Physical Sciences and Mathematics Commons^™

Technology Integration In Secondary Mathematics Classrooms: Effect On Students’ Understanding, Megan Sheehan, Leah Nillas

Scholarly Publications

Technology use in secondary mathematics courses has the potential to bring about broad changes in learning environment and teaching pedagogy, allowing students to communicate and collaborate in new ways and to conjecture, justify, and generalize findings. However, this potential is only realized when teachers use technology in ways encouraging these outcomes (Galbraith, 2006). The purpose of this study is to examine the integration of technology in secondary mathematics classrooms and to evaluate the effectiveness of its use in relation to students’ learning outcomes. This self study research was conducted in honors geometry and AP calculus classes. Data sources included transcripts …

Powerpack: Energy Profiling And Analysis Of High-Performance Systems And Applications, Rong Ge, Xizhou Feng, Shuaiwen Song, Hung-Ching Chang, Dong Li, Kirk W. Cameron

Mathematics, Statistics and Computer Science Faculty Research and Publications

Energy efficiency is a major concern in modern high-performance computing system design. In the past few years, there has been mounting evidence that power usage limits system scale and computing density, and thus, ultimately system performance. However, despite the impact of power and energy on the computer systems community, few studies provide insight to where and how power is consumed on high-performance systems and applications. In previous work, we designed a framework called PowerPack that was the first tool to isolate the power consumption of devices including disks, memory, NICs, and processors in a high-performance cluster and correlate these measurements …

On Semigroups With Lower Semimodular Lattice Of Subsemigroups, Peter R. Jones

Mathematics, Statistics and Computer Science Faculty Research and Publications

The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green's relation J is trivial have this property, a description in such terms is natural. In the case of periodic semigroups—a case that turns out to include all eventually regular semigroups—the characterization becomes quite explicit and yields interesting consequences. In the general case, …

The Politics Before The Politics: Census 2010, Reapportionment, And Redistricting, Karen Saxe, T. Ratliff

Karen Saxe

No abstract provided.

An Attempt To Get And Keep Women Involved In Physics, Jim Crumley, Kristen Nairn, Lynn Ziegler

MapCores Faculty Publications

In this talk I will briefly review some of the obstacles to the full participation of women in the STEM disciplines. In order to increase the number of women in physics, computer science, and mathematics we have started a cohort-based program with curricular and scholarship components for women in these majors. I will present the results of our program so far and offer some advice based on our experiences.

Entanglement-Assisted Quantum Low-Density Parity-Check Codes, Yuichiro Fujiwara, David C. Clark, Peter Vandendriessche, Maarten De Boeck, Vladimir Tonchev

Department of Mathematical Sciences Publications

This article develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error-correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error-correction performance and many other codes with better block error rates in simulations over the …

Topics In Random Knots And R-Matrices From Frobenius Algebras, Enver Karadayi

USF Tampa Graduate Theses and Dissertations

In this dissertation, we study two areas of interest in knot theory: Random knots in the unit cube, and the Yang-Baxter solutions constructed from Frobenius algebras.

The study of random knots can be thought of as a model of DNA strings situated in confinement. A random knot with n vertices is a polygonal loop formed by selecting n distinct points in the unit cube, for a positive integer n, and connecting these points by straight line segments successively, such that the last point selected is joined with the first one. We present a step by step description of our algorithm …

A Characterization Of Weingarten Surfaces In Hyperbolic 3-Space, Nikos Georgiou, Brendan Guilfoyle

Preprints

We study 2-dimensional submanifolds of the space L(H3) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian if there exists a surface in H3 orthogonal to the geodesics of Σ. We prove that the induced metric on a Lagrangian surface in L(H3) has zero Gauss curvature if the orthogonal surfaces in H3 are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in L(H3) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in H3.

Flat Zipper-Unfolding Pairs For Platonic Solids, Joseph O'Rourke

Computer Science: Faculty Publications

We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.

