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 Title
 A Variablelength manyobjective optimization approach in image segmentation problems
 Creator
 Huang, Xuhui
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

Image segmentation is a technique of dividing an image space into a number of meaningful homogeneous regions. Various data clustering techniques have been adapted in solving segmentation problems. In particular, data clustering is often posed as multioptimization problem so that characteristics of data could be caught by different objectives simultaneously. Traditional multioptimization methods often require some prior knowledge or assumptions about data, performance is poor if these...
Show moreImage segmentation is a technique of dividing an image space into a number of meaningful homogeneous regions. Various data clustering techniques have been adapted in solving segmentation problems. In particular, data clustering is often posed as multioptimization problem so that characteristics of data could be caught by different objectives simultaneously. Traditional multioptimization methods often require some prior knowledge or assumptions about data, performance is poor if these assumptions do not hold. Limitations with established multioptimization methods are caused by their inadequacy in handling a large number of objectives. Nondominated sorting genetic algorithm III (NSGAIII) is proposed to alleviate this issue. However, NSGAIII is inefficient in removing some bad solutions in highdimensional searching space during evolution. In this article, we propose a variable string length manyobjective genetic algorithm(VMOGA) whose framework has evolved from NSGAIII and its encoding strategy, genetic and evolutionary operator have been redesigned. Performance of VMOGA in image segmentation problems is further enhanced by an appropriate selection of objectives. In the end, we conduct unsupervised segmentation by proposed clustering technique on magnetic resonance image(MRI) of human brain. Comparisons with other evolutionary algorithms are presented and dominance of VMOGA has been demonstrated quantitatively. VMOGA is also performed on detection of delamination area caused by fatigue loading in Mode I glass fiber reinforced polymer (GFRP) samples. Results are compared with fast marching algorithm(FMA) and superiority of VMOGA suggests future potential application in fatigue detection.
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 Title
 A comparative evaluation of programmed and lecture instruction in college business mathematics
 Creator
 Swartz, Manfred E.
 Date
 1985
 Collection
 Electronic Theses & Dissertations
 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
 ALGEBRAIC TOPOLOGY AND GRAPH THEORY BASED APPROACHES FOR PROTEIN FLEXIBILITY ANALYSIS AND B FACTOR PREDICTION
 Creator
 Bramer, David
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

Protein fluctuation, measured by B factors, has been shown to highly correlate to protein flexibility and function. Several methods have been developed to predict protein B factoras well as related applications such as docking pose ranking, domain separation, entropycalculation, hinge detection, hot spot detection, stability analysis, etc. While many B factormethods exist, reliable B factor prediction continues to be an ongoing challenge and there ismuch room for improvement.This work...
Show moreProtein fluctuation, measured by B factors, has been shown to highly correlate to protein flexibility and function. Several methods have been developed to predict protein B factoras well as related applications such as docking pose ranking, domain separation, entropycalculation, hinge detection, hot spot detection, stability analysis, etc. While many B factormethods exist, reliable B factor prediction continues to be an ongoing challenge and there ismuch room for improvement.This work introduces a paradigm shifting geometric graph based model called the multiscale weighted colored graph (MWCG) model. The MWCG model is a new generation of computational algorithms that signicantly improves the current landscape of protein structural fluctuation analysis. The MWCG model treats each protein as a colored graph where colored nodes correspond to atomic element types and edges are weighted by a generalized centrality metric. Each graph contains multiple subgraphs based on interaction typesbetween graphic nodes, then protein rigidity is represented by generalized centralities of subgraphs. MWCGs predict the B factors of protein residues and accurately analyze the flexibility of all atoms in a protein simultaneously. The MWCG model presented in thiswork captures element specific interactions across multiple scales and is a novel visual tool for identifying various protein secondary structures. This work also demonstrates MWCG protein hinge detection using a variety of proteins.Cross protein prediction of protein B factors has previously been an unsolved problem in terms of B factor prediction methods. Since many proteins are dicult to crystallize, and for some it is likely impossible, models that can cross predict protein B factor are absolutelynecessary. By integrating machine learning and the advanced graph theory MWCG method, this work provides a robust cross protein B factor prediction solution using a set of known proteins to predict the B factors of a protein previously unseen to the algorithm. Thealgorithm connects different proteins using global protein features such as the resolution of the Xray crystallography data. The combination of global and local features results in successful cross protein B factor prediction. To test and validate these results this work considers several machine learning approaches such as random forest, gradient boosted trees, and deep convolutional neural networks.Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of topological representations of the molecule. However, persistent homology is rarely employed for the analysis of atomic properties, such as biomolecular flexibility analysis or B factor prediction. This work introduces atom specific persistent homology (ASPH) to provide a local atomic level representation of a molecule via a global topological tool. This is achieved through the construction of a pair of conjugated sets of atoms and corresponding conjugated simplicial complexes, as well as conjugated topological spaces. The difference between the topological invariants of the pair of conjugated sets is measured by Bottleneck and Wasserstein metrics and leads to anatom specic topological representation of individual atomic properties in a molecule. Atom specific topological features are integrated with various machine learning algorithms, including gradient boosting trees and convolutional neural network for protein thermal fluctuation analysis and blind cross protein B factor prediction.Extensive numerical testing indicates the proposed methods provide novel and powerful graph theory and algebraic topology based tools for analyzing and predicting atom specific, localized protein flexibility information.
