Commutator theory for relatively modular quasivarieties
Keith
Kearnes;
Ralph
McKenzie
465-502
Abstract: We develop a commutator theory for relatively modular quasivarieties that extends the theory for modular varieties. We characterize relatively modular quasivarieties, prove that they have an almost-equational axiomatization and we investigate the lattice of subquasivarieties. We derive the result that every finitely generated, relatively modular quasivariety of semigroups is finitely based.
On the positive solutions of semilinear equations $\Delta u+\lambda u-hu\sp p=0$ on the compact manifolds
Tiancheng
Ouyang
503-527
Abstract: In this paper, we study the existence, nonexistence, and uniqueness of positive solutions of semilinear equations $ \Delta u + \lambda u - h{u^p}= 0$ on compact Riemannian manifolds as well as on bounded smooth domains in ${R^n}$ with homogeneous Dirichlet or Neumann boundary conditions.
The Bergman projection on Hartogs domains in ${\bf C}\sp 2$
Harold P.
Boas;
Emil J.
Straube
529-540
Abstract: Estimates in $ {L^2}$ Sobolev norms are proved for the Bergman projection in certain smooth bounded Hartogs domains in ${{\mathbf{C}}^2}$. In particular, (1) if the domain is pseudoconvex and "nonwormlike" (the normal vector does not wind on a critical set in the boundary), then the Bergman projection is regular; and (2) Barrett's counterexample domains with irregular Bergman projection nevertheless admit a priori estimates.
Invariant affine connections on Lie groups
H. Turner
Laquer
541-551
Abstract: The space of bi-invariant affine connections is determined for arbitrary compact Lie groups. In particular, there is a surprising new family of such connections on $ SU(n)$.
Finite determination on algebraic sets
L.
Kushner
553-561
Abstract: The concept of finite relative determination was introduced by Porto and Loibel $ [$P-L$]$ in 1978 and it deals with subspaces of $ {{\mathbf{R}}^n}$. In this paper we generalize this concept for algebraic sets, and relate it with finite determination on the right. We finish with an observation between Lojasiewicz ideals and finite relative determination.
Galois groups and the multiplicative structure of field extensions
Robert
Guralnick;
Roger
Wiegand
563-584
Abstract: Let $K/k$ be a finite Galois field extension, and assume $k$ is not an algebraic extension of a finite field. Let ${K^{\ast} }$ be the multiplicative group of $K$, and let $ \Theta (K/k)$ be the product of the multiplicative groups of the proper intermediate fields. The condition that the quotient group $ \Gamma = {K^{\ast} }/\Theta (K/k)$ be torsion is shown to depend only on the Galois group $G$. For algebraic number fields and function fields, we give a complete classification of those $ G$ for which $ \Gamma$ is nontrivial.
Approximation of Jensen measures by image measures under holomorphic functions and applications
Shang Quan
Bu;
Walter
Schachermayer
585-608
Abstract: We show that Jensen measures defined on $ {\mathbb{C}^n}$ or more generally on a complex Banach space $X$ can be approximated by the image of Lebesgue measure on the torus under $X$-valued polynomials defined on $\mathbb{C}$. We give similar characterizations for Jensen measures in terms of analytic martingales and Hardy martingales. The results are applied to approximate plurisubharmonic martingales by Hardy martingales, which enables us to give a characterization of the analytic Radon-Nikodym property of Banach spaces in terms of convergence of plurisubharmonic martingales, thus solving a problem of G. A. Edgar.
Isoparametric submanifolds of hyperbolic spaces
Bingle
Wu
609-626
Abstract: In this paper we prove a decomposition theorem for isoparametric submanifolds of hyperbolic spaces. And as a consequence we obtain all polar actions on hyperbolic spaces. We also prove that any isoparametric submanifold of infinite dimensional hyperbolic space is either totally geodesic, or finite dimensional.
On the analyticity of solutions of first-order nonlinear PDE
Nicholas
Hanges;
François
Trèves
627-638
Abstract: Let $(x,t) \in {R^m} \times R$ and $u \in {C^2}\,({R^m} \times R)$. We discuss local and microlocal analyticity for solutions $u$ to the nonlinear equation $\displaystyle {u_t}= f(x,t,u,{u_x})$ . Here $f(x,t,{\zeta _0},\zeta)$ is complex valued and analytic in all arguments. We also assume $f$ to be holomorphic in $ ({\zeta _0},\zeta) \in C \times {C^m}$. In particular we show that $\displaystyle {\text{WF}}_A\,u \subset \operatorname{Char}({L^u})$ where $ {\text{WF}}_A$ denotes the analytic wave-front set and $ \operatorname{Char}({L^u})$ is the characteristic set of the linearized operator $\displaystyle {L^u}= \partial /\partial t - \sum \partial \,f/\partial \,{\zeta _j}(x,t,u,{u_x})\;\partial /\partial \,{x_j}$ . If we assume $u \in {C^3}\;({R^m} \times R)$ then we show that the analyticity of $u$ propagates along the elliptic submanifolds of $ {L^u}$.
