| dc.contributor.author | Taylor, Tara D. | en_US |
| dc.date.accessioned | 2014-10-21T12:35:55Z | |
| dc.date.available | 2005 | |
| dc.date.issued | 2005 | en_US |
| dc.identifier.other | AAINR08403 | en_US |
| dc.identifier.uri | http://hdl.handle.net/10222/54733 | |
| dc.description | This thesis presents a study of symmetric binary fractal trees using methods of computational topology. Fractal trees can be used to model various natural systems, such as the cardiovascular system or river drainage networks. | en_US |
| dc.description | Symmetric binary fractal trees were first introduced by Mandelbrot in [30]. A symmetric binary fractal tree is defined by two parameters: the branching angle theta (between 0 and 180 degrees) and a scaling ratio r (between 0 and 1). A trunk of length 1 splits into two branches, one on the left and one on the right, with lengths equal to the scaling ratio and forming an angle theta with the extension of the trunk. Each of these branches splits into two new branches, and the branching is continued ad infinitum . The resulting object is the fractal tree, which can be seen as a representation of the free monoid MLR on two generators L and R. | en_US |
| dc.description | We study the self-avoiding and self-contacting trees. Motivated by techniques from shape theory and computational topology, we will be considering these trees along with their closed epsilon-neighbourhoods as epsilon ranges over the non-negative real numbers. We investigate various features of the closed epsilon-neighbourhoods, based on the holes in these neighborhoods. | en_US |
| dc.description | Due to the nice geometric nature of the trees, we can refine our approach by classifying holes according to their shape and location in the tree. The action of MLR on the tree brings a natural grading by level to these holes. We will see that the level 0 holes form a kind of fundamental domain, and we can restrict our attention to the level 0 holes. To describe the location of a hole, we have generalized the notion of contact address (for self-contacting trees) to hole locator address and hole locator pairs. | en_US |
| dc.description | We determine the hole sequence of these trees together with the persistence intervals of the holes as the 'topological barcodes' (as defined by Carlsson et al.) of these trees. We find that the notion of persistence has some interesting and perhaps unexpected properties in this context. | en_US |
| dc.description | From various notions and properties of holes we derive several classifications of the symmetric binary fractal trees. These are the complexity, location, type and hole sequence classifications. They lead to the determination of certain critical values for the angle theta with respect to location, the scaling ratio r as a function of theta and with respect to complexity, and epsilon as a function of both r and theta and with respect to the hole sequence. | en_US |
| dc.description | We illustrate the theory with a presentation of a collection of specific trees and their closed epsilon-neighbourhoods. We discuss four particularly interesting trees which scale according to the golden ratio. | en_US |
| dc.description | Thesis (Ph.D.)--Dalhousie University (Canada), 2005. | en_US |
| dc.language | eng | en_US |
| dc.publisher | Dalhousie University | en_US |
| dc.publisher | | en_US |
| dc.subject | Statistics. | en_US |
| dc.title | Computational topology and fractal trees. | en_US |
| dc.type | text | en_US |
| dc.contributor.degree | Ph.D. | en_US |