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 Title
 Abelian varieties associated to Clifford algebras
 Creator
 Machen, Casey
 Date
 2016
 Collection
 Electronic Theses & Dissertations
 Description

The KugaSatake construction is a construction in algebraic geometry which associates an abelian variety to a polarized K3surface X. This abelian variety, A, is created from the Clifford algebra arising from the quadratic space H^2(X,Z)/torsion with its natural cohomology pairing. Furthermore, there is an inclusion of Hodge structures H^2(X,Q)\hookrightarrow H^1(A,Q)\otimes H^1(A,Q) relating the cohomology of the original K3surface with that of the abelian variety. We investigate when this...
Show moreThe KugaSatake construction is a construction in algebraic geometry which associates an abelian variety to a polarized K3surface X. This abelian variety, A, is created from the Clifford algebra arising from the quadratic space H^2(X,Z)/torsion with its natural cohomology pairing. Furthermore, there is an inclusion of Hodge structures H^2(X,Q)\hookrightarrow H^1(A,Q)\otimes H^1(A,Q) relating the cohomology of the original K3surface with that of the abelian variety. We investigate when this construction can be generalized to both arbitrary quadratic forms as well as higher degree forms. Specifically, we associate an abelian variety to the Clifford algebra of an arbitrary quadratic form in a way which generalizes the KugaSatake construction. When the quadratic form arises as the intersection pairing on the middledimensional cohomology of an algebraic variety Y, we investigate when the cohomology of the abelian variety can be related to that of Y. Additionally, we explore when families of algebraic varieties give rise to families of abelian varieties via this construction. We use these techniques to build an analogous method for constructing an abelian variety from the generalized Clifford algebra of a higher degree form. We find certain families of complex projective 3folds and 4folds for which an abelian variety can be constructed from the respective cubic and quartic forms on H^2. The relations between the cohomology of the abelian variety and the original variety are also discussed.
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 Title
 Adaptive independent component analysis : theoretical formulations and application to CDMA communication system with electronics implementation
 Creator
 Albataineh, Zaid
 Date
 2014
 Collection
 Electronic Theses & Dissertations
 Description

Blind Source Separation (BSS) is a vital unsupervised stochastic area that seeks to estimate the underlying source signals from their mixtures with minimal assumptions about the source signals and/or the mixing environment. BSS has been an active area of research and in recent years has been applied to numerous domains including biomedical engineering, image processing, wireless communications, speech enhancement, remote sensing, etc. Most recently, Independent Component Analysis (ICA) has...
