\chapter{Classification of Finite Simple Groups}
\section{Early Results}
{\bf Feit-Thompson:}  The only finite simple groups or odd order are
${\mathbb Z}_p, p\ne 2$. The proof follows the CN classification.\\
\\
{\bf Glauberman $ZJ$:}  If $C_G(O_p(G)) \le O_p(G)$ and the action of
$G$ on its chief factors of $G$ is $p-$stable then $G=N_G(Z(J(S)))$. \\
\\
{\bf Glauberman's $Z^*$ Theorem:}
Let $G$ be a finite group and $t$ and involution in $G$ which is weakly closed in
$C(t)$.  Then $t^* \in Z(G^*)$  where $G^*= G/O_{2'} (G)$.
\section{Major Classification Results}
{\bf Brauer, Suzuki:} Suppose the Sylow $2$-subgroups of $G$ are quaternion then
${\mathbb Z}^*(G)/O_{2'}(G) \cong C_2$.
\\
\\
{\bf Gorenstein, Walter:} Suppose $O_{2'}(G)=1$ and that the sylow $2$ subgroups of $G$ are
dihedral then $F^*(G)$ is isomorphic to $PSL_2(q), q=1 \jmod{2}$ or $A_7$.
\\
\\
{\bf Alperin, Brauer, Gorenstein:} Suppose $G$ is simple and the Sylow
$2$-subgroups are semi-dihedral then $G$ is isomorphic to
$PSL_3(q), q= -1 \jmod{4}$,
$PSU_3(q), q= 1 \jmod{4}$,
or $M_{11}$.
\\
\\
{\bf Bender:} Suppose $G$ possesses a strongly embedded subgroup then the Sylow
$2$-subgroups of $G$ are cyclic or quaternion or $G$ possesses a normal series
$1 \lhd M \lhd L \lhd G$ such that $M$, and $G/L$ have odd order and
$L/M$ is isomorphic to
$PSL_2(2^n)$, $Sz(2^{2n-1})$, or $PSU_3(2^n), n \ge 2$.
\\
\\
{\bf Goldschmidt:} Suppose $S \in S_2(G)$ and $A$ and Abelian subgroup of $S$
such that $a \in A, a^g \in S \rightarrow a^g \in A$.  Suppose that $G= \langle A^G \rangle $ and
$O_{2'}(G)=1$ then $G=F^*(G)$, $A=O_2(G) \Omega(T)$ and for every component
$K$ of $G$, the factor group $K/{\mathbb Z}(K)$ is isomorphic to
$PSL_2(2^n)$, $Sz(2^{2n-1})$, $PSU_3(2^n), n \ge 2$, $PSL_2(q), q = 3, 5 \jmod {8}$,
$R(3^{2n+1})$, or $J_1$.
\\
\\
{\bf Thompson:} Suppose $G$ is a non-solvable group all of whose
$2$-locals are solvable $\forall p \in \pi(G)$, then $F^*(G)$ is isomorphic to
$PSL_2(q), q>3$, $Sz(2^{2n-1})$, $PSU_3(2^n), n \ge 2$, $A_7$, $M_{11}$, $PSL_3(3)$,
$PSU_3(3)$
or $^2F_4(2)'$.
\\
\\
{\bf Gorenstein, Lyons, Janko, Smith:} Suppose $G$ is a non-solvable group all of whose
$2$-locals are solvable, then $F^*(G)$ is isomorphic to
$PSL_2(q), q > 3$, $Sz(2^{2n-1})$, $PSU_3(2^n), n \ge 2$, $A_7$, $M_{11}$, $PSL_3(3)$,
$PSU_3(3)$ or $^2F_4(2)'$.
\\
\\
{\bf Walter:} Let $G$ be a group with 2 rank $\ge 5$ and $O_{2'}(G)=1$ with the property
that the centralizer of every involution is $2-$constrained then $O_{2'}(C(x))=1$ for
every involution $x$.
\\
\\
{\bf Example of groups of semi-simple type:} $GL_n(q)$, $q$, odd, $t \in Inv(G)$ has
form
$t= \left(
\begin{array}{cc}
I_m & 0\\
0 & I_r\\
\end{array}
\right), n= m+r$ and
$C_G(t)= \left(
\begin{array}{cc}
A & 0\\
0 & B\\
\end{array}
\right)$,
$A \cong GL_m(q)$ and
$B \cong GL_r(q)$. \\
\\
{\bf Definition:}
$e(G)= max \{ m_p(H),
H \le G  \textnormal{ is a } 2-\textnormal{local and } p \ne 2 \}$.
A group $G$ is \emph{quasi-thin} if
$m_2(G) \ge 3$ but every $e(G) \le 2$.
\\
\\
{\bf Classification decomposes as follows:} (1) Minimal counterexample has characteristic
$2$-type ($C_H(O_p(H)) \le O_p(H)$) where $H$ is a $2$-local (Lie type $q= 2^n$) small
groups are quasi-thin (2) $G$ is not of characteristic $2$-type, small groups have
$m_2(G) \le 2$.
\section{Connected Groups}
{\bf Definition:}
Let $\Omega_G$ be a collection of subgroups of $G$ \emph{define}
${\cal D}(\Omega_G)$ as the graph with points $\Omega_G$; two points
$A,B \in \Omega_G$ are joined by an edge if $[A,B]=1$.  Note that $G$ acts as a
group of automorphisms on ${\cal D}(\Omega_G)$ via conjugation.
\\
\\
{\bf Theorem:}
Let $\Delta$ be a $G$-invariant collection of subgroups of $G$ and $H < G$.
The following are equivalent:
(1) $H$ controls fusion in $H \cap \Delta$ and $N_G(X) \le H$ for $x \in H \cap \Delta$;
(2) $H \cap H^g \cap \Delta= \emptyset$ if $g \in G \setminus H$;
(3) the members of $H \cap \Delta$ fix a unique point in the permutation representation
of $G$ on $G/H$ by right multiplication;
(4) $H \cap \Delta$ is the uniion of a set, $\Gamma$, of connected components of
${\cal D}(\Delta)$ and $\Gamma \cap \Gamma^g= \emptyset$ if $g \in G \setminus H$.
\begin{quote}
\emph{Proof:}
\\
\\
$1 \rightarrow 2$:
Let $g \in G$ with $H \cap H^g \cap \Delta \ne \emptyset$.  $\exists X \in H \cap \Delta$
with $X^g \le H$ so $H$ controls fusion in $H \cap \Delta$ and
$X^{gh}=X, h \in H$.  $gh \in N_G(X) \le H$ so $g \in H$.
\\
\\
$2 \rightarrow 3$:
Consider the representation $G \rightarrow G/H$.  $X \in H \cap \Delta$ fixes
$Hg$ iff $X \le H^g$ which happens iff $g \in H$.
\\
\\
$3 \rightarrow 4$:
For $A \in H \cap \Delta$, $\{ H \} = Fix(A)$ so $N_G(A) \le H$.  If $B \in \Delta$ is incident
to $A$ in ${\cal D}(\Delta)$ then $B \le C(A) \le H$ so $B \in H \cap \Delta$ and
$H \cap \Delta$ is the union of some set $\Gamma$ of connected components of ${\cal D}( \Delta )$.
Further, if $A \in \theta \in \Gamma$ and $\theta^g \in \Gamma$ then
$A^g \le H$ so $\{ H \} = Fix(A^g ) = \{ Hg \}$ and $g \in H$.
\\
\\
$4 \rightarrow 1$:
If $X \in H \cap \Delta$ and $g \in G$ with $X^g \le H$ then
$X \in \theta \in \Gamma$ and
$X^g \in \theta' \in \Gamma$, so $\theta' = \theta^g \in \Gamma \cap \Gamma^g$ hence
$g \in H$.
\end{quote}
{\bf Definition:}
Define ${\cal E}_k^p(G)$ to be the set of all elementary abelian
$p$-subgroups of $G$ of $p$-rank at least $k$. $G$ is said to be $k$\emph{-connected}
for the prime $p$ if ${\cal D}({\cal E}_k^p(G))$ is connected.
Define $\Gamma_{P,k}(G) = \langle N_G(X), X \rangle \le P, m(X) \ge k \rangle$ and
$\Gamma_{P,k}^0(G) = \langle N_G(X), X \le P, m(X) \ge k, m(XC_P(X)) > k \rangle$.
If $P \in S_p(G)$ then
$\Gamma_{P,k}(G)$ is called the \emph{$k$-generated $p$-core of $G$}.
\\
\\
{\bf Theorem:}
Let $P$ be a $p$-group.  Then
(1) $P$ is $1$-connected for the prime $p$;
(2) If $m(P) > 2$, $\exists E_{p^2} \cong U \lhd P$ and $X \in {\cal E}_2^p(P)$ is in
the same connected component of ${\cal D}({\cal E}_k^p(P))$ as $U$ when
$m(C_P(X)) > 2$;
(3) If $p=2$ and $\exists E_{8} \cong U \lhd P$ then $P$ is $2$-connected for the prime $2$;
(4) If $p=3$ and $m(P) > 3$ then $P$ is $2$-connected for the prime $3$.
\begin{quote}
\emph{Proof:}
(1) follows from ${\mathbb Z}(G) \ne 1$.  Assume $m(G) > 2$ then
there is an $E_{p^2} \cong  U \lhd G$ and $G/C_G(U) \le SL_2(p)$ and
$SL_2(p)$ is of $p$-rank $1$.  If $A \in {\cal E}^2_3(G)$ then
$m(C_A(U) )  \ge 2$
so $m(UC_A(U)) > 2$ and
$m(C_A(U) ) > 2$.  If
$U \ne E \in {\cal E}^p_2(G)$ is in the same connected component
as $U$, then
$\exists D: E \ne D \in {\cal E}^p_2(G)$ and $E$ is adjacent to $D$ and
further,  $m(C_G(E)) \ge m(DE) \ge 3$.  All we need to show is that
$A \in {\cal E}^p_2(G)$ is in the same connected component as $U$.
But $\langle A, C_A(U), U C_A(U), U \rangle$ is in a path of
${\cal D}({\cal E}^p_2(G))$ completing 2.
Assume $p = 2$ and $E_8 \cong V \lhd G$ and let
$E \in {\cal E}^p_2(G)$, we must show there is a path from $E$ to $V$.
For $e \in E: m(C_V(e)) \ge 2$ and so we can assume $e \in E \setminus V$ so
$\langle E, \langle e, C_V(E) \rangle, C_V(e), V \rangle$ is a path in
${\cal D}({\cal E}^p_2(G))$.
$\exists E_8 \cong V \lhd G$
and the argument above establishes 4.
\end{quote}
{\bf Theorem:}
Let $H \le G, P \in S_p(H)$ and $k \in {\mathbb Z}^+$.  Then
(1) If $m(p) \ge k$ and $\Gamma_{P,k}(G) \le H$ then $P \in S_p(G)$; and
(2) If $m(p) > k$ and $\Gamma_{P,k}^0(G) \le H$ then $P \in S_p(G)$.
\begin{quote}
\emph{Proof:}
Follows from previous result.
\end{quote}
{\bf Theorem:}
Let $H \le G$, $P \in S_p(G)$ and $m(p) \ge k$.  Then the following are equivalent:
(1) $\Gamma_{P,k}(G)<H$;
(2) $H$ controls fusion in ${\cal E}_k^p(H)$ and $N_G(X) \le H$ for
$x \in {\cal E}_k^p(H)$;
(3) $m(H \cap H^g)<k$ for $g \in G \setminus H$;
(4) each member of
${\cal E}_k^p(H)$ fixes a unique point in the permutation representation of $G$ on $G/H$.
\begin{quote}
\emph{Proof:}
Parts 2, 3, 4 are equivalent by the above result except
${\cal E}^p_k(G)$ should be
${\cal E}^p_k(H)$.  It remains to show 1 implies 2.
Assume
$\Gamma_{P, k}(G) \le H$ and $X \in {\cal E}^p_k(H)$.  It STS
that if $g \in G$ with $X^g \le H$ then $g \in H$.  By Sylow,
$\langle P, P^g \rangle \le P$ and by the above
$P \in S_p(G)$.  By Alperin, $\exists P_i \in S_p(G), 1 \le i \le n$ and
$g_i \in N_G(P \cap P_i )$ with $g= g_1 g_2 \ldots g_n$, $X \le P_1$
and $X^{g_1 g_2 \ldots g_i} \le P \cap P_i$ and $m(P \cap P_i) \ge k$
