Primarit\'e de $L\sp{p}(X)$
Michèle
Capon
431-487
Abstract: Soit $X$ un espace de Banach à base symétrique. Nous étudions les opérateurs de ${L^p}(X)$ dans lui-même, en leur associant une "matrice" d'opérateurs de $ {L^p}$. Cette technique nous permet de démontrer que pour tout $p$ réel dans
Topological semiconjugacy of piecewise monotone maps of the interval
Bill
Byers
489-495
Abstract: This paper establishes a topological semiconjugacy between two piecewise monotone maps of the interval which have the same kneading sequences and do not map one turning point into another, whenever itineraries under the second map are given uniquely by their invariant coordinate. Various examples are given and consequences obtained.
Steenrod and Dyer-Lashof operations on $B{\bf U}$
Timothy
Lance
497-510
Abstract: This paper describes a simple, fast algorithm for the computation of Steenrod and Dyer-Lashof operations on $BU$. The calculations are carried out in $ H^{\ast}(BU,{{\mathbf{Z}}_{(p)}})$ and $ {H_{\ast}}(BU,{{\mathbf{Z}}_{(p)}})$ where $p$ local lifts are determined by the values on primitives and Cartan formulas. This algorithm also provides a description of Steenrod and Dyer-Lashof operations on the fiber of any $H$ map (or infinite loop map) $BU \to BU$, and applications to the classifying spaces of surgery which arise in this fashion will appear shortly.
Some conjectures on elliptic curves over cyclotomic fields
D.
Goldfeld;
C.
Viola
511-515
Abstract: We give conjectures for the mean values of Hasse-Weil type $ L$-functions over cyclotomic fields. In view of the Birch-Swinnerton-Dyer conjectures, this translates to interesting arithmetic information.
Neighborhoods of algebraic sets
Alan H.
Durfee
517-530
Abstract: In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighborhoods will be defined as level sets of polynomial or semialgebraic functions.
A characterization of bounded symmetric domains by curvature
J. E.
D’Atri;
I.
Dotti Miatello
531-540
Abstract: This paper will prove that a bounded homogeneous domain is symmetric if and only if, in the Bergman metric, all sectional curvatures are nonpositive.
Weakening the topology of a Lie group
T. Christine
Stevens
541-549
Abstract: With any topological group $ (G, \mathcal{U})$ one can associate a locally arcwise-connected group $ (G, {\mathcal{U}}^{\ast})$, where $ {\mathcal{U}}^{\ast}$ is stronger than $ \mathcal{U}$. $(G, \mathcal{U})$ is a weakened Lie $ (WL)$ group if $(G, {\mathcal{U}}^{\ast})$ is a Lie group. In this paper the author shows that the WL groups with which a given connected Lie group $(L,\mathcal{J})$ is associated are completely determined by a certain abelian subgroup $ H$ of $L$ which is called decisive. If $ L$ has closed adjoint image, then $H$ is the center $Z(L)$ of $L$; otherwise, $H$ is the product of a vector group $V$ and a group $J$ that contains $Z(L)$. $J/Z(L)$ is finite (trivial if $L$ is solvable). We also discuss the connection between these theorems and recent results of Goto.
Interpolating sequences for $QA\sb{B}$
Carl
Sundberg;
Thomas H.
Wolff
551-581
Abstract: Let $B$ be a closed algebra lying between ${H^\infty}$ and $ {L^\infty}$ of the unit circle. We define $QA_B = H^\infty \cap \bar{B}$, the analytic functions in $ Q_B = B \cap \bar{B}$. By work of Chang, ${Q_B}$ is characterized by a vanishing mean oscillation condition. We characterize the sequences of points $\left\{{{z_n}} \right\}$ in the open unit disc for which the interpolation problem $ f({z_n}) = {\lambda _n}, n = 1, 2,\ldots$, is solvable with $f \in {Q_B}$ for any bounded sequence of numbers $ \left\{{{\lambda _n}} \right\}$. Included as a necessary part of our proof is a study of the algebras $Q{A_B}$ and ${Q_B}$.
