Applied Mathematics Commons^™

Development Of Scoring Rubrics And Pre-Service Teachers Ability To Validate Mathematical Proofs, Timothy J. Middleton

Mathematics & Statistics ETDs

The basic aim of this exploratory research study was to determine if a specific instructional strategy, that of developing scoring rubrics within a collaborative classroom setting, could be used to improve pre-service teachers facility with proofs. During the study, which occurred in a course for secondary mathematics teachers, the primary focus was on creating and implementing a scoring rubric, rather than on direct instruction about proofs. In general, the study had very mixed results. Statistically, the quantitative data indicated no significant improvement occurred in participants' ability to validate proofs. However, the qualitative results and the considerable improvement by some participants …

Finite Sample Properties Of Minimum Kolmogorov-Smirnov Estimator And Maximum Likelihood Estimator For Right-Censored Data, Jerzy Wieczorek

Dissertations and Theses

MKSFitter computes minimum Kolmogorov-Smirnov estimators (MKSEs) for several different continuous univariate distributions, using an evolutionary optimization algorithm, and recommends the distribution and parameter estimates that best minimize the Kolmogorov-Smirnov (K-S) test statistic. We modify this tool by extending it to use the Kaplan-Meier estimate of the cumulative distribution function (CDF) for right-censored data. Using simulated data from the most commonly-used survival distributions, we demonstrate the tool's inability to consistently select the correct distribution type with right-censored data, even for large sample sizes and low censoring rates. We also compare this tool's estimates with the right-censored maximum likelihood estimator (MLE). While …

Pain As A Predictor Of Depression Treatment Outcomes In Women With Childhood Sexual Abuse, Ellen L. Poleshuck, Nancy L. Talbot, Haiyan Su, Xin Tu, Linda Chaudron, Stephanie Gamble, Donna E. Giles

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

Objectives: Childhood sexual abuse (CSA) increases risk for both depression and pain in women. Pain is associated with worse depression treatment response. The contribution of pain to depression treatment outcomes in women with histories of CSA is unknown. This study examined whether clinically significant pain would be associated with worse depression and functioning outcomes among women with CSA histories treated with interpersonal psychotherapy. Method: Participants were 66 women with major depression and CSA who presented to a community mental health center. An interpersonal psychotherapy protocol planned for 14 weekly sessions followed by 2 biweekly sessions. Patients were classified as experiencing …

A Berry-Esseen Theorem For Sample Quantiles Under Weak Dependence, S. N. Lahiri, Shuxia Sun

Mathematics and Statistics Faculty Publications

This paper proves a Berry-Esseen theorem for sample quantiles of strongly-mixing random variables under a polynomial mixing rate. The rate of normal approximation is shown to be O(n^-1/2) as n -> infinity, where n denotes the sample size. This result is in sharp contrast to the case of the sample mean of strongly-mixing random variables where the rate O(n^-1/2) is not known even under an exponential strong mixing rate. The main result of the paper has applications in finance and econometrics as financial time series important data often are heavy-tailed and quantile …

A Direct Solution Of The Robin Inverse Problem, Weifu Fang, Suxing Zeng

Mathematics and Statistics Faculty Publications

We present a direct, linear boundary integral equation method for the inverse problem of recovering the Robin coefficient from a single partial boundary measurement of the solution to the Laplace equation.

A Note On The Positive Solutions Of An Inhomogeneous Elliptic Equation On Rn, Yinbin Deng, Yi Li, Fen Yang

Mathematics and Statistics Faculty Publications

This paper is contributed to the elliptic equation

(0.1) Δu+K(|x|)up+μf(|x|)=0,

where p>1, x∈Rn, n⩾3, and μ⩾0 is a constant. We study the structure of positive radial solutions of (0.1) and obtain the uniqueness of solution decaying faster than r−m at ∞ if μ is small enough under some assumptions on K and f, where m is the slow decay rate.

