The structure of Legendre foliations
Myung-Yull
Pang
417-455
Abstract: The local and global structure of Legendre foliations of contact manifolds is analysed. The main invariant of a Legendre foliation is shown to be a quadratic form on the tangent bundle to the foliation--the fundamental quadratic form. The equivalence problem is solved in the case when the fundamental quadratic form is nondegenerate and a generalization of Chern's solution to the equivalence problem for Finsler manifolds is obtained. A normal form for Legendre foliations is given which is closely related to Weinstein's structure theorem for Lagrangian foliations. It is shown that every compact, simply connected leaf of a Legendre foliation is diffeomorphic to a sphere.
Second order theta functions and vector bundles over Jacobi varieties
David S.
Yuen
457-492
Abstract: We consider the Picard vector bundles defined over Jacobi varieties. The rank $g + 1$ Picard bundle imbeds in the rank $ {2^g}$ Clifford bundle, so the second order theta functions, viewed appropriately, span the dual of the Picard bundle over each fiber. We prove a result on the minimum number of such second order theta functions required to span the whole bundle at each point. We give an application of using these functions to describe subvarieties of the Jacobian. There follow comments on which functions we could use, and generalizations to higher order theta functions.
Polynomial flows on ${\bf C}\sp n$
Brian A.
Coomes
493-506
Abstract: We show that polynomial flows on $ {\mathbb{R}^n}$ extend to functions holomorphic on ${\mathbb{C}^{n + 1}}$ and that the group property holds after this extension. Then we give some methods, based on power series, for determining when a vector field has a polynomial flow.
Smooth great circle fibrations and an application to the topological Blaschke conjecture
C. T.
Yang
507-524
Abstract: We study great smooth circle fibrations of round spheres and Blaschke manifolds of the homotopy type of complex projective spaces.
Matroid representations and free arrangements
Günter M.
Ziegler
525-541
Abstract: We show that Terao's Conjecture ("Freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid") holds over fields with at most four elements. However, an example demonstrates that the field characteristic has to be fixed for this.
Leavable gambling problems with unbounded utilities
A.
Maitra;
R.
Purves;
W.
Sudderth
543-567
Abstract: The optimal return function $U$ of a Borel measurable gambling problem with a positive utility function is known to be universally measurable. With a negative utility function, however, $ U$ may not be so measurable. As shown here, the measurability of $ U$ for all Borel gambling problems with negative utility functions is equivalent to the measurability of all PCA sets, a property of such sets known to be independent of the usual axioms of set theory. If the utility function is further required to satisfy certain uniform integrability conditions, or if the gambling problem corresponds to an optimal stopping problem, the optimal return function is measurable. Another return function $W$ is introduced as an alternative to $U$. It is shown that $W$ is always measurable and coincides with $ U$ when the utility function is positive.
Boundedness versus periodicity over commutative local rings
Vesselin N.
Gasharov;
Irena V.
Peeva
569-580
Abstract: Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period $2$. It is proved however, that the conjecture holds over rings of small length.
Semialgebraic expansions of ${\bf C}$
David
Marker
581-592
Abstract: We prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers.
The Dirichlet problem for radially homogeneous elliptic operators
Richard F.
Bass
593-614
Abstract: The Dirichlet problem in the unit ball is considered for the strictly elliptic operator $L = \sum {{a_{ij}}{D_{ij}}} $, where the $ {a_{ij}}$, are smooth away from the origin and radially homogeneous: ${a_{ij}}(rx) = {a_{ij}}(x),\;r > 0,\;x \ne 0$. Existence and uniqueness are proved for solutions in a certain space of functions. Necessary and sufficient conditions are given for an extended maximum principle to hold.
