Mathematics Commons^™

The Constructs Of Phd Students About Infinity: An Application Of Repertory Grids, Serdar Aztekin, Ahmet Arikan, Bharath Sriraman

The Mathematics Enthusiast

Infinity has been one of the more difficult concepts for humanity to grasp. A major component of the research on mathematics education related to infinity has been the study of student’s conceptions and reasoning about calculus subjects, particularly limits and series. Some related studies are about Cantor’s ordinal and cardinal infinity. However since most students at the high school and college level are unfamiliar with symbolic representations and terminology, such as a set theoretic approach, a context (generally geometric) is used for investigating notions of infinity indirectly. In this paper we report on a study on the constructs of PhD …

When Is .999... Less Than 1?, Karin Usadi Katz, Mikhail G. Katz

The Mathematics Enthusiast

We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's "semicolon" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999... be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student.

Bringing Toric Codes To The Next Dimension, Ivan Soprunov, Jenya Soprunova

Mathematics and Statistics Faculty Publications

This paper is concerned with the minimum distance computation for higher dimensional toric codes defined by lattice polytopes in $\mathbb{R}^n$. We show that the minimum distance is multiplicative with respect to taking the product of polytopes, and behaves in a simple way when one builds a k-dilate of a pyramid over a polytope. This allows us to construct a large class of examples of higher dimensional toric codes where we can compute the minimum distance explicitly.

Alternative Technical Efficiency Measures: Skew, Bias, And Scale, William Clinton Horrace, Qu Feng

Center for Policy Research

In the fixed-effects stochastic frontier model an efficiency measure relative to the best firm in the sample is universally employed. This paper considers a new measure relative to the worst firm in the sample. We find that estimates of this measure have smaller bias than those of the traditional measure when the sample consists of many firms near the efficient frontier. Moreover, a two-sided measure relative to both the best and the worst firms is proposed. Simulations suggest that the new measures may be preferred depending on the skewness of the inefficiency distribution and the scale of efficiency differences.

Large Cardinals, Oliver Pechenik

Honors Papers

Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of …

The Fundamental Gap For Hyperbolic Triangles, Dennis Stuart Fillebrown

Honors Theses

In 1983, M. van den Berg made his Fundamental Gap Conjecture about the difference between the first two Dirichlet eigenvalues (the fundamental gap) of any convex domain in the Euclidean plane. Recently, progress has been made in the case where the domains are polygons and, in particular, triangles. We examine the conjecture for triangles in hyperbolic geometry, though we seek an for an upper bound for the fundamental gap rather than a lower bound.

Gliomapredict: A Clinically Useful Tool For Assigning Glioma Patients To Specific Molecular Subtypes, Aiguo Li, Serdar Bozdag, Yuri Kotliarov, Howard A. Fine

Mathematics, Statistics and Computer Science Faculty Research and Publications

Background: Advances in generating genome-wide gene expression data have accelerated the development of molecular-based tumor classification systems. Tools that allow the translation of such molecular classification schemas from research into clinical applications are still missing in the emerging era of personalized medicine.

Results: We developed GliomaPredict as a computational tool that allows the fast and reliable classification of glioma patients into one of six previously published stratified subtypes based on sets of extensively validated classifiers derived from hundreds of glioma transcriptomic profiles. Our tool utilizes a principle component analysis (PCA)-based approach to generate a visual representation of the analyses, quantifies …

A Class Of Discontinuous Petrov–Galerkin Methods. Ii. Optimal Test Functions, Leszek Demkowicz, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through …

Multigrid In A Weighted Space Arising From Axisymmetric Electromagnetics, Dylan M. Copeland, Jay Gopalakrishnan, Minah Oh

Mathematics and Statistics Faculty Publications and Presentations

Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex then the multigrid Vcycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a …

Edge Coloring Bibds And Constructing Moelrs , John S. Asplund

Dissertations, Master's Theses and Master's Reports - Open

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Some historical uses and background are touched upon as well. The majority of the definitions are contained within this chapter as well.

