Applied Mathematics Commons^™

Results In Lattices, Ortholattices, And Graphs, Jianning Su

Doctoral Dissertations

This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce "π-versions" of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π, if a ✶ (b ✶ c) ≤ (a ✶ b) ✶ c for all 3-element antichains { a, b, c}. We consider …

Quantitative Stability Of Linear Infinite Inequality Systems Under Block Perturbations With Applications To Convex Systems, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra

Mathematics Research Reports

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is loo(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn …

A “Peak” At The Algorithm Behind “Peaklet Analysis” Software, Bruce Kessler

Mathematics Faculty Publications

In response to a problem posed by faculty at the Applied Physics Institute at Western Kentucky University, the speaker has developed an algorithm for providing an automated analysis of spectrum data for the purpose of determining the elemental composition of the item generating the data. A full, non-provisional patent application has been filed on the idea, and a full marketing campaign has started to license software implementing the algorithm. This presentation will give a brief explanation of the mathematics in use in the algorithm, and will give some examples of the software in action.

Quantitative Interpretation Of A Genetic Model Of Carcinogenesis Using Computer Simulations, Donghai Dai, Brandon Beck, Xiaofang Wang, Cory Howk, Yi Li

Mathematics and Statistics Faculty Publications

The genetic model of tumorigenesis by Vogelstein et al. (V theory) and the molecular definition of cancer hallmarks by Hanahan and Weinberg (W theory) represent two of the most comprehensive and systemic understandings of cancer. Here, we develop a mathematical model that quantitatively interprets these seminal cancer theories, starting from a set of equations describing the short life cycle of an individual cell in uterine epithelium during tissue regeneration. The process of malignant transformation of an individual cell is followed and the tissue (or tumor) is described as a composite of individual cells in order to quantitatively account for intra-tumor …

A Primer On Chaos And Fractals, Bruce Kessler

Mathematics Faculty Publications

This is a prelude to a performance of the play "Arcadia" exclusively for the science and math majors of Lipscomb University. One of the main characters of the story is a mathematical genius, and has realized the power and limitations of iterations in generating mathematical models and structures, although she is living in the early 1800's. This talk gives an introduction to the ideas of chaos theory, fractals, and randomness.

Complete Characterizations Of Local Weak Sharp Minima With Applications To Semi-Infinite Optimization And Complementarity, Boris S. Mordukhovich, Naihua Xiu, Jinchuan Zhou

Mathematics Research Reports

In this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only significantly extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to semi-infinite complementarity problems.

Global Well-Posedness And Asymptotic Behavior Of A Class Of Initial-Boundary-Value Problems Of The Kdv Equation On A Finite Domain, Ivonne Rivas, Muhammad Usman, Bingyu Zhang

Mathematics Faculty Publications

In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg- de Vries equation posed on a ?nite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global L2 a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially.

Projective-Planar Graphs With No K3,4-Minor, John Maharry, Dan Slilaty

Mathematics and Statistics Faculty Publications

An exact structure is described to classify the projective‐planar graphs that do not contain a K3, 4‐minor.

Sharp Asymptotics Of The Lp Approximation Error For Interpolation On Block Partitions, Yuliya Babenko, Tatyana Leskevich, Jean-Marie Mirebeau

Faculty Articles

Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic …

Tangential Extremal Principles For Finite And Infinite Systems Of Sets, Ii: Applications To Semi-Infinite And Multiobjective Optimization, Boris S. Mordukhovich, Hung M. Phan

Mathematics Research Reports

This paper contains selected applications of the new tangential extremal principles and related results developed in [20] to calculus rules for infinite intersections of sets and optimality conditions for problems of semi-infinite programming and multiobjective optimization with countable constraints.

Which Chessboards Have A Closed Knight's Tour Within The Rectangular Prism?, Joseph Demaio, Mathew Bindia

Faculty Articles

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the m x n rectangular chessboards that admit a closed knight's tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk's paper, this article extends his result by classifying the i x j x k rectangular prisms that admit a closed knight's tour.

Tangential Extremal Principles For Finite And Infinite Systems Of Sets, I: Basic Theory, Boris S. Mordukhovich, Hung M. Phan

Mathematics Research Reports

In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.