Show moreBlind Source Separation (BSS) is a vital unsupervised stochastic area that seeks to estimate the underlying source signals from their mixtures with minimal assumptions about the source signals and/or the mixing environment. BSS has been an active area of research and in recent years has been applied to numerous domains including biomedical engineering, image processing, wireless communications, speech enhancement, remote sensing, etc. Most recently, Independent Component Analysis (ICA) has become a vital analytical approach in BSS. In spite of active research in BSS, however, many foundational issues still remain in regards to convergence speed, performance quality and robustness in realistic or adverse environments. Furthermore, some of the developed BSS methods are computationally expensive, sensitive to additive and background noise, and not suitable for a real4time or real world implementation. In this thesis, we first formulate new effective ICA4based measures and their corresponding robust adaptive algorithms for the BSS in dynamic "convolutive mixture" environments. We demonstrate their superior performance to present competing algorithms. Then we tailor their application within wireless (CDMA) communication systems and Acoustic Separation Systems. We finally explore a system realization of one of the developed algorithms among ASIC or FPGA platforms in terms of real time speed, effectiveness, cost, and economics of scale. Firstly, we propose a new class of divergence measures for Independent Component Analysis (ICA) for estimating sources from mixtures. The Convex Cauchy4Schwarz Divergence (CCS4DIV) is formed by integrating convex functions into the Cauchy4Schwarz inequality. The new measure is symmetric and convex with respect to the joint probability, where the degree of convexity can be tuned by a (convexity) parameter. A non4parametric (ICA) algorithm generated from the proposed divergence is developed exploiting convexity parameters and employing the Parzen window4based distribution estimates. The new contrast function results in effective parametric and non4parametric ICA4based computational algorithms. Moreover, two pairwise iterative schemes are proposed to tackle the high dimensionality of sources. Secondly, a new blind detection algorithm, based on fourth order cumulant matrices, is presented and applied to the multi4user symbol estimation problem in Direct Sequence Code Division Multiple Access (DS4CDMA) systems. In addition, we propose three new blind receiver schemes, which are based on the state space structures. These so4called blind state4space receivers (BSSR) do not require knowledge of the propagation parameters or spreading code sequences of the users but relies on the statistical independence assumption among the source signals. Lastly, system realization of one of the developed algorithms has been explored among ASIC or FPGA platforms in terms of cost, effectiveness, and economics of scale. Based on our findings of current stat4of4the4art electronics, programmable FPGA designs are deemed to be the most effective technology to be used for ICA hardware implementation at this time.In this thesis, we first formulate new effective ICAbased measures and their corresponding robust adaptive algorithms for the BSS in dynamic "convolutive mixture" environments. We demonstrate their superior performance to present competing algorithms. Then we tailor their application within wireless (CDMA) communication systems and Acoustic Separation Systems. We finally explore a system realization of one of the developed algorithms among ASIC or FPGA platforms in terms of real time speed, effectiveness, cost, and economics of scale.We firstly investigate several measures which are more suitable for extracting different source types from different mixing environments in the learning system. ICA for instantaneous mixtures has been studied here as an introduction to the more realistic convolutive mixture environments. Convolutive mixtures have been investigated in the time/frequency domains and we demonstrate that our approaches succeed in resolving the standing problem of scaling and permutation ambiguities in present research. We propose a new class of divergence measures for Independent Component Analysis (ICA) for estimating sources from mixtures. The Convex CauchySchwarz Divergence (CCSDIV) is formed by integrating convex functions into the CauchySchwarz inequality. The new measure is symmetric and convex with respect to the joint probability, where the degree of convexity can be tuned by a (convexity) parameter. A nonparametric (ICA) algorithm generated from the proposed divergence is developed exploiting convexity parameters and employing the Parzen windowbased distribution estimates. The new contrast function results in effective parametric and nonparametric ICAbased computational algorithms. Moreover, two pairwise iterative schemes are proposed to tackle the high dimensionality of sources. These wo pairwise nonparametric ICA algorithms are based on the new highperformance Convex CauchySchwarz Divergence (CCSDIV). These two schemes enable fast and efficient demixing of sources in realworld applications where the dimensionality of the sources is higher than two.Secondly, the more challenging problem in communication signal processing is to estimate the source signals and their channels in the presence of other cochannel signals and noise without the use of a training set. Blind techniques are promising to integrate and optimize the wireless communication designs i.e. equalizers/ filters/ combiners through its potential in suppressing the intersymbol interference (ISI), adjacent channel interference, cochannel and the multi access interference MAI. Therefore, a new blind detection algorithm, based on fourth order cumulant matrices, is presented and applied to the multiuser symbol estimation problem in Direct Sequence Code Division Multiple Access (DSCDMA) systems. The blind detection is to estimate multiple symbol sequences in the downlink of a DSCDMA communication system using only the received wireless data and without any knowledge of the user spreading codes. The proposed algorithm takes advantage of higher cumulant matrix properties to reduce the computational load and enhance performance. In addition, we address the problem of blind multiuser equalization in the wideband CDMA system, in the noisy multipath propagation environment. Herein, we propose three new blind receiver schemes, which are based on the state space structures. These socalled blind statespace receivers (BSSR) do not require knowledge of the propagation parameters or spreading code sequences of the users but relies on the statistical independence assumption among the source signals. We then develop and derive three updatelaws in order to enhance the performance of the blind detector. Also, we upgrade three semiblind adaptive detectors based on the incorporation of the RAKE receiver and the stochastic gradient algorithms which are used in several blind adaptive signal processing algorithms, namely FastICA, RobustICA, and principle component analysis PCA. Through simulation evidence, we verify the significant bit error rate (BER) and computational speed improvements achieved by these algorithms in comparison to other leading algorithms.Lastly, system realization of one of the developed algorithms has been explored among ASIC or FPGA platforms in terms of cost, effectiveness, and economics of scale. Based on our findings of current statoftheart electronics, programmable FPGA designs are deemed to be the most effective technology to be used for ICA hardware implementation at this time.
