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University of Colorado
1.
DeSalvo, Jason A.
A Cross Section of Oscillator Dynamics.
Degree: MS, Applied Mathematics, 2010, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/11 
► The goal of this research is to explore criteria sufficient to produce oscillations, sample some dynamical systems that oscillate, and investigate synchronization. A discussion…
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▼ The goal of this research is to explore criteria sufficient to produce oscillations, sample some dynamical systems that oscillate, and investigate synchronization. A discussion on linear oscillators attempts to demonstrate why autonomous oscillators are inherently nonlinear in nature. After describing some criteria on second-order dynamics that ensure periodic orbits, we explore the dynamics of two second-order oscillators in both autonomous and periodically driven fashion. Finally, we investigate the phenomena of synchronization with the nonlinear phase-locked loop. Methods of analysis are exemplified as they become relevant including Poincaré; maps and the Zero-One test for chaos.
The Poincaré-Bendixson theorem is used to demonstrate the existence of periodic orbits in R2 under extraordinarily general conditions. Liénard's equation and theorem are introduced, which provide an intuitive parameterization for a class of oscillators. Liénard's equation is a second- order, ordinary differential equation that characterizes an oscillator with respect a state dependent damping function and a restoring force function. Liénard's theorem establishes sufficient criteria under which the Liénard's equation has a unique, stable, limit cycle.
The Duffing equation conforms with the Liénard equation, yet produces limit cycles without satisfying Liénard's theorem. Our Duffing dynamics are explained in the context of a nonlinear spring model. We survey the parameter space, which form both pitchfork and hyperbolic potential wells with respect to the displacement. These two wells characterize the bifurcations between the four fundamental undamped dynamical modes. One interesting result is that chaotic trajectories of the Duffing equation are able to quickly shed light on a multitude of quasi-periodic trajectories at the boundaries of the Poincaré map.
Next we introduce an oscillator that is similar to many engineered oscillators. The Van der Pol (VDP) oscillator model is presented in the context of a nonlinear current source in parallel with an inductor, a capacitor, and a resistor. It provides a net negative conductance destabilizing the equilibrium, and is tamed into global stability by increasing damping by the square of the voltage. The VDP oscillator is the opposite of the Duffing equation in that its nonlinearity is in the damping function, with a linear restoring force function. Like the VDP oscillator, many engineered oscillators are self-excited, autonomous systems that produce limit cycles.
Finally, we investigate the process of synchronization with the phase-locked loop (PLL). Synchronization is a nonlinear process in which systems entrain their frequencies to external signals or other systems. Naturally occurring PLLs lie at the foundation of synchronization. We describe the basic topology of the PLL. Interestingly, the phase model introduced conforms with Liénard's equation and is similar to the model used for the Josephson junction and the driven pendulum. Perhaps explaining the…
Advisors/Committee Members: James Meiss, James Curry.
Subjects/Keywords: Duffing; Dynamics; Oscillators; Synchronization; Van der Pol; Applied Mathematics
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APA (6th Edition):
DeSalvo, J. A. (2010). A Cross Section of Oscillator Dynamics. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/11
Chicago Manual of Style (16th Edition):
DeSalvo, Jason A. “A Cross Section of Oscillator Dynamics.” 2010. Masters Thesis, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/appm_gradetds/11.
MLA Handbook (7th Edition):
DeSalvo, Jason A. “A Cross Section of Oscillator Dynamics.” 2010. Web. 26 Jan 2021.
Vancouver:
DeSalvo JA. A Cross Section of Oscillator Dynamics. [Internet] [Masters thesis]. University of Colorado; 2010. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/appm_gradetds/11.
Council of Science Editors:
DeSalvo JA. A Cross Section of Oscillator Dynamics. [Masters Thesis]. University of Colorado; 2010. Available from: https://scholar.colorado.edu/appm_gradetds/11
University of Colorado
2.
Aguiar, Izabel Pirimai.
Dynamic Active Subspaces: a Data-Driven Approach to Computing Time-Dependent Active Subspaces in Dynamical Systems.
