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University of Colorado
1.
Broido, Anna.
Characterizing the tails of degree distributions in real-world networks.
Degree: PhD, Applied Mathematics, 2019, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/143 
► This is a thesis about how to characterize the statistical structure of the tails of degree distributions of real-world networks. The primary contribution is…
(more)
▼ This is a thesis about how to characterize the statistical structure of the tails of degree distributions of real-world networks. The primary contribution is a statistical test of the prevalence of scale-free structure in real-world networks. A central claim in modern network science is that real-world networks are typically "scale free," meaning that the fraction of nodes with degree k follows a power law, decaying like k
-a, often with 2 < a< 3. However, empirical evidence for this belief derives from a relatively small number of real-world networks. In the first section, we test the universality of scale-free structure by applying state-of-the-art statistical tools to a large corpus of nearly 1000 network data sets drawn from social, biological, technological, and informational sources. We fit the power-law model to each degree distribution, test its statistical plausibility, and compare it via a likelihood ratio test to alternative, non-scale-free models, e.g., the log-normal. Across domains, we find that scale-free networks are rare, with only 4% exhibiting the strongest-possible evidence of scale-free structure and 52% exhibiting the weakest-possible evidence. Furthermore, evidence of scale-free structure is not uniformly distributed across sources: social networks are at best weakly scale free, while a handful of technological and biological networks can be called strongly scale free. These results undermine the universality of scale-free networks and reveal that real-world networks exhibit a rich structural diversity that will likely require new ideas and mechanisms to explain. A core methodological component of addressing the ubiquity of scale-free structure in real-world networks is an ability to fit a power law to the degree distribution. In the second section, we numerically evaluate and compare, using both synthetic data with known structure and real-world data with unknown structure, two statistically principled methods for estimating the tail parameters for power-law distributions, showing that in practice, a method based on extreme value theory and a sophisticated bootstrap and the more commonly used method based an empirical minimization approach exhibit similar accuracy.
Advisors/Committee Members: Aaron Clauset, Jem Corcoran, Daniel Larremore, Manuel Lladser, Juan Restrepo.
Subjects/Keywords: networks; power law; scale free; Applied Statistics; Other Applied Mathematics; Probability; Statistical Methodology; Statistical Models
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APA (6th Edition):
Broido, A. (2019). Characterizing the tails of degree distributions in real-world networks. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/143
Chicago Manual of Style (16th Edition):
Broido, Anna. “Characterizing the tails of degree distributions in real-world networks.” 2019. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/143.
MLA Handbook (7th Edition):
Broido, Anna. “Characterizing the tails of degree distributions in real-world networks.” 2019. Web. 23 Jan 2021.
Vancouver:
Broido A. Characterizing the tails of degree distributions in real-world networks. [Internet] [Doctoral dissertation]. University of Colorado; 2019. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/143.
Council of Science Editors:
Broido A. Characterizing the tails of degree distributions in real-world networks. [Doctoral Dissertation]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/143
University of Colorado
2.
Krishnan, Nikhil.
Foraging in Stochastic Environments.
Degree: MS, Applied Mathematics, 2019, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/145
► For many organisms, foraging for food and resources is integral to survival. Mathematical models of foraging can provide insight into the benefits and drawbacks…
(more)
▼ For many organisms, foraging for food and resources is integral to survival. Mathematical models of foraging can provide insight into the benefits and drawbacks of different foraging strategies. We begin by considering the movement of a memoryless starving forager on a one-dimensional periodic lattice, where each location contains one unit of food. As the forager lands on sites with food, it consumes the food leaving the sites empty. If the forager lands consecutively on a certain number of empty sites, then it starves. The forager has two modes of movement: it can either diffuse by moving with equal probability to adjacent lattice sites, or it can jump uniformly randomly amongst the lattice sites. The lifetime of the forager can be approximated in either paradigm by the sum of the cover time plus the number of empty sites it can visit before starving. The lifetime of the forager varies nonmontonically according to the probability of jumping. The tradeoff between jumps and diffusion is explored using simpler systems as well as numerical simulation, and we demonstrate that the best strategy is one that incorporates both jumps and diffusion. When long range jumps are time-penalized, counterintuitively, this shifts the optimal strategy to pure jumping. We next consider optimal strategies for a group of foragers to search for a target (such as food in an environment where it is sparsely located). There is a single target in one of several patches, with a greater penalty if the foragers decide to switch their positions among the patches. Both in the case of a single searcher, and in the case of a group of searchers, efficient deterministic strategies can be found to locate the target.
Advisors/Committee Members: Zachary Kilpatrick, Nancy Rodriguez, Juan Restrepo.
