Browsing by Subject "Mathematics"

Now showing 1 - 20 of 73
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    Adjoint recovery of superconvergent linear functionals from Galerkin approximations.
    (2010-02) Ichikawa, Ryuhei
    The thesis is concerned with superconvergent approximations of linear functionals. We extend the adjoint error correction technique of Pierce and Giles [SIAM Review, 42 (2000), pp. 247-264] for obtaining superconvergent approximations of functionals to Galerkin methods. We illustrate the technique in the framework of discontinuous Galerkin methods for problems in one dimension and two dimensions. In one dimension our focus is on ordinary differential and convection-diffusion equations. It is well known that approximations to linear functionals obtained by discontinuous Galerkin methods with polynomials of degree k can be proven to converge with order 2 k + 1 and 2 k for ordinary differential and convection-diffusion equations, respectively. In contrast, the order of convergence can be proven to be 4 k + 1 and 4 k, using our technique. Since both approaches have a computational complexity of the same order, the adjoint error correction method is clearly a competitive alternative. In two dimensions we deal with a simple second-order elliptic model problem. We show that approximate functionals converge with order 4 k with our method. Numerical results which confirm the theoretical predictions are presented.
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    Analysis and numerics of the mechanics of gels
    (2009-06) Chabaud, Brandon Michael
    In this thesis a mathematical model of polymer gel dynamics is proposed and analyzed. This work is motivated by problems in biomedical device manufacturing. The goal of this thesis is to develop and analyze models of gels consisting of balance laws in the form of systems of partial differential equations and boundary conditions. The model based on mixture theory accounts for nonlinear elasticity, viscoelasticity, transport, and diffusion. The derived model includes as limiting cases incompressible elasticity, viscous incompressible fluid, and Doi's stress diffusion equations. Two classes of problems are considered. The first class addresses nonlinear problems in special domains and the second class addresses linear problems in arbitrary domains. Special emphasis is placed on linear problems with the goal of studying and implementing finite element methods. The first class of problems includes a one dimensional free boundary problem analyzed in terms of one dimensional hyperbolic theory. The second class includes coupled elasticity and fluid flow problems. One challenging issue is accounting for the fact that, although the gel may be incompressible, the polymer may experience large changes of volume. Numerical analysis of elastic solids and polymer gels is carried out. The Taylor-Hood algorithm for Stokes flow is applied to linearly visco-hyperelastic polymers. The simulations show the presence of stress concentrations at the boundary which relax over time. In the case of a gel, the conditionally stable mixed finite element method proposed by Feng and He for Doi's stress-diffusion coupling model is modified to handle the case of polymer viscosity. The modified numerical scheme is shown to be unconditionally stable and convergent.
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    Analytic Composition Expansions About Functional Equation Fixed Points
    (2021) Vogt, Cameron;
    Given holomorphic functions satisfying the functional equation φ = σ ◦ φ ◦ τ where τ has an attracting fixed point paired with a repelling fixed point of σ, we prove φ can be expressed as a composition expansion limn→∞ σ ◦ n ◦ ψ ◦ τ ◦ n where ψ approximates φ in some sense. With certain restrictions, φ is the unique function satisfying the functional equation. Conversely, given a functional equation of the specified form, we construct a function which satisfies it. The idea behind the proof is to view the transformation f → σ ◦ f ◦ τ as a contraction mapping on a particular space of holomorphic functions. As a basic example, the functional equation cos z = 2 cos2 (z/2) − 1 generates a composition expansion for cos z.
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    Anonymity and privacy in public key cryptography.
    (2012-05) Saraswat, Vishal
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    Applications of moving frames to lie pseudo-groups.