so $g_i \in N_G(P \cap P_i) \le
\Gamma_{P, k}(G) \le H $ and so
$g= g_1 g_2 \ldots g_n \in H$ and 2 holds.
\end{quote}
{\bf Definition:}  If $k=1$ in any of the above equivalent conditions we say $H$ is
\emph{strongly $p$-embedded in $G$}.
\\
\\
{\bf Theorem:}
Let $P \in S_p(G)$ and suppose $P$ is $k$-connected then
(1) ${\cal E}_k^p(\Gamma_{P,k}(G))$ is a connected component of
${\cal D}({\cal E}_k^p(G))$ and
$\Gamma_{P,k}(G))$ is the stabiliser of that component;
(2) $G$ is $k$-disconnected for the prime $p$ iff $G$ has a proper $k$-generated $p$-core.
\begin{quote}
\emph{Proof:}
1 implies 2.  Let $H= \Gamma_{P, k}(G)$.  By the earlier resutls,
${\cal E}^p_k(G)$
is a union of connected components of
${\cal D}({\cal E}^p_2(G))$
while
${\cal E}^p_k(P)$ is contained in some component since
$P$ is $k$-connected and $H \le N_G( \Delta )$.  Hence
${\cal E}^p_k(H) \subseteq \Delta$ by Sylow. Thus
$\Delta = {\cal E}^p_k(H)$ and $H= N_G( \Delta )$ and 2 holds.
\end{quote}
{\bf Theorem:}
$G$ posseses a strongly $p$-embedded subgroup iff $G$ is $1$-disconnected for the prime $p$.
\begin{quote}
\emph{Proof:}
Follows from previous result.
\end{quote}
{\bf Theorem:}
Let $m_p(G) > 2, P \in S_p(G)$ and
${\cal E}_k^p(G)^0$ be the set of subgroups
$X \in {\cal E}_k^p(G)$ with $m_p(XC_G(X)) > 2$.  Then
(1) ${\cal E}_k^p(G)^0$ is the set of points that are not isolated in
${\cal D}({\cal E}_k^p(G))$;
(2) ${\cal E}_2^p(\Gamma_{P,2}^0(G))^0$ is a connected component of
${\cal D}({\cal E}_2^p(G))$ and
$\Gamma_{P,2}^0(G))$ is the stabilizer of this connected component;
(3)
${\cal D}({\cal E}_2^p(G))^0$ is connected iff
$G= \Gamma_{P,2}^0(G))$.
\begin{quote}
\emph{Proof:}
1 is trivial and 3 easily follows from 2.   By the previous result,
${\cal E}^p_2(G)^0$ is contained in a connected component $\Delta$ of
${\cal D}({\cal E}^p_2(G))$.  Thus
$H=
\Gamma_{P, 2}^0(G) \le N_G( \Delta )$.
Let $\Gamma= {\cal E}^p_2(G)^0$.
Since ${\cal E}^p_2(G)^0 \subseteq \Delta$ and $H$ acts on $\Delta$,
$\Gamma \subseteq \Delta$ by Sylow.  If $\Delta \ne \Gamma$, $\exists x \in \Gamma$,
$Y \in \Delta \setminus \Gamma$ with $X$ and $Y$ are adjacent
${\cal D}( \Delta )$.   WLOG,
$X \in {\cal E}^p_2(G)^0$, so $Y \le C_G(X) \le H$ and hence
$m_P(C_H(Y)) \ge m_P(XY) \ge 3$, $Y \in \Gamma$, a contradiction.
So $\Delta = \Gamma$, it remains to show that if
$X, X^g \in \Gamma$ then $g \in G$.
Suppose
$X, X^g \in \Gamma$ then $N_G(X) \le H \ge N_G(X^g )$ so there is a
$E_{p^3} \cong A \le C_G(X) \le H$ and $A^g \le H$.  So by Sylow take $A$,
$A^g \le P$.  Now apply Alperin, using
$ \Gamma_{P, 3}(G) \le \Gamma_{P, 2}(G)^0 \le H $ to conclude $g \in H$.
\end{quote}
{\bf Theorem:}
Let $\Gamma$ be the elements $a \in G$ of order $p$ with $m_p(C_G(a)) > 2$ and
$\theta: \Gamma \rightarrow p'(G)$ such that $\forall a,b \in \Gamma, [a,b]=1$ and
all $g \in G: \theta(a^g)=\theta(a)^g$ and $\theta(a) \cap C_G(b) \le \theta(b)$.
Let $P \in S_p(G)$ and assume $G=\Gamma_{P,2}(G)$ and $O_{p'}(G)=1$ and finally assume
that either $\theta(a)$ is solvable for each $a \in \Gamma$ or the Signalizer Functor Theorem
holds on $G$ then $\theta(a)=1, \forall a \in \Gamma$.
\begin{quote}
\emph{Proof:}
For $A \in {\cal E}^p_3(G)$, $\theta$ is an $A$-signalizer functor.  For
$B \in {\cal E}^p_2(G)$, define $W_B= \langle \theta(b): b \in B^{\#} \rangle$.  Then
$\exists A \in {\cal E}^p_3(G)$ with $B \le A$ and hence $W_B = W_A$ and
$\Gamma_{A,2}(G) \le N_G(W_A)$.  In particular, if $B, D$ are distinct members of
${\cal E}^p_2(G)$ adjacent in
${\cal D}({\cal E}^p_2(G))$ , $BD \in
{\cal E}^p_2(G)$ so $W_B = W_{BD} = W_D$.  Thus
$G = \Gamma^0_{P, k}(G) \le N_G(W)$ since
$N_G(B) \le \Gamma_{A,2}(G) \le N_G(W)$.  But by 1 and the solvable signalizer functor theorem,
$W$ is a $p'$ group so $W \le O_{p'}(G)$.
Since $O_{p'}(G) = 1$ and $\theta(a) \le W, \forall a \in \Gamma$, the lemma is proved.
\end{quote}
{\bf Theorem:}
Let $\Gamma$ be the elements $a \in G$ of order $p$ with $m_p(C_G(a)) > 2$ and
$P \in S_p(G)$ and assume $G=\Gamma_{P,2}(G)$ and $O_{p'}(G)=1$, $C_G(a)$ is balenced
for the prime $p$ and for each $a \in \Gamma$ either
$O_{p'}(C_G(a))$ is solvable or the Signalizer Functor Theorem holds on $G$ then
$O_{p'}(C_G(a))=1, \forall a \in \Gamma$.
\begin{quote}
\emph{Proof:}
For $a \in \Gamma$, let $\theta(a)= O_{p'}(C_G(a))$ and we know
$\theta(a) \cap C_G(b) \le \theta(b), \forall a, b \in \Gamma$ with $[a,b]= 1$ and applying the
previous result, we're done.
\end{quote}
\section{Aschbacher's Plan}
{\bf Definition:}
$m_{2,p}(G)= max_H \{ m_p(H) \}$, where $H$ is a $2$-local of $G$.
$e(G)= max \{m_{2,p}(G) \}, p$ odd.
\\
\\
{\bf Definition:}
$O_{p', E}(G)/O_{p'}(G)= E(G/O_{p'}(G))$.
\\
\\
{\bf Theorem:} $P \in p(G)$ then
(1) $O_{p', E}(N_G(P)) \le C_G(O_p(G))$ and
(2) if $P \le O_p(G)$ then $O^p(F^*(N_G(P))= O^p(F^*(G))$.
\begin{quote}
\emph{Proof:}
Todo.
\end{quote}
{\bf Definition:}
$G$ is \emph{balenced} for $p$, prime if $O_{p'}(C_G(X)) \le O_{p'}(G), \forall X < G: |X|=p$.
$m_{2,p}(G)= max_H \{ m_p(H) \}$, where $H$ is a $2$-local of $G$.
$e(G)= max \{m_{2,p}(G) \}, p$ odd.
\\
\\
{\bf Theorem:} Let $O_{p'}(G)=1$ and $Aut_H(L)$ is balenced for the prime $p$
for each $L \in Comp(G)$ and for each $L \in Comp(G)$ and each $H \le G$ with $L \lhd H$
then $G$ is balenced for the prime $p$.
\begin{quote}
\end{quote}
{\bf Theorem:} If $G$ is a non-abelian simple group with $m_2(G) \le 2$ then either
(1) a Sylow $2$-group of $G$ is dihedrael, semi-dihedral, or
${\mathbb Z}_{2^n} \wr {\mathbb Z}_2$ and $G \cong L_2(q), L_3(q), U_3(q), q$, odd, or
$M_{11}$; or (2) $G \cong U_3(4)$.
\begin{quote}
\end{quote}
{\bf Theorem:} If $G$ is a non-abelian simple group with $m_2(G) > 2$  and
$G$ has a proper $2$-generated $2$-core then either $G$ is of Lie type of characteristic 2
and Lie rank 1 ($L_2(2^n), L_2(2^n), Sz(2^n)$ or $J_1$).
\begin{quote}
\end{quote}
{\bf Definition of $B_p$ property:}   Let $p$ be a prime and $O_{p'}(G)=1$ and $x$ and element of
order $p$ in $G$ then $O_{p',E}(C_G(x))= O_{p'}(C_G(x))E(C_G(x))$.
\\
\\
{\bf Theorem:}
Let $p \ne 2$ and suppose $G$ contains no normal abelian subgroups of rank $3$ then
$G$ has $p$-rank $\le 2$.
\begin{quote}
\end{quote}
{\bf Theorem:} If $G$ is a non-abelian simple group with $m_2(G) \le 2$ then either
(1) a Sylow $2$-group of $G$ is dihedrael, semi-dihedral, or
${\mathbb Z}_{2^n} \wr {\mathbb Z}$ and $G \cong L_2(q), L_3(q), U_3(q), q$, odd, or
$M_{11}$; or (2) $G \cong U_3(4)$.
\begin{quote}
\end{quote}
{\bf Theorem:} If $G$ is a non-abelian simple group with $m_2(G) > 2$  and
$G$ has a proper $2$-generated $2$-core then either $G$ is of Lie type of characteristic 2
and Lie rank 1 ($L_2(2^n), L_2(2^n), Sz(2^n)$ or $J_1$.
\begin{quote}
\end{quote}
{\bf Definition of $B_p$ property:}   Let $p$ be a prime and $O_{p'}(G)=1$ and $x$ and element of
order $p$ in $G$ then $O_{p',E}(C_G(x))= O_{p'}(C_G(x))E(C_G(x))$.
\\
\\
{\bf Unbalenced Group Theorem:} If $G$ is a group with $F^*(G)$, simple which is unbalenced for
the prime $2$, then $F^*(G)$ is a group of Lie type and odd characteristic $A_{2n+1}$,
$L_3(4)$ or $He$.
\\
\\
{\bf Component Theorem:} Let $G$ be a group with $F^*(G)$ simple satisfying the $B_2$ property
and possessing an involutions, $t$, such that $O_{2',E}(C_G(t)) \ne O_{2'}(C_G(t))$ then
$G$ posseses a standard subgroup for the prime $2$.
\\
\\
{\bf Standard Form Problem for $(L,r)$:}   Determine all finite groups $G$ possessing a
standard subgroup $H$ for the prime $r$: $E(H) \cong L$.
\\
\\
{\bf Theorem:} Let $G$ be a minimal counterexample to the classification theorem and assume $G$
is generic of even characteristic then one of the following holds:
(1) $G$ possesses a standard subgroup  for some $p \in \sigma(G)$,
(2) there is a symplectic involution, $t$, such that $F^*(C_G(t))$ is a $2$-group
of symplectic type or
(3) $G$ is in the uniqueness case.
\\
\\
{\bf Aschbacher:} Suppose $G$ is a finite simple group, the $B$-Theorem holds, and $E(C_G(t)) \neq 1$
for some $t \in Inv(G)$.  $\exists x \in Inv(G)$ and a quasi-simple group,
$K$ of $C(z)$ such that $C_{C(z)}(K)$ either has $2$-rank 1 or is solvable with elementary abelian
or dihedral $2$-Sylow subgroups.
\\
\\
{\bf Standard form:} Suppose $x \in Inv(G)$ and  $H = C(x)$ has a normal quasi-simple subgroup $L$,
of Lie type with odd characteristic such that $C_H(L)$ has $2$-rank 1.   We say $H$ is in standard
form with standard component $L$.