On genus $2$ Heegaard diagrams for the $3$-sphere
Takeshi
Kaneto
583-597
Abstract: Let $D$ be any genus $2$ Heegaard diagram for the $ 3$-sphere and $\left\langle {{a_1}, {a_2}; {{\tilde r}_1}, {{\tilde r}_2}} \right\rangle$ be the cyclically reduced presentation associated with $D$. We shall show that ${{\tilde{r}}_1}$ contains ${{\tilde{r}}_2}$ or ${\tilde{r}}_2^{-1}$ as a subword in cyclic sense if $ \left\{{\tilde r}_1, {\tilde r}_2 \right\} \ne \left\{{a_1}^{\pm 1}, {a_2}^{\pm 1} \right\}$ holds, and that, using this property, $\left\langle {a_1}, {a_2};{r_1}, {r_2} \right\rangle$ can be transformed to the trivial one $\left\langle {{a_1}, {a_2};{a_1}^{\pm 1}, a_2^{\pm 1}} \right\rangle$. By the recent positive solution of genus $2$ Poincaré conjecture, our result implies the purely algebraic, algorithmic solution to the decision problem; whether a given $3$-manifold with a genus $2$ Heegaard splitting is simply connected or not, equivalently, is homeomorphic to the $ 3$-sphere or not.
The derived functors of the primitives for ${\rm BP}\sb\ast (\Omega S\sp{2n+1})$
Martin
Bendersky
599-619
Abstract: Formulas for the Hopf invariant, and the $P$ map in the Novikov double suspension sequence are derived. The formulas allow an effective inductive computation of the ${E_2}$-term of the unstable Adams-Novikov spectral sequence. The $3$ primary ${E_2}$-term through the $54$ stem is displayed.
A correction and some additions to: ``Fundamental solutions for differential equations associated with the number operator''
Yuh Jia
Lee
621-624
Abstract: Let $(H,B)$ be an abstract Wiener pair and $\mathfrak{N}$ the operator defined by $\mathfrak{N}u(x) = - {\text{trace}}_H{D^2}u(x) + (x,Du(x))$, where $x \in B$ and $ (\cdot, \cdot )$ denotes the $ B$-$B^{\ast}$ pairing. In this paper, we point out a mistake in the previous paper concerning the existence of fundamental solutions of ${\mathfrak{N}^k}$ and intend to make a correction. For this purpose, we study the fundamental solution of the operator ${(\mathfrak{N} + \lambda I)^k}\,(\lambda > 0)$ and investigate its behavior as $\lambda \to 0$. We show that there exists a family $ \{{Q_\lambda}(x,dy)\}$ of measures which serves as the fundamental solution of $ {(\mathfrak{N} + \lambda I)^k}$ and, for a suitable function $f$, we prove that the solution of $ {\mathfrak{N}^k}u = f$ can be represented by $u(x) = {\lim _{\lambda \to 0}}\int_B f(y){Q_\lambda}(x,dy) + C$, where $C$ is a constant.
Smooth type ${\rm III}$ diffeomorphisms of manifolds
Jane
Hawkins
625-643
Abstract: In this paper we prove that every smooth paracompact connected manifold of dimension $ \geqslant 3$ admits a smooth type $ {\text{III}}_\lambda$ diffeomorphism for every $0 \leqslant \lambda \leqslant 1$. (Herman proved the result for $ \lambda = 1$ in [7].) The result follows from a theorem which gives sufficient conditions for the existence of smooth ergodic real line extensions of diffeomorphisms of manifolds.