    (2009-05) Valiquette, Francis
    Recently, Olver and Pohjanpelto have successfully extended the theory of equivariant moving frames to infinite-dimensional Lie pseudo-groups. Based on its finite-dimensional counterpart, this new theory promises to be a source of interesting new results and applications. In this thesis, we look at two applications of this new theory. By combining the powerful theories of Lie groupoids and variational bicomplexes, Olver and Pohjanpelto have developed a practical algorithm for determining the Maurer--Cartan structure equations of Lie pseudo-groups. The structure equations obtained with this new theory are compared with those derived by Cartan. It is shown that for transitive Lie pseudo-groups the two structure theories are isomorphic while for intransitive Lie pseudo-groups the two sets of structure equations do not agree. To make the two structure theory isomorphic we argue that Cartan's structure equations need to be slightly modified. The effect of this modification on Cartan's definition of essential invariants is analyzed. In 1965, Singer and Sternberg gave an infinitesimal interpretation of Cartan's structure equations for transitive Lie pseudo-groups. This interpretation is extended to intransitive Lie pseudo-groups and the result is used to state a symmetry-based linearization theorem for systems of nonlinear partial differential equations which does not require the integration of the infinitesimal determining equations of the symmetry group. The theory of equivariant moving frames is a powerful tool for determining a generating set of the differential invariant algebra of Lie pseudo-groups. After reviewing this theory, the method is illustrated with three applications. In the first two applications, generating sets of differential invariant algebra for the symmetry groups of the Infeld--Rowlands equation and the Davey--Stewartson equations are determined. Then we show that for two and three dimensional Riemannian manifolds the sectional curvatures generate the differential invariant algebra of the pseudo-group of locally invertible changes of variables.
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    Asymptotic properties of positive solutions of parabolic equations and cooperative systems with Dirichlet boundary data.
    (2009-07) Foldes, Juraj.
    We study symmetry properties of non-negative bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains with Dirichlet boundary conditions. First we consider scalar case, and we propose sufficient conditions on the equation and domain, which guarantee asymptotic symmetry of solutions. Then we consider fully nonlinear weakly coupled systems of parabolic equations. Assuming the system is cooperative we prove the asymptotic symmetry of positive bounded solutions. To facilitate an application of the method of moving hyperplanes, we derive several estimates for linear parabolic equations and systems, such as maximum principle on small domains, Alexandrov- Krylov estimate and Harnack type estimates.
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    Automorphic forms on certain affine symmetric spaces.
    (2011-05) Zhang, Lei
    In this thesis, we consider automorphic periods associated to certain affine symmetric spaces such as the symmetric pairs. In this thesis, we consider automorphic periods associated to certain affine symmetric spaces such as the symmetric pairs (Sp4n; ResK=kSp2n) and (GSp4n; ResK=kGSp2n); where k is a number field and K is an Etale algebra over k of dimension 2. We consider the period integral of a cusp forms of Sp4n(Ak) against with an Eisenstein series of the symmetric subgroup ResK=kSp2n. We expect to establish an identity between this period integrals and the special value of the spin L-function of the symplectic group. In the local theory, using Aizenbud and Gourevitch's generalized Harish-Chandra method and traditional methods, i.e. the Gelfand-Kahzdan theorem, we can prove that these symmetric pairs are Gelfand pairs when Kv is a quadratic extension field over kv for any n, or Kv is isomorphic to kv x kv for n <_ 2. Since (U(J2n; kv(p #28; )); Sp2n(kv)) is a descendant of (Sp4n(kv); Sp2n(kv) #2; Sp2n(kv)), we prove that it is a Gelfand pair for both archimedean and non-archimedean fields. According to the Yu' construction in [76] of irreducible tame supercuspidal representations, we give a parametrization of the distinguished tame supercuspidal representation of symplectic groups in this thesis. Applying the dimension formula of the space HomH(#25;; 1) given by Hakim and Murnaghan [28], we prove that if (G;H) is the symmetric pair (U(J2n;Kv); Sp2n(kv)) there is no H-distinguished tame supercuspidal representation, where Kv is a quadratic extension over kv. In addition, for the symmetric pair (Sp4n(kv); Sp2n(Kv)), we give the sufficient and necessary conditions of generic cuspidal data such that the corresponding tame supercuspidal representations are H-distinguished. Note that our case is the first case worked out with none of G and H being the general linear groups. Furthermore, motivated by a sub-question, we also give an example for the distinguished representations of finite groups of Lie Type in a low rank case. In particular, we show that #18;10 is the unique SL2(Fq2) distinguished cuspidal representation of Sp4(Fq). Note: See PDF abstract for the correct interpretation of the mathematical symbols.