Subjects/Keywords: Foraging; Dynamic Systems
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APA (6th Edition):
Krishnan, N. (2019). Foraging in Stochastic Environments. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/145
Chicago Manual of Style (16th Edition):
Krishnan, Nikhil. “Foraging in Stochastic Environments.” 2019. Masters Thesis, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/145.
MLA Handbook (7th Edition):
Krishnan, Nikhil. “Foraging in Stochastic Environments.” 2019. Web. 23 Jan 2021.
Vancouver:
Krishnan N. Foraging in Stochastic Environments. [Internet] [Masters thesis]. University of Colorado; 2019. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/145.
Council of Science Editors:
Krishnan N. Foraging in Stochastic Environments. [Masters Thesis]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/145
University of Colorado
3.
Broido, Anna D.
Characterizing the Tails of Degree Distributions in Real-World Networks.
Degree: PhD, 2019, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/153
► This is a thesis about how to characterize the statistical structure of the tails of degree distributions of real-world networks. The primary contribution is a…
(more)
▼ This is a thesis about how to characterize the statistical structure of the tails of degree distributions of real-world networks. The primary contribution is a statistical test of the prevalence of scale-free structure in real-world networks. A central claim in modern network science is that real-world networks are typically "scale free," meaning that the fraction of nodes with degree k follows a power law, decaying like k^-a, often with 2 < a < 3. However, empirical evidence for this belief derives from a relatively small number of real-world networks. In the first section, we test the universality of scale-free structure by applying state-of-the-art statistical tools to a large corpus of nearly 1000 network data sets drawn from social, biological, technological, and informational sources. We fit the power-law model to each degree distribution, test its statistical plausibility, and compare it via a likelihood ratio test to alternative, non-scale-free models, e.g., the log-normal. Across domains, we find that scale-free networks are rare, with only 4% exhibiting the strongest-possible evidence of scale-free structure and 52% exhibiting the weakest-possible evidence. Furthermore, evidence of scale-free structure is not uniformly distributed across sources: social networks are at best weakly scale free, while a handful of technological and biological networks can be called strongly scale free. These results undermine the universality of scale-free networks and reveal that real-world networks exhibit a rich structural diversity that will likely require new ideas and mechanisms to explain. A core methodological component of addressing the ubiquity of scale-free structure in real-world networks is an ability to fit a power law to the degree distribution. In the second section, we numerically evaluate and compare, using both synthetic data with known structure and real-world data with unknown structure, two statistically principled methods for estimating the tail parameters for power-law distributions, showing that in practice, a method based on extreme value theory and a sophisticated bootstrap and the more commonly used method based an empirical minimization approach exhibit similar accuracy.
Advisors/Committee Members: Aaron Clauset, Jem Corcoran, Daniel Larremore, Manuel Lladser, Juan Restrepo.
Subjects/Keywords: networks; power law; scale free; Applied Mathematics
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APA (6th Edition):
Broido, A. D. (2019). Characterizing the Tails of Degree Distributions in Real-World Networks. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/153
Chicago Manual of Style (16th Edition):
Broido, Anna D. “Characterizing the Tails of Degree Distributions in Real-World Networks.” 2019. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/153.
MLA Handbook (7th Edition):
Broido, Anna D. “Characterizing the Tails of Degree Distributions in Real-World Networks.” 2019. Web. 23 Jan 2021.
Vancouver:
Broido AD. Characterizing the Tails of Degree Distributions in Real-World Networks. [Internet] [Doctoral dissertation]. University of Colorado; 2019. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/153.
Council of Science Editors:
Broido AD. Characterizing the Tails of Degree Distributions in Real-World Networks. [Doctoral Dissertation]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/153
University of Colorado
4.
Mitchell, Rebecca Amelia.
Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.
Degree: PhD, Applied Mathematics, 2017, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/94
► Mixing of a passive scalar in a fluid flow results from a two part process in which large gradients are first created by advection and…
(more)
▼ Mixing of a passive scalar in a fluid flow results from a two part process in which large gradients are first created by advection and then smoothed by diffusion. We investigate methods of designing efficient stirrers to optimize mixing of a passive scalar in a two-dimensional nonautonomous, incompressible flow over a finite time interval. The flow is modeled by a sequence of area-preserving maps whose parameters change in time, defining a mixing protocol. Stirring efficiency is measured by the mix norm, a negative Sobolev seminorm; its decrease implies creation of fine-scale structure. A Perron-Frobenius operator is used to numerically advect the scalar for three examples: compositions of Chirikov standard maps, of Harper maps, and of blinking vortex maps. In the case of the standard maps, we find that a protocol corresponding to a single vertical shear composed with horizontal shearing at all other steps is nearly optimal. For the Harper maps, we devise a predictive, one-step scheme to choose appropriate fixed point stabilities and to control the Fourier spectrum evolution to obtain a near optimal protocol. For the blinking vortex model, we devise two schemes: A one-step predictive scheme to determine a vortex location, which has modest success in producing an efficient stirring protocol, and a scheme that finds the true optimal choice of vortex positions and directions of rotation given four possible fixed vortex locations. The results from the numerical experiments suggest that an effective stirring protocol must include not only steps devoted to decreasing the mix norm, but also steps devoted to preparing the density profile for future steps of mixing.
