Mathematics Commons^™

Simulation Study Of Estimation And Inference In Factor Analysis: Normal And Non-Normal Noise Distributions, Ping Zhang

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

Objective: To study the estimation and inference m factor analyses when the data have normal or non-normal noise distributions.

Methods: Population data were created in package R with a specified number of factors, factor structure and observable variables with known loadings. Then, repeated simple random samples (SRS's) were taken from the population, independently. The maximum likelihood method with varimax rotation was used to perform factor analysis and inference on each sampled dataset. Factor loadings were estimated to determine if the estimation of the loadings was (approximately) unbiased and/or efficient for each specified population and chi-square x2-statistics were obtained to test …

Exterior Differential Systems With Symmetry, Ian M. Anderson, Mark E. Fels

Mathematics and Statistics Faculty Publications

We use the theory of reduction of exterior differential systems with symmetry to study the problem of using a symmetry group of a differential equation to find noninvariant solutions.

Coffee To Go! Modeling Thermoclines In Multivariable Calculus, Andrea Bruder, Brynja R. Kohler

Mathematics and Statistics Faculty Publications

While mathematical modeling is an integral process in applied mathematics, students rarely encounter genuine modeling opportunities in their calculus courses. Here we introduce a laboratory experience as a natural starting point for calculus students to investigate multivariable functions. A layered system of coffee and milk serves as a physical model for temperature gradients in lakes or the atmosphere, where temperature depends on both a temporal and spatial variable. Students create, observe, and collect temperature data of their own, graph the data, and develop mathematical models to fit the data. We require students to write a report about their findings. This …

The Robustness Of Factor Analyses When The Data Does Not Conform To Standard Parametric Requirements, Haisong Peng

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

Objective: To access the robustness of factor analyses when the data does not conform to standard parametric requirements.

Methods: Data were simulated in package R. Maximum likelihood was used to fit and assess the factor models. Chi-square statistics were obtained to test hypotheses about the correct number of factors in simulated settings where the true number of factors was known. The number of true factors varied between 1 and 3; the number of observed variables was either 6 (for 1 factor) or 3 per factor for 2 or more factors.

Results: With standard normal factor populations, and normal errors added …

Semilinear Elliptic Equations In Unbounded Domains, Francois A. Van Heerden

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We studied some semilinear elliptic equations on the entire space R^N. Our approach was variational, and the major obstacle was the breakdown in compactness due to the unboundedness of the domain. First, we considered an asymptotically linear Scltrodinger equation under the presence of a steep potential well. Using Lusternik-Schnirelmann theory, we obtained multiple solutions depending on the interplay between the linear, and nonlinear parts. We also exploited the nodal structure of the solutions. For periodic potentials, we constructed infinitely many homoclinic-type multibump solutions. This recovers the analogues result for the superlinear case. Finally, we introduced weights on the linear and …

Four-Dimensional Non-Reductive Homogeneous Manifolds With Neutral Metrics, Andrew Renner

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A method due to É. Cartan was used to algebraically classify the possible four-dimensional manifolds that allow a (2, 2)-signature metric with a transitive group action which acts by isometries. These manifolds are classified according to the Lie algebra of the group action. There are six possibilities: four non-parameterized Lie algebras, one discretely parameterized family, and one family parameterized by R.

Lorentz Homogeneous Spaces And The Petrov Classification, Adam Bowers

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A. Z. Petrov gave a complete list of all local group actions on a four-dimensional space-time that admit an invariant Lorentz metric, up to an equivalence relation. His list was compiled by directly constructing all possible Lie algebras of infinitesimal generators of group actions that preserve a Lorentz metric. The goal of this paper was to verify that classification by algebraically constructing a list of all possible three-dimensional homogeneous spaces and calculating which among them have a non-degenerate invariant metric.

Technology And Mathematics Standards: An Integrated Approach, Chris Merrill, Mark Comerford

Publications

The article focuses on the use of standards-based teaching and learning that has been gaining significant attention in the education world. State and national associations now base their specific subject area or discipline solely on standards, i.e., International Technology Education Association (ITEA), National Council of Teachers of Mathematics (NCTM), National Science Education Association (NSEA). Moreover, at the public school level, state boards of education are holding school districts accountable for teaching standards-based curricula. It is with the latter definition in mind that the authors created a standards-based, integrated technology and mathematics lesson using the design and construction of stair systems.

