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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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Title: Modulational stability of multi-pulses within the functionalized Cahn-Hilliard gradient flow
Creator: Guckir Cakir, Hayriye
Date: 2019
Collection: Electronic Theses & Dissertations
Description: The Functionalized Cahn-Hilliard (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 Euler-Lagrange equation for the mass constrained Functionalized Cahn-Hilliard(FCH) free energy with zero functionalization terms is derived and a large family of multi-pulse critical points is constructed. We show that the FCH energy with no functiona-lization terms subject to a mass constraint has global...
Show moreThe Functionalized Cahn-Hilliard (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 Euler-Lagrange equation for the mass constrained Functionalized Cahn-Hilliard(FCH) free energy with zero functionalization terms is derived and a large family of multi-pulse critical points is constructed. We show that the FCH energy with no functiona-lization terms subject to a mass constraint has global minimizers over a variety of admissible sets. We introduce a multi-pulse ansatz as the extensions of the periodic multi-pulse critical points to R and establish the H^2-coercivity of the second variation of the energy about multi-pulse ansatz. Modulational stability and the dynamic evolution of the multi-pulse ansatz with respect to the Pi_0-gradient flow are also addressed.
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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 multi-scale weighted colored graph (MWCG) model. The MWCG model is a new generation of computational algorithms that signicantly improves the current landscape of protein struc-tural 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 X-ray 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: M-level rook placements
Creator: Barrese, Kenneth
Date: 2015
Collection: Electronic Theses & Dissertations
Description: Rook theory focuses on placements of non-attacking rooks on boards of various shapes. An important role is played by the rook numbers which count the number of non-attacking placements of a given number of rooks on a board. Ferrers boards,which are boards indexed by integer partitions, are of particular interest. Briggs and Remmel introduced a generalization of rook placements, called m-level rook placements, where a rook is able to attack a subset of the rows.This manuscript presents...
Show moreRook theory focuses on placements of non-attacking rooks on boards of various shapes. An important role is played by the rook numbers which count the number of non-attacking placements of a given number of rooks on a board. Ferrers boards,which are boards indexed by integer partitions, are of particular interest. Briggs and Remmel introduced a generalization of rook placements, called m-level rook placements, where a rook is able to attack a subset of the rows.This manuscript presents generalizations of many of the central results regarding rook placements to the case of m-level rook placements. Goldman, Joichi, and White defined the rook polynomial of a board to be the generating function for the rook numbers of that board in the falling factorial basis. By doing so, they were able to give an elegant factorization of the rook polynomial of a Ferrers board in terms of the various column heights. Briggs and Remmel were able to generalize this factorization to the m-level rook polynomial of a subset of Ferrers boards called singleton boards.We give two factorization theorems for the m-level rook polynomial of a Ferrers board. The first is a generalization of the factorization theorem of Briggs and Remmel, working from similar principles. The second relies on a generalization of transposition which we present, called the l-operator. We are also able to use the factorization to describe a unique representative in any m-level equivalence class of Ferrers boards and count the number of singleton boards in the class..When generalizing the factorization from singleton boards to all Ferrers boards, we preserve the definition of the m-level rook polynomial and alter the factorization to apply to all Ferrers boards. We also consider the dual of this problem, applying the factorization of Briggs and Remmel to all Ferrers boards, then trying to determine what is counted by the coefficients of the polynomial in the m-falling factorial basis. It turns out that the coefficients count weighted file placements on a Ferrers board. We also describe a unique representative in each weighted file placement equivalence class of Ferrers boards, as well as count of the number of Ferrers boards in a given weighted file placement equivalence class.Foata and Schü}tzenberger presented explicit bijections between rook placements on any two rook equivalent Ferrers boards as part of their construction of a unique representative in each equivalence class of Ferrers boards. A key tool in their construction was local transposition. We present analogous bijections between m-level rook placements on any two $m$-level rook equivalent Ferrers boards using the local l-operator.The Garsia-Milne Involution Principle was first used in Garsia and Milne's bijective proof of the Rogers-Ramanujan identities. We use it to construct two types of explicit bijections. The first is an explicit bijection between m-level rook placements on any two m-level rook equivalent singleton boards. The second bijection is between the sets counted by the m-level analogue of hit numbers of any two m-level rook equivalent Ferrers boards, providing a bijective proof that $m$-level equivalent Ferrers boards have the same hit numbers.
