The Creation Of A Video Review Guide For The Free-Response Section Of The Advanced Placement Calculus Exam, 2016 Western Michigan University

#### The Creation Of A Video Review Guide For The Free-Response Section Of The Advanced Placement Calculus Exam, Jeffrey Brown

*Honors Theses*

*The Creation of a Video Review Guide for the Free-Response Section of the Advanced Placement Calculus Exam *follows the creation of a resource to help students prepare for the College Board’s Advanced Placement Calculus Exam. This project originated out of the authors personal experiences in preparing for this exam. The goal of the project was to create an accessible resource that reviews content, provides insights into the Advanced Placement exam, and creates successful habits in student responses. This paper, chronologically, details the development of the resource and a reflection on the final product and future uses.

Multilevel Models For Longitudinal Data, 2016 East Tennessee State University

#### Multilevel Models For Longitudinal Data, Aastha Khatiwada

*Electronic Theses and Dissertations*

Longitudinal data arise when individuals are measured several times during an ob- servation period and thus the data for each individual are not independent. There are several ways of analyzing longitudinal data when different treatments are com- pared. Multilevel models are used to analyze data that are clustered in some way. In this work, multilevel models are used to analyze longitudinal data from a case study. Results from other more commonly used methods are compared to multilevel models. Also, comparison in output between two software, SAS and R, is done. Finally a method consisting of fitting individual models for each ...

Applications Of Discrete Mathematics For Understanding Dynamics Of Synapses And Networks In Neuroscience, 2016 University of Nebraska - Lincoln

#### Applications Of Discrete Mathematics For Understanding Dynamics Of Synapses And Networks In Neuroscience, Caitlyn Parmelee

*Dissertations, Theses, and Student Research Papers in Mathematics*

Mathematical modeling has broad applications in neuroscience whether we are modeling the dynamics of a single synapse or the dynamics of an entire network of neurons. In Part I, we model vesicle replenishment and release at the photoreceptor synapse to better understand how visual information is processed. In Part II, we explore a simple model of neural networks with the goal of discovering how network structure shapes the behavior of the network.

Vision plays an important role in how we interact with our environments. To fully understand how visual information is processed requires an understanding of the way signals are ...

Hybrid Chebyshev Polynomial Scheme For The Numerical Solution Of Partial Differential Equations, 2016 University of Southern Mississippi

#### Hybrid Chebyshev Polynomial Scheme For The Numerical Solution Of Partial Differential Equations, Balaram Khatri Ghimire

*Dissertations*

In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including ...

An Algorithm For The Machine Calculation Of Minimal Paths, 2016 East Tennessee State University

#### An Algorithm For The Machine Calculation Of Minimal Paths, Robert Whitinger

*Electronic Theses and Dissertations*

Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R^{3}, but also to the general case of finding minimal functionals on hypersurfaces in R^{n} associated with an arbitrary metric.

Octahedral Dice, 2016 Butler University

#### Octahedral Dice, Todd Estroff, Jeremiah Farrell

*Jeremiah Farrell*

All five Platonic solids have been used as random number generators in games involving chance with the cube being the most popular. Martin Gardenr, in his article on dice (MG 1977) remarks: "Why cubical?... It is the easiest to make, its six sides accomodate a set of numbers neither too large nor too small, and it rolls easily enough but not too easily."

Gardner adds that the octahedron has been the next most popular as a randomizer. We offer here several problems and games using octahedral dice. The first two are extensions from Gardner's article. All answers will be ...

The Most-Perfect 4*4 Puzzle Game, 2016 Butler University

#### The Most-Perfect 4*4 Puzzle Game, Jeremiah Farrell, Karen Farrell

*Jeremiah Farrell*

Draft of the 'Solution Page' for Jeremiah's puzzle "The Most Perfect 4*4 Puzzle Game", which was exchanged at the 2014 London International Puzzle Party. 100 puzzle designers create 100 copies of their puzzle and pass it out at the party and exhange them. Includes e-mail correspondence with editor of the book.

Seven Roads To Roam: A Magical Journey, 2016 Butler University

#### Seven Roads To Roam: A Magical Journey, Jeremiah Farrell, Judith H. Morrel

*Jeremiah Farrell*

Farrell and Morrel's contribution to "Homage to a Pied Puzzler"

**Related Works: **

See "The ASTEROID Puzzle and Games" online at:

http://digitalcommons.butler.edu/facsch_papers/913/

The Magic Octagon, 2016 Butler University

#### The Magic Octagon, Jeremiah Farrell, Tom Rodgers

*Jeremiah Farrell*

The black nodes mark the corners of an octagon and each of these nodes in connected to four others by lines. The (rather hard) puzzle is to assign the sixteen numbers 0 through 15 to each of the sixteen lines so that each black node has a sum of 30 when the line numbers leading into it are added.

The word version of the puzzle was described in the article "Most-Perfect Word Magic", Oscar Thumpbindle, *Word Ways* Vol. 40(4). Nov. 2007.

Some Curious Cut-Ups, 2016 Butler University

#### Some Curious Cut-Ups, Jeremiah Farrell, Ivan Moscovich

*Jeremiah Farrell*

We have noticed a certain kind of n-gon dissection into triangles that has a wonderful property of interest to most puzzlists. Namely that any two triangles have at least one edge in common yet no two triangles need be congruent. In an informal poll of specialists at a recent convention, none of them saw immediately how this could be accomplished. But in fact it is very straightforward.

Farrell's Spider, 2016 Butler University

#### Farrell's Spider, Jeremiah Farrell, Ivan Moscovich

*Jeremiah Farrell*

Puzzle game featured in Ivan Moscovich's magnetic puzzle pack:

Place the 18 discs on the web so that the sum of the numbers on each of the three hexagons and on each of the three ribs equals 57.

