Physical Sciences and Mathematics Commons^™

Fractured Branched Circle Packings On The Plane, James Russell Ashe

Masters Theses

William Thurston first proposed that real circles could be used to approximate the underlying infinitesimal circles of conformal maps in 1985. Inspired pioneers developed Circle Packing into a very rich and deep field that can be used as a method for constructing discrete conformal maps of surfaces on different types of geometries. Offering the advantages of a computational method that lends itself to experimentation and the easy creation of visual models, Circle Packing has proven itself as valuable new tool in approaching both old and new problems.

In particular, Circle Packing has been used to make discrete analogues of continuous …

Mathematical Representations Of The Architecture Of Graphical User Interfaces, Eric Freeman Mann

Masters Theses

This text is an introduction to mathematical representations of graphical user interfaces, GUIs. Vectors are employed as a means to represent the state of a GUI and the user interaction with a GUI. These representations form a model of the behavior of a GUI over time. The usefulness of the model in testing and developing well-behaved GUIs is discussed and demonstrated by example.

A Study Of Hidden Markov Model, Yang Liu

Masters Theses

The purpose of this thesis is fourfold: Introduce the definition of Hidden Markov Model. Present three problems about HMM that must be solved for the model to be useful in real-world application. Cover the solution of the three basic problems; explain some algorithms in detail as well as the mathematical proof. And finally, give some examples to show how these algorithms work.

Modeling The Cascade Of Reactions In Visual Transduction, Philip Torry Patton

Masters Theses

Phototransduction in the process by which light energy is transferred to an electrical potential by photoreceptors in the eye. Recently, Hamer et al proposed a mechanism for the cascade of reactions that occurs when a rhodopsin molecule on the surface of a disc in the rod photoreceptor becomes activated. From these reactions, one can derive a system of ordinary differential equations to simulate phototransduction.

These ODEs are solved numerically using forward Euler’s method. The products of this cascade are then used to predict the drop in photocurrent that occurs. A kinetic Monte Carlo simulation with the Gillespie algorithm is implemented …

Decidability In Algebraic Geometry, John James Iskra

Doctoral Dissertations

The central theme of our investigation is the concept of Decidability in Algebra/Algebraic Geometry. To the best of our knowledge this seems to be novel in the sense that there is no work known to isolate or to focus on the concept of Decidability in the context of Commutative Algebra. Decidability is more restrictive than Grothendieck's concept of formally unramified, but weaker than the concept of étale. In this article we study these relationships by characterizing Decidability for ring-extensions of essentially finite type. In the absence of essential finiteness we can only show, at present, that a separable algebraic …

Fibrator Properties Of Pl Manifolds, Violeta Vasilevska

Doctoral Dissertations

In the early 90s, R.Daverman defined the concept of the PL fibrator ([12]). PL fibrators, by definition, provide detection of PL approximate fibrations. Daver- man defines a closed, connected, orientable PL n-manifold to be a codimension-k PL orientable fibrator if for all closed, connected, orientable PL (n + k)-manifolds M and PL maps p : M → B, where B is a polyhedron, such that each fiber collapses to an n-complex homotopy equivalent to Nⁿ, p is always an approximate fibration.

If N is a codimension-k PL orientable fibrator for all k > 0, …

Enhancements To Crisp Possibilistic Reconstructability Analysis, Anas Al-Rabadi, Martin Zwick

Systems Science Faculty Publications and Presentations

Modified Reconstructibility Analysis (MRA), a novel decomposition within the framework of set-theoretic (crisp possibilistic) Reconstructibility Analysis, is presented. It is shown that in some cases while 3-variable NPN-classified Boolean functions are not decomposable using Conventional Reconstructibility Analysis (CRA), they are decomposable using Modified Reconstructibility Analysis (MRA). Also, it is shown that whenever a decomposition of 3-variable NPN-classified Boolean functions exists in both MRA and CRA, MRA yields simpler or equal complexity decompositions. A comparison of the corresponding complexities for Ashenhurst-Curtis decompositions, and Modified Reconstructibility Analysis (MRA) is also presented. While both AC and MRA decompose some but …

Wavelet Estimation Of Partially Linear Models, Xiao-Wen Chang, Leming Qu

Leming Qu

A wavelet approach is presented for estimating a partially linear model (PLM). We find an estimator of the PLM by minimizing the square of the l₂ norm of the residual vector while penalizing the l₁ norm of the wavelet coefficients of the nonparametric component. This approach, an extension of the wavelet approach for nonparametric regression problems, avoids the restrictive smoothness requirements for the nonparametric function of the traditional smoothing approaches for PLM, such as smoothing spline, kernel and piecewise polynomial methods. To solve the optimization problem, an efficient descent algorithm with an exact line search is presented. Simulation …