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Title: Jones-type link invariants and applications to 3-manifold topology
Creator: Lee, Christine Ruey Shan
Date: 2015
Collection: Electronic Theses & Dissertations
Description: "It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semi-adequate link has a well-defined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semi-adequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new...
Show more"It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semi-adequate link has a well-defined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semi-adequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new family of pretzel knots which are not adequate. We also study the relationship between the Jones polynomial and the topology of the knot complement by relating its coefficients to the non-orientable genus of an alternating knot. The two-sided bound we obtain can often determine the non-orientable genus. Lastly, we develop the connection between the Jones polynomial and a knot invariant coming from Heegaard Floer homology by using it to classify 3-braids which are L-space knots." -- Abstract.
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Title: Estimates on singular values of functions of perturbed operators
Creator: Liu, Qinbo
Date: 2016
Collection: Electronic Theses & Dissertations
Description: In this thesis we study the behavior of functions of operators under perturbations. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \myeq \{f: \omega_{f}(\delta)\leq \text{const} \; \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$, then%the following inequality holds for all $ s_j(f(A)-f(B))\leq c\cdot \omega_{\ast}\big((1+j)^{-\frac{1}{p}}\Vert A-B \Vert_{S_{p}^l}\big) \cdot \Vert f \Vert_{\Lambda_{\omega}}$ for arbitrary self-adjoint operators $A$, $B$...
Show moreIn this thesis we study the behavior of functions of operators under perturbations. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \myeq \{f: \omega_{f}(\delta)\leq \text{const} \; \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$, then%the following inequality holds for all $ s_j(f(A)-f(B))\leq c\cdot \omega_{\ast}\big((1+j)^{-\frac{1}{p}}\Vert A-B \Vert_{S_{p}^l}\big) \cdot \Vert f \Vert_{\Lambda_{\omega}}$ for arbitrary self-adjoint operators $A$, $B$ and all $1\leq j\leq l$, where $\omega_{\ast}(x) \myeq x \int_{x}^{\infty}\frac{\omega(t)}{t^2}dt \;( x>0) $. The result is then generalized to contractions, maximal dissipative operators, normal operators and $n$-tuples of commuting self-adjoint operators.
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Title: The berry connection and other aspects of the Ginzburg-Landau theory in dimension 2
Creator: Nagy, Akos
Date: 2016
Collection: Electronic Theses & Dissertations
Description: In the first chapter, we analyze the 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large area limit.In the second chapter, we generalize Bradlow's theorem about existence of irreducible absolute minimizers of the Ginzburg-Landau functional.

Title: On minimization of some non-smooth convex functionals arising in micromagnetics
Creator: Gao, Hongli
Date: 2015
Collection: Electronic Theses & Dissertations
Description: This thesis is motivated by studying the properties of ferromagnetic materials usingthe Landau-Lifshitz theory of micromagnetics. In this theory the state of a ferromagneticmaterial is described by the magnetization vector m in terms of a total micromagnetic energythat consists of several competing sub-energies: exchange energy, anisotropy energy, externalinteraction energy and magnetostatic energy. For large ferromagnetic materials and undersome limiting regimes of the model, the exchange...
Show moreThis thesis is motivated by studying the properties of ferromagnetic materials usingthe Landau-Lifshitz theory of micromagnetics. In this theory the state of a ferromagneticmaterial is described by the magnetization vector m in terms of a total micromagnetic energythat consists of several competing sub-energies: exchange energy, anisotropy energy, externalinteraction energy and magnetostatic energy. For large ferromagnetic materials and undersome limiting regimes of the model, the exchange energy can be negligible and the totalenergy becomes a reduced model. Our investigations focus on the study of such a reducedmodel of Landau-Lifshitz theory.The primary focus of the thesis includes two parts: the minimization (static) study andthe evolution (dynamic) study. We investigate a new method for the existence of minimizersof the reduced micromagnetic energy based on a duality method. In this method, the reducedmicromagnetic energy is closely related to a convex functional (the dual functional) on thecurl-free vector functions. Our minimization and dynamics studies are based on the studyof the minimization and gradient ow of this dual functional. Much of the thesis is focusedon the minimization problem of two special cases: soft case and uniaxial case on the annulusdomain; in particular, in the soft case, for some range of the parameter, the energy minimizersof the original micromagnetic energy are constructed through the Euler-Lagrange equationof the dual functional using the characteristics method for a reduced Eikonal type equation.The second direction of our study of this thesis is an attempt to obtain certain reasonabledynamic process for the evolution of m, where the asymptotic behavior of the gradient owof the reduced energy functional is investigated.