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    Automorphic partial differential equations and spectral theory with applications to number theory.
    (2011-05) DeCelles, Amy Therese
    Note: See PDF abstract
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    Awi-wewebanaabiin! Go Fish Game
    (2010) Baker, Nashay
    Lesson plan for counting and communicating in Ojibwe.
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    The BV formalism for homotopy Lie algebras
    (2014-11) Bashkirov, Denis Aleksandrovich
    The present work concerns certain aspects of homotopy Lie and homotopy Batalin-Vilkovisky structures. In the first part we characterize a class of homotopy BV-algebras canonically associated to strongly homotopy Lie algebras and show that a category of strongly homotopy Lie algebras embeds into a certain subcategory of particularly simple homotopy Batalin-Vilkovisky algebras. In the second part of the work we introduce the notions of coboundary and triangular homotopy Lie bialgebras and discuss a possible a framework for quantization of such bialgebras.
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    Chaos, attractors and the Lorenz conjecture: Noninvertible transitive maps of ivariant sets are sensitive.
    (2010-06) Taft, Garrett Thomas
    In 1989, Edward Lorenz published a paper entitled, “Computational chaos- a prelude to computational instability” [L]. His paper looked at Euler approximations to differential equations. If the time increment of the approximating function was increased, he found that computational chaos set in. Since the numerics suggested transitivity and noninvertibility, he conjectured that transitive, noninvertible maps of an attractor were chaotic. To set the stage for investigating this conjecture, this thesis looked to examine the relationships between some of the standard definitions of chaos and attractor used throughout the literature. In addition to offering a proof of the Lorenz conjecture, a review of a number of related results was conducted. A side product of the work done was a partial result that tried to address whether topological transitivity carries sensitivity at a point to sensitivity on a set.
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    Combination topological field theory
    (2009-07) Hanhart, Alexander Louis
    This work contains the construction of a topological quantum field theory (TQFT, or TFT) based on combinatorial information which consists of directed metric graphs with vertices labeled by metric ribbon graphs. A canonical map between such objects and smooth Riemann surfaces is established using the theory of quadratic differentials investigated by Strebel and others. The surfaces derived have natural decomposition into finite and infinite length cylinders enumerated by the edges of the directed metric graph. Moreover, the surfaces have a gluing operation which agrees with a natural connecting operation on the level of graphs. Finally, the cylindrical decomposition gives the surfaces the structure of a model surface originally investigated by M. Schwartz. He offers a functor on the category of such surfaces which satisfies the properties of a TFT. Combining this functor with the combinatorial information gives the construction presented herein.
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    Convergence analysis of the immersed boundary method.
    (2011-12) Liu, Yang
    Many problems involving internal interfaces can be formulated as partial differential equations with singular source terms. Numerical approximation to such problems on a regular grid necessitates suitable regularizations of delta functions. We study the convergence properties of such discretizations for constant coefficient elliptic problems using the Immersed Boundary (IB) method, which is both a mathematical formulation and a numerical scheme widely used to solve fluid-structure interaction problems, as an example. IB schemes use a uniform Cartesian mesh for the fluid, a Lagrangian curvilinear mesh for the immersed structure, and discrete delta functions for communication between these two grids. We show how the order of the differential operator, order of the finite difference discretization, and properties of the discrete delta function all influence the local convergence behavior. In particular, we show how a recently introduced property of discrete delta functions - the smoothing order - is important in the determination of local convergence rates. We apply our theories to stationary Stokes flow problem and obtain both local and L^P convergence results. We examine the predicted results with numerical simulations.
