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Dependence relations on homogeneous groups and homogeneous expansions of Hilbert spaces.

机译:希尔伯特空间齐次群和齐次展开的依存关系。

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摘要

A notion of independence called non-forking was introduced by Shelah to study problems related to classification theory. Forking has been a very useful tool in first-order theories for analyzing the structure of models. For example, this analysis has lead to characterizing models in a classifiable theory through a collection of invariants.; Buechler and Lessmann defined a notion of independence, called freeness, working inside a large strongly homogeneous structure. When this notion satisfies the expected properties, the structure is called simple. When a simple structure has a bound on the size of independent extensions of types over sets, the structure is called simple stable.; Working in a simple stable strongly homogeneous model, we use tools from first-order logic to study groups. We define the notions of generics, stabilizers and connected components. We generalize results from first-order stability theory and show that connected 1-based groups are abelian and that the geometry of an SU-rank one 1-based group is that of a module over a ring. We define internality and we prove the existence of a strongly homogeneous group. When M is |M|-strongly homogeneous, the group we get is the binding group.; We study freeness in strongly homogeneous expansions of Hilbert spaces. We prove the following theorems:; Theorem 0.1. Let (H, +, ⟨, ⟩, {lcub}Eλ{rcub}) be a strongly δ-homogeneous δ-saturated Hilbert space where {lcub}Eλ{rcub} is the resolution of the identity of a self-adjoint operator T, δ > ( 2ℵ0 )+. This structure is 1 -simple stable and for fH, AH, A = BDD( A), tp(f/A) is stationary and the projection of f over A can be identified with the canonical base of f over A.; Theorem 0.2. Let X be a set with | X| > 2ℵ0 .{09}Let l2 (X) be the set of square integrable functions from X to R and + and × their pointwise addition and multiplication. Then (l2(X), +, ⟨, ⟩, ×) is 1 -simple stable. Furthermore, the structure is 1-based and for fH, A = acl( A) a closed subspace, tp(f/A) is stationary and the projection of f over A can be taken to be the canonical base of f over A.
机译:Shelah引入了一种称为非分叉的独立性概念,以研究与分类理论有关的问题。分叉一直是一阶理论中用于分析模型结构的非常有用的工具。例如,这种分析导致通过不变量的集合在可分类的理论中表征模型。 Buechler和Lessmann定义了一个独立概念,即自由,它在大型的强烈均质结构中工作。当此概念满足期望的特性时,该结构称为简单结构。当简单结构对类型对集合的独立扩展的大小有限制时,该结构称为简单稳定。在简单的稳定的强同质模型中工作,我们使用一阶逻辑中的工具来研究小组。我们定义泛型,稳定器和连接的组件的概念。我们对一阶稳定性理论的结果进行了归纳,并表明相连的基于1的基团是abelian,并且 SU -基于1的基团的几何形状是环上模块的几何形状。我们定义内部性,并证明存在一个高度同质的组。当 M 是| M |-非常均一时,我们得到的基团是结合基团。我们研究希尔伯特空间的高度均匀扩张中的自由度。我们证明以下定理: 定理0.1 。令( H ,+,〈,〉,{lcub} E λ {rcub})为强δ均质δ饱和的希尔伯特其中{lcub} E λ {rcub}是自伴算子,δ>( 2 &aleph; 0 + 。此结构为 &aleph; 1 -简单稳定,对于 f H A H A = BDD A ), tp f / A )是固定的,并且 f A 上的投影可以通过的规范基础来识别f 超过 定理0.2 。设 X 为| X |的集合。 > 2 &aleph; 0 。{09}让 l 2 X )是从 X R < / math>和+和×的逐点加法和乘法。然后( l 2 X ),+,〈,〉,×)是 &aleph; 1 -简单稳定。此外,该结构基于1,对于 f H A = acl A )是一个封闭的子空间, tp f / A )是固定的,并且 f A上的投影可以被认为是 f 相对于 的规范基础。

著录项

  • 作者单位

    University of Notre Dame.;

  • 授予单位 University of Notre Dame.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2002
  • 页码 83 p.
  • 总页数 83
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

  • 入库时间 2022-08-17 11:46:08

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