Advisors/Committee Members: James D. Meiss, Keith Julien, Juan Restrepo, John Crimaldi, Roseanna Neupauer.
Subjects/Keywords: area-preserving maps; chaotic mixing; dynamical sytems; Perron-Frobenius; Applied Mathematics
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APA (6th Edition):
Mitchell, R. A. (2017). Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/94
Chicago Manual of Style (16th Edition):
Mitchell, Rebecca Amelia. “Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/94.
MLA Handbook (7th Edition):
Mitchell, Rebecca Amelia. “Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.” 2017. Web. 23 Jan 2021.
Vancouver:
Mitchell RA. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/94.
Council of Science Editors:
Mitchell RA. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/94
University of Colorado
5.
Ali, Ashar Fawad.
ULF Waves and Diffusive Radial Transport of Charged Particles.
Degree: PhD, Applied Mathematics, 2016, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/71
► The Van Allen radiation belts contain highly energetic particles which interact with a variety of plasma and magnetohydrodynamic (MHD) waves. Waves in the ultra…
(more)
▼ The Van Allen radiation belts contain highly energetic particles which interact with a variety of plasma and magnetohydrodynamic (MHD) waves. Waves in the ultra low-frequency (ULF) range play an important role in the loss and acceleration of energetic particles. Considering the geometry of the geomagnetic field, charged particles trapped in the inner magnetosphere undergo three distinct types of periodic motions; an adiabatic invariant is associated with each type of motion. The evolution of the phase space density of charged particles in the magnetosphere in the coordinate space of the three adiabatic invariants is modeled by the Fokker-Planck equation. If we assume that the first two adiabatic invariants are conserved while the third invariant is violated, then the general Fokker-Planck equation reduces to a radial diffusion equation with the radial diffusion coefficient quantifying the rate of the radial diffusion of charged particles, including contributions from perturbations in both the magnetic and the electric fields.
This thesis investigates two unanswered questions about ULF wave-driven radial transport of charged particles. First, how important are the ULF fluctuations in the magnetic field compared with the ULF fluctuations in the electric field in driving the radial diffusion of charged particles in the Earth's inner magnetosphere? It has generally been accepted that magnetic field perturbations dominate over electric field perturbations, but several recently published studies suggest otherwise. Second, what is the distribution of ULF wave power in azimuth, and how does ULF wave power depend upon radial distance and the level of geomagnetic activity? Analytic treatments of the diffusion coefficients generally assume uniform distribution of power in azimuth, but in situ measurements suggest that this may not be the case. We used the magnetic field data from the Combined Release and Radiation Effects Satellite (CRRES) and the electric and the magnetic field data from the Radiation Belt Storm Probes (RBSP) to compute the electric and the magnetic component of the radial diffusion coefficient using the Fei et al. [2006] formulation. We conclude that contrary to prior notions, the electric component is dominant in driving radial diffusion of charged particles in the Earth's inner magnetosphere instead of the magnetic component. The electric component can be up to two orders of magnitude larger than the magnetic component. In addition, we see that ULF wave power in both the electric and the magnetic fields has a clear dependence on
Kp with wave power decreasing as radial distance decreases. For both fields, the noon sectors generally contain more ULF wave power than the dawn, dusk, and the midnight magnetic local time (MLT) sectors. There is no significant difference between ULF wave power in the dawn, dusk, and the midnight sectors.
Advisors/Committee Members: Scot R. Elkington, Jem Corcoran, Howard Singer, Juan Restrepo, William Kleiber.
Subjects/Keywords: CRRES; Magnetospheric Physics; Radial Diffusion; RBSP; ULF Waves; Van Allen Radiation Belts; Applied Mathematics; Plasma and Beam Physics
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APA (6th Edition):
Ali, A. F. (2016). ULF Waves and Diffusive Radial Transport of Charged Particles. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/71
Chicago Manual of Style (16th Edition):
Ali, Ashar Fawad. “ULF Waves and Diffusive Radial Transport of Charged Particles.” 2016. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/71.
MLA Handbook (7th Edition):
Ali, Ashar Fawad. “ULF Waves and Diffusive Radial Transport of Charged Particles.” 2016. Web. 23 Jan 2021.
Vancouver:
Ali AF. ULF Waves and Diffusive Radial Transport of Charged Particles. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/71.
Council of Science Editors:
Ali AF. ULF Waves and Diffusive Radial Transport of Charged Particles. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/appm_gradetds/71
University of Colorado
6.
