Mathematical Sciences: Measure and Dimension, Scaling, Random Objects and Selections

数学科学：测量和尺寸、缩放、随机对象和选择

• 批准号: 9007035
• 负责人: R. Daniel Mauldin
• 金额: $ 9.33万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-01 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9007035&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Measure; Dimension; Scaling

Professor Mauldin will conduct research in various aspects of the measure theory of fractal objects. One major theme centers on Hausdorff measure and its variants. Topics include (i) further development of deterministic and random recursions yielding objects in n-space and determining their Hausdorff and packing dimension functions, (ii) finding the exact Hausdorff dimension of the graphs of general Hardy-Weierstrass functions, and (iii) characterizing Hausdorff measures which scale. A second theme centers on the production of random maps and finding their properties. Most smooth objects in space have an integer dimension. For example a solid object has dimension three and a surface dimension two. Hausdorff dimension and measure are capable of describing fractal objects which have non-integer dimension.

莫尔丁教授将在分形物的测度论的各个方面进行研究。一个主要主题集中在Hausdorff度量及其变体上。主题包括(I)在n-空间中产生对象的确定性递归和随机递归的进一步发展以及确定它们的Hausdorff和填充维度函数，(Ii)寻找一般Hardy-Weerstrass函数图的确切Hausdorff维度，以及(Iii)刻画尺度的Hausdorff度量。第二个主题集中在随机地图的产生和寻找它们的属性。空间中的大多数平滑对象都具有整数维。例如，一个实体物体有三个维度和一个表面维度两个维度。Hausdorff维数和Hausdorff测度能够描述非整数维的分形体。