Mathematics Commons^™

Improving The Development Of The I-Chart For Use In Biopharmaceutical Manufacturing Operation, Kimberly Schveder

Undergraduate Research Posters 2017

The Shewhart control chart is a statistical tool used by pharmaceutical companies, as well as chemical and other batch manufacturers, to help detect errors in the manufacturing process and ensure control of product quality. One particular type of control chart is the I-chart. The average run length (ARL) statistic of the I-chart can easily be determined when output from the manufacturing process is normally distributed with known population parameters. This paper investigates the impact on the ARL statistic when the I-chart is based on mean and standard deviation estimates obtained from small sample sizes of less than 50 batches. The …

P1: How High Does The Lower Atmosphere Go?, Vladimir Sworski, Justin Flaherty

Undergraduate Research Posters 2017

The Atmospheric Boundary Layer (ABL), consisting of the bottom few kilometers of the troposphere, is a region with strong mixing of moisture and winds. This region's activity has a large impact on weather and climate models. In this study, we use a high resolution computer model: Large Eddy Simulation (LES). Statistics produced require a strong understanding of the height of the ABL. The purpose of this study was to create a method for determining this height accurately and consistently, as previous models demonstrated significant error.

Rational Number Operations On The Number Line, Christopher Yakes

The Mathematics Enthusiast

The Common Core State Standards make clear that teachers should use the number line to represent integer operations in Grades 6, 7 and 8, continuing the development of number line understanding begun in earlier grades. In this paper, we will investigate the basics of the colored chip and number line models for representing integers and their operations. We will then draw connections between the two models. Finally, we will argue that the number line model is an important one that should be taught alongside or in lieu of the chip model in order to deepen students’ understanding operations with integers, …

Perihelion Precession In The General Theory Of Relativity, Charles G. Torre

Tutorials on... in 1 hour or less

This is a relatively quick and informal sketch of a demonstration that general relativistic corrections to the bound Kepler orbits introduce a perihelion precession. Any decent textbook on the general theory of relativity will derive this result. My analysis aligns with that found in the good old text "Introduction to General Relativity", by Adler, Bazin and Schiffer.

On The Definition Of Linear Independence, Yonah Cherniavsky, Artour Mouftakhov

The Mathematics Enthusiast

We discuss a certain very common flaw in the definition of linear independence, which is one of the most important concepts taught in any college or university course of Linear Algebra. This note may be useful to lecturers and students which teach and study Linear Algebra of any level and like the mathematically rigorous approach.

Numberlines: Hockey Line Nicknames Based On Jersey Numbers, Egan J. Chernoff

The Mathematics Enthusiast

The purpose of this article, in general, is to expound Chernoff’s (2016) notion of numberlines, that is, hockey line nicknames based on jersey numbers. The article begins with a brief discussion of the history of hockey line nicknames, which allows for the parsing of numberlines and quasi-numberlines (nicknames based on numbers associated with hockey players). Focusing, next, on jersey number restrictions for the National Hockey League (NHL), a repeated calculation of the number of possible numberlines winnows down the number from a theoretical upper bound to a practical upper bound. Moving beyond the numbers, the names of natural numbers – …

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.