Twisted sums of sequence spaces and the three space problem
N. J.
Kalton;
N. T.
Peck
1-30
Abstract: In this paper we study the following problem: given a complete locally bounded sequence space Y, construct a locally bounded space Z with a subspace X such that both X and $Z/X$ are isomorphic to Y, and such that X is uncomplemented in Z. We give a method for constructing Z under quite general conditions on Y, and we investigate some of the properties of Z. In particular, when Y is $ {l_p}\,(1\, < \,p\, < \,\infty )$, we identify the dual space of Z, we study the structure of basic sequences in Z, and we study the endomorphisms of Z and the projections of Z on infinite-dimensional subspaces.
Injectivity, projectivity, and the axiom of choice
Andreas
Blass
31-59
Abstract: We study the connection between the axiom of choice and the principles of existence of enough projective and injective abelian groups. We also introduce a weak choice principle that says, roughly, that the axiom of choice is violated in only a set of different ways. This principle holds in all ordinary Fraenkel-Mostowski-Specker and Cohen models where choice fails, and it implies, among other things, that there are enough injective abelian groups. However, we construct an inner model of an Easton extension with no nontrivial injective abelian groups. In the presence of our weak choice principle, the existence of enough projective sets is as strong as the full axiom of choice, and the existence of enough free projective abelian groups is nearly as strong. We also prove that the axiom of choice is equivalent to ``all free abelian groups are projective'' and to ``all divisible abelian groups are injective."
Algebraic description of homogeneous cones
Josef
Dorfmeister
61-89
Abstract: This paper finishes the author's investigations on homogeneous cones. As a result a classification of homogeneous cones is derived. The most important tool to get insight into the structure of homogeneous cones are J-morphisms. Therefore, in this paper we mainly deal with morphisms of homogeneous cones. The main result gives an algebraic description of J-morphisms. It includes a description of ``Linear imbeddings of self-dual homogeneous cones'' and the above mentioned classification of homogeneous cones. In a subsequent paper it will be used to describe homogeneous Siegel domains.
On the asymptotic behavior of a fundamental set of solutions
Charles
Powder
91-110
Abstract: We consider nth order homogeneous linear ordinary differential equations whose coefficients have an asymptotic expansion as $x \to \infty$ in terms of real powers of x and are analytic in sectors of the complex plane. In earlier work Bank (Funkcial. Ekvac. 11 (1968), 87-100) developed a method for reading off the asymptotic behavior of solutions directly from the equation, except in certain cases where roots asymptotically coalesce. For our results, we consider coefficients in a field of the type developed by Strodt (Trans. Amer. Math. Soc. 105 (1962), 229-250). By successive algebraic transforms, we show that an equation in the exceptional case can be reduced to the nonexceptional case and so the asymptotic behavior of the solutions can be read from the equation. This generalizes the classical results when $\infty$ is a singular point and the coefficients are analytic in neighborhoods of $\infty$. The strength of our results is that the coefficients need not be defined in a full neighborhood of $ \infty$, and that the asymptotic behavior can be read directly from the equation.
Structure mappings, coextensions and regular four-spiral semigroups
John
Meakin
111-134
Abstract: The structure mapping approach to regular semigroups developed by K. S. S. Nambooripad and J. Meakin is used to describe the $ \mathcal{K}$-coextensions of the fundamental four-sprial semigroup and hence to describe the structure of all regular semigroups whose idempotents form a four-spiral biordered set. Isomorphisms between regular four-spiral semigroups are studied. The notion of structural uniformity of a regular semigroup is defined and exploited.
Ergodic behaviour of nonstationary regenerative processes
David
McDonald
135-152
Abstract: Let ${V_t}$ be a regenerative process whose successive generations are not necessarily identically distributed and let A be a measurable set in the range of ${V_t}$. Let ${\mu _n}$ be the mean length of the nth generation and $ {\alpha _n}$ be the mean time ${V_t}$ is in A during the nth generation. We give conditions ensuring $ {\lim _{t \to \infty }}\,\operatorname{prob} \{ \,{V_t}\, \in \,A\,\} \, = \,\alpha /\mu$ where $\mathop {\lim }\limits_{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\mu _j}\, = \mu $ and $\mathop {\lim }\limits_{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\alpha _j}\, = \,\alpha$.
CR functions and tube manifolds
M.