Anderson, Paul Vincent.
Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime.
Degree: PhD, Aerospace Engineering Sciences, 2015, University of Colorado
URL: https://scholar.colorado.edu/asen_gradetds/91
► The geosynchronous orbit (GEO) is a unique commodity of the satellite industry that is becoming increasingly contaminated with orbital debris, but is heavily populated…
(more)
▼ The geosynchronous orbit (GEO) is a unique commodity of the satellite industry that is becoming increasingly contaminated with orbital debris, but is heavily populated with high-value assets from the civil, commercial, and defense sectors. The GEO arena is home to hundreds of communications, data transmission, and intelligence satellites collectively insured for an estimated 18.3 billion USD. As the lack of natural cleansing mechanisms at the GEO altitude renders the lifetimes of GEO debris essentially infinite, conjunction and risk assessment must be performed to safeguard operational assets from debris collisions. In this thesis, longitude-dependent debris congestion is characterized by predicting the number of near-miss events per day for every longitude slot at GEO, using custom debris propagation tools and a torus intersection metric. Near-miss events with the present-day debris population are assigned risk levels based on GEO-relative position and speed, and this risk information is used to prioritize the population for debris removal target selection. Long-term projections of debris growth under nominal launch traffic, mitigation practices, and fragmentation events are also discussed, and latitudinal synchronization of the GEO debris population is explained via node variations arising from luni-solar gravity. In addition to characterizing localized debris congestion in the GEO ring, this thesis further investigates the conjunction risk to operational satellites or debris removal systems applying low-thrust propulsion to raise orbit altitude at end-of-life to a super-synchronous disposal orbit. Conjunction risks as a function of thrust level, miss distance, longitude, and semi-major axis are evaluated, and a guidance method for evading conjuncting debris with continuous thrust by means of a thrust heading change via single-shooting is developed.
Advisors/Committee Members: Hanspeter Schaub, Brandon Jones, Jeffrey Parker, Juan Restrepo, Darren McKnight.
Subjects/Keywords: active debris removal; geostationary orbit; geosynchronous orbit; orbital debris; Aerospace Engineering
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APA (6th Edition):
Anderson, P. V. (2015). Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/asen_gradetds/91
Chicago Manual of Style (16th Edition):
Anderson, Paul Vincent. “Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime.” 2015. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/asen_gradetds/91.
MLA Handbook (7th Edition):
Anderson, Paul Vincent. “Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime.” 2015. Web. 23 Jan 2021.
Vancouver:
Anderson PV. Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime. [Internet] [Doctoral dissertation]. University of Colorado; 2015. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/asen_gradetds/91.
Council of Science Editors:
Anderson PV. Characterizing Longitude-Dependent Orbital Debris Congestion in the Geosynchronous Orbit Regime. [Doctoral Dissertation]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/asen_gradetds/91
University of Colorado
7.
Monnig, Nathan D.
From Nonlinear Embedding to Graph Distances: A Spectral Perspective.
Degree: PhD, Applied Mathematics, 2015, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/64
► In this thesis, we explore applications of spectral graph theory to the analysis of complex datasets and networks. We consider spectral embeddings of general…
(more)
▼ In this thesis, we explore applications of spectral graph theory to the analysis of complex datasets and networks. We consider spectral embeddings of general graphs, as well as data sampled from smooth manifolds in high dimension. We specifically focus on the development of algorithms that require minimal user input. Given the inherent difficulty in parameterizing these types of complex datasets, an ideal algorithm should avoid poorly-defined user-selected parameters.
A significant limitation of nonlinear dimensionality reduction embeddings computed from datasets is the absence of a mechanism to compute the inverse map. We address the problem of computing a stable inverse using radial basis functions (RBFs) to interpolate the inverse map everywhere on the low-dimensional image of the forward map. We demonstrate that the scale-free cubic RBF kernel performs better than the Gaussian kernel: it does not suffer from ill-conditioning, and does not require the choice of a scale.
The definition of metrics to compare networks is an open research area. In this thesis, we define the resistance perturbation distance on the space of weighted connected undirected graphs with the same vertex set. This novel distance quantifies changes in the network connectivity, as measured by the respective effective resistance matrices in each graph. We introduce a fast algorithm to approximate this novel distance in linear time. We demonstrate on simple graph models and real networks that the resistance perturbation distance is able to ignore random fluctuation of edges that are irrelevant to the network connectivity, while remaining sensitive to changes triggered by edges that play a fundamental role in the graph connectivity.
Advisors/Committee Members: Francois G. Meyer, Bengt Fornberg, Jem N. Corcoran, Thomas Manteuffel, Juan Restrepo.
