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Error Estimates For A Mixed Finite Element Method For The Maxwell's Transmission Eigenvalue Problem, Chao Wang, Jintao Cui, Jiguang Sun

Michigan Tech Publications

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell's transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the …

Approximation Via Degree Reduction Of Nonlinearities With Applications To Turbulent Flows, Flame Fronts, And Magnetohydrodynamics, Matthew Enlow

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

We perform an analytical and computational investigation on the effectiveness of a locally bounded truncation function, which we call a calming function, when applied to the nonlinear terms of several dissipative partial differential equations. In particular, the 3D Navier-Stokes equations of incompressible fluid flow, the 2D Kuramoto-Sivashinsky equations of laminar flame fronts, and the 2D MHD-Boussinesq equations of magnetohydrodynamics. Each of these equations have open questions about the global existence and uniqueness of their solutions. These calming functions effectively reduce the algebraic degree of select nonlinear terms, thus one can verify global wellposedness for these "calmed systems." More specifically, in …

On Distortion Of Surface Groups In Right-Angled Artin Groups, Lucas Bridges

Mathematical Sciences Undergraduate Honors Theses

Surfaces have long been a topic of interest for scholars inside and outside of mathe- matics. In a topological sense, surfaces are spaces which appear flat on a local scale. Surfaces in this sense have a restricted set of properties, including the behavior of loops around a surface, codified in the fundamental group.

All but 3 surface groups have been shown to embed into a class of groups called right-angled Artin groups. The method through which these embeddings are created places large restrictions on all homomorphisms from surface groups to right-angled Artin groups.

One such restriction on these homomorphisms is …

Derived Jet Schemes And Arc Spaces, And Arithmetic Arc Space Representability, C. Eric Overton-Walker

Graduate Theses and Dissertations

Associated to a given scheme X one can define geometric and arithmetic notions of jet schemes and arc spaces. We develop a construction of the geometric jets and arcs in the setting of derived schemes and explore consequences thereof. In particular, we prove an analogous theorem to that of one by Tommaso de Fernex and Roi Docampo concerning the cotangent sheaves of geometric jets and arcs. Our version then produces many subsequent results which allow us to prove stronger versions of results concerning geometric jets and arcs by removing unnecessary hypotheses. Separately, we explore evidence as to why, in contrast …

Tasks For Learning Trigonometry, Sydnee Andreasen

All Graduate Reports and Creative Projects, Fall 2023 to Present

Many studies have been done using task-based learning within different mathematics courses. Within the field of trigonometry, task-based learning is lacking. The following research aimed to create engaging, mathematically rich tasks that meet the standards for the current trigonometry course at Utah State University and align with the State of Utah Core Standards for 7th through 12th grades. Four lessons were selected and developed based on the alignment of standards, the relevance to the remainder of the trigonometry course, and the relevance to courses beyond trigonometry. The four lessons that were chosen and developed were related to trigonometric ratios, graphing …

On The Existence Of Periodic Traveling-Wave Solutions To Certain Systems Of Nonlinear, Dispersive Wave Equations, Jacob Daniels

All Graduate Theses and Dissertations, Fall 2023 to Present

A variety of physical phenomena can be modeled by systems of nonlinear, dispersive wave equations. Such examples include the propagation of a wave through a canal, deep ocean waves with small amplitude and long wavelength, and even the propagation of long-crested waves on the surface of lakes. An important task in the study of water wave equations is to determine whether a solution exists. This thesis aims to determine whether there exists solutions that both travel at a constant speed and are periodic for several systems of water wave equations. The work done in this thesis contributes to the subfields …

Vectors And Vector Borne Disease: Models For The Spread Of Curly Top Disease And Culex Mosquito Abundance, Rachel M. (Frantz) Georges

All Graduate Theses and Dissertations, Fall 2023 to Present

Mathematical models are useful tools in managing infectious disease. When designed appropriately, these models can provide insight into disease incidence patterns and transmission rates. In this work, we present several models that provide information that is useful in monitoring diseases spread by insects.