% Created: Enze Chen, May 2017
% Last edited: Enze Chen, May 2017
%
% Appendix A of the MSE 142 coursereader. This chapter serves as a reference sheet for physical constants, SI prefixes, the Schrodinger equation, and other useful formulas.

% Uncomment the following three lines and last line to individually compile this chapter
%\documentclass[12pt, english]{book}
%\usepackage{142crstyle}
%\begin{document}

\chapter{Reference} \label{ch:ref}
%{ \doublespacing
Here are some fundamental constants and equations that you might need to refer to. \par
\section{Fundamental constants}
\begin{table}[!h]
	\centering
	\begin{tabular}{lll}
	\textbf{Name} & \textbf{Symbol} & \textbf{Value} \\ \toprule
	Planck's constant & $h$ & \SI{6.626070e-34}{\joule \second} \\
	Reduced Planck's constant & $\hbar = h/2\pi$ & \SI{1.054572e-34}{\joule\second} \\
	Speed of light & $c$ & \SI{2.997925e8}{\meter/\second} \\
	Mass of an electron & $m_e$ & \SI{9.109384e-31}{\kilogram} \\
	Mass of a proton & $m_p$ & \SI{1.672622e-27}{\kilogram} \\
	Charge of an electron & $e$ & \SI{1.602177e-19}{\coulomb} \\
	Boltzmann's constant & $k_B$ & \SI{1.380649e-23}{\joule/\kelvin} \\
	Permittivity of free space & $\varepsilon_0$ & \SI{8.854188e-12}{\farad/\meter}
	\end{tabular}
\end{table}

\begin{table}[!h]
	\centering
	\begin{tabular}{lccccccccc}
	\textbf{SI Prefix}: & tera & giga & mega & kilo & milli & micro & nano & pico & femto \\ \midrule
	\textbf{Value}: & $10^{12}$ & $10^{9}$ & $10^{6}$ & $10^{3}$ & $10^{-3}$ & $10^{-6}$ & $10^{-9}$ & $10^{-12}$ & $10^{-15}$
	\end{tabular}
\end{table}

\section{The \Sch\ equation}
In its most general form, the \textbf{time-dependent} \Sch\ equation is given by
\begin{tcolorbox}[title=Time-dependent \Sch\ equation] \vspace{-2ex}
\begin{equation*}
	i\hbar \pdv{t} \Psi(\textbf{r}, t) = \hat{H}\Psi(\textbf{r}, t)
\end{equation*}
\end{tcolorbox}

where $i$ is the imaginary number $i=\sqrt{-1}$, $\hbar$ is the reduced Planck's constant, $\pdv{t}$ is the partial derivative with respect to time, $\Psi$ is the wavefunction, $\mathbf{r}$ is the position vector, $t$ is time, and $\hat{H}$ is the Hamiltonian operator. Often times, we make the simplification to obtain the \textbf{time-independent} \Sch\ equation for stationary states. It is given generally by
\begin{equation*}
\hat{H}\Psi = E\Psi
\end{equation*}

where $E$ is a proportionality constant equal to the energy of state $\Psi$. For a non-relativistic particle, the time-independent \Sch\ equation in three-dimensional space is given by
\begin{equation*}
\left[ -\dfrac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) \right]\Psi(\mathbf{r}) = E\Psi(\mathbf{r})
\end{equation*}

where $\nabla^2$ is the Laplacian operator and $V$ is the potential. In just one dimension, say along the $x$ direction, this expression simplifies down to the canonical
\begin{tcolorbox}[title=Time-independent \Sch\ equation] \vspace{-2ex}
\[ -\dfrac{\hbar^2}{2m} \dv[2]{\psi(x)}{x} + V(x)\psi(x) = E\psi(x) \]
\end{tcolorbox}

%}
%\end{document}


\subsection{Bivariate Statistics}

\noindent{\bf Description}
\smallskip

Bivariate statistics are used to quantitatively describe the association between
two features, such as test their statistical (in-)dependence or measure
the accuracy of one data feature predicting the other feature, in a sample.
The \BivarScriptName{} script computes common bivariate statistics,
such as \NameStatR{} and \NameStatChi{}, in parallel for many pairs
of data features.  For a given dataset matrix, script \BivarScriptName{} computes
certain bivariate statistics for the given feature (column) pairs in the
matrix.  The feature types govern the exact set of statistics computed for that pair.
For example, \NameStatR{} can only be computed on two quantitative (scale)
features like `Height' and `Temperature'.
It does not make sense to compute the linear correlation of two categorical attributes
like `Hair Color'.

\smallskip
\noindent{\bf Usage}
\smallskip

{\hangindent=\parindent\noindent\it%\tolerance=0
{\tt{}-f }path/\/\BivarScriptName{}
{\tt{} -nvargs}
{\tt{} X=}path/file
{\tt{} index1=}path/file
{\tt{} index2=}path/file
{\tt{} types1=}path/file
{\tt{} types2=}path/file
{\tt{} OUTDIR=}path
% {\tt{} fmt=}format

}

\smallskip
\noindent{\bf Arguments}
\begin{Description}
\item[{\tt X}:]
Location (on HDFS) to read the data matrix $X$ whose columns are the features
that we want to compare and correlate with bivariate statistics.
\item[{\tt index1}:] % (default:\mbox{ }{\tt " "})
Location (on HDFS) to read the single-row matrix that lists the column indices
of the \emph{first-argument} features in pairwise statistics.
Its $i^{\textrm{th}}$ entry (i.e.\ $i^{\textrm{th}}$ column-cell) contains the
index $k$ of column \texttt{X[,$\,k$]} in the data matrix
whose bivariate statistics need to be computed.
% The default value means ``use all $X$-columns from the first to the last.''
\item[{\tt index2}:] % (default:\mbox{ }{\tt " "})
Location (on HDFS) to read the single-row matrix that lists the column indices
of the \emph{second-argument} features in pairwise statistics.
Its $j^{\textrm{th}}$ entry (i.e.\ $j^{\textrm{th}}$ column-cell) contains the
index $l$ of column \texttt{X[,$\,l$]} in the data matrix
whose bivariate statistics need to be computed.
% The default value means ``use all $X$-columns from the first to the last.''
\item[{\tt types1}:] % (default:\mbox{ }{\tt " "})
Location (on HDFS) to read the single-row matrix that lists the \emph{types}
of the \emph{first-argument} features in pairwise statistics.
Its $i^{\textrm{th}}$ entry (i.e.\ $i^{\textrm{th}}$ column-cell) contains the type
of column \texttt{X[,$\,k$]} in the data matrix, where $k$ is the $i^{\textrm{th}}$
entry in the {\tt index1} matrix.  Feature types must be encoded by
integer numbers: $1 = {}$scale, $2 = {}$nominal, $3 = {}$ordinal.
% The default value means ``treat all referenced $X$-columns as scale.''
\item[{\tt types2}:] % (default:\mbox{ }{\tt " "})
Location (on HDFS) to read the single-row matrix that lists the \emph{types}
of the \emph{second-argument} features in pairwise statistics.
Its $j^{\textrm{th}}$ entry (i.e.\ $j^{\textrm{th}}$ column-cell) contains the type
of column \texttt{X[,$\,l$]} in the data matrix, where $l$ is the $j^{\textrm{th}}$
entry in the {\tt index2} matrix.  Feature types must be encoded by
integer numbers: $1 = {}$scale, $2 = {}$nominal, $3 = {}$ordinal.
% The default value means ``treat all referenced $X$-columns as scale.''
\item[{\tt OUTDIR}:]
Location path (on HDFS) where the output matrices with computed bivariate
statistics will be stored.  The matrices' file names and format are defined
in Table~\ref{table:bivars}.
% \item[{\tt fmt}:] (default:\mbox{ }{\tt "text"})
% Matrix file output format, such as {\tt text}, {\tt mm}, or {\tt csv};
% see read/write functions in SystemML Language Reference for details.
\end{Description}

\begin{table}[t]\hfil
\begin{tabular}{|lll|}
\hline\rule{0pt}{12pt}%
Ouput File / Matrix         & Row$\,$\# & Name of Statistic   \\[2pt]
\hline\hline\rule{0pt}{12pt}%
\emph{All Files}            &     1     & 1-st feature column \\
\rule{1em}{0pt}"            &     2     & 2-nd feature column \\[2pt]
\hline\rule{0pt}{12pt}%
bivar.scale.scale.stats     &     3     & \NameStatR          \\[2pt]
\hline\rule{0pt}{12pt}%
bivar.nominal.nominal.stats &     3     & \NameStatChi        \\
\rule{1em}{0pt}"            &     4     & Degrees of freedom  \\
\rule{1em}{0pt}"            &     5     & \NameStatPChi       \\
\rule{1em}{0pt}"            &     6     & \NameStatV          \\[2pt]
\hline\rule{0pt}{12pt}%
bivar.nominal.scale.stats   &     3     & \NameStatEta        \\
\rule{1em}{0pt}"            &     4     & \NameStatF          \\[2pt]
\hline\rule{0pt}{12pt}%
bivar.ordinal.ordinal.stats &     3     & \NameStatRho        \\[2pt]
\hline
\end{tabular}\hfil
\caption{%
The output matrices of \BivarScriptName{} have one row per one bivariate
statistic and one column per one pair of input features.  This table lists
the meaning of each matrix and each row.%
% Signs ``+'' show applicability to scale or/and to categorical features.
}
\label{table:bivars}
\end{table}

\pagebreak[2]

\noindent{\bf Details}
\smallskip

Script \BivarScriptName{} takes an input matrix \texttt{X} whose columns represent
the features and whose rows represent the records of a data sample.
Given \texttt{X}, the script computes certain relevant bivariate statistics
for specified pairs of feature columns \texttt{X[,$\,i$]} and \texttt{X[,$\,j$]}.
Command-line parameters \texttt{index1} and \texttt{index2} specify the files with
column pairs of interest to the user.  Namely, the file given by \texttt{index1}
contains the vector of the 1st-attribute column indices and the file given
by \texttt{index2} has the vector of the 2nd-attribute column indices, with
``1st'' and ``2nd'' referring to their places in bivariate statistics.
Note that both \texttt{index1} and \texttt{index2} files should contain a 1-row matrix
of positive integers.

The bivariate statistics to be computed depend on the \emph{types}, or
\emph{measurement levels}, of the two columns.
The types for each pair are provided in the files whose locations are specified by
\texttt{types1} and \texttt{types2} command-line parameters.
These files are also 1-row matrices, i.e.\ vectors, that list the 1st-attribute and
the 2nd-attribute column types in the same order as their indices in the
\texttt{index1} and \texttt{index2} files.  The types must be provided as per
the following convention: $1 = {}$scale, $2 = {}$nominal, $3 = {}$ordinal.

The script orgainizes its results into (potentially) four output matrices, one per
each type combination.  The types of bivariate statistics are defined using the types
of the columns that were used for their arguments, with ``ordinal'' sometimes
retrogressing to ``nominal.''  Table~\ref{table:bivars} describes what each column
in each output matrix contains.  In particular, the script includes the following
statistics:
\begin{Itemize}
\item For a pair of scale (quantitative) columns, \NameStatR;
\item For a pair of nominal columns (with finite-sized, fixed, unordered domains),
the \NameStatChi{} and its p-value;
\item For a pair of one scale column and one nominal column, \NameStatF{};
\item For a pair of ordinal columns (ordered domains depicting ranks), \NameStatRho.
\end{Itemize}
Note that, as shown in Table~\ref{table:bivars}, the output matrices contain the
column indices of the features involved in each statistic.
Moreover, if the output matrix does not contain
a value in a certain cell then it should be interpreted as a~$0$
(sparse matrix representation).

Below we list all bivariate statistics computed by script \BivarScriptName.
The statistics are collected into several groups by the type of their input
features.  We refer to the two input features as $v_1$ and $v_2$ unless
specified otherwise; the value pairs are $(v_{1,i}, v_{2,i})$ for $i=1,\ldots,n$,
where $n$ is the number of rows in \texttt{X}, i.e.\ the sample size.

\paragraph{Scale-vs-scale statistics.}
Sample statistics that describe association between two quantitative (scale) features.
A scale feature has numerical values, with the natural ordering relation.
\begin{Description}
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it\NameStatR]:
A measure of linear dependence between two numerical features:
\begin{equation*}
r \,\,=\,\, \frac{\Cov(v_1, v_2)}{\sqrt{\Var v_1 \Var v_2}}
\,\,=\,\, \frac{\sum_{i=1}^n (v_{1,i} - \bar{v}_1) (v_{2,i} - \bar{v}_2)}%
{\sqrt{\sum_{i=1}^n (v_{1,i} - \bar{v}_1)^{2\mathstrut} \cdot \sum_{i=1}^n (v_{2,i} - \bar{v}_2)^{2\mathstrut}}}
\end{equation*}
Commonly denoted by~$r$, correlation ranges between $-1$ and $+1$, reaching ${\pm}1$ when all value
pairs $(v_{1,i}, v_{2,i})$ lie on the same line.  Correlation near~0 means that a line is not a good
way to represent the dependence between the two features; however, this does not imply independence.
The sign indicates direction of the linear association: $r > 0$ ($r < 0$) if one feature tends to
linearly increase (decrease) when the other feature increases.  Nonlinear association, if present,
may disobey this sign.
\NameStatR{} is symmetric: $r(v_1, v_2) = r(v_2, v_1)$; it does not change if we transform $v_1$ and $v_2$
to $a + b v_1$ and $c + d v_2$ where $a, b, c, d$ are constants and $b, d > 0$.

