A fractal-like algebraic splitting of the classifying space for vector bundles
V.
Giambalvo;
David J.
Pengelley;
Douglas C.
Ravenel
433-455
Abstract: The connected covers of the classifying space $BO$ induce a decreasing filtration $\{ {B_n}\}$ of ${H_{\ast}}(BO;\,Z/2)$ by sub-Hopf algebras over the Steenrod algebra $A$. We describe a multiplicative grading on $ {H_{\ast}}(BO;\,Z/2)$ inducing a direct sum splitting of ${B_n}$ over ${A_n}$, where $ \{ {A_n}\}$ is the usual (increasing) filtration of $A$. The pieces in the splittings are finite, and the grading extends that of ${H_{\ast}}{\Omega ^2}{S^3}$ which splits it into Brown-Gitler modules. We also apply the grading to the Thomifications $ \{ {M_n}\}$ of $\{ {B_n}\}$, where it induces splittings of the corresponding cobordism modules over the entire Steenrod algebra. These generalize algebraically the previously known topological splittings of the connective cobordism spectra $MO$, $MSO$ and $M\,Spin$.
Applications of nonstandard models and Lebesgue measure to sequences of natural numbers
Steven C.
Leth
457-468
Abstract: By use of a nonstandard model, sequences of natural numbers are associated with a collection of closed subsets of reals in a natural way. The topological and measure-theoretic properties of the associated closed sets are used to prove standard theorems and define new density functions on sequences.
Asymptotic periodicity of the iterates of positivity preserving operators
M.
Miklavčič
469-479
Abstract: Assume that (A1) $X$ is a real Banach space. (A2) $ {X^ + }$ is a closed subset of $X$ with the following properties: (i) if $x \in {X^ + }$, $y \in {X^ + }$, $\alpha \in [0,\,\infty )$ then $x + y \in {X^ + }$ and $\alpha x \in {X^ + }$; (ii) there exists ${M_0} \in (0,\,\infty )$ such that for each $x \in X$ there exist ${x_ + } \in {X^ + }$ and ${x_ - } \in {X^ + }$ which satisfy $\displaystyle x = {x_ + } - {x_ - },\qquad \vert\vert{x_ + }\vert\vert \leqslan... ...t\vert,\qquad \vert\vert{x_ - }\vert\vert \leqslant {M_0}\vert\vert x\vert\vert$ and if $x = {y_ + } - {y_ - }$ for some ${y_ + } \in {X^ + }$, ${y_ - } \in {X^ + }$ then ${y_ + } - {x_ + } \in {X^ + }$; (iii) if $x \in {X^ + }$, $ y \in {X^ + }$ then $\vert\vert x\vert\vert \leqslant \vert\vert x + y\vert\vert$. (A3) $B$ is a bounded linear operator on $ X$. (A4) $B{X^ + } \subset {X^ + }$. (A5) $ {F_0}$ is a nonempty compact subset of $X$ and ${\lim _{n \to \infty }}\operatorname{dist} ({B^n}x,\,{F_0}) = 0$ whenever $x \in {X^ + }$ and $\vert\vert x\vert\vert = 1$. Then $ {B^n}x$ is asymptotically periodic for every $x \in X$. This, and other properties of $ B$, are proven in the paper.
A classification of a class of $3$-branchfolds
Yoshihiro
Takeuchi
481-502
Abstract: An $n$-orbifold is a topological space provided with a local modelling on (an open set in ${{\mathbf{R}}^n}$)/(a finite group action). Mainly, we deal with $3$-branchfolds (i.e. $3$-orbifolds with $1$-dimensional singular locus). We define a map between two $3$-branchfolds. With respect to this map, we prove some facts parallel to $3$-manifold theorems. Using the facts, we classify a class of $3$-branchfolds, analogous to Waldhausen's classification theorem of Haken manifolds.
Total linking number modules
Oziride Manzoli
Neto
503-533
Abstract: Given a codimension two link $L$ in a sphere ${S^k}$ with complement $X = {S^k} - L$, the total linking number covering of $ L$ is the covering $\hat X \to X$ associated to the surjection ${\pi _1}(X) \to Z$ defined by sending the meridians to $1$. The homology ${H_{\ast}}(\hat X)$ define weaker invariants than the homology of the universal abelian covering of $L$. The groups $ {H_i}(\hat X)$ are modules over $Z\left[ {t,\,{t^{ - 1}}} \right]$ and this work gives an algebraic characterization of these modules for $ k \geqslant 4$ except for the pseudo null part of $ {H_1}(\hat X)$.
