U_n})T({t_n}) \cdots E({U_1})T({t_1})\phi .$ To each $\phi$ contained in the extreme points ${\Phi _{ + + }}$ of $ {\Phi _ + }$ and each time t we find a probability measure $P_t^ \ast (\phi, \cdot )$ on ${\Phi _{ + + }}$ such that $T(t)\phi = {\smallint _{{\Phi _{ + + }}}}\psi P_t^ \ast (\phi ,d\psi )$. $P_t^ \ast$ is the transition probability function of a temporally homogeneous Markov process ${X^ \ast }$ on $ {\Phi _{ + + }}$ for which there exists a function f such that $X = f({X^ \ast })$. We show that in a certain sense ${X^ \ast }$ is the smallest of all Markov processes Y for which there exists a function g with $X = g(Y)$. We then apply these results to a class of stochastic process in which future and past are independent given the present and the conditional distribution, on the past, of a collection of random variables in the future.
Nonlinear approximation in uniformly smooth Banach spaces
Edward R.
Rozema;
Philip W.
Smith
199-211
Abstract: John R. Rice [Approximation of functions. Vol. II, Addison-Wesley, New York, 1969] investigated best approximation from a nonlinear manifold in a finite dimensional, smooth, and rotund space. The authors define the curvature of a manifold by comparing the manifold with the unit ball of the space and suitably define the ``folding'' of a manifold. Rice's Theorem 11 extends as follows: Theorem. Let X be a uniformly smooth Banach space, and $ F:{R^n} \to X$ be a homeomorphism onto $M = F({R^n})$. Suppose $\nabla F(a)$ exists for each a in X, $\nabla F$ is continuous as a function of a, and $ \nabla F(a) \cdot {R^n}$ has dimension n. Then, if M has bounded curvature, there exists a neighborhood of M each point of which has a unique best approximation from M. A variation theorem was found and used which locates a critical point of a differentiable functional defined on a uniformly rotund space Y. [See M. S. Berger and M. S. Berger, Perspectives in nonlinearity, Benjamin, New York, 1968, p. 58ff. for a similar result when $Y = {R^n}$.] The paper is concluded with a few remarks on Chebyshev sets.
Symmetric norm ideals and relative conjugate ideals
Norberto
Salinas
213-240
Abstract: In this paper some aspects of the algebraic structure of the ring of all bounded linear operators on an infinite dimensional separable complex Hilbert space are discussed. In particular, a comparison criterion for maximal and minimal norm ideals is established. Also, a general notion of the conjugate of an ideal relative to another ideal is studied and some questions concerning joins and intersections of ideals are solved.
Semirings and $T\sb{1}$ compactifications. I
Douglas
Harris
241-258
Abstract: With each infinite cardinal ${\omega _\mu }$ is associated a topological semiring $ {{\mathbf{F}}_\mu }$, whose underlying space is finite complement topology on the set of all ordinals less than ${\omega _\mu }$, and whose operations are the natural sum and natural product defined by Hessenberg. The theory of the semirings $ {C_\mu }(X)$ of maps from a space X into ${{\mathbf{F}}_\mu }$ is developed in close analogy with the theory of the ring $C(X)$ of continuous real-valued functions; the analogy is not on the surface alone, but may be pursued in great detail. With each semiring a structure space is associated; the structure space of ${C_\mu }(X)$ for sufficiently large ${\omega _\mu }$ will be the Wallman compactification of X. The classes of ${\omega _\mu }$-entire and ${\omega _\mu }$-total spaces, which are respectively analogues of realcompact and pseudocompact spaces, are examined, and an $ {\omega _\mu }$-entire extension analogous to the Hewitt realcompactification is constructed with the property (not possessed by the Wallman compactification) that every map between spaces has a unique extension to their ${\omega _\mu }$-entire extensions. The semiring of functions of compact-small support is considered, and shown to be related to the locally compact-small spaces in the same way that the ring of functions of compact support is related to locally compact spaces.
A discontinuous intertwining operator
Allan M.
Sinclair
259-267
Abstract: If T and R are continuous linear operators on Banach spaces X and Y with the spectrum of R countable, we obtain necessary and sufficient conditions on the pair T, R that imply the continuity of every linear operator S from X into Y satisfying $ST = RS$.
Local norm convergence of states on the zero time Bose fields
Ola
Bratteli
269-280
Abstract: For a sequence of vector states on the Boson Fock space which are norm convergent on the Newton-Wigner local algebras, conditions are given which guarantee norm convergence on the relativistic local algebras also. These conditions are verified for the cutoff physical vacuum states of the $P{(\phi )_2}$ field theory, and yield a simplification of the proof of the locally normal property of the physical vacuum in that theory.
A partition property characterizing cardinals hyperinaccessible of finite type
James H.