Subjects/Keywords: effective resistance; graph distances; graph theory; nonlinear dimension reduction; radial basis functions; spectral algorithms; Numerical Analysis and Computation; Set Theory
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APA (6th Edition):
Monnig, N. D. (2015). From Nonlinear Embedding to Graph Distances: A Spectral Perspective. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/64
Chicago Manual of Style (16th Edition):
Monnig, Nathan D. “From Nonlinear Embedding to Graph Distances: A Spectral Perspective.” 2015. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/64.
MLA Handbook (7th Edition):
Monnig, Nathan D. “From Nonlinear Embedding to Graph Distances: A Spectral Perspective.” 2015. Web. 23 Jan 2021.
Vancouver:
Monnig ND. From Nonlinear Embedding to Graph Distances: A Spectral Perspective. [Internet] [Doctoral dissertation]. University of Colorado; 2015. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/64.
Council of Science Editors:
Monnig ND. From Nonlinear Embedding to Graph Distances: A Spectral Perspective. [Doctoral Dissertation]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/appm_gradetds/64
University of Colorado
8.
Kaufman, Adam M.
Laser-Cooling Atoms to Indistinguishability: Atomic Hong-Ou-Mandel Interference and Entanglement Through Spin-Exchange.
Degree: PhD, Physics, 2015, University of Colorado
URL: https://scholar.colorado.edu/phys_gradetds/140
► In this thesis, I describe the development of and scientific results from a new platform for creating ultracold atoms via single-atom control. We employ…
(more)
▼ In this thesis, I describe the development of and scientific results from a new platform for creating ultracold atoms via single-atom control. We employ Raman-sideband cooling to isolated bosonic
87Rb atoms confined within sub-micron optical tweezers, yielding single particle three- dimensional ground-state fractions of 90%. We create multiple, independent, mobile optical tweezers, which simultaneously allows multi-particle studies with single-atom microscopy and highly tunable length-scales. We employ this toolset in both of the main experiments discussed in this thesis. In one experiment, we observe Hong-Ou-Mandel interference of two bosonic atoms, each of which is independently prepared in spatially separated optical tweezers. The interference we observe is a direct consequence of the purity of the single particle quantum states produced, and the indistinguishability of the atoms. In a second experiment, we introduce a spin-degree of freedom and exploit spin-exchange dynamics, driven by the quantum-statistics of the particles, to create a spin-entangled pair of spatially separated atoms.
Advisors/Committee Members: Cindy A. Regal, Ana Maria Rey, Deborah S. Jin, David J. Wineland, Juan Restrepo.
Subjects/Keywords: Laser cooling; Quantum control; Single atoms; ultracold atoms; Atomic, Molecular and Optical Physics; Quantum Physics
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APA (6th Edition):
Kaufman, A. M. (2015). Laser-Cooling Atoms to Indistinguishability: Atomic Hong-Ou-Mandel Interference and Entanglement Through Spin-Exchange. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/phys_gradetds/140
Chicago Manual of Style (16th Edition):
Kaufman, Adam M. “Laser-Cooling Atoms to Indistinguishability: Atomic Hong-Ou-Mandel Interference and Entanglement Through Spin-Exchange.” 2015. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/phys_gradetds/140.
MLA Handbook (7th Edition):
Kaufman, Adam M. “Laser-Cooling Atoms to Indistinguishability: Atomic Hong-Ou-Mandel Interference and Entanglement Through Spin-Exchange.” 2015. Web. 23 Jan 2021.
Vancouver:
Kaufman AM. Laser-Cooling Atoms to Indistinguishability: Atomic Hong-Ou-Mandel Interference and Entanglement Through Spin-Exchange. [Internet] [Doctoral dissertation]. University of Colorado; 2015. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/phys_gradetds/140.
Council of Science Editors:
Kaufman AM. Laser-Cooling Atoms to Indistinguishability: Atomic Hong-Ou-Mandel Interference and Entanglement Through Spin-Exchange. [Doctoral Dissertation]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/phys_gradetds/140
University of Colorado
9.
Brutz, Michael Joseph.
Mathematical Modelling and Analysis of Several Diffusive Processes.
Degree: PhD, Applied Mathematics, 2014, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/60
► The underlying theme of this research is using numerical methods to develop computationally efficient algorithms for three separate problems driven by diffusive processes. The…
(more)
▼ The underlying theme of this research is using numerical methods to develop computationally efficient algorithms for three separate problems driven by diffusive processes. The problems under consideration are: contaminant dispersal through fracture networks, modelling the flow of glacial ice, and community detection on networks.
