Applied Mathematics Commons^™

Cyclic Universe With An Inflationary Phase From A Cosmological Model With Real Gas Quintessence, Rossen Ivanov, Emil Prodanov

Articles

Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction uid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing in ationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.

How To Define Relative Approximation Error Of An Interval Estimate: A Proposal, Vladik Kreinovich

Departmental Technical Reports (CS)

The traditional definition of a relative approximation error of an estimate X as the ratio |X - x|/|x| does not work when the actual value x is 0. To avoid this problem, we propose a new definition |X - x|/|X|. We show how this definition can be naturally extended to the case when instead of a numerical estimate X, we have an interval estimate [x], i.e., an interval that is guaranteed to contain the actual (unknown) value x.

A Unified Approach To Generalized Stirling Functions, Tian-Xiao He

Scholarship

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in \cite{He11}. Previous well-known Stirling functions introduced by Butzer and Hauss \cite{BH93}, Butzer, Kilbas, and Trujilloet \cite{BKT03} and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in \cite{HS98} to the defined Stirling functions.

Quantifying Performance Bias In Label Fusion, Alexander M. Venzin

Theses and Dissertations

Classification systems are employed to remotely assess whether an element of interest falls into a target class or non-target class. These systems have uses in fields as far ranging as biostatistics to search engine keyword analysis. The performance of the system is often summarized as a trade-off between the proportions of elements correctly labeled as target plotted against the number of elements incorrectly labeled as target. These are empirical estimates of the true positive and false positive rates. These rates are often plotted to create a receiver operating characteristic (ROC) curve that acts as a visual tool to assess classification …

The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson

Research Vignettes

The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

270: How To Win The Presidency With Just 17.56% Of The Popular Vote, Charles D. Wessell

Math Faculty Publications

With the U.S. presidential election fast approaching we will often be reminded that the candidate who receives the most votes is not necessarily elected president. Instead, the winning candidate must receive a majority of the 538 electoral votes awarded by the 50 states and the District of Columbia. Someone with a curious mathematical mind might then wonder: What is the small fraction of the popular vote a candidate can receive and still be elected president? [excerpt]

Higher Order Symmetries Of The Kdv Equation, Ian M. Anderson

Research Vignettes

In this worksheet we symbolically construct the formal inverse of the total derivative operator and use it to construct the recursion operator for the higher-order symmetries of the KdV equation. Using this recursion operator we generate the first 5 generalized symmetries of the KdV equation and verify that they all commute.

PDF and Maple worksheets can be downloaded from the links below.

An Analysis Of The Career Length Of Professional Basketball Players, Kwame D. Fynn, Morgan Sonnenschein

The Macalester Review

An interesting problem in professional basketball is predicting how long a player remains in the NBA League. Previous research on this problem has focused on factors such as race, performance in games, and size. We propose to analyze career duration in the NBA based on awards won, position played and biological variables such as height. Using Accelerated Failure Time models, Cox Proportional Hazards models and Kaplan-Meier analyses, we determine that both height and number of awards won lengthen career duration; however, only certain player positions significantly affect career length of a player.

Nasa Flight Opportunities Program (Fop) Platform Tradeoffs Analysis, Stephanie Kugler, Dougal Maclise

STAR Program Research Presentations

The Flight Opportunities Program (FOP) exemplifies NASA’s shift in policy from a public driven space industry towards an emphasis on public-private partnerships. The Payloads team, as part of FOP, is responsible for soliciting, selecting and shepherding payloads that require flight testing in order to mature technologies, not only to reduce risk in a deep space or manned space missions, but also to develop critical technologies with multiple applications in space. Several companies have been awarded contracts to provide these flight opportunities and each have unique capabilities to fly payloads in environments that closely imitate the environment of space missions. As …

Local Polynomial Regression With Application To Sea Surface Temperatures, Michael Finney

All Theses

Our problem involves methods for determining the times of a maximum or minimum for a general mean function in time series data. The methods explored here involve polynomial smoothing. In theory, the methods calculate a general number of derivatives of the estimated polynomial. Using these techniques, we wish to find a balance between error, variance, and complexity and apply it to a time series of sea surface temperatures. We will first explore the theory behind the method and then find a way to optimally apply it to our data.

