Differentials of complex interpolation processes for K\"othe function spaces
N. J.
Kalton
479-529
Abstract: We continue the study of centralizers on Köthe function spaces and the commutator estimates they generate (see [29]). Our main result is that if $X$ is a super-reflexive Köthe function space then for every real centralizer $\Omega$ on $X$ there is a complex interpolation scale of Köthe function spaces through $X$ inducing $\Omega$ as a derivative, up to equivalence and a scalar multiple. Thus, in a loose sense, all real centralizers can be identified with derivatives of complex interpolation processes. We apply our ideas in an appendix to show, for example, that there is a twisted sum of two Hilbert spaces which fails to be a $ ({\text{UMD}})$-space.
On automorphisms of Markov chains
Wolfgang
Krieger;
Brian
Marcus;
Selim
Tuncel
531-565
Abstract: We prove several theorems about automorphisms of Markov chains, using the weight-per-symbol polytope.
Equivariant cohomology and lower bounds for chromatic numbers
Igor
Kříž
567-577
Abstract: We introduce a general topological method for obtaining a lower bound of the chromatic number of an $n$-graph. We present numerical lower bounds for intersection $n$-graphs.
Lannes' $T$ functor on summands of $H\sp *(B({\bf Z}/p)\sp s)$
John C.
Harris;
R. James
Shank
579-606
Abstract: Let $H$ be the $\bmod$-$p$ cohomology of the classifying space $B({\mathbf{Z}}/p)$ thought of as an object in the category, $ \mathcal{U}$, of unstable modules over the Steenrod algebra. Lannes constructed a functor $T:\mathcal{U} \to \mathcal{U}$ which is left adjoint to the functor $A \mapsto A \otimes H$. In this paper we evaluate $ T$ on the indecomposable $\mathcal{U}$-summands of ${H^{ \otimes s}}$, the tensor product of $ s$ copies of $ H$. Our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring $ {{\mathbf{F}}_p}[{{\mathbf{M}}_{s,}}_s({\mathbf{Z}}/p)]$. The main application is to determine the homotopy type of the space of maps from $ B({\mathbf{Z}}/p)]$ to $ X$ when $X$ is a wedge summand of the space $\Sigma (B{({\mathbf{Z}}/p)^s})$.
A Fatou theorem for the solution of the heat equation at the corner points of a cylinder
Kin Ming
Hui
607-642
Abstract: In this paper the author proves existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain $D \times (0,\infty )$ where $ D$ is a bounded smooth domain in ${R^n}$ with zero lateral values. A unique representation of the strong solution is given in terms of measures $\mu$ on $D$ and $\lambda$ on $ \partial D$. We also show that the strong solution $u(x,t)$ of the heat equation in a cylinder converges a.e. ${x_0} \in \partial D \times \{ 0\}$ as $(x,t)$ converges to points on $\partial D \times \{ 0\}$ along certain nontangential paths.
On the $\Theta$-function of a Riemannian manifold with boundary
Pei
Hsu
643-671
Abstract: Let $\Omega$ be a compact Riemannian manifold of dimension $n$ with smooth boundary. Let ${\lambda _1} < {\lambda _2} \leq \cdots$ be the eigenvalues of the Laplace-Beltrami operator with the boundary condition $ [\partial /\partial n + \gamma ]\phi = 0$ . The associated $\Theta $-function ${\Theta _\gamma }(t) = \sum\nolimits_{n = 1}^\infty {\exp [ - {\lambda _n}t]}$ has an asymptotic expansion of the form $\displaystyle {(4\pi t)^{n/2}}{\Theta _\gamma }(t) = {a_0} + {a_1}{t^{1/2}} + {a_2}t + {a_3}{t^{3/2}} + {a_4}{t^2} + \cdots .$ The values of $ {a_0}$ , ${a_1}$ are well known. We compute the coefficients ${a_2}$ and ${a_3}$ in terms of geometric invariants associated with the manifold by studying the parametrix expansion of the heat kernel $p(t,x,y)$ near the boundary. Our method is a significant refinement and improvement of the method used in [McKean-Singer, J. Differential Geometry 1 (1969), 43-69].
