deletions | additions       diff --git a/Math.tex b/Math.tex  index 42ad90e..da247ef 100644  --- a/Math.tex  +++ b/Math.tex 
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In this case, $ω_j ∈ {\Bbb R}$, $ω_0=x$, $|ω_j − ω_{j−1}| = 1$, $j = 1, 2, . . . , n$, $ω_i \neq ω_j$ for $i \neq j$, and $0 ≤ i < j ≤ n$. In other words, no new movement (each of which is a length of $1$ in a discrete direction in the real plane) can occur such that the path coincides with itself in two distinct events. Put in context, this means that two tumorous cells cannot occupy the same space.  A pseudorandom integer was calculated for the ratio of the volume of the SAP, expressed in the following pseudocode:  \begin{lstlisting}[language=python]  tumor_length = randint(int(0.1 * matrix.area), int(0.9 * matrix.area))  coord = [randint(1, matrix.m), random.randint(1, matrix.n)]  for i in range(0, matrix.tumor_length):  direction = randint(1, 4)  if direction == UP:  coord[1] -= 1  elif direction == DOWN:  coord[1] += 1  elif direction == LEFT:  coord[0] -= 1  elif direction == RIGHT:  coord[0] += 1  \end{lstlisting}  The equation defines at which position the path will lie after $n$ number of steps.  \hspace{2000 mm} To model the distribution of $S_{n}$ we need to set it to $x$ and set the number of steps to the right and left by $r$ and $l$, respectively. Given this, $x = r - l$ and $n$, the total number of steps actually executed, $= r + l$. By simply solving for $r$ and $l$ we deduce that $r = (1/2)(x+n)$ and $l = (1/2)(n-x)$.  Given this, there are now\textbf{ (n choose l)} ways that l given steps can occur in a total of n steps. This also denotes the number of ways of arriving at point x (the end point), with each way having a probabilty of p^{r}q^{l}. One thing that must be noted is that x and n must be either odd or even.