Periodic Traveling Waves In Sirs Endemic Models, Tong Li, Yi Li, Herbert W. Hethcote

Mathematics and Statistics Faculty Publications

Mathematical models are used to determine if infection wave fronts could occur by traveling geographically in a loop around a region or continent. These infection wave fronts arise by Hopf bifurcation for some spatial models for infectious disease transmission with distributed-contacts. Periodic traveling waves are shown to exist for the spatial analog of the SIRS endemic model, in which the temporary immunity is described by a delay, but they do not exist in a similar spatial SIRS endemic model without a delay. Specifically, we found that the ratio of the delay ω in the recovered class and the average infectious …

On The Estimation Of Averages Over Infinite Intervals With An Application To Average Persistence In Population Models, Sean F. Ellermeyer

Faculty Articles

We establish a general result for estimating the upper average of a continuous and bounded function over an infinite interval. As an application, we show that a previously studied model of microbial growth in a chemostat with time–varying nutrient input admits solutions (populations) that exhibit weak persistence but not weak average persistence.

Statistical Inferences For Functions Of Parameters Of Several Pareto And Exponential Populations With Application In Data Traffic, Sumith Gunasekera

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this dissertation, we discuss the usability and applicability of three statistical inferential frameworks--namely, the Classical Method, which is sometimes referred to as the Conventional or the Frequentist Method, based on the approximate large sample approach, the Generalized Variable Method based on the exact generalized p -value approach, and the Bayesian Method based on prior densities--for solving existing problems in the area of parametric estimation. These inference procedures are discussed through Pareto and exponential distributions that are widely used to model positive random variables relevant to social, scientific, actuarial, insurance, finance, investments, banking, and many other types of observable phenomena. …

On The Mixed Sum Of Doubly Infinite And Finite Independent Random Variables, Mridula Garg

Applications and Applied Mathematics: An International Journal (AAM)

The aim of the present paper is to study the distribution of the mixed sum of two random variables. Here we establish a theorem which gives the probability density function (pdf) of sum of doubly infinite and finite independent random variables. The distribution of the infinite and finite independent random variables is given in the form of corollary. As an application of these results we have obtained a distribution of sum of bilateral exponential variate with triangular, Rayleigh with uniform and Weibull with triangular variate. Some graphs of these distributions have also been given.

Asymptotic Behavior Of Linearized Viscoelastic Flow Problem, Yinnian He, Yi Li

Mathematics and Statistics Faculty Publications

In this article, we provide some asymptotic behaviors of linearized viscoelastic flows in a general two-dimensional domain with certain parameters small and the time variable large.

Topological Dynamics Of Two-Piece Eventually Expanding Maps, Youngna Choi

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

In this work we show that two-piece eventually expanding maps have the same topological dynamics as two-piece expanding maps. A two-piece eventually expanding map possesses an invariant set that is either a topological attractor or can be perturbed to become one.

Existence Of Multiple-Stable Equilibria For A Multi-Drug-Resistant Model Of Mycobacterium Tuberculosis, Abba B. Gumel, Baojun Song

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

The resurgence of multi-drug-resistant tuberculosis in some parts of Europe and North America calls for a mathematical study to assess the impact of the emergence and spread of such strain on the global effort to effectively control the burden of tuberculosis. This paper presents a deterministic compartmental model for the transmission dynamics of two strains of tuberculosis, a drug-sensitive (wild) one and a multi-drug-resistant strain. The model allows for the assessment of the treatment of people infected with the wild strain. The qualitative analysis of the model reveals the following. The model has a disease-free equilibrium, which is locally asymptotically …

Model-Based Clustering Of Methylation Array Data: A Recursive-Partitioning Algorithm For High-Dimensional Data Arising As A Mixture Of Beta Distributions, E. Andres Houseman, Brock C. Christensen, Ru-Fang Yeh, Carmen J. Marsit, Margaret R. Karagas, Margaret Wrensch, Heather H. Nelson, Joseph Wiemels, Shichun Zheng, John K. Wiencke, Karl T. Kelsey

Harvard University Biostatistics Working Paper Series

No abstract provided.