Robust Parameter Estimation In The Weibull And The Birnbaum-Saunders Distribution, Jing Zhao

All Theses

This paper concerns robust parameter estimation of the two-parameter Weibull distribution and the two-parameter Birnbaum-Saunders distribution. We use the proposed method to estimate the distribution parameters from (i) complete samples with and without contamination (ii) type-II censoring samples, in both distributions. Also, we consider the maximum likelihood estimation and graphical methods to compare the maximum likelihood estimation and graphical method with the proposed method based on quantile. We find the advantages and disadvantages for those three different methods.

Modeling The Curvature Of A Ferrofluid Interface Using A Height Function Method, Holly Timme

Theses, Dissertations and Culminating Projects

The behavior of an interface embedded in a fluid is central to a wide range of biological, chemical, environmental and physical problems and engineering processes. Modeling the evolution of a fluid interface is thus a critical and important problem. In many instances, including two-phase (e.g. liquid-gas) flows, the interface is an internal boundary within a PDE model. A model of the interface properties and its evolution is then typically performed by numerical computation, within the framework of the PDE solution method, such as finite differences (FD). Volume of Fluid (VOF) is a simple FD based method which exhibits excellent volume …

Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager

Department of Mathematics: Dissertations, Theses, and Student Research

Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to …

Degree Constrained Triangulation, Roshan Gyawali

UNLV Theses, Dissertations, Professional Papers, and Capstones

Triangulation of simple polygons or sets of points in two dimensions is a widely investigated problem in computational geometry. Some researchers have considered variations of triangulation problems that include minimum weight triangulation, de-launay triangulation and triangulation refinement. In this thesis we consider a constrained version of the triangulation problem that asks for triangulating a given domain (polygon or point sites) so that the resulting triangulation has an increased number of even degree vertices. This problem is called Degree Constrained Triangulation (DCT). We propose four algorithms to solve DCT problems. We also present experimental results based on the implementation of the …

A Set Of Tournaments With Many Hamiltonian Cycles, Hayato Ushijima-Mwesigwa

All Theses

For a random tournament on $3^n$ vertices, the expected number of Hamiltonian cycles is known to be $(3^n -1)!/2^{3^n}$. Let $T_1$ denote a tournament of three vertices $ {v_1, v_2, v_3}$. Let the orientation be such that there are directed edges from $v_1 $to $v_2$ , from $v_2$ to $v_3$ and from $v_3$ to $ v_1$. Construct a tournament $T_i$ by making three copies of $T_{i-1}$, $T_{i-1}'$, $T_{i-1}''$ and $T_{i-1}'''$. Let each vertex in $T_{i-1}'$ have directed edges to all vertices in $T_{i-1}''$, similarly place directed edges from each vertex in $T_{i-1}''$ to all vertices in $T_{i-1}'''$ and from $T_{i-1}'''$ …

Branching Rules For Minimum Congestion Multi-Commodity Flow Problems, Cameron Megaw

All Theses

In this paper, we examine various branch and bound algorithms for a minimum congestion origin-destination integer multi-commodity flow problem.
The problem consists of finding a routing such that the congestion of the most congested arc is minimum. For our implementation, we assume that all demands are known a priori.
We provide a mixed integer linear programming formulation of our problem and propose various new branching rules to solve the model. For each rule, we provide theoretical and experimental proof of their effectiveness.
In order to solve large instances, that more accurately portray real-world applications, we outline a path formulation model …

Physicic-Based Algorithms And Divergence Free Finite Elements For Coupled Flow Problems, Nicholas Wilson

