Mathematics Commons^™

Model Reduction For Simulation, Optimization And Control, Oleg Edward Roderick

Dissertations and Theses

Many tasks of simulation, optimization and control can be performed more efficiently if the intermediate complexity of the numerical model is reduced. In our work, we investigate model reduction, as applied to reaction-transport systems of atmospheric chemistry. We use a Proper Orthogonal Decomposition-based approach to extract information from a set of model observations, and to project the model equations onto a reduced order space chosen in such a way that the essential model behavior is preserved in the solution of the reduced version. We examine and improve many features of the method. In particular, we show how to measure sensitivities …

Variational Theory Of Balance Systems, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work we apply the Poincare-Cartan formalism of the Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles of the configurational bundle and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system is realized as a mapping between a (partial) k-jet bundle and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system corresponding to a constitutive relation C is studied. Special cases of balance systems -Lagrangian systems …

Quantum Multiplexers, Parrondo Games, And Proper Quantization, Faisal Shah Khan

Dissertations and Theses

A quantum logic gate of particular interest to both electrical engineers and game theorists is the quantum multiplexer. This shared interest is due to the facts that an arbitrary quantum logic gate may be expressed, up to arbitrary accuracy, via a circuit consisting entirely of variations of the quantum multiplexer, and that certain one player games, the history dependent Parrondo games, can be quantized as games via a particular variation of the quantum multiplexer. However, to date all such quantizations have lacked a certain fundamental game theoretic property.

The main result in this dissertation is the development of quantizations of …

Quaternions, Octonions, And The Quantization Of Games, Aden Omar Ahmed

Dissertations and Theses

We present an effect on classical games that is obtained by replacing the notion of probability distribution with the notions of quantum superposition and measurement. Our particular focus will be on two and three player games where each player has precisely two pure strategic choices. Games in normal form are represented as "payoff" functions.

Game quantization requires the extension of these functions to much larger domains. The main result of this work is the co-ordinatization of these extended functions by either the quaternions or octonions in order to obtain computationally friendly versions of these functions. This computational capability is then …

Large-Scale Assembly Of Periodic Nanostructures With Metastable Square Lattices, Chih-Hung Sun, Wei-Lun Min, Nicholas C. Linn, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

This article reports a simple and scalable spin-coating technique for assembling non-close-packed colloidal crystals with metastable square lattices over wafer-scale areas. The authors observe the alternate formation of hexagonal and square diffraction patterns when the thickness of the colloidal crystals is gradually reduced during spin coating. No prepatterned templates are needed to induce the formation of the resulting metastable crystals with square arrangement. This bottom-up technology also enables the large-scale production of a variety of squarely ordered nanostructures that are consistent with the industry-standard rectilinear coordinate system for simplified addressing and circuit interconnection. Broadband moth-eye antireflection gratings with square lattices …

Data Assimilation, Adaptive Observations And Applications, Humberto Carlos Godinez Vasquez

Dissertations and Theses

Sensitivity analysis, data assimilation and targeting observation strategies are methods that are applied to various complex mathematical models of fluid dynamics. In this research we investigate new directions to improve on the current strategies used to deploy additional observational resources (targeting strategies) for data assimilation in dynamical systems of fluid mechanics.

Targeting strategies aim to determine optimal locations where additional observations will improve the solution of the data assimilation process by identifying regions where state errors in the model have a high potential to grow.

Properly accounting for nonlinear error growth is an unresolved issue in targeted observations for numerical …

A New Elasticity Element Made For Enforcing Weak Stress Symmetry, Bernardo Cockburn, Jay Gopalakrishnan, Johnny Guzmán

Mathematics and Statistics Faculty Publications and Presentations

We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer k, the row-wise divergence of the element space spans the set of polynomials of total degree k. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress. This is achieved using certain "bubble matrices", which are special divergence-free matrix-valued polynomials. We prove that the approximation error is of order k + …

Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such extensions are a fundamental ingredient of high order finite element analysis.