In Chapter 2 we consider the question whether one can decompose λ copies of monochromatic K_v into copies of K_k such that each copy of the K_k contains at most one edge from each K_v. This is called a proper edge coloring (Hurd, Sarvate, [29]). The majority of the content in this section is a wide variety of …

Super Special Codes Using Super Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

The new classes of super special codes are constructed in this book using the specially constructed super special vector spaces. These codes mainly use the super matrices. These codes can be realized as a special type of concatenated codes. This book has four chapters. In chapter one basic properties of codes and super matrices are given. A new type of super special vector space is constructed in chapter two of this book. Three new classes of super special codes namely, super special row code, super special column code and super special codes are introduced in chapter three. Applications of these …

Fusion Of Imprecise Qualitative Information, Florentin Smarandache, Xinde Li, Xianzhong Dai, Jean Dezert

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we present a new 2-tuple linguistic representation model, i.e. Distribution Function Model (DFM), for combining imprecise qualitative information using fusion rules drawn from Dezert-Smarandache Theory (DSmT) framework. Such new approach allows to preserve the precision and efficiency of the combination of linguistic information in the case of either equidistant or unbalanced label model. Some basic operators on imprecise 2-tuple labels are presented together with their extensions for imprecise 2-tuple labels. We also give simple examples to show how precise and imprecise qualitative information can be combined for reasoning under uncertainty. It is concluded that DSmT can deal …

Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers Of Sciences), Vol. Iv, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors - all collected in the previous three decades (1980-2010) – but most of them are from the last decade (2000-2010), some of them being lost and found, yet others are extended, diversified, improved versions. This is an eclectic tome of 800 pages with papers …

Morphing Of Triangular Meshes In Shape Space, Stefanie Wuhrer, Prosenjit Bose, Chang Shu, Joseph O'Rourke, Alan Brunton

Computer Science: Faculty Publications

We present a novel approach to morph between two isometric poses of the same non-rigid object given as triangular meshes. We model the morphs as linear interpolations in a suitable shape space S. For triangulated 3D polygons, we prove that interpolating linearly in this shape space corresponds to the most isometric morph in R³ . We then extend this shape space to arbitrary triangulations in 3D using a heuristic approach and show the practical use of the approach using experiments. Furthermore, we discuss a modified shape space that is useful for isometric skeleton morphing. All of the newly presented …

A Class Of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation, Leszek Demkowicz, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size h and polynomial degree p, and outperforms the standard upwind DG method.

A Projection-Based Error Analysis Of Hdg Methods, Jay Gopalakrishnan, Bernardo Cockburn, Francisco-Javier Sayas

Mathematics and Statistics Faculty Publications and Presentations

We introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods. The technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods. This renders the analysis of the projections of the discretization errors simple and concise. By showing that these projections of the errors are bounded in terms of the distance between the solution and its projection, our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection. We illustrate the technique on a specific HDG …

Hybridization And Postprocessing Techniques For Mixed Eigenfunctions, Bernardo Cockburn, Jay Gopalakrishnan, F. Li, Ngoc Cuong Nguyen, Jaume Peraire

Mathematics and Statistics Faculty Publications and Presentations

We introduce hybridization and postprocessing techniques for the Raviart–Thomas approximation of second-order elliptic eigenvalue problems. Hybridization reduces the Raviart–Thomas approximation to a condensed eigenproblem. The condensed eigenproblem is nonlinear, but smaller than the original mixed approximation. We derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates. An element-by-element postprocessing technique to improve accuracy of computed eigenfunctions is also presented. We prove that a projection of the error in the eigenspace approximation by the mixed method (of any order) superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions. Numerical experiments using …

On The Integrability Of Orthogonal Distributions In Poisson Manifolds, Daniel Fish, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provided.

Phase-Linking And The Perceived Motion During Off-Vertical Axis Rotation, Jan Holly, Scott Wood, Gin Mccollum

Mathematics and Statistics Faculty Publications and Presentations

Human off-vertical axis rotation (OVAR) in the dark typically produces perceived motion about a cone, the amplitude of which changes as a function of frequency. This perception is commonly attributed to the fact that both the OVAR and the conical motion have a gravity vector that rotates about the subject. Little-known, however, is that this rotating-gravity explanation for perceived conical motion is inconsistent with basic observations about self-motion perception: (a) that the perceived vertical moves toward alignment with the gravito-inertial acceleration (GIA) and (b) that perceived translation arises from perceived linear acceleration, as derived from the portion of the GIA …

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, …