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 Title
 Algebraic topology and machine learning for biomolecular modeling
 Creator
 Cang, Zixuan
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

Data is expanding in an unprecedented speed in both quantity and size. Topological data analysis provides excellent tools for analyzing high dimensional and highly complex data. Inspired by the topological data analysis's ability of robust and multiscale characterization of data and motivated by the demand of practical predictive tools in computational biology and biomedical researches, this dissertation extends the capability of persistent homology toward quantitative and predictive data...
Show moreData is expanding in an unprecedented speed in both quantity and size. Topological data analysis provides excellent tools for analyzing high dimensional and highly complex data. Inspired by the topological data analysis's ability of robust and multiscale characterization of data and motivated by the demand of practical predictive tools in computational biology and biomedical researches, this dissertation extends the capability of persistent homology toward quantitative and predictive data analysis tools with an emphasis in biomolecular systems. Although persistent homology is almost parameter free, careful treatment is still needed toward practically useful prediction models for realistic systems. This dissertation carefully assesses the representability of persistent homology for biomolecular systems and introduces a collection of characterization tools for both macromolecules and small molecules focusing on intra and intermolecular interactions, chemical complexities, electrostatics, and geometry. The representations are then coupled with deep learning and machine learning methods for several problems in drug design and biophysical research.In realworld applications, data often come with heterogeneous dimensions and components. For example, in addition to location, atoms of biomolecules can also be labeled with chemical types, partial charges, and atomic radii. While persistent homology is powerful in analyzing geometry of data, it lacks the ability of handling the nongeometric information. Based on cohomology, we introduce a method that attaches the nongeometric information to the topological invariants in persistent homology analysis. This method is not only useful to handle biomolecules but also can be applied to general situations where the data carries both geometric and nongeometric information.In addition to describing biomolecular systems as a static frame, we are often interested in the dynamics of the systems. An efficient way is to assign an oscillator to each atom and study the coupled dynamical system induced by atomic interactions. To this end, we propose a persistent homology based method for the analysis of the resulting trajectories from the coupled dynamical system.The methods developed in this dissertation have been applied to several problems, namely, prediction of protein stability change upon mutations, proteinligand binding affinity prediction, virtual screening, and protein flexibility analysis. The tools have shown top performance in both commonly used validation benchmarks and communitywide blind prediction challenges in drug design.
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 Title
 Applications of Geometric Measure Theory to Complex and Quasiconformal Analysis
 Creator
 Bongers, Tyler Charles
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

There are many intersections between complex analysis, geometric measure theory, and harmonic analysis; the interactions between these fields yield many important results and applications. In this work, we focus on two aspects of these connections: the regularity theory of quasiconformal maps and the quantitative study of rectifiable sets. Quasiconformal maps are orientationpreserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses...
