Mathematics Commons^™

A Critical Centre-Stable Manifold For The Cubic Focusing Schroedinger Equation In Three Dimensions, Marius Beceanu

Mathematics and Statistics Faculty Scholarship

Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t \psi + \Delta \psi + |\psi|^2 \psi = 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(\Gamma + vx - t|v|^2 + \alpha^2 t)} \phi(x-2tv-D, \alpha), where \phi(x, \alpha) is a positive ground state solution of the semilinear elliptic equation -\Delta \phi + \alpha^2\phi = \phi^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive …

A Berry-Esseen Theorem For Sample Quantiles Under Weak Dependence, S. N. Lahiri, Shuxia Sun

Mathematics and Statistics Faculty Publications

This paper proves a Berry-Esseen theorem for sample quantiles of strongly-mixing random variables under a polynomial mixing rate. The rate of normal approximation is shown to be O(n^-1/2) as n -> infinity, where n denotes the sample size. This result is in sharp contrast to the case of the sample mean of strongly-mixing random variables where the rate O(n^-1/2) is not known even under an exponential strong mixing rate. The main result of the paper has applications in finance and econometrics as financial time series important data often are heavy-tailed and quantile …

Second-Order Analysis Of Polyhedral Systems In Finite And Infinite Dimensions With Applications To Robust Stability Of Variational Inequalities, René Henrion, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order …

Loss Of Synchrony In An Inhibitory Network Of Type-I Oscillators, Myongkeun Oh

Dissertations

Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris-Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation …

Innovation Imitation And North South Trade: Economic Theory And Policy., Debasis Mondal Dr.

Doctoral Theses

No abstract provided.

Constant Sign And Nodal Solutions For Problems With The P-Laplacian And A Nonsmooth Potential Using Variational Techniques, Ravi P. Agarwal, Michael E. Filippakis, Donal O'Regan, Nikolaos S. Papageorgiou

Mathematics and System Engineering Faculty Publications

We consider a nonlinear elliptic equation driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality) and Dirichlet boundary condition. Using a variational approach based on nonsmooth critical point theory together with the method of upper and lower solutions, we prove the existence of at least three nontrivial smooth solutions: one positive, the second negative, and the third sign changing (nodal solution). Our hypotheses onthe nonsmooth potential incorporate in our framework of analysis the so-called asymptotically p-linear problems.

Multiwavelets For Quantitative Pattern Matching, Bruce Kessler

Mathematics Faculty Publications

This was my presentation in Hawaii that accompanied my paper on pattern matching, published in the conference proceedings.

Parts Of The Whole: Is Everything Equally Important?, Dorothy Wallace

Numeracy

Two strategies are proposed for sorting out priorities and goals in education. At the college level the column argues that goals, and therefore requirements, should be set based on the desired knowledge base of the entire population rather than the individual. At the scale of a single K-12 classroom the column argues that priority be given to content requiring most instructor intervention, describing how the structure of class time and school activities would change as a result. Implementation of these strategies would lead to more efficient use of teacher’s time and effort, better allocation of systemic resources, and clarity of …

Review Of The Chicago Guide To Writing About Numbers By Jane E. Miller, Neil Lutsky

Numeracy

Miller, J. E. 2004. The Chicago Guide to Writing about Numbers. Chicago, IL: University of Chicago Press. 304 pp. Cloth $45 ISBN: 9780226526300, Softcover $17. ISBN: 9780226526317.

The Chicago Guide to Writing about Numbers is a reference work suitable for anyone interested in understanding, using, or promoting quantitative thinking. Its primary aim is to identify and illustrate ways in which information associated with numbers can be conveyed most effectively given a particular communication purpose and context. The book is directed at writers who incorporate numbers in verbal or visual displays in documents, in oral presentations, or on the Web. The …

Quantitative Literacy On The Web Of Science, 2 – Mining The Health Numeracy Literature For Assessment Items, H L. Vacher, Todd Chavez

Numeracy

A topic search of the Web of Science (WoS) database using the term “numeracy” produced a bibliography of 293 articles, reviews and editorial commentaries (Oct 2008). The citation graph of the bibliography clearly identifies five benchmark papers (1995-2001), four of which developed numeracy assessment instruments. Starting with the 80 papers that cite these benchmarks, we identified a set of 25 papers (1995-2008) in which the medical research community reports the development and/or application of health-numeracy assessments. In all we found 10 assessment instruments from which we have compiled a total of 48 assessment items. There are both general and context-specific …

The Case For Infusing Quantitative Literacy Into Introductory Geoscience Courses, Jennifer M. Wenner, Eric M. Baer, Cathryn A. Manduca, R. Heather Macdonald, Samuel Patterson, Mary Savina

