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Pages
 Title
 Evaluating content validity in crossnational achievement tests
 Creator
 Jakwerth, Pamela M. (Pamela Marie)
 Date
 1996
 Collection
 Electronic Theses & Dissertations
 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
 Some properties of backward forward parabolic equations from population dynamics
 Creator
 Bao, Lianzhang
 Date
 2013
 Collection
 Electronic Theses & Dissertations
 Description

In this dissertation, we developed a new population dynamics model from probability theory and logistic model:\begin{eqnarray} u_t&=&(D(u)u_x))_x,\label{eq:0001} \\ D(u)&<&0, \quad \mbox{for} \quad u\in (0,\alpha);\quad D(u)>0 \quad \mbox{for} \quad u\in (\alpha,1).\label{eq:0002}\end{eqnarray} These equations describe population aggregation when the population density is small and diffusion when the population is large. When the net birth is also considered, the equation becomes:\begin...
Show moreIn this dissertation, we developed a new population dynamics model from probability theory and logistic model:\begin{eqnarray} u_t&=&(D(u)u_x))_x,\label{eq:0001} \\ D(u)&<&0, \quad \mbox{for} \quad u\in (0,\alpha);\quad D(u)>0 \quad \mbox{for} \quad u\in (\alpha,1).\label{eq:0002}\end{eqnarray} These equations describe population aggregation when the population density is small and diffusion when the population is large. When the net birth is also considered, the equation becomes:\begin{equation}\label{eq:0003} u_t=(D(u)u_x))_x+g(u).\end{equation}We assume that the net birth $g(u)$ satisfies\begin{equation}\label{eq:0004}g(0)=g(1)=0,\quad g(u)>0 \quad \forall u\in(0,1).\end{equation}Because of the singularity at point $u=\alpha$, we can not obtain the traveling wave solution easily by methods used for purely diffusion model (which means $D(u)\geq 0$ for all $u\in [0,1]$). In order to overcome this difficulty, we introduce a suitable definition of weak traveling wave solution and obtain the existence of the traveling wave solutions depending on the traveling wave speed $c$.\par Secondly, we consider some properties of the weak solution of Equation \eqref{eq:0001} with nonflux boundary conditions such as existence and nonexistence of the weak solution when $x\in \Omega\subseteq \mathbb{R}^1$ and the asymptotic behavior of the solution:\begin{equation}\label{eq:0005} \lim_{t\rightarrow+\infty}u(x,t)=\frac{1}{\Omega}\int_{\Omega}u(x,0)dx, \quad u(x,0)\geq \alpha.\end{equation}\par In the third part, we consider the original discrete model which has solution naturally under general initial condition. We first prove the discrete population density $0\leq u(j,t)\leq 1$ which satisfies our assumptions before deriving our new population model. Secondly, we obtain the asymptotic behavior of this discrete model when $u(j,0)\geq \alpha=1/2, j=0,1,2,\dots,N$ and $u(j,0)$ is monotone:\begin{equation}\label{eq:0006} \lim_{t\rightarrow +\infty}u(j,t)=\frac{1}{N1}\sum_{j=1}^{N1}u(j,0).\end{equation}When $N\leq4$ and $u(0,t)=u(N,t)=0$, we obtain the convergence results of the solutions as $t\rightarrow +\infty$ under very general initial conditions.One more very interesting question is: How does the forward region $Q_d^+(t)=\{(i,t) u(i,t)\geq \alpha,i\in [1,2,3,\dots,N]\}$ and backward region $Q_d^(t)=\{(i,t) u(i,t)\leq \alpha,i\in [1,2,3,\dots,N]\}$ change as time progresses? We can prove under very general initial conditions:\begin{equation}\label{eq:0007} Q_d^+(t)\subseteq Q_d^+(t_1), \quad\forall t\leq t_1.