Some results on increments of the partially observed empirical process
Zacharie
Dindar
427-440
Abstract: The author investigates the almost sure behaviour of the increments of the partially observed, uniform empirical process. Some functional laws of the iterated logarithm are obtained for this process. As an application, new laws of the iterated logarithm are established for kernel density estimators.
Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds
Lei
Ni;
Huaiyu
Ren
441-456
Abstract: In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.
Analysis and geometry on manifolds with integral Ricci curvature bounds. II
Peter
Petersen;
Guofang
Wei
457-478
Abstract: We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding's volume convergence results and extend the Cheeger-Colding splitting theorem.
Coding into $K$ by reasonable forcing
Ralf-Dieter
Schindler
479-489
Abstract: We present a technique for coding sets ``into $K$,'' where $K$ is the core model below a strong cardinal. Specifically, we show that if there is no inner model with a strong cardinal then any $X\subset\omega_1$ can be made $\boldsymbol{\Delta}^1_3$ (in the codes) in a reasonable and stationary preserving set generic extension.
The completeness of the isomorphism relation for countable Boolean algebras
Riccardo
Camerlo;
Su
Gao
491-518
Abstract: We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF $C^*$-algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.
Tame and Wild Coordinates of $K[z][x,y]$
Vesselin
Drensky;
Jie-Tai
Yu
519-537
Abstract: Let $K$ be a field of characteristic zero. We characterize coordinates and tame coordinates in $K[z][x,y]$, i.e. the images of $x$ respectively under all automorphisms and under the tame automorphisms of $K[z][x,y]$. We also construct a new large class of wild automorphisms of $K[z][x,y]$ which maps $x$ to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of $K[z][x,y,t]$.
An arithmetic property of Fourier coefficients of singular modular forms on the exceptional domain
Shou-Te
Chang;
Minking
Eie
539-556
Abstract: We shall develop the theory of Jacobi forms of degree two over Cayley numbers and use it to construct a singular modular form of weight 4 on the 27-dimensional exceptional domain. Such a singular modular form was obtained by Kim through the analytic continuation of a nonholomorphic Eisenstein series. By applying the results in a joint work with Eie, A. Krieg provided an alternative proof that a function with a Fourier expansion obtained by Kim is indeed a modular form of weight 4. This work provides a systematic and general approach to deal with the whole issue.
Comparing Heegaard and JSJ structures of orientable 3-manifolds
Martin
Scharlemann;
Jennifer
Schultens
557-584
Abstract: The Heegaard genus $g$ of an irreducible closed orientable $3$-manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if $p$ of the complementary components are not Seifert fibered, then $p \leq g-1$. This generalizes work of Kobayashi. The Heegaard genus $g$ also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the Seifert pieces has base space $P$ and $f$ exceptional fibers, then $f - \chi(P) \leq 3g - 3 - p$.
Central extensions and generalized plus-constructions
G.
Mislin;
G.
Peschke
585-608
Abstract: We describe the effect of homological plus-constructions on the homotopy groups of Eilenberg-MacLane spaces in terms of universal central extensions.
Asymptotic convergence of the Stefan problem to Hele-Shaw
Fernando
Quirós;
Juan
Luis
Vázquez
609-634
Abstract: We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as $t\to \infty$, and give the precise asymptotic growth rate for the radius.
Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators
Dirk
Buschmann;
Günter
Stolz
635-653
Abstract: We prove exponential localization at all energies for two types of one-dimensional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-parameter version of spectral averaging, as used in localization proofs for Anderson models, breaks down. We use the new method of two-parameter spectral averaging and apply it to the Poisson as well as the displacement case. In addition, we apply results from inverse spectral theory, which show that two-parameter spectral averaging works for sufficiently many energies (all but a discrete set) to conclude localization at all energies.
Uniqueness of solution to a free boundary problem from combustion
C.
Lederman;
J.
L.
Vázquez;
N.
Wolanski
655-692
Abstract: We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function $u(x,t)\geq 0,$ defined in a domain $\mathcal{D} \subset {\mathbb{R}}^{N}\times (0,T)$ and such that \begin{displaymath}\Delta u+\sum a_{i}\,u_{x_{i}}-u_{t}=0\quad \text{in}\quad \mathcal{D}\cap \{u>0\}. \end{displaymath} We also assume that the interior boundary of the positivity set, $\mathcal{D} \cap \partial \{u> \nobreak 0\}$, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied: \begin{displaymath}u=0 ,\quad -\partial u/\partial \nu = C. \end{displaymath} Here $\nu$ denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of $\mathcal{D}$. This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit). The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.