Predicting Mountain Pine Beetle Development With The Extended Von Foerster Model, Jeffrey Tullis Leek

Undergraduate Honors Capstone Projects

The mountain pine beetle (Dendroctonus ponderosae Hopkins) represents a significant threat to ponderosa pine and lodgepole pine stands in the western United States, and has the potential to threaten commercially valuable jack pine in both the United States and Canada. The success of the mountain pine beetle is based on synchronization of developmental events to time cold-hardened life stages for extreme winter temperatures and to facilitate mass attack and overwhelm the defenses of the host. This paper presents a solution methodology for an extended McKendrick - von Foerster model for the development of the mountain pine beetle in varying …

Computer Simulation And Homogenization In Heating Design Optimization, Daniel K. Balls

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

The ability to ensure uniformity of temperature within a given finite physical region is an essential element in the success of many scientific processes, especially those that involve extreme fluctuation in temperature. Such a process is performed in an instrument called the LightTyper developed by Idaho Technology, Inc. of Salt Lake City Utah. This paper details the development and results of a scheme intended to obtain a heating design that ensures a high degree of temperature uniformity within the Idaho Technology instrument. Due to the experiments performed during this project, we were able to answer many questions that concerned finding …

Identification Of Optimal Conditions For Dry Drilling (Analytical Approach To Prediction Of The Occurrence Of Bue), Prasad Gali

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Lubrication is used during the drilling of aluminum to counter the formation of a built-up-edge (BUE), among other reasons. The elimination of the use of lubricants in drilling of aluminum is important because of the associated high costs of cleaning and disassembly involved in lubrication. The optimal conditions sought in this work include the elimination of the use of lubricants along with the possible attainment of a high material removal rate, which could help in reduction of cost and increase productivity at the same time. BUE has been found to be almost always present in the process of metal cutting …

Bootstrap Unit Root Tests For Heavy-Tailed Observations, Andrejus Parfionovas

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We explore the application of the bootstrap unit root test to time series with heavy-tailed errors. The size and power of the tests are estimated for two different autoregressive models (AR(1)) using computer simulated data. Real-data examples are also presented. Two different bootstrap methods and the subsampling approach are compared. Conclusions on the optimal bootstrap parameters, the range of applicability, and the performance of the tests are made.

Finite Element Studies In Metal Cutting, Suhail Ahmed

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

AdvantEdge is a finite element software package that integrates advanced dynamics, thermo-mechanically coupled finite element numerics and material modelling appropriate for machining processes. AdvantEdge allows users to specify the workpiece material, tool geometry and cutting conditions. It then provides accurate estimates of thermo-mechanical properties of machining processes such as cutting forces, chip morphology, machined surface residual stresses and temperature behavior of the tool and the workpiece. We will use AdvantEclge to investigate two areas of interest in metal cutting: process clamping via crushing of workpiece material and drilling of metal stacks.

Test Suite For Multiobjective Optimization And Results Using Normal Boundary Intersection (Nbi) In Design Explorer, Siva Kumar Natarajan

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

Several methods have been developed to solve multiobjective optimization problems (MOP's). One of these, Normal Boundary Intersection (NBI), is a method developed by John Dennis and Indraneel Das. NBI is used at The Boeing Company as a tool to solve MOP's. This report presents a test suite of MOP's that I developed for Boeing during me internship in summer 2003.

The problems in the test suite were chosen to represent the different types of multiobjective optimization problems that could arise in practice and the complexities involved in solving them. These problems range from those that have nice convex Pareto surfaces …

Optimal Path Planning And The Fast Marching Method, J. J. Clark

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

The problem of determining an optimal path for an object moving through some obstacle space presents several nontrivial subproblems. The foremost being the computational complexity that is involved and how to best deal with the associated large data volume. For example, a non-symmetric object moving in three dimensions possesses six degrees of freedom. This can lead to a computational grid that may easily be on the order of 10¹². Furthermore, for every point in the computational domain, several complex calculations must be performed. These include performing tests to determine if the object and obstacles intersect, and numerically solving …

Transverse Group Actions On Bundles, Ian M. Anderson, Mark E. Fels

Mathematics and Statistics Faculty Publications

An action of a Lie group G on a bundle is said to be transverse if it is projectable and if the orbits of G on E are diffeomorphic under π to the orbits of G on M. Transverse group actions on bundles are completely classified in terms of the pullback bundle construction for G-invariant maps. This classification result is used to give a full characterization of the G invariant sections of E for projectable group actions.

