The Dirac operator on spaces with conical singularities and positive scalar curvatures
Arthur Weichung
Chou
1-40
Abstract: We study, in the spirit of Jeff Cheeger, the Dirac operator on a space with conical singularities. We obtain a Bochner-type vanishing theorem and prove an index theorem in the singular case. Also, the relationship with manifolds with boundary is explored. In the Appendix two methods of deforming the metric near the boundary are established and applied to obtain several new results on constructing complete metrics with positive scalar curvature.
Degrees of indiscernibles in decidable models
H. A.
Kierstead;
J. B.
Remmel
41-57
Abstract: We show that the problem of finding an infinite set of indiscernibles in an arbitrary decidable model of a first order theory is essentially equivalent to the problem of finding an infinite path through a recursive $\omega$-branching tree. Similarly, we show that the problem of finding an infinite set of indiscernibles in a decidable model of an $\omega$-categorical theory with decidable atoms is essentially equivalent to finding an infinite path through a recursive binary tree.
Spaces with coarser minimal Hausdorff topologies
Jack
Porter;
Johannes
Vermeer
59-71
Abstract: A technique is developed, using $H$-closed extensions, for determining when certain Hausdorff spaces are Katetov, i.e., have a coarser minimal Hausdorff topology. Our technique works for Čech-complete Lindelöf spaces, complete metrizable spaces, and many other spaces. Also, a number of interesting examples are presented; the most striking is an example of a Katetov space whose semiregularization is not Katetov.
Regularity properties of solutions to the basic problem in the calculus of variations
F. H.
Clarke;
R. B.
Vinter
73-98
Abstract: This paper concerns the basic problem in the calculus of variations: minimize a functional $J$ defined by $\displaystyle J(x) = \int_a^b {L(t,x(t),\dot x(t))\;dt}$ over a class of arcs $ x$ whose values at $ a$ and $b$ have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution $x$, there exists a relatively open subset $ \Omega$ of $[a,b]$, of full measure, on which $ x$ is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions. Three such classes are identified, and results of Bernstein, Tonelli, and Morrey are extended. One of these classes is of a novel nature, and its study implies the new result that when $L$ is independent of $t$, the solution has essentially bounded derivative.
General defect relations of holomorphic curves
Kiyoshi
Niino
99-113
Abstract: Let $ x:{\mathbf{C}} \to {P_n}{\mathbf{C}}$ be a holomorphic curve of finite lower order $ \mu$, and let $A = \{ \alpha \}$ be an arbitrary finite family of holomorphic curves $\alpha :{\mathbf{C}} \to {({P_n}{\mathbf{C}})^\ast}$ satisfying $ T(r,\alpha ) = o(T(r,x))\;(r \to \infty )$. Suppose $x$ is nondegenerate with respect to $A$, and $A$ is in general position. We show the following general defect relations: (1) $ x$ has at most $ n$ deficient curves in $ A$ if $\mu = 0$. (2) $\sum\nolimits_{\alpha \in A} {\delta (\alpha ) \leq n\;{\text{if}}\;0 < \mu \leq 1/2}$. (3) $ \sum\nolimits_{\alpha \in A} {\delta (\alpha ) \leq [2n\mu ] + 1\;{\text{if}}\;1/2 < \mu < + \infty } $.
Anosov diffeomorphisms and expanding immersions. I
Lowell
Jones
115-131
Abstract: The purpose of this paper is to develop a theory for representing Anosov diffeomorphisms by expanding immersions on compact branched manifolds. This theory was motivated by R. F. Williams' study of expanding attractors [15,17].
Abstract theory of abelian operator algebras: an application of forcing
Thomas J.
Jech
133-162
Abstract: The abstract abelian operator theory is developed from a general standpoint, using the method of forcing and Boolean-valued models.
Uniform operators
Hsiao Lan
Wang;
Joseph G.
Stampfli
163-169
Abstract: A general representation theorem for uniform operators is obtained which enables one to verify a conjecture of Cowen and Douglas in the presence of a mild additional restriction.
