28 citations · 270 across the 43 of their papers we have counts for
9 papers · 1 filter
Geometric Approaches to Matrix Normalization and Graph Balancing
Tom Needham, Clayton Shonkwiler
Normal matrices, or matrices which commute with their adjoints, are of fundamental importance in pure and applied mathematics. In this paper, we study a natural functional on the s…
On the existence of Parseval frames for vector bundles
Samuel A. Ballas, Tom Needham, Clayton Shonkwiler
Frames in finite-dimensional vector spaces are spanning sets of vectors which provide redundant representations of signals. The Parseval frames are particularly useful and importan…
Distributions of Distances and Volumes of Balls in Homogeneous Lens Spaces
Brenden Balch, Chris Peterson, Clayton Shonkwiler
Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-…
Expected Distances on Manifolds of Partially Oriented Flags
Brenden Balch, Chris Peterson, Clayton Shonkwiler
Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are…
The symplectic geometry of closed equilateral random walks in 3-space
Jason Cantarella, Clayton Shonkwiler
A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. Th…
The Expected Total Curvature of Random Polygons
Jason Cantarella, Alexander Y Grosberg, Robert B. Kusner +1
We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference be…
Probability Theory of Random Polygons from the Quaternionic Viewpoint
Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler
We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the comp…
The complete Dirichlet-to-Neumann map for differential forms
Vladimir Sharafutdinov, Clayton Shonkwiler
The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold with boundary is a generalization of the classical Dirichlet-to-Neumann map which arises in the problem…
Poincare duality angles for Riemannian manifolds with boundary
Clayton Shonkwiler
On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concr…