Theses and Dissertations at Montana State University (MSU)
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Item Searching and reconstruction: algorithms with topological descriptors(Montana State University - Bozeman, College of Engineering, 2020) Micka, Samuel Adam; Chairperson, Graduate Committee: Brittany FasyTopological data analysis and, more specifically, persistent homology have received significant attention as a method of describing the shape of complex data. Persistent homology measures the persistence (i.e., 'relative size') of topological features such as connected components, holes, voids, etc. as a space is filtered. The persistence is often plotted in what is referred to as a persistence diagram. Persistence diagrams encode both topological and geometric information about shapes. Moreover, certain parameterized sets of persistence diagrams are sufficient for representing particular classes of shapes. In other words, a set of persistence diagrams can be substituted for the shape. Shape representation using persistence diagrams has shown promise in several learning and classification tasks on shape data. However, choosing a sufficient parameterized set of persistence diagrams is challenging. Previous solutions utilize exhaustive sampling approaches based on assumptions on the geometry of the shape and algebraic results. We consider the inverse problem of 'reconstructing' the shape, using (augmented) persistence diagrams. Developing algorithms for reconstruction provides an alternative method of finding descriptors for representation that is optimized to minimize the number of diagrams and time complexity. In this work, we provide deterministic algorithms for reconstructing simplicial complexes in Rd and discuss challenges in reconstruction using other topological descriptors. Shape reconstruction, and several other areas of research utilizing the persistence diagram, are predisposed to generating large numbers of diagrams. As such, developing efficient methods for searching in the space of persistence diagrams is also of great importance. We consider the bottleneck distance in the space of persistence diagrams. The bottleneck distance is used often in practice for comparing one diagram to another. However, the cost of computing the bottleneck distance can grow prohibitive for large sets of diagrams using a brute-force approach. We offer a data structure and algorithm for identifying the approximate nearest neighbor (and approximate k-nearest neighbors) in the space of persistence diagrams in less time than the brute-force approach for large sets of diagrams.Item Effect of the use of an Internet-based 'problem of the week' on high school geometry student problem-solving achievement and attitudes toward mathematics(Montana State University - Bozeman, College of Education, Health & Human Development, 1999) Goudelock, Clifford P.Item Perplexities and immediate proximity(Montana State University - Bozeman, College of Arts & Architecture, 1992) Ott, Charles RichardPainting provides me with a structure to address perplexities found within my reality. My artistic vision resides neither in an optimistic or pessimistic point of view; rather, it is rooted in the belief that one is held in limbo, suspended between two points of reference. The extremes of history and the future, bridged by nothing more than one’s continually evolving existence, have the potential to act as obstacles to one’s reality. I have found myself at an immediate proximity to reality. The significance of the creative act is found in the resulting object which satisfies the desire to establish an equilibrium between history and the future. The diverse processes and materials incorporated within my work play a vital role in exploring peculiar circumstances which are a part of my reality. The use of geometry, painted passages and objects creates a dynamic dialogue. This dialogue acts as a means of linking together contrasting elements found within the paintings. The four paintings "Light," "Breathing," "Thought," and "Pulse Mechanism," utilize geometry and physical objects to establish the mechanics of a theoretical machine. The tar covered gear system of "Light Mechanism," and the tar covered bicycle wheel of "Breathing Mechanism" are metaphorical symbols for movement within these mechanisms. The plausibility of these works is dependent upon the integrity of particular elements working together to create a theoretical machine.Item The effect on final achievement in solid geometry of an introductory unit based upon developing visualization and understanding through the use of models(Montana State University - Bozeman, College of Education, Health & Human Development, 1962) Picton, John O.Item Unity' : a site specific public sculpture(Montana State University - Bozeman, College of Arts & Architecture, 2001) Mueller, Greg L.My thesis proposes that sculpture can provide a focus for community gathering and individual contemplation. I embrace the Benedictine philosophy that values time for contemplation and encourages the individual to contribute to his community. This personal vision is made manifest in Unity, a site-specific public sculpture. My thematically unified complex consists of a series of forms which define a path, a portal and a resting place. I use archetypal forms to represent the search for spiritual discovery. I chose to install Unity on the front lawn of Haynes Hall to bring vitality to a dispirited site, and to address the social need for a School of Art gathering space. My forms are based on two primary influences: my fascination with the principles of sacred geometry and metal’s ability to celebrate the inherent beauty of structure.Item Three methods of instruction in high school geometry and the effects they have on achievement, retention, and attitude(Montana State University - Bozeman, College of Education, Health & Human Development, 1992) Thompson, Edward OtisItem Analysis of group problem-solving tasks in a geometry course for teachers using computer-mediated conferencing(Montana State University - Bozeman, College of Education, Health & Human Development, 2000) Beaudrie, Brian PatrickItem The preparation of perspective teachers in the geometry content of elementary school mathematics texts(Montana State University - Bozeman, College of Education, Health & Human Development, 1971) Banning, Margaret Neoma Botkin