Erratum to: ``Remarks on classical invariant theory''
Roger
Howe
An invariant of regular isotopy
Louis H.
Kauffman
417-471
Abstract: This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted ${L_K}$ for a link $K$, and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2. $ {L_{[{\text{unk}}]}} = 1$. 3. $ {L_{[{\text{unk}}]}} = aL,\qquad {L_{[{\text{unk}}]}} = {a^{ - 1}}L$. 4. $ {L_{[{\text{unk}}]}} + {L_{[{\text{unk]}}}} = z({L_{[{\text{unk]}}}} + {L_{[{\text{unk]}}}})$. Small diagrams indicate otherwise identical parts of larger diagrams. Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from $L$ by writhe-normalization.
Generalized balanced tournament designs
E. R.
Lamken
473-490
Abstract: A generalized balanced tournament design, $GBTD(n,k)$, defined on a $kn$-set $V$, is an arrangement of the blocks of a $(kn,k,k - 1)$-$BIBD$ defined on $V$ into an $ n \times (kn - 1)$ array such that (1) every element of $V$ is contained in precisely one cell of each column, and (2) every element of $V$ is contained in at most $k$ cells of each row. In this paper, we introduce $ GBTD(n,k)s$ and describe connections between these designs and several other types of combinatorial designs. We also show how to use $GBTDs$ to construct resolvable, near resolvable, doubly resolvable and doubly near resolvable $ BIBDs$.
Two differential-difference equations arising in number theory
Ferrell S.
Wheeler
491-523
Abstract: We survey many old and new results on solutions of the following pair of adjoint differential-difference equations: (1) $\displaystyle \sum\limits_{\begin{array}{*{20}{c}} {1 < n \leqslant x} {{P_2... ... x)}^\alpha }} \qquad (x \to \infty ,\;u \geqslant 1,\;\alpha \in {\mathbf{R}})$ where $ {P_1}(n)$ and $ {P_2}(n)$ are the first and second largest prime divisors of $n$ and $f(u)$ satisfies (2) with $(a,b) = (1 - \alpha , - 1)$.
Large deviations in dynamical systems
Lai-Sang
Young
525-543
Abstract: We prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics, rate of escape problems, and to shift spaces.
On infinite-dimensional manifold triples
Katsuro
Sakai;
Raymond Y.
Wong
545-555
Abstract: Let $Q$ denote the Hilbert cube $ {[ - 1,1]^\omega },\;s = {( - 1,1)^\omega }$ the pseudo-interior of $Q,\;\Sigma = \{ ({x_i}) \in s\vert\sup \vert{x_i}\vert < 1\}$ and $\sigma = \{ ({x_i}) \in s\vert{x_i} = 0\;{\text{except for finitely many}}\;i\} $. A triple $(X,M,N)$ of separable metrizable spaces is called a $ (Q,\Sigma ,\sigma )$- (or $(s,\Sigma ,\sigma )$-)manifold triple if it is locally homeomorphic to $(Q,\Sigma ,\sigma )$ (or $(s,\Sigma ,\sigma )$). In this paper, we study such manifold triples and give some characterizations.
Bounded polynomial vector fields
Anna
Cima;
Jaume
Llibre
557-579
Abstract: We prove that, for generic bounded polynomial vector fields in ${{\mathbf{R}}^n}$ with isolated critical points, the sum of the indices at all their critical points is ${( - 1)^n}$. We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in ${{\mathbf{R}}^2}$. We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane.
Invariant Radon transforms on a symmetric space
Jeremy
Orloff
581-600
Abstract: Injectivity and support theorems are proved for a class of Radon transforms, $ {R_\mu }$, for $ \mu$ a smooth family of measures defined on a certain space of affine planes in ${\mathbb{X}_0}$, where ${\mathbb{X}_0}$ is the tangent space, of a Riemannian symmetric space of rank one. The transforms are defined by integrating against $\mu$ over these planes. We show that if $ {R_\mu }f$ is supported inside a ball of radius $R$ then so is $f$. This is true for $f \in L_c^2({\mathbb{X}_0})$ or
Ergodic and mixing properties of equilibrium measures for Markov processes
Enrique D.