Group Actions In Number Theory, Tyler J. Evans

Tyler Evans

A Notion Of Rectifiability Modeled On Carnot Groups, Scott D. Pauls

Dartmouth Scholarship

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and the existence of approximate tangent cones isometric to N almost everywhere in E. Second, we prove that, under a stronger condition concerning the existence of approximate tangent cones …

A Numerical Scheme For Mullins-Sekerka Flow In Three Space Dimensions, Sarah Marie Brown

Theses and Dissertations

The Mullins-Sekerka problem, also called two-sided Hele-Shaw flow, arises in modeling a binary material with two stable concentration phases. A coarsening process occurs, and large particles grow while smaller particles eventually dissolve. Single particles become spherical. This process is described by evolving harmonic functions within the two phases with the moving interface driven by the jump in the normal derivatives of the harmonic functions at the interface. The harmonic functions are continuous across the interface, taking on values equal to the mean curvature of the interface. This dissertation reformulates the three-dimensional problem as one on the two-dimensional interface by using …

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N-Dimensions, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We consider the inverse problem of determining the shape of some inacces­ sible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples

Determining The Length Of A One-Dimensional Bar, Natalya Yarlikina, Holly Walrath

Mathematical Sciences Technical Reports (MSTR)

In this paper we examine the inverse problem of determining the length of a one-dimensional bar from thermal measurements (temperature and heat flux) at one end of the bar (the "accessible" end); the other inaccessible end of the bar is assumed to be moving. We develop two different approaches to estimating the length of the bar, and show how one approach can also be adapted to find unknown boundary conditions at the inaccessible end of the bar.

Psl(2,7)-Extensions With Certain Ramification At Two Primes, Glen E. Simpson

Theses and Dissertations

We conduct a parallel Hunter search in order to find a degree 7 number field K ramified at primes q and p with discriminant d(K)=q^6 p^2 where q=11 and 2

Large Prandtl Number Behavior Of The Boussinesq System Of Rayleigh-Bénard Convection, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We establish the validity of the infinite Prandtl number model as an approximation of the Boussinesq system at large Prandtl number on finite and infinite time interval, as well as in some statistical sense. © 2004 Elsevier Ltd. All rights reserved.

Asymptotic Solutions Of Semilinear Stochastic Wave Equations, Pao-Liu Chow

Mathematics Research Reports

Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in mean-square and the almost sure sense, are studied. Then, under some sufficient conditions, the existence of a unique invariant measure is proved. Two examples are presented to illustrate some applications of the theorems.

A Liouville-Gelfand Equation For K-Hessian Operators, Jon T. Jacobsen

All HMC Faculty Publications and Research

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the space dimension and the value of k. The choice of exponential nonlinearity is motivated by the classical Liouville-Gelfand problem from combustible gas dynamics and prescribed curvature problems.

A High -Order Finite Difference Method For Solving Bioheat Transfer Equations In Three-Dimensional Triple -Layered Skin Structure, Haofeng Yu

Doctoral Dissertations

Investigations on instantaneous skin burns are useful for an accurate assessment of burn-evaluation and for establishing thermal protections for various purposes. Meanwhile, hyperthermia with radiation is important in the treatment of cancer, and it is essential for developers and users of hyperthermia systems to predict, and interpret correctly the biomass thermal and vascular response to heating. In this dissertation, we employ the well-known Pennes' bioheat transfer equation to predict the degree of skin burn and the temperature distribution in hyperthermia cancer treatment.

A fourth-order compact finite difference scheme is developed to solve Pennes' bioheat transfer equation in a three-dimensional single …

Oscillation Of Symplectic Dynamic Systems, Martin Bohner, Ondřej Došlý

Mathematics and Statistics Faculty Research & Creative Works

We investigate oscillatory properties of a perturbed symplectic dynamic system on a time scale that is unbounded above. the unperturbed system is supposed to be Non oscillatory, and we give conditions on the perturbation matrix, which guarantee that the perturbed system becomes oscillatory. Examples illustrating the general results are given as well. © Australian Mathematical Society 2004.

Evolutionary Convergence To Ideal Free Dispersal Strategies And Coexistence, Richard Gejji, Yuan Lou, Daniel Munther, Justin Peyton

Mathematics and Statistics Faculty Publications

We study a two species competition model in which the species have the same population dynamics but different dispersal strategies and show how these dispersal strategies evolve. We introduce a general dispersal strategy which can result in the ideal free distributions of both competing species at equilibrium and generalize the result of Averill et al. (2011). We further investigate the convergent stability of this ideal free dispersal strategy by varying random dispersal rates, advection rates, or both of these two parameters simultaneously. For monotone resource functions, our analysis reveals that among two similar dispersal strategies, selection generally prefers the strategy …

Model Parameters And Outbreak Control For Sars, Christopher Kribs, Gerardo Chowell, Carlos Castillo-Chavez, Paul W. Fenimore, Leon Arriola, James M. Hyman