In the first part of this dissertation, we present two models that predict disease incidence patterns for Curly Top disease (CT) in tomato crops. CT affects a wide variety of plants and is spread through the bite of the Beet Leafhopper. This disease is particularly devastating to tomato crops. When infected, tomato plants present with stunted growth …

A Comprehensive Uncertainty Quantification Methodology For Metrology Calibration And Method Comparison Problems Via Numeric Solutions To Maximum Likelihood Estimation And Parametric Bootstrapping, Aloka B. S. N. Dayarathne

All Graduate Theses and Dissertations, Fall 2023 to Present

In metrology, the science of measurements, straight line calibration models are frequently employed. These models help understand the instrumental response to an analyte, whose chemical constituents are unknown, and predict the analyte’s concentration in a sample. Techniques such as ordinary least squares and generalized least squares are commonly used to fit these calibration curves. However, these methods may yield biased estimates of slope and intercept when the calibrant, substance used to calibrate an analytical procedure with known chemical constituents (x-values), carries uncertainty. To address this, Ripley and Thompson (1987) proposed functional relationship estimation by maximum likelihood (FREML), which considers uncertainties …

Formalization Of A Security Framework Design For A Health Prescription Assistant In An Internet Of Things System, Thomas Rolando Mellema

Electronic Theses and Dissertations

Security system design flaws will create greater risks and repercussions as the systems being secured further integrate into our daily life. One such application example is incorporating the powerful potential of the concept of the Internet of Things (IoT) into software services engineered for improving the practices of monitoring and prescribing effective healthcare to patients. A study was performed in this application area in order to specify a security system design for a Health Prescription Assistant (HPA) that operated with medical IoT (mIoT) devices in a healthcare environment. Although the efficiency of this system was measured, little was presented to …

Ramanujan Type Congruences For Quotients Of Klein Forms, Timothy Huber, Nathaniel Mayes, Jeffery Opoku, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients …

Local Existence Of Solutions To A Nonlinear Autonomous Pde Model For Population Dynamics With Nonlocal Transport And Competition, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Highlights

• Partial differential equation models are ubiquitous in applied sciences.

• A partial differential equation based in ecology is studied for solution existence.

• Energy methods and convergence analysis lead to local classical solutions.

Abstract

In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to qualitatively describe temporal changes in population densities over space through accounting for location desirability and fast, long-range travel. Beginning with sufficiently regular initial conditions, through smoothing the PDE and employing energy arguments, we obtain a sequence …

Weighted Ehrhart Theory: Extending Stanley's Nonnegativity Theorem, Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, Josephine Yu

School of Mathematical & Statistical Sciences Faculty Publications

We generalize R. P. Stanley's celebrated theorem that the h∗-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h∗-polynomial as a real-valued function for a larger family of weights.

We generalize R. P. Stanley's celebrated theorem that the h ⁎ -polynomial of …

Inference For Time Series In Change Points And Statistical Learning, Jiaqi Li

Arts & Sciences Graduate Student Theses and Dissertations

This study aims to develop a comprehensive framework for inference theory in time series, with a focus on change-point detection and online learning. First, we introduce an $\ell^2$-based inference approach for multiple change-point detection in high-dimensional time series, which targets dense or spatially clustered signals by a novel Two-Way MOSUM (moving sum) test statistic. We derive the limiting distribution of this $\ell^2$-aggregated statistic by extending the high-dimensional Gaussian approximation to non-stationary spatial-temporal processes. Simulation exhibits promising performance of our test in detecting non-sparse weak signals, and the application to COVID-19 analysis shows the real-world relevance of our method. Next, we …

Art And Math Via Cubic Polynomials, Polynomiography And Modulus Visualization, Bahman Kalantari