A Comparison Of Several Algorithms And Models For Analyzing Multivariate Normal Data With Missing Responses, Mojtaba Ganjali, H. Ranji

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we compare some modern algorithms i.e. Direct Maximization of the Likelihood (DML), the EM algorithm, and Multiple Imputation (MI) for analyzing multivariate normal data with missing responses. We also compare two approaches for modeling incomplete data (1) ignoring missing data and (2) joint modeling of response and non-response mechanisms. Several types of Software which can be used to implement the above algorithms are also mentioned. We used these algorithms for a simulation study and to analyze a data set where outliers affect the parameter estimates and final conclusion. As the variance of the estimates cannot be obtained …

Some Applications Of Dirac's Delta Function In Statistics For More Than One Random Variable, Santanu Chakraborty

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we discuss some interesting applications of Dirac's delta function in Statistics. We have tried to extend some of the existing results to the more than one variable case. While doing that, we particularly concentrate on the bivariate case.

The Weak Euler Scheme For Stochastic Delay Equations, Evelyn Buckwar, Rachel Kuske, Salah-Eldin A. Mohammed, Tony Shardlow

Articles and Preprints

We study weak convergence of an Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

Methods Of Assessing And Ranking Probable Sources Of Error, Nataniel Greene

Publications and Research

A classical method for ranking n potential events as sources of error is Bayes' theorem. However, a ranking based on Bayes' theorem lacks a fundamental symmetry: the ranking in terms of blame for error will not be the reverse of the ranking in terms of credit for lack of error. While this is not a flaw in Bayes' theorem, it does lead one to inquire whether there are related methods which have such symmetry. Related methods explored here include the logical version of Bayes' theorem based on probabilities of conditionals, probabilities of biconditionals, and ratios or differences of credit to …

A Monte Carlo Power Analysis Of Traditional Repeated Measures And Hierarchical Multivariate Linear Models In Longitudinal Data Analysis, Hua Fang, Gordon P. Brooks, Maria L. Rizzo, Kimberly A. Espy, Robert S. Barcikowski

Developmental Cognitive Neuroscience Laboratory: Faculty Publications

The power properties of traditional repeated measures and hierarchical linear models have not been clearly determined in the balanced design for longitudinal studies in the current literature. A Monte Carlo power analysis of traditional repeated measures and hierarchical multivariate linear models are presented under three variance-covariance structures. Results suggest that traditional repeated measures have higher power than hierarchical linear models for main effects, but lower power for interaction effects. Significant power differences are also exhibited when power is compared across different covariance structures. Results also supplement more comprehensive empirical indexes for estimating model precision via bootstrap estimates and the approximate …

Connectivity In Frame Matroids, Dan Slilaty, Hongxun Qin

Mathematics and Statistics Faculty Publications

We discuss the relationship between the vertical connectivity of a biased graph Ω and the Tutte connectivity of the frame matroid of Ω (also known as the bias matroid of Ω).

The Signed-Graphic Representations Of Wheels And Whirls, Dan Slilaty, Hongxun Qin

Mathematics and Statistics Faculty Publications

We characterize all of the ways to represent the wheel matroids and whirl matroids using frame matroids of signed graphs. The characterization of wheels is in terms of topological duality in the projective plane and the characterization of whirls is in terms of topological duality in the annulus.

Delay-Induced Instabilities In Self-Propelling Swarms, Eric Forgoston, Ira B. Schwartz

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

We consider a general model of self-propelling particles interacting through a pairwise attractive force in the presence of noise and communication time delay. Previous work by Erdmann [Phys. Rev. E 71, 051904 (2005)] has shown that a large enough noise intensity will cause a translating swarm of individuals to transition to a rotating swarm with a stationary center of mass. We show that with the addition of a time delay, the model possesses a transition that depends on the size of the coupling amplitude. This transition is independent of the initial swarm state (traveling or rotating) and is characterized by …

An Overview Of Conditionals And Biconditionals In Probability, Nataniel Greene

Publications and Research

Conditional and biconditional statements are a standard part of symbolic logic but they have only recently begun to be explored in probability for applications in artificial intelligence. Here we give a brief overview of the major theorems involved and illustrate them using two standard model problems from conditional probability.

