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\title{MAS115: Homework 2}
\author{Sam Marsh}
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\maketitle
\section{Mathematics and Statistics at the University of Sheffield}
The School of Mathematics and Statistics has the following main research groups.
\begin{itemize}
\item Algebra
\item Analysis
\item Category Theory
\item Differential Geometry
\item Environmental Dynamics
\item Fluid Dynamics
\item Mathematical Biology
\item Nonlinear Control
\item Number Theory
\item Particle Astrophysics and Gravitation
\item Probability and Statistics
\item Solar Physics and Space Plasma Research Centre
\item Topology
\end{itemize}
\section{Dr Sam Marsh's research interests}
Sam Marsh gained his PhD in \emph{algebraic topology}, which is a branch of pure mathematics involved in the study of spaces by algebraic means. His PhD thesis concerned using a collection of so-called \emph{cohomology theories} known as the Morava $E$-theories to better understand spaces related to the general linear groups, and was carried out under the supervision of Professor Neil Strickland.
Now employed as a teaching fellow, Sam is currently more interested in logic and set theory whose aim is to understand the nature of the foundations of mathematics. He has a soft spot for the work of Kurt G\"odel, whose incompleteness theorem is simultaneously one the greatest results of the twentieth century and a complete irrelevance.
\section{Solution re-write}
\begin{quote}\emph{A line $L$ passes through the points $A = (8,1)$ and $B = (2, 3)$.
Find the equation of $L$.}\end{quote}
The line $L$ has equation $y=mx+c$, where $m$ is the gradient and $c$ is the $y$-intercept. Since points $A$ and $B$ both lie on $L$, we have
$$m=\frac{3-1}{2-8}=\frac{2}{-6}=-\frac{1}{3}.$$
It follows, using point $A$, that $c=1-8.(-1/3)=1+8/3=11/3$. Hence the equation of $L$ is
$$y=\frac{11}{3}-\frac{x}{3}.$$
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