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Pages
 Title
 Continuity of weighted estimates in harmonic analysis with respect to the weight
 Creator
 Pattakos, Nikolaos
 Date
 2012
 Collection
 Electronic Theses & Dissertations
 Description
 Given the class of Ap weights, 1Given the class of Ap weights, 1
 Title
 Distance Preserving Graphs
 Creator
 Zahedi, Emad
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

The computational complexity of exploring distance properties of large graphs such as realworld social networks which consist of millions of nodes is extremely expensive. Recomputing distances in subgraphs of the original graph will add to the cost. One way to avoid this is to use subgraphs where the distance between any pair of vertices is the same as in the original graph. Such a subgraph is called {\em isometric}. A connected graph is {\em distance preserving}, for which we use the...
Show moreThe computational complexity of exploring distance properties of large graphs such as realworld social networks which consist of millions of nodes is extremely expensive. Recomputing distances in subgraphs of the original graph will add to the cost. One way to avoid this is to use subgraphs where the distance between any pair of vertices is the same as in the original graph. Such a subgraph is called {\em isometric}. A connected graph is {\em distance preserving}, for which we use the abbreviation dp, if it has an isometric subgraph of every order. In this framework we study dp graphs from both the structural and algorithmic perspectives. First, we study the structural nature of dp graphs. This involves classifying graphs based on the dp property and the relation between dp graphs to other graph classes. Second, we study the recognition problem of dp graphs. We intend to develop efficient algorithms for finding isometric subgraphs as well as deciding whether a graph is dp or not.
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 Title
 Field Modeling, Symplectic Tracking, and Spin Decoherence for EDM and Muon g2 Lattices
 Creator
 Valetov, Eremey Vladimirovich
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

While the first particle accelerators were electrostatic machines, and several electrostatic storage rings were subsequently commissioned and operated, electrostatic storage rings pose a number of challenges. Unlike motion in the magnetic field, where particle energy remains constant, particle energy generally changes in electrostatic elements. Conservation of energy in an electrostatic element is, in practice, only approximate, and it requires careful and accurate design, manufacturing,...
Show moreWhile the first particle accelerators were electrostatic machines, and several electrostatic storage rings were subsequently commissioned and operated, electrostatic storage rings pose a number of challenges. Unlike motion in the magnetic field, where particle energy remains constant, particle energy generally changes in electrostatic elements. Conservation of energy in an electrostatic element is, in practice, only approximate, and it requires careful and accurate design, manufacturing, installation, and operational use. Electrostatic deflectors require relatively high electrostatic fields, tend to introduce nonlinear aberrations of all orders, and are more challenging to manufacture than homogeneous magnetic dipoles. Accordingly, magnetic storage rings are overwhelmingly prevalent.The search for electric dipole moments (EDMs) of fundamental particles is of key importance in the study of C and CP violations and their sources. C and CP violations are part of the Sakharov conditions that explain the matter–antimatter asymmetry in the universe. Determining the source of CP violations would provide valuable empirical insight for beyondStandardModel physics. EDMs of fundamental particles have not to this date been experimentally observed. The search for fundamental particle EDMs has narrowed the target search region; however, an EDM signal is yet to be discovered.In 2008, Brookhaven National Laboratory (BNL) had proposed the frozen spin (FS) concept for the search of a deuteron EDM. The FS concept envisions launching deuterons through a storage ring with combined electrostatic and magnetic fields. The electrostatic and magnetic fields are in a proportion that would, without an EDM, freeze the deuteron's spin along its momentum as the deuteron moves around the lattice. The radial electrostatic field would result in a torque on the spin vector, proportional to a deuteron EDM, rotating the spin vector out of the midplane.The principle of an anomalous magnetic dipole moment (MDM) measurement using a storage ring, shared by BNL's completed E821 Experiment and the ongoing E989 Experiment operated by Fermi National Accelerator Laboratory (FNAL), requires injecting muons into a magnetic ring at the socalled magic momentum. The magic momentum, as defined in this context, would freeze the muon's spin vector along its momentum if the anomalous MDM was zero. The spin precession in the horizontal plane relative to the momentum is proportional to the anomalous MDM.Storage rings for measurement of EDM and anomalous MDM present a new frontier in tracking code accuracy requirements. For accurate tracking of storage rings with electrostatic particle optical elements, it is necessary to model the fringe fields of such elements accurately, in particular, because not doing so provides a mechanism for energy conservation violation. However, the previous research on fringe fields tended to focus on magnetic rather than electrostatic particle optical elements. We will study and model the fringe fields of several electrostatic deflectors. Field falloffs of electrostatic deflectors are slower than exponential, and Enge functions are not suitable for accurate modeling of these falloffs. We will propose an alternative function to model field falloffs of electrostatic deflectors. We will use conformal mapping methods to obtain the main field of the Muon g2 storage ring high voltage quadrupole, and we will calculate its fringe field and effective field boundary (EFB) using Fourier analysis.Furthermore, we will study tracking of storage rings with electrostatic elements using map methods. We will find that, for simultaneous symplecticity and energy conservation, it is only necessary to enforce symplecticity in COSY INFINITY. We will model and track several benchmark lattices – an electrostatic spherical deflector, a homogeneous magnetic dipole, and a proton EDM lattice – in COSY INFINITY and MSURK89, our inhouse eighth order Runge–Kutta–Verner tracking code. Finally, we will investigate spin decoherence and systematic errors in FS and quasifrozen spin (QFS) lattices. Spin decoherence effects are similar in FS and QFS lattices, and spin decoherence in said lattices often remains in the same range over time, indicating the feasibility of EDM measurement using FS and QFS lattices.
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 Title
 SIMULATION ANALYSES OF INTEGRATED TAGGING AND CATCHATAGE ANALYSIS MODELS AND APPLICATION TO LAKE ERIE WALLEYE
 Creator
 Vincent, Matthew T.
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