On the index of a number field
Enric
Nart
171-183
Abstract: Arithmetic invariants are found which determine the index $i(K)$ of a number field $K$. They are used to obtain an explicit formula under certain restrictions on $ K$. They provide also a complete explanation of a phenomenon conjectured by Ore [ ${\mathbf{8}}$] and showed by Engstrom in a particular case [ $ {\mathbf{2}}$].
On contractions of semisimple Lie groups
A. H.
Dooley;
J. W.
Rice
185-202
Abstract: A limiting formula is given for the representation theory of the Cartan motion group associated to a Riemannian symmetric pair $(G,K)$ in terms of the representation theory of $ G$.
Some structure theory for a class of triple systems
Nora C.
Hopkins
203-212
Abstract: This paper deals with a class of triple systems satisfying two generalized five linear identities and having nondegenerate bilinear forms with certain properties. If $(M,\{ ,,\} )$ is such a triple system with bilinear form $\phi (,)$, it is shown that if $M$ is semisimple, then $ M$ is the direct sum of simple ideals if $\phi$ is symmetric or symplectic or if $ M$ is completely reducible as a module for its right multiplication algebra $\mathcal{L}$. It is also shown that if $M$ is a completely reducible $\mathcal{L}$-module, $M$ is the direct sum of a semisimple ideal and the center of $M$. Such triple systems can be embedded into certain nonassociative algebras and the results on the triple systems are extended to these algebras.
Complete linear systems on rational surfaces
Brian
Harbourne
213-226
Abstract: We determine the dimension, fixed components and base points of complete linear systems on blowings-up of ${{\mathbf{P}}^2}$ having irreducible anticanonical divisor.
A nonshrinkable decomposition of $S\sp 3$ whose nondegenerate elements are contained in a cellular arc
W. H.
Row;
John J.
Walsh
227-252
Abstract: A decomposition $ G$ of ${S^3}$ is constructed with the following properties: (1) The set ${N_G}$ of all nondegenerate elements consists of a null sequence of arcs and $ J = {\text{CL}}( \cup \{ g \in {N_G}\} )$ is a simple closed curve. (2) Each arc contained in $J$ is cellular. (3) $J$ is the boundary of a disk $Q$ that is locally flat except at points of $J$. (4) The decomposition $G$ is not shrinkable; that is, the decomposition space is not homeomorphic to $ {S^3}$.
The splittability and triviality of $3$-bridge links
Seiya
Negami;
Kazuo
Okita
253-280
Abstract: A method to simplify $3$-bridge projections of links and knots, called a wave move, is discussed in general situation and it is shown what kind of properties of $3$-bridge links and knots can be recognized from their projections by wave moves. In particular, it will be proved that every $3$-bridge projection of a splittable link or a trivial knot can be transformed into a disconnected one or a hexagon, respectively, by a finite sequence of wave moves. As its translation via the concept of $2$-fold branched coverings of ${S^3}$, it follows that every genus $ 2$ Heegaard diagram of ${S^2} \times {S^2}\char93 L(p,q)$ or $ {S^3}$ can be transformed into one of specific standard forms by a finite sequence of operations also called wave moves.
On the asphericity of ribbon disc complements
James
Howie
281-302
Abstract: The complement of a ribbon $n$-disc in the $(n + 2)$-ball has a $2$-dimensional spine which shares some of the combinatorial properties of classical knot complement spines. It is an open question whether such $2$-complexes are always aspherical. To any ribbon disc we associate a labelled oriented tree, from which the homotopy type of the complement can be recovered, and we prove asphericity in certain special cases described by conditions on this tree. Our main result is that the complement is aspherical whenever the associated tree has diameter at most $ 3$.
Deductive varieties of modules and universal algebras
Leslie
Hogben;
Clifford
Bergman
303-320
Abstract: A variety of universal algebras is called deductive if every subquasivariety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above.
Filtering cohomology and lifting vector bundles
E. Graham
Evans;
Phillip
Griffith
321-332
Abstract: For a module $ M$ over a local Cohen-Macaulay ring $R$ we develop a (finite) sequence of presentations of $ M$ which facilitates the study of invariants arising from the cohomology modules of $M$. As an application we use this data, in case $ R$ is regular and $ M$ represents a vector bundle on the punctured spectrum of $R$ with a vanishing cohomology module, to obtain bounds on how far $M$ can be lifted as a vector bundle.
