Mathematics Commons^™

Review: Pythagoras’ Revenge: A Mathematical Mystery; The Housekeeper And The Professor, Susan Jane Colley

Faculty & Staff Scholarship

No abstract provided.

Bringing Toric Codes To The Next Dimension, Ivan Soprunov, Jenya Soprunova

Mathematics and Statistics Faculty Publications

This paper is concerned with the minimum distance computation for higher dimensional toric codes defined by lattice polytopes in $\mathbb{R}^n$. We show that the minimum distance is multiplicative with respect to taking the product of polytopes, and behaves in a simple way when one builds a k-dilate of a pyramid over a polytope. This allows us to construct a large class of examples of higher dimensional toric codes where we can compute the minimum distance explicitly.

Alternative Technical Efficiency Measures: Skew, Bias, And Scale, William Clinton Horrace, Qu Feng

Center for Policy Research

In the fixed-effects stochastic frontier model an efficiency measure relative to the best firm in the sample is universally employed. This paper considers a new measure relative to the worst firm in the sample. We find that estimates of this measure have smaller bias than those of the traditional measure when the sample consists of many firms near the efficient frontier. Moreover, a two-sided measure relative to both the best and the worst firms is proposed. Simulations suggest that the new measures may be preferred depending on the skewness of the inefficiency distribution and the scale of efficiency differences.

[Introduction To] Ordinary And Particial Differential Equations: An Introduction To Dynamical Systems, John W. Cain, Angela M. Reynolds

Bookshelf

Differential equations arise in a variety of contexts, some purely theoretical and some of practical interest. As you read this textbook, you will find that the qualitative and quantitative study of differential equations incorporates an elegant blend of linear algebra and advanced calculus. This book is intended for an advanced undergraduate course in differential equations. The reader should have already completed courses in linear algebra, multivariable calculus, and introductory differential equations.

Large Cardinals, Oliver Pechenik

Honors Papers

Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of …

The Fundamental Gap For Hyperbolic Triangles, Dennis Stuart Fillebrown

Honors Theses

In 1983, M. van den Berg made his Fundamental Gap Conjecture about the difference between the first two Dirichlet eigenvalues (the fundamental gap) of any convex domain in the Euclidean plane. Recently, progress has been made in the case where the domains are polygons and, in particular, triangles. We examine the conjecture for triangles in hyperbolic geometry, though we seek an for an upper bound for the fundamental gap rather than a lower bound.

Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, William T. Ross, Stephan Ramon Garcia, Warren R. Wogen

Department of Math & Statistics Faculty Publications

We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.

The Norm Of A Truncated Toeplitz Operator, William T. Ross, Stephan Ramon Garcia

Department of Math & Statistics Faculty Publications

We prove several lower bounds for the norm of a truncated Toeplitz operator and obtain a curious relationship between the H² and H^∞ norms of functions in model spaces.

Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger

CMC Senior Theses

Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.

Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, And Similarity, William T. Ross, Joseph A. Cima, Stephan Ramon Garcia, Warren R. Wogen

Department of Math & Statistics Faculty Publications

A truncated Toeplitz operator Aᵩ : K_Ɵ → K_Ɵ is the compression of a Toeplitz operator Tᵩ : H² → H² to a model space K_Ɵ := H² ⊖ ^ƟH². For Ɵ inner, let T_Ɵ denote the set of all bounded truncated Toeplitz operators on K_Ɵ. Our main result is a necessary and sufficient condition on inner functions Ɵ₁ and Ɵ₂ which guarantees that T_Ɵ1 and T_Ɵ2 are spatially isomorphic. (i.e., UT_Ɵ1 = T_Ɵ2 U for some unitary U : K_{Ɵ1 …}

Numerical Solution Of The Five-Moment Ideal Two-Fluid Equations In One Dimension, Marcus Scott

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Plasmas are frequently treated as a single conducting fluid and modeled using the equations of magnetohydrodynamics. However, this regime works better for low-frequency plasmas. High-frequency plasmas may be modeled using the principles of kinetic theory. For plasmas with frequencies between these two extremes, a two-fluid approach can yield better results. In 2006, Ammar Hakim mathematically modeled a plasma with a set of equations called the five-moment ideal two-fluid equations. An attempt is made reproduce those results. A derivation of this set of equations by taking moments of the Boltzmann equation is presented. Electric and magnetic fields contribute to the source …

A Comparison Of Prediction Methods Of Functional Autoregressive Time Series, Devin Didericksen

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Functional data analysis (FDA) is a relatively new branch of statistics that has seen a lot of expansion recently. With the advent of computer processing power and more efficient software packages we have entered the beginning stages of applying FDA methodology and techniques to data. Part of this undertaking should include an empirical assessment of the effectiveness of some of the tools of FDA, which are sound on theoretical grounds. In a small way, this project helps advance this objective.

