Mathematics Commons^™

Use Of Cognitive Constructs In Linear Algebra, Azucena Zamora

Open Access Theses & Dissertations

Thesis analyzed the presence of two cognitive entities-- modes of thinking and metonymies and metaphors- in the reasoning of three students enrolled in a first year matrix algebra course at a southwest university via the responses given to a set of eight questions asked during one-on-one interviews. The findings of our analysis shed light on the cognitive constructs that students employ to form their understanding of linear algebra concepts. Furthermore, our findings provide clues about the ways in which students move from one mode of thinking to another in the context of varying levels of exposure to graphical, algebraic, and …

Signature Matrices: The Eigenvalue Problem, Valeria Aguirre Holguin

Open Access Theses & Dissertations

Dealing with matrices can give us a hard time, especially when their dimension is too big, but we also well know how valuable information a matrix may carry, and that is why we study them. When a matrix has a significant number of zeroes we realize how much easier all calculations are. For instance, the product will be simpler to calculate, the determinant, the inverse and even the eigenvalue problem. This thesis provides the description and behavior of a very special kind of matrices which we call signature matrices. A particular feature of these matrices lies in the fact that …

Multiscale Modeling Of Cellular Systems In Biology, J. C. Dallon

Faculty Publications

Here we review eight different multiscale modeling efforts dealing with cellular systems in biology. The first two models focus on collagen based tissue, one dealing with the biomechanical properties of the tissue and the other focusing on how the dermis is remodeled in scar tissue formation. The next two models deal with first avascular tumor growth and then the role of the vasculature in tumor growth. We then consider two models which use the Immersed Boundary method to model tissue properties and cell-cell adhesion. Finally we conclude with two models with treatments of the Cellular Potts Model. The first models …

Asymptotic Properties Of Markov Modulated Sequences With Fast And Slow Time Scales, Son Luu Nguyen

Wayne State University Dissertations

In this dissertation we investigate asymptotic properties of Markov modulated random processes having two-time scales. The model contains a number of mixing sequences modulated by a switching process that is a discrete-time Markov chain. The motivation of our study stems from applications in manufacturing systems, communication networks, and economic systems, in which regime-switching models are used.

This thesis focuses on asymptotic properties of the Markov modulated processes under suitable scaling. Our main effort focuses on obtaining weak convergence and strong approximation results.

The Effect Of Learning Styles Strategies On Benchmark Eighth Grade Middle School Mathematics Achievement, Jean Ferrara

Walden Dissertations and Doctoral Studies

Low standardized mathematics scores resulted in a suburban middle school not reaching adequate yearly progress (AYP) for the 2 previous years. There were many possible factors contributing to this problem, among them the design of instruction. The purpose of this study was to identify learning styles of students and implement differentiated instructional strategies that address the learners' needs. The study was based on the Silver and Hanson's theory of learning style instruction and Gardner's multiple intelligences as a model for differentiating instruction. This sequential mixed methods quasi-experimental causal comparative design study investigated the effect of classroom intervention based on learning …

Ethnomathematics For Capacity Building In Mathematics Education, Catherine P. Vistro-Yu

Mathematics Faculty Publications

The Mathematics Framework for Philippine Basic Education (MATHTED and SEI, in press), a document that aims to guide the development of curricular contents in mathematics, identifies cultural-rootedness as one of the cognitive values that mathematics education in the Philippines must inculcate. Cultural-rootedness is defined as ?appreciating the cultural value of mathematics and its origins in many cultures, its rich history and how it has grown and continues to evolve?. Ethnomathematics, described as the ?mathematics which is practiced among identifiable cultural groups, such as national tribal societies, labor groups, children of a certain age bracket, professional classes and so on? (D?Ambrosio, …

On The Characteristics Of A Class Of Gaussian Processes Within The White Noise Space Setting, Daniel Alpay, Haim Attia, David Levanony

Mathematics, Physics, and Computer Science Faculty Articles and Research

Using the white noise space framework, we define a class of stochastic processes which include as a particular case the fractional Brownian motion and its derivative. The covariance functions of these processes are of a special form, studied by Schoenberg, von Neumann and Krein.

Discrete-Time Multi-Scale Systems, Daniel Alpay, Mamadou Mboup

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce multi-scale filtering by the way of certain double convolution systems. We prove stability theorems for these systems and make connections with function theory in the poly-disc. Finally, we compare the framework developed here with the white noise space framework, within which a similar class of double convolution systems has been defined earlier.

Linear Stochastic State Space Theory In The White Noise Space Setting, Daniel Alpay, David Levanony, Ariel Pinhas

Mathematics, Physics, and Computer Science Faculty Articles and Research

We study state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely its average behavior.