On local structures of the singularities $A\sb k\;D\sb k$ and $E\sb k$ of smooth maps
Yoshifumi
Ando
639-651
Abstract: In studying the singularities of type ${A_k}$ of smooth maps between manifolds $ N$ and $P$ the Boardman manifold ${\sum ^{i,1, \ldots,10}}$ in ${J^\infty }\,(N,P)$ has been very useful. We will construct the submanifolds $ \sum {D_k}$ and $\Sigma {E_k}$ in ${J^\infty }\,(N,P)$ playing the similar role for singularities ${D_k}$ and ${E_k}$ and study their properties in its process.
On twistor spaces of anti-self-dual Hermitian surfaces
Massimiliano
Pontecorvo
653-661
Abstract: We consider a complex surface $M$ with anti-self-dual hermitian metric $ h$ and study the holomorphic properties of its twistor space $Z$. We show that the naturally defined divisor line bundle $[X]$ is isomorphic to the $- \frac{1} {2}$ power of the canonical bundle of $ Z$, if and only if there is a Kähler metric of zero scalar curvature in the conformal class of $h$. This has strong consequences on the geometry of $M$, which were also found by C. Boyer $ [3]$ using completely different methods. We also prove the existence of a very close relation between holomorphic vector fields on $ M$ and $Z$ in the case that $M$ is compact and Kähler.
Products of commutative rings and zero-dimensionality
Robert
Gilmer;
William
Heinzer
663-680
Abstract: If $R$ is a Noetherian ring and $ n$ is a positive integer, then there are only finitely many ideals $I$ of $R$ such that the residue class ring $R/I$ has cardinality $\leq n$. If $R$ has Noetherian spectrum, then the preceding statement holds for prime ideals of $R$. Motivated by this, we consider the dimension of an infinite product of zero-dimensional commutative rings. Such a product must be either zero-dimensional or infinite-dimensional. We consider the structure of rings for which each subring is zero-dimensional and properties of rings that are directed union of Artinian subrings. Necessary and sufficient conditions are given in order that an infinite product of zero-dimensional rings be a directed union of Artinian subrings.
A bounded mountain pass lemma without the (PS) condition and applications
Martin
Schechter
681-703
Abstract: We present a version of the mountain pass lemma which does not require the ( $\mathbf{PS}$) condition. We apply this version to problems where the ( $ \mathbf{PS}$) condition is not satisfied.
A restriction theorem for modules having a spherical submodule
Nicolás
Andruskiewitsch;
Juan A.
Tirao
705-725
Abstract: We introduce the following notion: a finite dimensional representation $ V$ of a complex reductive algebraic group $G$ is called spherical of rank one if the generic stabilizer $M$ is reductive, the pair $(G,M)$ is spherical and $\dim \;{V^M}= 1$. Let $U$ be another finite dimensional representation of $G$; we denote by $ S^{\prime}(U)\;(S^{\prime}{(U)^G})$ the ring of polynomial functions on $ U$ (the ring of $ G$-invariant polynomial functions on $U$). We characterize the image of $S^{\prime}{(U \oplus V)^G}$ under the restriction map into $S^{\prime}\,(U \oplus {V^M})$ as the $W= {N_G}(M)/M$ invariants in the Rees ring associated to an ascending filtration of $S^{\prime}{(U)^M}$. Furthermore, under some additional hypothesis, we give an isomorphism between the graded ring associated to that filtration and $S^{\prime}{(U)^P}$, where $P$ is the stabilizer of an unstable point whose $G$-orbit has maximal dimension.
The Gauss map of a genus three theta divisor
Clint
McCrory;
Theodore
Shifrin;
Robert
Varley
727-750
Abstract: A smooth complex curve is determined by the Gauss map of the theta divisor of the Jacobian variety of the curve. The Gauss map is invariant with respect to the $(- 1)$-map of the Jacobian. We show that for a generic genus three curve the Gauss map is locally $ {\mathbf{Z}}/2$-stable. One method of proof is to analyze the first-order $ {\mathbf{Z}}/2$-deformations of the Gauss map of a hyperelliptic theta divisor.
Growth series of some wreath products
Walter
Parry
751-759
Abstract: The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions.
$3$-manifold groups with the finitely generated intersection property
Teruhiko
Soma
761-769
Abstract: In this paper, first we consider whether the fundamental groups of certain geometric $3$-manifolds have FGIP or not. Next we give the sufficient conditions that FGIP for $3$-manifold groups is preserved under torus sums or annulus sums and connect this result with a conjecture by Hempel $[4]$.