Monge-Amp\`ere measures associated to extremal plurisubharmonic functions in ${\bf C}\sp n$
Norman
Levenberg
333-343
Abstract: We consider the extremal plurisubharmonic functions $L_E^\ast$ and $U_E^\ast$ associated to a nonpluripolar compact subset $E$ of the unit ball $B \subset {{\mathbf{C}}^n}$ and show that the corresponding Monge-Ampère measures ${(d{d^c}L_E^\ast )^n}$ and ${(d{d^c}U_E^\ast )^n}$ are mutually absolutely continuous. We then discuss the polynomial growth condition $({L^\ast})$, a generalization of Leja's polynomial condition in the plane, and study the relationship between the asymptotic behavior of the orthogonal polynomials associated to a measure on $E$ and the $ ({L^\ast})$ condition.
Points fixes d'applications holomorphes dans un domaine born\'e convexe de ${\bf C}\sp n$
Jean-Pierre
Vigué
345-353
Abstract: Let $D$ be a bounded convex domain in $ {{\mathbf{C}}^n}$. We prove that the set $V$ of fixed points of a holomorphic map $ f:D \to D$ is a complex submanifold of $D$ and, if $V$ is not empty, $V$ is a holomorphic retract of $D$.
Chaos, periodicity, and snakelike continua
Marcy
Barge;
Joe
Martin
355-365
Abstract: The results of this paper relate the dynamics of a continuous map $ f$ of the interval and the topology of the inverse limit space with bonding map $ f$. These inverse limit spaces have been studied by many authors, and are examples of what Bing has called "snakelike continua". Roughly speaking, we show that when the dynamics of $ f$ are complicated, the inverse limit space contains indecomposable subcontinua. We also establish a partial converse.
The fine structure of transitive Riemannian isometry groups. I
Carolyn S.
Gordon;
Edward N.
Wilson
367-380
Abstract: Let $M$ be a connected homogeneous Riemannian manifold, $G$ the identity component of the full isometry group of $M$ and $H$ a transitive connected subgroup of $G$. $G = HL$, where $L$ is the isotropy group at some point of $ M$. $M$ is naturally identified with the homogeneous space $ H/H \cap L$ endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of $ M$ as a Riemannian homogeneous space of a connected Lie group $H$, describe the structure of the full connected isometry group $G$ in terms of $H$. This problem has already been studied in case $ H$ is compact, semisimple of noncompact type, or solvable. We use the fact that every Lie group is a product of subgroups of these three types in order to study the general case.
On the ideals of a Noetherian ring
J. T.
Stafford
381-392
Abstract: We construct various examples of Noetherian rings with peculiar ideal structure. For example, there exists a Noetherian domain $ R$ with a minimal, nonzero ideal $I$, such that $R/I$ is a commutative polynomial ring in $ n$ variables, and a Noetherian domain $S$ with a (second layer) clique that is not locally finite. The key step in the construction of these rings is to idealize at a right ideal $I$ in a Noetherian domain $ T$ such that $ T/I$ is not Artinian.
Ergodic semigroups of epimorphisms
Daniel
Berend
393-407
Abstract: The conditions for ergodicity of semigroups of epimorphisms of compact groups are studied. In certain cases ergodic semigroups are shown to contain small ergodic subsemigroups. Properties related to ergodicity, such as that of admitting no infinite closed invariant proper subset of the group, are discussed for semigroups of epimorphisms and of affine transformations.
On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces
S.
Argyros;
V.
Farmaki
409-427
Abstract: A characterization of weakly compact subsets of a Hilbert space, when they are considered as subsets of $B$-spaces with an unconditional basis, is given. We apply this result to renorm a class of reflexive $ B$-spaces by defining a norm uniformly convex in every direction. We also prove certain results related to the factorization of operators. Finally, we investigate the structure of weakly compact subsets of $ {L^1}(\mu )$.
Erratum to: ``On wave fronts propagation in multicomponent media'' [Trans. Amer. Math. Soc. {\bf 276} (1983), no. 1, 181--191; MR0684501 (84a:35027)]
M. I.
Freidlin
429