Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.

This study compares the scholastic achievement of 2,811 students enrolled in Fairfax County, Virginia, Public Schools for the 1983-1984 school year. Scholastic achievement of an experimental group of 1,068 students enrolled in combination or split/grade classes is compared with a control group of 1,743 students enrolled in regular graded classes. Five research questions were developed, three of which related directly to grade level student scholastic achievement by comparing test results for combination and regular grade classes, and two which attempted to identify any significance resulting from differences used by principals to select teachers and students for placement in combination classes. / Ed. D.

This thesis documents the optimized design of a touch screen using infrared technology as a three dimensional problem. The framework is fundamentally built on laser diode technology and introduces mirrors for signal reflection. The rising popularity of touch screens are credited to the naturally intuitive control of display interfaces, extensive data presentation, and the improved manufacturing process of various touch screen implementations. Considering the demands on touch screen technology, the design for a large scaled touch panel is inevitable, and signal reduction techniques become a necessity to facilitate signal processing and accurate touch detection. The developed research model seeks to capture realistic touch screen design limitations to create a deploy-able configuration. The motivation of the problem stems from the significant reduction of representation achieved by combinatorial group theory. The research model is of difficulty NP-complete. Additional exclusive-or functions for uniqueness, strengthening model search space, symmetry eliminating constraints, and implementation constraints are incorporated for enhanced performance. The computational results and analysis of objectives, valuing the emphasis on diodes and layers are evaluated. The evaluation of trade-off between diodes and layers is also investigated.

This thesis documents the optimized design of a touch screen using infrared technology as a three dimensional problem. The framework is fundamentally built on laser diode technology and introduces mirrors for signal reflection. The rising popularity of touch screens are credited to the naturally intuitive control of display interfaces, extensive data presentation, and the improved manufacturing process of various touch screen implementations. Considering the demands on touch screen technology, the design for a large scaled touch panel is inevitable, and signal reduction techniques become a necessity to facilitate signal processing and accurate touch detection. The developed research model seeks to capture realistic touch screen design limitations to create a deploy-able configuration. The motivation of the problem stems from the significant reduction of representation achieved by combinatorial group theory. The research model is of difficulty NP-complete. Additional exclusive-or functions for uniqueness, strengthening model search space, symmetry eliminating constraints, and implementation constraints are incorporated for enhanced performance. The computational results and analysis of objectives, valuing the emphasis on diodes and layers are evaluated. The evaluation of trade-off between diodes and layers is also investigated.

Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.