Numeracy

We present the case for introductory geoscience courses as model venues for increasing the quantitative literacy (QL) of large numbers of the college-educated population. The geosciences provide meaningful context for a number of fundamental mathematical concepts that are revisited several times in a single course. Using some best practices from the mathematics education community surrounding problem solving, calculus reform, pre-college mathematics and five geoscience/math workshops, geoscience and mathematics faculty have identified five pedagogical ideas to increase the QL of the students who populate introductory geoscience courses. These five ideas include techniques such as: place mathematical concepts in context, use multiple …

Establishing The Quantitative Thinking Program At Macalester, David Bressoud

Numeracy

In November 2005, the faculty of Macalester College voted to institute a graduation requirement in Quantitative Thinking (QT) that is truly interdisciplinary. It currently draws on courses from thirteen departments including Anthropology, Economics, Geography, Political Science, Theater, Mathematics, Environmental Science, and Geology. This article describes the process that led to the creation of this program. It explains how we were able to get broad buy-in at the beginning and the long process of trial and error—informed by formative assessment—that was needed to refine the initial vision and shape it into a viable program that would be accepted by most of …

Confronting Challenges, Overcoming Obstacles: A Conversation About Quantitative Literacy, Bernard L. Madison, Lynn A. Steen

Numeracy

An edited transcript of the opening session of a workshop on quantitative literacy held Oct. 10-12, 2008 at Carleton College, Northfield, Minnesota. The workshop, which brought together interdisciplinary teams from two dozen colleges and universities, was sponsored by the Quantitative Inquiry, Reasoning, and Knowledge (QuIRK) Initiative at Carleton and the Washington-based Project Kaleidoscope. Two mathematicians in the forefront of quantitative literacy initiatives over the period 1997-2008, Lynn Arthur Steen and Bernard L. Madison, converse about attitudes, obstacles, changes and accomplishments. The conversation, structured as an interview, begins with the relationship between mathematics and quantitative literacy and moves through issues central …

All The More Reason For Qr Across The Curriculum, Bernard L. Madison

Numeracy

No abstract provided.

Wavelet Decompositions For Quantitative Pattern Matching, Bruce Kessler

Mathematics Faculty Publications

The purpose of this paper is to provide an introduction to the concepts of wavelets and multiwavelets, and explain how these tools can be used by the analyst community to find patterns in quantitative data. Three multiwavelet bases are introduced, the GHM basis from \cite{GHM}, a piecewise polynomial basis with approximation order 4 from \cite{DGH}, and a smoother approximation-order-4 basis developed by the author in previous work \cite{K}. The technique of using multiwavelets to find patterns is illustrated in a traffic-analysis example. Acknowledgements: This work supported in part by the NACMAST consortium under contract EWAGSI-07-SC-0003.

An Isomorphism Between The Fusion Algebras Of Vl+ And Type D(1) Level 2, Michael Cuntz, Christopher D. Goff

College of the Pacific Faculty Presentations

This talk will show that the fusion algebra of the vertex operator algebra V_L⁺ (where L is a rank 1 even lattice) is isomorphic to the fusion algebra of the affine Kac-Moody Lie algebra of type D⁽¹⁾ at level 2.

Introduction To The "Prisoners And Guards" Game, Eugen J. Ionascu, Tim Howard

Faculty Bibliography

We study the half-dependent problem for the king graph Kn. We give proofs to establish the values h(Kn) for n ∈ {1, 2, 3, 4, 5, 6} and an upper bound for h(Kn) in general. These proofs are independent of computer assisted results. Also, we introduce a two-player game whose winning strategy is tightly related with the values h(Kn). This strategy is analyzed here for n = 3 and some facts are given for the case n = 4. Although the rules of the game are very simple, the winning strategy is highly complex even for n = 4.

Solution Of Problem 878, Eugen J. Ionascu

Faculty Bibliography

No abstract provided.

Galois Structure And De Rhan Invariants Of Elliptic Curves, Darren B. Glass, Sonin Kwon

Math Faculty Publications

Let K be a number field with ring of integers OK. Suppose a finite group G acts numerically tamely on a regular scheme X over OK. One can then define a de Rham invariant class in the class group Cl(OK[G]), which is a refined Euler characteristic of the de Rham complex of X. Our results concern the classification of numerically tame actions and the de Rham invariant classes. We first describe how all Galois etale G-covers of a K-variety may be built up from finite Galois extensions of K and from geometric covers. When X is a curve of positive …

Reliability Confidence Intervals For Oil Spills In The Gulf Of Mexico, William V. Harper, Thomas R. James, Ted G. Eschenbach

Mathematics Faculty Scholarship

An extensive study [Eschenbach and Harper (2006)] of offshore oil spills in the Gulf of Mexico with extensions to the northern seas of Alaska involved the estimation of the likelihood of oil spill volumes in the Gulf of Mexico for both pipeline and platform spills. This paper develops both maximum likelihood based reliability and percentile confidence intervals for the 3- parameter Weibull distribution. The statistical aspects are discussed along with applications of developed Excel VBA functions. The Excel routines are available free on the web at http://faculty.otterbein.edu/WHarper/. The functions are illustrated with Gulf of Mexico oil spill data.