Endomorphism rings of formal $A\sb 0$-modules
Shuji
Yamagata
615-623
Abstract: Let ${A_0}$ be the valuation ring of a finite extension ${K_0}$ of ${Q_p}$ and $ A \supset {A_0}$ be a complete discrete valuation ring with the perfect residue field. We consider the endomorphism rings of $ n$-dimensional formal $ {A_0}$-modules $ \Gamma$ over $ A$ of finite $ {A_0}$-height with reduction absolutely simple up to isogeny. Especially we prove commutativity of ${\operatorname{End} _{A,{A_0}}}(\Gamma )$. Given an arbitrary finite unramified extension $ {K_1}$ of ${K_0}$, a variety of examples (different dimensions and different ${A_0}$-heights) is constructed whose absolute endomorphism rings are isomorphic to the valuation ring of ${K_1}$.
On a theorem of Stein
Steven G.
Krantz
625-642
Abstract: In this paper the Kobayashi metric on a domain in ${{\mathbf{C}}^n}$ is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if $f$ is holomorphic on the domain and in the classical Lipschitz space ${\Lambda _\alpha }$ then in fact $f$ is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces.
Butler groups of infinite rank. II
Manfred
Dugas;
Paul
Hill;
K. M.
Rangaswamy
643-664
Abstract: A torsion-free abelian group $G$ is called a Butler group if $\operatorname{Bext} (G,T) = 0.$ for any torsion group $ T$. We show that every Butler group $G$ of cardinality $ {\aleph _1}$ is a $ {B_2}$-group; i.e., $ G$ is a union of a smooth ascending chain of pure subgroups ${G_\alpha }$ where $ {G_{\alpha + 1}} = {G_\alpha } + {B_\alpha },{B_\alpha }$ a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding $ {\aleph _\omega }$ is a $ {B_2}$-group. Moreover, we are able to prove that the derived functor $ {\operatorname{Bext} ^2}(A,T) = 0$ for any torsion group $T$ and any torsion-free $A$ with $\vert A\vert \leqslant {\aleph _\omega }$. This implies that under CH all balanced subgroups of completely decomposable groups of cardinality $\leqslant {\aleph _\omega }$ are $ {B_2}$-groups.
Symmetry properties of the solutions to Thomas-Fermi-Dirac-von Weizs\"acker type equations
Rafael D.
Benguria;
Cecilia
Yarur
665-675
Abstract: We consider a semilinear elliptic equation with a spherically symmetric potential (specifically, Thomas-Fermi-Dirac-von Weizsäcker type equations without electronic repulsion). Assuming some regularity properties of the solutions at the origin and at infinity, we prove that the solutions have spherical symmetry.
Comparison of commuting one-parameter groups of isometries
Ola
Bratteli;
Hideki
Kurose;
Derek W.
Robinson
677-694
Abstract: Let $\alpha ,\;\beta$ be two commuting strongly continuous one-parameter groups of isometries on a Banach space $\mathcal{A}$ with generators ${\delta _\alpha }$ and ${\delta _\beta }$, and analytic elements $\mathcal{A}_\omega ^\alpha ,\;\mathcal{A}_\omega ^\beta$, respectively. Then it is easy to show that if ${\delta _\alpha }$ is relatively bounded by ${\delta _\beta }$, then $\mathcal{A}_\omega ^\beta \subseteq \mathcal{A}_\omega ^\alpha$, and in this paper we establish the inverse implication for unitary one-parameter groups on Hilbert spaces and for one-parameter groups of $ ^{\ast}$-automorphisms of abelian $ {C^{\ast}}$-algebras. It is not known in general whether the inverse implication holds or not, but it does not hold for one-parameter semigroups of contractions.
On the sparsity of representations of rings of pure global dimension zero
Birge
Zimmermann-Huisgen;
Wolfgang
Zimmermann
695-711
Abstract: It is shown that the rings $R$ all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition: For each positive integer $ n$ there are only finitely many isomorphism types of (a) indecomposable left $ R$-modules of length $ n$; (b) finitely presented indecomposable right $R$-modules of length $n$; (c) indecomposable right $R$-modules having minimal projective resolutions with $n$ relations. Moreover, our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right $R$-modules entails finite representation type.
The orderability and closed images of scattered spaces
S.