Integrated tagging and catchatage analysis (ITCAAN)models incorporate tagrecovery data within statistical catchatage models for spatiallyexplicit assessment of fish stocks. I investigated two tagrecovery frameworks for ITCAAN models that assumed natal homing spawning behavior: releaseconditioned and recoveryconditioned. In Chapter 1, I investigated the performance of a releaseconditioned ITCAANmodel under varying levels of parameter complexity, movement rates, data quality, and...
Show moreIntegrated tagging and catchatage analysis (ITCAAN)models incorporate tagrecovery data within statistical catchatage models for spatiallyexplicit assessment of fish stocks. I investigated two tagrecovery frameworks for ITCAAN models that assumed natal homing spawning behavior: releaseconditioned and recoveryconditioned. In Chapter 1, I investigated the performance of a releaseconditioned ITCAANmodel under varying levels of parameter complexity, movement rates, data quality, and misspecification of natural mortality or reporting rates. The releaseconditioned ITCAAN model simultaneously estimated movement rates, natural mortality, and tag reporting rates, though accuracy and precision of model estimates decreased with greater model complexity and fewer tags released. In Chapter 2, I investigated a recoveryconditioned ITCAAN model under a range of model complexities, different movement rates, misspecification of natural mortality or reporting rates, and spatially varying reporting rates. The recoveryconditioned ITCAAN model accurately estimated relative reporting rates at low intermixing rates. Biased estimates of individual population abundance resulted when intermixing was high and the reporting rate was assumed to be spatially constant, but the true reporting rates were not. I recommend recoveryconditioned ITCAAN model only be used for assessment of intermixed fish stocks when there is high certainty of spatially constant reporting rates. Both release and recoveryconditioned ITCAAN models had difficulty estimating individual population abundances and reporting rates under high rates of movement and large differences in population sizes. For both ITCAAN frameworks, estimation of both natural mortality and reporting rates reduced the accuracy and precision of model estimates, but estimates were less biased than misspecifying one of these parameters by 50%. In all investigated simulations, parameter estimates from the releaseconditioned ITCAAN model were more accurate and precise compared to the recoveryconditioned ITCAAN model. In Chapter 3, I examined the ability of a releaseconditioned ITCAAN model to estimate timevarying natural mortality and reporting rate parameters and sensitivity to tagshedding, highreward tagrecovery data, and seasonal movement dynamics. Natural mortality and reporting rate estimates were most precise and accurate when estimated in 5year time blocks. Estimation of natural mortality and reportingrates as temporally constant when reporting rates decreased linearly over time caused severe bias in abundance estimates, especially as the frequency of highreward tag release events decreased. I recommend that highreward tags be released annually to increase the precision and accuracy of ITCAAN model estimates. The releaseconditioned ITCAAN model was sensitive to whether the operating model simulated tagshedding and seasonal movement dynamics. In Chapter 4, a releaseconditioned ITCAAN model was applied to Lakes Erie and Huron walleye (Sander vitreus) data and estimates were compared to current assessment models. Reasonable fits to all data sources were obtained for an ITCAAN model that estimated a single reporting rate for all recreational fisheries and assumed the effective sample size of the tagrecovery data was equal to the number of tags released divided by 10. Estimates of population abundance from the ITCAAN model were similar to the model currently used to manage Lake Huron walleye. Conversely, estimates ofabundance and natural mortality for the western/central basin of Lake Erie were lower compared to the assessment model currently used to manage the stock. Estimates of natural mortality and abundance for the eastern basin of Lake Erie from the ITCAAN model were larger than estimates for the current assessment model. Based on results from Chapters 1 and 3 the estimates of abundancefor the eastern basin of Lake Erie may be overestimated. Given the level of movement within and between Lakes Erie and Huron, ITCAAN models may be a beneficial assessment methodology for management. I recommend future tagging studies of walleye on Lakes Erie and Huron be designed such that highreward tags and standardreward tags be released annually so that ITCAAN modelscan provide accurate abundances, natural mortality, and reporting rates estimates. Additionally, I recommend creel surveys be conducted on the HuronErie corridor by the Michigan Department of Natural Resources and Ontario Ministry of Natural Resources and Forestry to measure the currently unaccounted for walleye harvest, which is likely a significant source of mortality.
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 Title
 Integration of Topological Fingerprints and Machine Learning for the Prediction of Chemical Mutagenicity
 Creator
 Cao, Yin
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

Toxicity refers to the interaction between chemical molecules that leads to adverse effects in biological systems, and mutagenicity is one of its most important endpoints. Prediction of chemical mutagenicity is essential to ensuring the safety of drugs, foods, etc. In silico modeling of chemical mutagenicity, as a replacement of invivo bioassays, is increasingly encouraged, due to its efficiency, effectiveness, lower cost and less reliance on animal tests.The quality of a good molecular...
Show moreToxicity refers to the interaction between chemical molecules that leads to adverse effects in biological systems, and mutagenicity is one of its most important endpoints. Prediction of chemical mutagenicity is essential to ensuring the safety of drugs, foods, etc. In silico modeling of chemical mutagenicity, as a replacement of invivo bioassays, is increasingly encouraged, due to its efficiency, effectiveness, lower cost and less reliance on animal tests.The quality of a good molecular representation is usually the key to building an accurate and robust in silico model, in that each representation provides a different way for the machine to look at the molecular structure. While most molecular descriptors were introduced based on the physiochemical and biological activities of chemical molecules, in this study, we propose a new topological representation for chemical molecules, the combinatorial topological fingerprints (CTFs) based on persistent homology, knowing that persistent homology is a suitable tool to extract global topological information from a discrete sample of points. The combination of the proposed CTFs and machine learning algorithms could give rise to efficient and powerful in silico models for mutagenic toxicity prediction. Experimental results on a developmental toxicity dataset have also shown the predictive power of the proposedCTFs and its competitive advantages of characterizing and representing chemical molecules over existing fingerprints.
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 Title
 Experiments and Modeling of a Turbulent Jet Ignition System for Internal Combustion Engines
 Creator
 Gholamisheeri, Masumeh
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