\end{equation}\par In the last part, we consider the traveling wave solution for a celltocell model with adhesion which is different to our first model. \begin{equation}\label{eq:0008} \rho_t= [ D(\rho)\rho_x]_x + g(\rho)\quad t\geq 0, \quad x\in \mathbb{R},\end{equation}with quadratic coefficient\begin{equation}\label{eq:0009} D(\rho) = 3\gamma \rho^24\gamma\rho+1,\end{equation}where $\rho(x,t)$ represent the cell density and $\gamma$ is the adhesive coefficient between cells. We obtain the following existence theorem:\\ $\mathbf{Theorem}$ Let $D(\rho)$ and $g(\rho)$ be given functions respectively satisfying \eqref{eq:0009} and \eqref{eq:0004} and $3/4\leq \gamma\leq 1$.There exists a value $c^*>0$, satisfying\begin{eqnarray*} &&\max\{D'(0)g(0), D'(\rho^{\sharp}(\gamma))g(\rho^{\sharp}(\gamma))\}\leq \frac{(c^*)^2}{4} \\ &&\leq \max\{\sup_{s\in(0, \rho^{\flat}(\gamma)]}\frac{D(s)g(s)}{s},\sup_{s\in(\rho^{\flat}(\gamma),1],\rho\neq\rho^{\sharp}(\gamma)}\frac{D(s)g(s)}{s\rho^{\sharp}(\gamma)}\},\end{eqnarray*}such that Equation \eqref{eq:0008} has\par i) no weak traveling wave solution for $cc^*$.\par Where $(\rho^{\flat}(\gamma),\rho^{\sharp}(\gamma)) := (\frac{2\gamma\sqrt{\gamma(4\gamma3)}}{3\gamma},\frac{2\gamma+\sqrt{\gamma(4\gamma3)}}{3\gamma})$.
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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
 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
 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
 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
 The relationship between item format and cognitive processes in widescale assessment of mathematics
 Creator
 Garavaglia, Diane R.
 Date
 2001
 Collection
 Electronic Theses & Dissertations
 Title
 The impact of test consequences and response format on performance
 Creator
 DeMars, Christine
 Date
 1998
 Collection
 Electronic Theses & Dissertations
 Title
 Computations of Floer homology and gauge theoretic invariants for Montesinos twins
 Creator
 Knapp, Adam C.
 Date
 2008
 Collection
 Electronic Theses & Dissertations
 Title
 Strategies in repeated games
 Creator
 Li, Mingfei
 Date
 2008
 Collection
 Electronic Theses & Dissertations
 Title
 Elementary teachers' mathematics textbook use in terms of cognitive demands and influential factors : a mixed method study
 Creator
 Son, JiWon
 Date
 2008
 Collection
 Electronic Theses & Dissertations
 Title
 Aristotle as secondary mathematics teacher educator : metaphors and strengths
 Creator
 Johnson, Whitney Pamela
 Date
 2005
 Collection
 Electronic Theses & Dissertations
 Title
 The origins of connectedness im kleinen
 Creator
 McGrew, John Michael, 1947
 Date
 1976
 Collection
 Electronic Theses & Dissertations
 Title
 A comparative evaluation of programmed and lecture instruction in college business mathematics
 Creator
 Swartz, Manfred E.
 Date
 1985
 Collection
 Electronic Theses & Dissertations
 Title
 Geometric evolution of singlelayer interfaces in the functionalized CahnHilliard equation
 Creator
 Hayrapetyan, Gurgen Ruben
 Date
 2011
 Collection
 Electronic Theses & Dissertations
 Description

We study the Functionalized CahnHilliard Energy (FCH), which is a higherorder reformulation of the CahnHilliard energy, as a model for network formation in polymersolvent mixtures. The model affords a finite interfacial width, accommodates merging and other topological reorganization, and couples naturally to momentum balance and other macroscopic mass transport equations.The corresponding constrained L
2 gradient flow has a rich family of approximately steadystate...