Show moreThere are many intersections between complex analysis, geometric measure theory, and harmonic analysis; the interactions between these fields yield many important results and applications. In this work, we focus on two aspects of these connections: the regularity theory of quasiconformal maps and the quantitative study of rectifiable sets. Quasiconformal maps are orientationpreserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of uniformly bounded eccentricity. Such maps have many useful geometric distortion properties, and yield a flexible and powerful generalization of conformal mappings. These maps arise naturally in the study of elasticity, in complex dynamics, and in the analysis of partial differential equations. We study the singularities of these maps; in particular, we consider the size and structure of the sets where a quasiconformal map can exhibit given stretching and rotation behavior. We improve the previously known results to give examples of stretching and rotation sets with nonsigmafinite measure at the critical Hausdorff dimension. We further improve this to give examples with positive Riesz capacity at the critical homogeneity, as well as positivity of measure for a broad class of gauged Hausdorff measures at the critical dimension.The local distortion properties of quasiconformal maps also give rise to a certain degree of global regularity and H\"older continuity. We give new lower bounds for the H\"older continuity of these maps, relating both the structure of the underlying partial differential equation for the maps and the geometric distortion they can exhibit; the analysis is based on combining the isoperimetric inequality with a study of the length of quasicircles. Furthermore, the extremizers for H\"older continuity are characterized, and we give a natural application to solutions of elliptic partial differential equations.Finally, given a set in the plane, the average length of its projections in all directions is called the Favard length of a set; it is closely related to the Buffon needle probability of the set. This quantity measures the size and structure of a set, and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. We develop new geometrically motivated techniques for estimating Favard length. We will give a new proof relating Hausdorff dimension to the decay rate of the Favard length of neighborhoods of a set. We will also show that, for a large class of selfsimilar onedimensional sets, the sequence of Favard lengths of the generations of the set is convex; this leads directly to lower bounds on Favard length for various fractal sets.
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 Title
 Aristotle as secondary mathematics teacher educator : metaphors and strengths
 Creator
 Johnson, Whitney Pamela
 Date
 2005
 Collection
 Electronic Theses & Dissertations
 Title
 Bijective Proofs For The Shuffle Compatibility Of Descent Statistics
 Creator
 BakerJarvis, Duff
 Date
 2019
 Collection
 Electronic Theses & Dissertations
 Description

Define a permutation to be any sequence of distinct positive integers. Given two permutations pi and sigma on disjoint underlying sets, we denote by pi \shuffle sigma the set of shuffles of pi and sigma, that is, the set of all permutations obtained by interleaving the two permutations. A permutation statistic is a function, St, whose domain is the set of permutations and has the property that St(pi) only depends on the relative order of the elements of pi. A permutation statistic is shuffle...
Show moreDefine a permutation to be any sequence of distinct positive integers. Given two permutations pi and sigma on disjoint underlying sets, we denote by pi \shuffle sigma the set of shuffles of pi and sigma, that is, the set of all permutations obtained by interleaving the two permutations. A permutation statistic is a function, St, whose domain is the set of permutations and has the property that St(pi) only depends on the relative order of the elements of pi. A permutation statistic is shuffle compatible if the distribution of St on pi \shuffle sigma depends only on the lengths of pi and sigma and St(pi) and St(sigma) rather than on the individual permutations themselves. This notion is implicit in the work of Stanley when he developed his theory of Ppartitions, where P is a partially ordered set.The definition was explicitly given by Gessel and Zhuang who proved that various permutation statistics were shuffle compatible using mainly algebraic means. This work was continued by Grinberg. The purpose of the present work is to use bijective techniques to give demonstrations of shuffle compatibility. In particular, we show how a large number of permutation statistics can be shown to be shuffle compatible using a few simple bijections. Our approach also leads to a method for constructing such bijective proofs rather than having to treat each one in an ad hoc manner. Finally, we are able to prove a conjecture of Gessel and Zhuang about the shuffle compatibility of a certain statistic.
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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
 CAYLEY GRASSMANNIAN AND DEFORMATIONS IN COMPLEX G2 MANIFOLDS
 Creator
 Yildirim, Ustun
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

Geometric objects related to the exceptional lie groups G_2 and Spin(7) have become increasingly popular in the recent years. Especially so after Bryant's (and others') work which showed the existence of riemannian manifolds with holonomy group equal to one of these groups [Bry87]. However, not much attention is given to the complex manifestations of these objects. This thesis consists of two parts which fills some of these gaps.In the first part of this thesis, we investigate the Cayley...
