Geometry

We introduce a novel geometric approach to characterize entanglement relations in large quantum systems. Our approach is inspired by Schumacher’s singlet state triangle inequality, which used an entropic-based distance to capture the strange properties of entanglement using geometric-based inequalities. Schumacher uses classical entropy and can only describe the geometry of bipartite states. We extend his approach by using von Neumann entropy to create an entanglement monotone that can be generalized for higher dimensional systems. We achieve this by utilizing recent definitions for entropic areas, volumes, and higher dimensional volumes for multipartite which we introduce in this thesis. This enables us to differentiate systems with high quantum correlation from systems with low quantum correlation and differentiate between different types of multi-partite entanglement. It also enable us to describe some of the strange properties of quantum entanglement using simple geometrical inequalities. Our geometrization of entanglement provides new insight into quantum entanglement. Perhaps by constructing well motivated geometrical structures (e.g. relations among areas, volumes ...), a set of trivial geometrical inequalities can reveal some of the complex properties of higher-dimensional entanglement in multi-partite systems. We provide numerous illustrative applications of this approach.

An algebraic surface defined by an equation of the form z2 = (x+a1y) ... (x + any) (x - 1) is studied, from both an algebraic and geometric point of view. It is shown that the surface is rational and contains a singular point which is nonrational. The class group of Weil divisors is computed and the Brauer group of Azumaya algebras is studied. Viewing the surface as a cyclic cover of the affine plane, all of the terms in the cohomology sequence of Chase, Harrison and Roseberg are computed.

The purpose of this research was to identify if 1) there is a difference in student achievement between students who use the GeoLeg manipulative tool and students who use a traditional compass, protractor, and ruler on the same geometry unit; 2) there is a difference in student achievement between the genders between those who use the GeoLeg manipulative tool and those students who do not; and 3) there is a relationship between identified learning styles and student achievement on a geometry unit posttest after using the GeoLeg manipulative tool. There were 317 students in the study. The research found that students using the GeoLeg manipulative tool produced significantly better student performance on a posttest in this particular school setting. Although these results cannot be generalized to other school sites, it is plausible that these results could generalize to school sites whose demographics are similar. The research findings revealed that there was no statistically significant difference between male and female students within the treatment group. The significant finding is that the GeoLeg manipulative tool appears to work equally well with both genders. None of the learning styles, as identified by the Honey and Mumford Learning Styles Questionnaire, were correlated with student posttest score achievement on the tested geometry unit. In addition, there was no evidence to suggest that a student's learning style moderates the effectiveness of the use of the GeoLeg manipulative tool. There is no evidence to suggest that the effectiveness of the GeoLeg manipulative tool is any different depending upon the student's gender or learning style. The results of this research provide strong support for the use of the GeoLeg manipulative tool for improving student performance. Further research is needed to confirm these results in similar and different populations.