Andjel
601-614
Abstract: Let $S(t)$ be the semigroup corresponding to a Markov process on a metric space $X$. Suppose $S(t)$ commutes with a homeomorphism $T$ of $X$. We prove that under certain conditions, an equilibrium measure for the process is ergodic under $ T$. We also show that, under stronger conditions this measure must be mixing under $ T$. Several applications of these results are given.
Domain-independent upper bounds for eigenvalues of elliptic operators
Stephen M.
Hook
615-642
Abstract: Let $\Omega \subseteq {\mathbb{R}^m}$ be a bounded open set, $ \partial \Omega$ its boundary and $\Delta$ the Laplacian on ${\mathbb{R}^m}$. Consider the elliptic differential equation: (1) $\displaystyle - \Delta u = \lambda u\quad {\text{in}}\;\Omega ;\qquad u = 0\quad {\text{on}}\;\partial \Omega .$ It is known that the eigenvalues, ${\lambda _i}$, of (1) satisfy (2) $\displaystyle \sum\limits_{i = 1}^n {\frac{{{\lambda _i}}} {{{\lambda _{n + 1}} - {\lambda _i}}}} \geqslant \frac{{mn}} {4}$ provided that ${\lambda _{n + 1}} > {\lambda _n}$. In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.
Hamilton-Jacobi equations with state constraints
I.
Capuzzo-Dolcetta;
P.-L.
Lions
643-683
Abstract: In the present paper we consider Hamilton-Jacobi equations of the form $H(x,u,\nabla u) = 0,\;x \in \Omega$, where $ \Omega$ is a bounded open subset of ${R^n},H$ is a given continuous real-valued function of $(x,s,p) \in \Omega \times R \times {R^n}$ and $ \nabla u$ is the gradient of the unknown function $u$. We are interested in particular solutions of the above equation which are required to be supersolutions, in a suitable weak sense, of the same equation up to the boundary of $\Omega$. This requirement plays the role of a boundary condition. The main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system, as well from related questions in constrained geodesics.
Hausdorff dimension of harmonic measures on negatively curved manifolds
Yuri
Kifer;
François
Ledrappier
685-704
Abstract: We show by probabilistic means that harmonic measures on manifolds, whose curvature is sandwiched between two negative constants have positive Hausdorff dimensions. A lower bound for harmonic measures of open sets is derived, as well. We end with the results concerning the Hausdorff dimension of harmonic measures on universal covers of compact negatively curved manifolds.
Symmetric derivates, scattered, and semi-scattered sets
Chris
Freiling
705-720
Abstract: We call a set right scattered (left scattered) if every nonempty subset contains a point isolated on the right (left). We establish the following monotonicity theorem for the symmetric derivative. If a real function $f$ has a nonnegative lower symmetric derivate on an open interval $I$, then there is a nondecreasing function $ g$ such that $f(x) > g(x)$ on a right scattered set and $f(x) < g(x)$ on a left scattered set. Furthermore, if $R$ is any right scattered set and $L$ is any left scattered set disjoint with $R$, then there is a function which is positive on $R$, negative on $L$, zero otherwise, and which has a zero lower symmetric derivate everywhere. We obtain some consequence including an analogue of the Mean Value Theorem and a new proof of an old theorem of Charzynski.
A conformal inequality related to the conditional gauge theorem
Terry R.
McConnell
721-733
Abstract: We prove the inequality $h{(x)^{ - 1}}G(x,y)h(y) \leqslant cG(x,y) + c$, where $G$ is the Green function of a plane domain $ D,\;h$ is positive and harmonic on $D$, and $c$ is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains $c$ may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant $c$ in the above inequality is finite.