Liouvillian first integrals of differential equations
Michael F.
Singer
673-688
Abstract: Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form. For example, we can show that if the system $dx/dz = P(x,y)$, $ dy/dz = Q(x,y)$ has a solution $(x(z),y(z))$ satisfying a liouvillian relation then either $x(z)$ and $y(z)$ are algebraically dependent or the system has a liouvillian first integral of the form $F(x,y) = \smallint RQ\,dx - RP\,dy$ where $R = \exp (\smallint U\,dx + V\,dy)$ and $U$ and $V$ rational functions of $x$ and $y$ . We can also reprove an old result of Ritt stating that a second order linear differential equation has a nonconstant solution satisfying a liouvillian relation if and only if all of its solutions are liouvillian.
Complemented ideals in the Fourier algebra and the Radon Nikod\'ym property
Brian
Forrest
689-700
Abstract: Necessary and sufficient conditions are given for an ideal $I(H)$ of the Fourier algebra to be complemented when $H$ is a closed subgroup of $G$ . Using the Radon Nikodym property, an example of a group $G$ with a normal abelian subgroup $H$ for which $I(H)$ is not complemented is presented.
Comparing periodic orbits of maps of the interval
C.
Bernhardt;
E.
Coven;
M.
Misiurewicz;
I.
Mulvey
701-707
Abstract: Let $\pi$ and $\theta$ be cyclic permutations of finite ordered sets. We say that $\pi$ forces $\theta$ if every continuous map of the interval which has a representative of $ \pi$ also has one of $ \theta$. We give a geometric version of Jungreis' combinatorial algorithm for deciding in certain cases whether $\pi$ forces $\theta$ .
A topological method for bounded solutions of nonautonomous ordinary differential equations
James R.
Ward
709-720
Abstract: The existence of bounded solutions to nonlinear nonautonomous ordinary differential equations is studied. This is done by associating the given equation to nonlinear autonomous ones by means of a family of skew-product flows related by homotopy. The existence of a bounded solution to the original differential equation is then related to the nontriviality of a certain Conley index associated with the autonomous differential equations. The existence of nontrivial bounded solutions is also considered. The differential equations studied are perturbations of homogeneous ones.
Galois groups of maximal $p$-extensions
Roger
Ware
721-728
Abstract: Let $p$ be an odd prime and $F$ a field of characteristic different from $p$ containing a primitive $p$th root of unity. Assume that the Galois group $G$ of the maximal $p$-extension of $F$ has a finite normal series with abelian factor groups. Then the commutator subgroup of $G$ is abelian. Moreover, $ G$ has a normal abelian subgroup with pro-cyclic factor group. If, in addition, $F$ contains a primitive ${p^2}$th root of unity then $G$ has generators ${\{ x,{y_i}\} _{i \in I}}$ with relations $ {y_i}{y_j} = {y_j}{y_i}$ and $ x{y_i}{x^{ - 1}} = y_i^{q + 1}$ where $q = 0$ or $q = {p^n}$ for some $n \geq 1$. This is used to calculate the cohomology ring of $G$, when $G$ has finite rank. The field $F$ is characterized in terms of the behavior of cyclic algebras (of degree $ p$) over finite $ p$-extensions.
The index of a Brauer class on a Brauer-Severi variety
Aidan
Schofield;
Michel
Van den Bergh
729-739
Abstract: Let $D$ and $E$ be central division algebras over $k$; let $K$ be the generic splitting field of $ E$; we show that the index of $D{ \otimes _k}K$ is the minimum of the indices of $D \otimes {E^{ \otimes i}}$ as $i$ varies. We use this to calculate the index of $D$ under related central extensions and to construct division algebras with special properties.
A spectral sequence for pseudogroups on ${\bf R}$
Solomon M.
