Mathematics Commons^™

Developing Mental Rotation Ability Through Engagement In Assignments That Involve Solids Of Revolution, Atara Shriki, Ruthi Barkai, Dorit Patkin

The Mathematics Enthusiast

Spatial ability is essential for succeeding in the STEM (Sciences, Technology, Engineering and Mathematics) disciplines, especially mental rotation. Research points out that spatial ability is malleable, and therefore calls for developing learners’ ability by engaging them in appropriate assignments, starting from kindergarten. Given this, our paper presents several assignments designed for mathematics prospective teachers with the aim of fostering their mental rotation skills. Specifically, these assignments deal with solids of revolution, three-dimensional shapes formed by revolving a planar shape about a given axis that lies on the same plane.

The Document Similarity Network: A Novel Technique For Visualizing Relationships In Text Corpora, Dylan Baker

HMC Senior Theses

With the abundance of written information available online, it is useful to be able to automatically synthesize and extract meaningful information from text corpora. We present a unique method for visualizing relationships between documents in a text corpus. By using Latent Dirichlet Allocation to extract topics from the corpus, we create a graph whose nodes represent individual documents and whose edge weights indicate the distance between topic distributions in documents. These edge lengths are then scaled using multidimensional scaling techniques, such that more similar documents are clustered together. Applying this method to several datasets, we demonstrate that these graphs are …

Sudoku Variants On The Torus, Kira A. Wyld

HMC Senior Theses

This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.

Pattern Recognition In Stock Data, Kathryn Dover

HMC Senior Theses

Finding patterns in high dimensional data can be difficult because it cannot be easily visualized. There are many different machine learning methods to fit data in order to predict and classify future data but there is typically a large expense on having the machine learn the fit for a certain part of a dataset. We propose a geometric way of defining different patterns in data that is invariant under size and rotation. Using a Gaussian Process, we find that pattern within stock datasets and make predictions from it.

Toric Ideals, Polytopes, And Convex Neural Codes, Caitlin Lienkaemper

HMC Senior Theses

How does the brain encode the spatial structure of the external world?

A partial answer comes through place cells, hippocampal neurons which

become associated to approximately convex regions of the world known

as their place fields. When an organism is in the place field of some place

cell, that cell will fire at an increased rate. A neural code describes the set

of firing patterns observed in a set of neurons in terms of which subsets

fire together and which do not. If the neurons the code describes are place

cells, then the neural code gives some information about the …

The Combinatorics Of Modified Macdonald Polynomials, Jacob Rodeheffer

Theses, Dissertations and Capstones

This paper is concerned with finding a combinatorial Schur expansion of the modified Macdonald polynomials. We will use the Robinson–Schensted–Knuth (RSK) algorithm to obtain the result for a limited set of partition shapes, and we will use Foata’s bijection to extend the result to conjugate shapes. We will explore possibilities for modifying the RSK algorithm in such a way that it could be applied to obtain the result for more general shapes.

Structure Formulas For Wave Operators Under A Small Scaling Invariant Condition, Marius Beceanu, Wilhelm Schlag

Mathematics and Statistics Faculty Scholarship

We obtain structure formulas for the intertwining wave operators of a Schroedinger operator with potential V in R^3. The difference from our previous submission arXiv:1612.07304 lies with the fact that here we impose a scaling invariant condition on the potential, albeit with a smallness requirement.

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications, Ren Zhao

Wayne State University Dissertations

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing,

a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems.

Our first target is to develop a boundary …

Eigenvalue Dependence Of Numerical Oscillations In Parabolic Partial Differential Equations, R. Corban Harwood

Faculty Publications - Department of Mathematics

This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations can create infeasible results. Since oscillation-free behavior is not ensured by stability conditions, a more precise condition would be useful for accurate solutions. Using Von Neumann and spectral analyses, we find and explore oscillation-free conditions for several finite difference schemes. Further relationships between oscillatory behavior and eigenvalues is supported with numerical evidence and proof. Also, evidence suggests that the oscillation-free stability condition for a consistent linearization may be sufficient to provide oscillation-free stability …

Steady And Stable: Numerical Investigations Of Nonlinear Partial Differential Equations, R. Corban Harwood

Faculty Publications - Department of Mathematics

Excerpt: "Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation."

05. Table Of Contents - Design And Analysis Of Experiments, Angela Dean, Dan Voss, Danel Draguljic

Design and Analysis of Experiments

The table of content of Design and Analysis of Experiments.

04. Chapters - Design And Analysis Of Experiments, Angela Dean, Dan Voss, Danel Draguljic

Design and Analysis of Experiments

A list of the chapters from Design and Analysis of Experiments.

02. Preface Of Design And Analysis Of Experiments - 2nd Edition, Angela Dean, Dan Voss, Danel Draguljic

Design and Analysis of Experiments

Preface of the second edition of Design and Analysis of Experiments.

09. R Data Files For Design And Analysis Of Experiments, Angela Dean, Dan Voss, Danel Draguljic

Design and Analysis of Experiments

R Data files for Design and Analysis of Experiments.

