The influence of boundary data on the number of solutions of boundary value problems with jumping nonlinearities
Greg A.
Harris
417-464
Abstract: This paper contains results concerning the number of solutions, as a function of the boundary data, for boundary value problems with jumping nonlinearities. An example seems to indicate that boundary data has a different influence on this number than does forcing data. Through approximating techniques this example leads to lower bounds on solution numbers for the more general case
Cardinal conditions for strong Fubini theorems
Joseph
Shipman
465-481
Abstract: If $ {\kappa _1},{\kappa _2}, \ldots ,{\kappa _n}$ are cardinals with ${\kappa _1}$ the cardinality of a nonmeasurable set, and for $i = 2,3, \ldots ,n$ ${\kappa _i}$ is the cardinality of a set of reals which is not the union of ${\kappa _{i - 1}}$ measure-0 sets, then for any nonnegative function $f:{{\mathbf{R}}^n} \to {\mathbf{R}}$ all of the iterated integrals $\displaystyle {I_\sigma } = \iint \cdots \int {f({x_1},{x_2}, \ldots ,{x_n})d{x_{\sigma (1)}}d{x_{\sigma (2)}} \cdots d{x_{\sigma (n)}},\quad \sigma \in {S_n}}$ , which exist are equal. If all $n!$ of the integrals exist, then the weaker condition of the case $n = 2$ implies they are equal. These cardinal conditions are consistent with and independent of ZFC, and follow from the existence of a real-valued measure on the continuum. Other necessary conditions and sufficient conditions for the existence and equality of iterated integrals are also treated.
Realization of the level one standard $\tilde{C}_{2k+1}$-modules
Kailash C.
Misra
483-504
Abstract: In this paper we study the level one standard (or irreducible integrable highest weight) modules for the affine symplectic Lie algebras. In particular, we give concrete realizations of all level one standard modules for the affine symplectic Lie algebras of even rank.
Large deviations in dynamical systems and stochastic processes
Yuri
Kifer
505-524
Abstract: The paper exhibits a unified approach to large deviations of dynamical systems and stochastic processes based on the existence of a pressure functional and on the uniqueness of equilibrium states for certain dense sets of functions. This enables us to generalize recent results from [OP, Y, and D] on large deviations for dynamical systems, as well, as to recover Donsker-Varadhan's [DV2] large deviation estimates for Markov processes.
The generalized Lusternik-Schnirelmann category of a product space
Mónica
Clapp;
Dieter
Puppe
525-532
Abstract: We continue to study the notions of $ \mathcal{A}$-category and strong $ \mathcal{A}$-category which we introduced in [2]. We give a characterization of them in terms of homotopy colimits and then use it to prove some product theorems in this context.
Zeta functions of formal languages
Jean
Berstel;
Christophe
Reutenauer
533-546
Abstract: Motivated by symbolic dynamics and algebraic geometry over finite fields, we define cyclic languages and the zeta function of a language. The main result is that the zeta function of a cyclic language which is recognizable by a finite automation is rational.
A short proof of principal kinematic formula and extensions
W.
Rother;
M.
Zähle
547-558
Abstract: Federer's extension of the classical principal kinematic formula of integral geometry to sets with positive reach is proved in a direct way by means of generalised unit normal bundles, associated currents, and the coarea theorem. This enables us to extend the relation to more general sets. At the same time we get a short proof for the well-known variants from convex geometry and differential geometry.
Curves on $K$-theory and the de Rham homology of associative algebras
John G.
Ryan
559-582
Abstract: This paper describes the generalization to arbitrary associative algebras of the complex of "typical curves on algebraic $ K$-theory" and shows, in particular, that for certain $ {\mathbf{Q}}$-algebras, $ A$, the complex is isomorphic to the "generalized de Rham complex," $(H{H_*}(A),B)$, in which $B$ is Connes' operator acting on the Hochschild homology groups of $A$.
On the integrals of a singular real analytic differential form in ${\bf R}\sp n$
A.
Meziani
583-594
Abstract: In this paper, we study the constancy on the fibers for the continuous integrals of a complex-valued real analytic differential form in ${R^n}$. Then we prove an isomorphism result between the space of smooth integrals and a space built from spaces of Whitney functions.
On the integrability of singular differential forms in two complex variables
A.
Meziani
595-620
Abstract: In this work we study the integrability of a germ at $0 \in {{\mathbf{C}}^2}$ of a singular differential form for which the closure of the integral curves are analytic varieties that pass through 0. The focii of this paper are the existence of pure meromorphic integrals, the linearization and the nonexistence of a topological criterion for transcendental integrability.
Classifying sets of measure zero with respect to their open covers
Winfried
Just;
Claude
Laflamme
621-645
Abstract: Developing ideas of Borel and Fréchet, we define a partial preorder which classifies measure zero sets of reals according to their open covers and study the induced partial order on the equivalence classes. The more "rarefied" a set of measure zero, the higher it will range in our partial order. Main results: The sets of strong measure zero form one equivalence class that is the maximum element of our order. There is a second highest class that contains all uncountable closed sets of measure zero. There is a minimum class that contains all dense ${G_\delta }$-subsets of the real line of measure zero. There exist at least four classes, and if Martin's axiom holds, then there are as many classes as subsets of the real line. It is also consistent with ZFC that there is a second lowest class.