Jekel
741-749
Abstract: Consider a pseudogroup $P$ of local homeomorphisms of $\mathbb{R}$ satisfying the following property: given points $ {x_0} < \cdots < {x_p}$ and $ {y_0} < \cdots < {y_p}$ in $ \mathbb{R}$ , there is an element of $P$, with domain an interval containing $[{x_0},{x_p}]$, taking each ${x_i}$ to ${y_i}$. The pseudogroup ${P^r}$ of local ${C^r}$ homeomorphisms, $0 \leq r \leq \infty$ , is of this type as is the pseudogroup $ {P^\omega }$ of local real-analytic homeomorphisms. Let $\Gamma$ be the topological groupoid of germs of elements of $P$. We construct a spectral sequence which involves the homology of a sequence of discrete groups $\{ {G_p}\}$. Consider the set $\{ f \in P\vert f(i) = i,i = 0,1, \ldots ,p\}$,; define $f\sim g$ if $f$ and $g$ agree on a neighborhood of $[0,p] \subset \mathbb{R}$. The equivalence classes under composition form the group $ {G_p}$. Theorem: There is a spectral sequence with $E_{p,q}^1 = {H_q}(B{G_p})$ which converges to $ {H_{p + q}}(B\Gamma )$. Our spectral sequence can be considered to be a version which covers the realanalytic case of some well-known theorems of J. Mather and G. Segal. The article includes some observations about how the spectral sequence applies to $ B\Gamma _1^\omega$. Further applications will appear separately.
Fourier inequalities with nonradial weights
C.
Carton-Lebrun
751-767
Abstract: Let $\mathcal{F}\;f(\gamma ) = {\smallint _{{\mathbb{R}^n}}}({e^{ - 2i\pi \gamma \bullet x}} - 1)f(x)\,dx,n > 1$, and $u$, $v$ be nonnegative functions. Sufficient conditions are found in order that $\left\Vert \mathcal{F}\;f\right\Vert _{q,u} \leq C\left\Vert f\right\Vert _{p,v}$ for all $f \in L_v^p({\mathbb{R}^n})$. Pointwise and norm approximations of $\mathcal{F}\;f$ are derived. Similar results are obtained when $u$ is replaced by a measure weight. In the case $v(x) = \vert x{\vert^{n(p - 1)}}$, a counterexample is given which shows that no Fourier inequality can hold for all $f$ in $ L_{c,0}^\infty$. Spherical restriction theorems are established. Further conditions for the boundedness of $ \mathcal{F}$ are discussed.
Radii of convergence and index for $p$-adic differential operators
Paul Thomas
Young
769-785
Abstract: We study the radii of $p$-adic convergence of solutions at a generic point of homogeneous linear differential operators whose coefficients are analytic elements. As an application we prove a conjecture of P. Robba (for a certain class of operators) concerning the relation between radii of convergence and index on analytic elements. We also give an explicit factorization theorem for $ p$-adic differential operators, based on the radii of generic convergence and the slopes of the associated Newton polygon.
The set of all iterates is nowhere dense in $C([0,1],[0,1])$
A. M.
Blokh
787-798
Abstract: We prove that if a mixing map $ f:[0,1] \to [0,1]$ belongs to the ${C^0}$-closure of the set of iterates and $f(0) \ne 0$, $ f(1) \ne 1$ then $ f$ is an iterate itself. Together with some one-dimensional techniques it implies that the set of all iterates is nowhere dense in $ C([0,1],[0,1])$ giving the final answer to the question of A. Bruckner, P. Humke and M. Laczkovich.
Bounded analytic functions on two sheeted discs
Mikihiro
Hayashi;
Mitsuru
Nakai;
Shigeo
Segawa
799-819
Abstract: Results of qualitative nature of both positive and negative directions on the point separation by bounded analytic functions of smooth subregions of two sheeted discs are given when two sheeted discs themselves are not separated by bounded analytic functions. We are, in particular, concerned about roles of branch points in two sheeted discs played in the point separation by bounded analytic functions.
Uniqueness in Cauchy problems for hyperbolic differential operators
Christopher D.