A common feature of containment facilities for nuclear waste is to use expansive geological formations as an added barrier to contaminant dispersal in the event of a leak. Although these formations are generally comprised of dense rock that is difficult to penetrate, fractures within them provide a potential means for contaminants to rapidly transport across the barrier. The typical width of such fractures is only on the order of millimeters whereas the typical scale of interest for contaminant transport is on the order of kilometers. When particle tracking methods are used to simulate the contaminant dispersal in fracture networks, this disparity of scales severely restricts maximum time step sizes because features at the millimeter scale need to be resolved. Our contribution to this problem is developing a coarse scale particle tracking method that allows for substantially larger time steps when particles are navigating straight fractures.
With global warming comes concerns as to how the changing temperature will impact glacial systems and their contribution to sea level rise. On glacial scales, ice behaves as a very slowly moving non-Newtonian fluid, and the primary problem for numerically simulating the evolution of ice masses comes with Glen's flow law for the effective viscosity. The flow law is empirically based, and its simple form has proven useful for analytical calculations. However, its simple form also allows for the effective viscosity to become unbounded in regions of low strain rate, and has proven to be very problematic for numerical simulations. Our contribution to this problem is re-examining the datasets the flow law was originally based on to develop an alternative model that fits the data with comparable accuracy, but without the problematic singularity.
When working with networks that represent real world systems, a common feature of interest is to find collections of vertices that form communities. Because the word "community" is an ambiguous term, our interpretation is that it is necessary to quantify what it means to be a community at a minimum of three scales for any given problem. These scales are at the level of: individual nodes, individual communities, and the network as a whole. Although our work focuses on detecting overlapping communities in the context of social networks, our primary contribution is developing a methodology that is highly modular and can easily be adapted to target other problem-specific notions of community.
Advisors/Committee Members: Francois Meyer, Tom Manteuffel, Harihar Rajaram, John Ruge, Juan Restrepo.
Subjects/Keywords: community; glacier; multiscale; network; flow; particle tracking; Applied Mathematics
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APA (6th Edition):
Brutz, M. J. (2014). Mathematical Modelling and Analysis of Several Diffusive Processes. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/60
Chicago Manual of Style (16th Edition):
Brutz, Michael Joseph. “Mathematical Modelling and Analysis of Several Diffusive Processes.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/60.
MLA Handbook (7th Edition):
Brutz, Michael Joseph. “Mathematical Modelling and Analysis of Several Diffusive Processes.” 2014. Web. 23 Jan 2021.
Vancouver:
Brutz MJ. Mathematical Modelling and Analysis of Several Diffusive Processes. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/60.
Council of Science Editors:
Brutz MJ. Mathematical Modelling and Analysis of Several Diffusive Processes. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/60
University of Colorado
10.
Romero, Henry Paul.
Fundamental Limits of Network Communication with General Message Sets: A Combinatorial Approach.
Degree: PhD, Applied Mathematics, 2014, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/61
► The classical theoretical framework for communication networks is based on the simplifying assumption that each message to be sent is known to a single…
(more)
▼ The classical theoretical framework for communication networks is based on the simplifying assumption that each message to be sent is known to a single transmitter and intended for a single receiver. Modern communication protocols reflect this framework by treating the physical layer as a network of individual links. However, this wireline view of wireless communications fails to account for the heterogeneous nature of network demands, consisting of both unicast and multicast services, and can fail to leverage the inherent broadcast advantage of the wireless medium.
This thesis extends the classical framework of a private-message interface to the physical layer to one with both private and common messages. A key difficulty, in both the description and analysis of a communication model with general messages sets, is that there are combinatorially many message possibilities. With order-theoretic language and tools from combinatorial optimization and graphical models, we develop a general framework for characterizing the fundamental limits of information transfer over large many-to-one (multiple access) and one-to-many (broadcast) communication channels with general message sets. In particular, achievable regions are proposed for arbitrary such channels. For the multiple-access channel, the achievable region is optimal, and the order-theoretic perspective both unifies and extends previous results. For the broadcast channel, the region is specialized to an inner bound to the Degree of Freedom region, a setting where it is provably optimal in select cases.
This thesis provides fresh insights into the long-standing random coding technique of superposition coding to arrive at these results. Governing constraints on reliable communication through superposition coding are shown to be polymatroidal over a lattice of subsets that may not be the boolean lattice of all subsets. Permissible input distributions for superposition coding are concisely characterized through directed graphical models of conditional dependencies. The two-user interference channel is also addressed, where the state-of-the art is extended from the case with two private messages to one with an additional common message.
Advisors/Committee Members: Mahesh Varanasi, Juan Restrepo, Vanja Dukic, Jem N. Corcoran, Clifford T. Mullis.
Subjects/Keywords: Information Theory; DM Multiple Access Channel; MIMO Multiple Access Channel; Broadcast Channel; Inteference Channel; Computer Sciences; Electrical and Computer Engineering; Mathematics
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Romero, H. P. (2014). Fundamental Limits of Network Communication with General Message Sets: A Combinatorial Approach. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/61
Chicago Manual of Style (16th Edition):
Romero, Henry Paul. “Fundamental Limits of Network Communication with General Message Sets: A Combinatorial Approach.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/61.