Schmerl
281-291
Abstract: Let ${\mathbf{P}}(n,\alpha )$ be the class of infinite cardinals which have the following property: Suppose for each $\nu < \kappa $ that ${C_\nu }$ is a partition of ${[\kappa ]^n}$ and card $({C_\nu }) < \kappa$; then there is $X \subset \kappa$ of length $\alpha$ such that for each $\nu < \kappa$, the set $X - (\nu + 1)$ is ${C_\nu }$-homogeneous. In this paper the classes ${\mathbf{P}}(n,\alpha )$ are studied and a nearly complete characterization of them is given. A principal result is that ${\mathbf{P}}(n + 2,n + 5)$ is the class of cardinals which are hyperinaccessible of type n.
A matrix representation for associative algebras. I
Jacques
Lewin
293-308
Abstract: Let F be a mixed free algebra on a set X over the field K. Let U, V be two ideals of F, and $ \{ \delta (x),(x \in X)\}$ a basis for a free $(F/U,F/V)$-bimodule T. Then the map $x \to (\begin{array}{*{20}{c}} {x + V} & 0 {\delta (x)} & {x + U} \end{array} )$ induces an injective homomorphism $ F/UV \to (\begin{array}{*{20}{c}} {F/V} & 0 T & {F/U} \end{array} )$. If $F/U$ and $F/V$ are embeddable in matrices over a commutative algebra, so is $F/UV$. Some special cases are investigated and it is shown that a PI algebra with nilpotent radical satisfies all identities of some full matrix algebra.
A matrix representation for associative algebras. II
Jacques
Lewin
309-317
Abstract: The results of part I of this paper are applied to show that if F is a free algebra over the field K and W is a subset of F which is algebraically independent modulo the commutator ideal [F, F], then W again generates a free algebra. On the way a similar theorem is proved for algebras that are free in the variety of K-algebras whose commutator ideal is nilpotent of class n. It is also shown that if L is a Lie algebra with universal enveloping algebra F, and U, V are ideals of L, then $FUF \cdot FVF \cap L = [U \cap V,U \cap V]$. This is used to extend the representation theorem of part I to free Lie algebras.
$K\sb{1}$ of a curve of genus zero
Leslie G.
Roberts
319-326
Abstract: We determine the structure of the vector bundles on a curve of genus zero and calculate the ``universal determinant'' $ {K_1}$ of such a curve.
Ultrafilter mappings and their Dedekind cuts
Andreas
Blass
327-340
Abstract: Let D be an ultrafilter on the set N of natural numbers. To each function $p:N \to N$ and each ultrafilter E that is mapped to D by p, we associate a Dedekind cut in the ultrapower D-prod N. We characterize, in terms of rather simple closure conditions, the cuts obtainable in this manner when various restrictions are imposed on E and p. These results imply existence theorems, some known and some new, for various special kinds of ultrafilters and maps.
Maximal $\alpha $-r.e. sets
Manuel
Lerman
341-386
Abstract: Various generalizations of maximal sets from ordinary recursion theory to recursion theory on admissible ordinals are considered. A justification is given for choosing one of these definitions as superior to the rest. For all the definitions considered to be reasonable, a necessary and sufficient condition for the existence of such maximal $ \alpha$-r.e. sets is obtained.
Some combinatorial principles
Jussi
Ketonen
387-394
Abstract: We extend some large cardinal axioms of Jensen to weakly inaccessible cardinals. Related problems regarding the saturatedness of certain filters are also studied.
Representation of functions as limits of martingales
Charles W.
Lamb
395-405
Abstract: In this paper we show that if $(\Omega ,\mathcal{F},P)$ is a probability space and if ${\{ \mathcal{F}{_n}\} _{n \geq 1}}$ is an increasing sequence of sub-$\sigma$-fields of $ \mathcal{F}$ which satisfy an additional condition, then every real valued, $ {\mathcal{F}_\infty }$-measurable function f can be written as the a.e. limit of a martingale ${\{ {f_n},{\mathcal{F}_n}\} _{n \geq 1}}$. The case where f takes values in the extended real line is also studied. A construction is given of a ``universal'' martingale ${\{ {f_n},{\mathcal{F}_n}\} _{n \geq 1}}$ such that any $ {\mathcal{F}_\infty }$-measurable function is the a.e. limit of a suitably chosen subsequence.
Symmetric integro-differential-boundary problems
Hyman J.
Zimmerberg
407-417
Abstract: Necessary and sufficient conditions for a linear vector integro-differential-boundary problem to be symmetric (selfadjoint) are developed, and then applied to obtain canonical forms of such symmetric problems. Moreover, the formulation of the integro-boundary conditions herein yields a simplification of one of the conditions for selfadjointness of a differential-boundary operator previously announced.
Spectra of polar factors of hyponormal operators
C. R.
Putnam
419-428
Abstract: An investigation is made of the interdependence and properties of the spectrum of a hyponormal operator T and of the spectra, and absolutely continuous spectra, of the factors in a polar factorization of T when the latter exists.