Sogge
821-833
Abstract: In this paper we prove a unique continuation theorem for second order strictly hyperbolic differential operators. Results also hold for higher order operators if the hyperbolic cones are strictly convex. These results are proved via certain Carleman inequalities. Unlike [6], the parametrices involved only have real phase functions, but they also have Gaussian factors. We estimate the parametrices and associated remainders using sharp $ {L^p}$ estimates for Fourier integral operators which are due to Brenner [1] and Seeger, Stein, and the author [5].
Unknotted solid tori and genus one Whitehead manifolds
Edward M.
Brown
835-847
Abstract: In this paper we study contractible open $3$-manifolds which are monotone unions of solid tori and which embed in a compact $3$-manifold. We show that the tori are unknotted in later tori. We then study pairs of unknotted solid tori, and prove a unique prime decomposition theorem. This is applied to the open $3$-manifolds above to get an essentially unique prime decomposition. A number of examples in the literature are analyzed, and some new examples are constructed.
Generalized Toda brackets and equivariant Moore spectra
Steven R.
Costenoble;
Stefan
Waner
849-863
Abstract: In this paper we develop a general theory of obstructions to the existence of equivariant Moore spectra. The obstructions we obtain coincide with higher order Toda brackets as defined by Spanier. We then apply the theory to show the existence of equivariant Moore spectra in various special cases.
A recurrent nonrotational homeomorphism on the annulus
Robbert J.
Fokkink;
Lex G.
Oversteegen
865-875
Abstract: We construct an area- and orientation-preserving recurrent diffeomorphism on the annulus without periodic points, which is not conjugate to a rotation. The mapping is, however, semiconjugate to an irrational rotation of a circle. Our example is a counterexample to the Birkhoff Conjecture.
The minimal degree of a finite inverse semigroup
Boris M.
Schein
877-888
Abstract: The minimal degree of an inverse semigroup $S$ is the minimal cardinality of a set $ A$ such that $ S$ is isomorphic to an inverse semigroup of one-to-one partial transformations of $ A$. The main result is a formula that expresses the minimal degree of a finite inverse semigroup $S$ in terms of certain subgroups and the ordered structure of $S$. In fact, a representation of $ S$ by one-to-one partial transformations of the smallest possible set $ A$ is explicitly constructed in the proof of the formula. All known and some new results on the minimal degree follow as easy corollaries.
Complete nonorientable minimal surfaces in ${\bf R}\sp 3$
Tōru
Ishihara
889-901
Abstract: We will study complete minimal immersions of nonorientable surfaces into $ {R^3}$. Especially, we construct a nonorientable surface ${P_2}$ which is homeomorphic to a Klein bottle and show that for any integer $m \geq 4$, there are complete minimal immersion of $ M = {P_2} - \{ q\}$, $q \in {P_2}$ in ${R^3}$ with one end and total curvature $C(M) = - 4m\pi$.
Nilpotence and torsion in the cohomology of the Steenrod algebra
Kenneth G.
Monks
903-912
Abstract: In this paper we prove the existence of global nilpotence and global torsion bounds for the cohomology of any finite Hopf subalgebra of the Steenrod algebra for the prime $2$. An explicit formula for computing such bounds is then obtained. This is used to compute bounds for ${H^{\ast} }({\mathcal{A}_n})$ for $n \leq 6$.
The uniqueness and stability of the solution of the Riemann problem of a system of conservation laws of mixed type
Hai Tao
Fan
913-938
Abstract: We establish the uniqueness and stability of the similarity solution of the Riemann problem for a $2 \times 2$ system of conservation laws of mixed type, with initial data separated by the elliptic region, which satisfies the viscosity-capillarity travelling wave admissibility criterion.
Corrigendum to: ``Characterizations of turbulent one-dimensional mappings via $\omega$-limit sets'' [Trans. Amer. Math. Soc. {\bf 326} (1991), no. 1, 261--280; MR1010884 (91j:58133)]
Michael J.
Evans;
Paul D.
Humke;
Cheng Ming
Lee;
Richard J.
O’Malley
939-940