$k$-flat structures and exotic characteristic classes
Lisa R.
Goldberg
433-453
Abstract: We generalize the concept of "foliation" and define $k$-flat structures; these are smooth vector bundles with affine connections whose characteristic forms vanish above a certain dimension. Using semisimplicial techniques we construct a classifying space for $k$-flat structures, and prove a classification theorem for these structures on smooth manifolds. Techniques from rational homotopy theory are used to relate the exotic characteristic classes of foliations to the rational homotopy groups and cohomology of the classifying space.
On the theory of biorthogonal polynomials
A.
Iserles;
S. P.
Nørsett
455-474
Abstract: Let $\varphi (x,\,\mu )$ be a distribution in $x \in {\mathbf{R}}$ for every $\mu$ in a real parameter set $ \Omega$. Subject to additional technical conditions, we study $m$th degree monic polynomials $ {p_m}$ that satisfy the biorthogonality conditions $\displaystyle \int_{ - \infty }^\infty {{p_m}(x)\,d\varphi (x,{\mu _l}) = 0,} \qquad l = 1,\,2, \ldots ,\,m,\;m \geqslant 1$ , for a distinct sequence ${\mu _1},\,{\mu _2},\, \ldots \; \in \Omega \,$. Necessary and sufficient conditions for existence and uniqueness are established, as well as explicit determinantal and integral representations. We also consider loci of zeros, existence of Rodrigues-type formulae and reducibility to standard orthogonality. The paper is accompanied by several examples of biorthogonal systems.
Classifying $1$-handles attached to knotted surfaces
Jeffrey
Boyle
475-487
Abstract: We study a method of obtaining knotted surfaces in the $4$-sphere ${S^4}$ by attaching embedded $ 2$-dimensional $ 1$-handles to a given knot. The main result is there is a one-to-one correspondence between the $1$-handles that can be attached to a knot and the double cosets of the peripheral subgroup in the group of the knot. Many examples and applications are given.
Geodesics and conformal transformations of Heisenberg-Reiter spaces
J. F.
Torres Lopera
489-498
Abstract: Generalized Heisenberg groups, in the sense of Reiter, can be endowed with left-invariant metrics whose geodesies and curvature are obtained. Using these curvature data it is also proved that on their nilmanifolds (compact or not), every conformal transformation is in fact an isometry. A large family of nonisometric examples is given.
Topological equivalence of flows on homogeneous spaces, and divergence of one-parameter subgroups of Lie groups
Diego
Benardete
499-527
Abstract: Let $\Gamma$ and $\Gamma '$ be lattices, and $\phi$ and $\phi '$ one-parameter subgroups of the connected Lie groups $G$ and $G'$. If one of the following conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces $G/\Gamma$ and $G' /\Gamma '$ are topologically equivalent, then they are topologically equivalent by an affine map. (a) $G$ and $G'$ are one-connected and nilpotent. (b) $ G$ and $G'$ are one-connected and solvable, and for all $X$ in $L(G)$ and $X'$ in $L(G' )$, $ \operatorname{ad} (x)$ and
On certain fibred ribbon disc pairs
Iain R.
Aitchison;
Daniel S.
Silver
529-551
Abstract: We prove that for any free group automorphism ${\phi ^{\ast}}$ having a specified form there exists an invertible ribbon disc pair $({B^4},\,{D^2})$ such that the closure of ${B_4} - \operatorname{nbd} ({D^2})$ fibres over the circle with fibre a handlebody and monodromy equal to ${\phi ^{\ast}}$. We apply this to obtain results about ribbon $1$- and $2$-knots.
Classification of continuous $JBW\sp *$-triples
G.
Horn;
E.
Neher
553-578
Abstract: We show that every $JB{W^{\ast}}$-triple without a direct summand of type I is isometrically isomorphic to an ${l^\infty }$-sum $\mathcal{R}{ \otimes ^\infty }H(A,\,\alpha )$ where $\mathcal{R}$ is a ${w^{\ast}}$-closed right ideal in a ${W^{\ast}}$-algebra $B$ and $ H(A,\,\alpha )$ are the elements of a $ {W^{\ast}}$-algebra $ A$ which are symmetric under a C-linear involution $\alpha$ of $A$. Both $A$ and $B$ do not have a direct ( $ {W^{\ast}}$-algebra) summand of type I. In order to refine the decomposition $\mathcal{R}{ \otimes ^\infty }H(A,\,\alpha )$ we define and characterize types of $ JB{W^{\ast}}$-triples.
The Casson-Gordon invariants in high-dimensional knot theory
Daniel
Ruberman
579-595
Abstract: The Casson-Gordon invariants of knots in all dimensions are interpreted in terms of surgery theory. Applications are given to finding non-doubly slice knots, and doubly slice knots which are not the double of a disk knot. In even dimensions, the property of being doubly slice is shown to be largely homotopy theoretic, while in odd dimensions the surgery-theoretic method shows such properties to depend on more than the homotopy type.