Degree: MS, 2018, University of Colorado
URL: https://scholar.colorado.edu/csci_gradetds/169
► Computational models are aiding in the advancement of science – from biological, to engineering, to social systems. To trust the predictions of computational models, however,…
(more)
▼ Computational models are aiding in the advancement of science – from biological, to engineering, to social systems. To trust the predictions of computational models, however, we must understand how the errors in the models’ inputs (i.e., through measurement error) affect the output of the systems: we must quantify the uncertainty that results from these input errors. Uncertainty quantification (UQ) becomes computationally complex when there are many parameters in the model. In such cases it is useful to reduce the dimension of the problem by identifying unimportant parameters and disregarding them for UQ studies. This makes an otherwise intractable UQ problem tractable. <i>Active subspaces</i> extend this idea by identifying important linear combinations of parameters, enabling more powerful and effective dimension reduction. Although active subspaces give model insight and computational tractability for scalar-valued functions, it is not enough. This analysis does not extend to time-dependent systems. In this thesis we discuss time-dependent, dynamic active subspaces. We develop a methodology by which to compute and approximate dynamic active subspaces, and introduce the analytical form of dynamic active subspaces for two cases. To highlight these methods we find dynamic active subspaces for a linear harmonic oscillator and a nonlinear enzyme kinetics system.
Advisors/Committee Members: Paul G. Constantine, Elizabeth Bradley, James Curry, Gianluca Iaccarino.
Subjects/Keywords: active subspaces; dimension reduction; dynamical systems; mathematical biology; sensitivity analysis; uncertainty quantification; Applied Mathematics; Computer Sciences
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APA (6th Edition):
Aguiar, I. P. (2018). Dynamic Active Subspaces: a Data-Driven Approach to Computing Time-Dependent Active Subspaces in Dynamical Systems. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/csci_gradetds/169
Chicago Manual of Style (16th Edition):
Aguiar, Izabel Pirimai. “Dynamic Active Subspaces: a Data-Driven Approach to Computing Time-Dependent Active Subspaces in Dynamical Systems.” 2018. Masters Thesis, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/csci_gradetds/169.
MLA Handbook (7th Edition):
Aguiar, Izabel Pirimai. “Dynamic Active Subspaces: a Data-Driven Approach to Computing Time-Dependent Active Subspaces in Dynamical Systems.” 2018. Web. 26 Jan 2021.
Vancouver:
Aguiar IP. Dynamic Active Subspaces: a Data-Driven Approach to Computing Time-Dependent Active Subspaces in Dynamical Systems. [Internet] [Masters thesis]. University of Colorado; 2018. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/csci_gradetds/169.
Council of Science Editors:
Aguiar IP. Dynamic Active Subspaces: a Data-Driven Approach to Computing Time-Dependent Active Subspaces in Dynamical Systems. [Masters Thesis]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/csci_gradetds/169
University of Colorado
3.
Damle, Anil.
Near Optimal Rational Approximations of Large Data Sets.
Degree: MS, Applied Mathematics, 2011, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/16
► We introduce a new computationally efficient algorithm for constructing near optimal rational approximations of large data sets. In contrast to wavelet-type approximations often used…
(more)
▼ We introduce a new computationally efficient algorithm for constructing near optimal rational approximations of large data sets. In contrast to wavelet-type approximations often used for the same purpose, these new approximations are effectively shift invariant. On the other hand, when dealing with large data sets the complexity of our current non-linear algorithms for computing near optimal rational approximations prevents their direct use. By using an intermediate representation of the data via B-splines, followed by a rational approximation of the B-splines themselves, we obtain a suboptimal rational approximation of data segments. Then, using reduction and merging algorithms for data segments, we arrive at an efficient procedure for computing near optimal rational approximations for large data sets. A motivating example is the compression of audio signals and we provide several examples of compressed representations produced by the algorithm.
Advisors/Committee Members: Gregory Beylkin, Lucas Monzon, James Curry.
Subjects/Keywords: Approximation by rational functions; Applied Mathematics
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APA (6th Edition):
Damle, A. (2011). Near Optimal Rational Approximations of Large Data Sets. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/16
Chicago Manual of Style (16th Edition):
Damle, Anil. “Near Optimal Rational Approximations of Large Data Sets.” 2011. Masters Thesis, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/appm_gradetds/16.
MLA Handbook (7th Edition):
Damle, Anil. “Near Optimal Rational Approximations of Large Data Sets.” 2011. Web. 26 Jan 2021.
Vancouver:
Damle A. Near Optimal Rational Approximations of Large Data Sets. [Internet] [Masters thesis]. University of Colorado; 2011. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/appm_gradetds/16.
Council of Science Editors:
Damle A. Near Optimal Rational Approximations of Large Data Sets. [Masters Thesis]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/16
University of Colorado
4.
