Digital Commons Network^™

The Schur Factorization Property As It Applies To Subsets Of The General Laguerre Polynomials, Christopher A. Gunnell

Senior Projects Spring 2016

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

Orthogonal Projections Of Lattice Stick Knots, Margaret Marie Allardice

Senior Projects Spring 2016

A lattice stick knot is a closed curve in R³ composed of finitely many line segments, sticks, that lie parallel to the three coordinate axes in R³, such that the line segments meet at points in the 3-dimensional integer lattice. The lattice stick number of a knot is the minimal number of sticks required to realize that knot as a lattice stick knot. A right angle lattice projection is a projection of a knot in R³onto the plane such that the edges of the projection lie parallel to the two coordinate axes in the plane, …

Abstractions And Analyses Of Grid Games, Taylor Rowan Boone

Senior Projects Spring 2016

In this paper, we define various combinatorial games derived from the NQueens Puzzle and scrutinize them, particularly the Knights Game, using combinatorial game theory and graph theory. The major result of the paper is an original method for determining who wins the Knights Game merely from the board's dimensions. We also inspect the Knights Game's structural similarities to the Knight's Tour and the Bishops Game, and provide some historical background and real-world applications of the material.

Constructing A Categorical Framework Of Metamathematical Comparison Between Deductive Systems Of Logic, Alex Gabriel Goodlad

Senior Projects Spring 2016

The topic of this paper in a broad phrase is “proof theory". It tries to theorize the general

notion of “proving" something using rigorous definitions, inspired by previous less general

theories. The purpose for being this general is to eventually establish a rigorous framework

that can bridge the gap when interrelating different logical systems, particularly ones

that have not been as well defined rigorously, such as sequent calculus. Even as far as

semantics go on more formally defined logic such as classic propositional logic, concepts

like “completeness" and “soundness" between the “semantic" and the “deductive system"

is too arbitrarily defined …

Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh

Senior Projects Spring 2016

Ordinary representation theory has been widely researched to the extent that there is a well-understood method for constructing the ordinary irreducible characters of a finite group. In parallel, John McKay showed how to associate to a finite group a graph constructed from the group's irreducible representations. In this project, we prove a structure theorem for the McKay graphs of products of groups as well as develop formulas for the graphs of two infinite families of groups. We then study the modular representations of these families and give conjectures for a modular version of the McKay graphs.

Winning Strategies In The Board Game Nowhere To Go, Najee Kahil Mcfarland-Drye

Senior Projects Spring 2016

Nowhere To Go is a two player board game played on a graph. The players take turns placing blockers on edges, and moving from vertex to vertex using unblocked edges and unoccupied vertices. A player wins by ensuring their opponent is on a vertex with all blocked edges. This project goes over winning strategies for Player 1 for Nowhere To Go on the standard board and other potential boards.

Lose Big, Win Big, Sum Big: An Exploration Of Ranked Voting Systems, Erin Else Stuckenbruck

Senior Projects Spring 2016

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

Envy-Free Fair Division With Two Players And Multiple Cakes, Justin J. Shin

Senior Projects Fall 2016

When dividing a valuable resource amongst a group of players, it is desirable to have each player believe that their allocation is at least as valuable as everyone else's allocation. This condition, where nobody is envious of anybody else's share in a division, is called envy-freeness. Fair division problems over continuous pools of resources are affectionately known as cake-cutting problems, as they resemble attempts to slice and distribute cake amongst guests as fairly as possible. Previous work in multi-cake fair division problems have attempted to prove that certain conditions do not allow for guaranteed envy-free divisions. In this paper, we …

Exploring Tournament Graphs And Their Win Sequences, Sadiki O. Lewis

Senior Projects Fall 2016

In this project we will be looking at tournaments on graphs and their win sequences. The main purpose for a tournament is to determine a winner amongst a group of competitors. Usually tournaments are played in an elimination style where the winner of a game advances and the loser is knocked out the tournament. For the purpose of this project I will be focusing on Round Robin Tournaments where all competitors get the opportunity to play against each other once. This style of tournaments gives us a more real life perspective of a fair tournament. We will model these Round …

Associativity Of Binary Operations On The Real Numbers, Samuel Joseph Audino

Senior Projects Spring 2016

It is known that there is an agreed upon convention of how to go about evaluating expressions in the real numbers. We colloquially call this PEMDAS, which is short for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It is also called the Order of Operations, since it is the order in which we execute the operators of a given expression. When we remove this convention and begin to execute the operators in every possible order, we begin to see that this allows for many different values based on the order in which the operations are executed. We will investigate this question …

Integer Generalized Splines On The Diamond Graph, Emmet Reza Mahdavi

Senior Projects Spring 2016

In this project we extend previous research on integer splines on graphs, and we use the methods developed on n-cycles to characterize integer splines on the diamond graph. First, we find an explicit module basis consisting of flow-up classes. Then we develop a determinantal criterion for when a given set of splines forms a basis.

A Variational Approach To The Moving Sofa Problem, Ningning Song

Senior Projects Spring 2016

The moving sofa problem is a two-dimensional idealisation of real-life furniture moving problems, and its goal is to find the biggest area that can be maneuvered around a L-shape hallway with unit width. In this project we will learn about Hammersly’s sofa ,Gerver’s sofa and adapt Hammersly’s sofa to non-right angle hallways. We will also use calculus of variations to maximize the area and find out Gerver’s sofa satisfied several conditions that the best sofa satisfies.