Homogeneous random measures and a weak order for the excessive measures of a Markov process
P. J.
Fitzsimmons
431-478
Abstract: Let $X = ({X_t},\,{P^x})$ be a right Markov process and let $ m$ be an excessive measure for $X$. Associated with the pair $(X,\,m)$ is a stationary strong Markov process $ ({Y_t},\,{Q_m})$ with random times of birth and death, with the same transition function as $X$, and with $m$ as one dimensional distribution. We use $({Y_t},\,{Q_m})$ to study the cone of excessive measures for $X$. A "weak order" is defined on this cone: an excessive measure $\xi$ is weakly dominated by $m$ if and only if there is a suitable homogeneous random measure $\kappa$ such that $({Y_t},\,{Q_\xi })$ is obtained by "birthing" $({Y_t},\,{Q_m})$, birth in $[t,\,t + dt]$ occurring at rate $\kappa (dt)$. Random measures such as $ \kappa$ are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over $({Y_t},\,{Q_m})$, including the moderate Markov property of $ ({Y_t},\,{Q_m})$ when the arrow of time is reversed. Applications to balayage and capacity are suggested.
Semistability at $\infty$, $\infty$-ended groups and group cohomology
Michael L.
Mihalik
479-485
Abstract: A finitely presented group $G$, is semistable at $\infty$ if for some (equivalently any) finite complex $X$, with $ {\pi _1}(X) = G$, any two proper maps $r,\,s:[0,\,\infty ) \to \tilde X$ ($\equiv$ the universal cover of $ X$) that determine the same end of $\tilde X$ are properly homotopic in $\tilde X$. If $G$ is semistable at $\infty$, then ${H^2}(G;\,ZG)$ is free abelian. 0- and $ 2$-ended groups are all semistable at $\infty$. Theorem. If $G = A{{\ast}_C}B$ where $C$ is finite and $ A$ and $ B$ are finitely presented, semistable at $\infty$ groups, then $G$ is semistable at $ \infty$. Theorem. If $\alpha :C \to D$ is an isomorphism between finite subgroups of the finitely presented semistable at $\infty$ group $H$, then the resulting $HNN$ extension is semistable at $\infty$. Combining these results with the accessibility theorem of M. Dunwoody gives Theorem. If all finitely presented $1$-ended groups are semistable at $ \infty$, then all finitely presented groups are semistable at $ \infty$.
The asymptotic behavior of the solutions of degenerate parabolic equations
Catherine
Bandle;
M. A.
Pozio;
Alberto
Tesei
487-501
Abstract: Existence of stationary states is established by means of the method of upper and lower solutions. The structure of the solution set is discussed and a uniqueness property for certain classes is proved by a generalized maximum principle. It is then shown that all solutions of the parabolic equation converge to a stationary state.
Squares of conjugacy classes in the infinite symmetric groups
Manfred
Droste
503-515
Abstract: Using combinatorial methods, we will examine squares of conjugacy classes in the symmetric groups ${S_\nu }$ of all permutations of an infinite set of cardinality $ {\aleph _\nu }$. For arbitrary permutations $p \in {S_\nu }$, we will characterize when each element $s \in {S_\nu }$ with finite support can be written as a product of two conjugates of $p$, and if $p$ has infinitely many fixed points, we determine when all elements of ${S_\nu }$ are products of two conjugates of $ p$. Classical group-theoretical theorems are obtained from similar results.
A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
J. M.
Borwein;
D.
Preiss
517-527
Abstract: We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Fréchet) smooth renorm is densely Gâteaux (weak Hadamard, Fréchet) differentiable. Our technique relies on a more powerful analogue of Ekeland's variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.
Conjugacy classes in algebraic monoids
Mohan S.
Putcha
529-540
Abstract: Let $M$ be a connected linear algebraic monoid with zero and a reductive group of units $ G$. The following theorem is established. Theorem. There exist affine subsets $ {M_1}, \ldots ,{M_k}$ of $M$, reductive groups ${G_1}, \ldots ,{G_k}$ with antiautomorphisms $^{\ast}$, surjective morphisms ${\theta _i}:{M_i} \to {G_i}$, such that: (1) Every element of $ M$ is conjugate to an element of some ${M_i}$, and (2) Two elements $ a$, $b$ in ${M_i}$ are conjugate in $M$ if and only if there exists $x \in {G_i}$ such that $x{\theta _i}(a){x^{\ast}} = {\theta _i}(b)$. As a consequence, it is shown that $M$ is a union of its inverse submonoids.
Nil $K$-theory maps to cyclic homology
Charles A.
Weibel
541-558
Abstract: Algebraic $ K$-theory breaks into two pieces: nil $K$-theory and Karoubi-Villamayor $ K$-theory. Karoubi has constructed Chern classes from the latter groups into cyclic homology. We construct maps from nil $K$-theory to cyclic homology which are compatible with Karoubi's maps, but with a degree shift. Several recent results show that in characteristic zero our map is often an isomorphism.