Davis, Christopher-Ian Raphael.
Numerical Tests of the Fokas Method for Helmholtz-type Partial Differential Equations: Dirichlet to Neumann Maps.
Degree: MS, Applied Mathematics, 2011, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/17
► A method for solving boundary value problems for linear partial differential equations in convex polygons developed by A.S. Fokas in the late 1990s is…
(more)
▼ A method for solving boundary value problems for linear partial differential equations in convex polygons developed by A.S. Fokas in the late 1990s is introduced. In order to solve well-posed boundary value problems using the novel Fokas approach, certain global relations must be derived. These global relations yield so-called Dirichlet to Neumann maps which not only allow us to solve Helmholtz-type PDEs using the Fokas method, but they are also of interest in their own right. Given a convex polygon and a prescribed set of boundary conditions associated with a PDE, the Dirichlet to Neumann map enables us to numerically recover unknown boundary conditions with relatively high accuracy without solving the PDE on the interior. The numerical implementation of the Dirichlet to Neumann map is shown to be an efficient and accurate method for resolving unknown boundary conditions. The map is also analyzed and certain parameters are optimized. With an accurate Dirichlet to Neumann map, solving the modified Helmholtz and the Helmholtz equations via the Fokas method becomes possible.
Advisors/Committee Members: Bengt Fornberg, James Curry, Natasha Flyer.
Subjects/Keywords: Boundary integral method; Dirichlet-Neumann map; Fokas transform method; Helmholtz equation; modified Helmholtz equation; polygonal domain; Applied Mathematics
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APA (6th Edition):
Davis, C. R. (2011). Numerical Tests of the Fokas Method for Helmholtz-type Partial Differential Equations: Dirichlet to Neumann Maps. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/17
Chicago Manual of Style (16th Edition):
Davis, Christopher-Ian Raphael. “Numerical Tests of the Fokas Method for Helmholtz-type Partial Differential Equations: Dirichlet to Neumann Maps.” 2011. Masters Thesis, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/appm_gradetds/17.
MLA Handbook (7th Edition):
Davis, Christopher-Ian Raphael. “Numerical Tests of the Fokas Method for Helmholtz-type Partial Differential Equations: Dirichlet to Neumann Maps.” 2011. Web. 26 Jan 2021.
Vancouver:
Davis CR. Numerical Tests of the Fokas Method for Helmholtz-type Partial Differential Equations: Dirichlet to Neumann Maps. [Internet] [Masters thesis]. University of Colorado; 2011. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/appm_gradetds/17.
Council of Science Editors:
Davis CR. Numerical Tests of the Fokas Method for Helmholtz-type Partial Differential Equations: Dirichlet to Neumann Maps. [Masters Thesis]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/17
University of Colorado
5.
Alexander, Zachary.
A Topology-Based Approach for Nonlinear Time Series with Applications in Computer Performance Analysis.
Degree: PhD, Applied Mathematics, 2012, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/24
► We present a topology-based methodology for the analysis of experimental data generated by a discrete-time, nonlinear dynamical system. This methodology has significant applications in…
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▼ We present a topology-based methodology for the analysis of experimental data generated by a discrete-time, nonlinear dynamical system. This methodology has significant applications in the field of computer performance analysis. Our approach consists of two parts. In the first part, we propose a novel signal separation algorithm that exploits the continuity of the dynamical system being studied. We use established tools from computational topology to test the connectedness of various regions of state space. In particular, a connected region of space that has a disconnected image under the experimental dynamics suggests the presence of multiple signals in the data. Using this as a guideline, we are able to model experimental data as an Iterated Function System (IFS). We demonstrate the success of our algorithm on several synthetic examples – including a Henon-like IFS. Additionally, we successfully model experimental computer performance data as an IFS. In the second part of the analysis, we represent an experimental dynamical system with an algebraic structure that allows for the computation of algebraic topological invariants. Previous work has shown that a cubical grid and the associated cubical complex are effective tools that can be used to identify isolating neighborhoods and compute the corresponding Conley Index – thereby rigorously verifying the existence of periodic orbits and/or chaotic dynamics. Our contribution is to adapt this technique by altering the underlying data structure – improving flexibility and efficiency. We represent the state space of the dynamical system with a simplicial complex and its induced simplicial multivalued map. This contains information about both geometry and dynamics, whereas the cubical complex is restricted by the geometry of the experimental data. This representation has several advantages; most notably, the complexity of the algorithm that generates the associated simplicial multivalued map is linear in the number of data points – as opposed to exponential in dimension for the cubical multivalued map. The synthesis of the two parts of our methodology results in a nonlinear time-series analysis framework that is particularly well suited for computer performance analysis. Complex computer programs naturally switch between `regimes' and are appropriately modeled as IFSs by part one of our program. Part two of our methodology provides the correct tools for analyzing each regime independently.