Travelling wave solutions to a gradient system
James F.
Reineck
535-544
Abstract: Given a system of reaction-diffusion equations where the nonlinearity is derived from a potential with certain restrictions, we use the Conley index and the connection matrix to show that there is a travelling wave solution connecting the maxima of the potential.
The dynamics of rotating waves in scalar reaction diffusion equations
S. B.
Angenent;
B.
Fiedler
545-568
Abstract: The maximal compact attractor for the RDE ${u_t} = {u_{xx}} + f(u,\,{u_x})$ with periodic boundary conditions is studied. It is shown that any $\omega$-limit set contains a rotating wave, i.e., a solution of the form $U(x - ct)$. A number of heteroclinic orbits from one rotating wave to another are constructed. Our main tool is the Nickel-Matano-Henry zero number. The heteroclinic orbits are obtained via a shooting argument, which relies on a generalized Borsuk-Ulam theorem.
Linear supergroup actions. I. On the defining properties
Oscar Adolfo
Sánchez-Valenzuela
569-595
Abstract: This paper studies the notions of linearity and bilinearity in the category of supermanifolds. Following the work begun by [OASV2], we deal with supermanifoldifications of supervector spaces. The ${{\mathbf{R}}^{1\vert 1}}$-module operations are defined componentwise. The linearity and bilinearity properties are stated by requiring commutativity of some appropriate diagrams of supermanifold morphisms. It is proved that both linear and bilinear supermanifold morphisms are completely determined by their underlying continuous maps, which in turn have to be linear (resp., bilinear) in the usual sense. It is observed that whereas linear supermanifold morphisms are vector bundle maps, bilinear supermanifold morphisms are not. A natural generalization of the bilinear evaluation map $\operatorname{Hom} (V,\,W) \times V \to W\;((F,\,v) \mapsto F(v))$ is given and some applications pointing toward the notions of linear supergroup actions and adjoint and coadjoint actions are briefly discussed.
Remarks on Grassmannian supermanifolds
Oscar Adolfo
Sánchez-Valenzuela
597-614
Abstract: This paper studies some aspects of a particular class of examples of supermanifolds; the supergrassmannians, introduced in [Manin]. Their definition, in terms of local data and glueing isomorphisms, is reviewed. Explicit formulas in local coordinates are given for the Lie group action they come equipped with. It is proved that, for those supergrassmannians whose underlying manifold is an ordinary grassmannian, their structural sheaf can be realized as the sheaf of sections of the exterior algebra bundle of some canonical vector bundle. This realization holds true equivariantly for the Lie group action in question, thus making natural in these cases the identification given in [Batchelor]. The proof depends on the computation of the transition functions of the supercotangent bundle as defined in a previous work [OASV 2]. Finally, it is shown that there is a natural supergroup action involved (in the sense of [OASV 3]) and hence, the supergrassmannians may be regarded as examples of superhomogeneous spaces--a notion first introduced in [Kostant]. The corresponding Lie superalgebra action can be realized as superderivations of the structural sheaf; explicit formulas are included for those supergrassmannians identifiable with exterior algebra vector bundles.
Infinitely many periodic solutions for the equation: $u\sb {tt}-u\sb {xx}\pm \vert u\vert \sp {p-1}u=f(x,t)$. II
Kazunaga
Tanaka
615-645
Abstract: Existence of forced vibrations of nonlinear wave equation: \begin{displaymath}\begin{array}{*{20}{c}} {{u_{tt}} - {u_{xx}} \pm \vert u{\ver... ...,t) \in (0,\,\pi ) \times {\mathbf{R}},} \end{array} \end{displaymath} is considered. For all $p \in (1,\,\infty )$ and $f(x,\,t) \in {L^{(p + 1)/p}}$, existence of infinitely many periodic solutions is proved. This improves the results of the author [29, 30]. We use variational methods to show the existence result. Minimax arguments and energy estimates for the corresponding functional play an essential role in the proof.