The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

Unified Hybridization Of Discontinuous Galerkin, Mixed, And Continuous Galerkin Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan, Raytcho Lazarov

Mathematics and Statistics Faculty Publications and Presentations

We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixed-dual finite element methods including hybridized mixed, continu- ous Galerkin, non-conforming and a new, wide class of hybridizable discontinuous Galerkin methods. The distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements. Since the associated matrix is sparse, symmetric and positive definite, these methods can be efficiently implemented. Moreover, the …

Biomimetic Subwavelength Antireflective Gratings On Gaas, Chih-Hung Sun, Brian J. Ho, Bin Jiang, Peng Jiang

Mathematics and Statistics Faculty Publications and Presentations

We have developed a simple and scalable bottom-up approach for fabricating moth-eye antireflective coatings on GaAs substrates. Monolayer, non-close-packed silica colloidal crystals are created on crystalline GaAs wafers by a spin-coating-based single-layer reduction technique. These colloidal monolayers can be used as etching masks during a BCl_3 dry-etch process to generate subwavelength-structured antireflective gratings directly on GaAs substrates. The gratings exhibit excellent broadband antireflective properties, and the specular reflection matches with the theoretical prediction using a rigorous coupled-wave analysis model. These bioinspired antireflection coatings have important technological applications ranging from efficient solar cells to IR detectors

Students' Reasoning About The Concept Of Limit In The Context Of Reinventing The Formal Definition, Craig Alan Swinyard

Dissertations and Theses

Many researchers (Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995) have noted the vital role limit plays as a foundational concept in analysis. The vast majority of topics encountered in calculus and undergraduate analysis are built upon understanding the concept of limit and being able to work flexibly with its formal definition (Bezuidenhout, 2001). The purpose of this study was to: (1) Develop insight into students' reasoning about limit in relation to their engagement in instruction designed to support their reinventing the formal definition of limit, and; (2) Inform the design of principled instruction that might support students' attempts to …

How Students Use Mathematical Resources In An Electrostatics Context, Dawn C. Meredith, Karen A. Marrongelle

Mathematics and Statistics Faculty Publications and Presentations

We present evidence that although students’ mathematical skills in introductory calculus-based physics classes may not be readily applied in physics contexts, these students have strong mathematical resources on which to build effective instruction. Our evidence is based on clinical interviews of problem solving in electrostatics, which are analyzed using the framework of Sherin’s symbolic forms. We find that students use notions of “dependence” and “parts-of-a-whole” to successfully guide their work, even in novel situations. We also present evidence that students’ naive conceptions of the limit may prevent them from viewing integrals as sums.

Bioinspired Broadband Antireflection Coatings On Gasb, Wei-Lun Min, Amaury P. Betancourt, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

We report an inexpensive yet scalable templating technique for fabricating moth-eye antireflection gratings on gallium antimonide substrates. Non-close-packed colloidal monolayers are utilized as etching masks to pattern subwavelength-structured nipple arrays on GaSb. The resulting gratings exhibit superior broadband antireflection properties and thermal stability than conventional multilayer dielectric coatings. The specular reflection of the templated nipple arrays match with the theoretical predictions using a rigorous coupled-wave analysis model. The effect of the nipple shape and size on the antireflection properties has also been investigated by the same model. These biomimetic coatings are of great technological importance in developing efficient thermophotovoltaic cells.

Templated Biomimetic Multifunctional Coatings, Chih-Hung Sun, Adriel Gonzalez, Nicholas C. Linn, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

We report a bioinspired templating technique for fabricating multifunctional optical coatings that mimic both unique functionalities of antireflective moth eyes and superhydrophobic cicada wings. Subwavelength-structured fluoropolymer nipple arrays are created by a soft-lithography-like process. The utilization of fluoropolymers simultaneously enhances the antireflective performance and the hydrophobicity of the replicated films. The specular reflectivity matches the optical simulation using a thin-film multilayer model. The dependence of the size and the crystalline ordering of the replicated nipples on the resulting antireflective properties have also been investigated by experiment and modeling. These biomimetic materials may find important technological application in self-cleaning antireflection coatings.