Advisors/Committee Members: Elizabeth Bradley, James D. Meiss, James Curry.
Subjects/Keywords: Applied Mathematics; Computer Sciences
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APA ·
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Export
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APA (6th Edition):
Alexander, Z. (2012). A Topology-Based Approach for Nonlinear Time Series with Applications in Computer Performance Analysis. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/24
Chicago Manual of Style (16th Edition):
Alexander, Zachary. “A Topology-Based Approach for Nonlinear Time Series with Applications in Computer Performance Analysis.” 2012. Doctoral Dissertation, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/appm_gradetds/24.
MLA Handbook (7th Edition):
Alexander, Zachary. “A Topology-Based Approach for Nonlinear Time Series with Applications in Computer Performance Analysis.” 2012. Web. 26 Jan 2021.
Vancouver:
Alexander Z. A Topology-Based Approach for Nonlinear Time Series with Applications in Computer Performance Analysis. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/appm_gradetds/24.
Council of Science Editors:
Alexander Z. A Topology-Based Approach for Nonlinear Time Series with Applications in Computer Performance Analysis. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/24
University of Colorado
6.
Bertrand, Nicholas.
Sparse Encoding of Observations from a Smooth Manifold via Locally Linear Approximations.
Degree: MS, Applied Mathematics, 2012, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/55
► We investigate the problem of finding a parameterization of a smooth, low-dimensional manifold based on noisy observations from a high-dimensional ambient space. The formulation…
(more)
▼ We investigate the problem of finding a parameterization of a smooth, low-dimensional manifold based on noisy observations from a high-dimensional ambient space. The formulation of such parameterizations sees applications in a variety of areas such as data denoising and image segmentation.
We introduce algorithms inspired by the existing k-svd algorithm for training dictionaries for sparse data representation, and the local best-fit at algorithm for hybrid linear modeling. The output of our algorithm is an assignment of input data points to locally linear models. To demonstrate the applicability of our algorithm, we discuss experiments performed on synthetic datasets.
Advisors/Committee Members: Francois Meyer, James Curry, Shannon Hughes.
Subjects/Keywords: k-svd algorithm; MLBF; Applied Mathematics
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APA (6th Edition):
Bertrand, N. (2012). Sparse Encoding of Observations from a Smooth Manifold via Locally Linear Approximations. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/55
Chicago Manual of Style (16th Edition):
Bertrand, Nicholas. “Sparse Encoding of Observations from a Smooth Manifold via Locally Linear Approximations.” 2012. Masters Thesis, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/appm_gradetds/55.
MLA Handbook (7th Edition):
Bertrand, Nicholas. “Sparse Encoding of Observations from a Smooth Manifold via Locally Linear Approximations.” 2012. Web. 26 Jan 2021.
Vancouver:
Bertrand N. Sparse Encoding of Observations from a Smooth Manifold via Locally Linear Approximations. [Internet] [Masters thesis]. University of Colorado; 2012. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/appm_gradetds/55.
Council of Science Editors:
Bertrand N. Sparse Encoding of Observations from a Smooth Manifold via Locally Linear Approximations. [Masters Thesis]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/55
University of Colorado
7.
Blakestad, Clifford.
On Generalizations of p-Adic Weierstrass Sigma and Zeta Functions.
Degree: PhD, 2018, University of Colorado
URL: https://scholar.colorado.edu/math_gradetds/56
► We generalize a paper of Mazur and Tate on p-adic sigma functions attached to elliptic curves of ordinary reduction over a <i>p</i>-adic field. We…
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▼ We generalize a paper of Mazur and Tate on p-adic sigma functions attached to elliptic curves of ordinary reduction over a <i>p</i>-adic field. We begin by generalizing the theory of division polynomials attached to an isogeny of elliptic curves, developed by Mazur and Tate, to isogenies of prinicipally polarized abelian varieties. As an application, we produce a notion of a <i>p</i>-adic sigma function attached to a prinicipally polarized abelian variety of good ordinary reduction over a complete non-archimedean field of residue characteristic <i>p</i>. Furthermore, we derive some the properties of the sigma function, many of which uniquely characterize the function. Independently, a notion of a pair of <i>p</i>-adic Weierstrass zeta functions is produced for a smooth projective curve <i>C</i> of genus two with invertible Hasse – Witt matrix over a <i>p</i>-adically complete field of characteristic zero. Using the explicit function theory afforded by Jacobians of genus two, general results about <i>p<i/>-adic sigma functions are made more descriptive and the zeta functions on <i>C</i> are compared to the second logarithmic derivatives of the sigma function on the Jacobian of <i>C</i>.
