A reimbedding algorithm for Casson handles
Žarko
Bižaca
435-510
Abstract: An algorithmic proof of Freedman's Reimbedding Theorem [F2] is given. This reimbedding algorithm produces an explicit description of an imbedded Casson tower with seven levels inside an arbitrary Casson tower with six levels. Our approach is similar to Freedman's original idea, but we also make essential use of the grope technology from [FQ]. The reimbedding algorithm is applied to obtain an explicitly described Casson handle inside an arbitrary six-level tower (Theorem A), a description of a family of exotic Casson handles (Theorem B) and an explicitly constructed exotic $ {\mathbb{R}^4}$.
Extremal properties of Green functions and A. Weitsman's conjecture
Alexander
Fryntov
511-525
Abstract: A new version of the symmetrization theorem is proved. Using a modification of the $\ast$-function of Baernstein we construct an operator which maps a family of $\delta$-subharmonic functions defined on an annulus into a family of subharmonic functions on an annular sector. Applying this operator to the Green function of special domains we prove A. Weitsman's conjecture linked with exact estimates of the Green functions of these domains.
Well-posedness and stabilizability of a viscoelastic equation in energy space
Olof J.
Staffans
527-575
Abstract: We consider the well-posedness and exponential stabilizability of the abstract Volterra integrodifferential system \begin{displaymath}\begin{array}{*{20}{c}} {v\prime (t) = - {D^\ast}\sigma (t) +... ... }^t {a(t - s)Dv(s)ds,\quad t \geq 0,} } \end{array} \end{displaymath} in ilbeubert space. In a typical viscoelastic interpretation of this equation one lets v represent velocit, $v\prime$ acceleratio $\sigma$, stres, $- {D^ \ast }\sigma$ the divergence of the stres, $v \geq 0$ pure viscosity (usually equal to zero) Dv the time derivative of the strain, and a the linear stress relaxation modulus of the material. The problems that we treat are one-dimensional in the sense that we require a to be scalar. First we prove well-posedness in a new semigroup setting, where the history component of the state space describes the absorbed energy of the system rather than the history of the function v. To get the well-posedness we need extremely weak assumptions on the kernel; it suffices if the system is "passive", i.e., a is of positive type; it may even be a distribution. The system is exponentially stabilizable with a finite dimensional continuous feedback if and only if the essential growth rate of the original system is negative. Under additional assumptions on the kernel we prove that this is indeed the case. The final part of the treatment is based on a new class of kernels. These kernels are of positive type, but they need not be completely monotone. Still, they have many properties similar to those of completely monotone kernels, and a number of results that have been proved earlier for completely monotone kernels can be extended to the new class.
Measurable quotients of unipotent translations on homogeneous spaces
Dave
Witte
577-594
Abstract: Let U be a nilpotent, unipotent subgroup of a Lie group G, and let $\Gamma$ be a closed subgroup of G. Marina Ratner showed that every ergodic U-invariant probability measure on the homogeneous space $\Gamma \backslash G$ is of a simple algebraic form. We use this fundamental new result to show that every measurable quotient of the U-action on $\Gamma \backslash G$ is of a simple algebraic form. Roughly speaking, any quotient is a double-coset space $ \Lambda \backslash G/K$.
Hilbert 90 theorems over division rings
T. Y.
Lam;
A.
Leroy
595-622
Abstract: Hilbert's Satz 90 is well-known for cyclic extensions of fields, but attempts at generalizations to the case of division rings have only been partly successful. Jacobson's criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of Satz 90, but the exact relationship between the two was apparently not known. In this paper, we study triples (K, S, D) where S is an endomorphism of the division ring K, and D is an S-derivation. Using the technique of Ore extensions $K[t,S,D]$, we characterize the notion of (S, D)-algebraicity for elements $a \in K$, and give an effective criterion for two elements $a,b \in K$ to be (S, D)-conjugate, in the case when the (S, D)-conjugacy class of a is algebraic. This criterion amounts to a general Hilbert 90 Theorem for division rings in the (K, S, D)-setting, subsuming and extending all known forms of Hilbert 90 in the literature, including the aforementioned Jacobson Criterion. Two of the working tools used in the paper, the Conjugation Theorem (2.2) and the Composite Function Theorem (2.3), are of independent interest in the theory of Ore extensions.