Broadband Moth-Eye Antireflection Coatings On Silicon, Chih-Hung Sun, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

We report a bioinspired templating technique for fabricating broadband antireflection coatings that mimic antireflective moth eyes. Wafer-scale, subwavelength-structured nipple arrays are directly patterned on silicon using spin-coated silica colloidal monolayers as etching masks. The templated gratings exhibit excellent broadband antireflection properties and the normal-incidence specular reflection matches with the theoretical prediction using a rigorous coupled-wave analysis (RCWA) model. We further demonstrate that two common simulation methods, RCWA and thin-film multilayer models, generate almost identical prediction for the templated nipple arrays. This simple bottom-up technique is compatible with standard microfabrication, promising for reducing the manufacturing cost of crystalline silicon solar cells.

Multisymplectic Theory Of Balance Systems, I, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this paper we are presenting the theory of balance equations of the Continuum Thermodynamics (balance systems) in a geometrical form using Poincare-Cartan formalism of the Multisymplectic Field Theory. A constitutive relation C of a balance system B_C is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle π : Y ͢ X and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system BC is presented in three different forms and the space of admissible variations is defined and studied. Action of automorphisms …

Order Restricted Inference For Comparing The Cumulative Incidence Of A Competing Risk Over Several Populations, Hammou El Barmi, Subhash C. Kochar, Hari Mukerjee

Mathematics and Statistics Faculty Publications and Presentations

There is a substantial literature on testing for the equality of the cumulative incidence functions associated with one specific cause in a competing risks setting across several populations against specific or all alternatives. In this paper we propose an asymptotically distribution-free test when the alternative is that the incidence functions are linearly ordered, but not equal. The motivation stems from the fact that in many examples such a linear ordering seems reasonable intuitively and is borne out generally from empirical observations. These tests are more powerful when the ordering is justified. We also provide estimators of the incidence functions under …

Correction To "Stochastic Comparisons Of Parallel Systems When Components Have Proportional Hazard Rates", Subhash C. Kochar, Maochao Xu

Mathematics and Statistics Faculty Publications and Presentations

In the article "Stochastic comparisons of parallel systems when component have proportional hazard rates" we found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order.

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory, …

Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H^½(∂K) into H¹(K), for any tetrahedron K.

Asymptotic And Numerical Techniques For Resonances Of Thin Photonic Structures, Jay Gopalakrishnan, Shari Moskow, Fadil Santosa

Mathematics and Statistics Faculty Publications and Presentations

We consider the problem of calculating resonance frequencies and radiative losses of an optical resonator. The optical resonator is in the form of a thin membrane with variable dielectric properties. This work provides two very different approaches for doing such calculations. The first is an asymptotic method which exploits the small thickness and high index of the membrane. We derive a limiting resonance problem as the thickness goes to zero, and for the case of a simple resonance, find a first order correction. The limiting problem and the correction are in one less space dimension, which can make the approach …

A Mixed Method For Axisymmetric Div-Curl Systems, Dylan M. Copeland, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

We present a mixed method for a three-dimensional axisymmetric div-curl system reduced to a two-dimensional computational domain via cylindrical coordinates. We show that when the meridian axisymmetric Maxwell problem is approximated by a mixed method using the lowest order Nédélec elements (for the vector variable) and linear elements (for the Lagrange multiplier), one obtains optimal error estimates in certain weighted Sobolev norms. The main ingredient of the analysis is a sequence of projectors in the weighted norms satisfying some commutativity properties.

The Indefinite Metric Of R. Mrugala And The Geometry Of The Thermodynamical Phase Space, Serge Preston, James Vargo

Mathematics and Statistics Faculty Publications and Presentations

We study an indefinite metric G which was introduced by R. Mrugala and is defined on the contact phase space (P, θ) of a homogeneous thermodynamical system. We describe the curvature properties and the isometry group of the metric G. We established an isomorphism of the space (P, θ, G) with the Heisenberg Lie group H_n, endowed with the right invariant contact structure and the right invariant indefinite metric. The lift of the metric G to the symplectization of contact space (P, θ) and its properties are studied. Finally we introduce the "hyperbolic projectivization" of the space …

