Plane autonomous systems with rational vector fields
Harold E.
Benzinger
465-483
Abstract: The differential equation $\dot z= R(z)$ is studied, where $ R$ is an arbitrary rational function. It is shown that the Riemann sphere is decomposed into finitely many open sets, on each of which the flow is analytic and, in each time direction, there is common long-term behavior. The boundaries of the open sets consist of those points for which the flow fails to be analytic in at least one time direction. The main idea is to express the differential equation as a continuous Newton method $\dot z = - f(z)/f^{\prime}\;(z)$, where $f$ is an analytic function which can have branch points and essential singularities. A method is also given for the computer generation of phase plane portraits which shows the correct time parametrization and which is noniterative, thereby avoiding the problems associated with the iteration of rational functions.
Fixed set systems of equivariant infinite loop spaces
Steven R.
Costenoble;
Stefan
Waner
485-505
Abstract: We develop machinery enabling us to show that suitable $G$-spaces, including the equivariant version of $BF$, are equivariant infinite loop spaces. This involves a "recognition principle" for systems of spaces which behave formally like the system of fixed sets of a $G$-space; that is, we give a necessary and sufficient condition for such a system to be equivalent to the fixed set system of an equivariant infinite loop space. The advantage of using the language of fixed set systems is that one can frequently replace the system of fixed sets of an actual $G$-space by an equivalent formal system which is considerably simpler, and which admits the requisite geometry necessary for delooping. We also apply this machinery to construct equivariant Eilenberg-Mac Lane spaces corresponding to Mackey functors.
Blow-up of straightening-closed ideals in ordinal Hodge algebras
Winfried
Bruns;
Aron
Simis;
Ngô Viêt
Trung
507-528
Abstract: We study a class of ideals $I$ in graded ordinal Hodge algebras $ A$. These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for $A/{I^n}$ which, under certain conditions, is sharp for $n$ large. Frequently one observes that ${I^n}= {I^{(n)}}$. Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".
Lie flows of codimension $3$
E.
Gallego;
A.
Reventós
529-541
Abstract: We study the following realization problem: given a Lie algebra of dimension $3$ and an integer $q,0 \leq q \leq 3$, is there a compact manifold endowed with a Lie flow transversely modeled on $\mathcal{G}$ and with structural Lie algebra of dimension $q$? We give here a quite complete answer to this problem but some questions remain still open $({\text{cf.}}\;\S2$.
Towards a functional calculus for subnormal tuples: the minimal normal extension
John B.
Conway
543-567
Abstract: In this paper the study of a functional calculus for subnormal $ n$-tuples is initiated and the minimal normal extension problem for this functional calculus is explored. This problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved. An analogous uniform approximation problem is also explored. In addition these general results are applied together with The Area and the The Coarea Formula from Geometric Measure Theory to operators on Bergman spaces and to the tensor product of two subnormal operators. The minimal normal extension of the tensor product of the Bergman shift with itself is completely determined.
Vanishing of $H\sp 2\sb w(M,K(H))$ for certain finite von Neumann algebras
Florin
Rădulescu
569-584
Abstract: We prove the vanishing of the second Hochschild cohomology group $H_w^2\,(M,K(H))$, whenever $M \subset B(H)$ is a finite countably decomposable von Neumann algebra not containing a non $ \Gamma$-factor or a factor without Cartan subalgebra as a direct summand. Here $ H$ is a Hubert space, and $ K(H)$ the compact operators.
On the distance between unitary orbits of weighted shifts
Laurent
Marcoux
585-612
Abstract: In this paper, we consider invertible bilateral weighted shift operators acting on a complex separable Hilbert space $\mathcal{H}$. They have the property that there exist a constant $\tau > 0$ and an orthonormal basis $ {\{ {{e_i}} \}_{i \in \mathbb{Z}}}$ for $ \mathcal{H}$ with respect to which a shift $V$ acts by $W{e_i}= {w_i}{e_{i + 1}},i \in \mathbb{Z}$ and $ {\mathbf{\vert}}{w_i}{\mathbf{\vert}} \geq \tau$. The equivalence class $\mathcal{U}(W)= \{ {U^{\ast}}\;WU:U \in \mathcal{B}(\mathcal{H}),U\;{\text{unitary}}\}$ of weighted shifts with weight sequence (with respect to the basis $ {\{ {U^{\ast}}{e_i}\} _{i \in \mathbb{Z}}}$ for $ \mathcal{H})$ identical to that of $W$ forms the unitary orbit of $W$. Given two shifts $W$ and $V$, one can define a distance $ d(\mathcal{U}(V),\mathcal{U}(W))= \inf \{\parallel \,X - Y\parallel :X \in \mathcal{U}(V),Y \in \mathcal{U}(W)\} $ between the unitary orbits of $W$ and $V$. We establish numerical estimates for upper and lower bounds on this distance which depend upon information drawn from finite dimensional restrictions of these operators.