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Title: Sharp estimates in harmonic analysis
Creator: Rey, Guillermo
Date: 2015
Collection: Electronic Theses & Dissertations
Description: We investigate certain sharp estimates related to singular integrals. In particular we give sharp level set estimates for sparse operators, we show how to reduce the problem of estimating Calder\'on-Zygmund operators by sparse operators, and we study some weighted inequalities for these operators.

Title: Statistical properties of some almost Anosov systems
Creator: Zhang, Xu
Date: 2016
Collection: Electronic Theses & Dissertations
Description: We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai-Ruelle-Bowen measures (SRB measures), where the almost Anosov diffeomorphism is a system which is hyperbolic everywhere except for one point. At the indifferent fixed point, the Jacobian matrix is an identity matrix. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the...
Show moreWe investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai-Ruelle-Bowen measures (SRB measures), where the almost Anosov diffeomorphism is a system which is hyperbolic everywhere except for one point. At the indifferent fixed point, the Jacobian matrix is an identity matrix. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and can be expressed by using coefficients of the third order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed points.We discuss the relationship between the existence of SRB measures and the differentia- bility of some almost Anosov diffeomorphisms near the indifferent fixed points in dimensions bigger than one. The eigenvalue of Jacobian matrix at the indifferent fixed point along the one-dimensional contraction subspace is less than one, while the other eigenvalues along the expansion subspaces are equal to one. As a consequence, there are twice-differentiable al- most Anosov diffeomorphisms that admit infinite SRB measures in two or three-dimensional spaces; there exist twice-differentiable almost Anosov diffeomorphisms with SRB measures in dimensions bigger than three. Further, we obtain the polynomial lower and upper bounds for the correlation functions of these almost Anosov maps that admit SRB measures.
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Title: Three-manifolds of higher rank
Creator: Lin, Samuel Zhong-En
Date: 2017
Collection: Electronic Theses & Dissertations
Description: Fixing K = −1, 0, or 1, a complete Riemannian manifold is said to have higher hyperbolic, Euclidean, or spherical rank if every geodesic admits a normal parallel field making curvature K with the geodesic. In this thesis, we establish rigidity results for three-manifolds of higher rank without a priori sectional curvature bounds. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are finite volume hyperbolic space forms. Complete three-manifolds have higher...
Show moreFixing K = −1, 0, or 1, a complete Riemannian manifold is said to have higher hyperbolic, Euclidean, or spherical rank if every geodesic admits a normal parallel field making curvature K with the geodesic. In this thesis, we establish rigidity results for three-manifolds of higher rank without a priori sectional curvature bounds. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are finite volume hyperbolic space forms. Complete three-manifolds have higher spherical rank if and only if they are spherical space forms.In addition to the rigidity results, we also provide constructions of non-homogeneous manifolds of higher hyperbolic rank of infinite volume. These examples show the necessity of the finite volume assumption in hyperbolic rank rigidity results.
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Title: High order finite difference WENO schemes for ideal magnetohydrodynamics
Creator: Feng, Xiao (Graduate of Michigan State University)
Date: 2017
Collection: Electronic Theses & Dissertations
Description: "In this dissertation we propose two high order finite difference numerical schemes for solving the ideal magnetohydrodynamic (MHD) equations." -- Abstract.

Title: Mathematical modeling and simulation of mechanoelectrical transducers and nanofluidic channels
Creator: Park, Jin Kyoung
Date: 2014
Collection: Electronic Theses & Dissertations
Description: Remarkable advances in nanotechnology and computational approaches enable researchers to investigate physical and biological phenomena in an atomic or molecular scale. Smaller-scale approaches are important to study the transport of ions and/or molecules through ion channels in living organisms as well as exquisitely fabricated nanofluidic channels. Both subjects have similar physical properties and hence they have common mathematical interests and challenges in modeling and simulating the...