Advisors/Committee Members: David Grant, Jonathan Wise, Sebastian Casalaina-Martin, Katherine Stange, James Curry.
Subjects/Keywords: abelian surface; abelian variety; arithmetic geometry; genus two curve; number theory; p-adic; Mathematics; Special Functions
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APA (6th Edition):
Blakestad, C. (2018). On Generalizations of p-Adic Weierstrass Sigma and Zeta Functions. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/56
Chicago Manual of Style (16th Edition):
Blakestad, Clifford. “On Generalizations of p-Adic Weierstrass Sigma and Zeta Functions.” 2018. Doctoral Dissertation, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/math_gradetds/56.
MLA Handbook (7th Edition):
Blakestad, Clifford. “On Generalizations of p-Adic Weierstrass Sigma and Zeta Functions.” 2018. Web. 26 Jan 2021.
Vancouver:
Blakestad C. On Generalizations of p-Adic Weierstrass Sigma and Zeta Functions. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/math_gradetds/56.
Council of Science Editors:
Blakestad C. On Generalizations of p-Adic Weierstrass Sigma and Zeta Functions. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/math_gradetds/56
University of Colorado
8.
Kumley, Kendra L.
Effect of Parameter Variations on the Electromagnetic Response of a Metafilm Comprised of Resonant Elements.
Degree: PhD, 2018, University of Colorado
URL: https://scholar.colorado.edu/ecen_gradetds/157
► A metafilm is a 2-dimensional version of a metamaterial which consists of a single layer of resonators. The elements resonate with changes in frequency and…
(more)
▼ A metafilm is a 2-dimensional version of a metamaterial which consists of a single layer of resonators. The elements resonate with changes in frequency and are arranged periodically in an array. In this thesis, a theory describing the characteristics of arrays of resonant scatterers is thoroughly developed using analytical techniques (which have the advantages of providing physical insight and being computationally inexpensive) and demonstrated by numerical analysis. The analysis is based on a dipole interaction model, and is validated using independent full-wave numerical simulations. The technique derived in this work, the interaction polarizability approximation, takes into account variations in the parameters of the elements in such a way that it can accurately predict both weak and strong coupling. For a metafilm, this leads to expressions for reflection and transmission coefficients that correctly predict the existence of Fano bands. Experimental measurements are carried out that confirm the existence of Fano resonances for a metafilm mounted in a waveguide, bearing out the practical significance of the predictions of this thesis. This thesis demonstrates that the Fano bands, or regions of rapid asymmetric frequency variation in an otherwise smooth curve, are not simply the result of measurement errors but are in fact an inherent behavior for a metafilm, one that simply results from manufacturing errors that cause variations in parameters that affect resonant behavior.
Advisors/Committee Members: Edward F. Kuester, Christopher Holloway, Dejan Filipovic, James Curry, Maxim Ignatenko.
Subjects/Keywords: array; fano; metafilm; metamaterial; periodic; resonator; Electrical and Computer Engineering; Electromagnetics and Photonics
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APA (6th Edition):
Kumley, K. L. (2018). Effect of Parameter Variations on the Electromagnetic Response of a Metafilm Comprised of Resonant Elements. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/ecen_gradetds/157
Chicago Manual of Style (16th Edition):
Kumley, Kendra L. “Effect of Parameter Variations on the Electromagnetic Response of a Metafilm Comprised of Resonant Elements.” 2018. Doctoral Dissertation, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/ecen_gradetds/157.
MLA Handbook (7th Edition):
Kumley, Kendra L. “Effect of Parameter Variations on the Electromagnetic Response of a Metafilm Comprised of Resonant Elements.” 2018. Web. 26 Jan 2021.
Vancouver:
Kumley KL. Effect of Parameter Variations on the Electromagnetic Response of a Metafilm Comprised of Resonant Elements. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/ecen_gradetds/157.