Suppose that we use simple linear regression to represent one feature given the other, say
represent $v_{2,i} \approx \alpha + \beta v_{1,i}$ by selecting $\alpha$ and $\beta$
to minimize the least-squares error $\sum_{i=1}^n (v_{2,i} - \alpha - \beta v_{1,i})^2$.
Then the best error equals
\begin{equation*}
\min_{\alpha, \beta} \,\,\sum_{i=1}^n \big(v_{2,i} - \alpha - \beta v_{1,i}\big)^2 \,\,=\,\,
(1 - r^2) \,\sum_{i=1}^n \big(v_{2,i} - \bar{v}_2\big)^2
\end{equation*}
In other words, $1\,{-}\,r^2$ is the ratio of the residual sum of squares to
the total sum of squares.  Hence, $r^2$ is an accuracy measure of the linear regression.
\end{Description}

\paragraph{Nominal-vs-nominal statistics.}
Sample statistics that describe association between two nominal categorical features.
Both features' value domains are encoded with positive integers in arbitrary order:
nominal features do not order their value domains.
\begin{Description}
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it\NameStatChi]:
A measure of how much the frequencies of value pairs of two categorical features deviate from
statistical independence.  Under independence, the probability of every value pair must equal
the product of probabilities of each value in the pair:
$\Prob[a, b] - \Prob[a]\,\Prob[b] = 0$.  But we do not know these (hypothesized) probabilities;
we only know the sample frequency counts.  Let $n_{a,b}$ be the frequency count of pair
$(a, b)$, let $n_a$ and $n_b$ be the frequency counts of $a$~alone and of $b$~alone.  Under
independence, difference $n_{a,b}{/}n - (n_a{/}n)(n_b{/}n)$ is unlikely to be exactly~0 due
to sample randomness, yet it is unlikely to be too far from~0.  For some pairs $(a,b)$ it may
deviate from~0 farther than for other pairs.  \NameStatChi{}~is an aggregate measure that
combines squares of these differences across all value pairs:
\begin{equation*}
\chi^2 \,\,=\,\, \sum_{a,\,b} \Big(\frac{n_a n_b}{n}\Big)^{-1} \Big(n_{a,b} - \frac{n_a n_b}{n}\Big)^2
\,=\,\, \sum_{a,\,b} \frac{(O_{a,b} - E_{a,b})^2}{E_{a,b}}
\end{equation*}
where $O_{a,b} = n_{a,b}$ are the \emph{observed} frequencies and $E_{a,b} = (n_a n_b){/}n$ are
the \emph{expected} frequencies for all pairs~$(a,b)$.  Under independence (plus other standard
assumptions) the sample~$\chi^2$ closely follows a well-known distribution, making it a basis for
statistical tests for independence, see~\emph{\NameStatPChi} for details.  Note that \NameStatChi{}
does \emph{not} measure the strength of dependence: even very weak dependence may result in a
significant deviation from independence if the counts are large enough.  Use~\NameStatV{} instead
to measure the strength of dependence.
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it Degrees of freedom]:
An integer parameter required for the interpretation of~\NameStatChi{} measure.  Under independence
(plus other standard assumptions) the sample~$\chi^2$ statistic is approximately distributed as the
sum of $d$~squares of independent normal random variables with mean~0 and variance~1, where $d$ is
this integer parameter.  For a pair of categorical features such that the $1^{\textrm{st}}$~feature
has $k_1$ categories and the $2^{\textrm{nd}}$~feature has $k_2$ categories, the number of degrees
of freedom is $d = (k_1 - 1)(k_2 - 1)$.
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it\NameStatPChi]:
A measure of how likely we would observe the current frequencies of value pairs of two categorical
features assuming their statistical independence.  More precisely, it computes the probability that
the sum of $d$~squares of independent normal random variables with mean~0 and variance~1
(called the $\chi^2$~distribution with $d$ degrees of freedom) generates a value at least as large
as the current sample \NameStatChi.  The $d$ parameter is \emph{degrees of freedom}, see above.
Under independence (plus other standard assumptions) the sample \NameStatChi{} closely follows the
$\chi^2$~distribution and is unlikely to land very far into its tail.  On the other hand, if the
two features are dependent, their sample \NameStatChi{} becomes arbitrarily large as $n\to\infty$
and lands extremely far into the tail of the $\chi^2$~distribution given a large enough data sample.
\NameStatPChi{} returns the tail ``weight'' on the right-hand side of \NameStatChi:
\begin{equation*}
P\,\,=\,\, \Prob\big[r \geq \textrm{\NameStatChi} \,\,\big|\,\, r \sim \textrm{the $\chi^2$ distribution}\big]
\end{equation*}
As any probability, $P$ ranges between 0 and~1.  If $P\leq 0.05$, the dependence between the two
features may be considered statistically significant (i.e.\ their independence is considered
statistically ruled out).  For highly dependent features, it is not unusual to have $P\leq 10^{-20}$
or less, in which case our script will simply return $P = 0$.  Independent features should have
their $P\geq 0.05$ in about 95\% cases.
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it\NameStatV]:
A measure for the strength of association, i.e.\ of statistical dependence, between two categorical
features, conceptually similar to \NameStatR.  It divides the observed~\NameStatChi{} by the maximum
possible~$\chi^2_{\textrm{max}}$ given $n$ and the number $k_1, k_2$~of categories in each feature,
then takes the square root.  Thus, \NameStatV{} ranges from 0 to~1,
where 0 implies no association and 1 implies the maximum possible association (one-to-one
correspondence) between the two features.  See \emph{\NameStatChi} for the computation of~$\chi^2$;
its maximum${} = {}$%
$n\cdot\min\{k_1\,{-}\,1, k_2\,{-}\,1\}$ where the $1^{\textrm{st}}$~feature
has $k_1$ categories and the $2^{\textrm{nd}}$~feature has $k_2$ categories~\cite{AcockStavig1979:CramersV},
so
\begin{equation*}
\textrm{\NameStatV} \,\,=\,\, \sqrt{\frac{\textrm{\NameStatChi}}{n\cdot\min\{k_1\,{-}\,1, k_2\,{-}\,1\}}}
\end{equation*}
As opposed to \NameStatPChi, which goes to~0 (rapidly) as the features' dependence increases,
\NameStatV{} goes towards~1 (slowly) as the dependence increases.  Both \NameStatChi{} and
\NameStatPChi{} are very sensitive to~$n$, but in \NameStatV{} this is mitigated by taking the
ratio.
\end{Description}

\paragraph{Nominal-vs-scale statistics.}
Sample statistics that describe association between a categorical feature
(order ignored) and a quantitative (scale) feature.
The values of the categorical feature must be coded as positive integers.
\begin{Description}
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it\NameStatEta]:
A measure for the strength of association (statistical dependence) between a nominal feature
and a scale feature, conceptually similar to \NameStatR.  Ranges from 0 to~1, approaching 0
when there is no association and approaching 1 when there is a strong association.
The nominal feature, treated as the independent variable, is assumed to have relatively few
possible values, all with large frequency counts.  The scale feature is treated as the dependent
variable.  Denoting the nominal feature by~$x$ and the scale feature by~$y$, we have:
\begin{equation*}
\eta^2 \,=\, 1 - \frac{\sum_{i=1}^{n} \big(y_i - \hat{y}[x_i]\big)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2},
\,\,\,\,\textrm{where}\,\,\,\,
\hat{y}[x] = \frac{1}{\mathop{\mathrm{freq}}(x)}\sum_{i=1}^n
\,\left\{\!\!\begin{array}{rl} y_i & \textrm{if $x_i = x$}\\ 0 & \textrm{otherwise}\end{array}\right.\!\!\!
\end{equation*}
and $\bar{y} = (1{/}n)\sum_{i=1}^n y_i$ is the mean.  Value $\hat{y}[x]$ is the average
of~$y_i$ among all records where $x_i = x$; it can also be viewed as the ``predictor''
of $y$ given~$x$.  Then $\sum_{i=1}^{n} (y_i - \hat{y}[x_i])^2$ is the residual error
sum-of-squares and $\sum_{i=1}^{n} (y_i - \bar{y})^2$ is the total sum-of-squares for~$y$.
Hence, $\eta^2$ measures the accuracy of predicting $y$ with~$x$, just like the
``R-squared'' statistic measures the accuracy of linear regression.  Our output $\eta$
is the square root of~$\eta^2$.
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it\NameStatF]:
A measure of how much the values of the scale feature, denoted here by~$y$,
deviate from statistical independence on the nominal feature, denoted by~$x$.
The same measure appears in the one-way analysis of vari\-ance (ANOVA).
Like \NameStatChi, \NameStatF{} is used to test the hypothesis that
$y$~is independent from~$x$, given the following assumptions:
\begin{Itemize}
\item The scale feature $y$ has approximately normal distribution whose mean
may depend only on~$x$ and variance is the same for all~$x$;
\item The nominal feature $x$ has relatively small value domain with large
frequency counts, the $x_i$-values are treated as fixed (non-random);
\item All records are sampled independently of each other.
\end{Itemize}
To compute \NameStatF{}, we first compute $\hat{y}[x]$ as the average of~$y_i$
among all records where $x_i = x$.  These $\hat{y}[x]$ can be viewed as
``predictors'' of $y$ given~$x$; if $y$ is independent on~$x$, they should
``predict'' only the global mean~$\bar{y}$.  Then we form two sums-of-squares:
\begin{Itemize}
\item \emph{Residual} sum-of-squares of the ``predictor'' accuracy: $y_i - \hat{y}[x_i]$;
\item \emph{Explained} sum-of-squares of the ``predictor'' variability: $\hat{y}[x_i] - \bar{y}$.
\end{Itemize}
\NameStatF{} is the ratio of the explained sum-of-squares to
the residual sum-of-squares, each divided by their corresponding degrees
of freedom:
\begin{equation*}
F \,\,=\,\,
\frac{\sum_{x}\, \mathop{\mathrm{freq}}(x) \, \big(\hat{y}[x] - \bar{y}\big)^2 \,\big/\,\, (k\,{-}\,1)}%
{\sum_{i=1}^{n} \big(y_i - \hat{y}[x_i]\big)^2 \,\big/\,\, (n\,{-}\,k)} \,\,=\,\,
\frac{n\,{-}\,k}{k\,{-}\,1} \cdot \frac{\eta^2}{1 - \eta^2}
\end{equation*}
Here $k$ is the domain size of the nominal feature~$x$.  The $k$ ``predictors''
lose 1~freedom due to their linear dependence with~$\bar{y}$; similarly,
the $n$~$y_i$-s lose $k$~freedoms due to the ``predictors''.

The statistic can test if the independence hypothesis of $y$ from $x$ is reasonable;
more generally (with relaxed normality assumptions) it can test the hypothesis that
\emph{the mean} of $y$ among records with a given~$x$ is the same for all~$x$.
Under this hypothesis \NameStatF{} has, or approximates, the $F(k\,{-}\,1, n\,{-}\,k)$-distribution.
But if the mean of $y$ given $x$ depends on~$x$, \NameStatF{}
becomes arbitrarily large as $n\to\infty$ (with $k$~fixed) and lands extremely far
into the tail of the $F(k\,{-}\,1, n\,{-}\,k)$-distribution given a large enough data sample.
\end{Description}

\paragraph{Ordinal-vs-ordinal statistics.}
Sample statistics that describe association between two ordinal categorical features.
Both features' value domains are encoded with positive integers, so that the natural
order of the integers coincides with the order in each value domain.
\begin{Description}
%%%%%%%%%%%%%%%%%%%% DESCRIPTIVE STATISTIC %%%%%%%%%%%%%%%%%%%%
\item[\it\NameStatRho]:
A measure for the strength of association (statistical dependence) between
two ordinal features, conceptually similar to \NameStatR.  Specifically, it is \NameStatR{}
applied to the feature vectors in which all values are replaced by their ranks, i.e.\
their positions if the vector is sorted.  The ranks of identical (duplicate) values
are replaced with their average rank.  For example, in vector
$(15, 11, 26, 15, 8)$ the value ``15'' occurs twice with ranks 3 and~4 per the sorted
order $(8_1, 11_2, 15_3, 15_4, 26_5)$; so, both values are assigned their average
rank of $3.5 = (3\,{+}\,4)\,{/}\,2$ and the vector is replaced by~$(3.5,\, 2,\, 5,\, 3.5,\, 1)$.

