An $L\sp{1}$-space for Boolean algebras and semireflexivity of spaces $L\sp{\infty }(X,\Sigma ,\mu )$
Dennis
Sentilles
1-37
Abstract: In this paper we suggest a measure free construction of $ {L^1}$-spaces using Boolean algebras and strict topologies and initiate a duality theory of $ ({L^\infty },{L^1})$ like that of the duality of continuous functions and Baire measures, showing that the Boolean context yields a formal link between uniform tightness, uniform $ \sigma$-additivity and uniform integrability.
Potential processes
R. V.
Chacon
39-58
Abstract: The prototype of a potential process is a stochastic process which visits the same points in the same order as a Markov process, but at a rate obtained from a nonanticipating time change. The definition of a potential process may be given intrinsically and most generally without mention of a Markov process, in terms of potential theory. The definition may be given more directly and less generally in terms of potentials which arise from Markov processes, or more directly than this, as suitably time-changed Markov processes. The principal purpose of studying the class of potential processes, which may be shown to include martingales as well as Markov processes themselves, is to give a unified treatment to a wide class of processes which has potential theory at its core. That it is possible to do so suggests that potential rather than martingale results are central to the study of Markov processes. Furthermore, this also suggests that it is not the Markov property itself which makes Markov processes tractable, but rather the potential structure which can be constructed with the assistance of the Markov property. The general theory of potential processes is developed in a forthcoming paper. It will be shown there that a Markov process subject to an ordinary continuous nonanticipating time change is a local potential process. It may be seen, by examining examples, that it is necessary to consider randomized stopping times and randomized nonanticipating time changes in the general case. In the forthcoming paper a more general notion than randomized nonanticipating time changes is used to obtain a characterization of potential processes. It is an open problem whether randomization itself is sufficient in the general case, and whether ordinary nonanticipating time changes are sufficient for continuous parameter martingales and Brownian motion on the line. The emphasis in the present paper will be on developing the theory of discrete parameter martingales as a special case of the general theory.
The orderability and suborderability of metrizable spaces
S.
Purisch
59-76
Abstract: A space is defined to be suborderable if it is embeddable in a (totally) orderable space. It is shown that a metrizable space X is suborderable iff (1) each component of X is orderable, (2) the set of cut points of each component of X is open, and (3) each closed subset of X which is a union of components has a base of clopen neighborhoods. Note that condition (1) and hence this result is topological since there are many good topological characterizations of connected orderable spaces. In a space X let Q denote the union of all nondegenerate components each of whose noncut points has no compact neighborhood. It is also shown that a metrizable space X is orderable iff (1) X is suborderable, (2) $X - Q$ is not a proper compact open subset of X, and (3) if W is a neighborhood of $p \in X$ and K is the component in X containing p such that $ (W - K) - Q$ has compact closure and $\{ p\}$ is the intersection of the closures of $(W - K) - Q$ and $(W - K) \cap Q$, then K is a singleton. Corollaries are given; every condition in each of these corollaries is concisely stated and sufficient for a space to be orderable when it is metrizable and suborderable. Both of these results are extended to a class properly containing the metrizable spaces.
The periodic points of Morse-Smale endomorphisms of the circle
Louis
Block
77-88
Abstract: Let $MS({S^1})$ denote the set of continuously differentiable maps of the circle with finite nonwandering set, which satisfy certain generic properties. For $f \in MS({S^1})$ let $ P(f)$ denote the set of positive integers which occur as the period of some periodic point of f. It is shown that for $f \in MS({S^1})$ there are integers $m \geqslant 1$ and $n \geqslant 0$ such that $ P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}$. Conversely, if m and n are integers, $m \geqslant 1,n \geqslant 0$, there is a map $f \in MS({S^1})$ with $P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}$.
On Lie algebras of vector fields
Akira
Koriyama;
Yoshiaki
Maeda;
Hideki
Omori
89-117
Abstract: This paper has two purposes. The first is a generalization of the theorem of Pursell-Shanks [10]. Our generalization goes by assuming the existence of a nontrivial core of a Lie algebra. However, it seems to be a necessary condition for the theorems of Pursell-Shanks type. The second is the classification of cores under the assumption that the core itself is infinitesimally transitive at every point. As naturally expected, we have the nonelliptic, primitive infinite-dimensional Lie algebras.
