Connected simple systems and the Conley index of isolated invariant sets
Dietmar
Salamon
1-41
Abstract: The object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows. Simple, explicit formulae are derived for the maps which play a central role in the theory. The presentation is self-contained.
Subellipticity of the $\bar \partial$-Neumann problem on nonpseudoconvex domains
Lop-Hing
Ho
43-73
Abstract: Following the work of Kohn, we give a sufficient condition for subellipticity of the $ \overline \partial$-Neumann problem for not necessarily pseudoconvex domains. We define a sequence of ideals of germs and show that if $1$ is in any of them, then there is a subelliptic estimate. In particular, we prove subellipticity under some specific conditions for $n - 1$ forms and for the case when the Levi-form is diagonalizable. For the necessary conditions, we use another method to prove that there is no subelliptic estimate for $q$ forms if the Levi-form has $n - q - 1$ positive eigenvalues and $ q$ negative eigenvalues. This was proved by Derridj. Using similar techniques, we prove a necessary condition for subellipticity for some special domains. Finally, we give a remark to Catlin's theorem concerning the hypoellipticity of the $\overline \partial$-Neumann problem in the case of nonpseudoconvex domains.
On Lipschitz homogeneity of the Hilbert cube
Aarno
Hohti
75-86
Abstract: The main contribution of this paper is to prove the conjecture of [Vä] that the Hilbert cube $Q$ is Lipschitz homogeneous for any metric $ {d_s}$, where $ s$ is a decreasing sequence of positive real numbers ${s_k}$ converging to zero, $ {d_s}(x,y) = \sup \{ {s_k}\vert{x_k} - {y_k}\vert:k \in N\}$, and $ R(s) = \sup \{ {s_k}/{s_{k + 1}}:k \in N\} < \infty $. In addition to other results, we shall show that for every Lipschitz homogeneous compact metric space $X$ there is a constant $\lambda < \infty$ such that $X$ is homogeneous with respect to Lipschitz homeomorphisms whose Lipschitz constants do not exceed $\lambda$. Finally, we prove that the hyperspace $ {2^I}$ of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.
Small zeros of quadratic forms
Wolfgang M.
Schmidt
87-102
Abstract: We give upper and lower bounds for zeros of quadratic forms in the rational, real and $p$-adic fields. For example, given $r > 0$, $s > 0$, there are infinitely many forms $\mathfrak{F}$ with integer coefficients in $r + s$ variables of the type $ (r,s)$ (i.e., equivalent over ${\mathbf{R}}$ to $ X_1^2 + \cdots + X_r^2 - X_{r + 1}^2 - \cdots - X_{r + s}^2$ such that every nontrivial integer zero $ {\mathbf{x}}$ has $ \vert{\mathbf{x}}\vert \gg {F^{r/2s}}$, where $F$ is the maximum modulus of the coefficients of $ \mathfrak{F}$.
The traction problem for incompressible materials
Y. H.
Wan
103-119
Abstract: The traction problem for incompressible materials is treated as a bifurcation problem, where the applied loads are served as parameters. We take both the variational approach and the classical power series approach. The variational approach provides a natural, unified way of looking at this problem. We obtain a count of the number of equilibria together with the determination of their stability. In addition, it also lays down the foundation for the Signorini-Stoppelli type computations. We find second order sufficient conditions for the existence of power series solutions. As a consequence, the linearization stability follows, and it clarifies in some sense the role played by the linear elasticity in the context of the nonlinear elasticity theory. A systematic way of calculating the power series solution is also presented.
Descriptive complexity of function spaces
D.
Lutzer;
J.
van Mill;
R.
Pol
121-128
Abstract: In this paper we show that ${C_\pi }(X)$, the set of continuous, real-valued functions on $X$ topologized by the pointwise convergence topology, can have arbitrarily high Borel or projective complexity in $ {{\mathbf{R}}^X}$ even when $X$ is a countable regular space with a unique limit point. In addition we show how to construct countable regular spaces $X$ for which $ {C_\pi }(X)$ lies nowhere in the projective hierarchy of the complete separable metric space $ {{\mathbf{R}}^X}$.
