Ergodic attractors
Charles
Pugh;
Michael
Shub
1-54
Abstract: Using the graph transform method, we give a geometric treatment of Pesin's invariant manifold theory. Beyond deriving the existence, uniqueness, and smoothness results by Fathi, Herman, and Yoccoz our method allows us to do four things: optimally conserve smoothness, deal with endomorphisms, prove absolute continuity of the Pesin laminations, and produce ergodic attractors.
Semicharacteristics, bordism, and free group actions
James F.
Davis;
R. James
Milgram
55-83
Abstract: In this paper we give characteristic class formulae for all semicharacteristic classes of all compact, closed manifolds with finite fundamental groups. These invariants are identified with elements in certain odd $L$-groups, and exactly which elements occur is specified. An appendix calculates the cohomology of the model groups needed. A second appendix determines the structure of the $L$-groups needed.
Determinacy of sufficiently differentiable maps
Alan M.
Selby
85-113
Abstract: Variants of the algebraic conditions of Mather are shown to be sufficient for the $k$-determinacy of ${C^u}$ maps with respect to $j$-flat, contact (or right) ${C^r}$ equivalence relations where $ u - k \leq r \leq u - k + j + 1$ and $ 0 \leq j < k \leq u$. The required changes of coordinates and matrix-valued functions are constructed from the variation of coefficients in polynomials. The main result follows from a finite-dimensional, polynomial pertubation argument which employs a parameter-dependent polynomial representation of functions based on Taylor's formula. For $r > k$, the algebraic conditions are seen to be necessary.
The spectrum of the Schr\"odinger operator
Martin
Schechter
115-128
Abstract: We describe the negative spectrum of the Schrödinger operator with a singular potential. We determine the exact value of the bottom of the spectrum and estimate it from above and below. We describe the dependence of a crucial constant on the eigenvalue parameter and discuss some of its properties. We show how recent results of others are simple consequences of a theorem proved by the author in 1972.
Rigidity for complete Weingarten hypersurfaces
M.
Dajczer;
K.
Tenenblat
129-140
Abstract: We classify, locally and globally, the ruled Weingarten hypersurfaces of the Euclidean space. As a consequence of the local classification and a rigidity theorem of Dajczer and Gromoll, it follows that a complete Weingarten hypersurface which does not contain an open subset of the form $ {L^3} \times {{\mathbf{R}}^{n - 3}}$, where ${L^3}$ is unbounded and $n \geq 3$, is rigid.
Quadrature and harmonic $L\sp 1$-approximation in annuli
D. H.
Armitage;
M.
Goldstein
141-154
Abstract: Open sets $ D$ in ${R^N}\;(N \geq 3)$ with the property that $ \bar D$ is a closed annulus $\{ x:{r_1} \leq \;\left\Vert x\right\Vert \; \leq {r_2}\}$ are characterized by quadrature formulae involving mean values of certain harmonic functions. One such characterization is used to give a criterion for the existence of a best harmonic $ {L^1}$ approximant to a function which is subharmonic (and satisfies some other conditions) in an annulus.
Deforming a PL submanifold of Euclidean space into a hyperplane
Jože
Vrabec
155-178
Abstract: Let $M$ be a closed, $k$-connected, $m$-dimensional $ {\text{PL}}$ submanifold of ${\mathbb{R}^{2m - k - 1}}\;(1 \leq k \leq m - 4)$. The main result of this paper states that if $ m - k$ is even, then every embedding of $M$ into $ {\mathbb{R}^{2m - k}}$ can be isotopically deformed into ${\mathbb{R}^{2m - k - 1}}$, and specifies which embeddings of $M$ into $ {\mathbb{R}^{2m - k}}$ can be deformed into ${\mathbb{R}^{2m - k - 1}}$ in case $ m - k$ is odd.
Holomorphic foliations in ruled surfaces
Xavier
Gómez-Mont
179-201
Abstract: We analyse the universal families of holomorphic foliations with singularities in a ruled surface. In terms of Chern classes we determine the general and the special families. We also classify all nonsingular foliations.
Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles
Vesselin M.
Petkov;
Luchezar N.
