tex: add NFDP math spec

latex/content.tex

@@ -147,8 +147,7 @@ NFBP algorithm. The yellow item is stored in bin 2, while it could fit in bin
 1, because the purple item is considered first and is too large to fit.
 
 \paragraph{} Each bin will have a fixed capacity of $ 1 $ and items and items
-will be of random sizes between $ 0 $ and $ 1 $. We will run X simulations % TODO
+will be of random sizes between $ 0 $ and $ 1 $.
-with 10 packets.
 
 \subsection{Variables used in models}
 
@@ -171,8 +170,8 @@ We use the following variables in our algorithms and models :
 Mathematically, the NFBP algorithm imposes the following constraint on the first box :
 
 \begin{align*}
-	T_1 = k \iff & U_1 + U_2 + \ldots + U_{k-1}  < 1                                  \\
+	T_1 = k \iff & U_1 + U_2 + \ldots + U_{k}  < 1                                      \\
-	\text{ and } & U_1 + U_2 + \ldots + U_{k}    \geq 1 \qquad \text{ with } k \geq 2
+	\text{ and } & U_1 + U_2 + \ldots + U_{k+1}    \geq 1 \qquad \text{ with } k \geq 1 \\
 \end{align*}
 
 
@@ -242,6 +241,14 @@ less bins than NFBP, due to less stringent constraints. The top of the bin is
 effectively removed, allowing for an extra item to be stored in the bin. We can
 easily see how with NFDBP each bin can at least contain two items.
 
+\paragraph{} The variables used are the same as for NFBP. Mathematically, the
+new constraints on the first bin can be expressed as follows :
+
+\begin{align*}
+	T_1 = k \iff & U_1 + U_2 + \ldots + U_{k-1}  < 1                                  \\
+	\text{ and } & U_1 + U_2 + \ldots + U_{k}    \geq 1 \qquad \text{ with } k \geq 2 \\
+\end{align*}
+
 \subsection{La giga demo}
 
 Let $ k \in \mathbb{N} $. Let $ (U_n)_{n \in \mathbb{N}} $ be a sequence of
@@ -288,7 +295,7 @@ $ U_1 $ and $ U_2 $ are independent, so
 	                   & = \begin{cases}
 		1 & \text{if } x \in [0, 1] \text{ and } y \in [0, 1] \\
 		0 & \text{otherwise}                                  \\
-	\end{cases}   \\
+	\end{cases}  \\
 \end{align*}
 
 Hence,