Physical Sciences and Mathematics Commons^™

Students' Reasoning About The Concept Of Limit In The Context Of Reinventing The Formal Definition, Craig Alan Swinyard

Dissertations and Theses

Many researchers (Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995) have noted the vital role limit plays as a foundational concept in analysis. The vast majority of topics encountered in calculus and undergraduate analysis are built upon understanding the concept of limit and being able to work flexibly with its formal definition (Bezuidenhout, 2001). The purpose of this study was to: (1) Develop insight into students' reasoning about limit in relation to their engagement in instruction designed to support their reinventing the formal definition of limit, and; (2) Inform the design of principled instruction that might support students' attempts to …

Double Domination Of Complementary Prisms., Lamont D. Vaughan

Electronic Theses and Dissertations

The complementary prism of a graph G is obtained from a copy of G and its complement G̅ by adding a perfect matching between the corresponding vertices of G and G̅. For any graph G, a set D ⊆ V (G) is a double dominating set (DDS) if that set dominates every vertex of G twice. The double domination number, denoted γ_×2(G), is the cardinality of a minimum double dominating set of G. We have proven results on graphs of small order, specific families and lower bounds on γ_{×2 …}

Some Conclusions Of Statistical Analysis Of The Spectropscopic Evaluation Of Cervical Cancer, Hailun Wang

Mathematics Theses

To significantly improve the early detection of cervical precancers and cancers, LightTouch™ is under development by SpectRx Inc.. LightTouch™ identifies cancers and precancers quickly by using a spectrometer to analyze light reflected from the cervix. Data from the spectrometer is then used to create an image of the cervix that highlights the location and severity of disease. Our research is conducted to find the appropriate models that can be used to generate map-like image showing disease tissue from normal and further diagnose the cervical cancerous conditions. Through large work of explanatory variable search and reduction, logistic regression and Partial Least …

Hiv/Aids Relative Survival And Mean Residual Life Analysis, Xinjian Zhang

Mathematics Theses

HIV/Aids Relative Survival and Mean Residual Life Analysis BY XINJIAN ZHANG Under the Direction of Gengsheng (Jeff) Qin and Ruiguang (Rick) Song ABSTRACT Generalized linear models with Poisson error were applied to investigate HIV/AIDS relative survival. Relative excess risk for death within 3 years after HIV/AIDS diagnosis was significantly higher for non-Hispanic blacks, American Indians and Hispanics compared with Whites. Excess hazard for death was also higher in men injection drug users compared with men who have sex with men (MSM). The relative excess hazard of old HIV/AIDS patients is significantly higher compared with younger patients. When CD4 increased, the …

Absolute Continuity Of Laws For Semilinear Stochastic Equations With Additive Noise, Benedetta Ferrario

Communications on Stochastic Analysis

No abstract provided.

Properties Of Certain Lévy And Geometric Lévy Processes, Vladimir Vinogradov

Communications on Stochastic Analysis

No abstract provided.

The Stochastic Heat Equation Driven By A Gaussian Noise: Germ Markov Property, Raluca Balan, Doyoon Kim

Communications on Stochastic Analysis

No abstract provided.

Solutions Of Backward Stochastic Differential Equations On Markov Chains, Samuel N Cohen, Robert J Elliott

Communications on Stochastic Analysis

No abstract provided.

Generic Fock Quantum Markov Semigroups With Instantaneous States, A Ben Ghorbal, F Fagnola, S Hachicha, H Ouerdiane

Communications on Stochastic Analysis

No abstract provided.

Singular Perturbation And Stationary Solutions Of Parabolic Equations In Gauss-Sobolev Spaces, Pao-Liu Chow

Communications on Stochastic Analysis

No abstract provided.

Backward Stochastic Differential Equations Associated With Lévy Processes And Partial Integro-Differential Equations, Mohamed El Otmani

Communications on Stochastic Analysis

No abstract provided.

Multilinear Function Series In Conditionally Free Probability With Amalgamation, Mihai Popa

Communications on Stochastic Analysis

No abstract provided.

Zeon Algebra, Fock Space, And Markov Chains, Philip Feinsilver

Communications on Stochastic Analysis

No abstract provided.