Our implementation of \NameStatRho{} is geared towards features having small value domains
and large counts for the values.  Given the two input vectors, we form a contingency table $T$
of pairwise frequency counts, as well as a vector of frequency counts for each feature: $f_1$
and~$f_2$.  Here in $T_{i,j}$, $f_{1,i}$, $f_{2,j}$ indices $i$ and~$j$ refer to the
order-preserving integer encoding of the feature values.
We use prefix sums over $f_1$ and~$f_2$ to compute the values' average ranks:
$r_{1,i} = \sum_{j=1}^{i-1} f_{1,j} + (f_{1,i}\,{+}\,1){/}2$, and analogously for~$r_2$.
Finally, we compute rank variances for $r_1, r_2$ weighted by counts $f_1, f_2$ and their
covariance weighted by~$T$, before applying the standard formula for \NameStatR:
\begin{equation*}
\rho \,\,=\,\, \frac{\Cov_T(r_1, r_2)}{\sqrt{\Var_{f_1}(r_1)\Var_{f_2}(r_2)}}
\,\,=\,\, \frac{\sum_{i,j} T_{i,j} (r_{1,i} - \bar{r}_1) (r_{2,j} - \bar{r}_2)}%
{\sqrt{\sum_i f_{1,i} (r_{1,i} - \bar{r}_1)^{2\mathstrut} \cdot \sum_j f_{2,j} (r_{2,j} - \bar{r}_2)^{2\mathstrut}}}
\end{equation*}
where $\bar{r}_1 = \sum_i r_{1,i} f_{1,i}{/}n$, analogously for~$\bar{r}_2$.
The value of $\rho$ lies between $-1$ and $+1$, with sign indicating the prevalent direction
of the association: $\rho > 0$ ($\rho < 0$) means that one feature tends to increase (decrease)
when the other feature increases.  The correlation becomes~1 when the two features are
monotonically related.
\end{Description}

\smallskip
\noindent{\bf Returns}
\smallskip

A collection of (potentially) 4 matrices.  Each matrix contains bivariate statistics that
resulted from a different combination of feature types.  There is one matrix for scale-scale
statistics (which includes \NameStatR), one for nominal-nominal statistics (includes \NameStatChi{}),
one for nominal-scale statistics (includes \NameStatF) and one for ordinal-ordinal statistics
(includes \NameStatRho).  If any of these matrices is not produced, then no pair of columns required
the corresponding type combination.  See Table~\ref{table:bivars} for the matrix naming and
format details.

\smallskip
\pagebreak[2]

\noindent{\bf Examples}
\smallskip

{\hangindent=\parindent\noindent\tt
\hml -f \BivarScriptName{} -nvargs
X=/user/biadmin/X.mtx
index1=/user/biadmin/S1.mtx
index2=/user/biadmin/S2.mtx
types1=/user/biadmin/K1.mtx
types2=/user/biadmin/K2.mtx
OUTDIR=/user/biadmin/stats.mtx

}


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\journal{Annals of Nuclear Energy}

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\begin{frontmatter}

\title{A nonstiff solution for the stochastic neutron point kinetics equations}

%% Group authors per affiliation:
\author[ufrgs]{M. Wollmann da Silva\fnref{milena}}
\author[ucb]{R. Vasques\corref{cor1}}
\author[ufrgs]{B.E.J. Bodmann\fnref{bodmann}}
\author[ufrgs]{M.T. Vilhena\fnref{vilhena}}

\address[ufrgs]{UFRGS - Federal University of Rio Grande do Sul, Av. Osvaldo
Aranha 99, 90046-900\\ Porto Alegre, RS, Brazil}
\address[ucb]{University of California, Berkeley, Department of Nuclear Engineering, 4155 Etcheverry Hall \\ Berkeley, CA 94720-1730}

\cortext[cor1]{Corresponding author: richard.vasques@fulbrightmail.org; Tel: (510) 340 0930\\
Postal address: University of California, Berkeley, Department of Nuclear Engineering, 4103 Etcheverry Hall, Berkeley, CA 94720-1730}
\fntext[milena]{milena.wollmann@ufrgs.br}
\fntext[bodmann]{bardobodmann@ufrgs.br}
\fntext[vilhena]{vilhena@mat.ufrgs.br}

\begin{abstract}

We propose an approach to solve the stochastic neutron point kinetics equations using an adaptation of the diagonalization-decomposition method (DDM).
This new approach (Double-DDM) yields a nonstiff solution for the stochastic formulation, allowing the calculation of the neutron and precursor densities at any time of interest without the need of using progressive time steps.
We use Double-DDM to compute results for stochastic problems with constant, linear, and sinusoidal reactivities.
We show that these results strongly agree with those obtained by other approaches established in the literature.
We also compute and analyze the first four statistical moments of the solutions.

\end{abstract}

\begin{keyword}
point kinetics \sep stochastic moments \sep stiffness \sep decomposition method
\end{keyword}

\end{frontmatter}

\section{Introduction}\label{sec1}
\setcounter{section}{1}
\setcounter{equation}{0}

The neutron point kinetics equations \citep{hetrick_71, kinard_04, hayes_05} are the coupled differential equations for the neutron density and for the delayed neutron precursor concentrations.
These equations model the time-dependent behavior of a nuclear reactor and  provide insight into the dynamics of its operation.
The time-dependent parameters in this system are the reactivity function and the neutron source term.

The neutron density and delayed neutron precursor concentrations vary randomly with time; however, the point kinetics equations are deterministic and can only be used to estimate average values.
Random fluctuations in the neutron density and precursor concentrations can be significant at low power levels \citep{hurwitz_63a,hurwitz_63b}, which points to the importance of estimating these variations.

\citet{hayes_05} have generalized the standard deterministic point kinetics equations, deriving a system of stochastic differential equations that model the random behavior of the neutron density and the precursor concentrations in a point reactor.
Due to the issue of stiffness, this system was implemented numerically using a stochastic piecewise constant approximation method (Stochastic PCA).
Work performed by \citet{saha_12} has shown that order 1.5 strong Taylor and Euler-Maruyama numerical methods are valid computational alternatives to Stochastic PCA in solving the stochastic point kinetics equations.
However, with the exception of cases modeled with either none or only one group of neutron precursors, the stiffness of the problem remains.

In this paper we propose to solve this stochastic formulation using a double decomposition approach based on the diagonalization-decomposition method (DDM) decribed by \citet{wollmann_14}.
This proposed method is the major novelty and principal contribution of this work, yielding a {\em nonstiff} solution for the stochastic point kinetics equations.
Specifically, this approach allows the calculation of the neutron and precursor densities at any time of interest without the need of using progressive time steps.
This solution is obtained with a minimal amount of numerical approximations of the model; the largest numerical effort lies in the truncation of the decomposition and  the integrations required by DDM.

The major caveat in this approach is that convergence of DDM is yet to be proven.
For this reason, a Lyapunov criterion \citep{boichenko_05} is used to guarantee convergence (cf. \citeauthor{petersen_11a}, \citeyear{petersen_11a}; \citeauthor{wollmann_14}, \citeyear{wollmann_14}).
We present computational results for problems with constant, linear, and sinusoidal reactivities.
The results of the proposed method are compared against those of other approaches established in the literature, showing strong agreement.
We also compute the first four statistical moments of the solutions.

This work is an expanded version of a recent conference paper \citep{wollmann_15b}.
The remainder of this paper is organized as follows.
In \cref{sec2} we present a brief review on the key aspects of DDM.
In \cref{sec3} we formulate the stochastic point kinetics equations.
We introduce the proposed double decomposition approach in \cref{sec4}.
Numerical results are given in \cref{sec5} for problems with constant (\cref{sec51}) and time-dependent (\cref{sec52}) reactivities.
The paper concludes in \cref{sec6} with a discussion of the work presented.

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\section{The Diagonalization-Decomposition Method (DDM)}
\label{sec2}
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Following \citet{wollmann_14}, one can obtain an analytical representation for the solution of the neutron point kinetics equations that is free of stiffness.
The neutron point kinetics equations with six groups of precursors and time-dependent reactivity $\rho(t)$ are written as:
\begin{subequations}\label[pluraleq]{eq2.1}
\begin{align}
\frac{d}{dt}n(t)&=\frac{\rho(t)-\beta}{\Lambda} n(t)
+\sum_{i=1}^{6}\lambda_{i}C_{i}(t),  \qquad n(0)=n_{0} \;,
\\
\frac{d}{dt}C_{i}(t)&=\frac{\beta_{i}}{\Lambda} n(t)-\lambda_{i}C_{i}(t),\hspace{72pt}
C_{i}(0)=\frac{\beta_{i}n_{0}}{\Lambda\lambda_{i}}\;,
\end{align}
\end{subequations}
for $i=1,2,...,6$.
Here, $n(t)$ is the neutron density; $C_{i}(t)$ is the density of the i$^{th}$ delayed neutron precursor group; $\lambda_i$ is the decay constant for a specific group $i$; $\Lambda$ represents the neutron mean generation time; and $\beta_i$ represents the delayed-neutron fraction in a specific group $i$.
The total fraction of delayed neutrons is given by $\beta = \sum\limits_{i=1}^6\beta_i$.

A recursive scheme with finite recursive depth $R$ is used to obtain a solution. The truncation index is determined with exponential convergence by the Lyapunov criterion \citep{boichenko_05, petersen_11a}, evaluated after each recursion step.
The neutron population and the precursors concentration are written in terms of the solution from a recursion initialization ($j=0$) and the respective correction terms ($j>0$) for an appropriate $R \in \mathbb{N}$:
\begin{subequations}\label[pluraleq]{eq2.2}
\begin{align}
n(t)&=\sum_{j=0}^{R}n_{j}(t)\,,\\
C_{i}(t)&=\sum_{j=0}^{R}C_{i,j}(t)\, .
\end{align}
\end{subequations}

The combination of \cref{eq2.1} and (\labelcref{eq2.2}) yields a system with $7\times R$ unknowns.
We define
\begin{subequations}\label[pluraleq]{eq2.3}
\begin{align}
\bm Y(t) &= \sum_{j=1}^R \bm Y_j(t)\,,\\
\bm Y_j(t) &=[n_j(t),C_{1,j}(t),C_{2,j}(t),C_{3,j}(t),C_{4,j}(t),C_{5,
j}(t),C_{6,j}(t)]^T\,,\\
\bm\Omega &=\text{diag}\left(\frac{\rho_{0}-\beta}{\Lambda},-\lambda_1, -\lambda_2, -\lambda_3, -\lambda_4, -\lambda_5, -\lambda_6 \right)\,,\label{eq2.3c}
\end{align}
and
\begin{align}
\bm\Xi &= \left[
\begin{array}{cc} \vspace{0.25cm}
\displaystyle{\frac{\rho_{1}(t)}{\Lambda}}&\{\lambda_i\} \\
\vspace{0.25cm}
 \displaystyle{\left\{\frac{\beta_i}{\Lambda}\right\}^T} & \bm 0\\
 \end{array}
 \right] \, , \label{eq2.3d}
\end{align}
\end{subequations}
where $\rho_0$ (constant) and $\rho_1(t)$ are such that $\rho(t) = \rho_0 + \rho_1(t)$. Given the recursive system
\begin{subequations}\label[pluraleq]{eq2.4}%\label{it1}
\begin{align}
& \frac{d}{dt}\bm Y_0(t)-\bm\Omega \bm Y_0(t)=0\,,\label{eq2.4a}\\
& \frac{d}{dt}\bm Y_j (t)-\bm\Omega \bm Y_j (t)=
\bm\Xi(t)\bm Y_{j-1}(t) \;, \qquad j>0\;\label{eq2.4b},
\end{align}
\end{subequations}
the solution of \cref{eq2.4a} is
\begin{subequations}\label[pluraleq]{eq2.5}%\label{solucao}
\begin{align}\label{eq2.5a}
\bm Y_0(t) = \exp(\bm\Omega t)\bm Y_0(0)\,,
\end{align}
with $\bm Y_0(0) = [n_0, C_1(0),C_2(0),C_3(0),C_4(0),C_5(0),C_6(0)]^T$.
\Cref{eq2.4b} may be formally solved by the Laplace transform:
\begin{align}\label{eq2.5b}
 \bm Y_j(t)&=\exp(\bm\Omega t)\cancelto{0}{\bm Y_j(0)}+\int_{0}^{t}\exp(\bm\Omega
\tau)\bm\Xi(t-\tau)\bm Y_{j-1}(t-\tau)d\tau \;, \qquad j>0\;,
\end{align}
\end{subequations}
since the initial condition from \cref{eq2.1} is fully absorbed in \cref{eq2.5a}.
The integral in \cref{eq2.5b} is evaluated using the Gauss-Legendre method.

