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This report explores several applications of iteration and the various connections that can be made to different areas of mathematics. The ties iteration has to the Wada Property, bifurcation diagram, root finding, and applications in geometry are all investigated. Finally, a rationale for incorporating iteration into secondary mathematics courses to support a more robust curriculum is discussed. / text

The purpose of the study was to compare the effectiveness of visualization of natural phenomena via computer simulation and manipulations of concrete objects, in a physics laboratory, on the development of students’ inquiry skills in mechanics. A quasi-experimental method that employed the 2 Learning Tools x 2 Time of learning split-plot factorial design was applied in the study. The sample consisted of 54 Grade 11 students from two physics classes of the university preparation section in the Ottawa-Catholic School Board. One class was assigned to interactive computer simulations (treatment) and the other to concrete objects in physics laboratory (control) as learning tools. Both tools were embedded in the general framework of the guided-inquiry cycle approach. The results showed that the interaction effect of the Learning Tools x Time of learning was not statistically significant. However, the results also showed a significant effect on the development of students’ inquiry skills (indicated by the pre- and post-inquiry skills test) regardless of the type of learning tool they had used. The findings suggested that these two strategies are effective in developing students’ inquiry skills in mechanics.

The purpose of the study was to compare the effectiveness of visualization of natural phenomena via computer simulation and manipulations of concrete objects, in a physics laboratory, on the development of students’ inquiry skills in mechanics. A quasi-experimental method that employed the 2 Learning Tools x 2 Time of learning split-plot factorial design was applied in the study. The sample consisted of 54 Grade 11 students from two physics classes of the university preparation section in the Ottawa-Catholic School Board. One class was assigned to interactive computer simulations (treatment) and the other to concrete objects in physics laboratory (control) as learning tools. Both tools were embedded in the general framework of the guided-inquiry cycle approach. The results showed that the interaction effect of the Learning Tools x Time of learning was not statistically significant. However, the results also showed a significant effect on the development of students’ inquiry skills (indicated by the pre- and post-inquiry skills test) regardless of the type of learning tool they had used. The findings suggested that these two strategies are effective in developing students’ inquiry skills in mechanics.

The purpose of the study was to compare the effectiveness of visualization of natural phenomena via computer simulation and manipulations of concrete objects, in a physics laboratory, on the development of students’ inquiry skills in mechanics. A quasi-experimental method that employed the 2 Learning Tools x 2 Time of learning split-plot factorial design was applied in the study. The sample consisted of 54 Grade 11 students from two physics classes of the university preparation section in the Ottawa-Catholic School Board. One class was assigned to interactive computer simulations (treatment) and the other to concrete objects in physics laboratory (control) as learning tools. Both tools were embedded in the general framework of the guided-inquiry cycle approach. The results showed that the interaction effect of the Learning Tools x Time of learning was not statistically significant. However, the results also showed a significant effect on the development of students’ inquiry skills (indicated by the pre- and post-inquiry skills test) regardless of the type of learning tool they had used. The findings suggested that these two strategies are effective in developing students’ inquiry skills in mechanics.

As groundwater models are being used increasingly in the area of resource allocation, there has been an increase in the level of complexity in an attempt to capture heterogeneity, complex geometries and detail in interaction between the model domain and the outside hydraulic influences. As models strive to represent the real world in ever increasing detail, there is a strong likelihood that the boundary conditions will become nonlinear. Nonlinearities exist in the groundwater flow equation even in simple models when watertable (unconfined) conditions are simulated. This thesis is concerned with how these nonlinearities are treated numerically, with particular focus on the MODFLOW groundwater flow software and the nonlinear nature of the unconfined condition simulation. One of the limitations of MODFLOW is that it employs a first order fixed point iterative scheme to linearise the nonlinear system that arises as a result of the finite difference discretisation process, which is well known to offer slow convergence rates for highly nonlinear problems. However, Newton's method can achieve quadratic convergence and is more effective at dealing with higher levels of nonlinearity. Consequently, the main objective of this research is to investigate the inclusion of Newton's method to the suite of computational tools in MODFLOW to enhance its flexibility in dealing with the increasing complexity of real world problems, as well as providing a more competitive and efficient solution methodology. Furthermore, the underpinning linear iterative solvers that MODFLOW currently utilises are targeted at symmetric systems and a consequence of using Newton's method would be the requirement to solve non-symmetric Jacobian systems. Therefore, another important aspect of this work is to investigate linear iterative solution techniques that handle such systems, including the newer Krylov style solvers GMRES and BiCGSTAB. To achieve these objectives a number of simple benchmark problems involving nonlinearities through the simulation of unconfined conditions were established to compare the computational performance of the existing MODFLOW solvers to the new solution strategies investigated here. One of the highlights of these comparisons was that Newton's method when combined with an appropriately preconditioned Krylov solver was on average greater than 40% more CPU time efficient than the Picard based solution techniques. Furthermore, a significant amount of this time saving came from the reduction in the number of nonlinear iterations due to the quadratic nature of Newton's method. It was also found that Newton's method benefited more from improved initial conditions than Picard's method. Of all the linear iterative solvers tested, GMRES required the least amount of computational effort. While the Newton method involves more complexity in its implementation, this should not be interpreted as prohibitive in its application. The results here show that the extra work does result in performance increase, and thus the effort is certainly worth it.

The purpose of the study was to compare the effectiveness of visualization of natural phenomena via computer simulation and manipulations of concrete objects, in a physics laboratory, on the development of students’ inquiry skills in mechanics. A quasi-experimental method that employed the 2 Learning Tools x 2 Time of learning split-plot factorial design was applied in the study. The sample consisted of 54 Grade 11 students from two physics classes of the university preparation section in the Ottawa-Catholic School Board. One class was assigned to interactive computer simulations (treatment) and the other to concrete objects in physics laboratory (control) as learning tools. Both tools were embedded in the general framework of the guided-inquiry cycle approach. The results showed that the interaction effect of the Learning Tools x Time of learning was not statistically significant. However, the results also showed a significant effect on the development of students’ inquiry skills (indicated by the pre- and post-inquiry skills test) regardless of the type of learning tool they had used. The findings suggested that these two strategies are effective in developing students’ inquiry skills in mechanics.

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<p> This thesis examines continued fraction expansions of the square root of nonsquare positive integers of periods one to six, and shows their relationships with Newton's method of approximation. It also contains known results concerning continued fractions.</p> / Thesis / Master of Science (MSc)

The human risk as a spatial component using Newton?s Gravity model is explored for the first time in this research along with fuels and road density variables to predict the fire occurrences for Southeast Mississippi fire district. The fire risk map indicated that fires occur in clusters and are dependent on fire size and distance from roads. The results proved that fire size increases as fire locations increase. Fire locations decrease significantly beyond 160 meters with increasing road distance. The Gravity model proved to be a better estimator of fire risk while a similar road density model proved to be better in very low and medium fire risk zones. Compared to road density, the gravity model significantly proved a better estimate of very low fire risk for all seasons and summer low fire risk. For all other fire risks, though, gravity model showed better results no significant differences were observed.