Mathematics Faculty Publications

Control of the 2002–2003 severe acute respiratory syndrome (SARS) outbreak was based on rapid diagnosis coupled with effective patient isolation. We used uncertainty and sensitivity analysis of the basic reproductive number R0 to assess the role that model parameters play in outbreak control. The transmission rate and isolation effectiveness have the largest fractional effect on R0. We estimated the distribution of the reproductive number R0 under perfect isolation conditions. The distribution lies in the interquartile range 0.19–1.08, with a median of 0.49. Even though the median of R0 is <1, we found that 25% of our R0 distribution lies at R0 > 1, even with perfect isolation. This implies the need to simultaneously …

Nice Polynomials, Jonathan Groves

Masters Theses & Specialist Projects

We consider the problem of finding, constructing, and classifying nice polynomials. After a short history of previous results, we present a general property of nice polynomials which leads to an important modification of the concept of equivalence classes of nice polynomials. We give several important results on nice symmetric or antisymmetric polynomials with an odd number of roots, which dramatically increase the speed of a computer search for examples. We present complete solutions to the symmetric three root case, the general three root case, and the symmetric four root case. We also give the relations between the roots and critical …

Contributions To Emerging Techniques In Survey Sampling., Sanghamitra Pal Dr.

Doctoral Theses

This dissertation contains seven Chapters. The contents in the respective Chapters may be briefly recounted as follows.A topic of classical interest in survey sampling is how to ensure the existence of a uniformly non-negative (UNN) unbiased estimator for the mean square error (MSE) of a homogeneous linear estimator (HLE) for a finite survey population total. Hájek (1958), Vijayan (1975), Rao and Vijayan (1977) and Rao (1979) developed a number of results which boil down to the folowing as narrated in the monograph by Chaudhuri and Stenger (1992).If there exist non-zero constants w, and the unknown values y, of the variable …

An Experimental Study Of Micron-Scale Droplet Aerosols Produced Via Ultrasonic Atomization, Thomas D. Donnelly, J. Hogan '03, A. Mugler '04, N. Schommer '04, M. Schubmehl '02, Andrew J. Bernoff, B. Forrest '02

All HMC Faculty Publications and Research

In the last 10 years, laser-driven fusion experiments performed on atomic clusters of deuterium have shown a surprisingly high neutron yield per joule of input laser energy. Results indicate that the optimal cluster size for maximizing fusion events should be in the 0.01–μm diameter range, but an appropriate source of droplets of this size does not exist. In an attempt to meet this need, we use ultrasonic atomization to generate micron-scale droplet aerosols of high average density, and we have developed and refined a reliable droplet sizing technique based on Mie scattering. Harmonic excitation of the fluid in …

How To Increase The Ability Of A Student To Learn, Srinivas R. Chakravarthy

Industrial & Manufacturing Engineering Presentations And Conference Materials

An instructor is always challenged when covering the materials in a course (according to the syllabus) and at the same time making sure that all students have the opportunity to learn and understand the materials presented in the classroom. In this paper we will present some ideas and tools that enable one to try to achieve a balance. These are based on the author’s experience and perspective in teaching deterministic and stochastic operations research courses.

A New Approach To Lie Symmetry Groups Of Minimal Surfaces, Robert D. Berry

Theses and Dissertations

The Lie symmetry groups of minimal surfaces by way of planar harmonic functions are determined. It is shown that a symmetry group acting on the minimal surfaces is isomorphic with H × H^2 — the analytic functions and the harmonic functions. A subgroup of this gives a generalization of the associated family which is examined.

Bounding The Firing Synchronization Problem On A Ring, André Berthiaume, Todd Bittner, Ljubomir Perković, Amber Settle, Janos Simon

Amber Settle

Absolutely Continuous Representations And A Kaplansky Density Theorem For Free Semigroup Algebras, Kenneth R. Davidson, Jiankui Li, David R. Pitts

Department of Mathematics: Faculty Publications

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a ∗-extendible represen- tation . A ∗-extendible representation of An is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous func- tionals are intimately related to maps which intertwine a given ∗-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is …

Assessment Of Surface Water Runoff And Groundwater Recharge Using Mathematical Models, Khaled Omar Mohamed Haroon

Accounting Dissertations

In and semi-arid regions, surface water resources are scarce and, in most cases, groundwater is the onlynatural resource of freshwater. Pumping of groundwater often exceeds natural recharge. Therefore,groundwater level are declining and its quality is deteriorating. Sustainable management of groundwater is thus a key Issue and requires implementation of appropriate technologies to augment ground water resources. Artificial recharge augments the natural movement of surface water into the underground form' illations using some means of construction whereby surface water from streams or lakes is made to infiltrate into the ground.

The UAE is known by its arid conditions and limited …

Multi-Scale Continuum Mechanics: From Global Bifurcations To Noise Induced High-Dimensional Chaos, Ira B. Schwartz, David S. Morgan, Lora Billings, Ying-Cheng Lai

Department of Mathematics Facuty Scholarship and Creative Works

Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural–mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between …