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    Culturally and Linguistically Responsive Mathematics Word Problem Solving with English Learners
    (2019-12) Jones, Leila
    Schema-based instruction is recognized as an effective practice to teach children word problem solving skills. The purpose of this study was to evaluate the effectiveness of a culturally and linguistically responsive adaptation to schema-based instruction with a sample of Spanish-speaking English learners. A multiple probe design across participants was used to evaluate the efficacy of the culturally and linguistically responsive schema-based instruction on word problem solving performance. Maintenance of intervention effects was assessed six weeks following intervention implementation. Student perceptions of the culturally and linguistically responsive schema-based intervention were also measured. Results indicate that the intervention was successful at improving and maintaining word problem solving performance with this sample. The students reported an overall positive attitude toward the intervention, providing evidence that they understood, enjoyed, and felt they benefited from the intervention.
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    Design of progressive additional lens with wavefront tracing method.
    (2010-09) Zhou, Fanhuan
    Progressive additional lenses is a new approach to compensate for the defects of presbyopia for the human visual system. While using two or more single-vision lenses with different power, progressive additional lenses require the specification of free surfaces. A progressive additional lens comprises a large distance zone with low power on the upper portion of the lens, a small near zone with higher power on the lower part, and a progressive corridor of increasing power connects these two zones smoothly and progressively. In this work, we proposed an optimization approach to the progressive additional lens design problem, in which, a cost function which balances the power distribution and other optical aberrations up to the second order aberrations. The goal is to minimize the cost function. In previous progressive additional lens design, the formulas for the power and astigmatism of the lens is an approximation of the refraction on the front and back lens surface. Also, the assumption of the thickness of the lens $d ll 1$ limits the design to be a thin lens design. To better evaluate the exact properties of the optical system, one should study the ray tracing though the lens and the wavefront passing through a lens. Therefore, geometric optics is employed to evaluate how a wavefront is deformed by a lens and how the curvature of a wavefront is transformed when propagating through a homogeneous medium and refracting on a lens surface. To understand the second order aberration of the optical system, a concept of Third Order Surface coefficients is also introduced. Wavefront tracing method can then be used to evaluate the exact property of the refracted wavefront of the lens system. We derived a series of formulas describing First Fundamental Form coefficients, Second Fundamental Form coefficients and Third Order Surface coefficients of the wavefront on propagation and on refraction. A full process of ray tracing through the lens and wavefront deformation is also derived explicitly. We also discuss the process of front lens surface design and back lens surface design. To numerically construct the progressive lens surface, tensor product B-spline functions is used to solve the optimization problem. Numerical Methods such as Gradient method, Newton's method and Quasi-Newton method are used. One numerical example of front surface design with the fixed back spherical lens surface is shown in this work. In summary, this thesis develops a model of progressive additional lens design up to second order aberration by wavefront tracing method.
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    The Differential Effects of Elaborated Task and Process Feedback on Multi-Digit Multiplication
    (2020-05) Edmunds, Rebecca
    Given persistent low achievement in mathematics for students in the United States, researchers and practitioners have a vested interest in identifying effective intervention components. This study explored the differential effects of elaborated task feedback (ETF) and elaborated process feedback (EPF) when combined with a cover, copy, compare (CCC) intervention as compared to a repeated practice control condition on students’ fluency and strategy use. The multi-digit multiplication class-wide intervention was implemented in 10-sessions with a sample of 101 students from two suburban schools in the Midwest. Due to an interest in the impact of feedback over time, hierarchical linear modeling (HLM) and hierarchical generalized linear modeling were used to examine changes in performance across the intervention. Despite an overall strong effect, the impact of feedback can vary by context, delivery, and purpose (Kluger & DeNisi, 1996). This study addressed gaps in the feedback literature by providing feedback on strategy use and testing the effects of feedback with elaboration to guide error correction. Non-significant effects were found for both types of feedback on fluency and strategy use. The observed increases in fluency over time across conditions provides additional support for the impact of deliberate, repeated practice in mathematics (e.g. Clarke et al., 2016; Fuchs et al., 2010). Implications of the bidirectional relationship observed between strategy use and fluency as well as the potential moderating effects of individual student characteristics are also explored; implications for practice and future research are discussed. Results underscore the importance of research on interventions targeting mathematics skills beyond single-digit computation.
