\documentclass[10pt]{article}
\usepackage{pslatex} 
\begin{document}
\author{Andre Masella}
\title{Mathematical Approximations for Mass Transit Passengers}
\maketitle
\section{Introduction}
All passengers of mass transit system will notice strange events that seem to reccur. This document attempts to mathematically model these situations.

\section{Schofield's Law of Thirds}
Schofield's Law of Thirds is that if the period of a Toronto bus is $T$ minutes, then 3 buses will show up every $3T$ minutes, $t$ seconds apart, where $t \ll T$.

\section{Masella's Distribution of Mass}
If $P$ represents the set of people on a subway car, then let $P\star \subset P$ such that any element of $P\star$ is carrying heavy objects. If $D$ is any useable door, then as $P\rightarrow P_{max} \Rightarrow \vert\overrightarrow{Dp}\vert \rightarrow 0\ : p\ \in\ P\star$.

\section{Masella's Distribution of Distance}
If $P$ represents the set of people on a subway car, then let $d(p,t)$ represent the distance that a person $p\ \in\ P$ wishes to travel at time $t$. If $D$ is any door, then as $d(p,t) \rightarrow \infty : p\ \in\ P \Rightarrow \vert\overrightarrow{DP\star}\vert \rightarrow 0\ \forall t : d(P,t) \neq 0$. Note that the final condition is necessary since this approximation tends to break down when a person arrives at their stop, since they must pass through the doors.

\section{The Capacity Paradox}
If a train $t$ enters a station such that the number of people on the platform $n_p$ is greater than the capacity of the train $C$, then $C-n_p$ will be left behind, but at each subsequent station where $n_p>0$ (since $C=0$), the train will pick up one more person per door than it lets off.

\end{document}