Asymptotic homotopy cycles for flows and $\Pi \sb 1$ de Rham theory
Diego
Benardete;
John
Mitchell
495-535
Abstract: We define the asymptotic homotopy of trajectories of flows on closed manifolds. These homotopy cycles take values in the $2$-step nilpotent Lie group which is associated to the fundamental group by means of Malcev completion. The cycles are an asymptotic limit along the orbit of the product integral of a Lie algebra valued $ 1$-form. Propositions 5.1-5.7 show how the formal properties of our theory parallel the properties of the asymptotic homology cycles of Sol Schwartzman. In particular, asymptotic homotopy is an invariant of topological conjugacy, and, in certain cases, of topological equivalence. We compute the asymptotic homotopy of those measure-preserving flows on Heisenberg manifolds which lift from the torus ${T^2}$ (Theorem 8.1), and then show how this invariant distinguishes up to topological equivalence certain of these flows which are indistinguishable homologically (Theorem 9.1). We also compute the asymptotic homotopy of those geodesic flows for Heisenberg manifolds which come from left invariant metrics on the Heisenberg group (Example 8.1), and then show how this invariant distinguishes up to topological conjugacy certain of these flows which are indistinguishable homologically.
An atriodic simple-$4$-od-like continuum which is not simple-triod-like
Piotr
Minc
537-552
Abstract: The paper contains an example of a continuum $K$ such that $K$ is the inverse limit of simple $4$-ods, $K$ cannot be represented as the inverse limit of simple triods and each proper subcontinuum of $ K$ is an arc.
Critical LIL behavior of the trigonometric system
I.
Berkes
553-585
Abstract: It is a classical fact that for rapidly increasing $({n_k})$ the sequence $(\cos {n_k}x)$ behaves like a sequence of i.i.d. random variables. Actually, this almost i.i.d. behavior holds if $({n_k})$ grows faster than ${e^{c\sqrt k }}$; below this speed we have strong dependence. While there is a large literature dealing with the almost i.i.d. case, practically nothing is known on what happens at the critical speed ${n_k} \sim {e^{c\sqrt k }}$ (critical behavior) and what is the probabilistic nature of $(\cos {n_k}x)$ in the strongly dependent domain. In our paper we study the critical LIL behavior of $(\cos {n_k}x)$ i.e., we investigate how classical fluctuational theorems like the law of the iterated logarithm and the Kolmogorov-Feller test turn to nonclassical laws in the immediate neighborhood of $ {n_k} \sim {e^{c\sqrt k }}$.
Fragments of bounded arithmetic and bounded query classes
Jan
Krajíček
587-598
Abstract: We characterize functions and predicates $ \Sigma _{i + 1}^b$-definable in $S_2^i$. In particular, predicates $\Sigma _{i + 1}^b$-definable in $ S_2^i$ are precisely those in bounded query class ${P^{\Sigma _i^p}}[O(\log n)]$ (which equals to $ \operatorname{Log}\;{\text{Space}}^{\Sigma _i^p}$ by [B-H, W]). This implies that $S_2^i \ne T_2^i$ unless $ {P^{\Sigma _i^p}}[O(\log n)] = \Delta _{i + 1}^p$. Further we construct oracle $ A$ such that for all $ i \geq 1$: ${P^{\Sigma _i^p(A)}}[O(\log n)] \ne \Delta _{i + 1}^p(A)$. It follows that $S_2^i(\alpha ) \ne T_2^i(\alpha )$ for all $i \geq 1$. Techniques used come from proof theory and boolean complexity.
L'espace des plongements d'un arc dans une surface
Robert
Cauty
599-614
Abstract: We prove that the space of embeddings of an arc into a surface without boundary $M$ is homeomorphic to the product $U(M) \times {l^2}$, where $U(M)$ is the unit tangent bundle of $M$.
Parabolic systems: the ${\rm GF}(3)$-case
Thomas
Meixner
615-637
Abstract: Parabolic systems defined over $GF(q)$ have been classified by Timmesfeld for $ q \geq 4$ and by Stroth for $q = 2$ (see references). We deal with the case $ q = 3$.
Wavelets in wandering subspaces
T. N. T.
Goodman;
S. L.
Lee;
W. S.
Tang
639-654
Abstract: Mallat's construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite $B$-splines is constructed.
