Motivated by quantum mechanics and geometric optics, it is a long-standing problem whether the length spectrum of a compact Riemannian manifold can be recovered from its Laplace spectrum. One route to proving that the length spectrum depends on the...
Spectral theory is the subfield of differential geometry which provided the solution to Kac''s famous question, "Can you hear the shape of a drum?" That is, can we use the Laplace spectrum of a manifold to draw conclusions about its geometry or...
Shape modeling is an important area in computer vision and medical imaging. Shape models have proved valuable in the tasks of object representation, recognition, and classification. This thesis focuses on shape models of 3D surfaces extracted from...
Magnetoencephalography -- Technique.; Magnetic susceptibility.
The measurement of human brain function is the goal of many noninvasive medical imaging modalities. Two sources of physiological contrast are commonly targeted to achieve this goal. The most direct approach to measuring neuronal activity is with...
Constructibility (Set theory). Trees (Graph theory).
This thesis investigates possible initial segments of the degrees of constructibility. Specifically, we completely characterize the structure of degrees in generic extensions of the constructible universe L via forcing with Souslin trees. Then we...
This thesis constitutes the first steps in the author's program to investigate the question of when a homotopy of 2-cocycles ω = {ω[subscript t]}[subscript t∊[0,1]] on a locally compact Hausdorff groupoid Ɠ induces an isomorphism of the...
The focus of this thesis is the study of nuclearity and exactness for groupoid crossed product C*-algebras. In particular, we present generalizations of two well-known facts from group dynamical systems and crossed products to the groupoid setting....
We prove the existence of nontrivial multiparameter isospectral deformations of metrics on the classical compact simple Lie groups SO (n) (n = 9, n ≥11), Spin(n) (n = 9, n ≥11), SU (n) (n ≥7), and Sp (n) (n ≥5). The proof breaks into three...
In this thesis we study 1=k-geodesics, those closed geodesics that minimize on any subinterval of length L=k, where L is the length of the geodesic. These curves arise as critical points of the uniform energy, a function introduced in Morse theory...
In this thesis, we study the dynamics of magnetic flows on compact nilmanifolds. Magnetic flows are generalizations of geodesic flows. They model the motion of a particle of unit mass and unit charge in a smooth manifold M in the presence of a...
'Orthogonal modular forms are algebraic modular forms arising from lattices in quadratic spaces. In this thesis, we define orthogonal modular forms, establish their basic properties, and then apply them to a case of ternary quadratic spaces to...
Prostacyclin (PGI2) is a major product of COX-2 derived arachidonic acid metabolism in the endothelium and uterine myometrium. Evidence indicates that PGI 2 protects against development and progression of vascular disease. While it has been known...
Riemannian manifolds. Singularities (Mathematics). Laplacian operator. Spectral theory (Mathematics). Riemann surfaces. Curves on surfaces. Geometry
Historically, inverse spectral theory has been concerned with the relationship between the geometry and the spectrum of compact Riemannian manifolds, where spectrum means the eigenvalue spectrum of the Laplace operator as it acts on smooth...
We present a number of findings concerning groupoid dynamical systems and groupoid crossed products. The primary result is an identification of the spectrum of the groupoid crossed product when the groupoid has continuously varying abelian...
This thesis contains some results concerning groupoid dynamical systems and crossed products. We introduce the notion of a proper groupoid dynamical system and of its generalized fixed point algebra. We show that our notion of proper groupoid...
Topological graph theory. Aperiodicity. Paths and cycles (Graph theory)
The condition ""every cycle has an entry"" first appeared in the literature in Kumjian, Pask, and Raeburn's paper on Cuntz-Krieger algebras of directed graphs, where it was called Condition (L). It provides a necessary condition for simplicity of...
Manifolds (Mathematics) Geodesics (Mathematics) Space and time.
We investigate weak and strong refocusing of light rays in a space-time and related concepts. A strongly causal space-time ( X^ n +1 , g ) is emphstrongly refocusing at x ∈ X if there is a point y ≠ x such that all null-geodesics through y pass...
This thesis investigates the embedding theory of orders in central simple algebras, placing a particular emphasis on the role that the phenomenon known as selectivity plays in the theory. Although the notion of selectivity is completely algebraic,...
Self-reconfiguring (SR) robots are modular systems that can autonomously change shape, or reconfigure , for increased versatility and adaptability in unknown environments. In this thesis, we investigate planning and control for systems of...
Renewable energy sources are becoming increasingly important as sustainable alternatives to fossil fuels. One of these renewable sources, cellulosic biomass, is a relatively inexpensive and abundant resource that can be converted into liquid fuels...