Embedding processes in Brownian motion in ${\bf R}\sp{n}$
Neil
Falkner
335-363
Abstract: We give a potential-theoretic characterization of the right-continuous processes which can be embedded in Brownian motion in ${{\mathbf{R}}^n}$ by means of an increasing family of standard stopping times. In general it is necessary to use a Brownian motion process whose filtration is richer than the natural one.
Curvature tensors on almost Hermitian manifolds
Franco
Tricerri;
Lieven
Vanhecke
365-398
Abstract: A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.
Counting divisors with prescribed singularities
Israel
Vainsencher
399-422
Abstract: Given a family of divisors $\{ {D_s}\}$ in a family of smooth varieties $\{ {Y_s}\}$ and a sequence of integers ${m_1}, \ldots ,{m_t}$, we study the scheme parametrizing the points $(s,{y_1}, \ldots ,{y_t})$ such that ${y_i}$ is a (possibly infinitely near) $ {m_i}$-fold point of $ {D_s}$. We obtain a general formula which yields, as special cases, the formula of de Jonquières and other classical results of Enumerative Geometry. We also study the questions of finiteness and the multiplicities of the solutions.
Self-maps of flag manifolds
Henry H.
Glover;
William D.
Homer
423-434
Abstract: Rationally, a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology. Two algebraic results for cohomology endomorphisms then serve (a) to determine those flag manifolds which have (nontrivial) self-maps that factor through a complex projective space, and (b) for a special class of flag manifolds, to classify the self-maps of their rationalizations up to homotopy.
Singular integrals and maximal functions associated with highly monotone curves
W. C.
Nestlerode
435-444
Abstract: Let $ \gamma :[ - 1,1] \to {{\mathbf{R}}^n}$ be an odd curve. Set $\displaystyle {H_\gamma }f(x) = {\text{PV}}\int {f(x - \gamma (t))\,(dt/t)}$ and $\displaystyle {M_\gamma }f(x) = \sup {h^{ - 1}}\int_0^h {\vert f(x - \gamma (t))\vert\,dt}$ . We introduce a class of highly monotone curves in $ {{\mathbf{R}}^n}$, $n \geqslant 2$, for which we prove that ${H_\gamma }$ and $ {M_\gamma }$ are bounded operators on ${L^2}({{\mathbf{R}}^n})$. These results are known if $ \gamma$ has nonzero curvature at the origin, but there are highly monotone curves which have no curvature at the origin. Related to this problem, we prove a generalization of van der Corput's estimate of trigonometric integrals.
Subspaces of $L\sp{1}$, via random measures
David J.
Aldous
445-463
Abstract: It is shown that every subspace of ${L^1}$ contains a subspace isomorphic to some $ {l_q}$. The proof depends on a fixed point theorem for random measures.
A relative Nash theorem
Selman
Akbulut;
Henry C.
King
465-481
Abstract: We prove that if $ M$ is a closed smooth manifold and ${M_i}$, $i = 1, \ldots ,k$, are transversally intersecting closed smooth submanifolds of $ M$, then there exist a nonsingular algebraic set $Z$ and nonsingular algebraic subsets $ {Z_i}$, $i = 1, \ldots ,k$, of $Z$ such that $(M;{M_1}, \ldots ,{M_k})$ is diffeomorphic to $ (Z;{Z_1}, \ldots ,{Z_k})$. We discuss a generalization and the consequences of this result.
Spherical means and geodesic chains on a Riemannian manifold
Toshikazu
Sunada
483-501
Abstract: Some spectral properties of spherical mean operators defined on a Riemannian manifold are given. As an application we deduce a statistic property of geodesic chains which is interesting from the view point of geometric probability.
The structure of tensor products of semilattices with zero
G.
Grätzer;
H.
Lakser;
R.
Quackenbush
503-515
Abstract: If $A$ and $B$ are finite lattices, then the tensor product $ C$ of $A$ and $B$ in the category of join semilattices with zero is a lattice again. The main result of this paper is the description of the congruence lattice of $ C$ as the free product (in the category of bounded distributive lattices) of the congruence lattice of $A$ and the congruence lattice of $B$. This provides us with a method of constructing finite subdirectly irreducible (resp., simple) lattices: if $A$ and $B$ are finite subdirectly irreducible (resp., simple) lattices then so is their tensor product. Another application is a result of E. T. Schmidt describing the congruence lattice of a bounded distributive extension of ${M_3}$.
Smooth perturbations of a function with a smooth local time
D.
Geman;
J.