Optimal natural dualities
B. A.
Davey;
H. A.
Priestley
655-677
Abstract: The authors showed previously that for each of the varieties $ {{\mathbf{B}}_n}(3 \leq n < \omega )$ of pseudocomplemented distributive lattices there exists a natural duality given by a set of $p(n) + 3$ binary algebraic relations, where $ p(n)$ denotes the number of partitions of $n$. This paper improves this result by establishing that an optimal set of $n + 3$ of these relations suffices. This is achieved by the use of "test algebras": it is shown that redundancy among the relations of a duality for a prevariety generated by a finite algebra may be decided by testing the duality on the relations, qua algebras.
Martin and end compactifications for non-locally finite graphs
Donald I.
Cartwright;
Paolo M.
Soardi;
Wolfgang
Woess
679-693
Abstract: We consider a connected graph, having countably infinite vertex set $ X$, which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix $P$ corresponding to a nearest neighbor random walk on $X$, we study the associated harmonic functions on $X$ and, in particular, the Martin compactification. We also study the end compactification of the graph. When the graph is a tree, we show that these compactifications coincide; they are a disjoint union of $ X$, the set of ends, and the set of improper vertices--new points associated with vertices of infinite degree. Other results proved include a solution of the Dirichlet problem in the context of the end compactification of a general graph. Applications are given to, e.g., the Cayley graph of a free group on infinitely many generators.
Product formulas and convolutions for angular and radial spheroidal wave functions
William C.
Connett;
Clemens
Markett;
Alan L.
Schwartz
695-710
Abstract: Product formulas for angular spheroidal wave functions on $[0,\pi ]$ and for radial spheroidal wave functions on $ [0,\infty)$ are presented, which generalize results for the ultraspherical polynomials and functions as well as for the Mathieu functions. Although these functions cannot be given in closed form, the kernels of the product formulas are represented in an explicit, and surprisingly simple way in terms of Bessel functions so that the exact range of positivity can easily be read off. The formulas are used to introduce two families of convolution structures on $ [0,\pi ]$ and $[0,\infty)$, many of which provide new hypergroups. We proceed from the fact that the spheroidal wave functions are eigenfunctions of Sturm-Liouville equations of confluent Heun type and employ a partial differential equation technique based on Riemann's integration method.
Loop space homology of spaces of small category
Yves
Félix;
Jean-Claude
Thomas
711-721
Abstract: Only little is known concerning ${H_\ast}(\Omega X;{\mathbf{k}})$, the loop space homology of a finite $ {\text{CW}}$ complex $ X$ with coefficients in a field $ {\mathbf{k}}$. A space $ X$ is called an $ r$-cone if there exists a filtration $\ast = {X_0} \subset {X_1} \subset \cdots \subset {X_r} = X$, such that ${X_i}$ has the homotopy type of the cofibre of a map from a wedge of sphere into ${X_{i - 1}}$. Denote by ${A_X}$ the sub-Hopf algebra image of ${H_\ast}(\Omega {X_1})$. We prove then that for a graded $r$-cone, $r \leq 3$, there exists an isomorphism ${A_X} \otimes T(U)\xrightarrow{ \cong }{H_\ast}(\Omega X)$.
On Dubrovin valuation rings in crossed product algebras
Darrell
Haile;
Patrick
Morandi
723-751
Abstract: Let $F$ be a field and let $V$ be a valuation ring in $ F$. If $A$ is a central simple $ F$-algebra then $ V$ can be extended to a Dubrovin valuation ring in $A$. In this paper we consider the structure of Dubrovin valuation rings with center $V$ in crossed product algebras $ (K/F,G,f)$ where $ K/F$ is a finite Galois extension with Galois group $G$ unramified over $V$ and $f$ is a normalized two-cocycle. In the case where $V$ is indecomposed in $K$ we introduce a family of orders naturally associated to $f$, examine their basic properties, and determine which of these orders is Dubrovin. In the case where $ V$ is decomposed we determine the structure in the case of certain special discrete, finite rank valuations.
Hyponormal Toeplitz operators and extremal problems of Hardy spaces
Takahiko
Nakazi;
Katsutoshi
Takahashi
753-767
Abstract: The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank.