MLA Handbook (7th Edition):
Romero, Henry Paul. “Fundamental Limits of Network Communication with General Message Sets: A Combinatorial Approach.” 2014. Web. 23 Jan 2021.
Vancouver:
Romero HP. Fundamental Limits of Network Communication with General Message Sets: A Combinatorial Approach. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/61.
Council of Science Editors:
Romero HP. Fundamental Limits of Network Communication with General Message Sets: A Combinatorial Approach. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/61
University of Colorado
11.
Wakefield, Nathan Paul.
Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.
Degree: PhD, Mathematics, 2013, University of Colorado
URL: https://scholar.colorado.edu/math_gradetds/25
► Let (<em>gi</em>)<em>i</em> ≥1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (<em>fi</em>) <em>i</em> ≥1 by the following recursion:…
(more)
▼ Let (<em>g
i</em>)<em>
i</em>
≥1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (<em>f
i</em>) <em>
i</em>
≥1 by the following recursion:
f1 =
g1, <em>f
n</em> = <em>g
n</em> ∘ <em> f
n</em>
–1, for
n ≥ 2. Choose α ∈ [special characters omitted] such that {[special characters omitted](α) :
n ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if <em>f
n</em>(α) = [special characters omitted] is written in lowest terms, then for all but finitely many
n > 0, the numerator, <em>A
n</em>, has a primitive divisor; that is, there is a prime
p which divides <em> A
n</em> but does not divide <em>A
i</em> for any
i <
n.
In addition to the main result, several of the tools developed to prove the main result may be of interest.
A key component of the main result was the development of a generalization of canonical height. Namely: If [
f] is a set of rational maps, all commuting with a common function
f, and f = [special characters omitted] is a generalized iteration of rational maps formed by <em>f
n</em>(
x) = <em>g
n</em>(<em> f
n</em>
–1(
x)) with <em> g
i</em> coming from [
f], then there is a unique canonical height funtion
ĥf :
K → [special characters omitted] which is identical to the canonical height function associated to
f.
Another key component of the main result was proving that under certain circumstances, being acted upon by a Chebyshev polynomial does not lead to significant differences between the size of the numerator and denominator of the result. Specifically, let γ ∈ {0, ±1, ±2} be fixed, and <em>g
i</em> be a sequence of Chebyshev polynomials. Let
f given by the following recurrence
f 1(
z) =
g1(
z), and <em>f
i</em> = <em>g
i</em>(<em> f
i</em>
–1(
z)) for
i ≥ 2. Pick any α ∈ [special characters omitted] with |α + γ| < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write <em>f
n</em>(α + γ) − γ = [special characters omitted] in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed.
Advisors/Committee Members: Su-Ion Ih, Katherine Stange, Robert Tubbs, Eric Stade, Juan Restrepo.
Subjects/Keywords: Arithmetic Dynamics; Chebyshev; Generalized Iteration; Primitive Divisors; Mathematics
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APA ·
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MLA ·
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CSE |
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Manager
APA (6th Edition):
Wakefield, N. P. (2013). Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/25
Chicago Manual of Style (16th Edition):
Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/math_gradetds/25.
MLA Handbook (7th Edition):
Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Web. 23 Jan 2021.
Vancouver:
Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/math_gradetds/25.
Council of Science Editors:
Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/math_gradetds/25
University of Colorado
12.
Perkins, Jerome Philippe.
Influence of Food Web Structure on Predator-Prey Dynamics in a Patchy Environment.
Degree: MS, Applied Mathematics, 2013, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/32
► In order to model the spatial distributions of predators and prey many investigators have used a simplified three-species system where a predator species consumes…
(more)
▼ In order to model the spatial distributions of predators and prey many investigators have used a simplified three-species system where a predator species consumes a prey species that consumes a resource. One of the recurring predictions from such models is that the spatial distribution of the predator will match the spatial distribution of the resource instead of that of the prey; this is known as ``leapfrogging". While it is interesting that leapfrogging is consistently predicted by models of three species, tritrophic systems, real biological communities are more complicated, being less like chains and more like multi-dimensional food webs (i.e., multiple prey and predator species interacting with each other). I ask: Are systems with more species and more connections among them well approximated by simpler, three-species single-chain models? I construct two different five-dimensional systems (a resource consumed by two prey species consumed by two predator species) and compare them to the single-chain system to see if more complicated systems yield the same predictions as a simpler single-chain system. I discovered that, in aggregate, the dynamics of predators in a multi-species web, is very similar to the dynamics of a simpler, single-chain system; yet individualy the dynamics of predator species in a multi-species web are very different from the dynamics of a simpler, single-chain system.