This thesis experimentally, computationally and analytically examines the transient jet used to ignite combustible mixtures during Turbulent Jet Ignition (TJI). The TJI system is a prechamber initiated combustion enhancement system that can be used in place of a spark plug in a spark ignition (SI) engine. In TJI the ignition source, which originates in the prechamber, enters the main chamber through a connecting nozzle(s) as a transient high temperature jet of reacted mixture, reacting...
Show moreThis thesis experimentally, computationally and analytically examines the transient jet used to ignite combustible mixtures during Turbulent Jet Ignition (TJI). The TJI system is a prechamber initiated combustion enhancement system that can be used in place of a spark plug in a spark ignition (SI) engine. In TJI the ignition source, which originates in the prechamber, enters the main chamber through a connecting nozzle(s) as a transient high temperature jet of reacted mixture, reacting mixture and active radicals. TJI is capable of enabling low temperature combustion, through either lean or dilute combustion. For this work, TJI experiments were performed in an optically accessible Rapid Compression Machine (RCM). High speed visualization was performed via an SA4 high speed color camera and the images were compared with Computational Fluid Dynamics (CFD) modeling results. Comparison was also made between the experimental and numerical pressure data.A significant portion of this work is dedicated to the CFD modeling of the TJI process and for the first time a theoretical study of the jet flow field, density gradients, turbulence intensity, and temperature fields in both the prechamber and the main chamber was performed. The influences of nozzle size and mixture stoichiometry on jet penetration speed and combustion performance were investigated. Experiments were completed for turbulent jet ignition system orifice diameters of 2.0, 2.5 and 3.0 mm each at leantostoichiometric equivalence ratios of =0.67, 0.8 and 1.0. The hot jet velocity at the orifice exit was calculated, for the first time, using mathematical correlations. The Mach number and Reynolds number were also computed. The high speed imaging shows the influence of orifice diameter on flame propagation and the shape and structure of vortices resulting from the turbulent jet. Results revealed a direct relationship between orifice exit area reduction and a decrease in hot jet penetration speed. There was also a reduction in hot jet penetration speed with an increase in the equivalence ratio. Moreover, the jet was turbulent with calculated Reynolds numbers of around 20,000 or greater. Normalized transient results are presented that produce good agreement between the various model predictions. A discussion is provided of a new correlation model for the transient TJI process.In a separate set of experiments, the impact of an auxiliary fueled prechamber on the burn rate and on the lean or dilute limit extension of the RCM was investigated. Nitrogen was used as the diluent and the nitrogen dilution limit was found to be 35% of system mass. Both experimental and numerical results confirmed the idea of combustion enhancement of diluted mixtures by the prechamber auxiliary injection events.To model the turbulent jet of the TJI system, Reynolds Averaged NavierStokes (RANS) and Large Eddy Simulation (LES) turbulence models and the SAGE chemistry solver were used. To determine the effect of mechanism reduction, the pressure traces were computed using four (4) comprehensive chemical kinetic mechanisms (San Diego, Aramco, GRI, and NUI) and one (1) reduced chemical kinetic mechanism, which are all compared with the experimental pressure data. Results indicate that none of the mechanisms are in complete agreement, however they are in good agreement with the experimental burn rate, peak pressure and ignition delay predictions. The numerical isosurface temperature contours (1200, 800, 2000, and, 2400 K) were obtained which enable 3D views of the flame propagation, the jet discharge, the ignition and extinction events, and the heat release process.
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 Title
 On superconvergent discontinuous Galerkin methods for Schrödinger equations and sparse grid central discontinuous Galerkin method
 Creator
 Chen, Anqi
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

In this thesis, we design and analyze a discontinuous Galerkin (DG) method for onedimensional Schrödinger equations under a general class of numerical fluxes, and another efficient DG method for highdimensional hyperbolic equations.In the first DG method, we develop an ultraweak discontinuous Galerkin (UWDG) method to solve the onedimensional nonlinear Schrödinger equation. Stability conditions and error estimates are derived for the scheme with a general class of numerical fluxes. The...
Show moreIn this thesis, we design and analyze a discontinuous Galerkin (DG) method for onedimensional Schrödinger equations under a general class of numerical fluxes, and another efficient DG method for highdimensional hyperbolic equations.In the first DG method, we develop an ultraweak discontinuous Galerkin (UWDG) method to solve the onedimensional nonlinear Schrödinger equation. Stability conditions and error estimates are derived for the scheme with a general class of numerical fluxes. The error estimates are based on detailed analysis of the projection operator associated with each individual flux choice. Depending on the parameters, we find out that in some cases, the projection can be defined elementwise, facilitating analysis. In most cases, the projection is global, and its analysis depends on the resulting 2×2 blockcirculant matrix structures. For a large class of parameter choices, optimal a priori L2 error estimates can be obtained. Numerical examples are provided verifying theoretical results.In addition to the stability and error analysis, we analyze the superconvergence properties of the UWDG method for onedimensional linear Schrödinger equation with various choices of flux parameters. Depending on the flux choices and if the polynomial degree k is even or odd, we prove 2k or (2k1)th order superconvergence rate for cell averages and numerical flux of the function, as well as (2k1) or (2k2)th order for numerical flux of the derivative. In addition, we prove superconvergence of (k+2) or (k+3)th order of the UWDG solution towards a special projection. At a class of special points, the function values and the first and second order derivatives of the UWDG solution are superconvergent with order k+2, k+1, k, respectively. The proof relies on the correction function techniques initiated in [12], and applied to [10] for direct DG (DDG) methods for diffusion problems. By negative norm estimates, we apply the post processing technique and show that the accuracy of our scheme can be enhanced to order 2k. Theoretical results are verified by numerical experiments.In the second DG method, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the dual mesh is introduced and analyzed, resulting in a sparse finite element space that can be used for nonperiodic problems. Theoretical results, such as L2 stability and error estimates are obtained for scalar problems. CFL conditions are studied numerically comparing discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods. Numerical results including scalar linear equations, acoustic and elastic waves are provided.
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 Title
 Prime Torsion in the Brauer Group of an Elliptic Curve
 Creator
 Ure, Charlotte
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

