John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.

John Napier, born at Merchiston in 1550, published The Whole Revelation of St. John in 1594; and he appears to have regarded that theological polemic as his most important achievement. Napier's invention of logarithms (with greatly advanced spherical trigonometry) was published in 1614 as Descriptio Canonis Logarithmorum; whereupon the mathematicians of Europe instantly acclaimed Napier as the greatest of them all. In 1617 he published Rabdologiae, which explained several devices for aiding calculation: (1) numbering rods to aid multiplication (known as 'Napier's bones'); (2) other rods to aid evaluation of square and cube roots; (3) the first publication of binary arithmetic, as far as square root extraction; and (4) the Promptuary for multiplication of numbers (up to 10 digits each), which has a strong claim to be regarded as the first calculating machine. Napier's explanation of the construction of his logarithms was published posthumously in 1619 as Constructio Canonis Logarithmorum, in which he developed much of the differential calculus in order to define his logarithms as the solution of a differential equation and then constructed strict upper and lower bounds for the solution. His incomplete manuscript on arithmetic and algebra (written in the early 1590s) was published in 1839 as De Arte Logistica. This thesis provides the first English translations of De Arte Logistica and of Rabdologiae, and it reprints Edward Wright's English translation (1616) of the Descriptio and W. R. Macdonald's English translation (1889) of the Constructio. Extensive commentaries are given on Napier's work on arithmetic, algebra, trigonometry and logarithms. The history of trigonometry is traced from ancient Babylonia and Greece through mediaeval Islam to Renaissance Europe. Napier's logarithms (and spherical trigonometry) resulted in an explosion of logarithms over most of the world, with European ships using logarithms for navigation as far as Japan by 1640. / Subscription resource available via Digital Dissertations only.

Doctor of Philosophy / Department of Mathematics / Chris Pinner and Todd Cochrane

Doctor of Philosophy / Department of Mathematics / Ivan Blank / Under the guidance of Dr. Ivan Blank, I study the obstacle problem with an elliptic operator in divergence form. First, I give all of the nontrivial details needed to prove a mean value theorem, which was stated by Caffarelli in the Fermi lectures in 1998. In fact, in 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. The formula stated by Caffarelli is much simpler, but he did not include the proof. Second, I study the obstacle problem with an elliptic operator in divergence form. I develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results allow us to begin the study of the regularity of the free boundary in the case where the coefficients are in the space of vanishing mean oscillation (VMO).

Doctor of Philosophy / Department of Mathematics / Marianne Korten / Charles N. Moore / We start with linear single variable conservation laws and examine the conditions under which a local extrapolation method (LEM) with upwinding underlying scheme is total variation diminishing TVD. The results are then extended to non-linear conservation laws. For this later case, we restrict ourselves to convex flux functions f, whose derivatives are positive, that is, fœ A0 and fœ A0. We next show that the Goodman-LeVeque flux satisfies the conditions for the LEM to be applied to it. We make heavy use of the CFL conditions, the geometric properties of convex functions apart from the martingle type properties of functions which are increasing, continuous, and differentiable.

Master of Science / Department of Mathematics / David R. Auckly / In this report we present the main concepts of probability theory: sample spaces, events, random variables, distributions, independence, central limit theorem. Most of the material may be found in the notes of Bass. The work is motivated by wide range of applications of probability theory in quantitative finance.

Master of Science / Department of Mathematics / Christopher G. Pinner / This thesis will look at Quadratic Diophantine Equations. Some well known proofs, including how to compute all Pythagorean triples and which numbers can be represented by the sum of two and four squares will be presented. Some concepts that follow from these theorems will also be presented. These include how to compute all Pythagorean Quadruples, which number can be represented by the difference of two squares and the Crossed Ladders problem. Then, Ramanujan's problem of finding which positive integers, a,b,c and d which allow aw^2+bx^2+cy^2+dz^2 to represent all natural numbers will be shown. The paper will conclude with a lengthy discussion of Uspensky's proof on which numbers can be represented by the sum three squares.

Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / This thesis establishes bounds (primarily upper bounds) on Waring's number in finite fields. Let $p$ be a prime, $q=p^n$, $\mathbb F_q$ be the finite field in $q$ elements, $k$ be a positive integer with $k|(q-1)$ and $t= (q-1)/k$. Let $A_k$ denote the set of $k$-th powers in $\mathbb F_q$ and $A_k' = A_k \cap \mathbb F_p$. Waring's number $\gamma(k,q)$ is the smallest positive integer $s$ such that every element of $\mathbb F_q$ can be expressed as a sum of $s$ $k$-th powers. For prime fields $\mathbb F_p$ we prove that for any positive integer $r$ there is a constant $C(r)$ such that $\gamma(k,p) \le C(r) k^{1/r}$ provided that $\phi(t) \ge r$. We also obtain the lower bound $\gamma(k,p) \ge \frac {(t-1)}ek^{1/(t-1)}-t+1$ for $t$ prime. For general finite fields we establish the following upper bounds whenever $\gamma(k,q)$ exists: $$ \gamma(k,q)\le 7.3n\left\lceil\frac{(2k)^{1/n}}{|A_k^\prime|-1}\right\rceil\log(k), $$ $$ \gamma(k,q)\le8n \left\lceil{\frac{(k+1)^{1/n}-1}{|A^\prime_k|-1}}\right\rceil, $$ and $$ \gamma(k,q)\ll n(k+1)^{\frac{\log(4)}{n\log|\kp|}}\log\log(k). $$ We also establish the following versions of the Heilbronn conjectures for general finite fields. For any $\ve>0$ there is a constant, $c(\ve)$, such that if $|A^\prime_k|\ge4^{\frac{2}{\ve n}}$, then $\gamma(k,q)\le c(\varepsilon) k^{\varepsilon}$. Next, if $n\ge3$ and $\gamma(k,q)$ exists, then $\gamma(k,q)\le 10\sqrt{k+1}$. For $n=2$, we have $\gamma(k,p^2)\le16\sqrt{k+1}$.

Master of Science / Department of Mathematics / Nathan Albin / The term stochastic is defined as having a random probability distribution or pattern that may be analyzed statistically but may not be predicted precisely. A stochastic model attempts to estimate outcomes while allowing a random variation in one or more inputs over time. These models are used across a number of fields from gene expression in biology, to stock, asset, and insurance analysis in finance. In this thesis, we will build up the basic probability theory required to make an ``optimal estimate", as well as construct the stochastic integral. This information will then allow us to introduce stochastic differential equations, along with our overall model. We will conclude with the "optimal estimator", the Kalman Filter, along with an example of its application.

Master of Science / Department of Mathematics / Andrew Bennett / In recent years, technology has improved and become a significant aspect in the classroom. Using technology has become a popular method of note-taking. This study investigates the effects of technology on note-taking by looking at the changes that can be shown between the traditional note-taking and taking notes by using different devices, such as the iPad and a smart pen. Modern technology, such as the smart pen which provides an automatic audio recording might improve student focus on important details. In addition, providing a standard note set along with note-taking tools such as an iPad might help student organize and access their notes. The result of this study showed that for all but one of the students, using technology did not affect their note-taking style or the amount of information in their notes. However, students were not satisfied with their notes when taken on the iPad.