On Gradings In Khovanov Homology And Sutured Floer Homology, J. Elisenda Grigsby, Stephan M. Wehrli

Mathematics - All Scholarship

We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories. These two gradings have simple representation-theoretic (resp., geometric) interpretations, which we also review.

Compressed Sensing With Coherent And Redundant Dictionaries, Emmanuel J. Candès, Yonina C. Eldar, Deanna Needell, Paige Randall

CMC Faculty Publications and Research

This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is viable in this context, but also that accurate recovery is possible via an ℓ₁-analysis optimization problem. We introduce a condition on the measurement/sensing matrix, which is a natural generalization of the now well-known restricted isometry property, and which guarantees accurate recovery of signals that are …

Extended Green-Liouville Asymptotics And Vacuum Polarization For Lukewarm Black Holes, Cormac Breen, Adrian Ottewill

Articles

We consider a quantum field on a lukewarm black hole spacetime. We introduce a new uniform approximation to the radial equation, constructed using an extension of Green-Liouville asymptotics. We then use this new approximation to construct the renormalized vacuum polarization in the Hartle-Hawking vacuum. Previous calculations of the vacuum polarization rely on the WKB approximation to the solutions of the radial equation, however the nonuniformity of the WKB approximations obscures the results of these calculations near both horizons. The use of our new approximation eliminates these obscurities, enabling us to obtain explicitly finite and easily calculable values of the vacuum …

Dissipation Scales And Anomalous Sinks In Steady Two-Dimensional Turbulence, Eleftherios Gkioulekas

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In previous papers I have argued that the fusion rules hypothesis, which was originally introduced by L’vov and Procaccia in the context of the problem of three-dimensional turbulence, can be used to gain a deeper insight in understanding the enstrophy cascade and inverse energy cascade of two-dimensional turbulence. In the present paper, we show that the fusion rules hypothesis, combined with nonperturbative locality, itself a consequence of the fusion rules hypothesis, dictates the location of the boundary separating the inertial range from the dissipation range. In so doing, the hypothesis that there may be an anomalous enstrophy sink …

Non-Invasive Prenatal Detection Of Trisomy 21 Using Tandem Single Nucleotide Polymorphisms, Sujana Ghanta, Michael Mitchell, Mary Ames, Mats Hidestrand, Pippa Simpson, Mary Goetsch, William Thilly, Craig Struble, Aoy Tomita-Mitchell

Mathematics, Statistics and Computer Science Faculty Research and Publications

Background: Screening tests for Trisomy 21 (T21), also known as Down syndrome, are routinely performed for the majority of pregnant women. However, current tests rely on either evaluating non-specific markers, which lead to false negative and false positive results, or on invasive tests, which while highly accurate, are expensive and carry a risk of fetal loss. We outline a novel, rapid, highly sensitive, and targeted approach to non-invasively detect fetal T21 using maternal plasma DNA.

Methods and Findings: Highly heterozygous tandem Single Nucleotide Polymorphism (SNP) sequences on chromosome 21 were analyzed using High-Fidelity PCR and Cycling Temperature Capillary …

Curvedland: An Applet For Illustrating Curved Geometry Without Embedding, Gary Felder, Stephanie Erickson

Physics: Faculty Publications

We have written a Java applet to illustrate the meaning of curved geometry. The applet provides a mapping interface similar to MapQuest or Google Maps; features include the ability to navigate through a space and place permanent point objects and/or shapes at arbitrary positions. The underlying two-dimensional space has a constant, positive curvature, which causes the apparent paths and shapes of the objects in the map to appear distorted in ways that change as you view them from different relative angles and distances.