Monogenic differential calculus
F.
Sommen
613-632
Abstract: In this paper we study differential forms satisfying a Dirac type equation and taking values in a Clifford algebra. For them we establish a Cauchy representation formula and we compute winding numbers for pairs of nonintersecting cycles in $ {\mathbb{R}^m}$ as residues of special differential forms. Next we prove that the cohomology spaces for the complex of monogenic differential forms split as direct sums of de Rham cohomology spaces. We also study duals of spaces of monogenic differential forms, leading to a general residue theory in Euclidean space. Our theory includes the one established in our paper [11] and is strongly related to certain differential forms introduced by Habetha in [4].
Harmonic measure versus Hausdorff measures on repellers for holomorphic maps
Anna
Zdunik
633-652
Abstract: This paper is a continuation of a joint paper of the author with F. Przytycki and M. Urbański. We study a harmonic measure on a boundary of so-called repelling boundary domain; an important example is a basin of a sink for a rational map. Using the results of the above-mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval, or else the harmonic measure is singular with respect to the Hausdorff measure corresponding to the function ${\phi _c}(t)= t\;\exp \,\left(c\sqrt {\log \frac{1} {t}\,\log \,\log \,\log } \frac{1} {t}\right)$ for some $c > 0$.
Recursive linear orders with incomplete successivities
Rodney G.
Downey;
Michael F.
Moses
653-668
Abstract: A recursive linear order is said to have intrinsically complete successivities if, in every recursive copy, the successivities form a complete set. We show (Theorem 1) that there is a recursive linear order with intrinsically complete successivities but (Theorem 2) that this cannot be a discrete linear oder. We investigate the related issues of intrinsically non-low and non-semilow successivities in discrete linear orders. We show also (Theorem 3) that no recursive linear order has intrinsically $wtt$-complete successivities.
The two-sided Stefan problem with a spatially dependent latent heat
Terry R.
McConnell
669-699
Abstract: We prove existence and uniqueness of solutions to a problem which generalizes the two-sided Stefan problem. The initial temperature distribution and variable latent heat may be given by positive measures rather than point functions, and the free boundaries which result are essentially arbitrary increasing functions which need not exhibit any degree of smoothness in general. Nevertheless, the solutions are "classical" in the sense that all derivatives and boundary values have the classical interpretation. We also study connections with the Skorohod embedding problem of probability theory and with a general class of optimal stopping problems.
Varieties of periodic attractor in cellular automata
Mike
Hurley
701-726
Abstract: We apply three alternate definitions of "attractor" to cellular automata. Examples are given to show that using the different definitions can give different answers to the question "Does this cellular automaton have a periodic attractor?" The three definitions are the topological notion of attractor as used by C. Conley, a more measure-theoretic version given by J. Milnor, and a variant of Milnor's definition that is based on the concept of the "center of attraction" of an orbit. Restrictions on the types of periodic orbits that can be periodic attractors for cellular automata are described. With any of these definitions, a cellular automaton has at most one periodic attractor. Additionally, if Conley's definition is used, then a periodic attractor must be a fixed point. Using Milnor's definition, each point on a periodic attractor must be fixed by all shifts, so the number of symbols used is an upper bound on the period; whether the actual upper bound is $1$ is unknown. With the third definition this restriction is removed, and examples are given of onedimensional cellular automata on three symbols that have finite "attractors" of arbitrarily large size (with the third definition, a finite attractor is not necessarily a single periodic orbit).
Classification of balanced sets and critical points of even functions on spheres
Charles V.
Coffman
727-747
Abstract: The Lyusternik-Schnirelman approach to the study of critical points of even functionals on the sphere ${S^N}$ employs min-max or max-min principles whose formulation uses a numerical invariant that is defined for compact balanced subsets of ${S^N}$. The Krasnosel'skii genus is an example. Here we study a general class of such invariants (which is quite large) with particular attention to the following questions: formulation of dual variational principles, multiplicity results for critical points, and determination of the Morse index of nondegenerate critical points.