Tangent cones to discriminant loci for families of hypersurfaces
Roy
Smith;
Robert
Varley
647-674
Abstract: A deformation of a variety with (nonisolated) hypersurface singularities, such as a projective hypersurface or a theta divisor of an abelian variety, determines a rational map of the singular locus to projective space and the resulting projective geometry of the singular locus describes how the singularities propagate in the deformation. The basic principle is that the projective model of the singular locus is dual to the tangent cone to the discriminant of the deformation. A detailed study of the method, which emerged from interpreting Andreotti-Mayer's work on theta divisors in terms of Schlessinger's deformation theory of singularities, is given along with examples, applications, and multiplicity formulas.
Automorphisms and isomorphisms of real Henselian fields
Ron
Brown
675-703
Abstract: Let $K$ and $L$ be ordered algebraic extensions of an ordered field $F$. Suppose $K$ and $L$ are Henselian with Archimedean real closed residue class fields. Then $K$ and $L$ are shown to be $F$-isomorphic as ordered fields if they have the same value group. Analogues to this result are proved involving orderings of higher level, unordered extensions, and, when $K$ and $L$ are maximal valued fields, transcendental extensions. As a corollary, generalized real closures at orderings of higher level are shown to be determined up to isomorphism by their value groups. The results on isomorphisms are applied to the computation of automorphism groups of $K$ and to the study of the fixed fields of groups of automorphisms of $K$. If $K$ is real closed and maximal with respect to its canonical valuation, then these fixed fields are shown to be exactly those real closed subfields of $ K$ which are topologically closed in $K$. Generalizations of this fact are proved. An example is given to illustrate several aspects of the problems considered here.
The differential operator ring of an affine curve
Jerry L.
Muhasky
705-723
Abstract: The purpose of this paper is to investigate the structure of the ring $ D(R)$ of all linear differential operators on the coordinate ring of an affine algebraic variety $X$ (possibly reducible) over a field $k$ (not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim $X \leqslant 1$. In this case, it is proved that $D(R)$ is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim $ X$, that the endomorphism ring of any simple (left or right) $D(R)$-module is finite dimensional over $ k$, that $D(R)$ has a unique smallest ideal $ L$ essential as a left or right ideal, and that $D(R)/L$ is finite dimensional over $ k$. The following ring-theoretic tool is developed for use in deriving the above results. Let $D$ be a subalgebra of a left noetherian $ k$-algebra $E$ such that $E$ is finitely generated as a left $D$-module and all simple left $ E$-modules have finite dimensional endomorphism rings (over $k$), and assume that $D$ contains a left ideal $I$ of $E$ such that $E/I$ has finite length. Then it is proved that $ D$ is left noetherian and that the endomorphism ring of any simple left $ D$-module is finite dimensional over $k$.
Strong homology is not additive
S.
Mardešić;
A. V.
Prasolov
725-744
Abstract: Using the continuum hypothesis (CH) we show that strong homology groups $\overline {{H_p}} (X)$ do not satisfy Milnor's additivity axiom. Moreover, CH implies that strong homology does not have compact supports and that $\overline {{H_p}} (X)$ need not vanish for $p < 0$.
Products of involution classes in infinite symmetric groups
Gadi
Moran
745-762
Abstract: Let $A$ be an infinite set. Denote by $ {S_A}$ the group of all permutations of $A$, and let ${R_i}$, denote the class of involutions of $ A$ moving $\vert A\vert$ elements and fixing $ i$ elements $(0 \leqslant i \leqslant \vert A\vert)$. The products $ {R_i}{R_j}$ were determined in [M1]. In this article we treat the products $ {R_{{i_1}}} \cdots {R_{{i_n}}}$ for $ n \geqslant 3$. Let INF denote the set of permutations in ${S_A}$ moving infinitely many elements. We show: (1) ${R_{{i_1}}} \cdots {R_{{i_n}}} = {S_A}$ for $n \geqslant 4$. (2)(a) ${R_i}{R_j}{R_k} = \operatorname{INF}$ if $\{ i,\,j,\,k\} $ contains two integers of different parity; (b) ${R_i}{R_j}{R_k} = {S_A}$ if $i + j + k > 0$ and all integers in $\{ i,\,j,\,k\}$ have the same parity. (3) $ R_0^3 = {S_A}\backslash E$, where $\theta \in E$ iff $\theta$ satisfies one of the following three conditions: (i) $\theta$ moves precisely three elements. (ii) $ \theta$ moves precisely five elements. (iii) $\theta$ moves precisely seven elements and has order $ 12$. These results were announced in 1973 in [MO]. (1) and part of (2)(a) were generalized recently by Droste [D1, D2].
