Browsing by Author "Shonkwiler, Clayton, advisor"

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Open Access
Hodge and Gelfand theory in Clifford analysis and tomography
(Colorado State University. Libraries, 2022) Roberts, Colin, author; Shonkwiler, Clayton, advisor; Adams, Henry, committee member; Bangerth, Wolfgang, committee member; Roberts, Jacob, committee member
There is an interesting inverse boundary value problem for Riemannian manifolds called the Calderón problem which asks if it is possible to determine a manifold and metric from the Dirichlet-to-Neumann (DN) operator. Work on this problem has been dominated by complex analysis and Hodge theory and Clifford analysis is a natural synthesis of the two. Clifford analysis analyzes multivector fields, their even-graded (spinor) components, and the vector-valued Hodge–Dirac operator whose square is the Laplace–Beltrami operator. Elements in the kernel of the Hodge–Dirac operator are called monogenic and since multivectors are multi-graded, we are able to capture the harmonic fields of Hodge theory and copies of complex holomorphic functions inside the space of monogenic fields simultaneously. We show that the space of multivector fields has a Hodge–Morrey-like decomposition into monogenic fields and the image of the Hodge–Dirac operator. Using the multivector formulation of electromagnetism, we generalize the electric and magnetic DN operators and find that they extract the absolute and relative cohomologies. Furthermore, those operators are the scalar components of the spinor DN operator whose kernel consists of the boundary traces of monogenic fields. We define a higher dimensional version of the Gelfand spectrum called the spinor spectrum which may be used in a higher dimensional version of the boundary control method. For compact regions of Euclidean space, the spinor spectrum is homeomorphic to the region itself. Lastly, we show that the monogenic fields form a sheaf that is locally homeomorphic to the underlying manifold which is a prime candidate for solving the Calderón problem using analytic continuation.
Open Access
Improved stick number upper bounds
(Colorado State University. Libraries, 2019) Eddy, Thomas D., author; Shonkwiler, Clayton, advisor; Adams, Henry, committee member; Chitsaz, Hamid, committee member
A stick knot is a mathematical knot formed by a chain of straight line segments. For a knot K, define the stick number of K, denoted stick(K), to be the minimum number of straight edges necessary to form a stick knot which is equivalent to K. Stick number is a knot invariant whose precise value is unknown for the large majority of knots, although theoretical and observed bounds exist. There is a natural correspondence between stick knots and polygons in R3. Previous research has attempted to improve observed stick number upper bounds by computationally generating such polygons and identifying the knots that they form. This thesis presents a new variation on this method which generates equilateral polygons in tight confinement, thereby increasing the incidence of polygons forming complex knots. Our generation strategy is to sample from the space of confined polygons by leveraging the toric symplectic structure of this space. An efficient sampling algorithm based on this structure is described. This method was used to discover the precise stick number of knots 935, 939, 943, 945, and 948. In addition, the best-known stick number upper bounds were improved for 60 other knots with crossing number ten and below.
Open Access
Normalizing Parseval frames by gradient descent
(Colorado State University. Libraries, 2024) Caine, Anthony, author; Peterson, Chris, advisor; Shonkwiler, Clayton, advisor; Adams, Henry, committee member; Neilson, Jamie, committee member
Equinorm Parseval Frames (ENPFs) are collections of equal-length vectors that form Parseval frames, meaning they are spanning sets that satisfy a version of the Parseval identity. As such, they have many of the desirable features of orthonormal bases for signal processing and data representation, but provide advantages over orthonormal bases in settings where redundancy is important to provide robustness to data loss. We give three methods for normalizing Parseval frames: that is, flowing a generic Parseval frame to an ENPF. This complements prior work showing that equal-norm frames could be "Parsevalized" and potentially provides new avenues for attacking the Paulsen problem, which seeks sharp upper bounds on the distance to the space of ENPFs in terms of norm and spectral data. This work is based on ideas from symplectic geometry and geometric invariant theory.
Open Access
Sparse matrix varieties, Daubechies spaces, and good compression regions of Grassmann manifolds
(Colorado State University. Libraries, 2024) Collery, Brian, author; Peterson, Chris, advisor; Shonkwiler, Clayton, advisor; Cavalieri, Renzo, committee member; Kirby, Michael, committee member; Pouchet, Louis-Nöel, committee member
The Grassmann manifold Gr(k, n) is a geometric object whose points parameterize k dimensional subspaces of Rn. The flag manifold is a generalization in that its points parameterize flags of vector spaces in Rn. This thesis concerns applications of the geometry of the Grassmann and flag manifolds, with an emphasis on image compression. As a motivating example, the discrete versions of Daubechies wavelets generate distinguished n-dimensional subspaces of R2n that can be considered as distinguished points on Gr(n, 2n). We show that geodesic paths between "Daubechies points" parameterize families of "good" image compression matrices. Furthermore, we show that these paths lie on a distinguished Schubert cell in the Grassmannian. Inspired by the structure of Daubechies wavelets, we define and explore sparse matrix varieties as a generalization. Keeping in that theme, we are interested in understanding geometric considerations that constrain the "good" compression region of a Grassmann manifold.