Kazlow
153-171
Abstract: Various generalizations of Bochner's theorem on the extension of holomorphic functions over tube domains are considered. It is shown that CR functions on tubes over connected, locally closed, locally starlike subsets of ${\textbf{R}^n}$ uniquely extend to CR functions on almost all of the convex hull of the tube set. A CR extension theorem on maximally stratified real submanifolds of $ {\textbf{C}^n}$ is proven. The above two theorems are used to show that the CR functions (resp. CR distributions) on tubes over a fairly general class of submanifolds of ${\textbf{R}^n}$ uniquely extend to CR functions (CR distributions) on almost all of the convex hull.
$\beta $-recursion theory
Sy D.
Friedman
173-200
Abstract: We define recursion theory on arbitrary limit ordinals using the J-hierarchy for L. This generalizes $\alpha $-recursion theory, where the ordinal is assumed to be $ {\Sigma _1}$-admissible. The notion of tameness for a recursively enumerable set is defined and the degrees of tame r.e. sets are studied. Post's Problem is solved when $ {\Sigma _1}\operatorname{cf} \beta \,\beta {\ast}$. Lastly, simple sets are constructed for all $\beta$ with the aid of a $\beta$-recursive version of Fodor's Theorem.
Moduli of punctured tori and the accessory parameter of Lam\'e's equation
L.
Keen;
H. E.
Rauch;
A. T.
Vasquez
201-230
Abstract: To solve the problems of uniformization and moduli for Riemann surfaces, covering spaces and covering mappings must be constructed, and the parameters on which they depend must be determined. When the Riemann surface is a punctured torus this can be done quite explicitly in several ways. The covering mappings are related by an ordinary differential equation, the Lamé equation. There is a constant in this equation which is called the ``accessory parameter". In this paper we study the behavior of this accessory parameter in two ways. First, we use Hill's method to obtain implicit relationships among the moduli of the different uniformizations and the accessory parameter. We prove that the accessory parameter is not suitable as a modulus-even locally. Then we use a computer and numerical techniques to determine more explicitly the character of the singularities of the accessory parameter.
Necessary conditions for the convergence of cardinal Hermite splines as their degree tends to infinity
T. N. T.
Goodman
231-241
Abstract: Let ${\mathcal{S}_{n,s}}$ denote the class of cardinal Hermite splines of degree n having knots of multiplicity S at the integers. In this paper we show that if $ {f_n}\, \to \,f$ uniformly on R, where $ {f_n}\, \in \,{\mathcal{S}_{{i_{n,s}}}}\,{i_n}\, \to \,\infty$ as $n\, \to \,\infty$, and f is bounded, then f is the restriction to R of an entire function of exponential type $ \leqslant \,S$. In proving this result, we need to derive some extremal properties of certain splines $ {\mathcal{E}_{n,s}}\, \in \,{\mathcal{S}_{n,s}}$, in particular that $ \vert\vert{\mathcal{E}_{n,s}}\vert{\vert _\infty }$ minimises $\vert\vert S\vert{\vert _\infty }$ over $ S\, \in \,{\mathcal{S}_{n,s}}$ with $ \vert\vert{S^{(n)}}\vert{\vert _\infty }\, = \,\vert\vert\mathcal{E}_{n,s}^{(n)}\vert{\vert _\infty }$.
On a class of transformations which have unique absolutely continuous invariant measures
Abraham
Boyarsky;
Manny
Scarowsky
243-262
Abstract: A class of piecewise ${C^2}$ transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.
Stability theory for functional-differential equations
T. A.
Burton
263-275
Abstract: We consider a system of functional differential equations $\mathcal{V}\,(t,\,x( \cdot ))$ with $ \mathcal{F}$ be bounded for $x( \cdot )$ bounded and that $\mathcal{F}$ depend on $x(s)$ only for $t\, - \,\alpha (t)\, \leqslant \,s\, \leqslant \,t$ where $\alpha$ is a bounded function in order to obtain stability properties. We show that if there is a function $H(t,\,x)$ whose derivative along $ \mathcal{F}$, and it improves the standard results on the location of limit sets for ordinary differential equations.
The variety of modular lattices is not generated by its finite members
Ralph
Freese
277-300
Abstract: This paper proves the result of the title. It shows that there is a five-variable lattice identity which holds in all finite modular lattices but not in all modular lattices. It is also shown that every free distributive lattice can be embedded into a free modular lattice. An example showing that modular lattice epimorphisms need not be onto is given.