A flowchart describing the implementation of this method is given in \cref{fig1}.
The solution is obtained in an analytical representation that may be evaluated for any time value (free of stiffness).

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\section{The Stochastic Formulation}
\label{sec3}
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\citet{hayes_05} derived a system of It\^o stochastic differential equations that model the dynamics of the neutron density and the delayed neutron precursors in a point nuclear reactor.
This formulation describes the variation of the population and can be interpreted as a balance between deaths, births, and transformations of neutrons in the system.
The probabilities of these events are determined by the physical parameters of the model, such as the total and partial delayed neutron fraction; the fraction of delayed neutrons of each precursor group; the decay constant of each group; and the average number of neutrons produced in each fission.

Assuming a time interval small enough such that only one event occurs, one can write
\begin{subequations}\label[pluraleq]{eq3.1}
\begin{align}\label{eq3.1a}
 \frac{d}{dt}\bm Y(t)=\bm{\hat{A}}(t) \bm Y(t)+
 \bm Q(t)
 +\bm{\hat{B}}^\frac{1}{2}(t)\frac{d}{dt}\bm{W}(t) \;,
\end{align}
where
\begin{align}
\bm Y(t) &=[n(t),C_{1}(t),C_{2}(t),C_{3}(t),C_{4}(t),C_{5}(t),C_{6}(t)]^T\,,\\
\bm Q(t) &=[q(t),0,0,0,0,0,0]^T\,,\\
\bm W(t) &=[W_0(t),W_1(t),W_2(t),W_3(t),W_4(t),W_5(t),W_6(t)]^T\,,\\
\bm{\hat{A}}(t)&=\bm\Omega + \bm\Xi(t)\,, \vspace{5pt}\\
\bm{\hat{B}}(t)&=\left[\begin{array}{ccccc}
 \zeta & a_1& a_2&\dots &a_6\\
 a_1 & b_{1,1} & b_{1,2} &\dots & b_{1,6}\\
 a_2 & b_{2,1} &b_{2,2} &\dots & b_{2,6}\\
 \vdots & \vdots & \ddots & \ddots & \vdots\\
 a_6&b_{6,1}&\dots&b_{6,5}&b_{6,6}
 \end{array}\right]\, .
\end{align}
Here, $\bm\Omega$ and $\bm\Xi(t)$ are given by \cref{eq2.3c,eq2.3d}, $W_i(t)$ are Wiener processes, and
\begin{align}
\zeta &=\left(\frac{2\beta-1-\rho(t)+(1-\beta)^2\nu}{\Lambda}\right) n(t) \sum_{i=1}^{6}\lambda_i C_i(t)+q(t)\;,\\
 a_i &=\frac{\beta_i}{\Lambda}\big(-1+(1-\beta)\nu\big)n(t)-\lambda_iC_i(t) \;,\\
 b_{i,j}&=\frac{\beta_i\beta_j\,\nu}{\Lambda}n(t)+\delta_{ij}\lambda_iC_i(t) \; ,
\end{align}
\end{subequations}
with $\nu =$ number of neutrons per fission.
Note that if $\bm{\hat{B}}(t)$ = 0, then \cref{eq3.1a} (with $\bm Q(t)=0$) reduces to the deterministic problem discussed in \cref{sec2}.

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\section{The Proposed Method (Double-DDM)}
\label{sec4}
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We propose to solve the stochastic formulation in \cref{sec3} by adapting the recursive method described in \cref{sec2}.
This approach yields a nonstiff solution to the stochastic system in \cref{eq3.1}.

Since the matrix $\bm{\hat{B}}$ depends on both the neutron populations
and the delayed neutron concentrations, we resort to a double decomposition to obtain a solution for this problem:
\begin{itemize}
\item[I.] Using DDM as described in \cref{sec2}, \cref{eq3.1a} is solved for $\bm{\hat{B}}=0$; this yields the deterministic solution $\bm Y(t)=\bm Y_d(t)$;
\item[II.] $\bm Y_d(t)$ is used to determine the matrix $\bm{\hat{B}}$ for a sequence of discrete time steps (its components are constant in each time step);
\item[III.] $\bm{\hat{B}}^\frac{1}{2}$ is obtained through diagonalization ($\bm{\hat{B}}$ is symmetric);
\item[IV.] Since $\bm Q(t)$ is known for a specific time interval, a decomposition scheme analogue to DDM is applied:
\begin{subequations}\label[pluraleq]{eq4.1}%\label{esto_it1}
\begin{align}
 \frac{d}{dt}\bm Y_0(t)-\bm\Omega \bm Y_0(t)&=0\,,\\
\frac{d}{dt}\bm Y_j (t)-\bm\Omega \bm Y_j (t)&=
\bm{\Xi}(t)\bm Y_{j-1}(t)+\bm{{\cal{F}}}(t) \; ,
\end{align}
\end{subequations}
where $\bm{{\cal{F}}}(t) =\bm Q(t)+\bm{\hat{B}}^\frac{1}{2}\frac{d}{dt} {\bm{W}}(t)$ are constants known in each time step.
\end{itemize}

The total number $K$ of stochastic components ``drawn" in this approach is defined by the Central Limit Theorem (CLM) to guarantee a small statistical error for the first four moments.
For all results shown in this paper, $K$ is large enough to guarantee a statistical error smaller than 0.05\% with 95\% confidence.
The flowchart in \cref{fig2} describes the implementation of the proposed method.

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\section{Numerical Results}
\label{sec5}
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In this section we present numerical results for the Double-DDM approach proposed in \cref{sec4} for examples with (i) constant and (ii) time-dependent reactivities.
We compare these results against those obtained with other approaches established in the literature.
We calculate the expected value $E(\bm Y)$ and variance Var$(\bm Y$), given by
\begin{subequations}
\begin{align}
E(\bm Y) &= \sum_{k=1}^K \frac{\bm Y^{(k)}}{K}\,,\\
\text{Var}(\bm Y) = \sigma^2(\bm Y) &= \sum_{k=1}^K \frac{\left(\bm Y^{(k)}-E(\bm Y)\right)^2}{K}\,,
\end{align}
where the index $k$ represents the different choices of stochastic components (histories).
We also calculate skewness and excess kurtosis for the neutron density $n$, defined as
\begin{align}
\text{Skew}(n) &= \gamma_1(n) =\sum_{k=1}^{K}\frac{\left(n^{(k)}- E(n)\right)^3}{K\sigma^3(n)}\,,\\
 \text{Kurt}(n)-3&=\gamma_2(n) = \sum_{k=1}^{K}\left[\frac{\left(n^{(k)}- E(n)\right)^4}{K\sigma^4(n)}\right]-3\,,
\end{align}
\end{subequations}
which gives us further insight on the behavior of the stochastic solutions.

\subsection{Constant Reactivity}\label{sec51}

In the following examples we present the results of four methods established in the literature: Monte Carlo and Stochastic PCA \citep{hayes_05}; order 1.5 strong Taylor and Euler-Maruyama \citep{saha_12}.
These results are reproduced here as they were reported in the aforementioned references.

We compare these results with those obtained with the deterministic diagonalization-decomposition method (DDM) and with the Double-DDM approach.
The solutions of the deterministic DDM are obtained by solving \cref{eq3.1a} with $\bm{\hat{B}}=0$; we point out that this is also the first step of Double-DDM.

\subsubsection{Step-reactivity insertion: one precursor}\label{sec511}

This example does not model an actual physical nuclear reactor problem.
Nevertheless, considering only one group of precursors implies that the stiffness of the problem disappears; this provides a simple computational solution that is useful for a first comparison with other methods.

The physical parameters are $\lambda_1=0.1$,  $\beta=\beta_1=0.05$, $\nu=2.5$, $q=200$, $\Lambda=\frac{2}{3}$, and $\rho(t)=-\frac{1}{3}$ for $t\geq 0$.
The initial condition is $\bm{Y}(0)=[400, 300]^T$.
The expected values and standard deviations of $n(t)$ and $C_1(t)$ at time $t=2$ are presented in \cref{tab1} for each of the methods.
\begin{table}[H]
\caption{Results for one group of precursors and step-reactivity insertion $\rho(t)=-1/3$.}\label{tab1}
\centering{\small{
\begin{tabular}{ccccccc} \hline\hline
 									& {\bf Monte} 	& {\bf Stochastic}	 & {\bf Euler}         & {\bf Taylor}		&  \multirow{2}{*}{{\bf DDM}}		&{\bf Double} \\
 									& {\bf Carlo}  	& {\bf PCA}			 & {\bf Maruyama}  & {\bf 1.5} 		&  	  		& {\bf DDM}\\ \hline
 ${E(n(2))}$ 				&400.03 	& 395.32     &412.23		  &412.10		&412.13  &402.35\\
${\sigma(n(2))}$ 		&27.311 	& 29.411     &34.391		  &34.519	& --			&28.610\\
${E(C_1(2))}$ 			& 300.00 & 300.67     &315.96		  &315.93	&315.93	&305.84\\
${\sigma(C_1 (2))}$ 	& 7.8073 & 8.3564     &8.2656 	  &8.3158	& --			&7.9240 \\ \hline\hline
\end{tabular}
}}
\end{table}
A total of $K=1,000$ histories were accumulated for the Double-DDM approach.
Skewness and excess kurtosis for the neutron density were found to be $\gamma_1(n(2)) = -1\times 10^{-10}$ and $\gamma_2(n(2)) = 3\times 10^{-11}$.
The fact that $\gamma_1$ and $\gamma_2$ are nearly zero implies that the distribution of stochastic solutions is symmetric and has Gaussian-like peak and tail.

It can be seen that there exists a close agreement between Double-DDM and the results obtained with Monte Carlo and Stochastic PCA.
Euler-Maruyama and order 1.5 strong Taylor (Taylor 1.5) yield slightly higher results, very close to those obtained with the deterministic DDM.

\subsubsection{Step-reactivity insertion: six precursors}\label{sec512}

The following two examples model step-reactivity insertions for a thermal nuclear reactor.
In this case, we consider a stiff system of equations with six precursor groups.
The set of physical parameters is taken from \citet{kinard_04}:
$\nu=2.5$, $\Lambda=0.00002$, $q=0$, and $\beta=7\times 10^{-3}$, with $\beta_i$ and $\lambda_i$ given in \cref{tab2}. The initial condition is given by
\begin{align}
\bm{Y}(0)=100 \left[1, \frac{\beta_1}{\lambda_1 \Lambda}, \frac{\beta_2}{\lambda_2 \Lambda}, \frac{\beta_3}{\lambda_3 \Lambda}, \frac{\beta_4}{\lambda_4 \Lambda}, \frac{\beta_5}{\lambda_5 \Lambda}, \frac{\beta_6}{\lambda_6 \Lambda}\right]^T\,.
\end{align}
\begin{table}[H]
\caption{Fraction of delayed neutrons and decay constants for the precursor groups.}\label{tab2}
\centering{\small{
\begin{tabular}{lllllll}
{\bf Group}							& {\bf 1}	&{\bf 2}	&{\bf 3}	&{\bf 4}	&{\bf 5}	&{\bf 6}\\\hline
$\beta_i\times 10^{-3}$	&0.266	&1.491	&1.316	&2.849	&0.896	&0.182 \\ \hline
$\lambda_i$						&0.0127	&0.0317	&0.115	&0.311	&1.4			&3.87 \\
\hline
\end{tabular}
}}
\end{table}

