A global theory of internal solitary waves in two-fluid systems
C. J.
Amick;
R. E. L.
Turner
431-484
Abstract: The problem analyzed is that of two-dimensional wave motion in a heterogeneous, inviscid fluid confined between two rigid horizontal planes and subject to gravity $g$. It is assumed that a fluid of constant density $ {\rho _ + }$ lies above a fluid of constant density ${\rho _ - } > {\rho _ + } > 0$ and that the system is nondiffusive. Progressing solitary waves, viewed in a moving coordinate system, can be described by a pair $ (\lambda ,w)$, where the constant $\lambda = g/{c^2}$, $c$ being the wave speed, and where $w(x,\eta ) + \eta$ is the height at a horizontal position $x$ of the streamline which has height $ \eta$ at $x = \pm \infty$. It is shown that among the nontrivial solutions of a quasilinear elliptic eigenvalue problem for $ (\lambda ,w)$ is an unbounded connected set in ${\mathbf{R}} \times (H_0^1 \cap {C^{0,1}})$. Various properties of the solution are shown, and the behavior of large amplitude solutions is analyzed, leading to the alternative that internal surges must occur or streamlines with vertical tangents must occur.
Hypoelliptic convolution equations in the space ${\scr K}'\sb e$
Dae Hyeon
Pahk
485-495
Abstract: We consider convolution equations in the space $ \exp ({e^{k\vert x\vert}})$ for some constant $k$. Our main results are to find conditions for convolution operators to be hypoelliptic in
Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees
C. J.
Ash
497-514
Abstract: We show that, under certain assumptions of recursiveness in $\mathfrak{A}$, the recursive structure $\mathfrak{A}$ is $ \Delta _\alpha ^0$-stable for $ \alpha < \omega _1^{CK}$ if and only if there is an enumeration of $\mathfrak{A}$ using a $\Sigma _\alpha ^0$ set of recursive ${\Sigma _\alpha }$ infinitary formulae and finitely many parameters from $ \mathfrak{A}$. This extends the results of [1]. To do this, we first obtain results concerning $ \Delta _\alpha ^0$ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a $\Delta _\alpha ^0$ function from the previous node and its label.
Explosion problems for symmetric diffusion processes
Kanji
Ichihara
515-536
Abstract: We discuss the explosion problem for a symmetric diffusion process. Hasminskii's idea cannot be applied to this case. Instead, the theory of Dirichlet forms is employed to obtain criteria for conservativeness and explosion of the process. The fundamental criteria are given in terms of the $\alpha$-equilibrium potentials and $ \alpha$-capacities of the unit ball centered at the origin. They are applied to obtain sufficient conditions on the coefficients of the infinitesimal generator for conservativeness and explosion.
Vector bundles on complex projective spaces and systems of partial differential equations. I
Peter F.
Stiller
537-548
Abstract: This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology ($ {H^1}$ in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.
The Euler characteristic as an obstruction to compact Lie group actions
Volker
Hauschild
549-578
Abstract: Actions of compact Lie groups on spaces $X$ with $ {H^{\ast}}(X,{\mathbf{Q}}) \cong {\mathbf{Q}}[{x_1}, \ldots ,{x_n}]/{I_0}$, $Q \in {I_0}$ a definite quadratic form, $\deg {x_i} = 2$, are considered. It is shown that the existence of an effective action of a compact Lie group $G$ on such an $X$ implies $\chi (X) \equiv O(\vert WG\vert)$, where $ \chi (X)$ is the Euler characteristic of $X$ and $\vert WG\vert$ means the order of the Weyl group of $ G$. Moreover the diverse symmetry degrees of such spaces are estimated in terms of simple cohomological data. As an application it is shown that the symmetry degree ${N_t}(G/T)$ is equal to $\dim G$ if $G$ is a compact connected Lie group and $T \subset G$ its maximal torus. Effective actions of compact connected Lie groups $K$ on $G/T$ with $ \dim K = \dim G$ are completely classified.
Around Effros' theorem
J. J.
Charatonik;
T.
Maćkowiak
579-602
Abstract: Some stronger versions of the Effros theorem are discussed in the paper, not only for homeomorphisms but also for some other mappings, e.g. for open ones. Equivalent formulations of the theorem are presented as the $\varepsilon$-push property and the existence of a so-called Effros' metric.