Situating Sotl Within The Disciplines: Mathematics In The United States As A Case Study, Jacqueline Dewar, Curtis Bennett

Mathematics, Statistics and Data Science Faculty Works

After two decades of work, many in the SoTL community are pondering the future of the SoTL movement. Will it sustain its influence? Will it continue to attract new participants? What role should the disciplines play? From the perspective of mathematics, this paper examines efforts by the Carnegie Academy and individuals within the mathematical community to build disciplinary support for the scholarship of teaching and learning. The authors, both mathematicians and Carnegie scholars, restrict their observations to the efforts undertaken in the United States during the last decade and examine the situation in mathematics in greater depth than has heretofore …

On The Construction Of Explicit Solutions To The Matrix Equation X2ax = Axa, Ed Mosteig, Aihua Li

Mathematics, Statistics and Data Science Faculty Works

In a previous article by Aihua Li and Duane Randall, the existence of solutions to certain matrix equations is demonstrated via nonconstructive methods. A recurring example appears in that work, namely the matrix equation AXA=X2AX, where A is a fixed, square matrix with real entries and X is an unknown square matrix. In this paper, the solution space is explicitly constructed for all 2×2 complex matrices using Gr ̈obner basis techniques. When A is a 2×2 matrix, the equation AXA=X2AXis equivalent to a system of four polynomial equations. The solution space then is the variety defined by the polynomials involved. …

Derived Categories And The Analytic Approach To General Reciprocity Laws. Part Iii, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

Building on the scaffolding constructed in the first two articles in this series, we now proceed to the geometric phase of our sheaf (-complex) theoretic quasidualization of Kubota's formalism for n-Hilbert reciprocity. Employing recent work by Bridgeland on stability conditions, we extend our yoga of t-structures situated above diagrams of specifically designed derived categories to arrangements of metric spaces or complex manifolds. This prepares the way for proving n-Hilbert reciprocity by means of singularity analysis.

Epimorphisms And Boundary Slopes Of 2–Bridge Knots, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

In this article we study a partial ordering on knots in S³ where K₁≥K₂ if there is an epimorphism from the knot group of K₁ onto the knot group of K₂ which preserves peripheral structure. If K₁ is a 2–bridge knot and K₁≥K₂, then it is known that K₂ must also be 2–bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2–bridge knot K_p∕q, produces infinitely many 2–bridge knots K_p′/q′ with K_p′∕q′≥K_p∕q. After characterizing all 2–bridge knots …

Spark-Induced Sparks As A Mechanism Of Intracellular Calcium Alternans In Cardiac Myocytes, Robert J. Rovetti, Xiaohua Cui, Alan Garfinkel, James N. Weiss, Zhilin Qu

Mathematics, Statistics and Data Science Faculty Works

Rationale: Intracellular calcium (Ca) alternans has been widely studied in cardiac myocytes and tissue, yet the underlying mechanism remains controversial.

Objective: In this study, we used computational modeling and simulation to study how randomly occurring Ca sparks interact collectively to result in whole-cell Ca alternans.

Methods and Results: We developed a spatially-distributed intracellular Ca cycling model in which Ca release units (CRUs) are locally coupled by Ca diffusion throughout the myoplasm and sarcoplasmic reticulum (SR) network. Ca sparks occur randomly in the CRU network when periodically paced with a clamped voltage waveform, but Ca alternans develops as the pacing speeds …

Upper Bounds On The Splitting Of The Eigenvalues, Phuoc L. Ho

University of Kentucky Doctoral Dissertations

We establish the upper bounds for the difference between the first two eigenvalues of the relative and absolute eigenvalue problems. Relative and absolute boundary conditions are generalization of Dirichlet and Neumann boundary conditions on functions to differential forms respectively. The domains are taken to be a family of symmetric regions in Rⁿ consisting of two cavities joined by a straight thin tube. Our operators are Hodge Laplacian operators acting on k-forms given by the formula Δ^(k) = dδ+δd, where d and δ are the exterior derivatives and the codifferentials respectively. A result …

Positive Solutions Of The (N-1, 1) Conjugate Boundary Value Problem, Bo Yang

Faculty Articles

We consider the (n - 1, 1) conjugate boundary value problem. Some upper estimates to positive solutions for the problem are obtained. We also establish some explicit sufficient conditions for the existence and nonexistence of positive solutions of the problem.

Review: Pythagoras’ Revenge: A Mathematical Mystery; The Housekeeper And The Professor, Susan Jane Colley

Faculty & Staff Scholarship

No abstract provided.

Proceedings Of The Fourth Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Introductory Texts

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 1st Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents

Investigating Mathematical Literacy Through Teacher Language, Alyson Lischka

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Communication about mathematical concepts using appropriate terminology is a standard established by the National Council of Teachers of Mathematics. However, international test results show that United States’ students are lagging in mathematical literacy. This case study analyzes the ways in which instructors use language to help students move toward conceptual understanding of mathematical vocabulary. Three mathematics education professors at a mid-size four year institution were observed teaching math classes to students enrolled in elementary or secondary certification programs. Collected data included: audio-recorded observations and field notes, lesson artifacts such as quizzes and handouts, and audio-recorded interviews with each participant. Findings …

Generalized Complex Hamiltonian Torus Actions: Examples And Constraints, Thomas Baird, Yi Lin

Mathematical Sciences: Faculty Publications

Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying rank(T)=n−2, using a surgery procedure on toric manifolds.