Statically tame periodic homeomorphisms of compact connected $3$-manifolds. I. Homeomorphisms conjugate to rotations of the $3$-sphere
Edwin E.
Moise
1-47
Abstract: Let f be a homeomorphism of the 3-sphere onto itself, of finite period n, and preserving orientation. Suppose that the fixed-point set F of f is a tame 1-sphere. It is shown that (1) the 3-sphere has a triangulation $ K({{\textbf{S}}^3})$ such that F forms a subcomplex of $K({{\textbf{S}}^3})$ and f is simplicial relative to $ K({{\textbf{S}}^3})$. Suppose also that F is unknotted. It then follows that (2) f is conjugate to a rotation.
Maximal inequalities related to generalized a.e. continuity
W. B.
Jurkat;
J. L.
Troutman
49-64
Abstract: An integral inequality of the classical Hardy-Littlewood type is obtained for the maximal function of positive convolution operators associated with approximations of the identity in ${R^n}$. It is shown that the (formally) rearranged maximal function can in general be estimated by an elementary integral involving the decreasing rearrangements of the kernel of the approximation and the function being approximated. (The estimate always holds when the kernel has compact support or a decreasing radial majorant integrable in a neighborhood of infinity; a one-dimensional counterexample shows that integrability alone may not suffice.) The finiteness of the integral determines a Lorentz space of functions which are a.e. continuous in the generalized sense of the approximation. Conversely, in dimension one it is established that this space is the largest strongly rearrangement invariant Banach space of such functions. In particular, the new inequality provides access to the study of Cesàro continuity of order less than one.
The free boundary for elastic-plastic torsion problems
Luis A.
Caffarelli;
Avner
Friedman
65-97
Abstract: Consider the variational inequality: find $ u\, \in \,K$ such that $ \int_Q {\nabla u\, \cdot \,\nabla \left( {v\, - \,u} \right)} \, \geqslant \,\mu \,\int_Q {\left( {v\, - \,u} \right)} \,\left( {\mu \, > \,0} \right)$ for any $ v\, \in \,K$, where $K\, = \,\left\{ {w\, \in \,H_0^1\left( Q \right),\,\left\vert {\nabla w} \right\vert\, \leqslant \,1\,} \right\}$ and Q is a simply connected domain whose boundary is piecewise ${C^3}$. The solution u represents the stress function in a torsion problem of an elastic bar with cross section Q; the sets $E\, = \,\left\{ {x\, \in \,Q;\,\left\vert {\nabla u\left( x \right)} \right\ve... ...x\, \in \,Q;\,\left\vert {\nabla u\left( x \right)} \right\vert = \,1} \right\}$ are the elastic and plastic subsets of Q. The ridge R of Q is, by definition, the set of points in Q where dist $\left( {x,\,\partial Q} \right)$ is not ${C^{1,1}}$. The paper studies the location and shape of E, P and the free boundary $ \Gamma \, = \,\partial E\, \cap \,Q$. It is proved that the ridge is elastic and that E is contained in a $\left( {c/\mu } \right)$-neighborhood of R, as $\mu \, \to \,\infty \,\left( {c\, > \,0} \right)$. The behavior of E and P near the vertices of $ \partial Q$ is studied in detail, as well as the nature of $\Gamma$ away from the vertices. Applications are given to special domains. The case where Q is multiply connected is also studied; in this case the definition of K is somewhat different. Some results on the ``upper plasticity'' and ``lower plasticity'' and on the behavior as $\mu \, \to \,\infty$ are obtained.
Continuity of the density of a gas flow in a porous medium
Luis A.
Caffarelli;
Avner
Friedman
99-113
Abstract: The equation of gas in a porous medium is a degenerate nonlinear parabolic equation. It is known that a unique generalized solution exists. In this paper it is proved that the generalized solution is continuous.
Sur les germes d'applications differentiables \`a singularit\'es isol\'ees
Jacek
Bochnak;
Wojciech
Kucharz
115-131
Abstract: Le but de cet article est d'étudier les germes d'applications différentiables $({{\textbf{R}}^n},0)\, \to \,({{\textbf{R}}^p},0)$, ou plus généralement les familles de telles applications, ayant en 0 une singularité isolée. Nous formulerons certains critères de ${C^0}$-suffisance de jets et nous démontrerons quelques théorèmes sur le nombre de types topologiques de germes qui apparaissent dans des familles de germes à singularité isolée.