We compute results for a prompt subcritical step-reactivity insertion $\rho(t)=0.003$ at time $t=0.1$, and for a prompt critical step-reactivity insertion $\rho(t)=0.007$ at time $t=0.001$.
We define
\begin{align}
C(t)=\sum_{i=1}^6 C_i(t)\,,
\end{align}
and present the expected values and standard deviations for each of the methods in \cref{tab3}.
\begin{table}[H]
\caption{Results for six groups of precursors with subcritical $(\rho(t)=0.003)$ and critical $(\rho(t)=0.007)$ step-reactivity insertions.}\label{tab3}
\centering{\small{
\begin{tabular}{cccccccc} \hline\hline
\multirow{2}{*}{{$\bm\rho$}}& 									& {\bf Monte} 	& {\bf Stochastic}	 & {\bf Euler}         & {\bf Taylor}		&  \multirow{2}{*}{{\bf DDM}}		&{\bf Double} \\
& 									& {\bf Carlo}  	& {\bf PCA}			 & {\bf Maruyama}  & {\bf 1.5} 		&  	  		& {\bf DDM}\\ \hline
\multirow{4}{*}{$0.003$}
&$E(n(0.1))$ 						&183.04 		& 186.31     &208.60		&199.41	&200.01  	&187.05\\
&$\sigma(n(0.1))$ 				&168.79 		& 164.16     &255.95		&168.55	& --				&167.83\\
&$E(C(0.1))\times 10^5$ 	& 4.478 		& 4.491  		&4.498		&4.497		&4.497		&4.488\\
&${\sigma(C(0.1))}$ 			& 1495.7 	& 1917.2    	&1233.4	 	&1218.8		& --				&1475.6 \\ \hline\hline
\multirow{4}{*}{$0.007$}
&$E(n(0.001))$ 						&135.67 		& 134.55     &139.57		&139.57	&139.61  	&135.86\\
&$\sigma(n(0.001))$ 				&93.376 		& 91.242     &92.042		&92.047	& --				&93.210\\
&$E(C(0.001))\times 10^5$ 	& 4.464 		& 4.464  		&4.463		&4.463		&4.463		&4.463\\
&${\sigma(C(0.001))}$ 			& 16.226 	& 19.444    	&6.071	 	&18.337		& --				&17.845 \\ \hline\hline
\end{tabular}
}
}\end{table}
We collected $K=1,000$ histories for the subcritical step-reactivity $\rho(t)=0.003$.
Skewness was found to be $\gamma_1(n(0.1)) = -1\times 10^{-7}$, and excess kurtosis $\gamma_2(n(0.1)) = 1.3\times 10^{-9}$.
For the critical step-reactivity $\rho(t)=0.007$, $K=5,000$ histories were collected.
Skewness and excess kurtosis were computed respectively as $\gamma_1(n(0.001))=-1.02\times 10^{-7}$ and $\gamma_2(n(0.001))=1.15\times 10^{-8}$.
These results for the third and fourth moments indicate that the distribution of the stochastic solutions is symmetric and has neither a sharp peak nor a heavy tail.

As in the previous example, the results obtained with Double-DDM are in close agreement to the results from Monte Carlo and Stochastic PCA. The results from Euler-Maruyama and order 1.5 strong Taylor are closer to those of deterministic DDM.

\subsection{Time-Dependent Reactivity}\label{sec52}

The current literature lacks numerical results for the stochastic system in \cref{eq3.1a} with time-dependent reactivities.
For this reason, the results collected from the literature and reproduced next represent only the deterministic solution (with $\bm{\hat{B}}=0$).
Although not ideal, this approach allows us to verify that the expected value obtained with Double-DDM closely agrees with well established models for problems with time-dependent reactivities.
All the following examples take into account six precursor groups.

\subsubsection{Linear reactivity $\rho(t)=at$}\label{sec521}

The following two examples model a ramp reactivity $\rho(t)=at$ for a thermal nuclear reactor.
The physical parameters considered are: $\Lambda=0.00001$, $n(0)=1.0$, and $\beta=6.403\times 10^{-3}$, with $\beta_i$ and $\lambda_i$ taken from \citet{lewins_78} and given in \cref{tab4}.
\begin{table}[H]
\caption{Fraction of delayed neutrons and decay constants for the precursor groups.}\label{tab4}
\centering{\small{
\begin{tabular}{lllllll}
{\bf Group}							& {\bf 1}	&{\bf 2}	&{\bf 3}	&{\bf 4}	&{\bf 5}	&{\bf 6}\\\hline
$\beta_i\times 10^{-3}$	&0.246	&1.363	&1.203	&2.605	&0.819	&0.167 \\ \hline
$\lambda_i$						&0.0127	&0.0317	&0.115	&0.311	&1.4			&3.87 \\
\hline
\end{tabular}
}}
\end{table}
We compute the neutron density $n(t)$ for two different choices of constant $a$: $0.25$ and $0.5$.
The results obtained with Double-DDM are given in \cref{tab5} for times $t=0.25, 0.5, 0.75,$ and $1.0$.
We compare these results with those obtained with the Pad\'e approximation \citep{aboanber_02} and the generalization of the analytical exponential model (GAEM), as reported by \citet{nahla_08}.
\begin{table}[H]
\caption{Neutron density $n(t)$ with ramp reactivity $\rho(t)=at$.}\label{tab5}
\centering{\small{
\begin{tabular}{cccccc} \hline\hline
\multirow{2}{*}{{$\bm a$}}& \multirow{2}{*}{{\bf Time}} & \multirow{2}{*}{{\bf Pad\'e}} &\multirow{2}{*}{{\bf GAEM}} &	\multirow{2}{*}{{\bf DDM}} & {\bf Double} \\
 & & & & & {\bf DDM} \\ \hline\
\multirow{4}{*}{$0.25$}
&0.25 & 1.069840 & 1.069541 & 1.069542 & 1.069763\\
&0.50 & 1.157065 & 1.156694 & 1.156695 & 1.157867\\
&0.75 & 1.265795 & 1.265331 & 1.265332 & 1.269374\\
&1.0   & 1.402562 & 1.401981 & 1.401982 & 1.403561\\\hline\hline
\multirow{4}{*}{$0.5$}
&0.25 & 1.149544 & 1.149200 & 1.149210 & 1.137216\\
&0.50 & 1.369438 & 1.368927 & 1.368928 & 1.356934\\
&0.75 & 1.708411 & 1.707600 & 1.707601 & 1.695607\\
&1.0   & 2.276692 & 2.275271 & 2.275272 & 2.263278\\\hline\hline
\end{tabular}
}
}\end{table}
The number of histories collected for the case $a=0.25$ was $K=15,738$.
We computed the higher moments for time $t=1.0$, finding the standard deviation $\sigma(n(1))=0.978$, skewness $\gamma_1(n(1))=-3.02\times 10^{-7}$, and excess kurtosis $\gamma_2(n(1))=-1 \times 10^{-7}$.
For the case $a=0.5$, we collected $K=27,523$ histories, and found
$\sigma(n(1))=1.11345$, $\gamma_1(n(1))=-2.15\times 10^{-6}$, $\gamma_2(n(1))=-3 \times 10^{-8}$.
This shows that, in both cases, the distribution of stochastic solutions is nearly normal.

Double-DDM shows good agreement with the other methods shown in \cref{tab5}.
The results obtained with Double-DDM are slightly larger for the first case ($a=0.25$), and slightly smaller for the second case.
The relative differences are around $1\%$ or smaller, being well within 1 standard deviation.

\subsubsection{Sinusoidal reactivity $\rho(t)=b\sin(t)$}\label{sec522}

The last example simulates a case with sinusoidal reactivity $\rho(t)=b\sin(t)$, with $b=0.00073$, $\Lambda = 0.00003$, and $n(0)=1.0$. The total fraction of delayed neutrons is given by $\beta=6.473\times 10^{-3}$, with $\beta_i$ and $\lambda_i$ shown in \cref{tab6}.
\begin{table}[H]
\caption{Fraction of delayed neutrons and decay constants for the precursor groups.}\label{tab6}
\centering{\small{
\begin{tabular}{lllllll}
{\bf Group}							& {\bf 1}	&{\bf 2}	&{\bf 3}	&{\bf 4}	&{\bf 5}	&{\bf 6}\\\hline
$\beta_i\times 10^{-3}$	&0.214	&1.423	&1.247	&2.568	&0.748	&0.273 \\ \hline
$\lambda_i$						&0.0124	&0.0305	&0.111	&0.301	&1.14			&3.01 \\
\hline
\end{tabular}
}}
\end{table}
The results obtained with Double-DDM are presented in \cref{tab7} for every whole second up until $t=10$.
We compare these results with those reported in \citep{wollmann_14}.
These were obtained with the method introduced by \citet{kang_73}, referred to as K \& H in \cref{tab7}, and with the method of Taylor series (cf. \citeauthor{nahla_11}, \citeyear{nahla_11}).
\begin{table}[H]
\caption{Neutron density $n(t)$ with sinusoidal reactivity $\rho(t)=0.00073\sin(t)$.}\label{tab7}
\centering{\small{
\begin{tabular}{ccccc} \hline\hline
\multirow{2}{*}{{\bf Time}} & \multirow{2}{*}{{\bf K \& H}} &\multirow{2}{*}{{\bf Taylor}} &	\multirow{2}{*}{{\bf DDM}} & {\bf Double} \\
& & & & {\bf DDM} \\ \hline
0   & 1.00000 & 1.00000 & 1.00000 & 1.0000000\\
1   & 1.12397 & 1.12394 & 1.12396 & 1.1119659\\
2   & 1.16881 & 1.16884 & 1.16889 & 1.1568959\\
3   & 1.07443 & 1.07442 & 1.07448 & 1.0624859\\
4   & 0.95381 & 0.95380 & 0.95382 & 0.9418259\\
5   & 0.90737 & 0.90737 & 0.90735 & 0.8953559\\
6   & 0.96151 & 0.96158 & 0.96153 & 0.9495359\\
7   & 1.08748 & 1.08749 & 1.08745 & 1.0754559\\
8   & 1.17168 & 1.17164 & 1.17167 & 1.1596759\\
9   & 1.11128 & 1.11124 & 1.11130 & 1.0993059\\
10 & 0.98464 & 0.98464 & 0.98468 & 0.9726859\\\hline\hline
\end{tabular}
}
}\end{table}
We collected $K=2,934,237$ histories to achieve the requirement imposed for the statistical error (less than 0.05\% with 95\% confidence).
The higher moments for time $t=10$ yield $\sigma(n(10))= 1.3242$, $\gamma_1(n(10))=-0.0048$, and $\gamma_2(n(10))=-0.013$. These results indicate (i) a very small asymmetry in the distribution of stochastic solutions, with a slightly larger left tail; and (ii) a very small yet noticeable  ``flatter" peak when compared to a normal.

In general, Double-DDM closely agrees with the other methods presented here for comparison.
Results displayed in \cref{tab7} show that Double-DDM  yields slightly smaller results than those attained with the other methods.
This can be confirmed in \cref{fig3}, which depicts the average behavior of the stochastic solution compared to the deterministic solution (DDM).
These differences are, once again, very small ($\approx 1\%$), and well within 1 standard deviation.

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\section{CONCLUSIONS}
\label{sec6}
\setcounter{section}{6}
\setcounter{equation}{0}

In this paper, we propose an approach to solve the stochastic neutron point kinetics equations with a solution procedure that is free of stiffness.
This is achieved through an adaptation of the diagonalization-decomposition method (DDM) introduced in \citet{wollmann_14}, wich provides a nonstiff solution for the deterministic point kinetics equations.
DDM uses the Laplace transform to obtain a formal solution, then applies a decomposition into a recursive scheme, using Gauss-Legendre integration.

The essential steps of the proposed approach (Double-DDM) are: (i) the deterministic problem is solved with DDM; (ii) the deterministic solution is used to build the stochastic component; (iii) another decomposition scheme analogue to DDM is used to solve the stochastic system.
This allows the calculation of the neutron density and precursor concentrations at any time of interest, without the need to resort to progressive time steps.
The elimination of stiffness comes from the fact that the evolution of the solution by recursion adds correction terms to the whole time interval of interest in each step, and simultaneously for each term that depends on a specific time scale.

Since convergence of DDM is yet to be proven, a Lyapunov criterion is used to guarantee convergence.
The results of the proposed method are compared against results obtained through other approaches established in the literature.
This comparison shows close agreement for problems with constant step-reactivity insertions, as well as time-dependent ramp reactivity and sinusoidal reactivity insertions.

In the current literature, the stochastic problem is mainly solved for constant reactivities; numerical solutions are limited to feasible time intervals due to the stiffness inherent to the problem.
The mitigation of the stiffness character in solving the stochastic formulation is the major novelty and principal contribution of this work.
Moreover, the analysis of the third and fourth moments of the stochastic solutions is, to the best of our knowledge, new.

The analyzed moments still depend on the size of the sample set and on the frequency with which the stochastic fluctuations are applied.
In principle, an adjustment such as variance reduction and its consequences for higher moments could yield more realistic results.
It would be necessary to find a reference scale in order to obtain such results independently of the sample size or frequency of application.
This still needs to be identified, as well as a necessary ingredient to mimic reactor fluctuations.
These tasks are left for future work.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Acknowledgments}
Milena Wollmann da Silva would like to thank the Coordination for the Improvement of Higher Education Personnel (CAPES, Brazil) for financial support.
Bardo E.J. Bodmann and Marco T\'ullio Vilhena thank the Federal University of Rio Grande do Sul (UFRGS) and also acknowledge financial support from the National Council of Scientific and Technological Development (CNPq, Brazil).
Richard Vasques prepared this paper under award number NRC-HQ-84-14-G-0052 from the Nuclear Regulatory Commission.
The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the view of the U.S. Nuclear Regulatory Commission.

