Mathematics Commons^™

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 …

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 …

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

School of Mathematical & Statistical Sciences Faculty Publications

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 …

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).

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.

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 …

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.

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 …

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.

"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 …

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.

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 …

A Class Of Discontinuous Petrov–Galerkin Methods. Part Iv: The Optimal Test Norm And Time-Harmonic Wave Propagation In 1d., Jeffrey Zitelli, Leszek Demkowicz, Jay Gopalakrishnan, D. Pardo, V. M. Calo

Mathematics and Statistics Faculty Publications and Presentations

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test …

On Circulant And Two-Circulant Weighing Matrices, K. T. Arasu, I. S. Kotsireast, C. Koukouvinos, Jennifer Seberry

Mathematics and Statistics Faculty Publications

We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we con- struct W (148, 144), W (152, 144), W (156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. We also fill a missing entry in Strassler’s table with answer “YES”, by constructing a circulant weighing matrix of order 142 with weight 100.

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 …

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 …

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 …

Economic Optimization Of Offshore Wind Farms Using The Geometric Algorithm, Mahidhar Nandigam

Electrical & Computer Engineering Theses & Dissertations

The research project related to this thesis focuses on the optimization of electrical systems for offshore wind farms for a given capacity. The optimal design and planning is a critical issue for developing cost effectively Offshore Wind Farms in energy systems. The Geometric Optimization Algorithms approach has been adopted to develop an optimization program, where the main components of the electrical system of an offshore wind farm and key technical specifications are used as parameters to be optimized for a minimum cost with necessary constraints. The effectiveness of the optimization program can be evaluated with real-time comparison between offshore wind …

"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 …