The Brauer group is an invariant in algebraic geometry and number theory, that can beassociated to a field, variety, or scheme. Let k be a field of characteristic different from 2 or3, and let E be an elliptic curve over k. The Brauer group of E is a torsion abelian groupwith elements given by Morita equivalence classes of central simple algebras over the functionfield k(E). The MerkurjevSuslin theorem implies that any such element can be describedby a tensor product of symbol algebras. We...
Show moreThe Brauer group is an invariant in algebraic geometry and number theory, that can beassociated to a field, variety, or scheme. Let k be a field of characteristic different from 2 or3, and let E be an elliptic curve over k. The Brauer group of E is a torsion abelian groupwith elements given by Morita equivalence classes of central simple algebras over the functionfield k(E). The MerkurjevSuslin theorem implies that any such element can be describedby a tensor product of symbol algebras. We give a description of elements in the dtorsionof the Brauer group of E in terms of these tensor products, provided that the dtorsion ofE is krational and k contains a primitive dth root of unity. Furthermore, if d = q is aprime, we give an algorithm to compute the qtorsion of the Brauer group over any field kof characteristic different from 2,3, and q containing a primitive qth root of unity.
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 Title
 Transition Path Theory and Transition State
 Creator
 Du, Jun
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

This thesis will mainly discuss the transition path theory and its extension to the transition state. The framework of transition path theory (TPT) is developed in the context of continuoustime Markov chains on discrete statespaces. Under the assumption of ergodicity,Transition path theory will first choose any two subsets (mostly metastable states) in the finite statespace based on the equilibrium distribution of the transition probability, and then it analyzes the statistical properties...
Show moreThis thesis will mainly discuss the transition path theory and its extension to the transition state. The framework of transition path theory (TPT) is developed in the context of continuoustime Markov chains on discrete statespaces. Under the assumption of ergodicity,Transition path theory will first choose any two subsets (mostly metastable states) in the finite statespace based on the equilibrium distribution of the transition probability, and then it analyzes the statistical properties of those associated reactive trajectories, for instance,those trajectories by which the random walker transits from one subset to another. Transition path theory gives properties of these trajectories, such as their probability distribution, their probability current and flux, and their rate of occurrence and finally the dominantreaction pathways. In this thesis, we will first introduce the framework of transition path theory for Markov chains, and then briefly discuss its relation to the electric resistor network theory and Laplacian eigenmaps, and also diffusion maps is discussed as well.Based on Transition Path Theory (TPT) for Markov jump processes, this thesis develops a general approach for identifying and calculating Transition States (TS) of stochastic chemical reacting networks. The thesis first extend the concept of probability current, originally defined on edges connecting different nodes in the configuration space, to each subnetwork. To locate subnetworks with maximal probability current on the separatrix between reactive and nonreactive events, which will give the Transition States of the reaction, constraint optimization is conducted. The thesis further introduce an alternative scheme to compute the transition pathways by topological sorting, which is shown to be highly efficient through analysis. Finally, the theory and algorithms are illustrated in several examples.
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 Title
 Green's Functions for Variants of the SchrammLoewner Evolution
 Creator
 Mackey, Benjamin
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

We prove upper bounds for the probability that a radial SLE$_{\kappa}$ curve comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for the probability that a wholeplane SLE$_{\kappa}$ passes near any $n$ points in the complex plane. We then use these estimates to show that the lower Minkowski content of both the radial and wholeplane SLE$_{\kappa}$ traces has finite moments of any order.For $\kappa \leq 4$, the...
Show moreWe prove upper bounds for the probability that a radial SLE$_{\kappa}$ curve comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for the probability that a wholeplane SLE$_{\kappa}$ passes near any $n$ points in the complex plane. We then use these estimates to show that the lower Minkowski content of both the radial and wholeplane SLE$_{\kappa}$ traces has finite moments of any order.For $\kappa \leq 4$, the reverse flow of the Loewner equation driven by $\sqrt{\kappa}B_t$ generates a random continuous function $\phi: \R^+ \to \R^+$ called the conformal welding. In studying backward SLE, this plays the roll of the global random object, rather than the SLE trace. Given any $x,y>0$ we use the Girsanov theorem to construct a family of probability measures, depending on some parameters, under which the conformal welding satisfies $\phi(x)=y$ almost surely. For one such law, we prove a onepoint estimate for the backward SLE welding and show how it coincides with the Green's function. In another case, we decompose the law of the welding conditioned to pass through $(x,y)$ into two pieces. Using this decomposition, we integrate this law over a set $U\subset [0,\infty)\times[0,\infty)$ to get a new measure on weldings which is absolutely continuous with respect to the original backward SLE welding. Moreover, the RadonNikodym derivative is given by the capacity time that the graph of $\phi$ spends in $U$. In the last chapter, we study a generalization of the chordal Loewner equation called chordal measure driven Loewner evolution. We show existence of a solution to the equation, and a onetoone correspondence between the appropriate measures and all continuously growing families of $\H$hulls. In \cite{MS}, the notion of measure driven Loewner evolution was first introduced in the radial setting, and a similar theorem was proven. This result is pure complex analysis, without any reference to probability theory.
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 Title
 Phase Retrieval from Continuous and Discrete Ptychographic Measurements
 Creator
 Merhi, Sami Eid
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