Domain Bloch constants
C. David
Minda
645-655
Abstract: The classical Bloch constant $ \mathcal{B}$ is defined for holomorphic functions $f$ defined on ${\mathbf{B}} = \{z:\vert z\vert < 1\}$ and normalized by $ \vert f^{\prime}(0)\vert = 1$. Let ${R_f}$ denote the Riemann surface of $f$ and ${B_f}$ the set of branch points. Then $\mathcal{B}$ can be regarded as a lower bound for the radius of the largest disk contained in ${R_f}\backslash {B_f}$. The metric on ${R_f}$ used to measure the size of disks on ${R_f}$ is obtained by lifting the euclidean metric from $ {\mathbf{C}}$ to $ {R_f}$. The surface $ {R_f}$ can also be regarded as spread over $ {\mathbf{B}}$ and the hyperbolic metric lifted to ${R_f}$. One may then ask for the radius of the largest hyperbolic disk on ${R_f}\backslash {B_f}$. A lower bound for this radius is called a domain Bloch constant. The determination of domain Bloch constants is nontrivial for nonconstant analytic functions $ f:{\mathbf{B}} \to X$, where $X$ is a hyperbolic Riemann surface. Upper and lower bounds for domain Bloch constants are given. Also, domain Bloch constants are given an interpretation as a radius of local schlichtness.
Generalized intersection multiplicities of modules
Sankar P.
Dutta
657-669
Abstract: In this paper we study intersection multiplicities of modules as defined by Serre and prove that over regular local rings of $ \dim \leqslant 5$, given two modules $M,N$ with $l(M\otimes_{R}N) < \infty$ and $\dim\;M + \dim \;N < \dim \;R,\chi (M,N) = \sum\nolimits_{i = 0}^{\dim\; R}( - 1)^i l(\operatorname{Tor}_i^R(M,N)) = 0$. We also study multiplicity in a more general set up. Finally we extend Serre's result from pairs of modules to pairs of finite free complexes whose homologies are killed by $ {I^n},{J^n}$, respectively, for some $n > 0$, with $\dim \,R/I + \dim \,R/J < \dim \,R$.
The splitting of $B{\rm O}\langle 8\rangle \wedge b{\rm o}$ and $M{\rm O}\langle 8\rangle \wedge b{\rm o}$
Donald M.
Davis
671-683
Abstract: Let $BO\left\langle 8 \right\rangle$ denote the classifying space for vector bundles trivial on the $7$-skeleton, and $MO\left\langle 8 \right\rangle$ the associated Thom spectrum. It is proved that, localized at $ 2$, $BO\left\langle 8 \right\rangle \wedge \,bo$ and $MO\left\langle 8 \right\rangle \wedge \,bo$ split as a wedge of familiar spectra closely related to $bo$, where $bo$ is the spectrum for connective $ KO$-theory.
Biholomorphic invariants of a hyperbolic manifold and some applications
B. L.
Fridman
685-698
Abstract: A biholomorphically invariant real function ${h_x}$ is defined for a hyperbolic manifold $ X$. Properties of such functions are studied. These properties are applied to prove the following theorem. If a hyperbolic manifold $ X$ can be exhausted by biholomorphic images of a strictly pseudoconvex domain $D \subset {{\mathbf{C}}^n}$ with $ \partial D\; \in \;{C^3}$, then $X$ is biholomorphically equivalent either to $ D$ or to the unit ball in $ {{\mathbf{C}}^n}$. The properties of ${h_D}$ are also applied to some questions concerning the group of analytical automorphisms of a strictly pseudoconvex domain and to similar questions concerning polyhedra.
Quotients of $L\sp{\infty }$ by Douglas algebras and best approximation
Daniel H.
Luecking;
Rahman M.
Younis
699-706
Abstract: We show that ${L^\infty}/A$ is not the dual space of any Banach space when $A$ is a Douglas algebra of a certain type. We do this by showing its unit ball has no extreme points. The method used requires that any function in ${L^\infty}$ has a nonunique best approximation in $A$. We therefore also show that the Douglas algebra $ {H^\infty} + L_F^\infty$, when $F$ is an open subset of the unit circle, permits best approximation. We use a method originating in Hayashi [6] and independently obtained by Marshall and Zame.