Propagation estimates for Schr\"odinger-type operators
Arne
Jensen
129-144
Abstract: Propagation estimates for a Schrödinger-type operator are obtained using multiple commutator techniques. A new method is given for obtaining estimates for powers of the resolvent. As an application, micro-local propagation estimates are obtained for two-body Schrödinger operators with smooth long-range potentials.
The Cauchy problem for $u\sb t=\Delta u\sp m$ when $0<m<1$
Miguel A.
Herrero;
Michel
Pierre
145-158
Abstract: This paper deals with the Cauchy problem for the nonlinear diffusion equation $\partial u/\partial t - \Delta (u\vert u{\vert^{m - 1}}) = 0$ on $(0,\infty ) \times {{\mathbf{R}}^N},u(0, \cdot ) = {u_0}$ when $0 < m < 1$ (fast diffusion case). We prove that there exists a global time solution for any locally integrable function ${u_0}$: hence, no growth condition at infinity for $ {u_0}$ is required. Moreover the solution is shown to be unique in that class. Behavior at infinity of the solution and $ L_{\operatorname{loc} }^\infty$-regularizing effects are also examined when $ m \in (\max \{ (N - 2)/N,0\} ,1)$.
Probabilistic square functions and a priori estimates
Andrew G.
Bennett
159-166
Abstract: We obtain a priori estimates for Riesz transforms and their variants, that is, estimates with bounds independent of the dimension of the space and/or the nature of the boundary. The key to our results is to give probabilistic definitions which do not depend on the geometry of the situation for the transformations in question. We then use probabilistic square functions to prove our a priori estimates.
Symmetric positive systems with boundary characteristic of constant multiplicity
Jeffrey
Rauch
167-187
Abstract: The theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity. No such hypothesis is needed on a neighborhood of the boundary. Both regularity theorems and mixed initial boundary value problems are discussed. Many classical ideas are sharpened in the process.
Baire sets of $k$-parameter words are Ramsey
Hans Jürgen
Prömel;
Bernd
Voigt
189-201
Abstract: We show that Baire sets of $k$-parameter words are Ramsey. This extends a result of Carlson and Simpson, A dual form of Ramsey's theorem, Adv. in Math. 53 (1984), 265-290. Employing the method established therefore, we derive analogous results for Dowling lattices and for ascending $k$-parameter words.
Constant term identities extending the $q$-Dyson theorem
D. M.
Bressoud;
I. P.
Goulden
203-228
Abstract: Andrews [1] has conjectured that the constant term in a certain product is equal to a $q$-multinomial coefficient. This conjecture is a $q$-analogue of Dyson's conjecture [5], and has been proved, combinatorically, by Zeilberger and Bressoud [15]. In this paper we give a combinatorial proof of a master theorem, that the constant term in a similar product, computed over the edges of a nontransitive tournament, is zero. Many constant terms are evaluated as consequences of this master theorem including Andrews' $q$-Dyson theorem in two ways, one of which is a $q$-analogue of Good's [6] recursive proof.
Hyponormal operators quasisimilar to an isometry
Pei Yuan
Wu
229-239
Abstract: An expression for the multiplicity of an arbitrary contraction is presented. It is in terms of the isometries which can be densely intertwined to the given contraction. This is then used to obtain a generalization of a result of Sz.-Nagy and Foiaş concerning the existence of a $C{._0}$ contraction which is a quasiaffine transform of a contraction. We then consider the problem when a hyponormal operator is quasisimilar to an isometry or, more generally, when two hyponormal contractions are quasisimilar to each other. Our main results in this respect generalize previous ones obtained by Hastings and the author. For quasinormal and certain subnormal operators, quasisimilarity or similarity to an isometry may even imply unitary equivalence.
Entropy and knots
John
Franks;
R. F.
Williams
241-253
Abstract: We show that a smooth flow on ${S^3}$ with positive topological entropy must possess periodic closed orbits in infinitely many different knot type equivalence classes.
