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- Genre: Academic theses
- Creators: Cochran, Douglas
- Creators: Ilkturk, Utku
Description
Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations.
ContributorsRobinson, Benjamin (Author) / Cochran, Douglas (Thesis advisor) / Moran, William (Thesis advisor) / Boggess, Albert (Committee member) / Milner, Fabio (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2014
Description
Modern measurement schemes for linear dynamical systems are typically designed so that different sensors can be scheduled to be used at each time step. To determine which sensors to use, various metrics have been suggested. One possible such metric is the observability of the system. Observability is a binary condition determining whether a finite number of measurements suffice to recover the initial state. However to employ observability for sensor scheduling, the binary definition needs to be expanded so that one can measure how observable a system is with a particular measurement scheme, i.e. one needs a metric of observability. Most methods utilizing an observability metric are about sensor selection and not for sensor scheduling. In this dissertation we present a new approach to utilize the observability for sensor scheduling by employing the condition number of the observability matrix as the metric and using column subset selection to create an algorithm to choose which sensors to use at each time step. To this end we use a rank revealing QR factorization algorithm to select sensors. Several numerical experiments are used to demonstrate the performance of the proposed scheme.
ContributorsIlkturk, Utku (Author) / Gelb, Anne (Thesis advisor) / Platte, Rodrigo (Thesis advisor) / Cochran, Douglas (Committee member) / Renaut, Rosemary (Committee member) / Armbruster, Dieter (Committee member) / Arizona State University (Publisher)
Created2015