Single-valued representation of set-valued mappings
A. D.
Ioffe
133-145
Abstract: It is shown that the graph of a set-valued mapping satisfying typical conditions which guarantee the existence of measurable selections can be represented as the union of graphs of measurable single-valued mappings depending continuously on a parameter running through some Polish space.
On the topology of the set of completely unstable flows
Zbigniew
Nitecki
147-162
Abstract: We show that: (1) on any open manifold other than the line or plane, there exist nonsingular flows with $\Omega \, \ne \,\emptyset $ which can be perturbed, in the strong ${C^r}$ topology (any r), to flows with $ \Omega \, \ne \,\emptyset$, and that (2) on certain open 3-manifolds there exist flows with $\Omega \, \ne \,\emptyset $ which cannot be approximated, in the strong $ {{\mathcal{C}}^1}$ topology, by flows satisfying both $\Omega \, \ne \,\emptyset $ and no ${{\mathcal{C}}^1}$ $\Omega$-explosions. These examples give partial negative answers to the conjecture of Takens and White, that the completely unstable flows with the strong $ {{\mathcal{C}}^r}$ topology equal the closure of their interior.
Complex-foliated structures. I. Cohomology of the Dolbeault-Kostant complexes
Hans R.
Fischer;
Floyd L.
Williams
163-195
Abstract: We study the cohomology of differential complexes, which we shall call Dolbeault-Kostant complexes, defined by certain integrable sub-bundles F of the complex tangent bundle of a manifold M. When M has a complex or symplectic structure and F is chosen to be the bundle of anti-holomorphic tangent vectors or, respectively, a ``polarization'' then the corresponding complexes are, respectively, the Dolbeault complex and (under further conditions) a complex introduced by Kostant in the context of geometric quantization. A simple condition on F insures that our complexes are elliptic. Assuming ellipticity and compactness of M, for example, one of our key results is a Hirzebruch-Riemann-Roch Theorem.
Zeros of Stieltjes and Van Vleck polynomials
Mahfooz
Alam
197-204
Abstract: The study of the polynomial solutions of the generalized Lamé differential equation gives rise to Stieltjes and Van Vleck polynomials. Marden has, under quite general conditions, established varied generalizations of the results proved earlier by Stieltjes, Van Vleck, Bocher, Klein, and, Pólya, concerning the location of the zeros of such polynomials. We study the corresponding problem for yet another form of the generalized Lamé differential equation and generalize some recent results due to Zaheer and to Alam. Furthermore, applications of our results to the standard form of this differential equation immediately furnish the corresponding theorems of Marden. Consequently, our main theorem of this paper may be considered as the most general result obtained thus far in this direction.
Structural stability and hyperbolic attractors
Artur Oscar
Lopes
205-219
Abstract: A necessary condition for structural stability is presented that in the two dimensional case means that the system has a finite number of topological attractors.
Wiman-Valiron theory for entire functions of finite lower growth
P. C.
Fenton
221-232
Abstract: A general method of Wiman-Valiron type for dealing with entire functions of finite lower growth is presented and used to obtain the lower-order version of a result of W. K. Hayman on the real part of entire functions of small lower growth.
Globally hypoelliptic and globally solvable first-order evolution equations
Jorge
Hounie
233-248
Abstract: We consider global hypoellipticity and global solvability of abstract first order evolution equations defined either on an interval or in the unit circle, and prove that it is equivalent to certain conditions bearing on the total symbol. We relate this to known results about hypoelliptic vector fields on the 2-torus.
Theorems of Fubini type for iterated stochastic integrals
Marc A.
Berger;
Victor J.
Mizel
249-274
Abstract: An extension of the Itô calculus which treats iterated Itô integration, as applied to a class of two-parameter processes, is introduced. This theory includes the integration of certain anticipative integrands and introduces a notion of stochastic differential for such integrands. Among the key results is a version of Fubini's theorem for iterated stochastic integrals, in which a ``correction'' term appears. Applications to stochastic integral equations and to the Itô calculus are given, and the relation of the present development to recent work of Ogawa is described.
Singular perturbations and nonstandard analysis
S.
Albeverio;
J. E.
Fenstad;
R.
Høegh-Krohn
275-295
Abstract: We study by methods of nonstandard analysis second order differential operators with zero order coefficients which are too singular to be defined by standard functions. In particular we study perturbations of the Laplacian in $ {R^3}$ given by potentials of the form $\lambda {\Sigma _j}\delta \left( {x\, - \,{x_j}} \right)$. We also study Sturm-Liouville problems with zero order coefficients given by measures and prove that they satisfy the same oscillation theorems as the regular Sturm-Liouville problems.