Horowitz
517-530
Abstract: A real Borel function on $[0,\,1]$ has a local time if its occupation measure up to each time $t$ (equivalently: its increasing, equimeasurable rearrangement on $[0,\,t]$) is absolutely continuous; the local time ${\alpha _t}(x)$ is then the density. An inverse relationship exists between the smoothness of the local time in $(t,\,x)$ and that of the original function. The sum of a function with a smooth local time and a well-behaved (e.g. absolutely continuous) function is shown to have a local time, which inherits certain significant properties from the old local time, and for which an explicit formula is given. Finally, using a probabilistic approach, examples are given of functions having local times of prescribed smoothness.
Hypersingular integrals and parabolic potentials
Sagun
Chanillo
531-547
Abstract: In this paper we characterize the potential spaces associated with the heat equation in terms of singular integrals of mixed homogeneity.
Partitions of products
David
Pincus;
J. D.
Halpern
549-568
Abstract: This paper extends some applications of a theorem of Halpern and Lauchli on partitions of products of finitary trees. The extensions are to weak infinite products of dense linear orderings, and ultrafilter preservation for finite product Sacks forcing.
Morse theory by perturbation methods with applications to harmonic maps
K.
Uhlenbeck
569-583
Abstract: There are many interesting variational problems for which the Palais-Smale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and the critical points converge, a Morse theory is valid. This theory applies to a class of variational problems consisting of the energy integral for harmonic maps with a lower order potential.
Twist maps, coverings and Brouwer's translation theorem
H. E.
Winkelnkemper
585-593
Abstract: We apply the Brouwer Translation Theorem to a class of twist maps of the annulus (which contains ${C^1}$ area preserving maps) to show that, if $ h$ belongs to this class, then a certain set $ {\mathcal{P}_0}$ of periodic points of $h$ cannot be dense. The definition of ${\mathcal{P}_0}$ does not impose any a priori restrictions on the periods of the points of ${\mathcal{P}_0}$.
Some universal sets of terms
Walter
Taylor
595-607
Abstract: For every $ \Pi _2^1$ class of cardinals containing 0 and $1$, there exists a finite set $T$ of terms, such that $X$ is precisely the class of cardinals in which $T$ is universal.
Compact groups of homeomorphisms on tree-like continua
J. B.
Fugate;
T. B.
McLean
609-620
Abstract: This paper is concerned with the fixed point sets of certain collections of homeomorphisms on a tree-like continuum. Extending a theorem of P. A. Smith, the authors prove that a periodic homeomorphism has a (nonvoid) continuum as its fixed point set. They then deduce possible periods for homeomorphisms on tree-like continua which satisfy certain decomposability or irreducibility conditions. The main result of the paper is that a compact group of homeomorphisms has a continuum as its fixed point set. This is applied to isometries. The paper concludes with sufficient conditions that a pointwise periodic homeomorphism have a fixed point.
KV-theory of categories
Charles A.
Weibel
621-635
Abstract: Quillen has constructed a $K$-theory $ {K_{\ast}}C$ for nice categories, one of which is the category of projective $ R$-modules. We construct a theory $ K{V_{\ast}}C$ for the nice categories parametrized by rings. When applied to projective modules we recover the Karoubi-Villamayor $ K$-theory $K{V_{\ast}}(R)$. As an application, we show that the Cartan map from $ {K_{\ast}}(R)$ to ${G_{\ast}}(R)$ factors through the groups $K{V_{\ast}}(R)$. We also compute $K{V_{\ast}}$ for the categories of faithful projectives and Azumaya algebras, generalizing results of Bass.
On the topological structure of even-dimensional complete intersections
A. S.
Libgober;
J. W.
Wood
637-660
Abstract: A topological connected sum decomposition into indecomposable pieces is given for complete intersections, and these pieces are described by plumbing constructions. The principal technical results are structure theorems for the intersection form on the middle dimensional homology and the submodule of spherical classes.
Conditional expectations in $C\sp{\ast} $-crossed products
Shigeru
Itoh
661-667
Abstract: Let $(A,\,G,\,\alpha )$ be a $ {C^{\ast}}$-dynamical system. Let $B$ be a $ {C^{\ast}}$-subalgebra of $ A$ and $P$ be a conditional expectation of $ A$ onto $B$ such that ${\alpha _t}P = P{\alpha _t}$ for each $ t \in G$. Then it is proved that there exists a conditional expectation of ${C^{\ast}}(G,\,A,\,\alpha )$ onto ${C^{\ast}}(G,\,B,\,\alpha )$. In particular, if $ G$ is amenable and $ A$ is unital, then there always exists a conditional expectation of $ {C^{\ast}}(G,\,A,\,\alpha )$ onto $ {C^{\ast}}(G)$. Some related results are also obtained.