Notes on ruled symplectic $4$-manifolds
Dusa
McDuff
623-639
Abstract: A symplectic 4-manifold $(V,\omega )$ is said to be ruled if it is the total space of a fibration whose fibers are 2-spheres on which the symplectic form does not vanish. This paper develops geometric methods for analysing the symplectic structure of these manifolds, and shows how this structure is related to that of a generic complex structure on V. It is shown that each V admits a unique ruled symplectic form up to pseudo-isotopy (or deformation). Moreover, if the base is a sphere or if V is the trivial bundle over the torus, all ruled cohomologous forms are isotopic. For base manfolds of higher genus this remains true provided that a cohomological conditon on the form is satisfied: one needs the fiber to be "small" relative to the base. These results correct the statement of Theorem 1.3 in The structure of rational and ruled symplectic manifolds, J. Amer. Math. Soc. 3 (1990), 679-712, and give more details of some of the proofs.
Foundations of BQO theory
Alberto
Marcone
641-660
Abstract: In this paper we study the notion of better-quasi-ordering (bqo) originally defined by Nash-Williams [14]. In particular we consider the approximation to this concept given by the notion of $\alpha$-wqo, for $\alpha$ a countable indecomposable ordinal [15]. We prove that if a quasi-ordering Q is $ \alpha$-wqo then ${Q^{ < \alpha }}$ is wqo, thereby obtaining a new proof of Nash-Williams' theorem that Q bqo implies $\tilde Q$ (the set of all countable transfinite sequences of elements of Q) bqo. We show that for $ \alpha < \alpha \prime ,\alpha \prime$-wqo is properly stronger than $ \alpha$-wqo. We also show that the definition of $\alpha$-wqo (and therefore also of bqo) can be modified by considering only barriers with a nice additional property. In the last part of the paper we establish a conjecture of Clote [3] by proving that the set of indices for recursive bqos is complete $ \Pi _2^1$.
Sensitivity analysis of solutions to generalized equations
A. B.
Levy;
R. T.
Rockafellar
661-671
Abstract: Generalized equations are common in the study of optimization through nonsmooth analysis. For instance, variational inequalities can be written as generalized equations involving normal cone mappings, and have been used to represent first-order optimality conditions associated with optimization problems. Therefore, the stability of the solutions to first-order optimality conditions can be determined from the differential properties of the solutions of parameterized generalized equations. In finite-dimensions, solutions to parameterized variational inequalities are known to exhibit a type of generalized differentiability appropriate for multifunctions. Here it is shown, in a Banach space setting, that solutions to a much broader class of parameterized generalized equations are "differentiable" in a similar sense.
Factoring $L$-functions as products of $L$-functions
Douglas
Grenier
673-692
Abstract: We will demonstrate two factorizations of L-functions associated with automorphic forms on $GL(n,\mathbb{R})$, where one factor is a Riemann zetafunction and the other is an L-function associated to an automorphic form for $GL(n - 1,\mathbb{R})$. These will be obtained by establishing the commutation of the Hecke operators and the $\Phi$-operator, a homomorphism from automorphic forms on $ GL(n,\mathbb{R})$ to automorphic forms on $ GL(n - 1,\mathbb{R})$.
Dynamics near the essential singularity of a class of entire vector fields
Kevin
Hockett;
Sita
Ramamurti
693-703
Abstract: We investigate the dynamics near the essential singularity at infinity for a class of zero-free entire vector fields of finite order, i.e., those of the form $f(z) = {e^{P(z)}}$ where $ P(z) = {z^d}$ or $P(z) = a{z^2} + bz + c$. We show that the flow generated by such a vector field has a "bouquet of flowers" attached to the point at infinity.
Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory
Walter
Philipp
705-727
Abstract: Let $ \mathcal{T} = \{ {q_1}, \ldots ,{q_\tau }\}$ be a finite set of coprime integers and let $ \{ {n_1},{n_2}, \ldots \}$ denote the mutiplicative semigroup generated by $\mathcal{T}$, and arranged in increasing order. Let $ {D_N}(\omega )$ denote the discrepancy of the sequence $\{ {n_k}\omega \} _{k = 1}^N\bmod 1$, $\omega \in [0,1)$. In this paper we solve a problem posed by R.C. Baker [3], by proving that for all $ \omega$ except on a set of Lebesgue measure 0 $\displaystyle \frac{1}{4} \leq \mathop {\lim \sup }\limits_{N \to \infty } \frac{{N{D_N}(\omega )}}{{\sqrt {N\log \log N} }} \leq C.$ Here the constant C only depends on the total number of primes involved in the prime factorization of ${q_1}, \ldots ,{q_\tau }$. The lower bound is obtained from a strong approximation theorem for the partial sums of the sequence $ \{ \cos 2\pi {n_k}\omega \} _{k = 1}^\infty$ by sums of independent standard normal random variables.
On the dependence of analytic solutions of partial differential equations on the right-hand side
Siegfried
Momm
729-752
Abstract: Given a nonzero polynomial $P(z) = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }{z^\alpha }}$ on ${\mathbb{C}^N}$, Martineau proved in the 1960s that for each convex domain G of ${\mathbb{C}^N}$ the partial differential operator $P(D)f = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }{f^{(\alpha )}}}$ acting on the Fréchet space $A(G)$ of all analytic functions on G is surjective. In the present paper it is investigated whether solutions f of the equation $P(D)f = g$ can be chosen as $ f = R(g)$ with a continuous linear operator $ R:A(G) \to A(G)$. For bounded G we give a necessary and sufficient condition for the existence of such an R.
Rigidity of ergodic volume-preserving actions of semisimple groups of higher rank on compact manifolds
Guillaume
Seydoux
753-776
Abstract: Let M be a compact manifold, H a semisimple Lie group of higher rank (e.g., $H = SL(n,{\mathbf{R}})$ with $n \geq 3$) and $a \in \mathcal{A}(H,M)$ an ergodic H-action on M which preserves a volume v. Such an H-action is conjectured to be "locally rigid": if $a \prime$ is a sufficiently $ {C^1}$-small perturbation of a, then there should exist a diffeomorphism $\Phi$ of the manifold M which conjugates $a \prime$ to a. This conjecture would imply that if $\omega$ is an a-invariant geometrical structure on M, then there should exist an. $a \prime$-invariant geometrical structure $ \omega \prime$ on M of the same type. Using Kazhdan property, superrigidity for cocycles, and Sobolev spaces techniques we prove, under suitable conditions, two such results with $\omega = v$ and with $\omega$ a Riemannian metric along the leaves of a foliation of M.
Centered bodies and dual mixed volumes
Gao Yong
Zhang
777-801
Abstract: We establish a number of characterizations and inequalities for intersection bodies, polar projection bodies and curvature images of projection bodies in $ {{\mathbf{R}}^n}$ by using dual mixed volumes. One of the inequalities is between the dual Quermassintegrals of centered bodies and the dual Quermassintegrals of their central $(n - 1)$-slices. It implies Lutwak's affirmative answer to the Busemann-Petty problem when the body with the smaller sections is an intersection body. We introduce and study the intersection body of order i of a star body, which is dual to the projection body of order i of a convex body. We show that every sufficiently smooth centered body is a generalized intersection body. We discuss a type of selfadjoint elliptic differential operator associated with a convex body. These operators give the openness property of the class of curvature functions of convex bodies. They also give an existence theorem related to a well-known uniqueness theorem about deformations of convex hypersurfaces in global differential geometry.
Boundary behavior of the Bergman kernel function on some pseudoconvex domains in ${\bf C}\sp n$
Sanghyun
Cho
803-817
Abstract: Let $\Omega$ be a bounded pseudoconvex domain in $ {\mathbb{C}^n}$ with smooth defining function r and let ${z_0} \in b\Omega$ be a point of finite type. We also assume that the Levi form $\partial \bar \partial r(z)$ of $b\Omega$ has $(n - 2)$-positive eigenvalues at $ {z_0}$. Then we get a quantity which bounds from above and below the Bergman kernel function in a small constant and large constant sense.
