Physical Sciences and Mathematics Commons^™

Relations Between Theta Functions Of Genus One And Two From Geometry, Thomas Hill

Undergraduate Honors Capstone Projects

Genus-two curves with special symmetries are related to pairs of genus-one curves by two and three-sheeted ramified coverings. This classical work dates back to early 20th century and is known as Jacobi and Hermite reduction. Jacobians of genus-two curves can be used to construct complex two-dimensional complex projective manifolds known as Kummer surfaces. On the other hand, the defining coordinates and parameters of both elliptic curves and Kummer surfaces can be related to Riemann Theta functions and Siegel Theta functions, respectively. This result goes back to the seminal work of Mumford in the 1980s. We use the geometric relation between …

Classification Of Spacetimes With Symmetry, Jesse W. Hicks

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Spacetimes with symmetry play a critical role in Einstein's Theory of General Relativity. Missing from the literature is a correct, usable, and computer accessible classification of such spacetimes. This dissertation fills this gap; specifically, we

i) give a new and different approach to the classification of spacetimes with symmetry using modern methods and tools such as the Schmidt method and computer algebra systems, resulting in ninety-two spacetimes;

ii) create digital databases of the classification for easy access and use for researchers;

iii) create software to classify any spacetime metric with symmetry against the new database;

iv) compare results of our …

Geometry And Electronic Structure Of Doped Clusters Via The Coalescence Kick Method, Boris Averkiev

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Developing chemical bonding models in clusters is one of the most challenging tasks of modern theoretical chemistry. There are two reasons for this. The first one is that clusters are relatively new objects in chemistry and have been extensively studied since the middle of the 20th century. The second reason is that clusters require high-level quantum-chemical calculations; while for many classical molecules their geometry and properties can be reasonably predicted by simpler methods.

The aim of this dissertation was to study doped clusters and explain their chemical bonding. The research was focused on three classes of compounds: aluminum clusters doped …

Geometry And Physical Properties Of The Chelungpu Fault, Taiwan, And Their Effect On Fault Rupture, Richard V. Heermance

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Rupture of the Chelungpu fault during the September 21, 1999, 7.6 M_wearthquake in Taiwan caused a 90-Jr,m-long surface rupture with variable displacement along strike. Analysis of core from two holes drilled through the fault zone, combined with geologic mapping and detailed investigation from three outcrops, define the fault geometry and physical properties of the Chelungpu fault in its northern and southern regions. In the northern region, the fault dips 45-60° east parallel to bedding and consists of a narrow (1-20 cm) core of dark-gray, sheared clay gouge at the base of a 30-50 m zone of increased fracture …

The Fundamental Groups Of The Complements Of Some Solid Horned Spheres, Norman William Riebe

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

One of the methods used for the construction of the classical Alexander horned sphere leads naturally to generalization to horned spheres of higher order. Let M₂, denote the Alexander horned sphere. This is a 2-horned sphere of order 2. Denote by M₃ and M₄, two 2-horned spheres of orders 3 and 4, respectively, constructed by such a generalization.

The fundamental groups of the complements of M₂, M₃, and M₄ are derived, and representations of these groups onto the Alternating Group, A₅, are found. The form of the presentations …