Show moreGeometric objects related to the exceptional lie groups G_2 and Spin(7) have become increasingly popular in the recent years. Especially so after Bryant's (and others') work which showed the existence of riemannian manifolds with holonomy group equal to one of these groups [Bry87]. However, not much attention is given to the complex manifestations of these objects. This thesis consists of two parts which fills some of these gaps.In the first part of this thesis, we investigate the Cayley Grassmannian (over C) which is the set of fourplanes that are closed under a threefold cross product in C^8. We define a torus action on the Cayley Grassmannian. Using this action, we prove that the minimal compactification is a singular variety. We also show that the singular locus is smooth and has the same cohomology ring as that of CP^5. Furthermore, we identify the singular locus with a quotient of G_2^C by a parabolic subgroup.In the second part of this thesis, we introduce the notion of (almost) G_2^Cmanifolds with compatible symplectic structures. Further, we describe "complexification" procedures for a G_2 manifold M inside M_C. As an application we show that isotropic deformations of an associative submanifold Y of a G_2 manifold inside of its complexification M_C is given by SeibergWitten type equations.
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 Title
 Certain summation & cubature formulas
 Creator
 Northam, Jack Irwin
 Date
 1939
 Collection
 Electronic Theses & Dissertations
 Title
 Combinatorial and Fourier Analytic L² Methods For Buffon's Needle Problem
 Creator
 Bond, Matthew Robert
 Date
 2011
 Collection
 Electronic Theses & Dissertations
 Description

For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this thesis.}\end{figure}In recent years, progress has been made on Buffon's needle problem, in which one considers a subset of the plane and asks how likely "Buffon's needle"  a long, straight needle with independent, uniform distributions on its position and orientation  is to intersect said set. The case in which the set is a small neighborhood of a one...
Show moreFor interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this thesis.}\end{figure}In recent years, progress has been made on Buffon's needle problem, in which one considers a subset of the plane and asks how likely "Buffon's needle"  a long, straight needle with independent, uniform distributions on its position and orientation  is to intersect said set. The case in which the set is a small neighborhood of a onedimensional unrectifiable Cantorlike set has been considered in recent years, and progress has been made, motivated in part by connections to analytic capacity.Call the set E, the radius of the neighborhood ε, and the neighborhood E_{ε}. Then in some special cases, it has been confirmed that Buffon's needle intersects E_{ε} with probability at most Clogεp , for p>0 small enough, C>0 large enough. In the special case of the socalled "four corner" Cantor set and Sierpinski's gasket, the lower bound C*loglogε/logε is known, replacing the previouslyknown lower bound C/logε which is good for more general onedimensional selfsimilar sets.In addition, the stronger lower bounds are still good if one "bends the needle" into the shape of a long circular arc, or "Buffon's noodle." The radius one uses can be as small as logεε_{0} $, for any ε_{0}, with the constant C depending on ε_{0}. It is unknown whether this condition or anything like it is necessary.Work continues on generalizing the upper bound results.
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 Title
 Common and textbook foil groupings : a social network approach to distractor analysis
 Creator
 Pearlman, Leslie
 Date
 2011
 Collection
 Electronic Theses & Dissertations
 Description

"This dissertation examines the patterns and types of mistakes students make on a largescale mathematics assessment, and puts these patterns into perspective based on the textbook used and the specific content covered in a student's classroom."From abstract.
 Title
 Computations of Floer homology and gauge theoretic invariants for Montesinos twins
 Creator
 Knapp, Adam C.
 Date
 2008
 Collection
 Electronic Theses & Dissertations
 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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 Title
 Connection Blocking in Lattice Quotients of Connected Lie Groups
 Creator
 Bidar, Mohammadreza
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

Finite blocking is an interesting concept originating as a problem in billiard dynamics and later in the context of Riemannian manifolds. Let $(M,g)$ be a complete connected, infinitely differentiable Riemannian manifold. To \textit{block} a pair of points $m_1,m_2 \in M$ is to find a \textit{finite} set $B \subset M\setminus \{m_1, m_2 \}$ such that every geodesic segment joining $m_1$ and $m_2$ intersects $B$. $B$ is called a \textit{blocking set} for the pair $m_1,m_2 \in M$. The manifold ...