On The Three-Dimensional Blaschke-Lebesgue Problem, Henri Anciaux, Brendan Guilfoyle

Publications

The width of a closed convex subset of n-dimensional Euclidean space is the distance between two parallel supporting hyperplanes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still open in dimension n ≥ 3. In this paper we describe a necessary condition that the minimizer of the Blaschke-Lebesgue must satisfy in dimension n = 3: we prove that the smooth components of the boundary of the minimizer have their smaller principal curvature constant and therefore are either spherical caps or pieces of tubes (canal surfaces).

One-Dimensional Wave Equations Defined By Fractal Laplacians, John Fun-Choi Chan, Sze-Man Ngai, Alexander Teplyaev

Sze-Man Ngai

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps. We prove the existence and uniqueness of weak solutions. We also study numerical computations of the solutions and prove the convergence of the approximation scheme. This is a joint work with John F. Chan and Alexander Teplyaev.

Analysis Of Nonlinear Spectral Eddy-Viscosity Models Of Turbulence, Max Gunzburger, Eunjung Lee, Yuki Saka, Catalin Trenchea, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

Fluid turbulence is commonly modeled by the Navier-Stokes equations with a large Reynolds number. However, direct numerical simulations are not possible in practice, so that turbulence modeling is introduced. We study artificial spectral viscosity models that render the simulation of turbulence tractable. We show that the models are well posed and have solutions that converge, in certain parameter limits, to solutions of the Navier-Stokes equations. We also show, using the mathematical analyses, how effective choices for the parameters appearing in the models can be made. Finally, we consider temporal discretizations of the models and investigate their stability. © 2009 Springer …

Generalized Newton's Method Based On Graphical Derivatives, T Hoheisel, C Kanzow, Boris S. Mordukhovich, Hung M. Phan

Mathematics Research Reports

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper …

Gaussian Brunn-Minkowski Inequalities, Richard J. Gardner, Artem Zvavitch

Mathematics Faculty Publications

A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality, the first of its type, is proved, together with precise equality conditions, and is shown to be the best possible from several points of view. A new Gaussian Brunn-Minkowski inequality is proposed and proved to be true in some significant special cases Throughout the study attention is paid to precise equality conditions and conditions on the coefficients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.

Non-Autonomous Periodic Systems With Allee Effects, Rafael Luís, Saber Elaydi, Henrique Oliveira

Mathematics Faculty Research

A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with tree fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper the properties and stability of the three fixed points are studied in the setting of nonautonomous periodic dynamical systems or difference equations. Finally we investigate the bifurcation of …

2010 (Fall), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2010 Fall Colloquium.

Mathematical Manipulative Models: In Defense Of "Beanbag Biology", John R. Jungck, Holly Gaff, Anton E. Weisstein

Biological Sciences Faculty Publications

Mathematical manipulative models have had a long history of influence in biological research and in secondary school education, but they are frequently neglected in undergraduate biology education. By linking mathematical manipulative models in a four-step process-1) use of physical manipulatives, 2) interactive exploration of computer simulations, 3) derivation of mathematical relationships from core principles, and 4) analysis of real data sets-we demonstrate a process that we have shared in biological faculty development workshops led by staff from the BioQUEST Curriculum Consortium over the past 24 yr. We built this approach based upon a broad survey of literature in mathematical educational …

"Toward Integration: From Quantitative Biology To Mathbio-Biomath?", Pat Marsteller, Lisette G. De Pillis, Ann Findley, Karl Joplin, John Pelesko, Karen Nelson, Katerina Thompson, David Usher, Joseph Watkins

All HMC Faculty Publications and Research

In response to the call of BIO2010 for integrating quantitative skills into undergraduate biology education, 30 Howard Hughes Medical Institute (HHMI) Program Directors at the 2006 HHMI Program Directors Meeting established a consortium to investigate, implement, develop, and disseminate best practices resulting from the integration of math and biology. With the assistance of an HHMI-funded mini-grant, led by Karl Joplin of East Tennessee State University, and support in institutional HHMI grants at Emory and University of Delaware, these institutions held a series of summer institutes and workshops to document progress toward and address the challenges of implementing a more quantitative …