All Dissertations

This thesis studies novel physics-based methods for
simulating incompressible fluid flow described by the Navier-Stokes equations (NSE) and
magnetohydrodynamics equations (MHD).
It is widely accepted in computational fluid dynamics (CFD) that numerical schemes which are more
physically accurate lead to more precise flow simulations especially over long time intervals.
A prevalent theme throughout will be the inclusion of as much
physical fidelity in numerical solutions as efficiently possible. In algorithm design, model
selection/development, and element choice, subtle changes can provide better physical accuracy,
which in turn provides better overall accuracy (in any measure). To this end we develop and study …

Sensitivity Anaylsis And Detectability For Magnetic Resonance Elastography, Catherine White

All Dissertations

This thesis is for a sensitivity analysis of magnetic resonance elastography, a hybrid imaging technique used in early-stage cancer screening. To quantitatively analyze the sensitivity, we introduce a notion of detectability, which is dened as a relative amplitude
drop in a small sti tumor region. This analysis is accomplished in both the full elastic and viscoelastic models and compared with that of the simpler scalar model which is frequently used in the actual application.
Some of the highlights are 1) a useful formula for detectability in terms of physical parameters, which will help the design of experiments; 2) the discrepancy …

Sparsity Regularization In Diffuse Optical Tomography, John Cooper

All Dissertations

The purpose of this dissertation is to improve image reconstruction in Diffuse Optical Tomography (DOT), a high contrast imaging modality that uses a near infrared light source. Because the scattering and absorption of a tumor varies significantly from healthy tissue, a reconstructed spatial representation of these parameters serves as tomographic image of a medium. However, the high scatter and absorption of the optical source also causes the inverse problem to be severely ill posed, and currently only low resolution reconstructions are possible, particularly when using an unmodulated direct current (DC) source.
In this work, the well posedness of the forward …

Determining Properties Of Metal By Analyzing Changes In Impedance, Chase Mathison, Laura Booton

Mathematical Sciences Technical Reports (MSTR)

In certain situations it is useful to identify an unknown sample of metal without contact or visual inspection. We wish to do this by inducing a current in a coil and placing the sample in the resulting magnetic field. For the special case in which the sample is an infinite slab, we have a model that gives the change in impedance of the coil based on the properties of the sample. In this paper we analyze the inverse problem of finding the metal properties from impedance measurements over a wide range of frequencies.

Robust Analysis Of Metabolic Pathways, Emily Gruber, Amy Ko, Michael Macgillvray, Miranda Sawyer

Mathematical Sciences Technical Reports (MSTR)

Flux Balance Analysis (FBA) is a widely used computational model for studying the metabolic pathways of cells and the role individual metabolites and reactions play in maintaining cell function. However, the successes of FBA have been limited by faulty biological assumptions and computational imperfections. We introduce Robust Analysis of Metabolic Pathways (RAMP) to provide a more theoretically sound and computationally accurate model of cellular metabolism. RAMP overcomes the faulty assumptions of traditional FBA by allowing deviation from steady-state and accounting for variability across a cellular culture. Computationally, RAMP more successfully predicts the lethality of gene knockouts and reduces degeneracy in …

Quantitative Reasoning And Sustainability, Corrine H. Taylor

Numeracy

Quantitative Reasoning and Sustainability have much in common. Both are complex, nuanced concepts with rather long definitions that have evolved over time. Both subjects are “everybody’s business” on college campuses, and must be approached in courses across the curriculum, not merely in one course on QR or in one course on Sustainability. The growing, wider presence of both QR and Sustainability on college campuses is due to their applicability in individuals’ personal, professional, and public lives. Moreover, QR and Sustainability support and enhance each other in and out of the classroom. Sustainability is an important, authentic, relevant context for lessons …

Parameter Estimation And Uncertainty Quantication For An Epidemic Model, Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd

Mathematics and Computer Science Faculty Publications

We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number (R0 )—an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of R0. Situations are highlighted in which this correlation allows R0 to be estimated with greater ease than its constituent parameters. Implications …

Parameter Estimation And Uncertainty Quantication For An Epidemic Model, Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun Lloyd