Hyperarithmetical index sets in recursion theory
Steffen
Lempp
559-583
Abstract: We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. An extension yields a $ \Pi _1^1$-complete index set. We also classify the index set of quasimaximal sets, of coinfinite r.e. sets not having an atomless superset, and of r.e. sets major in a fixed nonrecursive r.e. set.
Singularly perturbed Dirichlet problems with subquadratic nonlinearities
Albert J.
DeSanti
585-593
Abstract: Boundary and interior layer theory is provided for a class of singularly perturbed Dirichlet problems with subquadratic nonlinearities in the derivative terms. The results obtained generalize and extend well-known results on the semilinear problem.
$p$-ranks and automorphism groups of algebraic curves
Shōichi
Nakajima
595-607
Abstract: Let $X$ be an irreducible complete nonsingular curve of genus $g$ over an algebraically closed field $k$ of positive characteristic $ p$. If $g \geqslant 2$, the automorphism group $ \operatorname{Aut} (X)$ of $ X$ is known to be a finite group, and moreover its order is bounded from above by a polynomial in $g$ of degree four (Stichtenoth). In this paper we consider the $p$-rank $\gamma$ of $X$ and investigate relations between $ \gamma$ and $\operatorname{Aut} (X)$. We show that $\gamma$ affects the order of a Sylow $ p$-subgroup of $ \operatorname{Aut} (X)\;(\S3)$ and that an inequality $ \vert\operatorname{Aut} (X)\vert \leqslant 84(g - 1)g$ holds for an ordinary (i.e. $\gamma = g$) curve $X\,(\S4)$.
On weak continuity and the Hodge decomposition
Joel W.
Robbin;
Robert C.
Rogers;
Blake
Temple
609-618
Abstract: We address the problem of determining the weakly continuous polynomials for sequences of functions that satisfy general linear first-order differential constraints. We prove that wedge products are weakly continuous when the differential constraints are given by exterior derivatives. This is sufficient for reproducing the Div-Curl Lemma of Murat and Tartar, the null Lagrangians in the calculus of variations and the weakly continuous polynomials for Maxwell's equations. This result was derived independently by Tartar who stated it in a recent survey article [7]. Our proof is explicit and uses the Hodge decomposition.
Towers and injective cohomology algebras
Paul
Goerss;
Larry
Smith
619-636
Abstract: Let $Y$ be a space of finite type such that ${H^{\ast}}Y$ is injective as an unstable algebra over the Steenrod algebra $A$ and such that ${\overline H ^{\ast}}Y$ is $A$-unbounded. Let $X$ be a simply connected $p$-complete space. Then any map of $A$-algebras $f:{H^{\ast}}\Omega X \to {H^{\ast}}Y$ can be realized as a map of spaces.
$K\sb {l+1}$-free graphs: asymptotic structure and a $0$-$1$ law
Ph. G.
Kolaitis;
H. J.
Prömel;
B. L.
Rothschild
637-671
Abstract: The structure of labeled ${K_{l + 1}}$-free graphs is investigated asymptotically. Through a series of stages of successive refinement the structure of "almost all" such graphs is found sufficiently precisely to prove that they are in fact $l$-colorable ($l$-partite). With the asymptotic information obtained it is shown also that in the class of ${K_{l + 1}}$-free graphs there is a first-order labeled 0-$1$ law. With this result, and those cases already known, we can say that any infinite class of finite undirected graphs with amalgamations, induced subgraphs and isomorphisms has a 0-$1$ law.
The Radon-Nikod\'ym property and the Kre\u\i n-Milman property are equivalent for strongly regular sets
Walter
Schachermayer
673-687
Abstract: The result announced in the title is proved. As corollaries we obtain that RNP and KMP are equivalent for subsets of spaces with an unconditional basis and for $K$-convex Banach spaces. We also obtain a sharpening of a result of R. Huff and P. Morris: A dual space has the RNP iff all separable subspaces have the KMP.
Branched coverings of $2$-complexes and diagrammatic reducibility
S. M.
Gersten
689-706
Abstract: The condition that all spherical diagrams in a $2$-complex be reducible is shown to be equivalent to the condition that all finite branched covers be aspherical. This result is related to the study of equations over groups. Furthermore large classes of $ 2$-complexes are shown to be diagrammatically reducible in the above sense; in particular, every $2$-complex has a subdivision which admits a finite branched cover which is diagrammatically reducible.
Group actions on the complex projective plane
Dariusz M.
Wilczyński
707-731
Abstract: Let $G$ be a finite or compact Lie group. It is shown that $G$ acts on the complex projective plane (resp. on the Chern manifold) if and only if $G$ is isomorphic to a subgroup (resp. a pseudofree subgroup) of $PU(3)$. All actions considered are effective, locally smooth, and trivial on homology.
Nilpotent spaces of sections
Jesper Michael
Møller
733-741
Abstract: The space of sections of a fibration is nilpotent provided the base is finite $CW$-complex and the fiber is nilpotent. Moreover, localization commutes with the formation of section spaces.