Show moreRemarkable advances in nanotechnology and computational approaches enable researchers to investigate physical and biological phenomena in an atomic or molecular scale. Smaller-scale approaches are important to study the transport of ions and/or molecules through ion channels in living organisms as well as exquisitely fabricated nanofluidic channels. Both subjects have similar physical properties and hence they have common mathematical interests and challenges in modeling and simulating the transport phenomena. In this work, we first propose and validate a molecular level prototype for mechanoelectrical transducer (MET) channel in mammalian hair cells.Next, we design three ionic diffusive nanofluidic channels with different types of atomic surface charge distribution, and explore the current properties of each channel. We construct the molecular level prototype which consists of a charged blocker, a realistic ion channel and its surrounding membrane. The Gramicidin A channel is employed to demonstrate the realistic channel structure, and the blocker is a positively charged atom of radius $1.5$\AA\, which is placed at the mouth region of the channel. Relocating this blocker along one direction just outside the channel mouth imitates the opening and closing behavior of the MET channel. In our atomic scale design for an ionic diffusive nanofluidic channel, the atomic surface charge distribution is easy to modify by varying quantities and signs of atomic charges which are equally placed slightly above the channel surface. Our proposed nanofluidic systems constitutes a geometrically well-defined cylindrical channel and two reservoirs of KCl solution. For both the mammalian MET channel and the ion diffusive nanofluidic channel, we employ a well-established ion channel continuum theory, Poisson-Nernst-Planck theory, for three dimensional numerical simulations. In particular, for the nano-scaled channel descriptions, the generalized PNP equations are derived by using a variational formulation and by incorporating non-electrostatic interactions. We utilize several useful mathematical algorithms, such as Dirichlet to Neumann mapping and the matched interface and boundary method, in order to validate the proposed models with charge singularities and complex geometry. Moreover, the second-order accuracy of the proposed numerical methods are confirmed with our nanofluidic system affected by a single atomic charge and eight atomic charges, and further study the channels with a unipolar charge distribution of negative ions and a bipolar charge distribution. Finally, we analyze electrostatic potential and ion conductance through each channel model under the influence of diverse physical conditions, including external applied voltage, bulk ion concentration and atomic charge. Our MET channel prototype shows an outstanding agreement with experimental observation of rat cochlear outer hair cells in terms of open probability. This result also suggests that the tip link, a connector between adjacent stereocilia, gates the MET channel. Similarly, numerical findings, such as ion selectivity, ion depletion and accumulation, and potential wells, of our proposed ion diffusive realistic nanochannels are in remarkable accordance with those from experimental measurements and numerical simulations in the literature. In addition, simulation results support the controllability of the current within a nanofluidic channel.
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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 struc-tures 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 struc-tures 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: Integration of topological fingerprints and machine learning for the prediction of chemical mutagenicity
Creator: Cao, Yin (Quantitative analyst)
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 in-vivo bioassays, is increasingly encouraged, due to its efficiency, effectiveness, lower cost and less reliance on animal tests. The quality of a good molecular...
Show more"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 in-vivo 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 physio-chemical 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 proposed CTFs and its competitive advantages of characterizing and representing chemical molecules over existing fingerprints."--Page ii.
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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 be associated to a field, variety, or scheme. Let k be a field of characteristic different from 2 or 3, and let E be an elliptic curve over k. The Brauer group of E is a torsion abelian group with elements given by Morita equivalence classes of central simple algebras over the function field k(E). The Merkurjev-Suslin theorem implies that any such element can be described by a tensor product of symbol algebras....
Show moreThe Brauer group is an invariant in algebraic geometry and number theory, that can be associated to a field, variety, or scheme. Let k be a field of characteristic different from 2 or 3, and let E be an elliptic curve over k. The Brauer group of E is a torsion abelian group with elements given by Morita equivalence classes of central simple algebras over the function field k(E). The Merkurjev-Suslin theorem implies that any such element can be described by a tensor product of symbol algebras. We give a description of elements in the d-torsion of the Brauer group of E in terms of these tensor products, provided that the d-torsion of E is k-rational and k contains a primitive d-th root of unity. Furthermore, if d = q is a prime, we give an algorithm to compute the q-torsion of the Brauer group over any field k of characteristic different from 2,3, and q containing a primitive q-th root of unity.
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Title: On the stability/sensitivity of recovering velocity fields from boundary measurements
Creator: Zhang, Hai
Date: 2013
Collection: Electronic Theses & Dissertations
Description: The thesis investigates the stability/sensitivity of the inverse problem of recovering velocity fields in a bounded domain from the boundary measurements. The problem has important applications in geophysics where people are interested in finding the inner structure (the velocity field in the elastic wave models) of earth from measurements on the surface. Two types of measurements are considered. One is the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation. The other is...
