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- Creators: Ravel, Maurice, 1875-1937
Description
This thesis describes an approach to system identification based on compressive sensing and demonstrates its efficacy on a challenging classical benchmark single-input, multiple output (SIMO) mechanical system consisting of an inverted pendulum on a cart. Due to its inherent non-linearity and unstable behavior, very few techniques currently exist that are capable of identifying this system. The challenge in identification also lies in the coupled behavior of the system and in the difficulty of obtaining the full-range dynamics. The differential equations describing the system dynamics are determined from measurements of the system's input-output behavior. These equations are assumed to consist of the superposition, with unknown weights, of a small number of terms drawn from a large library of nonlinear terms. Under this assumption, compressed sensing allows the constituent library elements and their corresponding weights to be identified by decomposing a time-series signal of the system's outputs into a sparse superposition of corresponding time-series signals produced by the library components. The most popular techniques for non-linear system identification entail the use of ANN's (Artificial Neural Networks), which require a large number of measurements of the input and output data at high sampling frequencies. The method developed in this project requires very few samples and the accuracy of reconstruction is extremely high. Furthermore, this method yields the Ordinary Differential Equation (ODE) of the system explicitly. This is in contrast to some ANN approaches that produce only a trained network which might lose fidelity with change of initial conditions or if facing an input that wasn't used during its training. This technique is expected to be of value in system identification of complex dynamic systems encountered in diverse fields such as Biology, Computation, Statistics, Mechanics and Electrical Engineering.
ContributorsNaik, Manjish Arvind (Author) / Cochran, Douglas (Thesis advisor) / Kovvali, Narayan (Committee member) / Kawski, Matthias (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2011
Description
In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen.
ContributorsPatani, Nura (Author) / Kaliszewski, Steven (Thesis advisor) / Quigg, John (Thesis advisor) / Bremner, Andrew (Committee member) / Kawski, Matthias (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2011
Description
Bacteriophage (phage) are viruses that infect bacteria. Typical laboratory experiments show that in a chemostat containing phage and susceptible bacteria species, a mutant bacteria species will evolve. This mutant species is usually resistant to the phage infection and less competitive compared to the susceptible bacteria species. In some experiments, both susceptible and resistant bacteria species, as well as phage, can coexist at an equilibrium for hundreds of hours. The current research is inspired by these observations, and the goal is to establish a mathematical model and explore sufficient and necessary conditions for the coexistence. In this dissertation a model with infinite distributed delay terms based on some existing work is established. A rigorous analysis of the well-posedness of this model is provided, and it is proved that the susceptible bacteria persist. To study the persistence of phage species, a "Phage Reproduction Number" (PRN) is defined. The mathematical analysis shows phage persist if PRN > 1 and vanish if PRN < 1. A sufficient condition and a necessary condition for persistence of resistant bacteria are given. The persistence of the phage is essential for the persistence of resistant bacteria. Also, the resistant bacteria persist if its fitness is the same as the susceptible bacteria and if PRN > 1. A special case of the general model leads to a system of ordinary differential equations, for which numerical simulation results are presented.
ContributorsHan, Zhun (Author) / Smith, Hal (Thesis advisor) / Armbruster, Dieter (Committee member) / Kawski, Matthias (Committee member) / Kuang, Yang (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2012
Description
In the 1980's, Gromov and Piatetski-Shapiro introduced a technique called "hybridization'' which allowed them to produce non-arithmetic hyperbolic lattices from two non-commensurable arithmetic lattices. It has been asked whether an analogous hybridization technique exists for complex hyperbolic lattices, because certain geometric obstructions make it unclear how to adapt this technique. This thesis explores one possible construction (originally due to Hunt) in depth and uses it to produce arithmetic lattices, non-arithmetic lattices, and thin subgroups in SU(2,1).
ContributorsWells, Joseph (Author) / Paupert, Julien (Thesis advisor) / Kotschwar, Brett (Committee member) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kawski, Matthias (Committee member) / Arizona State University (Publisher)
Created2019
Description
The purpose of this senior thesis is to explore the abstract ideas that give rise to the well-known Fourier series and transforms. More specifically, finite group representations are used to study the structure of Hilbert spaces to determine under what conditions an element of the space can be expanded as a sum. The Peter-Weyl theorem is the result that shows why integrable functions can be expressed in terms of trigonometric functions. Although some theorems will not be proved, the results that can be derived from them will be briefly discussed. For instance, the Pontryagin Duality theorem states that there is a canonical isomorphism between a group and the second dual of the group, and it can be used to prove $Plancherel$ theorem which essentially says that the Fourier transform is itself a unitary isomorphism.
ContributorsReyna De la Torre, Luis E (Author) / Kaliszewski, Steven (Thesis director) / Rainone, Timothy (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
The theory of frames for Hilbert spaces has become foundational in the study of wavelet analysis and has far-reaching applications in signal and image-processing. Originally, frames were first introduced in the early 1950's within the context of nonharmonic Fourier analysis by Duffin and Schaeffer. It was then in 2000, when M. Frank and D. R. Larson extended the concept of frames to the setting of Hilbert C*-modules, it was in that same paper where they asked for which C*-algebras does every Hilbert C*-module admit a frame. Since then there have been a few direct answers to this question, one being that every Hilbert A-module over a C*-algebra, A, that has faithful representation into the C*-algebra of compact operators admits a frame. Another direct answer by Hanfeng Li given in 2010, is that any C*-algebra, A, such that every Hilbert C*-module admits a frame is necessarily finite dimensional. In this thesis we give an overview of the general theory of frames for Hilbert C*-modules and results answering the frame admittance property. We begin by giving an overview of the existing classical theory of frames in Hilbert spaces as well as some of the preliminary theory of Hilbert C*-modules such as Morita equivalence and certain tensor product constructions of C*-algebras. We then show how some results of frames can be extended to the case of standard frames in countably generated Hilbert C*-modules over unital C*-algebras, namely the frame decomposition property and existence of the frame transform operator. We conclude by going through some proofs/constructions that answer the question of frame admittance for certain Hilbert C*-modules.
ContributorsJaime, Arturo (Author) / Kaliszewski, Steven (Thesis director) / Spielberg, Jack (Committee member) / Aguilar, Konrad (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout.
ContributorsEikenberry, Keenan (Author) / Quigg, John (Thesis advisor) / Kaliszewski, Steven (Thesis advisor) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2016
ContributorsRavel, Maurice, 1875-1937 (Composer) / Goulart, Renato (Arranger)
Description
The role of technology in shaping modern society has become increasingly important in the context of current democratic politics, especially when examined through the lens of social media. Twitter is a prominent social media platform used as a political medium, contributing to political movements such as #OccupyWallStreet, #MeToo, and #BlackLivesMatter. Using the #BlackLivesMatter movement as an illustrative case to establish patterns in Twitter usage, this thesis aims to answer the question “to what extent is Twitter an accurate representation of “real life” in terms of performative activism and user engagement?” The discussion of Twitter is contextualized by research on Twitter’s use in politics, both as a mobilizing force and potential to divide and mislead. Using intervals of time between 2014 – 2020, Twitter data containing #BlackLivesMatter is collected and analyzed. The discussion of findings centers around the role of performative activism in social mobilization on twitter. The analysis shows patterns in the data that indicates performative activism can skew the real picture of civic engagement, which can impact the way in which public opinion affects future public policy and mobilization.
ContributorsTutelman, Laura (Author) / Voorhees, Matthew (Thesis director) / Kawski, Matthias (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Computer Science and Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2021-05
ContributorsRavel, Maurice, 1875-1937 (Composer)