Alexander modules of links with all linking numbers zero
M. L.
Platt
597-605
Abstract: In this paper we characterize the Alexander modules of links resulting from a surgical modification on the trivial link of any number of components. Using the presentation matrix obtained, we derive some properties of the Alexander polynomials of such links.
Exceptional boundary sets for solutions of parabolic partial differential inequalities
G. N.
Hile;
R. Z.
Yeh
607-621
Abstract: Let $\mathcal{M}$ be a second order, linear, parabolic partial differential operator with coefficients defined in a domain $\mathcal{D} = \Omega \times (0,\,T)$ in $ {{\mathbf{R}}^n} \times {\mathbf{R}}$, with $\Omega$ a domain in $ {{\mathbf{R}}^n}$. Let $ u$ be a suitably regular real function in $ \mathcal{D}$ such that $ u$ is bounded below and $\mathcal{M}u$ is bounded above in $\mathcal{D}$. If $ u \geqslant 0$ on $\Omega \times \{ 0\}$ except on a set $\Gamma \times \{ 0\}$, with $\Gamma$ a subset of $\Omega$ of suitably restricted Hausdorff dimension, then necessarily $u \geqslant 0$ also on $\Gamma \times \{ 0\}$. The allowable Hausdorff dimension of $\Gamma$ depends on the coefficients of $\mathcal{M}$. For example, if $\mathcal{M}$ is the heat operator $\Delta - \partial /\partial t$, the Hausdorff dimension of $\Gamma$ needs to be smaller than the number of space dimensions $n$. Analogous results are valid for exceptional boundary sets on the lateral boundary, $\partial \Omega \times (0,\,T)$, of $\mathcal{D}$.
Flows on vector bundles and hyperbolic sets
Dietmar
Salamon;
Eduard
Zehnder
623-649
Abstract: This note deals with C. Conley's topological approach to hyperbolic invariant sets for continuous flows. It is based on the notions of isolated invariant sets and Morse decompositions and it leads to the concept of weak hyperbolicity.
On the vanishing of homology and cohomology groups of associative algebras
Rolf
Farnsteiner
651-665
Abstract: This paper establishes sufficient conditions for the vanishing of the homology and cohomology groups of an associative algebra with coefficients in a two-sided module.
Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras
B. N.
Allison
667-695
Abstract: In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra $(\mathcal{A},\, - )$, we associate a quadratic form $Q$ called the Albert form of $(\mathcal{A},\, - )$. The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index $F_{4,1}^{21}$, $ {}^2E_{6,1}^{29}$, $E_{7,1}^{48}$ and $ E_{8,1}^{91}$. That description is used to obtain a classification of the indicated Lie algebras over $ {\mathbf{R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3$.
Outer functions in function algebras on the bidisc
Håkan
Hedenmalm
697-714
Abstract: Let $f$ be a function in the bidisc algebra $ A({{\mathbf{D}}^2})$ whose zero set $Z(f)$ is contained in $\{ 1\} \times \overline {\mathbf{D}}$. We show that the closure of the ideal generated by $f$ coincides with the ideal of functions vanishing on $Z(f)$ if and only if $f( \cdot ,\,\alpha )$ is an outer function for all $ \alpha \in \overline {\mathbf{D}}$, and $ f(1,\, \cdot )$ either vanishes identically or is an outer function. Similar results are obtained for a few other function algebras on $ {{\mathbf{D}}^2}$ as well.
Riemannian $4$-symmetric spaces
J. A.
Jiménez
715-734
Abstract: The main purpose of this paper is to classify the compact simply connected Riemannian $4$-symmetric spaces. As homogeneous manifolds, these spaces are of the form $G/L$ where $G$ is a connected compact semisimple Lie group with an automorphism $\sigma$ of order four whose fixed point set is (essentially) $L$. Geometrically, they can be regarded as fiber bundles over Riemannian $2$-symmetric spaces with totally geodesic fibers isometric to a Riemannian $2$-symmetric space. A detailed description of these fibrations is also given. A compact simply connected Riemannian $4$-symmetric space decomposes as a product $ {M_1} \times \ldots \times {M_r}$ where each irreducible factor is: (i) a Riemannian $2$-symmetric space, (ii) a space of the form $ \{ U \times U \times U \times U\} /\Delta U$ with $U$ a compact simply connected simple Lie group, $\Delta U =$ diagonal inclusion of $U$, (iii) $\{ U \times U\} /\Delta {U^\theta }$ with $ U$ as in (ii) and ${U^\theta }$ the fixed point set of an involution $ \theta$ of $U$, and (iv) $U/K$ with $U$ as in (ii) and $K$ the fixed point set of an automorphism of order four of $U$. The core of the paper is the classification of the spaces in (iv). This is accomplished by first classifying the pairs $(\mathfrak{g},\,\sigma )$ with $\mathfrak{g}$ a compact simple Lie algebra and $ \sigma$ an automorphism of order four of $ \mathfrak{g}$. Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For $ U$ "classical," the automorphisms $\sigma$ are explicitly constructed using their matrix representations. The idea of duality for $ 2$-symmetric spaces is extended to $4$-symmetric spaces and the duals are determined. Finally, those spaces that admit invariant almost complex structures are also determined: they are the spaces whose factors belong to the class (iv) with $ K$ the centralizer of a torus.