Three-dimensional manifolds with finitely generated fundamental groups
Robert
Messer
119-145
Abstract: Recent results of G. P. Scott and T. W. Tucker indicate that a three-dimensional manifold with a finitely generated fundamental group is, in various senses, close to being compact. In this paper the structure of such a manifold M is described in terms of a certain compact, incompressible submanifold of M. This result is used to show that the product of M with the real line is essentially the interior of a compact 4-manifold. Finally, when M is $ {P^2}$-irreducible, a necessary and sufficient condition is given for M to be homeomorphic to the complement of a closed subset of the boundary of a compact 3-manifold.
A development of contraction mapping principles on Hausdorff uniform spaces
Cheng Ming
Lee
147-159
Abstract: Certain generalized Banach's contraction mapping principles on metric spaces are unified and/or extended to Hausdorff uniform spaces. Also given are some relationships between the set of all cluster points of the Picard iterates and the set of all fixed points for the mapping. These are obtained by assuming that the latter set is nonempty and by considering certain ``quasi"-contractive conditions. The ("quasi") contractive conditions are defined by using a suitable family of pseudometrics on the uniform space.
The independence ratio and genus of a graph
Michael O.
Albertson;
Joan P.
Hutchinson
161-173
Abstract: In this paper we study the relationship between the genus of a graph and the ratio of the independence number to the number of vertices.
Some smooth maps with infinitely many hyperbolic periodic points
John M.
Franks
175-179
Abstract: If a smooth map of the two-disk to itself has only hyperbolic periodic points and has no source or sink whose period is a power of two then it has infinitely many periodic points. This and similar results are proved.
Discrete analytic functions of exponential growth
Doron
Zeilberger
181-189
Abstract: Analogues of classical representation formulas for entire functions of exponential type are proved in the class of discrete analytic functions.
Cohomological dimension of a group with respect to finite modules
Juan José
Martínez
191-201
Abstract: The purpose of this paper is to compare the cohomological dimension of a group, relative to finite modules, with the cohomological dimension, in the usual sense, of its profinite completion. The basic tool used to perform this comparison is certain stable cohomology of the group. The reason is that there exists a spectral sequence which relates the continuous cohomology of the profinite completion, with coefficients in this stable cohomology, to the ordinary cohomology of the group. Moreover, the direct method of connecting the cohomology of the group with the profinite cohomology of its completion arises from the edge effects on the base of this spectral sequence.
Hausdorff measure functions in the space of compact subsets of the unit interval
P. R.
Goodey
203-208
Abstract: The work done in this paper is the result of an attempt to classify those functions h for which the corresponding Hausdorff measure of $ \mathcal{F}[0,1]$ is zero. A partial characterization is achieved and in doing this some problems of E. Boardman are solved.
Higher algebraic $K$-theories
D.
Anderson;
M.
Karoubi;
J.
Wagoner
209-225
Abstract: A homotopy fibration is established relating the Volodin or BN-pair definition of algebraic K-theory to the theory defined by Quillen. In [2] we outlined the construction of natural homomorphisms $\displaystyle K_ \ast ^Q \to K_ \ast ^{BN} \to K_ \ast ^V \to K_ \ast ^{KV}$ between higher algebraic K-theories $K_ \ast ^Q$ of [10] and [11], $K_ \ast ^{BN}$ of [17], $ K_ \ast ^V$ of [16], and $K_ \ast ^{KV}$ of [7] and [8]. This was one of the steps in proving the various definitions of higher K-theory are equivalent. It turns out they all agree-including the theory $ K_ \ast ^S$ of [14], [5], and [8]-provided one restricts to the category of regular rings when using $ K_\ast^{KV}$. See [1], [2], [5], [8] and [18]. The purpose of this paper is to prove the following theorem, announced in [2], which yields the construction of $K_ \ast ^Q \to K_ \ast ^{BN}$. Theorem. For any associative ring with identity A $\displaystyle G{L^{BN}}(A) \to B{\{ {U_F}\} ^ + } \to BGL{(A)^ + }$ is a homotopy fibration. For the reader's convenience and because the presentation of the BN-pair K-theory $K_ \ast ^{BN}$ used here is slightly different from that of [17], we shall briefly recall the definition of $G{L^{BN}}$ and $ B\{ {U_F}\}$ in the first section.
Second-order differential equations with fractional transition points
F. W. J.
Olver
227-241
Abstract: An investigation is made of the differential equation $\displaystyle {d^2}w/d{x^2} = \{ {u^2}{(x - {x_0})^\lambda }f(u,x) + g(u,x)/{(x - {x_0})^2}\} w,$ in which u is a large real (or complex) parameter, $\lambda$ is a real constant such that $\lambda > -2$, x is a real (or complex) variable, and $f(u,x)$ and $g(u,x)$ are continuous (or analytic) functions of x in a real interval (or complex domain) containing ${x_0}$. The interval (or domain) need not be bounded. Previous results of Langer and Riekstins giving approximate solutions in terms of Bessel functions of order $1/(\lambda + 2)$ are extended and error bounds supplied.