In this dissertation, we present and study two novel approaches to phase retrieval  an inverse problem in which one attempts to reconstruct a complexvalued function (or vector) from phaseless (or magnitudeonly) measurements. Phase retrieval arises in several scientific areas including biochemistry, optics, astronomy, quantum mechanics, and speech signal processing. Early solutions to phase retrieval, although practical, lacked robustness guarantees. To this day, practitioners in...
Show moreIn this dissertation, we present and study two novel approaches to phase retrieval  an inverse problem in which one attempts to reconstruct a complexvalued function (or vector) from phaseless (or magnitudeonly) measurements. Phase retrieval arises in several scientific areas including biochemistry, optics, astronomy, quantum mechanics, and speech signal processing. Early solutions to phase retrieval, although practical, lacked robustness guarantees. To this day, practitioners in scientific imaging are still seeking demonstrably stable and robust recovery algorithms.Ptychography is a form of coherent diffractive imaging where diffraction patterns are processed by algorithms to recover an image of a specimen. More specifically, small regions of a specimen are illuminated oneatatime, and a detector captures the intensities of the resulting diffraction patterns. As such, the measurements are local and phaseless. In this work, we present two algorithms to recover signals from ptychographic measurements. The first algorithm aims to recover a discrete onedimensional signal from discrete spectrogram measurements via a modified Wigner distribution deconvolution (WDD) method. While the method is known to practitioners of scientific imaging, robustness and recovery guarantees are lacking, if not absent; our contribution is to supply such guarantees. The second algorithm aims to approximately recover a compactly supported function from continuous spectrogram measurements via lifting and angular synchronization. This setup can be interpreted as the infinitedimensional equivalent of discrete ptychographic imaging. Our contribution is a model which assumes infinitedimensional signals and measurements ab initio, as opposed to most recent algorithms in which discrete models are a necessity.Finally, we consider the worstcase noise robustness of any phase retrieval algorithm which aims to reconstruct all nonvanishing vectors from the magnitudes of an arbitrary collection of local correlation measurements. The robustness results provided therein apply to a wide range of ptychographic imaging scenarios. In particular, our contribution is to show that stable recovery of highresolution images of extremely large samples is likely to require a vast number of measurements, independent of the recovery algorithm employed.The first chapter introduces the phase retrieval problem and presents historical context, as well as applications in which phase retrieval manifests. In addition, we introduce ptychography, discuss existing WDD formulations, and compare these to our contribution in the discrete setting. Chapter 2 provides recovery guarantees for using aliased WDD methods to solve the phase retrieval problem in a discrete setting with subsampled measurements. In Chapter 3 we provide lower Lipschitz bounds for generic phase retrieval algorithms from locally supported measurements. Finally, Chapter 4 presents a numerical method to recover compactly supported functions from local measurements via lifting and angular synchronization.
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 Title
 BlowUp Problems for the Heat Equation with Local Nonlinear Neumann Boundary Conditions
 Creator
 Yang, Xin
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

This thesis studies the blowup problem for the heat equation $u_t=\Delta u$ in a $C^{2}$ bounded open subset $\Omega$ of $\m{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a local nonlinear Neumann boundary condition: $\frac{\p u}{\p n}=u^{q}$ on partial boundary $\Gamma_1\subseteq\p\O$ for some $q>1$ and $\frac{\p u}{\p n}=0$ on the rest of the boundary. The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying...
Show moreThis thesis studies the blowup problem for the heat equation $u_t=\Delta u$ in a $C^{2}$ bounded open subset $\Omega$ of $\m{R}^{n}(n\geq 2)$ with positive initial data $u_{0}$ and a local nonlinear Neumann boundary condition: $\frac{\p u}{\p n}=u^{q}$ on partial boundary $\Gamma_1\subseteq\p\O$ for some $q>1$ and $\frac{\p u}{\p n}=0$ on the rest of the boundary. The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. First, we establish the local existence and uniqueness of the classical solution for such a problem. Secondly, we show the finitetime blowup of the solution and estimate both upper and lower bounds of the blowup time $T^{*}$. In addition, the asymptotic behaviour of $T^{*}$ on $q$, $M_{0}$ (the maximum of the initial data) and $\Gamma_{1}$ (the surface area of $\Gamma_{1}$) are studied. \begin{itemize}\item As $q\searrow 1$, the order of $T^{*}$ is exactly $(q1)^{1}$.\item As $M_{0}\searrow 0$, the order of $T^{*}$ is at least $\ln(M_{0}^{1})$; if the region near $\Gamma_{1}$ is convex, then the order of $T^{*}$ is at least $M_{0}^{(q1)}/\ln(M_{0}^{1})$; if $\O$ is convex, then the order of $T^{*}$ is at least $M_{0}^{(q1)}$. On the other hand, if the initial data $u_{0}$ does not oscillate too much, then the order of $T^{*}$ is at most $M_{0}^{(q1)}$. \item As $\Gamma_{1}\searrow 0$, the order of $T^{*}$ is at least $\ln(\Gamma_{1}^{1})$ and at most $\Gamma_{1}^{1}$.If the region near $\Gamma_{1}$ is convex, then the order of $T^{*}$ is at least $\Gamma_{1}^{\frac{1}{n1}}\Big/\ln\big(\Gamma_{1}^{1}\big)$ for $n\geq 3$ and $\Gamma_{1}^{1}\big/\big[\ln\big(\Gamma_{1}^{1}\big)\big]^{2}$ for $n=2$. If $\O$ is convex, then the order of $T^{*}$ is at least $\Gamma_{1}^{\frac{1}{n1}}$ for $n\geq 3$ and $\Gamma_{1}^{1}\big/\ln\big(\Gamma_{1}^{1}\big)$ for $n=2$.\end{itemize} Finally, we provide two strategies from engineering point of view (which means by changing the setup of the original problem) to prevent the finitetime blowup. Moreover, if the region near $\Gamma_{1}$ is convex, then one of the strategies is applied to bound the solution from above by $M_{1}$ for any $M_{1}>M_{0}$. For the space shuttle mentioned in the motivation of this thesis, $\Gamma_{1}$ is on its left wing of the shuttle, so the region near $\Gamma_{1}$ is indeed convex. In addition, the relation between $T^{*}$ and small surface area $\Gamma_{1}$ is of particular interest for this problem. As an application of the above estimates to this problem, let $n=3$ and $\Gamma_{1}\searrow 0$, then the order of $T^{*}$ is between $\Gamma_{1}^{\frac{1}{2}}\Big/\ln\big(\Gamma_{1}^{1}\big)$ and $\Gamma_{1}^{1}$. On the other hand, one of the strategies can be applied to prevent the temperature from being too high.This thesis seems to be the first to systematically study the heat equation with piecewise continuous Neumann boundary conditions. It also seems to be the first to investigate the relation between $T^{*}$ and $\Gamma_{1}$, especially when $\Gamma_{1}\searrow 0$. The key innovative part of this thesis is Chapter 4. First, the new method developed in Chapter 4 is able to derive a lower bound for $T^{*}$ without the convexity assumption of the domain which was a common requirement in the historical works. Secondly, even for the convex domains, the lower bound estimate obtained by this new method improves the previous results significantly. Thirdly, this method does not involve any differential inequality argument which was an essential technique in the past on the blowup time estimate.
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 Title
 LogCanonical Poisson Structures and NonCommutative Integrable Systems
 Creator
 Ovenhouse, Nicholas
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

