On the Kummer congruences and the stable homotopy of $B$U
Andrew
Baker;
Francis
Clarke;
Nigel
Ray;
Lionel
Schwartz
385-432
Abstract: We study the torsion-free part of the stable homotopy groups of the space $ BU$, by considering upper and lower bounds. The upper bound is furnished by the ring $ P{K_{\ast}}(BU)$ of coaction primitives into which $\pi _{\ast}^S(BU)$ is mapped by the complex $ K$-theoretic Hurewicz homomorphism $\displaystyle \pi _{\ast}^S(BU) \to P{K_{\ast}}(BU).$ We characterize $P{K_{\ast}}(BU)$ in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical Kummer congruences among the Bernoulli numbers. For a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in $ BU$. From among these we define families of classes constructed from regular neighborhoods of embeddings of iterated Thom complexes in Euclidean space. Employing techniques of duality theory, we deduce that these two families correspond, except possibly in the lowest dimensions, under the Hurewicz homomorphism, which thus provides a link between the algebra and the geometry. In the course of this work we greatly extend certain $e$-invariant calculations of J. F. Adams.
Negative scalar curvature metrics on noncompact manifolds
John
Bland;
Morris
Kalka
433-446
Abstract: In this paper we prove that every noncompact smooth manifold admits a complete metric of constant negative scalar curvature.
Multiplicity of the adjoint representation in simple quotients of the enveloping algebra of a simple Lie algebra
Anthony
Joseph
447-491
Abstract: Let $\mathfrak{g}$ be a complex simple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra and $U(\mathfrak{g})$ the enveloping algebra of $\mathfrak{g}$. We calculate for each maximal two-sided ideal $ {J_{\max }}(\lambda ):\lambda \in {\mathfrak{h}^{\ast}}$ of $ U(\mathfrak{g})$ the number of times the adjoint representation occurs in $U(\mathfrak{g})/{J_{\max }}(\lambda )$. This is achieved by reduction via the Kazhdan-Lusztig polynomials to the case when $\lambda$ lies on a corner, i.e. is a multiple of a fundamental weight. Remarkably in this case one can always present $U(\mathfrak{g})/{J_{\max }}(\lambda )$ as a (generalized) principal series module and here we also calculate its Goldie rank as a ring which is a question of independent interest. For some of the more intransigent cases it was necessary to use recent very precise results of Lusztig on left cells. The results are used to show how a recent theorem of Gupta established for "nonspecial" $\lambda$ can fail if $\lambda$ is singular. Finally we give a quite efficient procedure for testing if an induced ideal is maximal.
Conjugacy classes whose square is an infinite symmetric group
Gadi
Moran
493-522
Abstract: Let ${X_\nu }$ be the set of all permutations $ \xi$ of an infinite set $ A$ of cardinality ${\aleph _\nu }$ with the property: every permutation of $A$ is a product of two conjugates of $\xi$. The set ${X_0}$ is shown to be the set of permutations $ \xi$ satisfying one of the following three conditions: (1) $ \xi$ has at least two infinite orbits. (2) $\xi$ has at least one infinite orbit and infinitely many orbits of a fixed finite size $n$. (3) $\xi$ has: no infinite orbit; infinitely many finite orbits of size $k,l$ and $k + l$ for some positive integers $k,l$; and infinitely many orbits of size $> 2$. It follows that $\xi \in {X_0}$ iff some transposition is a product of two conjugates of $\xi$, and $\xi$ is not a product $\sigma i$, where $\sigma$ has a finite support and $i$ is an involution. For $\nu > 0,\;\xi \in {X_\nu }$ iff $ \xi$ moves ${\aleph _\nu }$ elements, and satisfies (1), (2) or $(3')$, where $(3')$ is obtained from (3) by omitting the requirement that $\xi$ has infinitely many orbits of size $ > 2$. It follows that for $ \nu > 0,\;\xi \in {X_\nu }\;$ iff $\xi$ moves $ {\aleph _\nu }$ elements and some transposition is the product of two conjugates of $ \xi$. The covering number of a subset $X$ of a group $G$ is the smallest power of $X$ (if any) that equals $G$ [AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.
Isometric dilations for infinite sequences of noncommuting operators
Gelu
Popescu
523-536
Abstract: This paper develops a dilation theory for $\{ {T_n}\} _{n = 1}^\infty $ an infinite sequence of noncommuting operators on a Hilbert space, when the matrix $[{T_1},{T_2}, \ldots ]$ is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for $\{ {T_n}\} _{n = 1}^\infty $ are obtained. Some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given. This results are used to extend the Sz.-Nagy-Foiaş lifting theorem to this noncommutative setting.
A new algebraic approach to microlocalization of filtered rings
Maria Jesus
Asensio;
Michel
Van den Bergh;
Freddy
Van Oystaeyen
537-553
Abstract: Using the construction of the Rees ring associated to a filtered ring we provide a description of the microlocalization of the filtered ring by using only purely algebraic techniques. The method yields an easy approach towards the study of exactness properties of the microlocalization functor. Every microlocalization at a regular multiplicative Ore set in the associated graded ring can be obtained as the completion of a localization at an Ore set of the filtered ring.
A note on local change of diffeomorphism
Mikiya
Masuda
555-566
Abstract: Let $D(M)$ be the group of pseudo-isotopy classes of orientation preserving diffeomorphisms of a compact manifold $M$ which restrict to the identity on $\partial M$. If a compact manifold $N$ of the same dimension as $M$ is embedded in $M$, then extending maps in $ D(N)$ as the identity on the exterior of $N$ defines a homomorphism $E:D(N) \to D(M)$. We ask if the kernel of $ E$ is finite and show that this is the case for special cases.
