Geometric transfer and the homotopy type of the automorphism groups of a manifold
D.
Burghelea;
R.
Lashof
1-38
Abstract: Lifting concordances (pseudo-isotopies) in a smooth fibre bundle gives a transfer of stable concordance groups. Properties of the transfer are proved and exploited to obtain the homotopy structure of the group of diffeomorphisms or homeomorphisms of a manifold in a stable range.
Lipschitz spaces on stratified groups
Steven G.
Krantz
39-66
Abstract: Let $G$ be a connected, simply connected nilpotent Lie group. Call $G$ stratified if its Lie algebra $\mathfrak{g}$ has a direct sum decomposition $\mathfrak{g} = {V_1} \oplus \cdots \oplus {V_m}$ with $[{V_i},{V_j}] = {V_{i + j}}$ for $i + j \leqslant m$, $[{V_{i,}}{V_j}] = 0$ for $i + j > m$. Let $\{ {X_1}, \ldots ,{X_n}\} $ be a vector space basis for ${V_1}$. Let $f \in C(G)$ satisfy $\vert\vert f(g\exp {X_i} \cdot )\vert\vert \in {\Lambda _\alpha }({\mathbf{R}})$, uniformly in $g \in G$, where ${\Lambda _\alpha }$ is the usual Lipschitz space and $0 < \alpha < \infty$. It is proved that, under these circumstances, it holds that $ f \in {\Gamma _\alpha }(G)$ where $ {\Gamma _\alpha }$ is the nonisotropic Lipschitz space of Folland. Applications of this result to interpolation theory, hypoelliptic partial differential equations, and function theory are provided.
Almost sure invariance principles for sums of $B$-valued random variables with applications to random Fourier series and the empirical characteristic process
Michael B.
Marcus;
Walter
Philipp
67-90
Abstract: We establish an almost sure approximation of the partial sums of independent, identically distributed random variables with values in a separable Banach space $B$ by a suitable $B$-valued Brownian motion under the hypothesis that the partial sums can be ${L^1}$-closely approximated by finite-dimensional random variables. We show that this hypothesis is satisfied if the given random variables are random Fourier series or related stochastic processes. As an application we obtain an almost sure approximation of the empirical characteristic process by a suitable $ {\mathbf{C}}(K)$-valued Brownian motion whenever the empirical characteristic process satisfies the central limit theorem.
Weighted Sobolev spaces and pseudodifferential operators with smooth symbols
Nicholas
Miller
91-109
Abstract: Let ${u^\char93 }$ be the Fefferman-Stein sharp function of $u$, and for $1 < r < \infty$, let $ {M_r}u$ be an appropriate version of the Hardy-Littlewood maximal function of $u$. If $A$ is a (not necessarily homogeneous) pseudodifferential operator of order 0, then there is a constant $ c > 0$ such that the pointwise estimate ${(Au)^\char93 }(x) \leqslant c{M_r}u(x)$ holds for all $x \in {R^n}$ and all Schwartz functions $ u$. This estimate implies the boundedness of 0-order pseudodifferential operators on weighted ${L^p}$ spaces whenever the weight function belongs to Muckenhoupt's class ${A_p}$. Having established this, we construct weighted Sobolev spaces of fractional order in $ {R^n}$ and on a compact manifold, prove a version of Sobolev's theorem, and exhibit coercive weighted estimates for elliptic pseudodifferential operators.
Canonical semi-invariants and the Plancherel formula for parabolic groups
Ronald L.
Lipsman;
Joseph A.
Wolf
111-131
Abstract: A parabolic subgroup of a reductive Lie group is called "good" if the center of the universal enveloping algebra of its nilradical contains an element that is semi-invariant of weight proportional to the modular function. The "good" case is characterized here by invariance of the set of simple roots defining the parabolic, under the negative of the opposition element of the Weyl group. In the "good" case, the unbounded Dixmier-Pukanszky operator of the parabolic subgroup is described, the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified, and an explicit Plancherel formula is derived for that parabolic.
