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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=659555
• 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.****

大多数人在某一时刻遇到了对分形的美丽绘画描述。 但是，对于许多人来说，分形在数据压缩，分析流网络和对其他物理现象进行建模的科学领域中起着有用的作用，这可能使人感到惊讶。 分形的定义特征是它的自相似性。 也就是说，一小块分形带与整个分形非常相似。 许多自然现象具有相同类型的缩放行为，但通常仅作为近似值。 但是，这通常是一个足够好的近似值，可以在描述或预测上有用，因此分形已经进入了科学的大多数领域。 在第一个主题中，将对Cantor套装的几何形状进行非常精美的分析。 这是一个持续的程序，以获得Hausdorff的完整表征以及非常通用的Cantor集的包装措施和尺寸，特别是真实行的子集。 对于线性插槽集，方法是将几何形状与“差距”的渐近学联系起来。 在第二个主题中，新的演算工具将是来自经典微积分（例如各种类型的衍生物和积分）的工具的分形类似物。 在此研究计划中，该方法将是将迭代功能系统（IFS）用作建设性工具。 在以前的工作中，IF已被证明可用于在许多不同情况下描述和构造分形。该方法将是将IFS框架调整为所需的演算结构，从分形曲线的曲率和分形表面的差异形式开始，然后再进行差异形式。 这些算法将使科学家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。****