Physical Sciences and Mathematics Commons^™

The History And Mathematics Behind The Construction Of The Islamic Astrolabe, Lyda P. Urresta

Honors Theses

In this paper, we examine the mathematical methods employed in the construction of the astrolabe, an ancient measuring device used to solve problems in the field of astronomy. Essentially, the astrolabe is a two dimensional representation of the heavens obtained by projecting the celestial sphere onto the plane. Though several different types of astrolabes exist, our primary focus is on the most popular deisgn, which is created by the stereographic projection of the celestial sphere onto the plane defined by the equator with the south pole as the projection point.

Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre

Honors Theses

It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that …

Euler E271 : A Link Between Mathematics Of Yesterday, Today, And Tomorrow, Sarah Ann Nelson

Honors Theses

The major focus of this departmental thesis was to complete t he first English translation of E271 Arithmetic Theorems Proven by a New Method, a mathematical treatise published by Leonhard Euler in Latin in 1761. Most importantly, E271 contains Euler's generalization of Fermat's Litt le Theorem and an exploration of the properties of (n). Altogether, this paper includes an Abstract, Introduction, Note to the Readers, Translation of Arithmetic Theorems Proven by a New Method, Epilogue, and References. More specifically, the Introduction is about the historical background of the mathematics and applications leading up to E271 and the key corresponding mathematicians. …

Row Reduction Of Macaulay Matrices, Lorrin Debenport

Honors Theses

A computer can use a matrix to represent a system of non-linear multivariate polynomial equations. The fastest known ways to transform this system into a form with desirable computational properties rely on transforming its matrix into upper-triangular form [8, 9]. The matrix for such a system will have mostly zero entries, which we call sparse [7]. We propose to analyze several methods of performing row-reduction, the process by which matrices are reduced into upper-triangular form [2].

What is special about row-reducing matrices in this context? When row-reducing a matrix, swapping rows or columns is typically acceptable. However, if the order …

Rank One Perturbations Of Self-Adjoint Operators, Haoxuan Zheng

Honors Theses

No abstract provided.

Equality Of P-Partition Generating Functions, Ryan Ward

Honors Theses

To every partially ordered set (poset), one can associate a generating function, known as the P-partition generating function. We find necessary conditions and sufficient conditions for two posets to have the same P-partition generating function. We define the notion of a jump sequence for a labeled poset and show that having equal jumpsequences is a necessary condition for generating function equality. We also develop multiple ways of modifying posets that preserve generating function equality. Finally, we are able to give a complete classification of equalities among partially ordered setswith exactly two linear extensions.