The Classification Of Low Dimensional Nilpotent Lie Algebras, Kimberli C. Tripp

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

Nilpotent Lie algebras are the fundamental building blocks for generic (not semi-simple) Lie algebras. In particular, the classification of nilpotent algebras is the first step in classifying and identifying solvable Lie Algebras. The problem of classifying nilpotent Lie algebras was first studied by Umlauf [9] in 1891. More recently, classifications have been given up to dimension six using different techniques by Morosov (1958) [7], Skjelbred and Sund (1977) [8], and up to dimension five by Dixmier (1958) [2]. Using Morosov's method of classification by maximal abelian ideals, Winternitz reproduced the Morosov classification obtaining different canonical forms for the algebras. The …

A Recommendation For Determining The Efficacy Of Weight Removal Estimates For The Pacific Cod Longline Cdq Fishery, Anna L. Furniss

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

In January 2000, the Alaska Department of Community and Economic Development contacted the National Marine Fisheries Service (NMFS) regarding concerns over the methods used to determine catch estimates for the Pacific Cod Community Development Quota (CDQ) fishery. Currently, NMFS determines catch estimates for the Pacific Cod CDQ fishery based on the data collected by observers from the North Pacific Groundfish Observer Program (NPGOP).

Observer estimates for catch are based on the random sampling methods for a longline fishing vessel as described in the North Pacific Groundfish Observer Manual. These sampling methods provide an official total catch (OTC) estimate for each …

Generalized Minimum Penalized Hellinger Distance Estimation And Generalized Penalized Hellinger Deviance Testing For Generalized Linear Models: The Discrete Case, Huey Yan

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In this dissertation, robust and efficient alternatives to quasi-likelihood estimation and likelihood ratio tests are developed for discrete generalized linear models. The estimation method considered is a penalized minimum Hellinger distance procedure that generalizes a procedure developed by Harris and Basu for estimating parameters of a single discrete probability distribution from a random sample. A bootstrap algorithm is proposed to select the weight of the penalty term. Simulations are carried out to compare the new estimators with quasi-likelihood estimation. The robustness of the estimation procedure is demonstrated by simulation work and by Hapel's α-influence curve. Penalized minimum Hellinger deviance tests …

The Modern Mathematics Classroom: A Collection Of Virtual Manipulatives For Teachers And Students, David Stowell

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

An important assumption in the implementation of the National Council of Teachers of Mathematics (NCTM) Standards 2000 is that the mathematics classroom is a place where students are actively involved in the learning process. One way to foster such a learning environment is by using manipulatives. By their nature, manipulatives make the learning of mathematics a discovery-based activity. As computer use increases in the classroom, virtual manipulatives will become more important as instructional tools. Virtual manipulatives offer advantages over their traditional versions. Most important is their dynamic nature. Their dynamic capabilities provide two main benefits. First, the number of potential …

Analysis Of A Non-Replicated Split-Split Plot Experiment, Emily Simmons Sim

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

A major obstacle in the analysis of experimental data, in many situations, is the lack of "true" or "complete" replication. In some disciplines, researchers are very aware of the importance of replication and the methods for correctly replicating an experiment. In other subject areas, however, researchers are less aware of what it means to properly replicate an experiment. Due to this lack of awareness, many non-replicated experiments are carried out every year. For many of these non-replicated experiments, there is no satisfactory statistical analysis.

The subject of this report is the analysis of two non-replicated experiments in environmental engineering. First, …

A New Perspective On Classification, Guohua Zhao

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The idea of voting multiple decision rules was introduced in to statistics by Breiman. He used bootstrap samples to build different decision rules, and then aggregated them by majority voting (bagging). In regression, bagging gives improved predictors by reducing the variance (random variation), while keeping the bias (systematic error) the same. Breiman introduced the idea of bias and variance for classification to explain how bagging works. However, Friedman showed that for the two-class situation, bias and variance influence the classification error in a very different way than they do in the regression case.

In the first part of …

Pooled And Individual Bycatch Quotas: Exploring Tradeoffs Between Observer Coverage Levels, Bycatch Frequency, Pool Size, And The Precision Of Bycatch Estimates, Landon S. Jensen

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

The North Pacific Ocean is highly productive, hosting many of the world's largest groundfish populations and supporting a thriving fishing industry. Numerous regulations have been implemented to control the incidental take of non-target bycatch. Individual and Pooled Bycatch Quotas have recently been proposed as instruments that could further encourage the avoidance of such bycatch and increase enforceability of bycatch caps at less-than-entire-fishery levels of operation. The recent advent of fishing cooperatives such as the Pacific Whiting Conservation Cooperative and the Pollock Conservation Cooperatives create an additional impetus for examining the characteristics of pool and vessel specific bycatch quotas.