Ergodicity of finite-energy diffusions
Timothy C.
Wallstrom
735-747
Abstract: Recently, the existence of a class of diffusion processes with highly singular drift coefficients has been established under the assumption of "finite energy." The drift singularities of these diffusions greatly complicate their ergodicity properties; indeed, they can render the diffusion nonergodic. In this paper, a method is given for estimating the relaxation time of a finite-energy diffusion, when it is ergodic. These results are applied to show that the set of $\operatorname{spin} - \tfrac{1} {2}$ diffusions of stochastic mechanics is uniformly ergodic.
The stochastic mechanics of the Pauli equation
Timothy C.
Wallstrom
749-762
Abstract: In stochastic mechanics, the Bopp-Haag-Dankel diffusions on $ {\mathbb{R}^3} \times \operatorname{SO} (3)$ are used to represent particles with spin. Bopp and Haag showed that in the limit as the particle's moment of inertia $I$ goes to zero, the solutions of the Bopp-Haag equations converge to that of the regular Pauli equation. Nelson has conjectured that in the same limit, the projections of the Bopp-Haag-Dankel diffusions onto $ {\mathbb{R}^3}$ converge to a Markovian limit process. In this paper, we prove this conjecture for spin $\operatorname{spin} \;\tfrac{1} {2}$ and regular potentials, and identify the limit process as the diffusion naturally associated with the solution to the regular Pauli equation.
Metrizable spaces where the inductive dimensions disagree
John
Kulesza
763-781
Abstract: A method for constructing zero-dimensional metrizable spaces is given. Using generalizations of Roy's technique, these spaces can often be shown to have positive large inductive dimension. Examples of $ {\mathbf{N}}$-compact, complete metrizable spaces with $\operatorname{ind} = 0$ and $\operatorname{Ind} = 1$ are provided, answering questions of Mrowka and Roy. An example with weight $\mathfrak{c}$ and positive Ind such that subspaces with smaller weight have $\operatorname{Ind} = 0$ is produced in ZFC. Assuming an additional axiom, for each cardinal $\lambda$ a space of positive Ind with all subspaces with weight less than $\lambda$ strongly zero-dimensional is constructed.
Simple Lie algebras of characteristic $p$ with dependent roots
Georgia
Benkart;
J. Marshall
Osborn
783-807
Abstract: We investigate finite dimensional simple Lie algebras over an algebraically closed field $ {\mathbf{F}}$ of characteristic $p \geqslant 7$ having a Cartan subalgebra $ H$ whose roots are dependent over $ {\mathbf{F}}$. We show that $H$ must be one-dimensional or for some root $\alpha$ relative to $H$ there is a $1$-section $ {L^{(\alpha )}}$ such that the core of $ {L^{(\alpha )}}$ is a simple Lie algebra of Cartan type $H{(2:\underline m :\Phi )^{(2)}}$ or $W(1:\underline n )$ for some $n > 1$. The results we obtain have applications to studying the local behavior of simple Lie algebras and to classifying simple Lie algebras which have a Cartan subalgebra of dimension less than $ p - 2$.
Immersions of positively curved manifolds into manifolds with curvature bounded above
Nadine L.
Menninga
809-821
Abstract: Let $M$ be a compact, connected, orientable Riemannian manifold of dimension $n - 1 \geqslant 2$, and let $x$ be an isometric immersion of $ M$ into an $n$-dimensional Riemannian manifold $ N$. Let $K$ denote sectional curvature and $ i$ denote the injectivity radius. Assume, for some constant positive constant $ c$, that $ K(N) \leqslant 1/(4{c^2}),\quad 1/{c^2} \leqslant K(M)$, and $\pi c \leqslant i(N)$. Then the radius of the smallest $N$-ball containing $x(M)$ is less than $\tfrac{1} {2}\pi c$ and $x$ is in fact an imbedding of $M$ into $N$, whose image bounds a convex body.