This work begins by introducing FDA, scalar prediction techniques, and the functional autoregressive model of order one - FAR(1). Two …

Assessing The Precision And Accuracy In A Small Sample Of Actical Devices, Peter Sherick

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Actigraphy is an increasingly popular approach in medicine to assess patient activity levels in a variety of scenarios. The devices are essentially accelerometers encased in a write-watch type assembly. This project sought to determine the device precision and accuracy for the Actical model. In a sample of four Acticals, it was found that intra-device variability was minimal. However, one device was found to be statistically biased in comparison to the other three. This bias could have adverse effects on aggregated or magnitude dependent data analysis. Also, inter-device comparisons may be problematic.

Assessment Of Utah Bankruptcies By Census Tracts: A Spatial Statistical Approach, Kenneth Pena

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

There are two questions raised when looking at the spatial pattern of the rate of bankruptcies in Utah: (i) are there similarities between the bankruptcy data in adjacent census tracts and (ii) can local cluster and outliers be identified within the data? Specifically, are there similar rates of bankruptcies in bordering census tracts and are there any localized areas of interest where we find extremely high or extremely low rates of bankruptcies? This study uses spatial statistics to perform tests for spatial autocorrelation to address these two questions. It also looks at commonalities in the clusters and differences in the …

Simulating Power For One-Way Anova By Using Non-Normal Error Distributions, Caixia Xu

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Usually we assume that the distribution of the additive errors in a one-way ANOVA linear model is normal. However, exceptions to this assumption about the error distribution may exist. In such cases, we might consider non-normal error distributions, but proceed with the "usual" ANOVA F-test analyses. This study focuses on simulating power for one-way ANOVA when using non-normal error distributions.

Character Degrees Of Normally Monomial Maximal Class 5-Groups, Michael Slattery

Mathematics, Statistics and Computer Science Faculty Research and Publications

This paper will impose limits on the possible sets of irreducible character degrees of a normally monomial 5-group of maximal class.

Developing Preservice Teachers’ Mathematical And Pedagogical Knowledge Using An Integrated Approach, Marta Magiera, Leigh A. Van Den Kieboom

Mathematics, Statistics and Computer Science Faculty Research and Publications

This paper describes how an integrated mathematics content and early field-experience course provides opportunities for preservice elementary teachers to develop understanding of mathematics and mathematics teaching. Engaging preservice teachers in solving and discussing mathematical tasks and providing opportunities to implement these tasks with elementary students creates an authentic context for the future teachers to reflect on their own understanding of mathematics, mathematics teaching, and students’ mathematical thinking. Essential elements of the cycle of events in the integrated model of instruction are discussed: preservice students’ acquisition of mathematical concepts in the context of selected tasks in the content course; subsequent posing …

Gliomapredict: A Clinically Useful Tool For Assigning Glioma Patients To Specific Molecular Subtypes, Aiguo Li, Serdar Bozdag, Yuri Kotliarov, Howard A. Fine

Mathematics, Statistics and Computer Science Faculty Research and Publications

Background: Advances in generating genome-wide gene expression data have accelerated the development of molecular-based tumor classification systems. Tools that allow the translation of such molecular classification schemas from research into clinical applications are still missing in the emerging era of personalized medicine.

Results: We developed GliomaPredict as a computational tool that allows the fast and reliable classification of glioma patients into one of six previously published stratified subtypes based on sets of extensively validated classifiers derived from hundreds of glioma transcriptomic profiles. Our tool utilizes a principle component analysis (PCA)-based approach to generate a visual representation of the analyses, quantifies …

Inverse Limits With Upper Semi-Continuous Set Valued Bonding Functions: An Example, Christopher David Jacobsen

Masters Theses

"While there is a wealth of information pertaining to inverse limits with single valued bonding maps, comparatively little is known about inverse limits with upper semi-continuous set valued bonding functions. In order to add somewhat to the communal knowledge on the subject, this paper provides an example of an inverse limit with a single upper semi-continuous set valued bonding function. It is then shown that the space is a continuum, and its structure is examined via its arc components and through various of its properties, such as dimension and decomposability"--Abstract, page iii.

On Orders And Types Of Dirichlet Series Of Slow Growth, Yinying Kong, Huilin Gan

Turkish Journal of Mathematics

The present paper has the object of showing some interesting relationship on the maximum modulus, the maximum term, the index of maximum term and the coefficients of entire functions defined by Dirichlet series of slow growth; some properties like Taylor entire functions are obtained.