Dirac quantum fields on a manifold
J.
Dimock
133-147
Abstract: On globally hyperbolic Lorentzian manifolds we construct field operators which satisfy the Dirac equation and have a causal anticommutator. Ambiguities in the construction are removed by formulating the theory in terms of ${C^{\ast}}$ algebras of local observables. A generalized form of the Haag-Kastler axioms is verified.
On the monodromy at isolated singularities of weighted homogeneous polynomials
Benjamin G.
Cooper
149-166
Abstract: Assume $ f:{{\mathbf{C}}^m} \to {\mathbf{C}}$ is a weighted homogeneous polynomial with isolated singularity, and define $\phi :{S^{2m - 1}} - {f^{ - 1}}(0) \to {S^1}$ by $\phi (\overrightarrow z ) = f(\overrightarrow z ) / \vert f(\overrightarrow z )\vert$. If the monomials of $f$ are algebraically independent, then the closure ${\overline F _0}$ of ${\phi ^{ - 1}}(1)$ in $ {S^{2m - 1}}$ admits a deformation into the subset $G$ where each monomial of $f$ has nonnegative real values. For the polynomial $f({z_1}, \ldots ,{z_m}) = z_1^{{a_1}}{z_2} + \cdots + z_{m - 1}^{{a_{m - 1}}}{z_m} + z_m^{{a_m}}{z_1}$, $ G$ is a cell complex of dimension $m - 1$, invariant under a characteristic map $ h$ of the fibration $ \phi$, and the inclusion $G \to {F_0}$ induces isomorphisms in homology. To compute the homology of the link $K = {f^{ - 1}}(0) \cap {S^{2m - 1}}$ it thus suffices to calculate the action of ${h_{\ast}}$ on $ {H_{m - 1}}(G)$. Let $d = {a_1}{a_2} \cdots {a_m} + {( - 1)^{m - 1}}$. Let ${w_1},\,{w_2}, \ldots ,{w_m}$ be the weights associated with $f$, satisfying ${a_j} / {w_j} + 1 / {w_{j + 1}} = 1$ for $j = 1,\,2, \ldots ,\,m - 1$ and ${a_m}/{w_m} + 1/{w_1} = 1$. Let $n = d/{w_1}$, $ q = \gcd (n,\,d)$, $r = q + {( - 1)^m}$. Then ${H_{m - 2}}(K) = {Z^r} \oplus {z_{d/q}}$ and $ {H_{m - 1}}(K) = {Z^r}$.
Full continuous embeddings of toposes
M.
Makkai
167-196
Abstract: Some years ago, G. Reyes and the author described a theory relating first order logic and (Grothendieck) toposes. This theory, together with standard results and methods of model theory, is applied in the present paper to give positive and negative results concerning the existence of certain kinds of embeddings of toposes. A new class, that of prime-generated toposes is introduced; this class includes M. Barr's regular epimorphism sheaf toposes as well as the so-called atomic toposes introduced by M. Barr and R. Diaconescu. The main result of the paper says that every coherent prime-generated topos can be fully and continuously embedded in a functor category. This result generalizes M. Barr's full exact embedding theorem. The proof, even when specialized to Barr's context, is essentially different from Barr's original proof. A simplified and sharpened form of Barr's proof of his theorem is also described. An example due to J. Malitz is adapted to show that a connected atomic topos may have no points at all; this shows that some coherence assumption in our main result is essential.
Coextensions of regular semigroups by rectangular bands. I
John
Meakin;
K. S. S.
Nambooripad
197-224
Abstract: This paper initiates a general study of the structure of a regular semigroup $S$ via the maximum congruence $\rho$ on $S$ with the property that each $\rho$-class $e\rho$, for $e = {e^2} \in S$, is a rectangular subband of $S$. Congruences of this type are studied and the maximum such congruence is characterized. A construction of all biordered sets which are coextensions of an arbitrary biordered set by rectangular biordered sets is provided and this is specialized to provide a construction of all solid biordered sets. These results are used to construct all regular idempotent-generated semigroups which are coextensions of a regular idempotent-generated semigroup by rectangular bands: a construction of normal coextensions of biordered sets is also provided.