Classifications of Baire-$1$ functions and $c\sb 0$-spreading models
V.
Farmaki
819-831
Abstract: We prove that if for a bounded function f defined on a compact space K there exists a bounded sequence $ ({f_n})$ of continuous functions, with spreading model of order $\xi$, $1 \leq \xi < {\omega _1}$, equivalent to the summing basis of ${c_0}$, converging pointwise to f, then ${r_{{\text{ND}}}}(f) > {\omega ^\xi }$ (the index $ {r_{{\text{ND}}}}$ as defined by A. Kechris and A. Louveau). As a corollary of this result we have that the Banach spaces ${V_\xi }(K)$, $1 \leq \xi < {\omega _1}$, which previously defined by the author, consist of functions with rank greater than ${\omega ^\xi }$. For the case $ \xi = 1$ we have the equality of these classes. For every countable ordinal number $\xi$ we construct a function S with $ {r_{{\text{ND}}}}(S) > {\omega ^\xi }$. Defining the notion of null-coefficient sequences of order $\xi$, $ 1 \leq \xi < {\omega _1}$, we prove that every bounded sequence $ ({f_n})$ of continuous functions converging pointwise to a function f with ${r_{{\text{ND}}}}(f) \leq {\omega ^\xi }$ is a null-coefficient sequence of order $\xi$. As a corollary to this we have the following ${c_0}$-spreading model theorem: Every nontrivial, weak-Cauchy sequence in a Banach space either has a convex block subsequence generating a spreading model equivalent to the summing basis of ${c_0}$ or is a null-coefficient sequence of order 1.
Stable vector bundles on algebraic surfaces
Wei-Ping
Li;
Zhenbo
Qin
833-852
Abstract: We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of a certain moduli space of stable bundles on a rational ruled surface.
The representation of binary quadratic forms by positive definite quaternary quadratic forms
A. G.
Earnest
853-863
Abstract: A quadratic $\mathbb{Z}$-lattice L of rank n is denned to be k-regular for a positive integer $k \leq n$ if L globally represents all quadratic $ \mathbb{Z}$-lattices of rank k which are represented everywhere locally by L. It is shown that there exist only finitely many isometry classes of primitive positive definite quadratic $ \mathbb{Z}$-lattices of rank 4 which are 2-regular.
On power subgroups of profinite groups
Consuelo
Martínez
865-869
Abstract: In this paper we prove that if G is a finitely generated pro-(finite nilpotent) group, then every subgroup $ {G^n}$, generated by nth powers of elements of G, is closed in G. It is also obtained, as a consequence of the above proof, that if G is a nilpotent group generated by m elements $ {x_1}, \ldots ,{x_m}$, then there is a function $f(m,n)$ such that if every word in $x_i^{ \pm 1}$ of length $ \leq f(m,n)$ has order n, then G is a group of exponent n. This question had been formulated by Ol'shansky in the general case and, in this paper, is proved in the solvable case and the problem is reduced to the existence of such function for finite simple groups.
Normal tree orders for infinite graphs
J.-M.
Brochet;
R.
Diestel
871-895
Abstract: A well-founded tree T denned on the vertex set of a graph G is called normal if the endvertices of any edge of G are comparable in T. We study how normal trees can be used to describe the structure of infinite graphs. In particular, we extend Jung's classical existence theorem for trees of height $ \omega$ to trees of arbitrary height. Applications include a structure theorem for graphs without large complete topological minors. A number of open problems are suggested.
A class of exceptional polynomials
Stephen D.
Cohen;
Rex W.
Matthews
897-909
Abstract: We present a class of indecomposable polynomials of non prime-power degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups $PS{L_2}({2^k})$, where $k \geq 3$ and odd. (The first member of this class was previously found by P. Müller [17]. This realises one of only two possibilities for such a class which remain following deep work of Fried, Guralnick and Saxl [7]. The other is associated with $PS{L_2}({3^k})$, $k \geq 3$ , and odd in fields of characteristic 3.