Stojanov
203-235
Abstract: Let $\Omega \subset {{\mathbf{R}}^n}$ be a domain such that ${{\mathbf{R}}^n}\backslash \Omega$ is a disjoint union of a finite number of compact strictly convex obstacles with $ {C^\infty }$ smooth boundaries. In this paper the singularities of the scattering kernel $ s(t,\theta ,\omega)$, related to the wave equation in ${\mathbf{R}} \times \Omega $ with Dirichlet boundary condition, are studied. It is proved that for every $\omega \in {S^{n - 1}}$ there exists a residual subset $ \mathcal{R}(\omega)$ of ${S^{n - 1}}$ such that for each $\theta \in \mathcal{R}(\omega),\theta \ne \omega$ $\displaystyle {\text{singsupp}}\,s(t,\theta ,\omega) = {\{ - {T_\gamma }\} _\gamma },$ where $ \gamma$ runs over the scattering rays in $\Omega$ with incoming direction $\omega$ and with outgoing direction $ \theta$ having no segments tangent to $ \partial \Omega$, and ${T_\gamma }$ is the sojourn time of $ \gamma$. Under some condition on $\Omega$, introduced by M. Ikawa, the asymptotic behavior of the sojourn times of the scattering rays related to a given configuration, as well as the precise rate of the decay of the coefficients of the main singularity of $ s(t,\theta ,\omega)$, is examined.
Some families of isoparametric hypersurfaces and rigidity in a complex hyperbolic space
Micheal H.
Vernon
237-256
Abstract: The geometric notion of equivalence for submanifolds in a chosen ambient space is that of congruence. In this study, a certain type of isoparametric hypersurface of a complex hyperbolic space form is shown to have a rigid immersion by utilizing the congruences of a Lorentzian hyperbolic space form that lies as an ${S^1}$-fiber bundle over the complex hyperbolic space. Several families of isoparametric hypersurfaces (namely tubes and horospheres) are constructed whose immersions are rigid.
Summation, transformation, and expansion formulas for bibasic series
George
Gasper
257-277
Abstract: An indefinite bibasic sum containing three parameters is evaluated and used to derive bibasic extensions of Euler's transformation formula and of a Fields and Wimp expansion formula. It is also used to derive a transformation formula involving four independent bases, a $ q$-Lagrange inversion formula, and some quadratic, cubic and quartic summation formulas.
Well-quasi-ordering infinite graphs with forbidden finite planar minor
Robin
Thomas
279-313
Abstract: We prove that given any sequence $ {G_1},{G_2}, \ldots$ of graphs, where ${G_1}$ is finite planar and all other ${G_i}$ are possibly infinite, there are indices $i,j$ such that $i < j$ and ${G_i}$ is isomorphic to a minor of ${G_j}$ . This generalizes results of Robertson and Seymour to infinite graphs. The restriction on $ {G_1}$ cannot be omitted by our earlier result. The proof is complex and makes use of an excluded minor theorem of Robertson and Seymour, its extension to infinite graphs, Nash-Williams' theory of better-quasi-ordering, especially his infinite tree theorem, and its extension to something we call tree-structures over ${\text{QO}}$-categories, which includes infinitary version of a well-quasi-ordering theorem of Friedman.
Nearly representable operators
R.
Kaufman;
Minos
Petrakis;
Lawrence H.
Riddle;
J. J.
Uhl
315-333
Abstract: Among Bourgain's many remarkable theorems is one from 1980 which states that if $T$ is a non-Dunford-Pettis operator from ${L_1}[0,1]$ into an arbitrary Banach space, then there is a Dunford-Pettis operator $ D$ from ${L_1}[0,1]$ into $ {L_1}[0,1]$ such that the composition $T \circ D$ is not Bochner representable. This theorem sets up the following question: What are the operators $T$ from $ {L_1}[0,1]$ into a Banach space $X$ such that $T \circ D$ is Bochner representable for all Dunford-Pettis operators $D:{L_1}[0,1] \to {L_1}[0,1]$ ? We call such an operator nearly representable. In view of Bourgain's theorem, all nearly representable operators are Dunford-Pettis. If $X$ is a Banach space such that all nearly representable operators are, in addition, Bochner representable, then we say $X$ has the near Radon-Nikodym property (NRNP) and ask which Banach spaces have the NRNP? This paper is an attempt to provide at least partial answers to these questions. The first section collects terminology, gives the introductory results and shows that the NRNP is a three space property. The second section studies a continuity property that implies near representability. Finally, the third section contains the main result of the paper, Theorem 15, which states that if $ T:{L_1}[0,1] \to {L_1}[0,1]$ is a nonrepresentable operator, there exists a Dunford-Pettis operator $D:{L_1}[0,1] \to {L_1}[0,1]$ such that $ T \circ D$ is also nonrepresentable. This implies that the ${\text{NRNP}}$ is shared by ${L_1}[0,1]$, lattices not containing $ {c_0}$, and $ {L_1}({\mathbf{T}})/H_0^1$.