An Efficient And Robust Method For Simulating Two-Phase Gel Dynamics, Grady Wright, Robert D. Guy, Aaron L. Fogelson

Mathematics Faculty Publications and Presentations

We develop a computational method for simulating models of gel dynamics where the gel is described by two phases: a networked polymer and a fluid solvent. The models consist of transport equations for the two phases, two coupled momentum equations, and a volume-averaged incompressibility constraint, which we discretize with finite differences/volumes. The momentum and incompressibility equations present the greatest numerical challenges since (i) they involve partial derivatives with variable coefficients that can vary quite significantly throughout the domain (when the phases separate), and (ii) their approximate solution requires the “inversion” of a large linear system of equations. For solving this …

Dead Cores Of Singular Dirichlet Boundary Value Problems With Φ-Laplacian, Ravi P. Agarwal, Donal O'Regan, Svatoslav Staněk

Mathematics and System Engineering Faculty Publications

The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable.

Conflations Of Probability Distributions, Theodore P. Hill

Research Scholars in Residence

The conflation of a finite number of probability distributions P₁,..., P_n is a consolidation of those distributions into a single probability distribution Q=Q(P₁,..., P_n), where intuitively Q is the conditional distribution of independent random variables X₁,..., X_n with distributions P₁,..., P_n, respectively, given that X₁= ... =X_n. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the …

Random Dynamics And Memory: Structure Within Chaos? (Maa Invited Address / David Blackwell Lecture), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

No abstract provided.

Imagine Math Day: Encouraging Secondary School Students And Teachers To Engage In Authentic Mathematical Discovery, Darryl H. Yong, Michael E. Orrison Jr.

All HMC Faculty Publications and Research

Research mathematicians and school children experience mathematics in profoundly different ways. Ask a group of mathematicians what it means to “do mathematics” and you are likely to get a myriad of responses: mathematics involves analyzing and organizing patterns and relationships, reasoning and drawing conclusions about the world, or creating languages and tools to describe and solve important problems. Students of mathematics often report “doing mathematics” as performing calculations or following rules. It’s natural that they see mathematics as monolithic rather than an evolving, growing, socially constructed body of knowledge, because most mathematical training in primary and secondary schools consists of …

On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky

CMC Faculty Publications and Research

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z², as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum …

Dynamical Behavior And Influence Of Stochastic Noise On Certain Generalized Boolean Networks, Gary L. Beck, Mihaela Teodora Matache

Mathematics Faculty Publications

This study considers a simple Boolean network with N nodes, each node’s state at time t being determined by a certain number of parent nodes. The network is analyzed when the connectivity k is fixed or variable. Making use of a Boolean rule that is a generalization of Rule 22 of elementary cellular automata, a generalized formula for providing the probability of finding a node in state 1 at a time t is determined. We show typical behaviors of the iterations, and we study the dynamics of the network through Lyapunov exponents, bifurcation diagrams, and fixed point analysis. We conclude …

Prep Workshop Report: Expository Writing, Darren B. Glass

Math Faculty Publications

A significant part of the job of a mathematician involves writing - between research papers, expository writing, grant applications, letters of recommendation, and materials for our teaching, I know that I spend much of my days writing something or other. Yet most of us are never really trained to write mathematics, and even in our jobs we rarely find time to talk about the actual writing of the mathematics which has taken place. With this in mind, I chose to attend a PREP workshop held by the Mathematical Association of America at their headquarters in Washington, DC dedicated to the …

Palmated Antlers Of Moose May Serve As A Parabolic Reflector Of Sounds, George A. Bubenik, Peter G. Bubenik

Mathematics and Statistics Faculty Publications

It has been postulated that the excellent sense of hearing in moose is mostly due to: (1) the large surface of the external ear, (2) better stereophony due to the large distance between ears, (3) independently movable, extremely adjustable pinna, and (4) the amplification of sounds reflected by the palms of the antlers. The last factor, possible reflection of sounds into pinna by the palm of the antlers, was tested in this study on a large antler trophy of Alaskan moose. The reception of a standard tone, broadcast from the frontally placed speaker, was recorded by a sound level meter …

A Novel Proof Of The Heine-Borel Theorem, Matthew Macauley, Brian Rabern, Landon Rabern

Publications

Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary.

A Poisson-Like Model Of Sub-Clinical Signs From The Examination Of Healthy Aging Subjects, Stephen Merrill, Barbara Myklebust, Joel B. Myklebust, Norman Reynolds, Edmund Duthie

Mathematics, Statistics and Computer Science Faculty Research and Publications

Background and aims: Our studies of the standard neurological examination on 66 middle-aged (50–64 yrs) and elderly subjects (65–84 yrs) demonstrate that healthy elders have neurological deficits (or “signs”) that are not associated with specific known neurological disease. The purpose of the current study is to describe this loss of neurological function in healthy aging subjects as seen through accumulated subclinical neurological signs present.