Invariants of the Lusternik-Schnirelmann type and the topology of critical sets
Mónica
Clapp;
Dieter
Puppe
603-620
Abstract: We introduce and study in detail generalizations of the notion of Lusternik-Schnirelmann category which give information about the topology of the critical set of a differentiable function. We also improve a result of T. Ganea about the equality of the strong category and the category (even in the classical case).
A characterization and another construction of Janko's group $J\sb 3$
Richard
Weiss
621-633
Abstract: Graphs $ \Gamma$ with the following properties are classified: (i) $\Gamma$ is $(G,s)$-transitive for some $s \geqslant 4$ and some group $G \leqslant \operatorname{aut} (\Gamma )$ such that each vertex stabilizer in $G$ is finite, (ii) $s \geqslant (g - 1)/2$, where $g$ is the girth of $\Gamma$, and (ii) $\Gamma$ is connected. A new construction of ${J_3}$ is given.
The Pontryagin maximum principle from dynamic programming and viscosity solutions to first-order partial differential equations
Emmanuel Nicholas
Barron;
Robert
Jensen
635-641
Abstract: We prove the Pontryagin Maximum Principle for the Lagrange problem of optimal control using the fact that the value function of the problem is the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. The proof here makes rigorous the formal proof of Pontryagin's principle known for at least three decades.
On the zeros of successive derivatives of even Laguerre-P\'olya functions
Li-Chien
Shen
643-652
Abstract: Using "method of steepest descent", we prove that the final set (in the sense of Polya) of a class of even Laguerre-Polya functions is the entire real axis.
On maximal functions and Poisson-Szeg\H o integrals
Juan
Sueiro
653-669
Abstract: We study a class of maximal functions of Hardy-Littlewood type defined on spaces of homogeneous type and we give necessary and sufficient conditions for the corresponding maximal operators to be of weak type $(1,1)$. As a consequence we show that Poisson-Szegö integrals of ${L^p}$ functions possess certain boundary limits which are not implied by Korányi's theorem.
Crossed products and inner actions of Hopf algebras
Robert J.
Blattner;
Miriam
Cohen;
Susan
Montgomery
671-711
Abstract: This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if $\pi :H \to \overline H$ is a Hopf algebra epimorphism which is split as a coalgebra map, then $ H$ is algebra isomorphic to $A{\char93 _\sigma }H$, a crossed product of $H$ with the left Hopf kernel $A$ of $\pi$. Given any crossed product $A{\char93 _\sigma }H$ with $H$ (weakly) inner on $A$, then $ A{\char93 _\sigma }H$ is isomorphic to a twisted product ${A_\tau }[H]$ with trivial action. Finally, if $ H$ is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of $A$ implies that of $ A{\char93 _\sigma }H$; in particular this is true if the (weak) action of $ H$ is inner.
Construction of high-dimensional knot groups from classical knot groups
Magnhild
Lien
713-722
Abstract: In this paper we study constructions of high dimensional knot groups from classical knot groups. We study certain homomorphic images of classical knot groups. Specifically, let $ K$ be a classical knot group and $w$ any element in $K$. We are interested in the quotient groups $ G$ obtained by centralizing $w$, i.e. $G = K/[K,w]$, and ask whether $ G$ is itself a knot group. For certain $K$ and $w$ we show that $G$ can be realized as the group of a knotted $ 3$-sphere in $ 5$-space, but $ G$ is not realizable by a $ 2$-sphere in $ 4$-space. By varying $ w$, we also obtain quotients that are groups of knotted $2$-spheres in $4$-space, but they cannot be realized as the groups of classical knots. We have examples of quotients $K/[K,w]$ that have nontrivial second homology. Hence these groups cannot be realized as knot groups of spheres in any dimension. However, we show that these groups are groups of knotted tori in ${S^4}$.
Doubly sliced knots which are not the double of a disk
L.
Smolinsky
723-732
Abstract: In this paper we show that double disk knots can be distinguished from general doubly sliced knots in dimensions $4n + 1$.
Singularly perturbed quadratically nonlinear Dirichlet problems
Albert J.