Toeplitz operators and related function algebras on certain pseudoconvex domains
Nicholas P.
Jewell;
Steven G.
Krantz
297-312
Abstract: Toeplitz operators are defined on pseudoconvex domains in ${{\textbf{C}}^n}$ and their spectral properties are studied. In addition, the linear space ${H^\infty }\, + \,C$ is discussed and is seen to be a closed algebra on a variety of domains.
Expansive homeomorphisms and topological dimension
Ricardo
Mañé
313-319
Abstract: Let K be a compact metric space. A homeomorphism $f:\,K\mid$ is expansive if there exists $\varepsilon \, > \,0$ such that if $x, y\, \in \,K$ satisfy $d\left( {{f^n}\left( x \right),\,{f^n}\left( y \right)} \right)\, < \,\varepsilon$ for all $ n\, \in \,{\textbf{Z}}$ (where $d\left( { \cdot ,\, \cdot } \right)$ denotes the metric on K) then $x\, = \,y$. We prove that a compact metric space that admits an expansive homeomorphism is finite dimensional and that every minimal set of an expansive homeomorphism is 0-dimensional.
Inductive construction of homogeneous cones
Josef
Dorfmeister
321-349
Abstract: A method is explained how to construct all homogeneous cones in a unique way out of lower dimensional ones. The infinitesimal automorphisms of such a cone and its associated left-symmetric algebras are described in terms of the lower dimensional constituents of the cone. It is characterized when a homogeneous cone is self-dual or a sum of homogeneous cones.
Expanding maps on sets which are almost invariant. Decay and chaos
Giulio
Pianigiani;
James A.
Yorke
351-366
Abstract: Let $A\, \subset \,{R^n}$ be a bounded open set with finitely many connected components ${A_j}$ and let $T:\,\overline A \to \,{R^n}$ be a smooth map with $ A\,\, \subset \,\,T\left( A \right)$. Assume that for each ${A_j}$, $A\,\, \subset \,\,{T^m}\left( {{A_j}} \right)$ for all m sufficiently large. We assume that T is ``expansive", but we do not assume that $ T\left( A \right) = A$. Hence for $x\, \in \,A,\,{T^i}\,\left( x \right)$ may escape A as i increases. Let ${\mu _0}$ be a smooth measure on A (with ${\operatorname{inf} _A}\,{{d{\mu _0}} \mathord{\left/ {\vphantom {{d{\mu _0}} {dx}}} \right. \kern-\nulldelimiterspace} {dx}}\, > \,0$) and let $x\, \in \,A$ be chosen at random (using $ {\mu _0}$). Since T is ``expansive'' we may expect ${T^i}\left( x \right)\,$ to oscillate chaotically on A for a certain time and eventually escape A. For each measurable set $E\, \subset \,A$ define ${\mu _m}\left( E \right)$ to be the conditional probability that ${T^m}\left( x \right) \in \,E$ given that $x,T\left( x \right),\ldots,{T^m}\left( x \right)$ are in A. We show that ${\mu _m}$ converges to a smooth measure $\mu$ which is independent of the choice of $ {\mu _0}$. One dimensional examples are stressed.
Codimension one isometric immersions between Lorentz spaces
L. K.
Graves
367-392
Abstract: The theorem of Hartman and Nirenberg classifies codimension one isometric immersions between Euclidean spaces as cylinders over plane curves. Corresponding results are given here for Lorentz spaces, which are Euclidean spaces with one negative-definite direction (also known as Minkowski spaces). The pivotal result involves the completeness of the relative nullity foliation of such an immersion. When this foliation carries a nondegenerate metric, results analogous to the Hartman-Nirenberg theorem obtain. Otherwise, a new description, based on particular surfaces in the three-dimensional Lorentz space, is required.
On an extremal property of Doob's class
J. S.
Hwang
393-398
Abstract: Recently, we have solved a long open problem of Doob (1935). To introduce the result proved here, we say that a function $ f(z)$ belongs to Doob's class D, if $f(z)$ is analytic in the unit disk U and has radial limit zero at an endpoint of some arc R on the unit circle such that $\operatorname{lim} \,{\operatorname{inf} _{n \to \infty }}\,\left\vert {f({P_n})} \right\vert$, where $\{ {P_n}\}$ is an arbitrary sequence of points in U tending to an arbitrary interior point of R. With this definition, our main result is the following extremal property of Doob's class. Theorem. $ {\operatorname{inf} _{f \in D}}\left\Vert f \right\Vert\, = \,{2 /e}$, where