Ideal theory in $f$-algebras
C. B.
Huijsmans;
B.
de Pagter
225-245
Abstract: The paper deals mainly with the theory of algebra ideals and order ideals in $f$-algebras. Necessary and sufficient conditions are established for an algebra ideal to be prime, semiprime or idempotent. In a uniformly complete $ f$-algebra with unit element every algebra ideal is an order ideal iff the $ f$-algebra is normal. This result is based on the fact that the range of every orthomorphism in a uniformly complete normal Riesz space is an order ideal.
Attractors: persistence, and density of their basins
Mike
Hurley
247-271
Abstract: An investigation of qualitative features of flows on manifolds, in terms of their attractors and quasi-attractors. A quasi-attractor is any nonempty intersection of attractors. It is shown that quasi-attractors other than attractors occur for a large set of flows. It is also shown that for a generic flow (for each flow in a residual subset of the set of all flows), each attractor "persists" as an attractor of all nearby flows. Similar statements are shown to hold with "quasi-attractor", "chain transitive attractor", and "chain transitive quasi-attractor" in place of "attractor". Finally, the set of flows under which almost all points tend asymptotically to a chain transitive quasi-attractor is characterized in terms of stable sets of invariant sets.
A two-cardinal theorem for homogeneous sets and the elimination of Malitz quantifiers
Philipp
Rothmaler;
Peter
Tuschik
273-283
Abstract: Sufficient conditions for the eliminability of Malitz quantifiers in a complete first order theory are given. Proving that certain superstable and not $\omega$-stable theories satisfy these conditions, a question of Baldwin and Kueker is answered negatively.
An improvement of the Poincar\'e-Birkhoff fixed point theorem
Patricia H.
Carter
285-299
Abstract: If $g$ is a twist homeomorphism of an annulus $ A$ in the plane which leaves at most one point in the interior of $A$ fixed, then there is an essential simple closed curve in the interior of $A$ which meets its image in at most one point; hence the annular region bounded by this simple closed curve and the inside component of the boundary of $ A$ is mapped onto either a proper subset or a proper superset of itself.
Curvature operators and characteristic classes
Irl
Bivens
301-310
Abstract: Given tensors $ A$ and $B$ of type $(k,\,k)$ on a Riemannian manifold $M$ we construct in a natural way a $ 2k$ form ${F_k}(A,\,B)$. If $A$ and $B$ satisfy the generalized Codazzi equations then this $2k$ form is closed. In particular if ${R_{2k}}$ denotes the $2k$th curvature operator then $ {F_{2k}}({R_{2k,\,}}{R_{2k}})$ is (up to a constant multiple) the $k$th Pontrjagin class of $ M$. By means of a theorem of Gilkey we give conditions sufficient to guarantee that a form constructed from more complicated expressions involving the curvature operators does in fact belong to the Pontrjagin algebra. As a corollary we obtain Thorpe's vanishing theorem for manifolds with constant $ 2p$th sectional curvature. If at each point in $M$ the tangent space contains a subspace of a particular type (similar to curvature nullity) we show that certain Pontrjagin classes must vanish. We generalize the result that submanifolds of Euclidean space with flat normal bundle have a trivial Pontrjagin algebra. The curvature operator, ${R_2}$, is interesting in that the components of ${R_2}$ with respect to any orthonormal frame are given by certain universal (independent of frame) homogeneous linear polynomials in the components of the curvature tensor. We characterize all such operators and using this characterization derive in a natural way the Weyl component of ${R_2}$.
Finite sublattices of a free lattice
J. B.
Nation
311-337
Abstract: Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattice of a free lattice.
On the critical degree of differentiability of a complex planar curve
Joseph
Becker
339-350
Abstract: An example of a pair of complex analytic curves in ${{\mathbf{C}}^2}$ is given which have the same characteristic pairs but which do not have the same critical degree of differentiability.