Behavior of polynomials of best uniform approximation
E. B.
Saff;
V.
Totik
567-593
Abstract: We investigate the asymptotic behavior of the polynomials $\{ {P_n}(f)\} _0^\infty$ of best uniform approximation to a function $f$ that is continuous on a compact set $K$ of the complex plane ${\mathbf{C}}$ and analytic in the interior of $ K$, where $K$ has connected complement. For example, we show that for "most" functions $ f$, the error $f - {P_n}(f)$ does not decrease faster at interior points of $K$ than on $K$ itself. We also describe the possible limit functions for the normalized error $(f - {P_n}(f))/{E_n}$, where ${E_n}: = \vert\vert f - {P_n}(f)\vert{\vert _K}$, and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to $ f$ on $K$ exist that converge more rapidly at the interior points of $K$.
On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains
Catherine
Bandle;
Howard A.
Levine
595-622
Abstract: In this paper we study the first initial-boundary value problem for $ {u_t} = \Delta u + {u^p}$ in conical domains $D = (0,\infty ) \times \Omega \subset {R^N}$ where $ \Omega \subset {S^{N - 1}}$ is an open connected manifold with boundary. We obtain some extensions of some old results of Fujita, who considered the case $D = {R^N}$. Let $\lambda = - {\gamma _ - }$ where ${\gamma _ - }$ is the negative root of $ \gamma (\gamma + N - 2) = {\omega _1}$ and where $ {\omega _1}$ is the smallest Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We prove: If $1 < p < 1 + 2/(2 + \lambda )$, there are no nontrivial global solutions. If $1 < p < 1 + 2/\lambda$, there are no stationary solutions in $ D - \{ 0\}$ except $u \equiv 0$. If $1 + 2/\lambda < p < (N + 1)/(N - 3)$ (if $ N > 3$, arbitrary otherwise) there are singular stationary solutions $ {u_s}$. If $u(x,0) \leqslant {u_s}(x)$, the solutions are global. If $1 + 2/\lambda < p < (N + 2)/(N - 2)$ and $u(x,0) \leqslant {u_s}$, with $ u(x,0) \in C(\overline D )$, the solutions decay to zero. If $1 + 2/N < p$, there are global solutions. For $1 < p < \infty$, there are ${L^\infty }$ data of arbitrarily small norm, decaying exponentially fast at $ r = \infty$, for which the solution is not global. We show that if $ D$ is the exterior of a bounded region, there are no global, nontrivial, positive solutions if $ 1 < p < 1 + 2/N$ and that there are such if $ p > 1 + 2/N$. We obtain some related results for ${u_t} = \Delta u + \vert x{\vert^\sigma }{u^p}$ in the cone.
Regular transition functions and regular superprocesses
E. B.
Dynkin
623-634
Abstract: The class of regular Markov processes is very close to the class of right processes studied by Meyer, Getoor and others. We say that a transition function $p$ is regular if it is the transition function of a well-defined regular Markov process. A characterization of regular transition functions is given which implies that, if $p$ is regular, then the Dawson-Watanabe and the Fleming-Viot supertransition functions over $ p$ belong to the same class.
Complex approximation of real functions by reciprocals of polynomials
Daniel
Wulbert
635-652
Abstract: Characterizations are given for local and global best rational approximations to a real function. The characterizations are specialized to reciprocals of polynomials, where they are used to settle some conjectures and questions.
Interpolation between Sobolev and between Lipschitz spaces of analytic functions on starshaped domains
Emil J.
Straube
653-671
Abstract: We show that on a starshaped domain $\Omega$ in $ {\operatorname{C} ^n}$ (actually on a somewhat larger, biholomorphically invariant class) the $ {\mathcal{L}^p}$-Sobolev spaces of analytic functions form an interpolation scale for both the real and complex methods, for each $p,\;0 < p \leqslant \infty$. The case $p = \infty$ gives the Lipschitz scale; here the functor $ {(,)^{[\theta ]}}$ has to be considered (rather than ${(,)_{[\theta ]}}$).
The complex bordism of groups with periodic cohomology
Anthony
Bahri;
Martin
Bendersky;
Donald M.
Davis;
Peter B.
Gilkey
673-687
Abstract: Is is proved that if $BG$ is the classifying space of a group $ G$ with periodic cohomology, then the complex bordism groups $M{U_{\ast}}(BG)$ are obtained from the connective $ K$-theory groups $k{u_{\ast}}(BG)$ by just tensoring up with the generators of $ M{U_{\ast}}$ as a polynomial algebra over $ k{u_{\ast}}$. The explicit abelian group structure is also given. The bulk of the work is the verification when $G$ is a generalized quaternionic group.
Processes disjoint from weak mixing
S.
Glasner;
B.
Weiss
689-703
Abstract: We show that the family $ {\mathcal{W}^ \bot }$ of ergodic measure preserving transformations which are disjoint from every weakly mixing m.p.t. properly contains the family $ \mathcal{D}$ of distal ergodic m.p.t. In the topological case we show that $ \mathcal{P}\mathcal{I}$, the family of proximally isometric flows is properly contained in the family $\mathcal{M}({\mathcal{W}^ \bot })$ of multipliers for ${\mathcal{W}^ \bot }$.