07: Tan Animation, Ruth Dover

Mathematica Notebooks for Pre-Calculus and Calculus

TanAnimation.nb is a nice little animation to illustrate similar triangles to show why the graph of y = tan(x) looks the way it does.

A Radial Basis Function (Rbf) Compact Finite Difference (Fd) Scheme For Reaction-Diffusion Equations On Surfaces, Erik Lehto, Varun Shankar, Grady B. Wright

Mathematics Faculty Publications and Presentations

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in ℝ^d. The novelty of the method is in the approximation of the Laplace-Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace-Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at …

The Motivic Cofiber Of Τ And Exotic Periodicities, Bogdan Gheorghe

Wayne State University Dissertations

Consider the Tate twist τ ∈ H 0,1 (S 0,0 ) in the mod 2 cohomology of the motivic sphere.

After 2-completion, the motivic Adams spectral sequence realizes this element as a map

τ : S 0,−1 GGA S 0,0 . This thesis begins with the study of its cofiber, that we denote by Cτ.

We first show that this motivic 2-cell complex can be endowed with a unique E ∞ ring

structure. This promotes the known isomorphism π ∗,∗ Cτ ∼= Ext ∗,∗ BP ∗ BP (BP ∗ ,BP ∗ )

to an isomorphism of rings which also preserves …

Reverse Mathematics Of Matroids, Jeffry L. Hirst, Carl Mummert

Mathematics Faculty Research

Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for \Sigma^0_2 formulas.

Introduction To Axiomatic Geometry, Mark Barsamian

OHIO Open Faculty Textbooks

This book presents Euclidean Geometry and was designed for a one-semester course preparing junior and senior level college students to teach high school Geometry. The book could also serve as a text for a junior level Introduction to Proofs course.

Martin Gardner Puzzle-Games, Stephen Bloom, Lacey Echols, Jeremiah Farrell, Shannon Lieb

Scholarship and Professional Work - LAS

No abstract provided.

Euler Entertainments, Jeremiah Farrell, Karen Farrell

Scholarship and Professional Work - LAS

No abstract provided.

A Gathering For Gardner Puzzle-Game, Jeremiah Farrell, Chris Morgan

Scholarship and Professional Work - LAS

Each different letter of "GATHERING FOR GARDNER" is used exactly three times in the following words: DIE, FAD, FIT, FOG, GIN, HAG, HER, HOD, NOR, RAT, TEN.

The Tea Party, Stephen Bloom, Jeremiah Farrell

Scholarship and Professional Work - LAS

No abstract provided.

The French Resolution, Jeremiah Farrell, Karen Farrell

Scholarship and Professional Work - LAS

Jeremiah's puzzle "The French Resolution", which was exchanged at the 2017 International Puzzle Party in Paris, France. 100 puzzle designers create 100 copies of their puzzle and pass it out at the party and exchange them. This puzzle is also manufactured by Kadon Enterprises, Inc. as "The French Resolution".

Two Games Displayed By Butler’S 2017 Celebration Of Mind, Jeremiah Farrell

Scholarship and Professional Work - LAS

Jeremiah's two games displayed by Butler's 2017 Celebration of Mind.

Spectrally Similar Incommensurable 3-Manifolds, David Futer, Christian Millichap

Faculty Publications

Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds N_n and N^µ_n whose volume is approximately n and whose length spectra agree up to length n.

Both N_n and N^µ_n are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that …

Mutations And Short Geodesics In Hyperbolic 3-Manifolds, Christian Millichap

Faculty Publications

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.

The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least …

Arithmetic | Algebra Homework, Samar Elhitti, Ariane Masuda, Lin Zhou

Open Educational Resources

Arithmetic | Algebra Homework book is a static version of the WeBWork online homework assignments that accompany the textbook Arithmetic | Algebra for the developmental math courses MAT 0630 and MAT 0650 at New York City College of Technology, CUNY.

Subtraction Involving Negative Numbers: Connecting To Whole Number Reasoning, Laura Bofferding, Nicole Wessman-Enzinger

Faculty Publications - College of Education

In this article, we explore how students attempt to bridge from their whole number reasoning to integer reasoning as they solve subtraction problems involving negative numbers. Based on interviews with students ranging from first graders to preservice teachers, we identify two overarching strategies: making connections to known problem types and leveraging conceptions of subtraction. Their initial connections suggest that rather than identifying the best instructional models to teach integer concepts, we should focus on identifying integer instructional models that build on the potentially productive connections that students’ already make; we propose an example of one such form of instruction.

A Study Of Fourth-Grade Students' Perceptions On Homework Environment And Academic Motivation In Mathematics, Stefanie Harmon

Walden Dissertations and Doctoral Studies

The problem at an elementary school is teachers' lack of knowledge and information on the perceptions and motivation of students to complete independent mathematics homework. The purpose of this study was to identify students' perceptions regarding their homework environment and academic motivation in mathematics. The study's conceptual framework, attribution theory, supported the examination of drivers of motivation for participants related to homework completion. Guiding research questions, supported by Keller's ARCS model, focused on the identification of students' perceptions of homework attention, relevance, curiosity, satisfaction, and their preferred homework environment. This qualitative research study obtained data from semistructured interviews with 44 …