Physical Sciences and Mathematics Commons^™

The Characterization Of Vectors In RN With The Haar Property, Terrence Kelleher

Undergraduate Review

Introduction:
As modern communication becomes more digital than ever, infrastructure tries to maintain an equal pace. Unfortunately, this is not always possible. Therefore, computer scientists and mathematicians alike have endeavored to invent ways to store, send and retrieve data, even if transmitted signals are severely damaged. One such way is by using what is called an error-correcting code, or an ECC. An ECC is a method of encoding information such that a signal that possesses a message can be significantly altered and still be decoded. There are many different types of ECCs. The type that is of interest in this …

The Mystery Of The Non-Transitive Grime Dice, Nicholas Pasciuto

Undergraduate Review

No abstract provided.

Dynamics Of Climate Change: Explaining Glacier Retreat Mathematically, Robert Guillette

Undergraduate Review

Climate change is an important topic that has become extremely relevant this day and age. The world’s climate is undergoing monumental shifts with over two-thirds of the estimated 150 glaciers existing in 1850 disappearing by 1980. The melting of glaciers offers tangible evidence of broader environmental changes as they respond directly to long-term trends in temperature, precipitation, and solar radiation. Since the study of glacier retreat provides a barometer of climate change, it is important to better understand the effects of climatic factors on glaciers. In my project I created a mathematical model for the melting of glaciers and used …

Who Wants To Play Sadisticube?, Danica Baker

Undergraduate Review

Logic puzzles and games are popular amongst many people for the purpose of entertainment. They also provide intriguing questions for mathematical research. One popular game that has inspired interesting research is Rubik’s Cube. Researchers at MIT have investigated the Rubik’s Cube to find the maximum number of moves, from any starting position, needed to win the game [6]. Another logic puzzle that has recently become very popular is Sudoku. Sudoku is a Japanese number game where a 9x9 grid is set up with a few numbers scattered on the grid. Mathematicians have been investigating Sudoku, exploring questions such as the …

A Mathematical Analysis Of A Game Of Craps, Yaqin Sun

Undergraduate Review

The game of craps is an extremely popular game offered by casino operators. There are some 40 different types of bets that one can place each time the game is played. One of the best bets from a player’s point of view is the Pass Line bet. The probability of winning a Pass Line bet is almost the same as the probability of losing (²⁴⁴⁄₄₉₅ versus ²⁵¹⁄₄₉₅) as we will derive rigorously in this article. Since the “house” has such a small advantage over the players, many players possess the illusion that they have …

The Archbishop's Odyssey, Leonard Sprague

Undergraduate Review

For centuries, scholars have analyzed a collection of problems that, nowadays, has been defined as NP-complete. Currently, NP-complete problems have no known efficient solutions. The Clay Mathematics Institute has offered a reward of one million dollars for a solution. The problem of finding Hamilton paths and cycles has been shown to be in this category. Knight’s tours, where the knight must visit every square of a chessboard exactly once, are examples of Hamilton paths and cycles.

This research revolves around the creation of a new branch of the tour problems, through a new piece: the Archbishop. Chess Grandmaster Jose Capablanca …

Exploring Diagonals In The Calkin-Wilf Tree, Matthew Gagne

Undergraduate Review

For centuries, people have been interested in patterns. Even in that which appears random, humans have been trying to understand the underlying order of things. Mathematicians throughout time have studied many phenomena, including infinite sequences of numbers and have been able, at times, to see structure. Many have found the satisfaction, even joy, of discovering patterns in sequences. A typical way to describe this is by a recursive formula. A recursive definition defines a term in the sequence using the previous terms in the sequence. Even more satisfying than a recursive formula is a closed formula. With this, one can …

Computer Programming To Advance Gravitational Lensing, Alex Roche

Undergraduate Review

The purpose of this research was to create a computer code that would numerically test a Poisson equation relating the mass distribution of a lens galaxy cluster to weak gravitational shear. Einstein’s theory of general relativity predicts that space-time is bent by massive objects, and in weak gravitational lensing, galaxy clusters act as lenses. The observable result is that galaxies far behind the gravitational lens will appear slightly more elliptical than they actually are. The ellipticity of the background galaxies is quantifiable and is directly related to the weak gravitational shear, and the shear is used to determine the mass …

Analysis Of The “Travelling Salesman Problem” And An Application Of Heuristic Techniques For Finding A New Solution, Mateusz Pacha-Sucharzewski

Undergraduate Review

In 1832, a German travelling salesman published a handbook describing his profession. Sadly, his name is unknown; he only stated that the book was written by “one old travelling salesman.” However, he has come down in history thanks to a rather simple and quite obvious observation. He pointed out that when one goes on a business trip, one should plan it carefully; by doing so, one can “win” a great deal of time and increase the trip’s “economy.” Two centuries later, mathematicians and scientists are still struggling with what is now known as the “Travelling Salesman Problem” (TSP).

Analyzing The Galois Groups Of Fifth-Degree And Fourth-Degree Polynomials, Jesse Berglund

Undergraduate Review

It is known that the general equations of fourth-degree or lower are solvable by formula and general equations of fifth-degree or higher are not. To get an understanding of the differences between these two types of equations, Galois theory and Field theory will be applied. The Galois groups of field extensions will be analyzed, and give the solution to the query “What is the difference between unsolvable fifth-degree equations and fourth-degree equations?”

EΠi + 1=0: The History & Development, Dawne Charters-Nelson

Undergraduate Review

I have on occasion run across the equation in books, articles and in conversation with other mathematicians. In each of these encounters the person alluded to a fascination with this equation which links the five most important constants in the whole of analysis:

• 0 = The additive identity
• 1 = The multiplicative identity
• π = The circular constant
• e = The base of the natural logarithms
• i = The imaginary unit

Being a novice mathematician, I wondered how all these fundamental constants could end up in one equation and what it meant. Along with this thought came the realization that …

A Philosophical Examination Of Proofs In Mathematics, Eric Almeida

Undergraduate Review

No abstract provided.