DeSanti
733-746
Abstract: The Dirichlet problem for singularly perturbed elliptic equations of the form $ \varepsilon \Delta u = A({\mathbf{x}},u)\nabla u \cdot \nabla u + {\mathbf{B}}({\mathbf{x}},u) \cdot \nabla u + C({\mathbf{x}},u)$ in $\Omega \in {E^n}$ is studied. Under explicit and easily checked conditions, solutions are shown to exist for $\varepsilon$ sufficiently small and to exhibit specified asymptotic behavior as $\varepsilon \to 0$. The results are obtained using a method based on the theory of partial differential inequalities.
Diffuse sequences and perfect $C\sp \ast$-algebras
Charles A.
Akemann;
Joel
Anderson;
Gert K.
Pedersen
747-762
Abstract: The concept of a diffuse sequence in a $ {C^{\ast}}$-algebra is introduced and exploited to complete the classification of separable, perfect $ {C^{\ast}}$-algebras. A ${C^{\ast}}$-algebra is separable and perfect exactly when the closure of the pure state space consists entirely of atomic states.
Tameness and local normal bases for objects of finite Hopf algebras
Lindsay N.
Childs;
Susan
Hurley
763-778
Abstract: Let $R$ be a commutative ring, $ S$ an $R$-algebra, $H$ a Hopf $R$algebra, both finitely generated and projective as $ R$-modules, and suppose $ S$ is an $H$-object, so that $ {H^{\ast}} = {\operatorname{Hom} _R}(H,R)$ acts on $S$ via a measuring. Let $I$ be the space of left integrals of ${H^{\ast}}$. We say $S$ has normal basis if $S \cong H$ as $ {H^{\ast}}$modules, and $ S$ has local normal bases if ${S_p} \cong {H_p}$ as $H_p^{\ast}$-modules for all prime ideals $ p$ of $R$. When $R$ is a perfect field, $H$ is commutative and cocommutative, and certain obvious necessary conditions on $S$ hold, then $S$ has normal basis if and only if $IS = R = {S^{{H^{\ast}}}}$. If $ R$ is a domain with quotient field $K$, $H$ is cocommutative, and $L = S \otimes {}_RK$ has normal basis as $({H^{\ast}} \otimes K)$-module, then $ S$ has local normal bases if and only if $ IS = R = {S^{{H^{\ast}}}}$.
Minimal submanifolds of a sphere with bounded second fundamental form
Hillel
Gauchman
779-791
Abstract: Let $h$ be the second fundamental form of an $n$-dimensional minimal submanifold $M$ of a unit sphere ${S^{n + p}}(p \geqslant 2)$, $S$ be the square of the length of $ h$, and $\sigma (u) = \vert\vert h(u,u)\vert{\vert^2}$ for any unit vector $u \in TM$. Simons proved that if $S \leqslant n/(2 - 1/p)$ on $M$, then either $ S \equiv 0$, or $S \equiv n/(2 - 1/p)$. Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying $S \equiv n/(2 - 1/p)$. In this paper the analogous results for $ \sigma (u)$ are obtained. It is proved that if $\sigma (u) \leqslant \tfrac{1} {3}$, then either $\sigma (u) \equiv 0$, or $\sigma (u) \equiv \tfrac{1} {3}$. All minimal submanifolds satisfying $\sigma (u)$ are determined. A stronger result is obtained if $M$ is odd-dimensional.
On the Hausdorff dimension of some graphs
R. Daniel
Mauldin;
S. C.
Williams
793-803
Abstract: Consider the functions $\displaystyle {W_b}(x) = \sum\limits_{n = - \infty }^\infty {{b^{ - \alpha n}}[\Phi ({b^n}x + {\theta _n}) - \Phi ({\theta _n})],} $ where $b > 1$, $0 < \alpha < 1$, each ${\theta _n}$ is an arbitrary number, and $ \Phi$ has period one. We show that there is a constant $C > 0$ such that if $b$ is large enough, then the Hausdorff dimension of the graph of ${W_b}$ is bounded below by $2 - \alpha - (C/\ln b)$. We also show that if a function $f$ is convex Lipschitz of order $\alpha$, then the graph of $ f$ has $\sigma $-finite measure with respect to Hausdorff's measure in dimension $2 - \alpha$. The convex Lipschitz functions of order $ \alpha$ include Zygmund's class $ {\Lambda _\alpha }$. Our analysis shows that the graph of the classical van der Waerden-Tagaki nowhere differentiable function has $ \sigma$-finite measure with respect to $ h(t) = t/\ln (1/t)$.