Show moreFinite blocking is an interesting concept originating as a problem in billiard dynamics and later in the context of Riemannian manifolds. Let $(M,g)$ be a complete connected, infinitely differentiable Riemannian manifold. To \textit{block} a pair of points $m_1,m_2 \in M$ is to find a \textit{finite} set $B \subset M\setminus \{m_1, m_2 \}$ such that every geodesic segment joining $m_1$ and $m_2$ intersects $B$. $B$ is called a \textit{blocking set} for the pair $m_1,m_2 \in M$. The manifold $M$ is \textit{secure} if every pair of points in $M$ can be blocked. $M$ is \textit{uniformly secure} if the cardinality of blocking sets for all pairs of points in $M$ has a (finite) upper bound. The main blocking conjecture states that a closed Riemannian manifold is secure if and only if it is flat.Gutkin \cite{Connection blocking} initiated a similar study of blocking properties of quotients $G/\Gamma$ of a connected Lie group $G$ by a lattice $\Gamma \subset G$. Here the connection curves are the orbits of one parameter subgroups of $G$. To \textit{block} a pair of points $m_1,m_2 \in M$ is to find a finite set $B \subset M\setminus \{m_1, m_2 \}$ such that every connection curve joining $m_1$ and $m_2$ intersects $B$. The lattice quotient $M=G/\Gamma$ is \textit{connection blockable} if every pair of points in $M$ can be blocked, otherwise we call it \textit{nonblockable}. The corresponding main blocking conjecture states that $M=G/\Gamma$ is blockable if and only if its universal cover $\tilde{G}=\mathds{R}^n$, i.e. $M$ is a torus. In this dissertation we investigate blocking properties for two classes of lattice quotients, which are lattice quotients of semisimple and solvable Lie groups.According to the Levi decomposition, every connected Lie group $G$ is a semidirect product of a solvable Lie group $R$, and a semisimple Lie group $S$. A Lie group $G=R \rtimes S$ satisfies \textit{Raghunathan's condition} if the kernel of the action of $S$ on $R$ has no compact factors in its identity component. For a such Lie group $G$, if quotients of $R$ are nonblockable then quotients of $G$ are also nonblockable. The special linear group $\textrm{SL}(n,\mathds{R})$ is a simple Lie group for $n>1$.Let $M_n= \textrm{SL}(n,\mathds{R})/\Gamma$, where $\Gamma=\textrm{SL}(n,\mathds{Z})$ is the integer lattice. We focus on $M_2$ and show that the set of blockable pairs is a dense subset of $M_2 \times M_2$, and we use this to conclude manifolds $M_n$ are nonblockable. Next, we review a quaternionic structure of $\textrm{SL}(2,\mathds{R})$ and a way for making cocompact lattices in this context. We show that the obtained lattice quotients are not finitely blockable. In the context of solvable Lie groups, we study lattice quotients of \textit{Sol}. \textit{Sol} is a unimodular solvable Lie group, with the left invariant metric $ds^2=e^{2z}dx^2+e^{2z}dy^2+dz^2$, and is one of the eight homogeneous Thurston 3geometries. We prove that all quotients of $Sol$ are nonblockable. In particular, we show that for any lattice $\Gamma \subset Sol$, the set of nonblockable pairs is a dense subset of $Sol/\Gamma \times Sol/\Gamma$.
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 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
 Controllability of Hyperbolic and Degenerate Parabolic Equations in One Dimension
 Creator
 Bohn, Jonathan Matthew
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

In this thesis, we study the controllability problem for two systems of partial differential equations. We will first consider the wave equation with variable coefficients and potential in one dimension, $u_{tt}  (a(x)u_x)_x + pu = 0$, with control function $v(t)$ acting on the boundary. We consider a class of functions corresponding to a special weight function that contains the variable coefficient $a(x)$. From here, we derive a global Carleman estimate for this system, and establish the...
Show moreIn this thesis, we study the controllability problem for two systems of partial differential equations. We will first consider the wave equation with variable coefficients and potential in one dimension, $u_{tt}  (a(x)u_x)_x + pu = 0$, with control function $v(t)$ acting on the boundary. We consider a class of functions corresponding to a special weight function that contains the variable coefficient $a(x)$. From here, we derive a global Carleman estimate for this system, and establish the controllability property. We then later extend the class of admissible functions $a(x)$ for which the controllability property holds true. We then study the controllability problem for the degenerate heat equation in one dimension. For $0\leq \alpha <1$, on $(0,1) \times (0,T)$, we consider $w_t  (x^{\alpha}w_x)_x = f$. This equation is degenerate because the diffusion coefficient $x^{\alpha}$ is positive in the interior of the domain and vanishes at the boundary. We consider this problem under the Robin boundary conditions. Again, we derive a Carleman estimate for this system, taking into account the new boundary terms that arise from the Robin conditions.
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