\section*{References}

\bibliography{ANE-16}

\pagebreak

\noindent

\begin{figure}[H]
 \centering
 \includegraphics[scale=1]{fig1.eps}
\caption{\footnotesize{DDM approach to solve the deterministic problem}}\label{fig1}
\end{figure}

\begin{figure}[H]
\centering
 \includegraphics[scale=1]{fig2.eps}
\caption{\footnotesize{Double-DDM approach to solve the stochastic problem}}\label{fig2}
\end{figure}

\begin{figure}[H]
 \centering
 \includegraphics[scale=1.0]{fig3.eps}
 \caption{\footnotesize{Neutron density for a sinusoidal reactivity $\rho(t) = 0.00073\sin(t)$}}\label{fig3}
\end{figure}

\end{document}


%!TEX root = ../thesis.tex
%*******************************************************************************
%****************************** Third Chapter **********************************
%*******************************************************************************
\chapter{Scene Surveying}\label{chapter:SceneSurveying}
In this chapter, we provide details of the techniques and software developed to address the problem of aiding autonomous scene surveying, which was identified as part of the first research question stated in Section \ref{sec:ResearchQuestions}:
\\
\\
"\textit{\input{Chapters/Introduction/ResearchQuestion1.tex}}"
\\
\par As with the rest of the work in this thesis, this chapter is motivated by the context of the ROCSAFE project, but we present the work in a general setting. In the initial phase of any crime scene investigation, it is desirable for the crime scene manager to visually scan the entire crime scene area to thoroughly assess the scene as early as possible \cite{TechnicalWorkingGrouponCrimeSceneInvestigation2013CrimeEnforcement}. We will henceforth refer to the area to be scanned as the \textit{region of interest} for the purpose of generality. In the ROCSAFE project, it is intended to address this problem using a fleet of RAVs, which are equipped with cameras and sensors which are highly suited to data-gathering. The RAVs can operate concurrently and can communicate with a centralised controller. We developed a system to autonomously survey a hazardous situation using the simulation environment described in Chapter \ref{chap:HighFidelitySim}. We published this work in a number of papers \cite{Smyth2018AInvestigation}, \cite{Smyth2018ASupport}, \cite{Smyth2018UsingDrones}.
%The information acquired by this initial sweep can be used for a multitude of purposes and generally speaking the sooner it can be gathered, the better, since it will often be subjected to time-consuming analysis.
%We used the simulation environment described in Section \ref{chap:HighFidelitySim} to test the developed system.
%In order to ensure that the solutions explored are not domain-specific, we make few assumptions regarding the nature of the hardware used to solve the problem.
\par We began by providing a description of the two main sub-problems identified as part of autonomous scene-surveying:
\begin{enumerate}
    \item Given a polygonal region defined by a set of points that lie on the earth's surface, generate a discrete set of cells of uniform area that partition the region of interest.
    %according to a user-specified height and width.
    The cells can be described by their centre points, as they are assumed to be of uniform height and width. We will refer to this set as $R$, the set of \textit{waypoints}.
    \item Find a set of routes for each of $K$ RAVs to be used in the data-gathering process, such that these sets partition $R$ and the cost of the system of RAVs traversing these points is minimized. %Objective functions which define the cost of traversal are discussed in <reference relevant section>.
\end{enumerate}

Hence there are two main goals: find a discrete representation of a continuous area which represents the region of interest, and then find routes that the RAVs can execute in order to visit each of the grid points exactly once, while minimising the cost of doing so.
%We then give a mathematical definition of the sub-problems in sections \ref{sec:GenerationOfWaypoints}, \ref{subsec:SimplifiedVRP} and \ref{sec:SceneSurveyingBatteryConstraints}.

\input{Chapters/MultiAgentCoverage/BoundingPolygon/GeneratingWaypointsInPolygon.tex}

\input{Chapters/MultiAgentCoverage/MultipleTravellingSalesman/MultipleTravellingSalesman.tex}

\input{Chapters/MultiAgentCoverage/ExecutingRoutesInSimulation/ExecutingRoutesInSimulation.tex}

\input{Chapters/MultiAgentCoverage/MultipleTravellingSalesman/ORToolsSolution.tex}

\input{Chapters/MultiAgentCoverage/FutureWork/ConclusionsAndFutureWork.tex}


\subsection{Hill climbing}

We initialise at some point in the parameter space.

We identifify nearby alternative points in parameter space, and move to the one with the most improvement.

Movement only occurs in one parameter at a time.


\documentclass[12pt, oneside]{article}
\usepackage{geometry}
\geometry{letterpaper}
\usepackage{graphicx}

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{dsfont}

\usepackage{physics}
\usepackage{siunitx}
\usepackage{slashed}

\usepackage{mathtools}

\renewcommand\thesection{Meeting \arabic{section}. Date:}

\title{Summary of Meetings}
\author{M. Vilucchio, A. Ouzzani, C. Lopetegui}
%\date{}

\begin{document}
\maketitle
\section{2020-11-18}

The method used is the following, where $\varepsilon$ is the time step:
\[
	e^{\frac{\varepsilon}{2} (\operatorname{A})} e^{\varepsilon \operatorname{B}}e^{\frac{\varepsilon}{2} (\operatorname{A})} = e^{\varepsilon (\operatorname{A} + \operatorname{B})} + \order{\varepsilon^3}
\]
in the actual coding one useful thing to do is making the successive application of the exponential of $\operatorname{A}$ as one single operation. Like
\[
	\underbracket{e^{\frac{\varepsilon}{2} (\operatorname{A})}} \underbracket{e^{\varepsilon \operatorname{B}} }\underbracket{e^{\frac{\varepsilon}{2} (\operatorname{A})} e^{\frac{\varepsilon}{2} (\operatorname{A})} }_{e^{\varepsilon \operatorname{A}}} \underbracket{e^{\varepsilon \operatorname{B}} } \underbracket{e^{\frac{\varepsilon}{2} (\operatorname{A})} e^{\frac{\varepsilon}{2} (\operatorname{A})} }_{e^{\varepsilon \operatorname{A}}} \underbracket{ e^{\varepsilon \operatorname{B}} } \underbracket{e^{\frac{\varepsilon}{2} (\operatorname{A})} }
\]

While using the fft in two dimensions one should always remember to split space in every direction in $2^n$ parts to utilise the complexity advantage of the fft implementation.

The definition of the width is, in every direction:
\[
	\sigma_x = \sqrt{ \big\langle \qty(x - \langle x\rangle)^2\big\rangle}\quad \text{ with } \quad\langle f \rangle = \int_{\mathbb{R}^2} f \abs{\psi} \dd[2]x
\]

The implementation, for the dimensions of the wave-function and the space matrix should be vectorial.

Then to check the code try the example 2.I .

\paragraph{Questions}
\begin{itemize}
	\item What is the error in the scheme for both the temporal and the spatial discretisation?
\end{itemize}

\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% CS622: Theory of Formal Languages
% Copyright 2014 Pejman Ghorbanzade <pejman@ghorbanzade.com>
% Creative Commons Attribution-ShareAlike 4.0 International License
% More info: https://github.com/ghorbanzade/beacon
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{Question 4}

Two words $u, v \in A^*$ are \textit{conjugate} if we can write $u=xy$ and $v=yx$ for some words $x,y \in A^*$.
Define the binary relation $\kappa$, where $\left( x,y \right) \in \kappa$, if they are conjugate.
Prove that $\kappa$ is an equivalence relation on $A^* $.

\subsection*{Solution}

Binary relation $\kappa$ is an equivalence relation on $A^*$ if and only if it is reflexive, symmetric and transitive. These properties are investigated and proven for $\kappa$.

\begin{itemize}
\item Reflexivity: $\kappa$ is reflexive if and only if $\left( u,u \right) \in \kappa$ for any $u \in A^*$.

Proof is straightforward by taking $x$ as $u$ and $y$ as $\lambda$.
\begin{eqnarray}
u = u\lambda = xy \label{4eq1}\\
u = \lambda u = yx \label{4eq2}
\end{eqnarray}
From \eqref{4eq1} and \eqref{4eq2},
\begin{equation}\label{4eq3}
\left( u,u \right) \in \kappa
\end{equation}
\item Symmetry: $\kappa$ is symmetric if and only if $\left(v,u\right) \in \kappa$ when $\left( u,v \right) \in \kappa$.
\begin{equation}\label{4eq4}
\left( u,v \right) \in \kappa \Rightarrow u = xy, v = yx
\end{equation}
where $x,y \in A^*$. Assuming $x = y^\prime$ and $y = x^\prime$,
\begin{equation}\label{4eq5}
v = x^\prime y^\prime, u = y^\prime x^\prime \Rightarrow \left( v,u \right) \in \kappa
\end{equation}
From \eqref{4eq4} and \eqref{4eq5},
\begin{equation}\label{4eq6}
\left( u,v \right) \in \kappa \Rightarrow \left( v,u \right) \in \kappa
\end{equation}
\item Transitivity: $\kappa$ is transitive if and only if $\left( u,w \right) \in \kappa$ when $\left( u,v \right)\in \kappa$ and $\left( v,w \right) \in \kappa$.
\begin{equation}\label{4eq7}
\left( u , v \right) \in \kappa \Rightarrow \exists x,y \in A^* \mid u = xy, v = yx
\end{equation}
\begin{equation}\label{4eq8}
\left( v , w \right) \in \kappa \Rightarrow \exists s,t \in A^* \mid v = st, w = ts\\
\end{equation}
From \eqref{4eq7} and \eqref{4eq8},
\begin{equation}\label{4eq9}
v = yx = st
\end{equation}
Possible cases will arise based on length of $y$ and $s$.
\begin{enumerate}
\item If $|y| = |s|$ it follows that $|x| = |t|$ and from \eqref{4eq9},
\begin{equation}
x = t, y = s \Rightarrow u = ts, w = ts
\end{equation}
Assuming $x^\prime = \lambda$ and $y^\prime = ts$,
\begin{equation} \label{4eq11}
u = \lambda ts = x^\prime y^\prime
\end{equation}
\begin{equation} \label{4eq12}
w = ts \lambda = y^\prime x^\prime
\end{equation}
From \eqref{4eq11} and \eqref{4eq12},
\begin{equation}\label{4eq13}
\left( u,w \right) \in \kappa
\end{equation}
\item Assuming that $|y| > |s|$ it follows that $|t| > |x|$. We suppose $y = sy^\prime$ and $t = t^\prime x$. Thus,
\begin{equation}\label{4eq14}
v = sy^\prime x = st^\prime x \Rightarrow y^\prime = t^\prime
\end{equation}
$u$ and $w$ can be represented with the new notations.
\begin{equation}
u = xsy^\prime
\end{equation}
\begin{equation}
w = t^\prime xs
\end{equation}
Assuming $x^\prime = xs$ and from \eqref{4eq14},
\begin{equation}\label{4eq17}
u = x^\prime y^\prime
\end{equation}
\begin{equation}\label{4eq18}
w = y^\prime x^\prime
\end{equation}
From \eqref{4eq17} and \eqref{4eq18},
\begin{equation}\label{4eq19}
\left(u,w \right) \in \kappa
\end{equation}
\item Assuming that $|y| < |s|$ it follows that $|t| < |x|$. We suppose $s = ys^\prime$ and $x = x^\prime t$. Thus,
\begin{equation} \label{4eq20}
v = yx^\prime t = ys^\prime t \Rightarrow x^\prime = s^\prime
\end{equation}
$u$ and $v$ can be represented with the new notations.
\begin{equation}
u = x^\prime ty
\end{equation}
\begin{equation}
w = ty s^\prime
\end{equation}
Assuming $y^\prime = ty$ and from \eqref{4eq20},
\begin{equation}\label{4eq23}
u = x^\prime y^\prime
\end{equation}
\begin{equation}\label{4eq24}
w = y^\prime x^\prime
\end{equation}
From \eqref{4eq23} and \eqref{4eq24},
\begin{equation}\label{4eq25}
\left( u, w \right) \in \kappa
\end{equation}
\end{enumerate}
As cases were inclusive, from \eqref{4eq13}, \eqref{4eq19} and \eqref{4eq25} we conclude
\begin{equation}\label{4eq26}
\left(u,v \right),\left(v,w\right) \in \kappa \Rightarrow \left(u,w\right) \in \kappa
\end{equation}
\end{itemize}

Based on \eqref{4eq3}, \eqref{4eq6} and \eqref{4eq26}, it is concluded that $\kappa$ is reflexive, symmetric and transitive thus is an equivalence relation.