The sharp form of Ole\u\i nik's entropy condition in several space variables
David
Hoff
707-714
Abstract: We investigate the conditions under which the Volpert-Kruzkov solution of a single conservation law in several space variables with flux $F$ will satisfy the simplified entropy condition $ \operatorname{div}\,F^{\prime}(u) \leqslant 1/t$, and when this condition guarantees uniqueness for given $ {L^\infty}$ Cauchy data. We show that, when $F$ is ${C^1}$, our condition guarantees uniqueness iff $ F$ is isotropic, and that, for such $F$, the Volpert-Kruzkov solution always satisfies our condition.
Efficient computation in groups and simplicial complexes
John C.
Stillwell
715-727
Abstract: Using HNN extensions of the Boone-Britton group, a group $ E$ is obtained which simulates Turing machine computation in linear space and cubic time. Space in $E$ is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACE-complete problem for a topological model of computation previously used to characterize NP-completeness and RE-completeness.
Weighted norm inequalities for the Fourier transform
Benjamin
Muckenhoupt
729-742
Abstract: Given $ p$ and $q$ satisfying $1 < p \leqslant q < \infty$, sufficient conditions on nonnegative pairs of functions $U,V$ are given to imply $\displaystyle {\left[ {\int_{{R^n}}^{} {\vert\hat f(x){\vert^q}U(x)\,dx}} \righ... ...qslant c{\left[ {\int_{{R^n}}^{} {\vert f(x){\vert^p}V(x)\,dx}} \right]^{1/p}},$ where $\hat f$ denotes the Fourier transform of $f$, and $c$ is independent of $f$. For the case $ q = p^{\prime}$ the sufficient condition is that for all positive $r$, $\displaystyle \left[ {\int_{U(x) > Br} {U(x)\;dx}} \right]\left[ {\int_{V(x) < {r^{p - 1}}} {V{{(x)}^{- 1/(p - 1)}}\;dx}} \right] \leqslant A,$ where $A$ and $B$ are positive and independent of $ r$. For $q \ne p^{\prime}$ the condition is more complicated but also is invariant under rearrangements of $ U$ and $V$. In both cases the sufficient condition is shown to be necessary if the norm inequality holds for all rearrangements of $U$ and $V$. Examples are given to show that the sufficient condition is not necessary for a pair $U,V$ if the norm inequality is assumed only for that pair.
Shrinking countable decompositions of $S\sp{3}$
Richard
Denman;
Michael
Starbird
743-756
Abstract: Conditions are given which imply that a countable, cellular use decomposition $G$ is shrinkable. If the embedding of each element in $ G$ has the bounded nesting property, defined in this paper, then ${S^3}/G$ is homeomorphic to $ {S^3}$. The bounded nesting property is a condition on the defining sequence of cells for an element of $G$ which implies that $G$ satisfies the Disjoint Disk criterion for shrinkability [ $ {\mathbf{S1}}$, Theorem 3.1]. From this result, one deduces that countable, star-like equivalent use decompositions of $ {S^3}$ are shrinkable--a result proved independently by E. Woodruff [ ${\mathbf{W}}$]. Also, one deduces the shrinkability of countable bird-like equivalent use decompositions (a generalization of the star-like result), and the recently proved theorem that if each element of a countable use decomposition $G$ of ${S^3}$ has a mapping cylinder neighborhood, then $ G$ is shrinkable [ ${\mathbf{E}}$; $ {\mathbf{S1}}$, Theorem 4.1; ${\mathbf{S}}$-${\mathbf{W}}$, Theorem 1].
Constructing approximate fibrations
T. A.
Chapman;
Steve
Ferry
757-774
Abstract: In this paper two results concerning the construction of approximate fibrations are established. The first shows that there are approximate fibrations $ p:M \to S^2$ which are homotopic to bundle maps but which cannot be approximated by bundle maps. Here $M$ can be a compact $Q$-manifold or some topological $n$-manifold, $n \geqslant 5$. The second shows how to construct approximate fibrations $p:M \to B$ whose fibers do not have finite homotopy type, for any $B$ of Euler characteristic zero. Here $ M$ can be a compact $ Q$-manifold and $ B$ only has to be an ANR, or $M$ can be an $n$-manifold, $ n \geqslant 6$, and $ B$ must then also be a topological manifold.