Logcanonical Poisson structures are a particularly simple type of bracket which are given byquadratic expressions in local coordinates. They appear in many places, including the study ofcluster algebras. A Poisson bracket is “compatible” with a cluster algebra structure if the bracketis logcanonical with respect to each cluster. In joint work with John Machacek, we prove astructural result about such Poisson structures, which justifies the use and significance of suchbrackets in cluster...
Show moreLogcanonical Poisson structures are a particularly simple type of bracket which are given byquadratic expressions in local coordinates. They appear in many places, including the study ofcluster algebras. A Poisson bracket is “compatible” with a cluster algebra structure if the bracketis logcanonical with respect to each cluster. In joint work with John Machacek, we prove astructural result about such Poisson structures, which justifies the use and significance of suchbrackets in cluster theory. The result says that no rational coordinatechanges can transformthese brackets into a simpler form.The pentagram map is a discrete dynamical system on the space of plane polygons first introduced by Schwartz in 1992. It was proved to be Liouville integrable by Schwartz, Ovsienko, andTabachnikov in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associatedto certain networks embedded in a disc or annulus, and its relation to cluster algebras. ThesePoisson structures are logcanonical. Later, Gekhtman et al. and Tabachnikov reinterpreted thepentagram map in terms of these networks, and used the associated Poisson structures to give anew proof of integrability.In 2011, Mari Beffa and Felipe introduced a generalization of the pentagram map to certainGrassmannians, and proved it had a Lax representation. We reinterpret this Grassmann pentagram map in terms of noncommutative algebra, in particular the double brackets of Van denBergh, and generalize the approach of Gekhtman et al. to establish a noncommutative versionof integrability. The integrability of the pentagram maps in both projective space and the Grassmannian follow from this more general algebraic system by projecting to representation spaces.
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 Title
 MODULATIONAL STABILITY OF MULTIPULSES WITHIN THE FUNCTIONALIZED CAHNHILLIARD GRADIENT FLOW
 Creator
 Guckir Cakir, Hayriye
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

The Functionalized CahnHilliard (FCH) energy is a model describing the interfacial energy in a phase separated mixture of amphiphilic molecules and a solvent. On a bounded domain in R, the EulerLagrange equation for the mass constrained Functionalized CahnHilliard(FCH) free energy with zero functionalization terms is derived and a large family of multipulse critical points is constructed. We show that the FCH energy with no functionalization terms subject to a mass constraint has global...
Show moreThe Functionalized CahnHilliard (FCH) energy is a model describing the interfacial energy in a phase separated mixture of amphiphilic molecules and a solvent. On a bounded domain in R, the EulerLagrange equation for the mass constrained Functionalized CahnHilliard(FCH) free energy with zero functionalization terms is derived and a large family of multipulse critical points is constructed. We show that the FCH energy with no functionalization terms subject to a mass constraint has global minimizers over a variety of admissible sets. We introduce a multipulse ansatz as the extensions of the periodic multipulse critical points to R and establish the H^2coercivity of the second variation of the energy about multipulse ansatz. Modulational stability and the dynamic evolution of the multipulse ansatz with respect to the Pi_0gradient flow are also addressed.
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 Title
 KNOT CONCORDANCES IN 3MANIFOLDS
 Creator
 Yıldız, Eylem Zeliha
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

