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Articles 1 - 9 of 9
Full-Text Articles in Geometry and Topology
Infinite Line, Infinite Knowledge: The 'Spera' And Organized Chaos In Lambert's Encyclopedia, The 'Liber Floridus', Ava Romano
Theses and Dissertations
The Liber Floridus is a medieval encyclopedia renowned for its program of circular diagrams, or sperae. Inside this manuscript of 190 chapters, these diagrams frame and embody written knowledge, revealing a connection between encyclopedism and life in the Benedictine monastery by creating a coherent visual organization of chapters.
Plumbed 3-Manifolds And Neumann Moves, Noah J. Pope
Plumbed 3-Manifolds And Neumann Moves, Noah J. Pope
Theses and Dissertations
We give a constructive proof that every weakly negative definite plumbing tree can be transformed into a negative definite one by a finite sequence of Neumann moves. The argument combines Neumann’s plumbing calculus with the diagonalization algorithm of Duchon, Eisenbud, and Neumann, which extracts the eigenvalues of the framing matrix directly from the combinatorics of the tree. We show that any positive eigenvalues are supported on linear branches and can be eliminated systematically via controlled applications of Neumann moves. This provides an explicit algorithm reducing weakly negative definite plumbing trees to negative definite ones.
A Preliminary Study Of Hilbert–Kunz Functions: Coefficient Behavior In A Normal Affine Semigroup Ring, Jesus A. Mendiola Herrera
A Preliminary Study Of Hilbert–Kunz Functions: Coefficient Behavior In A Normal Affine Semigroup Ring, Jesus A. Mendiola Herrera
Theses and Dissertations
In 1890, David Hilbert published a set of notes on what now constitutes one of the bases of Commutative Algebra; his work would eventually influence the efforts of mathematicians like Ernst Kunz. In 1969, Ernst Kunz introduced a particular mapping regarding modules of regular local rings. His goal was to characterize Noetherian local rings of prime characteristic by computing the length of the composition series under Frobenius power transformations. In this thesis, the focus will be on stating the initial steps on finding the coefficients of the Hilbert-Kunz function of the normal affine semigroup ring of the form R = …
The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.
The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.
Theses and Dissertations
Gradient descent is a popular optimization method that utilizes a model’s prediction error to iteratively improve its parameters for a given task. The functions that measure this error can be defined to align with the user’s goals and sometimes satisfy metric or norm properties. It is common for these functions to measure over Rn, but any differentiable space allows for gradient descent to occur. There has been some research investigating the influence of topological spaces on optimization methods, but it is a limited field of study. This thesis further explores this phenomenon by applying a transformation prediction model to multiple …
Invariance And Invertibility In Deep Neural Networks, Han Zhang
Invariance And Invertibility In Deep Neural Networks, Han Zhang
Theses and Dissertations
Machine learning is concerned with computer systems that learn from data instead of being explicitly programmed to solve a particular task. One of the main approaches behind recent advances in machine learning involves neural networks with a large number of layers, often referred to as deep learning. In this dissertation, we study how to equip deep neural networks with two useful properties: invariance and invertibility. The first part of our work is focused on constructing neural networks that are invariant to certain transformations in the input, that is, some outputs of the network stay the same even if the input …
3-Maps And Their Generalizations, Kevin J. Mccall
3-Maps And Their Generalizations, Kevin J. Mccall
Theses and Dissertations
A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.
Series Solutions Of Polarized Gowdy Universes, Doniray Brusaferro
Series Solutions Of Polarized Gowdy Universes, Doniray Brusaferro
Theses and Dissertations
Einstein's field equations are a system of ten partial differential equations. For a special class of spacetimes known as Gowdy spacetimes, the number of equations is reduced due to additional structure of two dimensional isometry groups with mutually orthogonal Killing vectors. In this thesis, we focus on a particular model of Gowdy spacetimes known as the polarized T3 model, and provide an explicit solution to Einstein's equations.
Classification Of Compact 2-Manifolds, George H. Winslow
Classification Of Compact 2-Manifolds, George H. Winslow
Theses and Dissertations
It is said that a topologist is a mathematician who can not tell the difference between a doughnut and a coffee cup. The surfaces of the two objects, viewed as topological spaces, are homeomorphic to each other, which is to say that they are topologically equivalent. In this thesis, we acknowledge some of the most well-known examples of surfaces: the sphere, the torus, and the projective plane. We then observe that all surfaces are, in fact, homeomorphic to either the sphere, the torus, a connected sum of tori, a projective plane, or a connected sum of projective planes. Finally, we …
Numerical Investigation Of Pre-Detonator Geometries For Pde Applications, Robert T. Fievisohn
Numerical Investigation Of Pre-Detonator Geometries For Pde Applications, Robert T. Fievisohn
Theses and Dissertations
A parametric study was performed to determine optimal geometries to allow the successful transition of a detonation from a pre-detonator into the thrust tube of a pulse detonation engine. The study was performed using a two-dimensional Euler solver with progress variables to model the chemistry. The geometrical configurations for the simulations look at the effect of shock reflections, flow obstructions, and detonation diffraction to determine successful geometries. It was observed that there are success and failure rates associated with pre-detonators. These success rates appear to be determined by the transverse wave structure of a stably propagating detonation wave and must …