Approximation by smooth multivariate splines
C.
de Boor;
R.
DeVore
775-788
Abstract: The degree of approximation achievable by piecewise polynomial functions of given total order on certain regular grids in the plane is shown to be adversely affected by smoothness requirements--in stark contrast to the univariate situation. For a rectangular grid, and for the triangular grid derived from it by adding all northeast diagonals, the maximum degree of approximation (as the grid size $1/n$ goes to zero) to a suitably smooth function is shown to be $ O({n^{- \rho - 2}})$ in case we insist that the approximating functions are in ${C^\rho}$. This only holds as long as $\rho \leqslant (r - 3)/2$ and $\rho \leqslant (2r - 4)/3$, respectively, with $r$ the total order of the polynomial pieces. In the contrary case, some smooth functions are not approximable at all. In the discussion of the second mesh, a new and promising kind of multivariate ${\text{B}}$-spline is introduced.
Scattering theory and the geometry of multitwistor spaces
Matthew L.
Ginsberg
789-815
Abstract: Existing results which show the zero rest mass field equations to be encoded in the geometry of projective twistor space are extended, and it is shown that the geometries of spaces of more than one twistor contain information concerning the scattering of such fields. Some general constructions which describe spacetime interactions in terms of cohomology groups on subvarieties in twistor space are obtained and are used to construct a purely twistorial description of spacetime propagators and of first order ${\phi ^4}$ scattering. Spacetime expressions concerning these processes are derived from their twistor counterparts, and a physical interpretation is given for the twistor constructions.
Conjugate Fourier series on certain solenoids
Edwin
Hewitt;
Gunter
Ritter
817-840
Abstract: We consider an arbitrary noncyclic subgroup of the additive group ${\mathbf{Q}}$ of rational numbers, denoted by $ {{\mathbf{Q}}_{\mathbf{a}}}$, and its compact character group ${\Sigma _{\mathbf{a}}}$. For $1 < p < \infty$, an abstract form of Marcel Riesz's theorem on conjugate series is known. For $ f$ in $ {\mathfrak{L}_p}({\Sigma _{\mathbf{a}}})$, there is a function $\tilde{f}$ in ${\mathfrak{L}_p}({\Sigma _{\mathbf{a}}})$ whose Fourier transform $ (\tilde{f})\hat{\empty}(\alpha )$ at $\alpha$ in $ {{\mathbf{Q}}_{\mathbf{a}}}$ is $- i\,\operatorname{sgn}\,\alpha \hat{f}(\alpha )$. We show in this paper how to construct $\tilde{f}$ explicitly as a pointwise limit almost everywhere on $ {\Sigma_{\mathbf{a}}}$ of certain harmonic functions, as was done by Riesz for the circle group. Some extensions of this result are also presented.
Are primitive words universal for infinite symmetric groups?
D. M.
Silberger
841-852
Abstract: Let $W = W({x_1}, \ldots ,{x_j})$ be any word in the $j$ free generators ${x_1}, \ldots ,{x_j}$, and suppose that $ W$ cannot be expressed in the form $W = {V^k}$ for $V$ a word and for $k$ an integer with $\left\vert k \right\vert \ne 1$. We ask whether the equation $f = W$ has a solution $ ({x_1}, \ldots ,{x_j}) = (a_{1}, \ldots, a_{j}) \in G^{j}$ whenever $ G$ is an infinite symmetric group and $f$ is an element in $G$. We establish an affirmative answer in the case that $ W(x,y) = {x^m}{y^n}$ for $ m$ and $n$ nonzero integers.
Bundle-like foliations with K\"ahlerian leaves
Richard H.
Escobales
853-859
Abstract: For bundle-like foliations with Kählerian leaves a certain function $ f$ is studied and its Laplacian along a leaf is computed. From this computation one obtains geometric conditions which guarantee the integrability of the distribution orthogonal to that determined by the leaves. When the leaves are compact, the key condition needed to guarantee the integrability of this orthogonal distribution can be interpreted as a condition on the first Chern class of each of the leaves.