School of Mathematics and Statistics
http://hdl.handle.net/11343/293
20190524T14:58:11Z

Transport equations and boundary conditions for oscillatory rarefied gas flows
http://hdl.handle.net/11343/224064
Transport equations and boundary conditions for oscillatory rarefied gas flows
Liu, Nicholas Zhixian
A wide range of flow phenomena in everyday life can be modelled accurately using classical continuum theory: the NavierStokes equations with associated noslip conditions. However, oscillatory flows generated by nanoscale devices violate the basic assumptions underpinning continuum theory. Study of such flows, under the assumption of small perturbations from equilibrium, requires analysis of the unsteady Boltzmann equation. At sufficiently high oscillation frequencies, the resulting flow can result in wave propagation. This thesis presents a rigorous asymptotic analysis of slightly rarefied wave motion. Particular emphasis is placed on the boundary layer structure, transport equations and associated slip boundary conditions, valid for a general curved oscillating boundary with a velocityindependent body force.
© 2018 Nicholas Zhixian Liu
20180101T00:00:00Z

A combinatorial curvature flow for ideal triangulations
http://hdl.handle.net/11343/222445
A combinatorial curvature flow for ideal triangulations
Tianyu, Yang
We investigate a combinatorial analogue of the Ricci curvature flow for 3dimensional hyperbolic cone structures, obtained by gluing together hyperbolic ideal tetrahedra. Our aim is to find a hyperbolic structure for triangulated 3dimensional cusped manifolds such as the figure eight knot complement, by fixing the ``no shearing'' condition around edges and the condition of ``completeness'' at each cusp and varying the edge lengths to adjust the cone angles around the edges. To achieve this goal, we first work out a combinatorial curvature flow which only deals with a single ideal tetrahedron. We then deal with this kind of flow on a bipyramid (or double tetrahedron)  two ideal tetrahedra glued together along a triangular face. Now using the single and double tetrahedron cases as building blocks, we study a general combinatorial curvature flow for the 3manifold case with the geometric metric as the equilibrium for the flow. We then apply theorems worked out in this general case to 2tetrahedron manifolds, such as the figure eight knot complement, and to the 3tetrahedron manifold, m011. Most of the tools, such as evolution equations for curvatures, the covolume function and the analysis of the flat boundary and degenerating cases, can be used repeatedly in different situations. The conclusion is that for geometric triangulations (with all dihedral angles positive for the complete hyperbolic structure), the flow can find the geometric metric exponentially fast as long as the flat boundary can be avoided, as in the m011 case in Chapter 6.
© 2019 Dr. Tianyu Yang
20190101T00:00:00Z

Degree bounded geometric spanning trees with a bottleneck objective function
http://hdl.handle.net/11343/221999
Degree bounded geometric spanning trees with a bottleneck objective function
Andersen, Patrick
We introduce the geometric $\delta$minimum bottleneck spanning tree problem ($\delta$MBST), which is the problem of finding a spanning tree for a set of points in a geometric space (e.g., the Euclidean plane) such that no vertex in the tree has a degree that exceeds $\delta$, and the length of the longest edge in the tree is minimum. We give complexity results for this problem and describe several approximation algorithms whose performances we investigate, both analytically and through computational experimentation.
© 2019 Dr Patrick Andersen
20190101T00:00:00Z

Promoting change in teacher practice through supported differentiation of instruction in mathematics
http://hdl.handle.net/11343/221830
Promoting change in teacher practice through supported differentiation of instruction in mathematics
Dermody, Bryce Gilchrist
Differentiated instruction has been shown to be effective in improving student learning outcomes; however, the resulting work load can be difficult for teachers to manage. A teaching package known as the NRP (Number Resource Package) was created to support teachers to differentiate their instruction, and used effectively in two classrooms. The package allows teachers to identify their students’ current understanding using a diagnostic test and a Guttman Chart, and then provides appropriate material for the area in which students need further consolidation. It assists teachers to identify, and provide instruction for, several different knowledge levels within the one classroom. Use of the NRP in the two experimental classes was compared with five classes that did not use the NRP and continued to follow their school’s mathematics curriculum.
This study involved a quasiexperimental approach, using qualitative and quantitative data. Involved were an experimental group (two teachers) and a control group (five teachers) and a total of 147 year 7 students. The research took place in a large school in western Melbourne, Australia. The qualitative data consisted of three surveys and provided information on the effectiveness of the components in the NRP. The quantitative data consisted of a pre and a posttest completed by students in both the experimental and control groups. These tests were completed at the beginning and the end of a nineweek teaching cycle and the learning gains were determined for each student (i.e. the difference between the pre and posttest). There was a statistically significant difference between the experimental group and the control group when these learning gains were analysed.
The results demonstrated that students in the experimental group who were taught using the NRP showed greater improvement on the posttest when compared to students in the control group. It was noted that those students who performed ‘below’ the expected level and those students who performed ‘above’ the expected level showed the most improvement in the experimental group, when compared with the control group.
© 2019 Dr. Bryce Gilchrist Dermody
20190101T00:00:00Z