Best possible approximation constants
A. M.
Fink
243-255
Abstract: We study inequalities between the norm of the best approximating polynomial and the nth derivative of the function. These inequalities are then related to inequalities that have been considered elsewhere in different contexts.
A comparison of various definitions of contractive mappings
B. E.
Rhoades
257-290
Abstract: A number of authors have defined contractive type mappings on a complete metric space X which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice ${x_0} \in X$. In this paper we compare this multitude of definitions. X denotes a complete metric space with distance function d, and f a function mapping X into itself.
Function fields with isomorphic Galois groups
Robert J.
Bond
291-303
Abstract: Let K be a local field or a global field of characteristic p. Let ${G_K}$ be the Galois group of the separable closure of K over K. In the local case we show that ${G_K}$, considered as an abstract profinite group, determines the characteristic of K and the number of elements in the residue class field. In the global case we show that ${G_K}$ determines the number of elements in the constant field of K as well as the zeta function, genus and class number of K. Let $K'$ be another global field of characteristic p and assume we have
Linear operators for which $T\sp*T$ and $T+T\sp*$ commute. II
Stephen L.
Campbell;
Ralph
Gellar
305-319
Abstract: Let $\theta$ denote the set of bounded linear operators T, acting on a separable Hilbert space $ \mathcal{K}$ such that $ {T^\ast}T$ and $T + {T^\ast}$ commute. It is shown that such operators are ${G_1}$. A complete structure theory is developed for the case when $ \sigma (T)$ does not intersect the real axis. Using this structure theory, several nonhyponormal operators in $\theta$ with special properties are constructed.
A resolvent for an iteration method for nonlinear partial differential equations
J. W.
Neuberger
321-343
Abstract: For each of m and n a positive integer denote by $ S(m,i)$ the space of all real-valued symmetric i-linear functions on $ {E_m},i = 1,2, \ldots ,n$. Denote by L a nonzero linear functional on $ S(m,n)$, denote by f a real-valued analytic function on ${E_m} \times R \times S(m,1) \times \cdots \times S(m,[n/2])$ and denote by $\alpha$ a member of $D(f)$. Denote by H the space of all real-valued functions U, analytic at the origin of $ {E_m}$, so that $ K(\lambda )U$ converges, as $\lambda \to \infty$, to a solution Y to the partial differential equation $L{Y^{(n)}} = {f_Y}$. A resolvent j for this semigroup is determined so that $J(\lambda )U$ also converges to y as $\lambda \to \infty$ and so that $J{(\lambda /n)^n}U$ converges to $K(\lambda )U$ as $ n \to \infty$. The solutions $Y \in H$ of $ L{Y^{(n)}} = {f_Y}$ are precisely the fixed points of the semigroup K.
Oscillation and a class of linear delay differential equations
David Lowell
Lovelady
345-364
Abstract: The differential equation ${u^{(m)}}(t) + p(t)u(g(t)) = 0$. where P is one-signed, is broken into four cases, according to the parity of m and the sign of p. In each case, an analysis is given of the effect g can have on oscillation properties, and oscillation and nonoscillation criteria are given.
Zero-one laws and the minimum of a Markov process
P. W.
Millar
365-391
Abstract: If $\{ {X_t},t > 0\}$ is a real strong Markov process whose paths assume a (last) minimum at some time M strictly before the lifetime, then conditional on I, the value of this minimum, the process $\{ X(M + t),t > 0\}$ is shown to be Markov with stationary transitions which depend on I. For a wide class of Markov processes, including those obtained from Lévy processes via time change and multiplicative functional, a zero-one law is shown to hold at M in the sense that ${ \cap _{t > 0}}\sigma \{ X(M + s),s \leqslant t\} = \sigma \{ X(M)\}$, modulo null sets. When such a law holds, the evolution of $\{ X(M + t),t \geqslant 0\} $ depends on events before M only through $X(M)$ and I.
Addendum to: ``Some polynomials defined by generating relations'' (Trans. Amer. Math. Soc. {\bf 205} (1975), 360--370)
H. M.
Srivastava;
R. G.
Buschman
393-394
Abstract: Certain constraints are explicitly specified for the validity of a recent result involving a multivariate generating function, due to the present authors [1, p. 369, Theorem 6]. It is also indicated how this result can be further generalized. See Theorem ${6^\ast}$ below.