Composition operators between algebras of differentiable functions
Joaquín M.
Gutiérrez;
José G.
Llavona
769-782
Abstract: Let $E$, $F$ be real Banach spaces, $U \subseteq E$ and $V \subseteq F$ nonvoid open subsets and ${C^k}(U)$ the algebra of real-valued $k$-times continuously Fréchet differentiable functions on $U$, endowed with the compact open topology of order $k$. It is proved that, for $m \geq p$, the nonzero continuous algebra homomorphisms $A:{C^m}(U) \to {C^p}(V)$ are exactly those induced by the mappings $g:V \to U$ satisfying $\phi \circ g \in {C^p}(V)$ for each $\phi \in {E^\ast}$, in the sense that $A(f) = f \circ g$ for every $f \in {C^m}(U)$. Other homomorphisms are described too. It is proved that a mapping $g:V \to {E^{\ast \ast}}$ belongs to $ {C^k}(V,({E^{\ast \ast}},{w^\ast}))$ if and only if $\phi \circ g \in {C^k}(V)$ for each $\phi \in {E^\ast}$. It is also shown that if a mapping $g:V \to E$ verifies $\phi \circ g \in {C^k}(V)$ for each $\phi \in {E^\ast}$, then $g \in {C^{k - 1}}(V,E)$.
Induced connections on $S\sp 1$-bundles over Riemannian manifolds
G.
D’Ambra
783-798
Abstract: Let $(V,g)$ and $(W,h)$ be Riemannian manifolds and consider two $ {S^1}$-bundles $X \to V$ and $Y \to W$ with connections $\Gamma$ on $X$ and $\nabla$ on $Y$ respectively. We study maps $X \to Y$ which induce both connections and metrics. Our study relies on Nash's implicit function theorem for infinitesimally invertible differential operators. We show, for the case when $ Y \to W = {\mathbf{C}}{P^q}$ is the Hopf bundle, that if $2q \geq n(n + 1)/2 + 3n$ then there exists a nonempty open subset in the space of $ {C^\infty }$-pairs $(g,\Gamma)$ on $V$ which can be induced from $(h,\nabla)$ on $ {\mathbf{C}}{P^Q}$.
Algebraic particular integrals, integrability and the problem of the center
Dana
Schlomiuk
799-841
Abstract: In this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields. We first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center: these curves control the changes in the systems as parameters vary. The bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered, namely the real four-dimensional projective space. Next, we consider the known four algebraic conditions for the center for quadratic vector fields. One of them says that the system is Hamiltonian, a condition which has a clear geometric meaning. We determine the geometric meaning of the remaining other three algebraic conditions (I), (II), (III). We show that a quadratic system with a weak focus $F$, possessing algebraic particular integrals not passing through $F$ of the following types, satisfies in some coordinate axes the condition (I), (II) or (III) respectively and hence has a center at $F$: either a parabola and an irreducible cubic particular integral having only one point at infinity, coinciding with the one of the parabola; or a straight line and an irreducible conic curve; or distinct straight lines (possibly with complex coefficients). We show that each one of these geometric properties is generic for systems satisfying the corresponding algebraic condition for the center. Another version of this result in terms of real algebraic curves is given. These results make clear the many facets of the problem of the center in the quadratic case, in particular the question of integrability and form a basis for analogous investigations for the general problem of the center for cubic systems.
On manifolds with nonnegative curvature on totally isotropic 2-planes
Walter
Seaman
843-855
Abstract: We prove that a compact orientable $2n$-dimensional Riemannian manifold, with second Betti number nonzero, nonnegative curvature on totally isotropic $2$-planes, and satisfying a positivity-type condition at one point, is necessarily Kähler, with second Betti number $1$. Using the methods of Siu and Yau, we prove that if the positivity condition is satisfied at every point, then the manifold is biholomorphic to complex projective space.
A tom Dieck theorem for strong shape theory
Bernd
Günther
857-870
Abstract: We consider an appropriate class of locally finite closed coverings of spaces, for which the strong shape of the elements of the covering and of their intersections determine the strong shape of the whole space. Conclusions concerning shape dimension and spaces having the strong shape of ${\text{CW}}$-complexes are drawn, and a Leray spectral sequence for strong homology is given.