Uniform Materials And The Multiplicative Decomposition Of The Deformation Gradient In Finite Elasto-Plasticity, Vincenzo Ciancio, Marina Dolfin, Mauro Francaviglia, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work we analyze the relation between the multiplicative decomposition F = F e F p of the deformation gradient as a product of the elastic and plastic factors and the theory of uniform materials. We prove that postulating such a decomposition is equivalent to having a uniform material model with two configurations – total φ and the inelastic φ1. We introduce strain tensors characterizing different types of evolutions of the material and discuss the form of the internal energy and that of the dissipative potential. The evolution equations are obtained for the configurations (φ, φ1) and the material …

Utilization Of Revoicing Based On Learners‘ Thinking In An Inquiry-Oriented Differential Equations Class, Oh Kwon Nam, Mi Kyung Ju, Chris Rasmussen, Karen A. Marrongelle, Jae Hee Park, Kyoung Hee Cho, Jung Sook Park, Jee Hyun Park

Mathematics and Statistics Faculty Publications and Presentations

Researchers of mathematics education are increasingly interested in a teacher's discursive moves, which refer to deliberate actions taken by a teacher to participate in or influence debate and discussion in the mathematics classroom. This study explored one teacher's discursive moves in an undergraduate inquiry-oriented mathematics class. The data for this study come from four class sessions in which students investigated initial value problems as represented by the phase portrait of a system of differential equations. Through the analysis and a review of the literature, we identified four categories of a teacher's discursive moves: revoicing, questioning/requesting, telling, and managing. This report …

Templated Fabrication Of Large Area Subwavelength Antireflection Gratings On Silicon, Chih-Hung Sun, Wei-Lun Min, Nicholas C. Linn, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

We report a cheap and scalable bottom-up technique for fabricating wafer-scale, subwavelength-structured antireflection coatings on single-crystalline silicon substrates. Spin-coated monolayer colloidal crystals are utilized as shadow masks to generate metallic nanohole arrays. Inverted pyramid arrays in silicon can then be templated against nanoholes by anisotropic wet etching. The resulting subwavelength gratings greatly suppress specular reflection at normal incidence. The reflection spectra for flat silicon and the templated gratings at long wavelengths agree well with the simulated results using a rigorous coupled wave analysis model. These subwavelength gratings are of great technological importance in crystalline silicon solar cells.

Self-Assembled Biomimetic Antireflection Coatings, Nicholas C. Linn, Chih-Hung Sun, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

The authors report a simple self-assembly technique for fabricating antireflection coatings that mimic antireflective moth eyes. Wafer-scale, nonclose-packed colloidal crystals with remarkable large hexagonal domains are created by a spin-coating technology. The resulting polymer-embedded colloidal crystals exhibit highly ordered surface modulation and can be used directly as templates to cast poly(dimethylsiloxane) (PDMS) molds. Moth-eye antireflection coatings with adjustable reflectivity can then be molded against the PDMS master. The specular reflection of replicated nipple arrays matches the theoretical prediction using a thin-film multilayer model. These biomimetic films may find important technological application in optical coatings and solar cells.

Graduate Teaching Assistants' Statistical Knowledge For Teaching, Jennifer Ann Noll

Dissertations and Theses

This dissertation explores graduate teaching assistants’ (TAs’) statistical knowledge for teaching. Data collection methods that enabled the exploration of TA s’ statistical knowledge for teaching include: (a) a task-based web survey administered to 68 TAs from 18 universities across the United States; and, (b) a series of three taskbased interviews with a subset of five TAs from the larger survey population. Through and Strauss, 1967), I investigated the ways in which TAs reason about sampling tasks, and how they think about teaching and student learning in relation to sampling ideas. Building on past research in statistics education on K-12 and …

A Qualitative Analysis Of Hybrid Control Systems, Anca Georgeta Williams

Dissertations and Theses

Hybrid Control Systems are increasingly investigated as models for control systems where the interaction between continuous and discrete processes is tightly integrated. As yet, there is no general agreement as to what constitutes a canonical hybrid system. Rather various subclasses of systems combining continuous evolution and discrete event characteristics are analyzed for each application. In this dissertation we first unify the presentation of hybrid systems with a rich model which encompasses most examples of interest. We classify the available results and show how they fit together within the different subcategories. In the main part of the dissertation we prove that …