Methods: Logistic regression is applied to the data on each of six signs. Parameters determined are used to describe the distribution of first occurrence times for each sign. The results are then used to construct a …

Understanding Similarity: Bridging Geometric And Numeric Contexts For Proportional Reasoning, Dana Christine Cox

Dissertations

The concept of similarity is uniquely situated at the crossroads of geometric and numerical proportional reasoning. Although studies have documented the existence and nature of student difficulties with this topic, there exists a gap between documented visual insights of younger children and the quantitative inadequacies of older ones. Using a revised version of the Similarity Perception Test followed by 21 clinical interviews, this study investigated the visual and analytical strategies that are used by middle-school students to differentiate and construct similar figures.

New strategies for construction and differentiation were identified, and three overarching conclusions were drawn from the work. First, …

High Order Weighted Compact Boundary Condition, Zhengjie Wang

Mathematics Theses

In multi-dimension flows, we expect to have problems at the boundaries when a shock hits or reflects at the boundary wall the remedy to this would be to develop weighted boundary conditions similar to the Weighted Compact Scheme for the interior nodes, by choosing candidate stencils around the boundary nodes and assigning weights to each of these stencils with the ENO reconstruction. This would avoid spurious oscillations when shocks are encountered at the boundaries. This thesis investigates higher order weighted compact boundary conditions for Weighted Compact Scheme (WCS). WCS is a combination of Essentially Non Oscillatory Scheme and Weighted Compact …

At Least Four Distinct Circadian Regulatory Mechanisms Required For All Phases Of Rhythms In Mrna Amount, Sigrid Jacobshagen, Bruce Kessler, Claire Rinehart

Mathematics Faculty Publications

Since the advent of techniques to investigate gene expression on a large scale, numerous circadian rhythms in mRNA abundance have been reported. These rhythms generally differ in amplitude and phase. First studies on circadian rhythms of transcription on a large scale are also emerging. We investigated to what extent the same circadian regulatory mechanism of transcription can give rise to rhythms in RNA amount that differ in phase solely based on a parameter that is not regulated by the circadian clock. Using a discrete-time approach, we modeled a sinusoidal rhythm in transcription with various constant exponential RNA decay rates. We …

Noetherian Filtrations And Finite Intersection Algebras, Sara Malec

Mathematics Theses

This paper presents the theory of Noetherian filtrations, an important concept in commutative algebra. The paper describes many aspects of the theory of these objects, presenting basic results, examples and applications. In the study of Noetherian filtrations, a few other important concepts are introduced such as Rees algebras, essential powers filtrations, and filtrations on modules. Basic results on these are presented as well. This thesis discusses at length how Noetherian filtrations relate to important constructions in commutative algebra, such as graded rings and modules, dimension theory and associated primes. In addition, the paper presents an original proof of the finiteness …

New Non-Parametric Confidence Interval For The Youden, Haochuan Zhou

Mathematics Theses

Youden index, a main summary index for the Receiver Operating Characteristic (ROC) curve, is a comprehensive measurement for the effectiveness of a diagnostic test. For a continuous-scale diagnostic test, the optimal cut-point for the positive of disease is the cut-point leading to the maximization of the sum of sensitivity and specificity. Finding the Youden index of the test is equivalent to maximize the sum of sensitivity and specificity for all the possible values of the cut-point. In this thesis, we propose a new non-parametric confidence interval for the Youden index. Extensive simulation studies are conducted to compare the relative performance …

Bootstrap And Empirical Likelihood-Based Semi-Parametric Inference For The Difference Between Two Partial Aucs, Xin Huang

Mathematics Theses

With new tests being developed and marketed, the comparison of the diagnostic accuracy of two continuous-scale diagnostic tests are of great importance. Comparing the partial areas under the receiver operating characteristic curves (pAUC) is an effective method to evaluate the accuracy of two diagnostic tests. In this thesis, we study the semi-parametric inference for the difference between two pAUCs. A normal approximation for the distribution of the difference between two pAUCs has been derived. The empirical likelihood ratio for the difference between two pAUCs is defined and its asymptotic distribution is shown to be a scaled chi-quare distribution. Bootstrap and …