Tracially AF $C^*$-algebras
Huaxin
Lin
693-722
Abstract: Inspired by a paper of S. Popa and the classification theory of nuclear $C^*$-algebras, we introduce a class of $C^*$-algebras which we call tracially approximately finite dimensional (TAF). A TAF $C^*$-algebra is not an AF-algebra in general, but a ``large'' part of it can be approximated by finite dimensional subalgebras. We show that if a unital simple $C^*$-algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated $K_0$-group. All nuclear simple $C^*$-algebras of real rank zero, stable rank one, with weakly unperforated $K_0$-group classified so far by their $K$-theoretical data are TAF. We provide examples of nonnuclear simple TAF $C^*$-algebras. A sufficient condition for unital nuclear separable quasidiagonal $C^*$-algebras to be TAF is also given. The main results include a characterization of simple rational AF-algebras. We show that a separable nuclear simple TAF $C^*$-algebra $A$ satisfying the Universal Coefficient Theorem and having $K_1(A)=0$ and $K_0(A)=\mathbf{Q}$ is isomorphic to a simple AF-algebra with the same $K$-theory.
Linear maps determining the norm topology
Krzysztof
Jarosz
723-731
Abstract: Let $A$ be a Banach function algebra on a compact space $X$, and let $a\in A$ be such that for any scalar $\lambda$ the element $a+\lambda e$ is not a divisor of zero. We show that any complete norm topology on $A$ that makes the multiplication by $a$ continuous is automatically equivalent to the original norm topology of $A$. Related results for general Banach spaces are also discussed.
Block diagonal polynomials
Verónica
Dimant;
Raquel
Gonzalo
733-747
Abstract: In this paper we introduce and study a certain class of polynomials in spaces with unconditional finite dimensional decomposition. Some applications to the existence of copies of $\ell _\infty$ in spaces of polynomials and to the stabilization of polynomial algebras are given.
Serre's generalization of Nagao's theorem: An elementary approach
A.
W.
Mason
749-767
Abstract: Let $C$ be a smooth projective curve over a field $k$. For each closed point $Q$ of $C$ let $\mathcal{C} = \mathcal{C}(C, Q, k)$be the coordinate ring of the affine curve obtained by removing $Q$from $C$. Serre has proved that $GL_2(\mathcal{C})$ is isomorphic to the fundamental group, $\pi_1(G, T)$, of a graph of groups $(G, T)$, where $T$ is a tree with at most one non-terminal vertex. Moreover the subgroups of $GL_2(\mathcal{C})$attached to the terminal vertices of $T$ are in one-one correspondence with the elements of $\operatorname{Cl}(\mathcal{C})$, the ideal class group of $\mathcal{C}$. This extends an earlier result of Nagao for the simplest case $\mathcal{C} = k[t]$. Serre's proof is based on applying the theory of groups acting on trees to the quotient graph $\overline{X} = GL_2(\mathcal{C}) \backslash X$, where $X$ is the associated Bruhat-Tits building. To determine $\overline{X}$ he makes extensive use of the theory of vector bundles (of rank 2) over $C$. In this paper we determine $\overline{X}$using a more elementary approach which involves substantially less algebraic geometry. The subgroups attached to the edges of $T$ are determined (in part) by a set of positive integers $\mathcal{S}$, say. In this paper we prove that $\mathcal{S}$ is bounded, even when Cl $(\mathcal{C})$ is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of $GL_2(\mathcal{C})$, involving unipotent and elementary matrices.
Isomorphism problems and groups of automorphisms for generalized Weyl algebras
V.
V.
Bavula;
D.
A.
Jordan
769-794
Abstract: We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to $U(\mathfrak{sl}_2)$introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of $U(\mathfrak{sl}_2)$ by finding sets of generators for the group of automorphisms.
Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation
Daniel
Tataru
795-807
Abstract: The aim of this article is twofold. First we consider the wave equation in the hyperbolic space $\mathbb H^n$ and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in $\mathbb R^{n} \times \mathbb R$ which extend the ones of Georgiev, Lindblad, and Sogge.
Boundary value problems for higher order parabolic equations
Russell
M.
Brown;
Wei
Hu
809-838
Abstract: We consider a constant coefficient parabolic equation of order $2m$ and establish the existence of solutions to the initial-Dirichlet problem in cylindrical domains. The lateral data is taken from spaces of Whitney arrays which essentially require that the normal derivatives up to order $m-1$ lie in $L^2$ with respect to surface measure. In addition, a regularity result for the solution is obtained if the data has one more derivative. The boundary of the space domain is given by the graph of a Lipschitz function. This provides an extension of the methods of Pipher and Verchota on elliptic equations to parabolic equations.
Correction to ``Optimal factorization of Muckenhoupt weights''
Michael
Brian
Korey
839-851
Abstract: Peter Jones' theorem on the factorization of $A_p$ weights is sharpened for weights with bounds near $1$, allowing the factorization to be performed continuously near the limiting, unweighted case. When $1<p<\infty$ and $w$ is an $A_p$ weight with bound $A_p(w)=1+\varepsilon$, it is shown that there exist $A_1$ weights $u,v$ such that both the formula $w=uv^{1-p}$ and the estimates $A_1(u), A_1(v)=1+\mathcal O(\sqrt\varepsilon)$ hold. The square root in these estimates is also proven to be the correct asymptotic power as $\varepsilon\to 0$.