Advisors/Committee Members: Samuel Flaxman, Anne Dougherty, Juan Restrepo.
Subjects/Keywords: Dynamics; Predator; Prey; Spatial Distributions; Biology; Demography, Population, and Ecology
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MLA ·
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CSE |
Export
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Manager
APA (6th Edition):
Perkins, J. P. (2013). Influence of Food Web Structure on Predator-Prey Dynamics in a Patchy Environment. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/32
Chicago Manual of Style (16th Edition):
Perkins, Jerome Philippe. “Influence of Food Web Structure on Predator-Prey Dynamics in a Patchy Environment.” 2013. Masters Thesis, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/appm_gradetds/32.
MLA Handbook (7th Edition):
Perkins, Jerome Philippe. “Influence of Food Web Structure on Predator-Prey Dynamics in a Patchy Environment.” 2013. Web. 23 Jan 2021.
Vancouver:
Perkins JP. Influence of Food Web Structure on Predator-Prey Dynamics in a Patchy Environment. [Internet] [Masters thesis]. University of Colorado; 2013. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/appm_gradetds/32.
Council of Science Editors:
Perkins JP. Influence of Food Web Structure on Predator-Prey Dynamics in a Patchy Environment. [Masters Thesis]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/32
University of Colorado
13.
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 ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
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 23, 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. 23 Jan 2021.
Vancouver:
Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 23].
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
14.
Mysore Balasubramanya, Naveen.
Interference Time Analysis for a Cognitive Radio on an Unmanned Aircraft.
Degree: MS, Electrical, Computer & Energy Engineering, 2010, University of Colorado
URL: https://scholar.colorado.edu/ecen_gradetds/14
► This thesis considers a model consisting of a cognitive radio (CR) on an unmanned aircraft (UA) and a network of licensed primary users on…
(more)
▼ This thesis considers a model consisting of a cognitive radio (CR) on an unmanned aircraft (UA) and a network of licensed primary users on the ground. The cognitive radio uses the same frequency spectrum as the primary users for its operation and hence acts as an interferer. This work analyzes the duration of interference in such a model. It defines two important metrics – the interference radius and the detection radius. The interference radius determines the boundary of the area within which a primary user might be subjected to harmful interference due to the operation of the CR. The detection radius determines the boundary of the area within which the presence of a primary user might be detected by the CR. The interference and detection radii might vary due to the dynamic nature of the radio environment. This thesis derives the dependence of these metrics on the radio propagation parameters like antenna gain, antenna height, path-loss exponents, etc. It uses these metrics and characterizes the model using an M/G/infinity queue to determine the statistics of the interference time for the entire excursion of the unmanned aircraft. The key statistics determined are the distribution of the duration of interference periods, the mean and the total interference time. Firstly, this work analyzes a 1D system model where the primary users are distributed randomly along a straight line. The results are then extended to a 2D system where the primary users are distributed randomly over an area. The analysis is carried out for both sparsely-dense and highly-dense primary user ground network. This work gives a new dimension to analyze the effects of interference in terms of duration of interference. It also shows how these interference effects can be minimized on enhancing the detection capability of the cognitive radio. The results from this work can be used to determine the optimum setting for the cognitive radio system such that it restrains the duration of interference below tolerable limits.
Advisors/Committee Members: Timothy X Brown, Youjian (Eujene) Liu, Juan Restrepo.
Subjects/Keywords: cognitive radio; detection capability; duration of interference; interference analysis; M/G/infinity queue; unmanned aircraft; Aviation; Electrical and Computer Engineering
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Mysore Balasubramanya, N. (2010). Interference Time Analysis for a Cognitive Radio on an Unmanned Aircraft. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/ecen_gradetds/14
Chicago Manual of Style (16th Edition):
Mysore Balasubramanya, Naveen. “Interference Time Analysis for a Cognitive Radio on an Unmanned Aircraft.” 2010. Masters Thesis, University of Colorado. Accessed January 23, 2021.
https://scholar.colorado.edu/ecen_gradetds/14.
MLA Handbook (7th Edition):
Mysore Balasubramanya, Naveen. “Interference Time Analysis for a Cognitive Radio on an Unmanned Aircraft.” 2010. Web. 23 Jan 2021.
Vancouver:
Mysore Balasubramanya N. Interference Time Analysis for a Cognitive Radio on an Unmanned Aircraft. [Internet] [Masters thesis]. University of Colorado; 2010. [cited 2021 Jan 23].
Available from: https://scholar.colorado.edu/ecen_gradetds/14.
Council of Science Editors:
Mysore Balasubramanya N. Interference Time Analysis for a Cognitive Radio on an Unmanned Aircraft. [Masters Thesis]. University of Colorado; 2010. Available from: https://scholar.colorado.edu/ecen_gradetds/14
. 
Open Access Theses and Dissertations