Borel measures and Hausdorff distance
Gerald
Beer;
Luzviminda
Villar
763-772
Abstract: In this article we study the restriction of Borel measures defined on a metric space $X$ to the nonempty closed subsets $\operatorname{CL} (X)$ of $X$, topologized by Hausdorff distance. We show that a $\sigma$-finite Radon measure is a Borel function on $ \operatorname{CL} (X)$, and characterize those $X$ for which each outer regular Radon measure on $ X$ is semicontinuous when restricted to $ \operatorname{CL} (X)$. A number of density theorems for Radon measures are also presented.
Automorphisms of hyperbolic dynamical systems and $K\sb 2$
Frank
Zizza
773-797
Abstract: Let $\sigma :\Sigma \to \Sigma $ be a subshift of finite type and $\operatorname{Aut} (\sigma )$ be the group of homeomorphisms of $\Sigma$ which commute with $\sigma$. In [Wl], Wagoner constructs an invariant for the group $\operatorname{Aut} (\sigma )$ using $K$-theoretic methods. Smooth hyperbolic dynamical systems can be modeled by subshifts of finite type over the nonwandering sets. In this paper we extend Wagoner's construction to produce an invariant on the group of homeomorphisms of a smooth manifold which commute with a fixed hyperbolic diffeomorphism. We then proceed to show that this dynamical invariant can be calculated (at least $\bmod 2$) from the homology groups of the manifold and the action of the diffeomorphism and the homeomorphisms on the homology groups.
Cauchy problem of hyperbolic conservation laws in multidimensional space with intersecting jump initial data
De Ning
Li
799-812
Abstract: Cauchy problem of hyperbolic conservation laws in multidimensional space is considered, where the initial data have several jump discontinuity surfaces which develop into shock fronts intersecting at a common submanifold. Local existence is proved, assuming compatible conditions and uniform stability. For isentropic flow in $ 2$-dimensional space, the interaction of two shock fronts and the nonexistence of three intersecting shock fronts are discussed.
A very singular solution of a quasilinear degenerate diffusion equation with absorption
L. A.
Peletier;
Jun Yu
Wang
813-826
Abstract: The object of this paper is to study the existence of a nonnegative solution of the Cauchy problem $\displaystyle {u_t} = \operatorname{div} (\vert\nabla u{\vert^{p - 2}}\nabla u) - {u^q},\qquad u(x,\,0) = 0\quad {\text{if}}\;x \ne 0,$ which is more singular at $ (0,\,0)$ than the fundamental solution of the corresponding equation without the absorption term.
Chains on CR manifolds and Lorentz geometry
Lisa K.
Koch
827-841
Abstract: We show that two nearby points of a strictly pseudoconvex CR manifold are joined by a chain. The proof uses techniques of Lorentzian geometry via a correspondence of Fefferman. The arguments also apply to more general systems of chain-like curves on CR manifolds.
Deformations of finite-dimensional algebras and their idempotents
M.
Schaps
843-856
Abstract: Let $B$ be a finite dimensional algebra over an algebraically closed field $K$. If we represent primitive idempotents by points and basis vectors in $ {e_i}B{e_j}$ by "arrows" from ${e_j}$ to ${e_i}$, then any specialization of the algebra acts on this directed graph by coalescing points. This implies that the number of irreducible components in the scheme parametrizing $n$-dimensional algebras is no less than the number of loopless directed graphs with a total of $ n$ vertices and arrows. We also show that the condition of having a distributive ideal lattice is open.
A remark on a theorem of Vo Van Tan
Mihnea
Colţoiu
857-859
Abstract: In this paper we consider the following problem: Let $(X,\,S)$ be a $1$-convex manifold with $1$-dimensional exceptional set $ S$. Does it follow that $ X$ is a Kähler manifold? Although this was answered in the affirmative by Vo Van Tan in two papers, we show that his proofs are wrong. It is also shown that the Kähler condition implies that any strongly pseudoconvex domain $D \Subset X$ is embeddable, i.e. can be realized as a closed analytic submanifold in some $ {{\mathbf{C}}^N} \times {{\mathbf{P}}_M}$. On the other hand it is known that under some additional assumptions on $ S$ ($S$ is not rational or $S \simeq {{\mathbf{P}}^1}$ and $ \operatorname{dim} X \ne 3$) it follows that $X$ is embeddable, in particular it is Kählerian.