Show moreWe study the Functionalized CahnHilliard Energy (FCH), which is a higherorder reformulation of the CahnHilliard energy, as a model for network formation in polymersolvent mixtures. The model affords a finite interfacial width, accommodates merging and other topological reorganization, and couples naturally to momentum balance and other macroscopic mass transport equations.The corresponding constrained L2 gradient flow has a rich family of approximately steadystate solutions that include not only the singlelayer heteroclinic front profile seen in gradient flows of the CahnHilliard energy, but also a novel one parameter family of homoclinic bilayer solutions. In this thesis we rigorously derive the geometric evolution of the singlelayer polymersolvent interface.We form a manifold of quasiequilbria by "dressing" a large family of codimension one interfaces immersed in Rd with heteroclinic solutions of a onedimensional equilibrium equation derived from the first variation of the FCH energy. We show that solutions of the gradient flow that start sufficiently close to the manifold remain close, and moreover the flow can be decomposed, at leading order, as a normal velocity for the underlying codimension one interface. Assuming the smoothness of the interface under this flow, we develop rigorous estimates on the proximity of the true solution to the manifold, in an appropriate norm, for long time.
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 Title
 Volumes, Determinants, and Meridian Lengths of Hyperbolic Links
 Creator
 Burton, Stephan D.
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

We study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of nonalternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi...
Show moreWe study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of nonalternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi log det(K)/c(K). We prove that the closure of the set of determinant densities contains the set [0, v_8]. We examine the conjecture, due to Champanerkar, Kofman, and Purcell that vol(K) < 2 pi log det (K) for alternating hyperbolic links, where vol(K) = vol(S^3\ K) is the hyperbolic volume and det(K) is the determinant of K. We prove that the conjecture holds for 2bridge links, alternating 3braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.We derive bounds on the length of the meridian and the cusp volumeof hyperbolic knots in terms of the topology of essential surfaces spanned by the knot.We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound.As applications we find knot diagrammatic upper bounds on the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots withmeridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.
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 Title
 Learning to anticipate students' mathematical responses in two contexts : the case of one preservice teacher in a university and school setting
 Creator
 Kasten, Sarah Elizabeth
 Date
 2009
 Collection
 Electronic Theses & Dissertations
 Title
 Development of discourse on limits : connecting history and classroom practice through a communicational approach to learning
 Creator
 Güçler, Beste
 Date
 2010
 Collection
 Electronic Theses & Dissertations
 Title
 On some aspects of cluster algebras and combinatorial Hopf algebras
 Creator
 Machacek, John
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

This dissertation deals with problems in cluster algebras and combinatorial Hopf algebras.Total positivity has been closely related to cluster algebras since their inception.Postnikov's totally nonnegative Grassmannian is a concrete example of total positivity with rich combinatorics.Our first problem is the computation of Plücker coordinates inside a generalization of the totally nonnegative Grassmannian.We provide a combinatorial formula in terms of edge weighted directed graphs embedded on...
Show moreThis dissertation deals with problems in cluster algebras and combinatorial Hopf algebras.Total positivity has been closely related to cluster algebras since their inception.Postnikov's totally nonnegative Grassmannian is a concrete example of total positivity with rich combinatorics.Our first problem is the computation of Plücker coordinates inside a generalization of the totally nonnegative Grassmannian.We provide a combinatorial formula in terms of edge weighted directed graphs embedded on a surface.The next problem we consider is the equality of a cluster algebra and its upper cluster algebra.Particular attention is paid to the coefficient ring of the cluster algebra.We give a sufficient condition for the cluster algebra and upper cluster algebra to coincide while allowing greater generality of coefficient ring than was previous known.The final problem we consider in cluster algebras is showing that logcanonical coordinates are as simple as possible (in a certain precise sense).Logcanonical coordinates are a fundamental part of the Poisson geometry approach to cluster algebras put forth by Gekhtman, Shapiro, and Vainshtein.In the theory of combinatorial Hopf algebras we compute a formula for the antipode in a Hopf algebra on simplicial complexes.This antipode formula generalizes Humpert and Martin's formula for graphs.We then use the character theory of Aguiar, Bergeron, and Sottile to realize a version of Stanley's chromatic symmetric function for simplicial complexes.We prove that the degree sequence of a uniform hypertree can be recovered from its chromatic symmetric function.We also show the chromatic symmetric function is not a complete invariant for uniform hypertrees.
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