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Fractals in complexity and geometry.

机译:分形的复杂性和几何形状。

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摘要

Fractal dimensions have been used by mathematicians and physicists to study properties of dynamic systems and geometry for around a century and have been used by computer scientists to study complexity classes for two decades. But in computer science, the usefulness of fractal dimensions was very limited before Lutz effectivized classical Hausdorff dimension to effective dimensions in 2000. With effective dimensions, a singleton set may have the full dimension of the entire space. This enables us to use dimension-theoretic tools to study the structure of complexity classes which are typically countable sets and to study fractal properties of individual points in Euclidean spaces.;Using these new dimension-theoretic tools, we study both dimensionality and fractal geometry in a computationally resource bounded setting. On the dimensionality front, we will discuss the power of nontrivial fractals in terms of derandomization, the fractal dimension of certain complexity classes, and some relationship between fractal dimension and normality. For fractal geometry, we will discuss some new results in computable analysis, which will include an effective extension of the famous analyst's Traveling Salesman Theorem from geometric measure theory and the existence of a wicked simple curve that forces a computer to retrace. These results are not dimension-theoretic in nature. But the geometric constructions involved do share some features of common fractals.
机译:分形维数已被数学家和物理学家用于研究动力系统和几何学的特性已有大约一个世纪的历史,并且已被计算机科学家用于研究复杂性类达二十年之久。但是在计算机科学中,在Lutz于2000年将经典Hausdorff尺寸实现为有效尺寸之前,分形尺寸的用途非常有限。有了有效尺寸,单例集可能具有整个空间的完整尺寸。这使我们能够使用维度理论工具来研究通常是可数集的复杂性类的结构,并研究欧几里得空间中单个点的分形特性。;使用这些新的维度理论工具,我们可以研究维数和分形几何计算资源限制的设置。在维数方面,我们将从去随机化,某些复杂度类别的分形维数以及分形维数与正态性之间的某些关系方面讨论非平凡分形的威力。对于分形几何,我们将在可计算分析中讨论一些新结果,其中包括从几何测度理论有效扩展著名分析师的Traveling Salesman定理,以及存在邪恶的简单曲线(迫使计算机回溯)。这些结果本质上不是维度理论的。但是涉及的几何构造确实具有共同分形的某些特征。

著录项

  • 作者

    Gu, Xiaoyang.;

  • 作者单位

    Iowa State University.;

  • 授予单位 Iowa State University.;
  • 学科 Computer Science.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 145 p.
  • 总页数 145
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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