Research In Mathematics Educational Technology: Current Trends And Future Demands, Shannon O. Driskell, Robert N. Ronau, Christopher R. Rakes, Sarah B. Bush, Margaret L. Niess, David K. Pugalee

Mathematics Faculty Publications

This systematic review of mathematics educational technology literature identified 1356 manuscripts addressing the integration of educational technology into mathematics instruction. The manuscripts were analyzed using three frameworks (Research Design, Teacher Knowledge, and TPACK) and three supplementary lenses (Data Sources, Outcomes, and NCTM Principles) to produce a database to support future research syntheses and meta-analyses. Preliminary analyses of student and teacher outcomes (e.g., knowledge, cognition, affect, and performance) suggest that the effects of incorporating graphing calculator and dynamic geometry technologies have been abundantly studied; however, the usefulness of the results was often limited by missing information regarding measures of validity, reliability, …

Operation Comics: The Story Continues, Bruce Kessler, Janet Tassell, Tressa Tullis

Mathematics Faculty Publications

During the 2008-2009 academic year, the author K. wrote three issues of Operation Comics, a comic book with embedded mathematics content appropriate for 4th through 6th grade students. Several printed comics were placed in Cumberland Trace Elementary in the Warren County School System in Bowling Green, Kentucky, US. The author Ta. was enlisted to measure the impact of the comics on the attitudes and motivation of the students using the comics. A preliminary report was given by K. at the 2009 Bridges Banff Conference, and the written report appeared in the proceedings. Since then, data has been collected on the …

Connecting Mathematics Students To Philosophy And Faith, Nathan Moyer

ACMS Journal 2010-2011

This paper describes an upper-division class project that helps students wrestle with the deep philosophical questions of mathematics in a relevant way. Through article readings and group discussions, various historical views of mathematical philosophy are examined and critiqued. An emphasis is placed upon students making meaningful connections between their own views of mathematics and their personal Christian faith or worldview.

Vascular Countercurrent Network For 3d Triple-Layered Skin Structure With Radiation Heating, Xiaoqi Zeng

Doctoral Dissertations

Heat transfer in living tissue has become more and more attention for researchers, because high thermal radiation produced by intense fire, such as wild fires, chemical fires, accidents, warfare, terrorism, etc, is often encountered in human's daily life. Living tissue is a heterogeneous organ consisting of cellular tissue and blood vessels, and heat transfer in cellular tissue and blood vessel is quite different, because the blood vessels provide channels for fast heat transfer. The metabolic heat generation, heat conduction and blood perfusion in soft tissue, convection and perfusion of the arterial-venous blood through the capillary, and interaction with the environment …

An Exploration Of Modeling Techniques For The Study Of The Dynamics Of E-Mail Viruses, Karin Weule

Theses, Dissertations and Culminating Projects

We analyze real data sets from two e-mail viruses, the Magistr.b and the Sircam.a to explore how we can use mathematical models to predict the behavior described by the data. Analysis of the data is conducted primarily with computer programming in MatLab. We focus mainly on the use of two continuous models commonly used in the study of biological diseases, the SIS and the SIR models. A discrete modeling approach using agent-based simulations is also explored and revealed to be potentially useful in developing a compartmentalized model that incorporates both SIS and SIR model behavior. The theory behind the continuous …

Riordan Arrays Associated With Laurent Series And Generalized Sheffer-Type Groups, Tian-Xiao He

Scholarship

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type …

How Do Neurons Work Together? Lessons From Auditory Cortex, Kenneth D. Harris, Peter Bartho, Paul Chadderton, Carina Curto, Jaime De La Rocha, Liad Hollender, Vladimir Itskov, Artur Luczak, Stephan Marguet, Alfonso Renart, Shuzo Sakata

Department of Mathematics: Faculty Publications

Recordings of single neurons have yielded great insights into the way acoustic stimuli are represented in auditory cortex. However, any one neuron functions as part of a population whose combined activity underlies cortical information processing. Here we review some results obtained by recording simultaneously from auditory cortical populations and individual morphologically identified neurons, in urethane-anesthetized and unanesthetized passively listening rats. Auditory cortical populations produced structured activity patterns both in response to acoustic stimuli, and spontaneously without sensory input. Population spike time patterns were broadly conserved across multiple sensory stimuli and spontaneous events, exhibiting a generally conserved sequential organization lasting approximately …

Asymptotic Expansions And Stability Of Hybrid Systems With Two-Time Scales, Dung Tien Nguyen

Wayne State University Dissertations

In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One

of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.

In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has …

Fully Nonlinear Boundary Value Problems With Impulse, Paul Eloe, Muhammad Usman

Mathematics Faculty Publications

An impulsive boundary value problem with nonlinear boundary conditions for a second order ordinary differential equation is studied. In particular, sufficient conditions are provided so that a compression- expansion cone theoretic fixed point theorem can be applied to imply the existence of positive solutions. The nonlinear forcing term is assumed to satisfy usual sublinear or superlinear growth as t → ∞ or t → 0 +. The nonlinear impulse terms and the nonlinear boundary terms are assumed to satisfy the analogous asymptotic behavior.

Prospective Teachers' Use Of Representations In Solving Statistical Tasks With Dynamic Statistical Software, Hollylynne Lee, Shannon O. Driskell, Suzanne R. Harper, Keith R. Leatham, Gladis Kersaint, Robin L. Angotti

Mathematics Faculty Publications

This study examined a random stratified sample (n=62) of prospective teachers' work across eight institutions on three tasks that utilized dynamic statistical software. Our work was guided by considering how teachers may utilize their statistical knowledge and technological statistical knowledge to engage in cycles of investigation. Although teachers did not tend to take full advantage of dynamic linking capabilities, they utilized a large variety of graphical representations and often added statistical measures or other augmentations to graphs as part of their analysis.

