Geometry of fractals and vector calculus on fractals

分形几何和分形矢量微积分

• 批准号: 238549-2012
• 负责人: Mendivil, Franklin
• 金额: $ 0.87万
• 依托单位: Acadia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570442
• 关键词: Geometry; fractals; vector; calculus

Most people have encountered beautiful pictorial depictions of fractals at one point or another. However, it may come as a surprise to many that fractals play a useful role in scientific areas such as data compression, analyzing stream networks and modelling other physical phenomena. The defining characteristic of a fractal is its self-similarity. That is, small bits of a fractal bear a striking resemblance to the entire fractal. Many natural phenomena have this same type of scaling behaviour, but usually only as an approximation. However, it is often a good enough approximation to be descriptively or predictively useful and so fractals have found their way into most areas of science. This research project has two main themes, both involving fractals, with the respective goals of a better understanding of the geometry of fractals and building new calculus tools to use when applying fractal-based models. In the first theme a very fine analysis of the geometry of Cantor sets will be undertaken. This is a continuing program to obtain a complete characterization of the Hausdorff and packing measures and dimensions of very general Cantor sets, in particular subsets of the real line. For linear Cantor sets, the approach is to relate the geometry to the asymptotics of the "gaps". In the second theme the new calculus tools will be fractal analogues of the tools from classical calculus such as various types of derivatives and integrals. In this research program, the approach will be to use iterated function systems (IFS) as a constructive tool. In previous work IFS have proven to be useful in describing and constructing fractals in many different contexts. The method will be to adapt the IFS framework to the desired calculus construction, starting with curvatures for fractal curves and differential forms for fractal surfaces and then higher-dimensional analogues of these. An important outcome of the project will be practical and efficient algorithms to be used in computations involving calculus on fractals. These algorithms will make it possible for scientists to use fractal models more efficiently. This is a primary reason for using IFS methods, as they naturally lead to computational algorithms.

大多数人都曾在某一时刻遇到过美丽的分形图画。 然而，令许多人惊讶的是，分形在数据压缩、分析流网络和建模其他物理现象等科学领域发挥着有用的作用。 分形的定义特征是它的自相似性。 也就是说，分形的一小部分与整个分形有着惊人的相似之处。 许多自然现象都具有这种相同类型的缩放行为，但通常仅作为近似值。 然而，它通常是一个足够好的近似值，可以用于描述或预测，因此分形已经进入了大多数科学领域。 该研究项目有两个主题，都涉及分形，各自的目标是更好地理解分形的几何形状，并构建新的微积分工具以在应用基于分形的模型时使用。 在第一个主题中，将对康托集的几何进行非常精细的分析。 这是一个持续的程序，旨在获得非常一般的康托集（特别是实线子集）的豪斯多夫和包装测量和维数的完整表征。 对于线性康托集，该方法是将几何形状与“间隙”的渐进性联系起来。 在第二个主题中，新的微积分工具将是经典微积分工具的分形类似物，例如各种类型的导数和积分。 在本研究计划中，方法将使用迭代函数系统（IFS）作为构建工具。 在之前的工作中，IFS 已被证明在许多不同环境中描述和构造分形方面非常有用。该方法将使 IFS 框架适应所需的微积分构造，从分形曲线的曲率和分形表面的微分形式开始，然后是这些的更高维类似物。 该项目的一个重要成果将是在涉及分形微积分的计算中使用实用且高效的算法。 这些算法将使科学家能够更有效地使用分形模型。 这是使用 IFS 方法的主要原因，因为它们自然会导致计算算法。