A Dual Fano, And Dual Non-Fano Matroidal Network, 2016 California State University - San Bernardino

#### A Dual Fano, And Dual Non-Fano Matroidal Network, Stephen Lee Johnson

*Electronic Theses, Projects, and Dissertations*

Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions. In this paper we examine the basic information on network coding and matroid theory. We then go over the method of creating matroidal networks. Finally we construct matroidal networks from the dual of the fano matroid and the dual of the non-fano matroid, and breifly discuss some coding solutions.

Realizing Tournaments As Models For K-Majority Voting, 2016 California State University - San Bernardino

#### Realizing Tournaments As Models For K-Majority Voting, Gina Marie Cheney

*Electronic Theses, Projects, and Dissertations*

A *k*-majority tournament is a directed graph that models a *k*-majority voting scenario, which is realized by 2*k* - 1 rankings, called linear orderings, of the vertices in the tournament. Every *k*-majority voting scenario can be modeled by a tournament, but not every tournament is a model for a *k*-majority voting scenario. In this thesis we show that all acyclic tournaments can be realized as 2-majority tournaments. Further, we develop methods to realize certain quadratic residue tournaments as *k*-majority tournaments. Thus, each tournament within these classes of tournaments is a model for a *k*-majority ...

The Kauffman Bracket And Genus Of Alternating Links, 2016 California State University, San Bernardino

#### The Kauffman Bracket And Genus Of Alternating Links, Bryan M. Nguyen

*Electronic Theses, Projects, and Dissertations*

Giving a knot, there are three rules to help us finding the Kauffman bracket polynomial. Choosing knot’s orientation, then applying the Seifert algorithm to find the Euler characteristic and genus of its surface. Finally finding the relationship of the Kauffman bracket polynomial and the genus of the alternating links is the main goal of this paper.

The Evolution Of Cryptology, 2016 California State University - San Bernardino

#### The Evolution Of Cryptology, Gwendolyn Rae Souza

*Electronic Theses, Projects, and Dissertations*

We live in an age when our most private information is becoming exceedingly difficult to keep private. Cryptology allows for the creation of encryptive barriers that protect this information. Though the information is protected, it is not entirely inaccessible. A recipient may be able to access the information by decoding the message. This possible threat has encouraged cryptologists to evolve and complicate their encrypting methods so that future information can remain safe and become more difficult to decode. There are various methods of encryption that demonstrate how cryptology continues to evolve through time. These methods revolve around different areas of ...

Development Of Utility Theory And Utility Paradoxes, 2016 Lawrence University

#### Development Of Utility Theory And Utility Paradoxes, Timothy E. Dahlstrom

*Lawrence University Honors Projects*

Since the pioneering work of von Neumann and Morgenstern in 1944 there have been many developments in Expected Utility theory. In order to explain decision making behavior economists have created increasingly broad and complex models of utility theory. This paper seeks to describe various utility models, how they model choices among ambiguous and lottery type situations, and how they respond to the Ellsberg and Allais paradoxes. This paper also attempts to communicate the historical development of utility models and provide a fresh perspective on the development of utility models.

Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, 2016 Portland State University

#### Math And Sudoku: Exploring Sudoku Boards Through Graph Theory, Group Theory, And Combinatorics, Kyle Oddson

*Student Research Symposium*

Encoding Sudoku puzzles as partially colored graphs, we state and prove Akman’s theorem [1] regarding the associated partial chromatic polynomial [5]; we count the 4x4 sudoku boards, in total and fundamentally distinct; we count the diagonally distinct 4x4 sudoku boards; and we classify and enumerate the different structure types of 4x4 boards.

Neighborhood-Restricted Achromatic Colorings Of Graphs, 2016 East Tennessee State Universtiy

#### Neighborhood-Restricted Achromatic Colorings Of Graphs, James D. Chandler Sr.

*Electronic Theses and Dissertations*

A (closed) neighborhood-restricted 2-achromatic-coloring of a graph G is an assignment of colors to the vertices of G such that no more than two colors are assigned in any closed neighborhood. In other words, for every vertex v in G, the vertex v and its neighbors are in at most two different color classes. The 2-achromatic number is defined as the maximum number of colors in any 2-achromatic-coloring of G. We study the 2-achromatic number. In particular, we improve a known upper bound and characterize the extremal graphs for some other known bounds.

Row And Column Distributions Of Letter Matrices, 2016 College of William and Mary

#### Row And Column Distributions Of Letter Matrices, Xiaonan Hu

*College of William & Mary Undergraduate Honors Theses*

A *letter matrix* is an *n*-by-*n* matrix whose entries are *n* symbols, each appearing *n* times. The row (column) distribution of a letter matrix is an *n*-by-*n* nonnegative integer matrix that tells how many of each letter are in each row (column). A row distribution *R* and a column distribution *C* are compatible if there exits a letter matrix *A* whose row distribution is *R* and whose column distribution is *C*. We show that the matrix *J* of all ones is compatible with any *C*, and we also consider the the problem of when *R* and ...

The Root Finite Condition On Groups And Its Application To Group Rings, 2016 University of Wisconsin-Milwaukee

#### The Root Finite Condition On Groups And Its Application To Group Rings, James Gollin

*Theses and Dissertations*

A group $G$ is said to satisfy the root-finite condition if for every $g \in G$, there are only finitely many $x \in G$ such that there exists a positive integer $n$ such that $x^n = g$. It is shown that groups satisfy the root-finite condition iff they satisfy three subconditions, which are shown to be independent. Free groups are root-finite. Ordered groups are shown to satisfy one of the subconditions for the root-finite condition. Finitely generated abelian groups satisfy the root-finite condition. If, in a torsion-free abelian group $G$, there exists a positive integer $r$ such that the subgroup ...