Show moreThe thesis investigates the stability/sensitivity of the inverse problem of recovering velocity fields in a bounded domain from the boundary measurements. The problem has important applications in geophysics where people are interested in finding the inner structure (the velocity field in the elastic wave models) of earth from measurements on the surface. Two types of measurements are considered. One is the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation. The other is the restricted Hamiltonian flow induced by the corresponding velocity field at a sufficiently large time and with domain the cosphere bundle of the boundary, or its equivalent form the scattering relation. Relations between these two type of data are explored. Three main results on the stability/sensitivity of the associated inverse problems are obtained: (1). The sensitivity of recovering scattering relations from their associated DDtN maps. (2). The sensitivity of recovering velocity fields from their induced boundary DDtN maps. (3). The stability of recovering velocity fields from their induced Hamiltonian flows.In addition, a stability estimate for the X-ray transform in the resence of caustics is established. The X-ray transform is introduced by linearizing the operator which maps a velocity field to its corresponding Hamiltonian flow. Micro-local analysis are used to study the X-ray transform and conditions on the background velocity field are found to ensure the stability of the inverse transform. The main results suggest that the DDtN map is very insensitive to small perturbations of the velocity field, namely, small perturbations of velocity field can result changes to the DDtN map at the same level of large perturbations. This differs from existing H$\ddot{o}$lder type stability results for the inverse problem in the case when the velocity fields are simple. It gives hint that the methodology of velocity field inversion by DDtN map is inefficient in some sense. On the other hands, the main results recommend the methodology of inversion by Hamiltonian flow (or its equivalence the scattering relation), where the associated inverse problem has Lipschitz type stability.
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Title: A combinatorial approach to knot theory : volume bounds for hyperbolic semi-adequate link complements
Creator: Giambrone, Adam Joseph
Date: 2014
Collection: Electronic Theses & Dissertations
Description: An interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semi-adequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semi-adequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed...
Show moreAn interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semi-adequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semi-adequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed braids, and families of links that have both plat and closed braid aspects. By more closely studying each of these families, we can often improve the lower bounds on volume provided by the main result. Furthermore, we show that the bounds on volume can be expressed in terms of a single stable coefficient of the colored Jones polynomial. By doing this, we provide new collections of links that satisfy a Coarse Volume Conjecture. The main approach of this entire work is to use a combinatorial perspective to study the connections among knot theory, hyperbolic geometry, and graph theory.
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Title: Topics in knot theory : on generalized crossing changes and the additivity of the Turaev genus
Creator: Balm, Cheryl Lyn Jaeger
Date: 2013
Collection: Electronic Theses & Dissertations
Description: We first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev...
Show moreWe first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev genus one.
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Title: Level Structures on Finite Group Schemes and Applications
Creator: Guan, Chuangtian
Date: 2022
Collection: Electronic Theses & Dissertations
Description: The notion of level structures originates from the study of the moduli of elliptic curves. In this thesis, we consider generalizing the notion of level structures and make explicit calculations on different moduli spaces. The first moduli space we consider is the moduli of finite flat (commutative) group schemes. We give a definition of $\Gamma(p)$-level structure (also called the ``full level structure") over group schemes of the form $G\times G$, where $G$ is a group scheme or rank $p$ over...
Show moreThe notion of level structures originates from the study of the moduli of elliptic curves. In this thesis, we consider generalizing the notion of level structures and make explicit calculations on different moduli spaces. The first moduli space we consider is the moduli of finite flat (commutative) group schemes. We give a definition of $\Gamma(p)$-level structure (also called the ``full level structure") over group schemes of the form $G\times G$, where $G$ is a group scheme or rank $p$ over a $\Z_p$-scheme. The full level structure over $G\times G$ is flat over the base of rank $|\GL_2(\F_p)|$. We also observe that there is no natural notion of full level structures over the stack of all finite flat commutative group schemes. The second moduli space we consider is the moduli of principally polarized abelian surfaces in characteristic $p>0$ with symplectic level-$n$ structure ($n\ge 3$), which is known as the Siegel threefold. By decomposing the Siegel threefold using the Ekedahl--Oort stratification, we analyze the $p$-torsion group scheme of the universal abelian surface over each stratum. To do this, we establish a machinery to produce group schemes from their Dieudonn\'e modules using a version of Dieudonn\'e theory due to de Jong. By using this machinery, we give explicit local equations of the Hopf algebras over the superspecial locus, the supersingular locus and ordinary locus. Using these local equations, we calculate explicit equations of the $\Gamma_1(p)$-covers over these strata using Kottwitz--Wake primitive elements. These equations can be used to prove geometric and arithmetic properties of the $\Gamma_1(p)$-cover over the Siegel threefold. In particular, we prove that the $\Gamma_1(p)$-cover over the Siegel threefold is not normal.
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