Council of Science Editors:
Kumley KL. Effect of Parameter Variations on the Electromagnetic Response of a Metafilm Comprised of Resonant Elements. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/ecen_gradetds/157
University of Colorado
9.
Fox, Adam Merritt.
Destruction of Invariant Tori in Volume-Preserving Maps.
Degree: PhD, Applied Mathematics, 2013, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/36
► Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and KAM theory…
(more)
▼ Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and KAM theory guarantees the persistence of some tori upon perturbation. When these tori have codimension-one they act as boundaries to transport, and therefore play a prominent role in the global stability of the system. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. This dissertation begins by extending Greene's residue criterion to three-dimensional, reversible, volume-preserving maps.
The application of Greene's residue criterion requires the repeated computation of periodic orbits, which is costly if the system is nonreversible. We describe a quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a torus and demonstrate how the growth of the Sobolev norm or singular values of this conjugacy can be used to predict criticality. We will then use this method to study both reversible and nonreversible volume-preserving maps in two and three dimensions. The near-critical conjugacies, and the gaps that form within them, will be explored in the context of Aubry-Mather and Anti-Integrability theory, when applicable. This dissertation will conclude by exploring the locally and globally most robust tori in area-preserving maps.
Advisors/Committee Members: James D. Meiss, Juan Restrepo, Keith Julien, Elizabeth Bradley, James Curry.
Subjects/Keywords: KAM theory; Greene's residue criterion; near-critical conjugacies; Applied Mathematics
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APA ·
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APA (6th Edition):
Fox, A. M. (2013). Destruction of Invariant Tori in Volume-Preserving Maps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/36
Chicago Manual of Style (16th Edition):
Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/appm_gradetds/36.
MLA Handbook (7th Edition):
Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Web. 26 Jan 2021.
Vancouver:
Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/appm_gradetds/36.
Council of Science Editors:
Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/36
University of Colorado
10.
Mullen, Zachary.
Theory and Methods for Large Spatial Data.
Degree: PhD, 2018, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/105
► Correlated Gaussian processes are of central importance to the study of time series, spatial statistics, computer experiments, and many machine learning models. Large spatially…
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▼ Correlated Gaussian processes are of central importance to the study of time series, spatial statistics, computer experiments, and many machine learning models. Large spatially or temporally indexed datasets bring with them a host of computational and mathematical challenges. Parameter estimation of these processes often relies on maximum likelihood, which for Gaussian processes involves manipulations of the covariance matrix including solving systems of equations and determinant calculations. The score function, on the other hand, avoids direct calculation of the determinant, but still requires solving a large number of linear equations. We propose an equivalent kernel approximation to the score function of a stationary Gaussian process. A nugget effect is required for the approximation. We suggest two approximations, and for large sample sizes, our proposals are fast, accurate, and compare well against existing approaches. We then present a method for simulating time series of high frequency wind data calibrated by real data. The method provides and fits a parametric model for local wind directions by embedding them into the angular projection of a bivariate normal. Incorporating a temporal autocorrelation structure in that normal induces a continuous angular correlation over time in the simulated wind directions. The final joint model for speed and direction can be decomposed into the simulation of a single multivariate normal and a series of transformations thereof, allowing for fast and easy repeated generations of long time series. This is compared to a state of the art approach for simulating angular time series of swapping between discrete regimes of wind direction, a method that does not fully translate to high frequency data.
Advisors/Committee Members: William Kleiber, Eric Vance, Balaji Rajagopalan, Jem Corcoran, James Curry.
Subjects/Keywords: circular data; equivalent kernel; gaussian processes; spatial statistics; stochastic weather generators; wind modeling; Applied Mathematics; Statistical Theory
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APA (6th Edition):
Mullen, Z. (2018). Theory and Methods for Large Spatial Data. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/105
Chicago Manual of Style (16th Edition):
Mullen, Zachary. “Theory and Methods for Large Spatial Data.” 2018. Doctoral Dissertation, University of Colorado. Accessed January 26, 2021.
https://scholar.colorado.edu/appm_gradetds/105.
MLA Handbook (7th Edition):
Mullen, Zachary. “Theory and Methods for Large Spatial Data.” 2018. Web. 26 Jan 2021.
Vancouver:
Mullen Z. Theory and Methods for Large Spatial Data. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Jan 26].
Available from: https://scholar.colorado.edu/appm_gradetds/105.
Council of Science Editors:
Mullen Z. Theory and Methods for Large Spatial Data. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/105
. 
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