Global solvability and regularity for $\bar \partial$ on an annulus between two weakly pseudoconvex domains
Mei-Chi
Shaw
255-267
Abstract: Let ${M_1}$ and ${M_2}$ be two bounded pseudo-convex domains in $ {{\mathbf{C}}^n}$ with smooth boundaries such that ${\overline M _1} \subset {M_2}$. We consider the Cauchy-Riemann operators $\overline \partial$ on the annulus $M = {M_2}\backslash {\overline M _1}$. The main result of this paper is the following: Given a $ \overline \partial$-closed $(p,q)$ form $\alpha$, $0 < q < n$, which is ${C^\infty }$ on $ \overline M$ and which is cohomologous to zero on $M$, there exists a $(p,q - 1)$ form $u$ which is $ {C^\infty }$ on $\overline M$ such that $\overline \partial u = \alpha$.
Cobordism of $(k)$-framed manifolds
E.
Micha
269-280
Abstract: The cobordism theories that arise by considering manifolds whose stable normal bundle has category $k$ are introduced. Using these theories, we define a new filtration of the homotopy groups of spheres. We study the filtration and obtain an upper bound for the filtration of elements in the stable $n$-stem.
The homotopy theory of cyclic sets
W. G.
Dwyer;
M. J.
Hopkins;
D. M.
Kan
281-289
Abstract: The aim of this note is to show that the homotopy theory of the cyclic sets of Connes [3] is equivalent to that of $ \operatorname{SO} (2)$-spaces (i.e. spaces with a circle action) and hence to that of spaces over $K(Z,2)$.
On monomial algebras of finite global dimension
David J.
Anick
291-310
Abstract: Let $G$ be an associative monomial ${\mathbf{k}}$-algebra. If $G$ is assumed to be finitely presented, then either $G$ contains a free subalgebra on two monomials or else $G$ has polynomial growth. If instead $G$ is assumed to have finite global dimension, then either $G$ contains a free subalgebra or else $ G$ has a finite presentation and polynomial growth. Also, a graded Hopf algebra with generators in degree one and relations in degree two contains a free Hopf subalgebra if the number of relations is small enough.
Uniqueness for a forward backward diffusion equation
Alan V.
Lair
311-317
Abstract: Let $\phi$ be continuous, have at most finitely many local extrema on any bounded interval, be twice continuously differentiable on any closed interval on which there is no local extremum and be strictly decreasing on any closed interval on which it is decreasing. We show that the initial-boundary value problem for $ {u_t} = \phi {({u_x})_x}$ with Neumann boundary conditions has at most one smooth solution.
On nonlinear scalar Volterra integral equations. I
Hans
Engler
319-336
Abstract: The scalar nonlinear Volterra integral equation $\displaystyle u(t) + \int_0^t {g(t,s,u(s))\,ds = f(t)\qquad (0 \leqslant t)}$ is studied. Conditions are given under which the difference of two solutions can be estimated by the variation of the difference of the corresponding right-hand sides. Criteria for the existence of $\lim u(t)$ (as $t \to \infty$) are given, and existence and uniqueness questions are also studied.
The degrees of r.e. sets without the universal splitting property
R. G.
Downey
337-351
Abstract: It is shown that every nonzero r.e. degree contains an r.e. set without the universal splitting property. That is, if $ \delta$ is any r.e. nonzero degree, there exist r.e. sets $\emptyset < {}_TB < {}_TA$ with $\deg (A) = \delta$ such that if ${A_0} \sqcup {A_1}$ is an r.e. splitting of $ A$, then ${A_0}\not \equiv {}_TB$. Some generalizations are discussed.
Remarks on the stability of shock profiles for conservation laws with dissipation
Robert L.
Pego
353-361
Abstract: Two remarks are made. The first is to establish the stability of monotone shock profiles of the KdV-Burgers equation, based on an energy method of Goodman. The second remark illustrates, specifically in Burgers' equation, that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms.
Divisions of space by parallels
G. L.
Alexanderson;
John E.
Wetzel
363-377
Abstract: An arrangement of hyperplanes in $ {\mathbb{E}^d}$ is a "plaid" provided its hyperplanes form no multiple flats of intersection and lie in parallel families that are in general position. We develop some geometrically natural formulas for the number of $r$-faces that are formed by such an arrangement.