We have …

Mean-Square Error Bounds And Perfect Sampling For Conditional Coding, Xiangchen Cui

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In this dissertation, new theoretical results are obtained for bounding convergence and mean-square error in conditional coding. Further new statistical methods for the practical application of conditional coding are developed.

Criteria for the uniform convergence are first examined. Conditional coding Markov chains are aperiodic, π-irreducible, and Harris recurrent. By applying the general theories of uniform ergodicity of Markov chains on general state space, one can conclude that conditional coding Markov chains are uniformly ergodic and further, theoretical convergence rates based on Doeblin's condition can be found.

Conditional coding Markov chains can be also viewed as having finite state space. …

Comparing Nonlinear And Nonparametric Modeling Techniques For Mapping And Stratification In Forest Inventories Of The Interior Western Usa, Gretchen Gengenbach Moisen

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Recent emphasis has been placed on merging regional forest inventory data with satellite-based information both to improve the efficiency of estimates of population totals, and to produce regional maps of forest variables. There are numerous ways in which forest class and structure variables may be modeled as functions of remotely sensed variables, yet surprisingly little work has been directed at surveying modem statistical techniques to determine which tools are best suited to the tasks given multiple objectives and logistical constraints. Here, a series of analyses to compare nonlinear and nonparametric modeling techniques for mapping a variety of forest variables, and …

Positive Solutions Obtained As Local Minima Via Symmetries, For Nonlinear Elliptic Equations, Florin Catrina

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In this dissertation, we establish existence and multiplicity of positive solutions for semilinear elliptic equations with subcritical and critical nonlinearities. We treat problems invariant under subgroups of the orthogonal group. Roughly speaking, we prove that if enough "mass " is concentrated around special orbits, then among the functions with prescribed symmetry, there is a solution for the original problem.

Our results can be regarded as a further development of the work of Z.-Q. Wang, where existence of local minima in the space of symmetric functions was studied for the Schrödinger equation. We illustrate the general theory with three examples, all …

Low Seasonal Temperatures Promote Life Cycle Synchronization, Janette Lee Jenkins

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

In this paper, we discuss how seasonal temperature variation and dormancy can synchronize the development of exothermic organisms. Using a simple aging model, it is shown that minimal seasonal temperature variation and periods of dormancy during extreme temperature conditions are sufficient to establish stable, univoltine ovipositional cycles. Dormancy, in fact, promotes synchronous oviposition emergence. The mountain pine beetle, an important insect living in extreme temperature conditions and showing no evidence of diapause, invites direct application of this model. Simulations using mountain pine beetle parameters are used to determine temperature regimes for which stable, ovipositional cycles exist.

A Survey Of The Taniyama-Shimura Conjecture, Kady Schneiter

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

Perhaps the most famous problem in all of mathematics is the theorem that states that the equation aⁿ + bⁿ = cⁿ has no non-trivial solutions for integers a, b, and c, and n ≥ 2. This theorem was proposed by a seventeenth century French mathematician named Pierre de Fermat. Though the theorem is easy to understand, the proof has been elusive. Over the past 350 years many mathematicians have attempted to prove Fermat's theorem. They have used a variety of methods and many have been successful in proving the theorem in specific cases. …

Vessiot: A Maple Package For Varational And Tensor Calculus In Multiple Coordinate Frames, Charles E. Miller

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The Maple V package Vessiot is an extensive set of procedures for performing computations in variational and tensor calculus. Vessiot is an extension of a previous package, Helmholtz, which was written by Cinnamon Hillyard for performing operations in the calculus of variations. The original set of commands included standard operators on differential forms, Euler-Lagrange operators, the Lie bracket operator, Lie derivatives, and homotopy operators. These capabilities are preserved in Vessiot, and enhanced so as to function in a multiple coordinate frame context. In addition, a substantial number of general tensor operations have been added to the package. These include standard …

Construction And Analysis Of A Family Of Numerical Methods For Hyperbolic Conservation Laws With Stiff Source Terms, Cinnamon Hillyard

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Numerical schemes for the partial differential equations used to characterize stiffly forced conservation laws are constructed and analyzed. Partial differential equations of this form are found in many physical applications including modeling gas dynamics, fluid flow, and combustion. Many difficulties arise when trying to approximate solutions to stiffly forced conservation laws numerically. Some of these numerical difficulties are investigated.

A new class of numerical schemes is developed to overcome some of these problems. The numerical schemes are constructed using an infinite sequence of conservation laws.

Restrictions are given on the schemes that guarantee they maintain a uniform bound and satisfy …