A finiteness theorem in the Galois cohomology of algebraic number fields
Wayne
Raskind
743-749
Abstract: In this note we show that if $k$ is an algebraic number field with algebraic closure $\overline k$ and $M$ is a finitely generated, free ${{\mathbf{Z}}_l}$-module with continuous $ \operatorname{Gal} (\overline k /k)$-action, then the continuous Galois cohomology group $ {H^1}(k,\,M)$ is a finitely generated $ {{\mathbf{Z}}_l}$-module under certain conditions on $M$ (see Theorem 1 below). Also, we present a simpler construction of a mapping due to S. Bloch which relates torsion algebraic cycles and étale cohomology.
An operator-theoretic formulation of asynchronous exponential growth
G. F.
Webb
751-763
Abstract: A strongly continuous semigroup of bounded linear operators $ T(t)$, $t \geqslant 0$, in the Banach space $X$ has asynchronous exponential growth with intrinsic growth constant ${\lambda _0}$ provided that there is a nonzero finite rank operator ${P_0}$ in $X$ such that ${\lim _{t \to \infty }}{e^{ - {\lambda _0}t}}T(t) = {P_0}$. Necessary and sufficient conditions are established for $T(t)$, $ t \geqslant 0$, to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth.
Pure subgroups of torsion-free groups
Paul
Hill;
Charles
Megibben
765-778
Abstract: In this paper, we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsion-free abelian groups. In particular, implications of $ {\ast}$-purity, a concept introduced in one of our recent papers, are investigated. We settle an open question (posed by Nongxa) by proving that the union of an ascending countable sequence of ${\ast}$-pure subgroups is completely decomposable provided the subgroups are. This result is false for ordinary purity. The principal result of the paper, however, deals with $\Sigma$-purity, a concept stronger than $ {\ast}$-purity but weaker than the usual notion of strong purity. Our main theorem, which has a number of corollaries including the recent result of Nongxa that strongly pure subgroups of separable groups are again separable, states that a $ \Sigma$-pure subgroup of a $k$-group is itself a $k$-group. Among other results is the negative resolution of the conjecture (valid in the countable case) that a strongly pure subgroup of a completely decomposable group is again completely decomposable.
Quasi $F$-covers of Tychonoff spaces
M.
Henriksen;
J.
Vermeer;
R. G.
Woods
779-803
Abstract: A Tychonoff topological space is called a quasi $F$-space if each dense cozero-set of $X$ is $ {C^{\ast}}$-embedded in $ X$. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi $F$-cover" $QF(X)$ of a compact space $X$ as an inverse limit space, and identify the ring $C(QF(X))$ as the order-Cauchy completion of the ring $ {C^{\ast}}(X)$. In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi $F$-cover of an arbitrary Tychonoff space. In this paper the minimal quasi $F$-cover of a compact space $X$ is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of $X$. The relationship between $ QF(X)$ and $QF(\beta X)$ is studied in detail, and broad conditions under which $\beta (QF(X)) = QF(\beta X)$ are obtained, together with examples of spaces for which the relationship fails. (Here $\beta X$ denotes the Stone-Čech compactification of $X$.) The role of $QF(X)$ as a "projective object" in certain topological categories is investigated.
Rings of differential operators on invariant rings of tori
Ian M.
Musson
805-827
Abstract: Let $k$ be an algebraically closed field of characteristic zero and $G$ a torus acting diagonally on ${k^s}$. For a subset $\beta$ of ${\mathbf{s}} = \{ 1,\,2, \ldots ,\,s\}$, set ${U_\beta } = \{ u \in {k^s}\vert{u_j} \ne 0\;{\text{if}}\;j \in \beta \}$. Then $G$ acts on $\mathcal{O}({U_\beta })$, the ring of regular functions on ${U_\beta }$, and we study the ring $ D(\mathcal{O}{({U_\beta })^G})$ of all differential operators on the invariant ring. More generally suppose that $\Delta$ is a set of subsets of s, such that each invariant ring $\mathcal{O}{({U_\beta })^G}$, $\beta \in \Delta$, has the same quotient field. We prove that ${ \cap _{\beta \in \Delta }}D(\mathcal{O}{({U_\beta })^G})$ is Noetherian and finitely generated as a $ k$-algebra. Now $ G$ acts on each $ D(\mathcal{O}({U_\beta }))$ and there is a natural map $\displaystyle \theta :\bigcap\limits_{\beta \in \Delta } {D{{(\mathcal{O}({U_\b... ..._{\beta \in \Delta } {D(\mathcal{O}{{({U_\beta })}^G}) = D({Y_\Delta } / G)} }$ obtained by restriction of the differential operators. We find necessary and sufficient conditions for $ \theta$ to be surjective and describe the kernel of $\theta$. The algebras ${ \cap _{\beta \in \Delta }}D{(\mathcal{O}({U_\beta }))^G}$ and ${ \cap _{\beta \in \Delta }}D(\mathcal{O}{({U_\beta })^G})$ carry a natural filtration given by the order of the differential operators. We show that the associated graded rings are finitely generated commutative algebras and are Gorensetin rings. We also determine the centers of ${ \cap _{\beta \in \Delta }}D{(\mathcal{O}({U_\beta }))^G}$ and ${ \cap _{\beta \in \Delta }}D(\mathcal{O}{({U_\beta })^G})$.