We deal with some questions regarding concordance of knots in arbitrary closed $3$manifolds. We first prove that, any nontrivial element in the fundamental group of a closed, oriented $3$manifold gives rise to infinitely many distinct smooth almostconcordance classes in the free homotopy class of the unknot. In particular, we consider these distinct smooth almostconcordance classes on the boundary of a Mazur manifold and we show none of these distinct classes bounds a PLdisk in the...
Show moreWe deal with some questions regarding concordance of knots in arbitrary closed $3$manifolds. We first prove that, any nontrivial element in the fundamental group of a closed, oriented $3$manifold gives rise to infinitely many distinct smooth almostconcordance classes in the free homotopy class of the unknot. In particular, we consider these distinct smooth almostconcordance classes on the boundary of a Mazur manifold and we show none of these distinct classes bounds a PLdisk in the Mazur manifold. On the other hand, all the representatives we construct are topologically slice. We also prove that all knots in the free homotopy class of $S^1 \times pt$ in $S^1 \times S^2$ are smoothly concordant.
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 Title
 A topological study of toroidal dynamics
 Creator
 Gakhar, Hitesh
 Date
 2020
 Collection
 Electronic Theses & Dissertations
 Description

This dissertation focuses on developing theoretical tools in the field of Topological Data Analysis and more specifically, in the study of toroidal dynamical systems. We make contributions to the development of persistent homology by proving Künnethtype theorems, to topological time series analysis by further developing the theory of sliding window embeddings, and to multiscale data coordinatization in topological spaces by proving stability theorems. First, in classical algebraic topology,...
Show moreThis dissertation focuses on developing theoretical tools in the field of Topological Data Analysis and more specifically, in the study of toroidal dynamical systems. We make contributions to the development of persistent homology by proving Künnethtype theorems, to topological time series analysis by further developing the theory of sliding window embeddings, and to multiscale data coordinatization in topological spaces by proving stability theorems. First, in classical algebraic topology, the Künneth theorem relates the homology of two topological spaces with that of their product. We prove Künneth theorems for the persistent homology of the categorical and tensor product of filtered spaces. That is, we describe the persistent homology of these product filtrations in terms of that of the filtered components. Using these theorems, we also develop novel methods for algorithmic and abstract computations of persistent homology. One of the direct applications of these results is the abstract computation of Rips persistent homology of the Ndimensional torus.Next, we develop the general theory of sliding window embeddings of quasiperiodic functions and their persistent homology. We show that the sliding window embeddings of quasiperiodic functions, under appropriate choices of the embedding dimension and time delay, are dense in higher dimensional tori. We also explicitly provide methods to choose these parameters. Furthermore, we prove lower bounds on Rips persistent homology of these embeddings. Using one of the persistent Künneth formulae, we provide an alternate algorithm to compute the Rips persistent homology of the sliding window embedding, which outperforms the traditional methods of landmark sampling in both accuracy and time. We also apply our theory to music, where using sliding windows and persistent homology, we characterize dissonant sounds as quasiperiodic in nature.Finally, we prove stability results for sparse multiscale circular coordinates. These coordinates on a data set were first created to aid nonlinear dimensionality reduction analysis. The algorithm identifies a significant integer persistent cohomology class in the Rips filtration on a landmark set and solves a linear least squares optimization problem to construct a circled valued function on the data set. However, these coordinates depend on the choice of the landmarks. We show that these coordinates are stable under Wasserstein noise on the landmark set.
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 Title
 KRational Preperiodic Points and Hypersurfaces on Projective Space
 Creator
 Troncoso Naranjo, Sebastian Ignacio
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

The present thesis has two main parts. In the first one, we study bounds for the number of rational preperiodic points of an endomorphism of $\PP^1$. Let $K$ be a number field and $\phi$ be an endomorphism of $\PP^1$ over $K$ of degree $d\geq 2$. Let $S$ be the set of places of bad reduction for $\phi$ (including the archimedean places). Let $\Per(\phi,K)$, $\PrePer(\phi, K)$, and $\Tail(\phi,K)$ be the set of $K$rational periodic, preperiodic, and purely preperiodic points of $\phi$,...
Show moreThe present thesis has two main parts. In the first one, we study bounds for the number of rational preperiodic points of an endomorphism of $\PP^1$. Let $K$ be a number field and $\phi$ be an endomorphism of $\PP^1$ over $K$ of degree $d\geq 2$. Let $S$ be the set of places of bad reduction for $\phi$ (including the archimedean places). Let $\Per(\phi,K)$, $\PrePer(\phi, K)$, and $\Tail(\phi,K)$ be the set of $K$rational periodic, preperiodic, and purely preperiodic points of $\phi$, respectively.If we assume that $\Per(\phi,K) \geq 4$ (resp.\ $\Tail(\phi,K) \geq 3$), we prove bounds for $\Tail(\phi,K)$ (resp.\ $\Per(\phi,K)$) that depend only on the number of places of bad reduction $S$ (and not on the degree $d$). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when $\Per(\phi,K) < 4$ (resp.\ $\Tail(\phi,K) < 3$). The key tool involved in these results is a bound for the number of solutions of $S$unit equations.Using bounds for the number of solutions of the celebrated ThueMahler equation, we obtain bounds for $\Per(\phi,K)$ and $\Tail(\phi,K)$ in terms of the number of places of bad reduction $S$ and the degree $d$ of the rational function $\phi$. Bounds obtained in this way are a significant improvement to previous result given by J. Canci and L. Paladino.In the second part of the thesis, we study the set of $K$rational purely preperiodic hypersurfaces of $\PP^n$ of a given degree for an endomorphism of $\PP^n$. Let $\phi$ be an endomorphism of $\PP^n$ over $K$, $S$ be the set of places of bad reduction for $\phi$ and $\HTail(\phi,K,e)$ be the set of $K$rational purely preperiodic hypersurfaces of $\PP^n$ of degree $e$. We give a strong arithmetic relation between $K$rational purely preperiodic hypersurfaces and $K$rational periodic points. If we consider $N=\binom{e+n}{e}1$ and assume that $\phi$ has at least $2N+1$ $K$rational periodic points such that no $N+1$ of them lie in a hypersurface of degree $e$ then we give an effective bound on a large subset of $\HTail(\phi,K,e)$ depending on $e$ and the number of places of bad reduction $S$. Finally, we prove that the set $\HTail(\phi,K,e)$ is finite if we assume that $\phi$ is an endomorphism of $\PP^2$.
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 Title
 SubLinear Sparse Fourier Transform Algorithm
 Creator
 Zhang, Ruochuan
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

