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

Mauldin教授将在分形物体的测量理论的各个方面进行研究。其中一个主要的主题是豪斯多夫小节及其变体。主题包括(i)进一步发展n空间中产生对象的确定性和随机递归，并确定它们的Hausdorff维数和包装维数函数，（ii）找到一般Hardy-Weierstrass函数图的确切Hausdorff维数，以及（iii）表征可缩放的Hausdorff测度。第二个主题集中于随机地图的生成和查找其属性。空间中大多数光滑物体的维度都是整数。例如，一个固体物体具有三维空间和二维空间。豪斯多夫维数和测度能够描述非整数维数的分形对象。

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.