Construction of cohomology of discrete groups
Y. L.
Tong;
S. P.
Wang
735-763
Abstract: A correspondence between Hermitian modular forms and vector valued harmonic forms in locally symmetric spaces associated to $U(p,\,q)$ is constructed and also shown in general to be nonzero. The construction utilizes Rallis-Schiffmann type theta functions and simplified arguments to circumvent differential geometric calculations used previously in related problems.
Derivatives of meromorphic functions of finite order
Werner P.
Kohs;
Jack
Williamson
765-772
Abstract: Let $F$ be a nonentire, meromorphic function of finite order with only real zeros and real poles such that $F'$ has no zeros. We classify all such real $ F$ and all such strictly nonreal $F$ whose poles are of bounded multiplicities. We also give examples of such $F$ which are strictly nonreal and whose poles are of unbounded multiplicities.
Primeness and sums of tangles
Mario
Eudave Muñoz
773-790
Abstract: We consider knots and links obtained by summing a rational tangle and a prime tangle. For a given prime tangle, we show that there are at most three rational tangles that will induce a composite or splittable link. In fact, we show that there is at most one rational tangle that will give a splittable link. These results extend Scharlemann's work.
Representations of anisotropic unitary groups
Donald G.
James
791-804
Abstract: Let $SU(f)$ be the special unitary group of an anisotropic hermitian form $f$ over a field $k$. Assume $f$ represents only one norm class in $k$. The representations $ \alpha :\,SU(f) \to SL(n,\,R)$ are characterized when $R$ is a commutative ring with $2$ not a zero divisor and $n = \dim f \geqslant 3$ with $n \ne 4,\,6$. In particular, a partial classification of the normal subgroups of $SU(f)$ is given when $k$ is the function field ${\mathbf{C}}(X)$.
Finite-codimensional invariant subspaces of Bergman spaces
Sheldon
Axler;
Paul
Bourdon
805-817
Abstract: For a large class of bounded domains in $ \mathbb{C}$, we describe those finite codimensional subspaces of the Bergman space that are invariant under multiplication by $ z$. Using different techniques for certain domains in ${\mathbb{C}^N}$, we describe those finite codimensional subspaces of the Bergman space that are invariant under multiplication by all the coordinate functions.
General gauge theorem for multiplicative functionals
K. L.
Chung;
K. M.
Rao
819-836
Abstract: We generalize our previous work on the gauge theorem and its various consequences and complements, initiated in [8] and somewhat extended by subsequent investigations (see [6]). The generalization here is two-fold. First, instead of the Brownian motion, the underlying process is now a fairly broad class of Markov processes, not necessarily having continuous paths. Second, instead of the Feynman-Kac functional, the exponential of a general class of additive functionals is treated. The case of Schrödinger operator $\Delta /2 + \nu $, where $\nu$ is a suitable measure, is a simple special case. The most general operator, not necessarily a differential one, which may arise from our potential equations is briefly discussed toward the end of the paper. Concrete instances of applications in this case should be of great interest.
Tauberian theorems and stability of one-parameter semigroups
W.
Arendt;
C. J. K.
Batty
837-852
Abstract: The main result is the following stability theorem: Let $ \mathcal{T} = {(T(t))_{t \geqslant 0}}$ be a bounded ${C_0}$-semigroup on a reflexive space $ X$. Denote by $ A$ the generator of $\mathcal{T}$ and by $ \sigma (A)$ the spectrum of $A$. If $ \sigma (A) \cap i{\mathbf{R}}$ is countable and no eigenvalue of $A$ lies on the imaginary axis, then $ {\lim _{t \to \infty }}T(t)x = 0$ for all $x \in X$.
A class of nonlinear Sturm-Liouville problems with infinitely many solutions
Renate
Schaaf;
Klaus
Schmitt
853-859
Abstract: This paper is concerned with the existence of solutions of nonlinear Sturm-Liouville problems whose linear part is at resonance. It is shown that such problems may have infinitely many solutions if the nonlinear perturbations are periodic.