"Beyond Bio2010: Celebration And Opportunities" At The Intersection Of Mathematics And Biology, John R. Jungck, Holly D. Gaff, Adam P. Fagen, Jay B. Labov

Biological Sciences Faculty Publications

With this special edition of CBE-LSE, which focuses on connections between and integration of the biological and mathematical sciences, it is especially fitting that we report on an important symposium, Beyond BIO2010: Celebration and Opportunities,1 which was held at the National Acad- emy of Sciences (NAS) in Washington, D.C. on May 21–22, 2010. This symposium was organized to assess what progress has been made in addressing the challenges and recommendations in the National Research Council’s (NRC) report: BIO2010: Transforming Undergraduate Education for Future Research Biologists (NRC, 2003a). Most of the presen- tations and posters at this event emphasized the increasing …

Existence Of Solutions For A Semilinear Wave Equation With Non-Monotone Nonlinearity, Alfonso Castro, Benjamin Preskill '09

All HMC Faculty Publications and Research

For double-periodic and Dirichlet-periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with asymptotically linear nonlinearity, no resonance, and non-monotone nonlinearity when the forcing term is not flat on characteristics. The solutions are in L^∞ when the forcing term is in L^∞ and continous when the forcing term is continuous. This is in contrast with the results in [4], where the non-enxistence of continuous solutions is established even when forcing term is of class C^∞ but is flat on a characteristic.

Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky

CMC Faculty Publications and Research

Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of K^N where N≥ 2. Let Z_K be a union of varieties defined over K such that V ⊈ Z_K. We prove the existence of a point of small height in V \ Z_K, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of hypersurface containing Z_K, where dependence on …

Impact Of Interdisciplinary Undergraduate Research In Mathematics And Biology On The Development Of A New Course Integrating Five Stem Disciplines, Lester Caudill, April L. Hill, Kathy Hoke, Ovidiu Z. Lipan

Department of Math & Statistics Faculty Publications

Funded by innovative programs at the National Science Foundation and the Howard Hughes Medical Institute, University of Richmond faculty in biology, chemistry, mathematics, physics, and computer science teamed up to offer first- and second-year students the opportunity to contribute to vibrant, interdisciplinary research projects. The result was not only good science but also good science that motivated and informed course development. Here, we describe four recent undergraduate research projects involving students and faculty in biology, physics, mathematics, and computer science and how each contributed in significant ways to the conception and implementation of our new Integrated Quantitative Science course, a …

Impact Of Interdisciplinary Undergraduate Research In Mathematics And Biology On The Development Of A New Course Integrating Five Stem Disciplines, Lester Caudill, April L. Hill, Kathy Hoke, Ovidiu Z. Lipan

Biology Faculty Publications

Funded by innovative programs at the National Science Foundation and the Howard Hughes Medical Institute, University of Richmond faculty in biology, chemistry, mathematics, physics, and computer science teamed up to offer first- and second-year students the opportunity to contribute to vibrant, interdisciplinary research projects. The result was not only good science but also good science that motivated and informed course development. Here, we describe four recent undergraduate research projects involving students and faculty in biology, physics, mathematics, and computer science and how each contributed in significant ways to the conception and implementation of our new Integrated Quantitative Science course, a …

Coxeter Groups And Asynchronous Cellular Automata, Matthew Macauley, Henning S. Mortveit

Publications

The dynamics group of an asynchronous cellular automaton (ACA) relates properties of its long term dynamics to the structure of Coxeter groups. The key mathematical feature connecting these diverse fields is involutions. Group-theoretic results in the latter domain may lead to insight about the dynamics in the former, and vice-versa. In this article, we highlight some central themes and common structures, and discuss novel approaches to some open and open-ended problems. We introduce the state automaton of an ACA, and show how the root automaton of a Coxeter group is essentially part of the state automaton of a related ACA.