A separable space with no remote points
Alan
Dow
335-353
Abstract: In the model obtained by adding $ {\omega _2}$ side-by-side Sacks reals to a model of ${\mathbf{CH}}$, there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size ${\omega _2}$ such that every uncountable subset has density ${\omega _1}$. Furthermore assuming the existence of a measurable cardinal $\kappa$ with ${2^\kappa } = {\kappa ^ + }$, a space $X$ is produced with no isolated points but with remote points in $\upsilon X - X$. It is also shown that a pseudocompact space does not have remote points.
Digital representations using the greatest integer function
Bruce
Reznick
355-375
Abstract: Let ${S_d}(\alpha)$ denote the set of all integers which can be expressed in the form $\sum {{\varepsilon _i}[{\alpha ^i}]}$, with ${\varepsilon _i} \in \{ 0, \ldots ,d - 1\}$, where $d \geq 2$ is an integer and $\alpha \geq 1$ is real, and let $ {I_d}$ denote the set of $ \alpha$ so that $ {S_d}(\alpha) = {{\mathbf{Z}}^ + }$. We show that ${I_d} = [1,{r_d}) \cup \{ d\}$, where $ {r_2} = {13^{1/4}},{r_3} = {22^{1/3}}$ and ${r_2} = {({d^2} - d - 2)^{1/2}}$ for $ d \geq 4$. If $\alpha \notin {I_d}$ we show that ${T_d}(\alpha)$, the complement of ${S_d}(\alpha)$, is infinite, and discuss the density of $ {T_d}(\alpha)$ when $\alpha < d$. For $d \geq 4$ and a particular quadratic irrational $\beta = \beta (d) < d$, we describe ${T_d}(\beta)$ explicitly and show that $\vert{T_d}(\beta) \cap [0,n]\vert$ is of order ${n^{e(d)}}$, where $e(d) < 1$.
Generalizations of the stacked bases theorem
Paul
Hill;
Charles
Megibben
377-402
Abstract: Let $H$ be a subgroup of the free abelian group $G$. In order for there to exist a basis ${\{ {x_i}\} _{i \in I}}$ of $G$ for which $H = { \oplus _{i \in I}}\langle {n_i}{x_i}\rangle$ for suitable nonnegative integers $ {n_i}$, it is obviously necessary for $G/H$ to be a direct sum of cyclic groups. In the 1950's, Kaplansky raised the question of whether this condition on $G/H$ is sufficient for the existence of such a basis. J. Cohen and H. Gluck demonstrated in 1970 that the answer is "yes"; their result is known as the stacked bases theorem, and it extends the classical and well-known invariant factor theorem for finitely generated abelian groups. In this paper, we develop a theory that contains and, in fact, generalizes in several directions the stacked bases theorem. Our work includes a complete classification, using numerical invariants, of the various free resolutions of any abelian group.
Harnack's inequality for degenerate Schr\"odinger operators
Cristian E.
Gutiérrez
403-419
Abstract: We prove a Harnack inequality for nonnegative weak solutions of certain Schrödinger equations of the form $Lu - Vu = 0$ where $L$ is a second order degenerate elliptic operator in divergence form and $V$ is a potential in certain class.
The uniform bound problem for local birational nonsingular morphisms
Bernard
Johnston
421-431
Abstract: It is known that any factorization of a local birational morphism $f:\operatorname{Spec}\;S \to \operatorname{Spec}\;R$ of nonsingular (affine) schemes of arbitrary dimension via other nonsingular schemes must be finite in length. This fact generalizes the classical Local Factorization Theorem of Zariski and Abhyankar, which states that there is a unique such factorization, that given by quadratic transformations, in the surface case. A much stronger generalization is given here, namely, that there exists a uniform bound on the lengths of all such factorizations, provided that $R$ is excellent. This bound is explicitly calculated for some concrete extensions and examples are given to show that this is the strongest generalization possible in some sense.