Algebras Having Bases Consisting Entirely Of Units, Jeremy Moore

Mathematics Faculty Scholarship

We introduce a hierarchy of notions about algebras having a basis B consisting entirely of units. Such a basis is called an invertible basis and algebras that have invertible bases are said to be invertible algebras...

Degree Spectra Of The Successor Relation Of Computable Linear Orderings, Jennifer Chubb, Andrey Frolov, Valentina Harizanov

Mathematics

We establish that for every computably enumerable (c.e.) Turing degree b, the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings, the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees.

Generalized Helmholtz-Kirchhoff Model For Two-Dimensional Distributed Vortex Motion, Raymond J. Nagem, Guido Sandri, David Uminsky, C. Eugene Wayne

Mathematics

The two-dimensional Navier-Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz-Kirchhoff model for the evolution of point vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the …

Cooperation In An Evolutionary Prisoner's Dilemma On Networks With Degree-Degree Correlations, Stephen Devlin, T Treloar

Mathematics

We study the effects of degree-degree correlations on the success of cooperation in an evolutionary prisoner's dilemma played on a random network. When degree-degree correlations are not present, the standardized variance of the network's degree distribution has been shown to be an accurate analytical measure of network heterogeneity that can be used to predict the success of cooperation. In this paper, we use a local-mechanism interpretation of standardized variance to give a generalization to graphs with degree-degree correlations. Two distinct mechanisms are shown to influence cooperation levels on these types of networks. The first is an intrinsic measurement of base-line …

Evolution Of Cooperation Through The Heterogeneity Of Random Networks, Stephen Devlin, T Treloar

Mathematics

We use the standardized variance (nu_{st}) of the degree distribution of a random network as an analytic measure of its heterogeneity. We show that nu_{st} accurately predicts, quantitatively, the success of cooperators in an evolutionary prisoner's dilemma. Moreover, we show how the generating functional expression for nu_{st} suggests an intrinsic interpretation for the heterogeneity of the network that helps explain local mechanisms through which cooperators thrive in heterogeneous populations. Finally, we give a simple relationship between nu_{st} , the cooperation level, and the epidemic threshold of a random network that reveals an appealing connection between epidemic disease models and the …

A Finite Element Splitting Extrapolation For Second Order Hyperbolic Equations, Xiaoming He, Tao Lü

Mathematics and Statistics Faculty Research & Creative Works

Splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel. This article discusses a finite element splitting extrapolation for second order hyperbolic equations with time-dependent coefficients. This method possesses a higher degree of parallelism, less computational complexity, and more flexibility than Richardson extrapolation while achieving the same accuracy. By means of domain decomposition and isoparametric mapping, some grid parameters are chosen according to the problem. The multiparameter asymptotic expansion of the d-quadratic finite element error is also established. The splitting extrapolation formulas are developed from this expansion. An approximation with higher accuracy on a …

The Dixmier-Douady Invariant For Dummies, Claude Schochet

Mathematics Faculty Research Publications

The Dixmier-Douady invariant is the primary tool in the classification of continuous trace C*-algebras. These algebras have come to the fore in recent years because of their relationship to twisted K-theory and via twisted K-theory to branes, gerbes, and string theory.

This note sets forth the basic properties of the Dixmier-Douady invariant using only classical homotopy and bundle theory. Algebraic topology enters the scene at once since the algebras in question are algebras of sections of certain fibre bundles.

Banach Algebras And Rational Homotopy Theory, Gregory Lupton, N. Christopher Phillips, Claude Schochet, Samuel B. Smith

Mathematics Faculty Research Publications

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GL_n(A), the group of invertible n x n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lc_n(A) denote the space of "last columns" of GL_n(A). We construct a natural isomorphism

Ȟ^s(Max(A);ℚ) ≅ π_2n-1-s(Lc_n(A)) ⊗ ℚ …

Hybrid Approximate Proximal Method With Auxiliary Variational Inequality For Vector Optimization, L C. Ceng, Boris S. Mordukhovich, Jen-Chih Yao

Mathematics Research Reports

This paper studies the general vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C C Y with nonempty interior. In order to find a solution of this problem, we introduce an auxiliary variational inequality problem for monotone, Lipschitz-continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem by the combination of extragradient method for finding a solution to the variational inequality problem and approximate proximal …

A Newton Root-Finding Algorithm For Estimating The Regularization Parameter For Solving Ill-Conditioned Least Squares Problems, Jodi Mead, Rosemary Renaut

Mathematics Faculty Publications and Presentations

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a X² distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value …