LASER Journal

Throughout history, both quadratic and cubic polynomials have been rich sources for the discovery and development of deep mathematical properties, concepts, and algorithms. In this article, we explore both classical and modern findings concerning three key attributes of polynomials: roots, fixed points, and modulus. Not only do these concepts lead to fertile ground for exploring sophisticated mathematics and engaging educational tools, but they also serve as artistic activities. By utilizing innovative practices like polynomiography—visualizations associated with polynomial root finding methods—as well as visualizations based on polynomial modulus properties, we argue that individuals can unlock their creative potential. From crafting captivating …

Classification Of Topological Defects In Cosmological Models, Abigail Swanson

Departmental Honors & Graduate Capstone Projects

In nature, symmetries play an extremely significant role. Understanding the symmetries of a system can tell us important information and help us make predictions. However, these symmetries can break and form a new type of symmetry in the system. Most notably, this occurs when the system goes through a phase transition. Sometimes, a symmetry can break and produce a tear, known as a topological defect, in the system. These defects cannot be removed through a continuous transformation and can have major consequences on the system as a whole. It is helpful to know what type of defect is produced when …

An Investigation Into The Causes Of Home Field Advantage In Professional Soccer, Paige E. Tomer

Mathematics, Statistics, and Computer Science Honors Projects

Home-field advantage is the sporting phenomenon in which the home team outperforms the away team. Despite its widespread occurrence across sports, the underlying reasons for home-field advantage remain uncertain. In this paper, we employ a range of statistical methods to explore the causal relationships of potential determinants of home-field advantage. We measure home-field advantage using match outcomes and differential metrics (e.g., differences in yellow cards received). In an attempt to narrow the research disparity between men’s and women’s sports, we utilize data from the National Women’s Soccer League (NWSL) and the English Premier League (EPL) to investigate potential causes of …

A Discussion On Estimation Of The Best Constant For Spherical Restriction Inequalities, Hongyi Liu

Mathematics, Statistics, and Computer Science Honors Projects

The restriction conjecture asks for a meaningful restriction of the Fourier transform of a function to a sufficiently curved lower dimensional manifold. It then conjectures certain size estimates for this restriction in terms of the size of the original function. It has been proven in 2 dimensions, but it is open in dimensions 3 and larger, and is an area of much recent active effort. In our study, instead of aiming to prove the restriction conjecture, we target understanding its worst-case scenarios within known estimates. Specifically, we investigate the extension operator applied to antipodally concentrating profiles, examining the ratio of …

Representation Theory And Burnside's Theorem, Nathan Fronk

Senior Seminars and Capstones

In this paper we give a brief introduction to the representation theory of finite groups, and by extension character theory. These tools are extensions of group theory into linear algebra, that can then be applied back to group theory to prove propositions that are based entirely in group theory. We discuss the importance of simple groups and the Jordan-Hölder theorem in order to prepare for the statement of Burnside’s pq theorem. Lastly, we provide a proof of Burnside’s theorem that utilizes the character theory we covered earlier in the paper.

Largeness And Accessibility Of Sparse Sets, Oscar Quester

Honors Program Theses and Projects

One of the main goals in the study of Ramsey Theory is to find “order” in seemingly “random” structures. For example, Van der Waerden’s Theorem tells us that given any r-coloring of the positive integers, there will exist arbitrarily long monochromatic arithmetic progressions. The theorem places no requirement on the gap (common difference), d, of the arithmetic progression – it can be any natural number. With this in mind, we ask if we are still guaranteed arbitrarily long monochromatic arithmetic progressions when we restrict the possible values of d to some subset D ⊆ N. We also ask a similar …

A Tale Of Two Toroidal Graphs, Akshat Gulgulia

Honors Theses

A graph is toroidal if it can be embedded on a torus which is a doughnut-shaped surface. Two well-known examples of toroidal graphs are the complete graph K5 and the complete bipartite graph K3,3. In this thesis we elucidate the association of the subject matter with two renowned enigmas in graph theory, namely the Five Princes Problem and the Three Utilities Problems. Additionally, we look at their association with several renowned theorems in topological graph theory. We explore the link between these two graphs and a contemporary labeling concept.