Exactly $k$-to-$1$ maps between graphs
Jo
Heath;
A. J. W.
Hilton
771-785
Abstract: Suppose $ k$ is a positive integer, $ G$ and $H$ are graphs, and $f$ is a $ k{\text{-to-}}1$ correspondence from a vertex set of $G$ onto a vertex set of $H$. Conditions on the adjacency matrices are given that are necessary and sufficient for $ f$ to extend to a continuous $ k{\text{-to-}}1$ map from $ G$ onto $H$.
Parametrization of a singular Lagrangian variety
Goo
Ishikawa
787-798
Abstract: We give stabilization and parametrization theorems for a class of singular varieties in the space of polynomials of one variable and generalize the results of Arnol'd and Givental'. The class contains the open swallowtails and the open Whitney umbrella. The parametrization is associated with the singularity of a stable mapping (in the sense of Thom and Mather) of kernel rank one.
Schubert calculus in complex cobordism
Paul
Bressler;
Sam
Evens
799-813
Abstract: We study the structure of the complex cobordism ring of the flag variety of a compact connected Lie group. An explicit procedure for determining products of basis elements is obtained, generalizing the work of Bernstein-Gel'fand-Gel'fand on ordinary cohomology and of Kostant-Kumar on $ K$-theory. Bott-Samelson resolutions are used to replace the classical basis of Schubert cells.
Weakly compact homomorphisms
J. E.
Galé;
T. J.
Ransford;
M. C.
White
815-824
Abstract: We study the structure of weakly compact homomorphisms between Banach algebras. In particular, it is shown that between many pairs of algebras, the only weakly compact homomorphisms are those of finite rank.
Compact actions commuting with ergodic actions and applications to crossed products
C.
Peligrad
825-836
Abstract: Let $(A,K,\beta)$ be a $ {C^{\ast}}$-dynamical system with $K$ compact. In this paper we prove a duality result for saturated actions (Theorem 3.3). The proof of this result can also be considered as an alternate proof of the corresponding result for von Neumann algebras due to Araki, Haag, Kastler and Takesaki $ [14]$. We also obtain results concerning the simplicity and the primeness of the crossed product $ A \times _\beta K$ in terms of the ergodicity of the commutant of $\beta$ (Propositions 5.3 and 5.4).
Subsequence ergodic theorems for $L\sp p$ contractions
Roger L.
Jones;
James
Olsen;
Máté
Wierdl
837-850
Abstract: In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations, or Dunford Schwartz operators, are extended to operators which are positive contractions on $ {L^p}$ for $p$ fixed.
Maximal triads and prime decompositions of surfaces embedded in $3$-manifolds
Michael
Motto
851-867
Abstract: In 1975, Suzuki proved that prime decompositions of connected surfaces in ${S^3}$ are unique up to stable equivalence of the factors. This paper extends his result to a large class of $3$-manifolds, and demonstrates that this result does not apply to all $3$-manifolds. It also answers a question he raised by showing that it is possible for inequivalent surfaces in ${S^3}$ of the same genus to be stably equivalent. The techniques used involve the notion of Heegaard splittings of $3$-manifold triads.
The union of compact subgroups of an analytic group
Ta Sun
Wu
869-879
Abstract: Let $G$ be an analytic group. Let $\Omega (G)$ be the union of all compact subgroups of $ G$. We give a necessary and sufficient condition for $ \Omega (G)$ to be dense in $ G$ in terms of the action of a maximal compact torus $T$ of $G$ on the nilradical $N$ of $G$.
Rotation sets for homeomorphisms and homology
Mark
Pollicott
881-894
Abstract: In this article we propose a definition of rotation sets for homeomorphisms of arbitrary compact manifolds. This approach is based on taking the suspended flow and using ideas of Schwartzmann on homology and winding cycles for flows. Our main application is to give a generalisation of a theorem of Llibre and MacKay for tori to the context of surfaces of higher genus.
Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds
Peter B.
Shalen;
Philip
Wagreich
895-917
Abstract: It is shown that if $ M$ is a closed orientable irreducible $3$-manifold and $n$ is a nonnegative integer, and if ${H_1}(M,{\mathbb{Z}_p})$ has rank $\geq n + 2$ for some prime $p$, then every $n$-generator subgroup of ${\pi _1}\,(M)$ has infinite index in ${\pi _1}\,(M)$, and is in fact contained in infinitely many finite-index subgroups of ${\pi _1}\,(M)$. This result is used to estimate the growth rates of the fundamental group of a $ 3$-manifold in terms of the rank of the $ {\mathbb{Z}_p}$-homology. In particular it is used to show that the fundamental group of any closed hyperbolic $3$-manifold has uniformly exponential growth, in the sense that there is a lower bound for the exponential growth rate that depends only on the manifold and not on the choice of a finite generating set. The result also gives volume estimates for hyperbolic $ 3$-manifolds with enough $ {\mathbb{Z}_p}$-homology, and a sufficient condition for an irreducible $ 3$-manifold to be almost sufficiently large.