Spectral theory for subnormal operators
R. G.
Lautzenheiser
301-314
Abstract: We give an example of a subnormal operator T such that $ {\text{C}}\,\backslash \,\sigma (T)$ has an infinite number of components, $ \operatorname{int} (\sigma (T))$ has two components U and V, and T cannot be decomposed with respect to U and V. That is, it is impossible to write $T\, = \,{T_1}\, \oplus \,{T_2}$ with $ \sigma ({T_1})\, = \,\overline U$ and $\sigma ({T_2})\, = \,\overline V$. This example shows that Sarason's decomposition theorem cannot be extended to the infinitely-connected case. We also use Mlak's generalization of Sarason's theorem to prove theorems on the existence of reducing subspaces. For example, if X is a spectral set for T and $ K\, \subset \,X$, conditions are given which imply that T has a nontrivial reducing subspace $ \mathcal{M}$ such that $\sigma (T\vert\mathcal{M})\, \subset \,K$. In particular, we show that if T is a subnormal operator and if $\Gamma$ is a piecewise ${C^2}$ Jordan closed curve which intersects $\sigma (T)$ in a set of measure zero on $ \Gamma$, then $T\, = \,{T_1}\, \oplus \,{T_2}$ with $\sigma ({T_1})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname{ext} (\Gamma )}$ and $ \sigma ({T_2})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname{int} (\Gamma )}$.
Markov cell structures for expanding maps in dimension two
F. T.
Farrell;
L. E.
Jones
315-327
Abstract: Let $f:\,{M^2}\, \to \,{M^2}$ be an expanding self-immersion of a closed 2-manifold, then for some positive integer n, ${f^n}$ leaves invariant a cell structure on $ {M^2}$. A similar result is true when M is a branched 2-manifold.
Notes on square-integrable cohomology spaces on certain foliated manifolds
Shinsuke
Yorozu
329-341
Abstract: We discuss some square-integrable cohomology spaces on a foliated manifold with one-dimensional foliation whose leaves are compact and with a complete bundle-like metric. Applications to a contact manifold are given.
Results on weighted norm inequalities for multipliers
Douglas S.
Kurtz;
Richard L.
Wheeden
343-362
Abstract: Weighted $ {L^p}$-norm inequalities are derived for multiplier operators on Euclidean space. The multipliers are assumed to satisfy conditions of the Hörmander-Mikhlin type, and the weight functions are generally required to satisfy conditions more restrictive than $ {A_p}$ which depend on the degree of differentiability of the multiplier. For weights which are powers of $ \left\vert x \right\vert$, sharp results are obtained which indicate such restrictions are necessary. The method of proof is based on the function ${f^\char93 }$ of C. Fefferman and E. Stein rather than on Littlewood-Paley theory. The method also yields results for singular integral operators.
Adding and subtracting jumps from Markov processes
Richard F.
Bass
363-376
Abstract: If ${X_t}$ is a continuous Markov process with infinitesimal generator A, if n is a kernel satisfying certain conditions, and if B is an operator given by $\displaystyle Bg(x)\, = \,\int {[ {g( y)\, - \,g(x)}]} \,n({x,\,dy}),$ then $ A\, + \,B$ will be the generator of a Markov process that has Lévy system $ (n,\,dt)$. Conversely, if $ {X_t}$ has Lévy system $(n,\,dt)$, n satisfies certain conditions, and B is defined as above, then $A\, - \,B$ will be the generator of a continuous Markov process.
The classification of two-dimensional manifolds
Edward M.
Brown;
Robert
Messer
377-402
Abstract: Invariants are constructed to classify all noncompact 2-manifolds including those with boundary. The invariants of a 2-manifold M are the space of ends of M and the subspaces of nonplanar ends, of nonorientable ends, and of ends that are limits of compact boundary components. Also the space of ends of the boundary components together with its natural map into the ends of M and the orientation of these ends induced by orientations of neighborhoods of the orientable ends of M are used in addition to the usual compact invariants. Special properties are established for the invariants of a 2-manifold, and a 2-manifold is constructed for each set of invariants with the special properties.
Diffeomorphisms and volume-preserving embeddings of noncompact manifolds
R. E.
Greene;
K.
Shiohama
403-414
Abstract: The theorem of J. Moser that any two volume elements of equal total volume on a compact manifold are diffeomorphism-equivalent is extended to noncompact manifolds: A necessary and sufficient condition (equal total and same end behavior) is given for diffeomorphism equivalence of two volume forms on a noncompact manifold. Results on the existence of embeddings and immersions with the property of inducing a given volume form are also given. Generalizations to nonorientable manifolds and manifolds with boundary are discussed.