Non-Normality Points Of Β X\X, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

We seek conditions implying that (β X\X) \ {y} is not normal. Our main theorem: Assume GCH and all uniform ultrafilters are regular. If X is a locally compact metrizable space without isolated points, then (β X\X) \ {y} is not normal for all y ∈ β X\X. In preparing to prove this theorem, we generalize the notions “uniform”, “regular”, and “good” from set ultrafilters to z-ultrafilters. We discuss non-normality points of the product of a discrete space and the real line. We topologically embed a nonstandard real line into the remainder of this product space.

Algorithms For Area Preserving Flows, Catherine Kublik, Selim Esedoglu, Jeffrey A. Fessler

Mathematics Faculty Publications

We propose efficient and accurate algorithms for computing certain area preserving geometric motions of curves in the plane, such as area preserving motion by curvature. These schemes are based on a new class of diffusion generated motion algorithms using signed distance functions. In particular, they alternate two very simple and fast operations, namely convolution with the Gaussian kernel and construction of the distance function, to generate the desired geometric flow in an unconditionally stable manner. We present applications of these area preserving flows to large scale simulations of coarsening.

Algebraic Structures Using Super Interval Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The advantage of using super interval matrices is that one can build only one vector space using m × n interval matrices, but in case of super interval matrices we can have several such spaces depending on the partition on the interval matrix.

This book has seven chapters. Chapter one is introductory in nature, just introducing the super interval matrices or interval super matrices. In chapter two essential operations on super interval matrices are defined. Further in this chapter algebraic structures are defined on these super interval matrices using these operation. Using these super interval matrices semirings and semivector spaces …

Studies In Sampling Techniques And Time Series Analysis, Florentin Smarandache, Rajesh Singh

Branch Mathematics and Statistics Faculty and Staff Publications

This book has been designed for students and researchers who are working in the field of time series analysis and estimation in finite population. There are papers by Rajesh Singh, Florentin Smarandache, Shweta Maurya, Ashish K. Singh, Manoj Kr. Chaudhary, V. K. Singh, Mukesh Kumar and Sachin Malik. First chapter deals with the problem of time series analysis and the rest of four chapters deal with the problems of estimation in finite population. The book is divided in five chapters as follows: Chapter 1. Water pollution is a major global problem. In this chapter, time series analysis is carried out …

Algebraic Structures Using Natural Class Of Intervals, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Authors in this book introduce a new class of intervals called the natural class of intervals, also known as the special class of intervals or as natural intervals. These intervals are built using increasing intervals, decreasing intervals and degenerate intervals. We say an interval [a, b] is an increasing interval if a < b for any a, b in the field of reals R. An interval [a, b] is a decreasing interval if a > b and the interval [a, b] is a degenerate interval if a = b for a, b in the field of reals R. The natural class of intervals consists of the collection of increasing intervals, decreasing intervals and the degenerate intervals. Clearly R is contained in the natural …

Study Of Natural Class Of Intervals Using (–∞,∞) And (∞, –∞), Florentin Smarandache, W.B. Vasantha Kandasamy, D. Datta, H.S. Kushwaha, P.A. Jadhav

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors study the properties of natural class of intervals built using (–∞, ∞) and (∞, –∞). These natural class of intervals behave like the reals R, as far as the operations of addition, multiplication, subtraction and division are concerned. Using these natural class of intervals we build interval row matrices, interval column matrices and m × n interval matrices. Several properties about them are defined and studied. Also all arithmetic operations are performed on them in the usual way. The authors by defining so have made it easier for operations like multiplication, addition, finding determinant and …

Dsm Super Vector Space Of Refined Labels, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of supermatrices of refined labels. Authors prove super row matrix of refined labels form a group under addition. However super row matrix of refined labels do not form a group under product; it only forms a semigroup under multiplication. In this book super column matrix of refined labels and m × n matrix of refined labels are introduced and studied. We mainly study this to introduce to super vector space of refined labels using matrices. We in this book introduce the notion of semifield of refined labels using which …

Dsm Vector Spaces Of Refined Labels, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of DSm linear algebra of refined labels have been done by Florentin Smarandache, Jean Dezert, and Xinde Li.

In this book the authors introduce the notion of DSm vector spaces of refined labels. The reader is requested to refer the paper as the basic concepts are taken from that paper [35]. This book has six chapters. The first one is introductory in nature just giving only the needed concepts to make this book a self contained one. Chapter two introduces the notion of refined plane of labels, the three dimensional space of refined labels DSm vector spaces. Clearly …