Aggregate and fractal tessellations

Consider a sequence of stationary tessellations {‹n}, n=0,1,..., of  d consisting of cells {Cn(xin)}with the nuclei {xin}. An aggregate cell of level one, C01(xi0), is the result of merging the cells of ‹1 whose nuclei lie in C0(xi0). An aggregate tessellation ‹0n consists of the aggregate cells of level n, C0n(xi0), defined recursively by merging those cells of ‹n whose nuclei lie in Cnm1(xi0). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as nMX and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {‹n}

Course-Prerequisite Networks for Analyzing and Understanding Academic Curricula

Understanding a complex system of relationships between courses is of great importance for the university's educational mission. This paper is dedicated to the study of course-prerequisite networks that model interactions between courses, represent the flow of knowledge in academic curricula, and serve as a key tool for visualizing, analyzing, and optimizing complex curricula. We show how course-prerequisite networks can be used by students, faculty, and administrators for detecting important courses, improving existing and creating new courses, navigating complex curricula, allocating teaching resources, increasing interdisciplinary interactions between departments, revamping curricula, and enhancing the overall students' learning experience. The proposed methodology is illustrated with a network of courses taught at the California Institute of Technology.Comment: 24 pages, 20 figures, 10 Table