Numerical Investigation To Produce A Fundamental Polygon, Elizabeth Sipes

Honors College Theses

There exist multiple types of geometry, differing in the postulates they are based on, and therefore the theorems and proofs that make up said geometry. Hyperbolic geometry differs from others by allowing there to exist multiple lines through a single point not on a given line, that are parallel to the given line. Every geometry has the idea of distance and isometries, distance preserving maps. By considering special collections of isometries called discrete groups, we can construct interesting surfaces, such as the torus and genus-g surface. The connection between the surface and the discrete group can be understood through …

Euler Archive Spotlight: Multiple Search Options, Christopher Goff

Euleriana

The Euler Archive houses PDF versions of almost all of Euler's original publications. While most visitors search the archive via a work's Eneström number, the Archive can be searched via source publication name, date written, or decade of publication. The Archive also provides context for Euler's publications through short pieces of historical information.

Euler And A Proof Of The Functional Equation For The Riemann Zeta-Function He Could Have Given, Alexander Aycock

Euleriana

We explain how Euler could have proved a functional equation, which is equivalent to the one for the Riemann zeta-function, that he conjectured in his paper {\it ``Remarques sur un beau rapport entre les series des puissances tant directes que reciproques"} \cite{E352} (E352: ``Remarks on the beautiful relation between the series of the direct and reciprocal powers").

Euler And The Gaussian Summation Formula For The Hypergeometric Series, Alexander Aycock

Euleriana

We show that in his paper {\it ``Plenior expositio serierum illarum memorabilium, quae ex unciis potestatum binomii formantur"} \cite{E663} (E663: ``A more thorough exposition of those memorable series that are formed from the binomial coefficients") Euler could have found the Gaussian summation formula for the hypergeometric series from his own formulas in that same paper, if he actually set the task for himself.

Euler And Homogeneous Difference Equations With Linear Coefficients, Alexander Aycock

Euleriana

We present a method outlined by Euler in his paper{\it ``De fractionibus continuis observationes"} \cite{E123} (E123: ``Observations on continued fractions") that can be used to solve homogeneous difference equations with linear coefficients. We will illustrate his ideas by applying it to two familiar examples and explain how it can be understood from a more modern point of view.

On The Cases In Which The Formula X^4+Kxxyy+Y^4 Can Be Reduced To A Square, Georg Ehlers

Euleriana

Euler’s key idea for equating the Quartic in the title to a square is to set k=P+surd(Q). From this he derives P=f·x^2 and Q=4f·y^2+4 and solves the Pell equation for y. He then discusses various extensions to rational numbers that leave k an integer. Euler provides incomplete tables for integers k with |k|square.

Research On Arithmetic, Erik R. Tou

Euleriana

In this English translation, some of Joseph-Louis Lagrange's early number theory is presented. Here, he laid out a theory of binary quadratic forms with special attention to the representation problem: determining those integers which may be represented by a given form, and cataloguing the possible forms of their divisors.

Number Theory And More, Christopher Goff, Erik Tou

Euleriana

An introduction to the contents in Issue 1, Volume 4 of Euleriana.

Finite State And Sequential Automata, Kriti Gulgulia

Honors Theses

No abstract provided.

Caterpillar, Lobster, X Graphs, Gerald Melin, Landon Seward, Will Mahowald, Xavier Jones

Celebrating Scholarship and Creativity Day (2018-)

We studied a combinatorial game played between two players ("Alpha", who goes first, and "Beta", who goes second). The idea is that there are a lot of lightbulbs in a large warehouse, and they take turns turning a light bulb on. When a light bulb is turned on, it illuminates the area directly by it as well as the areas immediately surrounding it. The player who is the one to make all of the warehouse illuminated is the winner. This can be modeled on a graph. The two players take turns (1) selecting a vertex that has not yet been …