The Discrete Fourier Transform (DFT) plays a crucial role in signal processing and scientific computing. The most famous algorithm for computing the DFT is the Fast Fourier Transform (FFT), which has runtime O(N log N) for an input vector with length N. However, with the increasing size of data set, the FFT is no longer fast enough and often becomes the major computational bottleneck in many applications. The Sparse Fourier Transform (SFT) tries to solve this problem by finding the best s...
Show moreThe Discrete Fourier Transform (DFT) plays a crucial role in signal processing and scientific computing. The most famous algorithm for computing the DFT is the Fast Fourier Transform (FFT), which has runtime O(N log N) for an input vector with length N. However, with the increasing size of data set, the FFT is no longer fast enough and often becomes the major computational bottleneck in many applications. The Sparse Fourier Transform (SFT) tries to solve this problem by finding the best s−term Fourier representation using only a subset of the input data, in time sublinear in the data set size O(poly(s;log N)). Some of the existing SFT algorithms are capable of working with equally spaced samples, while others just assume that the algorithms can sample anywhere they want, which is an unrealistic assumption in many realworld applications. In this thesis, we propose a generic method of transforming any noise robust SFT algorithm into a sublineartime sparse DFT algorithm which rapidly approximates Ff from a given input vector f 2 CN, where F is the DFT matrix. Our approach is based on filter function and fast discrete convolution. We prove that with an appropriate filter function g (periodic Gaussian function in this thesis), one can always approximate the value of the convolution function g ∗ f at the desired point rapidly and accurately even when f is a high oscillating function. We then construct several new sublineartime sparse DFT algorithms from existing sparse Fourier algorithms which utilize unequally spaced function samples. Besides giving the theoretical runtime and error guarantee, we also show empirically that the best of these new discrete SFT algorithms outperforms both FFTW and sFFT2.0 in the sense of runtime and robustness when the vector length N is large. At the end of the thesis, we present a deterministic sparse Fourier transform algorithm which breaks the quadraticinsparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. We show empirically that this structured SFT algorithm outperforms standard sparse Fourier transforms in the rapid recovery of block frequency sparse functions.
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 Title
 Integration of Topological Data Analysis and Machine Learning for Small Molecule Property Predictions
 Creator
 Wu, Kedi
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

Accurate prediction of small molecule properties is of paramount importance to drug design and discovery. A variety of quantitative properties of small molecules has been studied in this thesis. These properties include solvation free energy, partition coefficient, aqueous solubility, and toxicity endpoints. The highlight of this thesis is to introduce an algebraic topology based method, called element specific persistent homology (ESPH), to predict small molecule properties. Essentially ESPH...
Show moreAccurate prediction of small molecule properties is of paramount importance to drug design and discovery. A variety of quantitative properties of small molecules has been studied in this thesis. These properties include solvation free energy, partition coefficient, aqueous solubility, and toxicity endpoints. The highlight of this thesis is to introduce an algebraic topology based method, called element specific persistent homology (ESPH), to predict small molecule properties. Essentially ESPH describes molecular properties in terms of multiscale and multicomponent topological invariants and is different from conventional chemical and physical representations. Based on ESPH and its modified version, elementspecific topological descriptors (ESTDs) are constructed. The advantage of ESTDs is that they are systematical, comprehensive, and scalable with respect to molecular size and composition variations, and are readily suitable for machine learning methods, rendering topological learning algorithms. Due to the inherent correlation between different small molecule properties, multitask frameworks are further employed to simultaneously predict related properties. Deep neural networks, along with ensemble methods such as random forest and gradient boosting trees, are used to develop quantitative predictive models. Physical based molecular descriptors and auxiliary descriptors are also used in addition to ESTDs. As a result, we obtain stateoftheart results for various benchmark data sets of small molecule properties. We have also developed two online servers for predicting properties of small molecules, TopPS and TopTox. TopPS is a software for topological learning predictions of partition coefficient and aqueous solubility, and TopTox is a software for computing elementspecific tological descriptors (ESTDs) for toxicity endpoint predictions. They are available at http://weilab.math.msu.edu/TopPS/ and http://weilab.math.msu.edu/TopTox/, respectively.
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 Title
 Concave Fillings and Branched Covers
 Creator
 Kasebian, Kaveh
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

This dissertation contains two results. The first result involves concave symplectic structures on a neighborhood of certain plumbing of symplectic surfaces, introduced by D. Gay.We draw the contact surgery diagram of the induced contact structure on boundary of aconcave filling, when the induced open book is planar. We show that every Brieskorn sphereadmits a concave Filling in the sense of D. Gay and the induced contact structure on it isovertwisted. We also show that in certain cases a (...
Show moreThis dissertation contains two results. The first result involves concave symplectic structures on a neighborhood of certain plumbing of symplectic surfaces, introduced by D. Gay.We draw the contact surgery diagram of the induced contact structure on boundary of aconcave filling, when the induced open book is planar. We show that every Brieskorn sphereadmits a concave Filling in the sense of D. Gay and the induced contact structure on it isovertwisted. We also show that in certain cases a (1)sphere in Gay's plumbing can beblown down to obtain a concave plumbing of the same type. The next result examines thecontact structure induced on the boundary of the cork W1, induced by the double branchedcover over a ribbon knot. We show this contact structure is overtwisted in a specific case.
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