Mathematics and Statistics Faculty Publications

We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number (R0 )—an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of R0. Situations are highlighted in which this correlation allows R0 to be estimated with greater ease than its constituent parameters. Implications …

Half-Life Learning Curves In The Defense Acquisition Life Cycle, Adedeji B. Badiru

Faculty Publications

Learning curves are useful for assessing performance improvement due to the positive impact of learning. In recent years, the deleterious effects of forgetting have also been recognized. Workers experience forgetting or decline in performance over time. Consequently, contemporary learning curves have attempted to incorporate forgetting components into learning curves. An area of increasing interest is the study of how fast and how far the forgetting impact can influence overall performance. This article introduces the concept of half-life analysis of learning curves using the concept of growth and decay, with particular emphasis on applications in the defense acquisition process. The computational …

Numerical Analysis For Interface Problems, Lin Mu

Theses and Dissertations

A large number of partial differential equations in geoscience and engineering can be described as interface problems. Three interface models will be considered in this dissertation. They are elliptic interface problems, Stokes interface problems, and Brinkman problems. The purpose of the dissertation is to develop and analyze robust finite element methods and finite volume methods for solving these problems. Firstly, we report the a posteriori error estimate for interface problems by conforming finite volume methods, nonconforming finite volume methods, and discontinuous finite volume methods. Recovery-based and residual-based a posteriori error estimators are constructed by using computable quantities. Secondly, we develop …

Adaptive Randomization Designs, Jenna Colavincenzo

Statistics

Adaptive design methodologies use prior information to develop a clinical trial design. The goal of an adaptive design is to maintain the integrity and validity of the study while giving the researcher flexibility in identifying the optimal treatment. An example of an adaptive design can be seen in a basic pharmaceutical trial. There are three phases of the overall trial to compare treatments and experimenters use the information from the previous phase to make changes to the subsequent phase before it begins.

Adaptive design methods have been in practice since the 1970s, but have become increasingly complex ever since. One …

Applying Differential Transform Method To Nonlinear Partial Differential Equations: A Modified Approach, Marwan T. Alquran

Applications and Applied Mathematics: An International Journal (AAM)

This paper proposes another use of the Differential transform method (DTM) in obtaining approximate solutions to nonlinear partial differential equations (PDEs). The idea here is that a PDE can be converted to an ordinary differential equation (ODE) upon using a wave variable, then applying the DTM to the resulting ODE. Three equations, namely, Benjamin-Bona-Mahony (BBM), Cahn-Hilliard equation and Gardner equation are considered in this study. The proposed method reduces the size of the numerical computations and use less rules than the usual DTM method used for multi-dimensional PDEs. The results show that this new approach gives very accurate solutions.

Distributional Properties Of Record Values Of The Ratio Of Independent Exponential And Gamma Random Variables, M. Shakil, M. Ahsanullah

Applications and Applied Mathematics: An International Journal (AAM)

Both exponential and gamma distributions play pivotal roles in the study of records because of their wide applicability in the modeling and analysis of life time data in various fields of applied sciences. In this paper, a distribution of record values of the ratio of independent exponential and gamma random variables is presented. The expressions for the cumulative distribution functions, moments, hazard function and Shannon entropy have been derived. The maximum likelihood, method of moments and minimum variance linear unbiased estimators of the parameters, using record values and the expressions to calculate the best linear unbiased predictor of record values, …

Geometric Programming Subject To System Of Fuzzy Relation Inequalities, Elyas Shivanian, Mahdi Keshtkar, Esmaile Khorram

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, an optimization model with geometric objective function is presented. Geometric programming is widely used; many objective functions in optimization problems can be analyzed by geometric programming. We often encounter these in resource allocation and structure optimization and technology management, etc. On the other hand, fuzzy relation equalities and inequalities are also used in many areas. We here present a geometric programming model with a monomial objective function subject to the fuzzy relation inequality constraints with maxproduct composition. Simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on …