Invariant arcs, Whitney levels, and Kelley continua
M.
van de Vel
749-771
Abstract: As an application of convexity in spaces of arcs, three results of a somewhat different nature have been obtained. The first one gives some simple conditions under which an arc of a semilattice is mapped back into itself by an order-preserving function. The second result states that certain Whitney levels are absolute retracts. Finally, Kelley continua are characterized by what we call approximating coselections.
A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map
Judy
Kennedy
773-793
Abstract: A powerful theorem and construction of Wayne Lewis are used to build two homeomorphisms on the pseudoarc, each of which is semiconjugate to the tent map on the unit interval. The first homeomorphism is transitive, thus answering a question of Marcy Barge as to whether such homeomorphisms exist. The second homeomorphism admits wandering points. Also, it is proven that any homeomorphism on the pseudoarc that is semiconjugate to the tent map and is irreducible with respect to the semiconjugacy must either be transitive or admit wandering points.
Representations of knot groups in ${\rm SU}(2)$
Eric Paul
Klassen
795-828
Abstract: This paper is a study of the structure of the space $R(K)$ of representations of classical knot groups into $ {\text{SU}}(2)$. Let $ \hat R(K)$ equal the set of conjugacy classes of irreducible representations. In $\S I$, we interpret the relations in a presentation of the knot group in terms of the geometry of ${\text{SU}}(2)$; using this technique we calculate $ \hat R(K)$ for $ K$ equal to the torus knots, twist knots, and the Whitehead link. We also determine a formula for the number of binary dihedral representations of an arbitrary knot group. We prove, using techniques introduced by Culler and Shalen, that if the dimension of $\hat R(K)$ is greater than $1$, then the complement in ${S^3}$ of a tubular neighborhood of $ K$ contains closed, nonboundary parallel, incompressible surfaces. We also show how, for certain nonprime and doubled knots, $ \hat R(K)$ has dimension greater than one. In $\S II$, we calculate the Zariski tangent space, ${T_\rho }(R(K))$, for an arbitrary knot $ K$, at a reducible representation $\rho$, using a technique due to Weil. We prove that for all but a finite number of the reducible representations, $\dim {T_\rho }(R(K))= 3$. These nonexceptional representations possess neighborhoods in $ R(K)$ containing only reducible representations. At the exceptional representations, which correspond to real roots of the Alexander polynomial, $\dim {T_\rho }(R(K)) = 3 + 2k$ for a positive integer $ k$. In those examples analyzed in this paper, these exceptional representations can be expressed as limits of arcs of irreducible representations. We also give an interpretation of these "extra" tangent vectors as representations in the group of Euclidean isometries of the plane.
On the existence and uniqueness of positive solutions for competing species models with diffusion
E. N.
Dancer
829-859
Abstract: In this paper, we consider strictly positive solutions of competing species systems with diffusion under Dirichlet boundary conditions. We obtain a good understanding of when strictly positive solutions exist, obtain new nonuniqueness results and a number of other results, showing how complicated these equations can be. In particular, we consider how the shape of the underlying domain affects the behaviour of the equations.
Actions of loop groups on harmonic maps
M. J.
Bergvelt;
M. A.
Guest
861-886
Abstract: We describe a general framework in which subgroups of the loop group $ \Lambda G{l_n}\mathbb{C}$ act on the space of harmonic maps from ${S^2}$ to $ G{l_n}\mathbb{C}$. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of $\Lambda G{l_n}\mathbb{C}$, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action ("completion") gives rise to her fundamental process of "adding a uniton". It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from ${S^2}$ to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton".
An equivariant torus theorem for involutions
W. H.
Holzmann
887-906
Abstract: A complete classification is given for equivariant surgery on incompressible tori with respect to involutions with possible $ 1$- or $ 2$-dimensional fixed sets.
Weighted inequalities for maximal functions associated with general measures
Kenneth F.
Andersen
907-920
Abstract: For certain positive Borel measures $\mu$ on $ {\mathbf{R}}$ and for $ {T_\mu }$ any of three naturally associated maximal function operators of Hardy-Littlewood type, the weight pairs $(u,\upsilon)$ for which ${T_\mu }$ is of weak type $(p,p),1 \leq p < \infty $, and of strong type $(p,p),1 < p < \infty$, are characterized. Only minimal assumptions are placed on $ \mu$; in particular, $ \mu$ need not satisfy a doubling condition nor need it be continuous.