Ricci flow, Einstein metrics and space forms
Rugang
Ye
871-896
Abstract: The main results in this paper are: (1) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3) ${L^2}$-pinched Riemannian manifolds can be deformed to space forms through Ricci flow.
The geometry of Julia sets
Jan M.
Aarts;
Lex G.
Oversteegen
897-918
Abstract: The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on intervals. Julia sets play a critical role in the understanding of the dynamics of families of mappings. In this paper we introduce another class of objects (called hairy objects) which share many properties with the Cantor set and the interval: they are topologically unique and admit only one embedding in the plane. These uniqueness properties explain the regular occurrence of hairy objects in pictures of Julia sets--hairy objects are ubiquitous. Hairy arcs will be used to give a complete topological description of the Julia sets of many members of the exponential family.
Parametrizing smooth compactly supported wavelets
Raymond O.
Wells
919-931
Abstract: In this paper a concrete parameter space for the compactly supported wavelet systems of Daubechies is constructed. For wavelet systems with $N$ (generic) nonvanishing coefficients the parameter space is a closed convex set in ${{\mathbf{R}}^{(N - 2)/2}}$, which can be explicitly described in the Fourier transform domain. The moment-free wavelet systems are subsets obtained by the intersection of the parameter space and an affine subspace of ${{\mathbf{R}}^{(N - 2)/2}}$.
Fibered products of homogeneous continua
Karen
Villarreal
933-939
Abstract: In this paper, we construct homogeneous continua by using a fibered product of a homogeneous continuum $X$ with itself. The space $ X$ must have a continuous decomposition into continua, and it must possess a certain type of homogeneity property with respect to this decomposition. It is known that the points of any one-dimensional, homogeneous continuum can be "blown up" into pseudo-arcs to form a new continuum with a continuous decomposition into pseudo-arcs. We will show that these continua can be used in the above construction. Finally, we will show that the continuum constructed by using the pseudo-arcs, the circle of pseudo-arcs, or the solenoid of pseudo-arcs is not homeomorphic to any known homogeneous continuum.
Extending the $t$-design concept
A. R.
Calderbank;
P.
Delsarte
941-952
Abstract: Let $\mathfrak{B}$ be a family of $k$-subsets of a $v$-set $V$, with $ 1 \leq k \leq v/2$. Given only the inner distribution of $\mathfrak{B}$, i.e., the number of pairs of blocks that meet in $j$ points (with $j = 0,1, \ldots ,k$), we are able to completely describe the regularity with which $\mathfrak{B}$ meets an arbitrary $t$-subset of $V$, for each order $t$ (with $ 1 \leq t \leq v/2$). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters $ v$, $k$, $t$. The main regularity parameter is the dimension of a well-defined subspace of ${\mathbb{R}^{t + 1}}$, called the $t$-form space of $\mathfrak{B}$. (This subspace coincides with ${\mathbb{R}^{t + 1}}$ if and only if $\mathfrak{B}$ is a $t$-design.) We show that the $t$-form space has the structure of an ideal, and we explain how to compute its canonical generator.
An inverse boundary value problem for Schr\"odinger operators with vector potentials
Zi Qi
Sun
953-969
Abstract: We consider the Schrödinger operator for a magnetic potential $ \vec A$ and an electric potential $q$, which are supported in a bounded domain in ${\mathbb{R}^n}$ with $n \geq 3$. We prove that knowledge of the Dirichlet to Neumann map associated to the Schrödinger operator determines the magnetic field $\operatorname{rot}(\vec A)$ and the electric potential $q$ simultaneously, provided $\operatorname{rot}(\vec A)$ is small in the ${L^\infty }$ topology.
Characterization of eigenfunctions of the Laplacian by boundedness conditions
Robert S.
Strichartz
971-979
Abstract: If $ {\{ {f_k}(x)\} _{k \in \mathbb{Z}}}$ is a doubly infinite sequence of functions on $ {\mathbb{R}^n}$ which are uniformly bounded and such that $\Delta {f_k} = {f_{k + 1}}$, then $\Delta {f_0} = - {f_0}$. This generalizes a theorem of Roe $(n = 1)$. The analogous statement is true on the Heisenberg group, but false in hyperbolic space.