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    Directional sensing and actin dynamics in dictyostelium discoideum amoebae.
    (2012-08) Khamviwath, Varunyu
    Abstract summary not available
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    Dynamical implications of network statistics
    (2014-08) Campbell, Patrick Ronald
    Dynamics on large networks can be highly complex. I present several methods for investigating the effects of network structure/statistics on rate dynamics and spike correlations. The dynamical models under consideration come from computational neuroscience, but these methods may generalize to other contexts. The thesis focuses on both network constructions and dynamics on networks.I present two approaches to network constructions: random networks, and networks with patterns. For random networks, I give a generalization of the expected degree model (EDM) and a formulation of the EDM and its generalization which is invariant of the number of nodes. This generalization allows one to produce random networks which have nontrivial second and third order correlations among edges. I also introduce a method for constructing networks with nontrivial structures/patterns at multiple scales. I investigate the spectral properties of the resulting networks and extend the method to include stochastic elements.The singular value decomposition (SVD) is a powerful tool with many applications. I review its application to network adjacency matrices, including a known result which relates the singular values of an adjacency matrix to a measure of its randomness. Further, I demonstrate that for several random network models the degree sequence is the most significant feature of the connectivity.The primary dynamical model I consider is the Poisson spiking model (PSM). I derive first and second order statistics for the PSM using a path integral formalism.The major contribution of this work is a dimension reduction method for dynamics on a network using the SVD. I demonstrate how one can use these low rank representations of the connectivity, together with the reduced equations to approximately recover node-specific activity. Thus, not only do I present methods that reduce the number of dynamical variables, but I show how the dynamics of the full system may be decomposed into the reduced variables and network structure.
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    An evaluation of blended instruction in terms of knowledge acquisition and attitude in an introductory Mathematics course
    (2014-01) Czaplewski, John
    A medium-sized accredited public university located in southeastern Minnesota has been offering an introductory undergraduate mathematics course with a consistent curriculum in two instructional formats: face-to-face and blended. Previously the course was offered only through a face-to-face instructional format while currently, it is only offered in a blended instructional format. This case study explored the influence that the method of instruction had on student achievement on common assessments, how a blended instruction course design impacted the attitude of students, and the amount of knowledge acquired in a blended instruction environment.A blended course is one taught by combining teacher-centered face-to-face instructional elements with online learning components and online course management tools. In more general terms, blended instruction is a term used to describe instruction or training events or activities where online learning, in its various forms, is combined with more traditional forms of instruction such as "classroom" learning. The terms hybrid and mixed mode are references to the same type of instruction and therefore used synonymously. An instrument developed by Martha Tapia and George Marsh measured changes in attitude toward mathematics related to a blended instructional course design. While one area of interest was the level of procedural knowledge acquired in a blended instructional environment versus that of a face-to-face setting, an additional interest was student comprehension beyond procedural knowledge. This study noted applications of the common knowledge students used to demonstrate their comprehension and sense-making ability. In order to evaluate the additional level of understanding, this study asked questions of students enrolled in a blended instructional environment via a series of interviews as well as observing classroom activities designed to allow for further exploration of content and demonstration of knowledge beyond that allowed for in a face-to-face setting. Results from this study indicated a statistically significant difference in comparing final course grades and final examination grades of the students enrolled in the blended instruction designed course versus the face-to-face lecture courses while the instructor was held constant. Students were less anxious working on assigned problems and assessments as they familiarized themselves with the design and instructional strategies. In addition, students were more engaged in discussions as the semester progressed, and students experienced the benefits of communicating with group members. The results also indicate that students enrolled in a blended instruction course perceive that the classroom environment promotes interactions, and they are involved in classroom discussions and activities.