Purisch
713-725
Abstract: A (totally) orderable scattered space and a space homeomorphic to a subspace of an ordinal space are characterized in terms of a neighborhood subbase for each of their points plus what corresponds to a neighborhood base for each of their non-$Q$-gaps. These generalize the characterizations in [ P$_{1}$] of an orderable compact scattered space and in [B] of a space homeomorphic to a compact ordinal space. Generalizing a result in [M] it is shown that a space is orderable and scattered iff it is the $ 2$ to $1$ image under a closed map of a subspace of an ordinal space. In response to a question of Telgarsky [T] a simple description is given of a closed map with discrete fibers from an orderable scattered space onto an orderable perfect space. Maps that preserve length conditions on a scattered space are touched upon.
Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions
Miguel A.
Ariño;
Benjamin
Muckenhoupt
727-735
Abstract: A characterization is given of a class of classical Lorentz spaces on which the Hardy Littlewood maximal operator is bounded. This is done by determining the weights for which Hardy's inequality holds for nonincreasing functions. An alternate characterization, valid for nondecreasing weights, is also derived.
A new proof of the strong partition relation on $\omega\sb 1$
Steve
Jackson
737-745
Abstract: Assuming the axiom of determinacy, we give a new proof of the strong partition relation on $ {\omega _1}$. The proof is direct and avoids appeal to complicated set-theoretic machinery.
A canonical extension for analytic functions on Banach spaces
Ignacio
Zalduendo
747-763
Abstract: Given Banach spaces $ E$ and $F$, a Banach space ${G_{EF}}$ is presented in which $ E$ is embedded and which seems a natural space to which extend $F$-valued analytic functions. Any $ F$-valued analytic function defined on a subset $U$ of $E$ may be extended to an open neighborhood of $ U$ in ${G_{EF}}$. This extension generalizes that of Aron and Berner. It is also related to the Arens product in Banach algebras, to the functional calculus for bounded linear operators, and to an old problem of duality in spaces of analytic functions. A characterization of the Aron-Berner extension is given in terms of continuity properties of first-order differentials.
Parametrization of domains in $\hat{\bf C}$: the logarithmic domains
Johannes
Michaliček;
Rodolfo
Wehrhahn
765-777
Abstract: We prove a generalization of Riemann's mapping theorem: Every $ n$-fold connected domain in $ \widehat{\mathbf{C}}$, whose boundary does not contain isolated points, is conformal equivalent to a logarithmic domain. The logarithmic domains are characterized by a Green's function consisting of a finite sum of logarithms.
Discontinuous ``viscosity'' solutions of a degenerate parabolic equation
Michiel
Bertsch;
Roberta
Dal Passo;
Maura
Ughi
779-798
Abstract: We study a nonlinear degenerate parabolic equation of the second order. Regularizing the equation by adding some artificial viscosity, we construct a generalized solution. We show that this solution is not necessarily continuous at all points.
Bounds for projection constants and $1$-summing norms
Hermann
König;
Nicole
Tomczak-Jaegermann
799-823
Abstract: It is shown that projection constants $ \lambda ({X_n})$ of $ n$-dimensional normed spaces ${X_n}$ satisfy $\lambda ({X_n}) \leqslant \sqrt n - c/\sqrt n$ where $c > 0$ is a numerical constant. Similarly, the $ 1$-summing norms of (the identity of) ${X_n}$ can be estimated by $ {\pi _1}({X_n}) \geqslant \sqrt n + c/\sqrt n$. These estimates are the best possible: for prime $n$, translation-invariant $ n$-dimensional spaces $ {X_n}$ such that $ \lambda ({X_n}) \geqslant \sqrt n - 2/\sqrt n$ and ${\pi _1}({X_n}) \leqslant \sqrt n + 2/\sqrt n$ can be constructed. For these spaces Gordon-Lewis constants and distances to Hilbert spaces are large as well: $\operatorname{gl} ({X_n}) \geqslant \tfrac{1} {3}\sqrt n ,d({X_n},l_2^n) = \sqrt n $.