A topological persistence theorem for normally hyperbolic manifolds via the Conley index
Andreas
Floer
647-657
Abstract: We prove that the cohomology ring of a normally hyperbolic manifold of a diffeomorphism $f$ persists under perturbation of $ f$. We do not make any quantitative assumptions on the expansion and contraction rates of $Df$ on the normal and the tangent bundles of $ N$.
On solvable groups of finite Morley rank
Ali
Nesin
659-690
Abstract: We investigate solvable groups of finite Morley rank. We find conditions on $ G$ for $G'$ to split in $G$. In particular, if $G'$ is abelian and $Z(G) = 1$ we prove that
Families of rational surfaces preserving a cusp singularity
Lee J.
McEwan
691-716
Abstract: Families of rational surfaces containing resolutions of cusp singularities are explicitly constructed. It is proved that the families constructed are universal deformations at each point. Two different monodromy formulas are established; one of these is shown to be connected to automorphisms of Inoue-Hirzebruch surfaces. Some evidence (but no proof) is offered for the conjecture that finite base changes of the families we construct are the versal-deformation spaces for singular Inoue-Hirzebruch surfaces.
Local orders whose lattices are direct sums of ideals
Jeremy
Haefner
717-740
Abstract: Let $R$ be a complete local Dedekind domain with quotient field $K$ and let $\Lambda$ be a local $R$-order in a separable $K$-algebra. This paper classifies those orders $ \Lambda$ such that every indecomposable $R$-torsionfree $\Lambda$-module is isomorphic to an ideal of $ \Lambda$. These results extend to the noncommutative case some results for commutative rings found jointly by this author and L. Levy.
A spanning set for ${\scr C}(I\sp n)$
Thomas
Bloom
741-759
Abstract: $\mathcal{C}({I^n})$ denotes the Banach space of continuous functions on the unit $n$-cube, ${I^n}$, in $ {{\mathbf{R}}^n}$. Let $\{ {a^i}\}$, $ i = 0,1,2, \ldots ,$, be a countable collection of $n$-tuples of positive real numbers satisfying $ {\operatorname{lim}_i}a_j^i = + \infty$ for $ j = 1, \ldots ,n$. We canonically enlarge the family of monomials $\{ {x^{{a^i}}}\}$ to a family of functions $\mathcal{F}(A)$. Conjecture. The linear span of $ \mathcal{F}(A)$ is dense in $ \mathcal{C}({I^n})$ if and only if $\Sigma _{i = 0}^\infty 1/\left\vert {{a^i}} \right\vert = + \infty$. For $n = 1$ this is equivalent to the Müntz-Szasz theorem. For $n > 1$ we prove the necessity in general and the sufficiency under the additional hypothesis that there exist constants $G$, $N > 1$ such that $\left\vert {{a^i}} \right\vert \leq G{\operatorname{exp}}({i^N})$ for all $i$.
Some global results on extension of CR-objects in complex manifolds
Guido
Lupacciolu
761-774
Abstract: We prove some results concerning the holomorphic extendability of CR-objects defined on real hypersurfaces of a complex manifold. After a preliminary generalization of the classic theorem on the extendability from the boundary of a relatively compact domain, we discuss the extendability from a part of the boundary of such a domain, the one side extendability from a piece of hypersurface and the extendability from the boundary of an unbounded domain.
Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type
Nicola
Garofalo;
Ermanno
Lanconelli
775-792
Abstract: In this paper we establish a uniform Harnack inequality for a class of degenerate equations whose prototype is Kolmogorov's equations in $ {{\mathbf{R}}^3}:{D_{{\text{yy}}}}u - {\text{y}}{D_z}u - {D_t}u = 0$. Our approach is based on mean value formulas for solutions of the equation under consideration on the level sets of the fundamental solution.
Hyperbolicity properties of $C\sp 2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions
Tomasz
Nowicki;
Sebastian
van Strien
793-810
Abstract: In this paper we study the dynamical properties of general ${C^2}$ maps $f:[0,1] \to [0,1]$ with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has (a) hyperbolicity on the set of periodic points; (b) nonexistence of wandering intervals; (c) sensitivity on initial conditions; and (d) exponential decay of branches (intervals of monotonicity) of ${f^n}$ as $ n \to \infty ;$ For these results we will not make any assumptions on the Schwarzian derivative $f$. We will also give an estimate of the return-time of points that start near critical points.
Existence and uniqueness of algebraic curvature tensors with prescribed properties and an application to the sphere theorem
Walter
Seaman
811-823
Abstract: An existence and uniqueness theorem is proved for algebraic curvature tensors and then applied to yield a global geometric theorem for locally weakly quarter pinched Riemannian manifolds whose second Betti number is nonzero.