A Method For Visualizing Multivariate Time Series Data, Roger D. Peng

Johns Hopkins University, Dept. of Biostatistics Working Papers

Visualization and exploratory analysis is an important part of any data analysis and is made more challenging when the data are voluminous and high-dimensional. One such example is environmental monitoring data, which are often collected over time and at multiple locations, resulting in a geographically indexed multivariate time series. Financial data, although not necessarily containing a geographic component, present another source of high-volume multivariate time series data. We present the mvtsplot function which provides a method for visualizing multivariate time series data. We outline the basic design concepts and provide some examples of its usage by applying it to a …

Mathematical Analysis Of The Transmission Dynamics Of Hiv/Tb Coinfection In The Presence Of Treatment, Oluwaseun Sharomi, Chandra N. Podder, Abba B. Gumel, Baojun Song

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

This paper addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic model, which incorporates many of the essential biological and epidemiological features of the two diseases. In the absence of TB infection, the model (IIIV-only model) is shown to have a globally asymptotically stable, disease-free equilibrium whenever the associated reproduction number is less than unity and has a unique endemic equilibrium whenever this number exceeds unity. On the other hand, the model with TB alone (TB-only model) undergoes the phenomenon of backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated …

An Order Model For Infinite Classical States, Joe Mashburn

Mathematics Faculty Publications

In 2002 Coecke and Martin (Research Report PRG-RR-02-07, Oxford University Computing Laboratory,2002) created a model for the finite classical and quantum states in physics. This model is based on a type of ordered set which is standard in the study of information systems. It allows the information content of its elements to be compared and measured. Their work is extended to a model for the infinite classical states. These are the states which result when an observable is applied to a quantum system. When this extended order is restricted to a finite number of coordinates, the model of Coecke and …

Qualitative Properties Of Nonlinear Volterra Integral Equations, Muhammad Islam, Jeffrey T. Neugebauer

Mathematics Faculty Publications

In this article, the contraction mapping principle and Liapunov's method are used to study qualitative properties of nonlinear Volterra equations of the form x(t)=a(t)−∫^t₀C(t,s)g(s,x(s))ds,t≥0. In particular, the existence of bounded solutions and solutions with various L^p properties are studied under suitable conditions on the functions involved with this equation.

Origin Of Conductive Surface Layer In Annealed Zno, David C. Look, B. Claflin, Helen Smith

Mathematics and Statistics Faculty Publications

The highly conductive surface layers found in nearly all as-grown or annealed bulk ZnO wafers are studied by temperature-dependent Hall-effect and secondary-ion mass spectroscopy (SIMS) measurements. In this work, we have used annealing in N₂ at 900 degrees C, and forming gas (5% H₂ in N₂) at 600 degrees C, to cause a large enough surface conduction that SIMS measurements can be reliably employed. The increased near-surface donor density, as determined from two-layer Hall-effect modeling, is consistent with an increased near-surface concentration of Al, Ga, and In atoms, resulting from diffusion. There is no evidence for …

Bayesian Analysis For Penalized Spline Regression Using Win Bugs, Ciprian M. Crainiceanu, David Ruppert, M.P. Wand

Johns Hopkins University, Dept. of Biostatistics Working Papers

Penalized splines can be viewed as BLUPs in a mixed model framework, which allows the use of mixed model software for smoothing. Thus, software originally developed for Bayesian analysis of mixed models can be used for penalized spline regression. Bayesian inference for nonparametric models enjoys the flexibility of nonparametric models and the exact inference provided by the Bayesian inferential machinery. This paper provides a simple, yet comprehensive, set of programs for the implementation of nonparametric Bayesian analysis in WinBUGS. MCMC mixing is substantially improved from the previous versions by using low{rank thin{plate splines instead of truncated polynomial basis. Simulation time …

The Substitution Theorem For Semilinear Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang

Articles and Preprints

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating …