%!TEX root = forallx.tex
\addcontentsline{toc}{chapter}{C\ Quick Reference}
\pagestyle{plain}
{\LARGE \bf Quick Reference}

%\section*{Characteristic Truth Tables}
\label{app.CharacteristicTTs}

%KB fordítása kezdet
Gyors Ajánló
\hfill
\begin{tabular}{c|c}
\script{A} & \enot\script{A}\\
\hline
T & F\\
F & T
\end{tabular}
\hfill
\begin{tabular}{c|c|c|c|c|c}
\script{A} & \script{B} & \script{A}\eand\script{B} & \script{A}\eor\script{B} & \script{A}\eif\script{B} & \script{A}\eiff\script{B}\\
\hline
T & T & T & T & T & T\\
T & F & F & T & F & F\\
F & T & F & T & T & F\\
F & F & F & F & T & T
\end{tabular}
\hfill

\vfill

\hfill
\begin{tabular}{c|c}
\script{A} & \enot\script{A}\\
\hline
1 & 0\\
0 & 1
\end{tabular}
\hfill
\begin{tabular}{c|c|c|c|c|c}
\script{A} & \script{B} & \script{A}\eand\script{B} & \script{A}\eor\script{B} & \script{A}\eif\script{B} & \script{A}\eiff\script{B}\\
\hline
1 & 1 & 1 & 1 & 1 & 1\\
1 & 0 & 0 & 1 & 0 & 0\\
0 & 1 & 0 & 1 & 1 & 0\\
0 & 0 & 0 & 0 & 1 & 1
\end{tabular}
\hfill

\vfill

\section*{Symbolization}
\begin{center}
\label{app.symbolization}
\begin{tabular*}{\textwidth}{rl}
\multicolumn{2}{c}{\textsc{Sentential Connectives} (chapter \ref{ch.SL})}\\ \\
Szimbolizáció
Tartozó kapcsolatok (2. fejezet)

It is not the case that $P$. & $\enot P$\\
Either $P$, or $Q$. & $(P \eor Q)$\\
Neither $P$, nor $Q$. & $\enot(P \eor Q)$\ or \ $(\enot P \eand \enot Q)$\\
Both $P$, and $Q$. & $(P \eand Q)$\\
If $P$, then $Q$. & $(P \eif Q)$\\
$P$ only if $Q$. & $(P \eif Q)$\\
$P$ if and only if $Q$. & $(P \eiff Q)$\\
Unless $P$, $Q$. $P$ unless $Q$. & $(P \eor Q)$\\
\\

Nem P. $\enot P$\\
Vagy P, vagy Q  $(P \eor Q)$\\
Nem P, vagy Q  $\enot(P \eor Q)$\ or \ $(\enot P \eand \enot Q)$\\
P és Q  $(P \eand Q)$\\
Ha P akkor Q $(P \eif Q)$\\
P csak akkor ha Q  $(P \eif Q)$\\
P ha és csak ha Q $(P \eiff Q)$\\
P nélkül Q. Q nélkül P $(P \eor Q)$\\

\\
\multicolumn{2}{c}{\label{SymbolizingPredicates}\textsc{Predicates} (chapter \ref{ch.QL})}\\ \\
All $F$s are $G$s. & $\forall x(Fx \eif Gx)$\\
Some $F$s are $G$s. & $\exists x(Fx \eand Gx)$\\
Not all $F$s are $G$s. & $\enot\forall x(Fx \eif Gx)$\ or\ $\exists x(Fx \eand \enot Gx)$\\
No $F$s are $G$s. & $\forall x(Fx \eif\enot Gx)$\ or\ $\enot\exists x(Fx \eand Gx)$\\
\\

Állítások (4. fejezet)

Minden F G is $\forall x(Fx \eif Gx)$\\
Létezik olyan F ami G $\exists x(Fx \eand Gx)$\\
Nem mindegyik F G $\enot\forall x(Fx \eif Gx)$\ or\ $\exists x(Fx \eand \enot Gx)$\\
F nem egyenlő G-vel $\forall x(Fx \eif\enot Gx)$\ or\ $\enot\exists x(Fx \eand Gx)$\\

\\
\multicolumn{2}{c}{\textsc{Identity} (section \ref{sec.identity})}\\ \\
Only $j$ is $G$. & $\forall x(Gx \eiff x=j)$\\
Everything besides $j$ is $G$. & $\forall x(x \neq j \eif Gx)$\\
%$j$ is more $R$ than anyone else. & $\forall x(x\neq j \eif Rjx)$\\
The $F$ is $G$. & $\exists x(Fx \eand \forall y(Fy \eif x=y) \eand Gx)$\\
\multicolumn{2}{l}{`The F is not G' can be translated two ways:} \\
It is not the case that the F is G. (wide)& $\enot\exists x(Fx \eand \forall y(Fy \eif x=y) \eand Gx)$\\
The $F$ is non-$G$. (narrow) & $\exists x(Fx \eand \forall y(Fy \eif x=y) \eand \enot Gx)$
\end{tabular*}
\end{center}

Azonosság (4.6. rész)

Csak j egyenlő G-vel $\forall x(Gx \eiff x=j)$\\
Csak j nem egyenlő G-vel $\forall x(x \neq j \eif Gx)$\\
F egyenlő G-vel $\exists x(Fx \eand \forall y(Fy \eif x=y) \eand Gx)$\\
Az F nem egyenlő G-vel két féle képpen értelmezhető:
Nem F egyenlő G-vel (tág) $\enot\exists x(Fx \eand \forall y(Fy \eif x=y) \eand Gx)$\\
F egyenlő nem G-vel (szűk)  $\exists x(Fx \eand \forall y(Fy \eif x=y) \eand \enot Gx)$

% BEGIN: symbolizing cardinality

\newpage
\section*{Using identity to symbolize quantities}

\subsection*{There are at least \blank\ $F$s.}
\label{summary.atleast}

\begin{ekey}
\item[one] $\exists xFx$
\item[two] $\exists x_1\exists x_2(Fx_1 \eand Fx_2 \eand x_1 \neq x_2)$
\item[three] $\exists x_1\exists x_2\exists x_3(Fx_1 \eand Fx_2 \eand Fx_3 \eand x_1 \neq x_2 \eand x_1 \neq x_3 \eand x_2 \neq x_3)$
\item[four] $\exists x_1\exists x_2\exists x_3\exists x_4 (Fx_1 \eand Fx_2 \eand Fx_3 \eand Fx_4 \eand x_1 \neq x_2 \eand x_1 \neq x_3 \eand x_1 \neq x_4 \eand x_2 \neq x_3 \eand x_2 \neq x_4 \eand x_3 \neq x_4)$
\item[n] $\exists x_1\cdots\exists x_n(Fx_1 \eand\cdots\eand Fx_n \eand x_1 \neq x_2 \eand\cdots\eand x_{n-1}\neq x_n)$
\end{ekey}

Azonosságok a mennyiségek bemutatására
Legalább  \blank\ $F$s.
Egy $\exists xFx$
Kettő $\exists x_1\exists x_2(Fx_1 \eand Fx_2 \eand x_1 \neq x_2)$
Három $\exists x_1\exists x_2\exists x_3(Fx_1 \eand Fx_2 \eand Fx_3 \eand x_1 \neq x_2 \eand x_1 \neq x_3 \eand x_2 \neq x_3)$
Négy $\exists x_1\exists x_2\exists x_3\exists x_4 (Fx_1 \eand Fx_2 \eand Fx_3 \eand Fx_4 \eand x_1 \neq x_2 \eand x_1 \neq x_3 \eand x_1 \neq x_4 \eand x_2 \neq x_3 \eand x_2 \neq x_4 \eand x_3 \neq x_4)$
N $\exists x_1\cdots\exists x_n(Fx_1 \eand\cdots\eand Fx_n \eand x_1 \neq x_2 \eand\cdots\eand x_{n-1}\neq x_n)$

\subsection*{There are at most \blank\ $F$s.}
\label{summary.atmost}

One way to say `at most $n$ things are $F$' is to put a negation sign in front of one of the symbolizations above and say $\enot$`at least $n+1$ things are $F$.' Equivalently:
\begin{ekey}
\item[one] $\forall x_1\forall x_2\bigl[(Fx_1 \eand Fx_2) \eif x_1=x_2\bigr]$
\item[two] $\forall x_1\forall x_2\forall x_3\bigl[(Fx_1 \eand Fx_2 \eand Fx_3) \eif (x_1=x_2 \eor x_1=x_3 \eor x_2=x_3)\bigr]$
\item[three] $\forall x_1\forall x_2\forall x_3\forall x_4\bigl[(Fx_1 \eand Fx_2 \eand Fx_3 \eand Fx_4) \eif (x_1=x_2 \eor x_1=x_3 \eor x_1=x_4 \eor x_2=x_3 \eor x_2=x_4 \eor x_3=x_4)\bigr]$
\item[n]$\forall x_1\cdots\forall x_{n+1}
\bigl[(Fx_1\eand \cdots \eand Fx_{n+1}) \eif (x_1=x_2 \eor \cdots \eor x_n=x_{n+1})\bigr]$
\end{ekey}

Legfeljebb \blank\ $F$s.

Az egyik módszer az, hogy azt mondjuk „legfeljebb $n$ dolog az $F$”, egy tagadási jel elhelyezése az egyik elé a fenti szimbólumok közül, és azt mondjuk, hogy $\enot$ „legalább n + 1 dolog F.” Egyenértékűen:

Egy $\forall x_1\forall x_2\bigl[(Fx_1 \eand Fx_2) \eif x_1=x_2\bigr]$
Kettő $\forall x_1\forall x_2\forall x_3\bigl[(Fx_1 \eand Fx_2 \eand Fx_3) \eif (x_1=x_2 \eor x_1=x_3 \eor x_2=x_3)\bigr]$
Három $\forall x_1\forall x_2\forall x_3\forall x_4\bigl[(Fx_1 \eand Fx_2 \eand Fx_3 \eand Fx_4) \eif (x_1=x_2 \eor x_1=x_3 \eor x_1=x_4 \eor x_2=x_3 \eor x_2=x_4 \eor x_3=x_4)\bigr]$
N $\forall x_1\cdots\forall x_{n+1}
\bigl[(Fx_1\eand \cdots \eand Fx_{n+1}) \eif (x_1=x_2 \eor \cdots \eor x_n=x_{n+1})\bigr]$

\subsection*{There are exactly \blank\ $F$s.}
\label{summary.exactly}

One way to say `exactly $n$ things are $F$' is to conjoin two of the symbolizations above and say `at least $n$ things are $F$' \eand\ `at most $n$ things are $F$.' The following equivalent formulae are shorter:
\begin{ekey}
\item[zero] $\forall x\enot Fx$
\item[one] $\exists x\bigl[Fx \eand \enot\exists y(Fy \eand x\neq y)\bigr]$
\item[two] $\exists x_1\exists x_2\bigl[Fx_1 \eand Fx_2 \eand x_1 \neq x_2 \eand \enot\exists y\bigl(Fy \eand y\neq x_1 \eand y \neq x_2\bigr) \bigr]$
\item[three] $\exists x_1\exists x_2\exists x_3\bigl[Fx_1 \eand Fx_2 \eand Fx_3 \eand x_1 \neq x_2 \eand x_1 \neq x_3 \eand x_2 \neq x_3 \eand\\
\enot\exists y(Fy \eand y \neq x_1 \eand y \neq x_2 \eand y\neq x_3) \bigr]$
\item[n] $\exists x_1\cdots\exists x_n\bigl[Fx_1 \eand\cdots\eand Fx_n  \eand x_1 \neq x_2 \eand\cdots\eand x_{n-1}\neq x_n \eand\\
 \enot\exists y(Fy \eand y\neq x_1 \eand \cdots \eand y\neq x_n)\bigr]$
%\item[one] $\exists x\forall y\bigl[Fx \eand (Fy \eif y = x)\bigr]$
%\item[two] $\exists x\exists y\forall z\Bigl(Fx \eand Fy \eand \bigl[Fz \eif (z=x \eor z=y)\bigr] \eand x \neq y\Bigr)$
%\item[three] $\exists x_1\exists x_2\exists x_3\forall y\Bigl(Fx_1 \eand Fx_2 \eand Fx_3 \eand [Fy \eif (y=x_1 \eor y=x_2 \eor y=x_3)] \eand x_1 \neq x_2 \eand x_1 \neq x_3 \eand x_2 \neq x_3\Bigr)$
%\item[n] $\exists x_1\cdots\exists x_n\forall y\Bigl(Fx_1 \eand\cdots\eand Fx_n \eand \bigl[Fy \eif (y=x_1 \eor \cdots \eor y=x_n)\bigr] \eand x_1 \neq x_2 \eand\cdots\eand x_{n-1}\neq x_n\Bigr)$
\end{ekey}

Pontosan \blank\ $F$s.
Az egyik módszer az, hogy azt mondjuk „pontosan n a dolgok F” a fenti szimbólumok közül kettőt összekapcsolunk és azt mondjuk „legalább n dolog F” és „legfeljebb n dolog F”. A következő ekvivalnes formulák rövidebbek :

Nulla $\forall x\enot Fx$
Egy $\exists x\bigl[Fx \eand \enot\exists y(Fy \eand x\neq y)\bigr]$
Kettő $\exists x_1\exists x_2\bigl[Fx_1 \eand Fx_2 \eand x_1 \neq x_2 \eand \enot\exists y\bigl(Fy \eand y\neq x_1 \eand y \neq x_2\bigr) \bigr]$
Három $\exists x_1\exists x_2\exists x_3\bigl[Fx_1 \eand Fx_2 \eand Fx_3 \eand x_1 \neq x_2 \eand x_1 \neq x_3 \eand x_2 \neq x_3 \eand\\
\enot\exists y(Fy \eand y \neq x_1 \eand y \neq x_2 \eand y\neq x_3) \bigr]$
N $\exists x_1\cdots\exists x_n\bigl[Fx_1 \eand\cdots\eand Fx_n  \eand x_1 \neq x_2 \eand\cdots\eand x_{n-1}\neq x_n \eand\\
 \enot\exists y(Fy \eand y\neq x_1 \eand \cdots \eand y\neq x_n)\bigr]$

\subsection*{Specifying the size of the UD}

Removing $F$ from the symbolizations above produces sentences that talk about the size of the UD. For instance, `there are at least 2 things (in the UD)' may be symbolized as $\exists x\exists y(x \neq y)$.

Az F eltávolítása a fenti szimbólumokból mondatokat eredményez amik az UD méretéről szólnak. Például: Legalább két dolog van (UD-ben) amit úgy szimbolizálnánk, hogy $\exists x\exists y(x \neq y)$.

%KB fordítása vége

%  BEGIN: Rules of proof
% change margins so that all the rules will fit
\setlength{\topmargin}{0 in}
\setlength{\headheight}{0 in}
\setlength{\headsep}{0 in}
\setlength{\textheight}{9 in}
\setlength{\evensidemargin}{0.25 in}
\setlength{\oddsidemargin}{0.25 in}
\setlength{\textwidth}{6 in}
\newpage
% This starts a new page and skips a page if necessary so as
% to start on an even numbered page.
% That way, the rules of proof will be on facing pages.
% It fills it in with a somewhat gratuitous reference table.
\ifthenelse{\isodd{\thepage}}{
%	\ \vspace{2 in}\par\centerline{[ This page intentionally left blank. ]}
\begin{table}
	Sometimes it is easier to show something by providing proofs than it is by providing models. Sometimes it is the other way round.
	\begin{center}
	\begin{tabular*}{\textwidth}{p{10em}|p{10em}|p{10em}|}
	\cline{2-3}
	 & {\centerline{YES}} & {\centerline{NO}}\\
	\cline{2-3}
	Is \script{A} a tautology? & prove $\vdash\script{A}$ & give a model in which \script{A} is false\\
	\cline{2-3}
	Is \script{A} a contradiction? &  prove $\vdash\enot\script{A}$ & give a model in which \script{A} is true\\
	\cline{2-3}
	Is \script{A} contingent? & give a model in which \script{A} is true and another in which \script{A} is false & prove $\vdash\script{A}$ or $\vdash\enot\script{A}$\\
	\cline{2-3}
	Are \script{A} and \script{B} equivalent? & prove \mbox{$\script{A}\vdash\script{B}$} and \mbox{$\script{B}\vdash\script{A}$}  & give a model in which \script{A} and \script{B} have different truth values\\
	\cline{2-3}
	Is the set \model{A} consistent? & give a model in which all the sentences in \model{A} are true & taking the sentences in \model{A}, prove \script{B} and \enot\script{B}\\
	\cline{2-3}
	Is the argument \mbox{`\script{P}, \therefore\ \script{C}'} valid? & prove $\script{P}\vdash\script{C}$ & give a model in which \script{P} is true and \script{C} is false\\
	\cline{2-3}
	\end{tabular*}
	\end{center}
\end{table}

	\newpage
}{}
% eliminate page numbers
\pagestyle{empty}
\twocolumn

\label{ProofRules}
{\LARGE \bf Basic Rules of Proof}

\textsc{Reiteration}

\begin{proof}
	\have[m]{a}{\script{A}}
	\have[\ ]{c}{\script{A}} \by{R}{a}
\end{proof}

\textsc{Conjunction Introduction}

\begin{proof}
	\have[m]{a}{\script{A}}
	\have[n]{b}{\script{B}}
	\have[\ ]{c}{\script{A}\eand\script{B}} \ai{a, b}
\end{proof}

\textsc{Conjunction Elimination}

\begin{proof}
	\have[m]{ab}{\script{A}\eand\script{B}}
	\have[\ ]{a}{\script{A}} \ae{ab}
\end{proof}

\begin{proof}
	\have[m]{ab}{\script{A}\eand\script{B}}
	\have[\ ]{b}{\script{B}} \ae{ab}
\end{proof}

\textsc{Disjunction Introduction}

\begin{proof}
	\have[m]{a}{\script{A}}
	\have[\ ]{ab}{\script{A}\eor\script{B}}\oi{a}
\end{proof}

\begin{proof}
	\have[m]{a}{\script{A}}
	\have[\ ]{ba}{\script{B}\eor\script{A}}\oi{a}
\end{proof}

\textsc{Disjunction Elimination}

\begin{proof}
	\have[m]{ab}{\script{A}\eor\script{B}}
	\have[n]{nb}{\enot\script{B}}
	\have[\ ]{a}{\script{A}} \oe{ab,nb}
\end{proof}

\begin{proof}
	\have[m]{ab}{\script{A}\eor\script{B}}
	\have[n]{na}{\enot\script{A}}
	\have[\ ]{b}{\script{B}} \oe{ab,nb}
\end{proof}

\textsc{Conditional Introduction}

\nopagebreak
\begin{proof}
	\open
		\hypo[m]{a}{\script{A}} \by{want \script{B}}{}
		\have[n]{b}{\script{B}}
	\close
	\have[\ ]{ab}{\script{A}\eif\script{B}}\ci{a-b}
\end{proof}

\pagebreak
\textsc{Conditional Elimination}

\begin{proof}
	\have[m]{ab}{\script{A}\eif\script{B}}
	\have[n]{a}{\script{A}}
	\have[\ ]{b}{\script{B}} \ce{ab,a}
\end{proof}

\textsc{Biconditional Introduction}

\begin{proof}
	\open
		\hypo[m]{a1}{\script{A}} \by{want \script{B}}{}
		\have[n]{b1}{\script{B}}
	\close
	\open
		\hypo[p]{b2}{\script{B}} \by{want \script{A}}{}
		\have[q]{a2}{\script{A}}
	\close
	\have[\ ]{ab}{\script{A}\eiff\script{B}}\bi{a1-b1,b2-a2}
\end{proof}

\textsc{Biconditional Elimination}

\begin{proof}
	\have[m]{ab}{\script{A}\eiff\script{B}}
	\have[n]{a}{\script{B}}
	\have[\ ]{b}{\script{A}} \be{ab,a}
\end{proof}

\begin{proof}
	\have[m]{ab}{\script{A}\eiff\script{B}}
	\have[n]{a}{\script{A}}
	\have[\ ]{b}{\script{B}} \be{ab,a}
\end{proof}

\textsc{Negation Introduction}

\begin{proof}
	\open
		\hypo[m]{a}{\script{A}} \by{for reductio}{}
		\have[n][-1]{b}{\script{B}}
		\have{nb}{\enot\script{B}}
	\close
	\have[\ ]{na}{\enot\script{A}}\ni{a-nb}
\end{proof}

\textsc{Negation Elimination}

\begin{proof}
	\open
		\hypo[m]{na}{\enot\script{A}} \by{for reductio}{}
		\have[n][-1]{b}{\script{B}}
		\have{nb}{\enot\script{B}}
	\close
	\have[\ ]{a}{\script{A}}\ne{na-nb}
\end{proof}

%PK fordítása kezdet

\newpage

{\LARGE \bf Quantifier Rules}

\textsc{Existential Introduction}

\begin{proof}
	\have[m]{a}{\script{A}\script{c}}
	\have[\ ]{c}{\exists \script{x}\script{A}\script{x}} \Ei{a}
\end{proof}

Note that \script{x} may replace some or all occurrences of \script{c} in \script{A}\script{c}.

\textsc{Existential Elimination}

\begin{proof}
	\have[m]{a}{\exists \script{x}\script{A}\script{x}}
	\open
		\hypo[n]{b}{\script{A}\script{c}^\ast}
		\have[p]{c}{\script{B}}
	\close
	\have[\ ]{d}{\script{B}} \Ee{a,b-c}
\end{proof}

$^\ast$ \script{c} must not appear in $\exists\script{x}\script{A}\script{x}$, in \script{B}, or in any undischarged assumption.

\textsc{Universal Introduction}

\begin{proof}
	\have[m]{a}{\script{A}\script{c}^\ast}
	\have[\ ]{c}{\forall \script{x}\script{A}\script{x}} \Ai{a}
\end{proof}

$^\ast$ \script{c} must not occur in any undischarged assumptions.

\textsc{Universal Elimination}

\begin{proof}
	\have[m]{a}{\forall \script{x}\script{A}\script{x}}
	\have[\ ]{c}{\script{A}\script{c}} \Ae{a}
\end{proof}

{\LARGE \bf Identity Rules}

\begin{proof}
	\have[\ \,\,\,]{x}{\script{c}=\script{c}} \by{=I}{}
\end{proof}

\begin{proof}
	\have[m]{e}{\script{c}=\script{d}}
	\have[n]{a}{\script{A}}
	\have[\ ]{ea1}{\script{A}{c}\circlearrowleft{d}} \by{=E}{e,a}
\end{proof}

One constant may replace some or all occurrences of the other.

\newpage

{\LARGE \bf Derived Rules}

\textsc{Dilemma}

\begin{proof}
	\have[m]{ab}{\script{A}\eor\script{B}}
	\have[n]{ac}{\script{A}\eif\script{C}}
	\have[p]{bc}{\script{B}\eif\script{C}}
	\have[\ ]{a}{\script{C}} \by{DIL}{ab,ac,bc}
\end{proof}

\textsc{Modus Tollens}

\begin{proof}
	\have[m]{ab}{\script{A}\eif\script{B}}
	\have[n]{a}{\enot\script{B}}
	\have[\ ]{b}{\enot\script{A}} \by{MT}{ab,a}
\end{proof}

\textsc{Hypothetical Syllogism}

\begin{proof}
	\have[m]{ab}{\script{A}\eif\script{B}}
	\have[n]{bc}{\script{B}\eif\script{C}}
	\have[\ ]{ac}{\script{A}\eif\script{C}}\by{HS}{ab,bc}
\end{proof}

{\LARGE \bf Replacement Rules}
{
\center

\textsc{Commutivity} (Comm)\\
$(\script{A}\eand\script{B}) \Longleftrightarrow (\script{B}\eand\script{A})$\\
$(\script{A}\eor\script{B}) \Longleftrightarrow (\script{B}\eor\script{A})$\\
$(\script{A}\eiff\script{B}) \Longleftrightarrow (\script{B}\eiff\script{A})$

\textsc{DeMorgan} (DeM)\\
$\enot(\script{A}\eor\script{B}) \Longleftrightarrow (\enot\script{A}\eand\enot\script{B})$\\
$\enot(\script{A}\eand\script{B}) \Longleftrightarrow (\enot\script{A}\eor\enot\script{B})$

\textsc{Double Negation} (DN)\\
$\enot\enot\script{A} \Longleftrightarrow \script{A}$

\textsc{Material Conditional} (MC)\\
$(\script{A}\eif\script{B}) \Longleftrightarrow (\enot\script{A}\eor\script{B})$\\
$(\script{A}\eor\script{B}) \Longleftrightarrow (\enot\script{A}\eif\script{B})$

\textsc{Biconditional Exchange} ({\eiff}{ex})\\
$[(\script{A}\eif\script{B})\eand(\script{B}\eif\script{A})] \Longleftrightarrow (\script{A}\eiff\script{B})$

\textsc{Quantifier Negation} (QN)\\
$\enot\forall\script{x}\script{A} \Longleftrightarrow \exists\script{x}\enot\script{A}$\\
$\enot\exists\script{x}\script{A} \Longleftrightarrow \forall\script{x}\enot\script{A}$

}


% specifies the documnt class. We usually use article but there are others.
\documentclass{article}

% these are standard packages used for the math symbols
\usepackage{amsmath,amssymb,amsthm, enumitem, hyperref, tabto}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{fancyhdr}
\usepackage{lastpage}
\usepackage{colortbl}
\usepackage[pdftex]{graphicx}
\usepackage{float}
\graphicspath{ {./Photos/} }

\definecolor{aqua}{rgb}{0.0, 1.0, 1.0}
\definecolor{orange}{rgb}{1.0, 0.65, 0.0}

% These commands below is to make sure the numbering of these are consistent with theorem
% If you are not sure what something means, delete them, build a new file and see the
% difference between the files. You can ignore this part for now.
\newtheorem{theorem}{Theorem}[section]
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}

% Title of your project
\title{%
  \Huge Ramanujan's Square \\
  \LARGE Detailed Analysis\\
  \Large A DV2136 Project}

% The author command places text right after title
\author{by \\
\Large Afzal (h1810003) \\
\Large Prannaya Gupta (h1810124) \\
\Large Yap Yuan Xi (h1810166) \\
}

\date{\Large July - September 2019}

\begin{document}

\maketitle

\tableofcontents
\newpage
\section{An Introduction}

\subsection{Inspiration}
We are working on Ramanujan's Magic Squares because it is a fascinating phenomenon. It is created using the date of the creator. Its an interesting concept and we really want to build upon it. We want to extend on this concept so we can see what may behold. We want to try. And we want to see if this paper bears any fruit at all.

\newpage
\subsection{Magic Squares}

\subsubsection{Abstract}
In real life situations, some problems relating to division of objects equal in numbers and value can be easily solved by constructing a semi magic square or a magic square in accordance with the given conditions. Mathematics is magic if we can either use one formula for a wide range of applications or the formula itself will produce magic properties.

\subsubsection{A Brief History of the Magic Square}
Magic squares were known to Chinese mathematicians, and Arab mathematicians, possibly as early as the $7^{th}$ century, when the Arabs conquered northwestern parts of the Indian subcontinent and to learned Indian mathematicians and astronomers, including other aspects of combinatorical mathematics. ‘Cornelius Agrippa’ (1486 B.C. to 1535 B.C.) of China is believed to be the first for construction of magic squares.

\subsubsection{The Properties of a general Magic Square}
A general Magic Square is the arrangement of random number within the cell such that sum of each row = each column = each diagonal. The common sum is known as ‘Magic Constant’ or ‘Magic Number’. If the above condition is valid only for the sum of elements of rows and columns and not for the diagonal elements, then that array is known as a semi magic square. All magic squares are semi magic squares. A normal magic square contains the integers from 1 to $n^2$. The constant sum in every row, column and diagonal is called the magic constant or magic sum, S. The magic constant of a normal magic square with continuous numbers depends only on n and has the value $S = \frac{n(n^2+1)}{2}$

\subsection{Variants of Magic Squares}
\begin{center}
    \includegraphics[width=0.3\textwidth]{Photos/MagicSquareHeirarchy.png}
\end{center}
\subsubsection{Semimagic Squares}
    In semimagic squares, the sums of the rows and columns add up to the same number.
\subsubsection{(Ordinary) Magic Squares}
    In ordinary magic squares, the sum of the rows, columns and the diagonals adds up to the same number.
\subsubsection{Panmagic Squares}
    In panmagic squares, the sum of the rows, columns, the diagonals and broken diagonals add up to the same number
\subsubsection{Complete Magic Squares}
    In complete magic squares, the corners, boxes and alternating rows and columns add up to the same sum as well.
\subsubsection{Most Perfect Squares}
    An example of the perfect magic square is the Ramanujan's Square, but including the sums of alternating rows and columns.

\subsection{The Area of Our Investigation}
We are trying to investigate The Ramanujan's Magic Square, a magic square devised by the genius mathematician Srinivasa Ramanujan which has a number of exciting properties.

\newpage
\section{Srinivasa Ramanujan, the creator}
\tab Born in 1887, Ramanujan was a young Indian student.
By the age of 23 Ramanujan was making important new discoveries in mathematics. Professor G.H. Hardy recognised a touch of pure genius in Ramanujan’s theorems. Hardy recognised his talents and arranged to bring Ramanujan to England. Ramanujan decided to accept Hardy’s offer, and on 17 March, 1914 he set out by ship to England.
\subsection{England}
In Cambridge, the young Indian set about working on hundreds of new theorems. Hardy said: “I have never met Ramanujan’s equal.”\cite{studyMagicSquare} However, Ramanujan did not fare so well in his private life. His fragile health suffered. He even became suicidal.
Sadly, Ramanujan never regained his health. He died on 20 April, 1920 in a care home near Madras (now Chennai). He continued working on new theorems even on his death bed. Showing his true Mathematical spirit.\cite{historyRamanujan}

\newpage
\section{Primary Basis of our Investigation}

\subsection{Ramanujan's Magic Square}
\tab We are investigating Srinivasa Ramanujan's Magic Square. It is noticed that in a normal magic square the columns and rows give the same sum. However in this Magic Square, there are other ways to get the same sum. Like the corners, the diagonals and many more.

\subsubsection{What we can derive from the square}
\tab This is very intriguing to us for its adaptability to any birthday date, and made us want to find out how it works and how this is possible. We want to explore the magic of this mathematical square.

\newpage
\section{How does it work?}

\subsection{Description}
\tab This square has a huge number of possible magic square possibilities. You can consider a huge amount magic square patterns like 'By Row', 'By Column', 'By Quarter'.  Below is a definitive list:

\begin{itemize}
  \item The sum of any column is a number y
  \item The sum of any row is the number y
  \item The sum of any diagonal is the number y
  \item The sum of any 2x2 is the number y (except for those in the middle two columns)
\end{itemize}

However, the real miracle of this square its ability to seemingly morph from one's birthday. The main magic square used as an example is actually derived from Ramanujan's birth date, 22 December 1887, thus amounting the sum y as 139. This is an interesting graphic and provides a majorly interesting phenomenon.

\subsection{Graphical Representation}
\begin{figure}[H]
\begin{center}
\includegraphics[scale = 0.4]{MagicSquare}
\cite{magicSquare}
\caption{These diagrams show the ways in which the numbers of the Square add up to 139 in the main Ramanujan Square}
\end{center}
\end{figure}

\newpage
\subsection{The Solution}
\begin{figure}[H]
\begin{center}
\includegraphics[scale = 1.0]{RamanujanSquare}
\cite{ramanujanSquare}
\caption{This diagram shows the algorithm by which the numbers of the Square add up to 139 in the main Ramanujan Square}
\end{center}
\end{figure}

\paragraph{
Let Me Explain: Consider the first row being the derivation.
}
\begin{equation*}
Sum_{given} = DD + MM + CC + YY
\end{equation*}
where DD is the date, MM is the month, CC is the century and YY is the year.

\paragraph{Now consider the first column.}
\begin{equation*}
Sum_{col_1} = DD + (MM-2) + (CC+1) + (YY+1) = DD + MM + CC + YY
\end{equation*}
\subparagraph{Therefore, $Sum_{col_1} = Sum_{given}$}

\paragraph{Try it yourself, this applies for every single property we showed, because the numbers are perfectly balanced.}

\newpage
\section{Extensions and Generalisations}

\subsection{Bigger Squares}
Can we make the square bigger.
    \subsubsection{5x5 Magic Square}
        Letting H, D, M, C, Y be the hour, the day, the month, the century and the year respectively. \\
        \def\arraystretch{2}
        \begin{center}
            \begin{tabular}{|c|c|c|c|c|}
                \hline
                 H + 0 & D + 0 & M + 0 & C + 0 & Y + 0  \\
                 \hline
                 D - 2 & C + 1 & Y + 1 & M + 3 & M - 3 \\
                 \hline
                 Y + 2 & M - 1 & D - 1 & H - 2 & C + 2 \\
                 \hline
                 M - 3 & H + 1 & C - 3 & Y + 2 & D + 3 \\
                 \hline
                 C + 3 & Y - 1 & H + 3 & D - 3 & M - 2 \\
                 \hline
            \end{tabular}\\
        \textbf{\\ Note: This magic square is only pandiagonal and has multiples of one of the numbers}
        \end{center}
    \subsubsection{6x6 Magic Square}
        Letting m, H, D, M, C and Y be the minute, the hour, the day, the month, the century and the year respectively. \\
        This table uses the minute of your birth, forming an even larger square. \\
        \def\arraystretch{2}
        \begin{center}
            \begin{tabular}{|c|c|c|c|c|c|}
                \hline
                 m + 0 & H + 0 & D + 0 & M + 0 & C + 0 & Y + 0  \\
                 \hline
                 M - 1 & Y + 1 & H - 3 & C + 3 & m - 2 & D + 2\\
                 \hline
                 D + 3 & m - 1 & C - 1 & H + 2 & Y - 2 & M - 1\\
                 \hline
                 Y + 2 & C - 4 & M - 3 & D + 1 & H + 3 & M + 1\\
                 \hline
                 H - 2 & D + 2 & m + 4 & Y - 2 & M - 1 & C - 1 \\
                 \hline
                 C - 2 & M + 2 & Y + 3 & m - 4 & D + 2 & H - 1 \\
                 \hline
            \end{tabular} \\
        \textbf{\\ Note: This is only a basic magic square with multiple repeated entries}
        \end{center}

\subsection{Other Forms}
Are there any other squares with the same properties?
There are many.\\
But first let us define some stuff.\\
We set D be Day, M be Month, C be century, Y be year. \\
For example, if your birthday was on 22 December 1887.\\
Then, D = 22, M = 12, C = 18, Y = 87.\\
    \subsubsection{Variation 1: Diagonal}
        \def\arraystretch{2}
        \begin{center}
            \begin{tabular}{|c|c|c|c|}
                \hline
                \cellcolor{yellow} D + 0 & Y + 2 & M - 1 & C - 1 \\
                \hline
                C - 2 & \cellcolor{green} M + 0 & Y - 1 & D + 3 \\
                \hline
                Y + 1 & D + 1 & \cellcolor{aqua} C + 0 & M - 2 \\
                \hline
                M + 1 & C - 3 & D + 2 & \cellcolor{orange} Y + 0 \\
                \hline
            \end{tabular}
        \end{center}
    \subsubsection{Variation 2: Box}
        \def\arraystretch{2}
        \begin{center}
            \begin{tabular}{|c|c|c|c|}
                \hline
                C - 2 & M + 1 & D + 2 & Y - 1 \\
                \hline
                Y + 1 & \cellcolor{yellow} D + 0 & \cellcolor{green} M + 0 & C - 1 \\
                \hline
                M - 1 & \cellcolor{aqua} C + 0 & \cellcolor{orange} Y + 0 & D + 1 \\
                \hline
                D + 2 & Y - 1 & C - 2 & M + 1 \\
                \hline
            \end{tabular}
        \end{center}

    \subsubsection{Variation 3: Column}
        \def\arraystretch{2}
        \begin{center}
            \begin{tabular}{|c|c|c|c|}
                \hline
                \cellcolor{yellow} D + 0 & C + 2 & Y - 1 & M - 1 \\
                \hline
                \cellcolor{green} M + 0 & Y - 2 & C - 1 & D + 3 \\
                \hline
                \cellcolor{aqua} C + 0 & D + 2 & M + 1 & Y - 3 \\
                \hline
                \cellcolor{orange} Y + 0 & M - 2 & D + 1 & C + 1 \\
                \hline
            \end{tabular}
        \end{center}

    \subsubsection{Method of making these squares}
        Looking at the above Variations. We believe that there is a method in the creation of these kinds of squares.

        This is the method we found while creating the squares

        \begin{enumerate}
            \item Set the beginning (Column, Row etc.)
            \item Set each box to base of one of the four given variables
            \item Set one number to differ
            \item Using that number set the others, keeping the beginning unchanged
            \item Check if it all add up and make changes if necessary. Repeat this till completion.
        \end{enumerate}

\subsection{Super Asymmetry}
Is there any form of derivation of the overall constant without any symmetrical significance?

Yes, yes there is.

\begin{center}
            \begin{tabular}{|c|c|c|c|}
                \hline
                \cellcolor{yellow} D + 0 & \cellcolor{yellow} M + 0 & \cellcolor{orange} C + 0 & \cellcolor{aqua} Y + 0 \\
                \hline
                \cellcolor{orange} Y + 1 & \cellcolor{aqua} C - 1 & \cellcolor{orange} M - 3 & \cellcolor{aqua} D + 3 \\
                \hline
                \cellcolor{aqua} M - 2 & \cellcolor{orange} D + 2 & \cellcolor{green} Y + 2 & \cellcolor{green} C - 2 \\
                \hline
                \cellcolor{green} D + 0 & \cellcolor{yellow} C + 2 & \cellcolor{yellow} Y - 1 & \cellcolor{green} M - 1 \\
                \hline
            \end{tabular}
        \end{center}

The sum of the numbers in the same coloured cell give the same sum.

\end{document}


\subsection{Drucker-Prager Plasticity Model with Ductile Damage}
The Drucker-Prager plasticity model was developed by Drucker \citet{drucker_implications_1950} for modelling frictional materials like granular soils and rock. An important aspect of this plasticity model is the use of a pressure dependant yield criterion to account for the increase in yield stress of geomaterials as the in-situ stresses increase. Specifically, the Drucker-Prager material model is formulated and used for materials with compressive yield strength much greater than the tensile yield strength such as one finds in soils and rocks. However, this material model is intended to simulate the material response under essentially monotonic loading which limits the capacity for modeling cyclic loading.

In addition, the Drucker-Prager model is suitable for using in conjunction with progressive damage and failure models. In this formulation, the Johnson-Cook Damage model is used to model the damage evolution of the rock mass \citep{Johnson_1985}. At a sufficiently large scale, the damage behaviour of NFR can be thought of as behaving in a ductile capacity.

Here, for the extended Drucker-Prager plasticity model, a linear yield function, $F\left(\bar{\sigma}_{ij}, \bar{\epsilon}^{pl}\right)$, is assumed to be a function of three stress invariants: the Von-Mises equivalent stress, $p\left(\bar{\sigma}_{ij}\right)$, the hydrostatic stress, $q\left(\bar{\sigma}_{ij}\right)$, and the third invariant of deviatoric stress, $r\left(\bar{\sigma}_{ij}\right)$. In addition, the yield function is written in terms of the compressive yield stress, $\sigma_c^y\left(\bar{\epsilon}^{pl}\right)$, which is defined by the hardening function and two material parameters: the friction angle, $\phi$, and a parameter $K$, defined as the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression:

\begin{equation}
\label{eqn:const8c}
\begin{split}
F\left(\bar{\sigma}_{ij}, \bar{\epsilon}^{pl}\right)=\frac{1}{2}q\left(\bar{\sigma}_{ij}\right)\left [ 1+\frac{1}{K}-\left ( 1-\frac{1}{K} \right )\left ( \frac{r\left(\bar{\sigma}_{ij}\right)}{q\left(\bar{\sigma}_{ij}\right)} \right )^3 \right ]- \\
p\left(\bar{\sigma}_{ij}\right)\tan\phi - \left[1-\frac{1}{3}\tan\phi \right]\sigma_c^y\left(\bar{\epsilon}^{pl}\right)
\end{split}
\end{equation}

%The hydrostatic stress:
%\begin{equation}
%p\left(\bar{\sigma}_{ij}\right)=\frac{1}{3}\bar{\sigma}_{kk}
%\label{eqn:druc3}
%\end{equation}

%The Von-Mises equivalent stress:
%\begin{equation}
%q\left(\bar{\sigma}_{ij}\right)=\sqrt{\frac{3}{2}S_{ij}S_{ji}}\label{eqn:druc4}
%\end{equation}

%Where $S_{ij}$ is known as the stress deviator with $\delta_{ij}$ being the Kronecker Delta:

%\begin{equation}
%S_{ij} = \bar{\sigma_{ij}} + p\left(\bar{\sigma}_{ij}\right)\delta_{ij}
%\label{eqn:druc4-1}
%\end{equation}

%The third invariant of deviatoric stress:

%\begin{equation}
%r\left(\bar{\sigma}_{ij}\right)= \sqrt[3]{\frac{9}{2}S_{ij} S_{jk} S_{ki}}
%\label{eqn:druc4-2}
%\end{equation}

The flow rule in this formulation is non-associated but the flow potential function, $G\left(\bar{\sigma}_{ij}\right)$, is written in a very similar form as the yield function with dilation angle, $\psi$, in place of the friction angle. As with the yield function, the flow potential function, is written in terms of three stress invariants and two material parameters, dilation angle and $K$:

\begin{equation}
G\left(\bar{\sigma}_{ij}\right)=\frac{1}{2}q\left(\bar{\sigma}_{ij}\right)\left [ 1+\frac{1}{K}-\left ( 1-\frac{1}{K} \right )\left ( \frac{r\left(\bar{\sigma}_{ij}\right)}{q\left(\bar{\sigma}_{ij}\right)} \right )^3 \right ]-p\left(\bar{\sigma}_{ij}\right)\tan\psi\label{eqn:const11}
\end{equation}

In addition to the yield function and the flow rule, the hardening rule is assumed to take the form of the Barcelona model \citep{lubliner_plastic-damage_1989}. The Barcelona model allows for material hardening before softening and approaches a yield stress of 0 as the plastic strain increases.  This form of the hardening function can be written in terms of three material parameters, initial compressive yield strength $\sigma_c^{iy}$, $\alpha$ and $\beta$:

\begin{equation}
%\sigma_{c}\left(\bar{\epsilon}^{in}\right)=\frac{\sigma_{c}^{iy}-\sigma_{c}^{p}}{\left(\epsilon_{c}^{pp}\right)^{2}}\left(\bar{\epsilon}^{in}-\epsilon_{c}^{pp}\right)^{2}+\sigma_{c}^{p}
\sigma_c=\sigma_c^{iy}\left [ \left ( 1+\alpha \right ) e^{-\beta\bar{\epsilon}^{pl}}-\alpha e^{-2\beta\bar{\epsilon}^{pl}}  \right ]
\label{eqn:param2-1}
\end{equation}

The damage initiation criterion for this material model is based on the Johnson-Cook model of ductile damage initiation \citep{Johnson_1985}. The standard Johnson-Cook model assumes the equivalent plastic strain when damage is initiated, $\bar{\epsilon}_{f}^{pl}\left(\eta\right)$, is a function of triaxiality, $\eta$, and is written in terms of five material parameters. However, assuming isothermal conditions, neglecting rate effects, and assuming a simplified form of the exponential relationship, the initiation criterion can be reduced to two material parameters, $D_2$ and $D_3$:

\begin{equation}
\bar{\epsilon}_{f}^{pl}\left(\eta\right)=D_{2}e^{D_{3}\eta}\label{eqn:druc8}
\end{equation}

After the material has experienced yield and material damage has occurred, the stress-strain relationship becomes strongly mesh-dependent because of strain localization due to the energy dissipation decreasing as the mesh is refined. As such, Hillerborg et al. \citet{Hillerborg_1976} proposed a stress-displacement response based on fracture energy after damage initiation assuming that evolving damage is a linear degradation of the material stiffness in compression. Assuming a linear form, the effective plastic displacement when the material is completely damaged, $\bar{u}^{pl}_f$, can be specified, and the damage evolution can then be written in terms of the effective plastic displacement, $\bar{u}^{pl}$:

\begin{equation}
\dot{D}=\frac{\dot{\bar{u}}^{pl}}{\bar{u}_{f}^{pl}}\label{eqn:druc9-1}
\end{equation}


\section{Mixture Models \& Secant Varieties}

\begin{frame}{ \emph{``Absence of evidence is not evidence of absence.''} }
    \begin{itemize}
    \item Suppose $\mc{P} \subset \Delta_{r-1}$ is a model for a random variable $X$ with state space $[r]$.
    \item Assume $Y$ is a \emph{hidden} or \emph{latent} random variable, with state space $[s]$; for each $j \in [s]$, the conditional distribution of $X$ given $Y = j$ is $p^{(j)} \in \mc{P}$.
    \item $Y$ also has some probability distribution $\pi \in \Delta_{s-1}$.
    \end{itemize}

    So the joint distribution of $Y$ and $X$ is given by the formula
    $$ P(Y = j; X = i) = \pi_{j} \cdot p_{i}^{(j)}. $$

    \begin{block}{\emph{Donald Rumsfeld, 21st US Secretary of Defense}, \cite{DR2002}}
        \emph{``[T]here are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know.''}
    \end{block}

\end{frame}

\begin{frame}{Mixture Models}

    \begin{itemize}
    \item But as $Y$ is hidden, we can only observe the marginal distribution of $X$, that is
    $$ P(X = i) = \sum_{j = 1}^{s} \pi_{j} \cdot p_{i}^{(j)}. $$
    \item In other words, the marginal distribution of $X$ is the convex combination of the $s$ distributions $p^{(1)}, \ldots, p^{(s)}$, with weights given by $\pi$.
    \end{itemize}

    \begin{block}{Definition \cite{BSSSMD2009}}
        Let $\mc{P} \subset \Delta_{r-1}$ be a statistical model. The \emph{$s$-th mixture model} is
        $$ \Mixt^{s}(\mc{P}) := \Set{ \sum_{j = 1}^{s} \pi_{j}\cdot p^{(j)} | \pi \in \Delta_{s-1},\ p^{(j)} \in \mc{P}, \text{ for all } j }. $$
    \end{block}

\end{frame}

\begin{frame}{Mixture Models}

    \begin{itemize}
    \item Mixture models provide ways to build complex models out of simpler ones.

    \item Basic assumption is that the underlying population to be modelled can be split into $s$ disjoint sub-populations.

    \item Restricted to each sub-population, the observable $X$ follows a probability distribution from the simple model $\mc{P}$.

    \item After marginalisation though, the structure becomes significantly more complex as it is now a convex combination of these simple distributions.

    \end{itemize}

\end{frame}

\begin{frame}{Phylogenetic Trees}

    \begin{itemize}
        \item Introduce \emph{phylogenetic trees}; describe the descent of species from a common ancestor:

        \begin{center}
        \includegraphics[height=0.75\textheight]{resources/phylogenetic-tree.pdf}
        \end{center}

    \end{itemize}

\end{frame}

\begin{frame}{Molecular Phylogenetics}

    \begin{itemize}
        \item Sequence of DNA molecules in a genome is represented as a sequence of letters from the four letter alphabet $\Sigma = \{ \texttt{A}, \texttt{C}, \texttt{G}, \texttt{T} \}$.

        \item \emph{Fix for now} an ancestral nucleotide $\texttt{Y} \in \Sigma$; we assume that the following evolution events occur independently \cite{EAsalmon}:

        $$ \texttt{Y} \overset{ \pi_{\texttt{Y}} \cdot p_{\texttt{A}}^{(\texttt{Y})}  }{\longmapsto}  \texttt{A}, \quad \texttt{Y} \overset{ \pi_{\texttt{Y}} \cdot p_{\texttt{C}}^{(\texttt{Y})}  }{\longmapsto} \texttt{C}, \quad \texttt{Y} \overset{ \pi_{\texttt{Y}} \cdot p_{\texttt{G}}^{(\texttt{Y})}  }{\longmapsto} \texttt{G}, \quad \texttt{Y} \overset{ \pi_{\texttt{Y}} \cdot p_{\texttt{T}}^{(\texttt{Y})}  }{\longmapsto} \texttt{T}, $$

        \item So \emph{given} $\texttt{Y}$, we have a joint distribution:

        $$ \pi_{\texttt{Y}} \cdot [ p_{\texttt{A}}^{(\texttt{Y})}, p_{\texttt{C}}^{(\texttt{Y})}, p_{\texttt{G}}^{(\texttt{Y})}, p_{\texttt{T}}^{(\texttt{Y})} ] \in \Delta_{3} = \Delta_{4-1}. $$

    \end{itemize}

\end{frame}

\begin{frame}{Example}
    \begin{itemize}
        \item  $\texttt{Y}$ is a hidden variable though; could have been anything from $\Sigma = \Set{\texttt{A}, \texttt{C}, \texttt{G}, \texttt{T}}$.

        \item For \emph{exactly one given choice} of \texttt{Y}, we had the distribution $\Delta_{3}$; need to consider \emph{all choices} of ancestral nucleotide \texttt{Y}.

        \item Hence, we get the mixture model \cite{EAsalmon}:
            \vspace*{-8pt}
        \begin{equation*}
            \begin{split}
                &\Mixt^{4}(\Delta_{3}) \\
                &= \Set{\sum_{ \texttt{Y} \in \Sigma  } \pi_{\texttt{Y}} \cdot p^{(\texttt{Y})} | \pi \in \Delta_{3},\ p^{(\texttt{Y})} \in \mc{P} \subseteq \Delta_{3}, \text{ for each \texttt{Y}} }.
            \end{split}
        \end{equation*}

    \end{itemize}

    \begin{block}{Question?}
    What is the analogue for mixture models in algebraic statistics?
    \end{block}

\end{frame}

\begin{frame}{Secant Varieties}
    \begin{block}{Answer!}
        Secant\footnote{from \emph{secare}, ``to cut'' in Latin; \emph{c.f. tangō}, ``to touch''.} varieties \cite{BSSSMD2009}!
    \end{block}

    \begin{block}{Definitions}
        \begin{itemize}
        \item Consider two varieties $V, W \subseteq \RR^{k}$. The \emph{join} of $V$ and $W$ is the variety
        $$ \mc{J}(V,W) := \{ \lambda v + (1-\lambda)w : v \in v, w \in W, \lambda \in [0,1] \}. $$

        \item If $V = W$, then this is the \emph{secant variety} of $V$, denoted $\Sec^{2}(V) = \mc{J}(V,V)$. The \emph{$s$-th higher secant variety} is:
        $$ \Sec^{1}(V) := V, \qquad \Sec^{s}(V) := \mc{J}(\Sec^{s-1}(V), V ). $$
        \end{itemize}
    \end{block}

\end{frame}

\begin{frame}{Secant Varieties}

    \begin{center}
        \includegraphics[height=0.25\textwidth, angle=0]{resources/secant-line.pdf}
    \end{center}

    \begin{center}
        \includegraphics[height=0.35\textwidth, angle=0]{resources/secant-circle.pdf}
    \end{center}

\end{frame}

\begin{frame}{More Complicated Phylogenetic Trees}

    \begin{itemize}
        \item Last example only had one extant species; what about if we had three extant species, all coming from the same ancestor?

    \begin{center}
        \includegraphics[width=0.6\textwidth, angle=0]{resources/three-extant.pdf}
    \end{center}

    \item Now we have to consider: $\Sec^{4}(\PP^{3} \times \PP^{3} \times \PP^{3})$; or equivalently $\Mixt^{4}(\Delta_{3} \times \Delta_{3} \times \Delta_{3})$ \cite{EAsalmon}.

    \item Finding the minimal set of polynomials defining $\Sec^{4}(\PP^{3} \times \PP^{3} \times \PP^{3})$ once gave rise to a very important application of algebraic statistics...

    \end{itemize}

\end{frame}

\begin{frame}{The \emph{Salmon Problem}}

\begin{block}{Statement}
    \emph{Determine the ideal\footnote{read this as ``set of defining polynomials''.} defining $\Sec^{4}(\PP^{3} \times \PP^{3} \times \PP^{3})$,} \cite{EAsalmon}.
\end{block}

\begin{block}{Prize}
    \begin{itemize}
    \item At an IMA workshop in 2007, Elizabeth Allman stated that she would personally catch and smoke copper river salmon from Alaska for whomever solved this problem.
    \item Solved in 2010 by Shmuel Friedland \& Elizabeth Gross \cite{SFEG2012} (see \cite{DBLO2011} too for an in-depth discussion).
    \end{itemize}
\end{block}

Solving this would then provide all polynomial invariants of the statistical model for any binary evolutionary tree, with any number of states \cite{EAJR2008, DBLO2011}.

\end{frame}

\begin{frame}{Revision}

Why $\Sec^{4}(\PP^{3} \times \PP^{3} \times \PP^{3})$ again?

\begin{itemize}
    \item Three independent variables (nucleotides in extant species) $\rightsquigarrow$ three factors in product;
    \item Each independently assumes one value from $\Sigma = \{ \texttt{A}, \texttt{C}, \texttt{G}, \texttt{T} \}$ $\rightsquigarrow$ distribution is a point in $\PP^{3} = \PP^{4-1}$;
    \item The ancestral nucleotide is unknown, but could assume any of the four values in $\Sigma$ $\rightsquigarrow$ mix four such independence models;
    \item The model for the three observed nucleotides is therefore
    \begin{equation*}
            \Sec^{4}(\PP^{3} \times \PP^{3} \times \PP^{3}),\quad \text{\emph{c.f.},} \quad \Mixt^{4}(\Delta_{3} \times \Delta_{3} \times \Delta_{3}).
    \end{equation*}
\end{itemize}
\end{frame}


\chapter{Sales pitches}
\label{ch:sales}
\newcommand{\pitch}[1]{\ii[\textsf{\color{blue}\ref{#1}}.] \textsf{\color{blue} \textbf{\nameref{#1}.}} \\[1ex]} % for now. . .
\newcommand{\buzzword}[1]{\textbf{\color{green!40!black} #1}}

This chapter contains a pitch for each part,
to help you decide what you want to read
and to elaborate more on how they are interconnected.

For convenience, here is again the dependency plot
that appeared in the frontmatter.
\input{tex/frontmatter/digraph}

\section{The basics}
\begin{itemize}
\pitch{part:startout}
I made a design decision that the first part
should have a little bit both of algebra and topology:
so this first chapter begins by defining a \buzzword{group},
while the second chapter begins by defining a \buzzword{metric space}.
The intention is so that newcomers get to see two different
examples of ``sets with additional structure''
in somewhat different contexts,
and to have a minimal amount of literacy as these sorts
of definitions appear over and over.\footnote{In particular,
	I think it's easier to learn
	what a homeomorphism is after seeing group isomorphism,
	and what a homomorphism is after seeing continuous map.}

\pitch{part:absalg}
The algebraically inclined can then delve into
further types of algebraic structures:
some more details of \buzzword{groups},
and then \buzzword{rings} and \buzzword{fields} ---
which will let you generalize $\ZZ$, $\QQ$, $\RR$, $\CC$.
So you'll learn to become familiar with all sorts of other nouns
that appear in algebra, unlocking a whole host of objects
that one couldn't talk about before.

We'll also come into \buzzword{ideals},
which generalize the GCD in $\ZZ$ that you might know of.
For example, you know in $\ZZ$ that any integer
can be written in the form $3a+5b$ for $a,b \in \ZZ$,
since $\gcd(3,5)=1$.
We'll see that this statement is really
a statement of ideals: ``$(3,5)=1$ in $\ZZ$'',
and thus we'll understand in what situations
it can be generalized, e.g.\ to polynomials.

\pitch{part:basictop}
The more analytically inclined can instead move into topology,
learning more about spaces.
We'll find out that ``metric spaces'' are actually too specific,
and that it's better to work with \buzzword{topological spaces},
which are based on the so-called \buzzword{open sets}.
You'll then get to see the buddings of some geometrical ideals,
ending with the really great notion of \buzzword{compactness},
a powerful notion that makes real analysis tick.

One example of an application of compactness to tempt you now:
a continuous function $f \colon [0,1] \to \RR$
always achieves a \emph{maximum} value.
(In contrast, $f \colon (0,1) \to \RR$ by $x \mapsto 1/x$ does not.)
We'll see the reason is that $[0,1]$ is compact.
\end{itemize}

\section{Abstract algebra}
\begin{itemize}
\pitch{part:linalg}
In high school, linear algebra is often really unsatisfying.
You are given these arrays of numbers,
and they're manipulated in some ways that don't really make sense.
For example, the determinant is defined as this
funny-looking sum with a bunch of products that seems
to come out of thin air. Where does it come from?
Why does $\det(AB) = \det A \det B$ with such a bizarre formula?

Well, it turns out that you \emph{can} explain all of these things!
The trick is to not think of linear algebra
as the study of matrices,
but instead as the study of \emph{linear maps}.
In earlier chapters we saw that we got great generalizations
by speaking of ``sets with enriched structure'' and ``maps between them''.
This time, our sets are \buzzword{vector spaces}
and our maps are \buzzword{linear maps}.
We'll find out that a matrix is actually just
a way of writing down a linear map as an array of numbers,
but using the ``intrinsic'' definitions
we'll de-mystify all the strange formulas from high school
and show you where they all come from.

In particular, we'll see \emph{easy} proofs
that column rank equals row rank,
determinant is multiplicative, trace is the sum of the diagonal entries.
We'll see how the dot product works,
and learn all the words starting with ``eigen-''.
We'll even have a bonus chapter for Fourier analysis
showing that you can also explain all the big buzz-words
by just being comfortable with vector spaces.

\pitch{part:groups}
Some of you might be interested in more about groups,
and this chapter will give you a way to play further.
It starts with an exploration of \buzzword{group actions},
then goes into a bit on \buzzword{Sylow theorems},
which are the tools that let us try to \emph{classify all groups}.

\pitch{part:repth}
If $G$ is a group, we can try to understand
it by implementing it as a \emph{matrix},
i.e.\ considering embeddings $G \injto \GL_n(\CC)$.
These are called \buzzword{representations} of $G$;
it turns out that they can be decomposed into \buzzword{irreducible} ones.
Astonishingly we will find that we can
\emph{basically characterize all of them}:
the results turn out to be short and completely unexpected.

For example, we will find out that there are finitely
many irreducible representations of a given finite group $G$;
if we label them $V_1$, $V_2$, \dots, $V_r$,
then we will find that $r$ is the number
of conjugacy classes of $G$, and moreover that
\[ |G| = (\dim V_1)^2 + \dots + (\dim V_r)^2 \]
which comes out of nowhere!

The last chapter of this part will show you some
unexpected corollaries.
Here is one of them:
let $G$ be a finite group and create variables $x_g$
for each $g \in G$.
A $|G| \times |G|$ matrix $M$ is defined by setting
the $(g,h)$th entry to be the variable $x_{g \cdot h}$.
Then this determinant will turn out to \emph{factor},
and the factors will correspond to the $V_i$ we described above:
there will be an irreducible factor of degree $\dim V_i$
appearing $\dim V_i$ times.
This result, called the \buzzword{Frobenius determinant},
is said to have given birth to representation theory.

\pitch{part:quantum}
If you ever wondered what \buzzword{Shor's algorithm} is,
this chapter will use the built-up linear algebra to tell you!
\end{itemize}

\section{Real and complex analysis}
\begin{itemize}
\pitch{part:calc}
In this part, we'll use our built-up knowledge of
metric and topological spaces to give short, rigorous definitions
and theorems typical of high school calculus.
That is, we'll really define and prove most everything you've seen about
\buzzword{limits}, \buzzword{series}, \buzzword{derivatives}, and \buzzword{integrals}.

Although this might seem intimidating,
it turns out that actually, by the time we start this chapter,
\emph{the hard work has already been done}:
the notion of limits, open sets, and compactness
will make short work of what was swept under the rug in AP calculus.
Most of the proofs will thus actually be quite short.
We sit back and watch all the pieces slowly come together
as a reward for our careful study of topology beforehand.

That said, if you are willing to suspend belief,
you can actually read most of the other parts
without knowing the exact details of all the calculus here,
so in some sense this part is ``optional''.

\pitch{part:cmplxana}
It turns out that \buzzword{holomorphic functions}
(complex-differentiable functions)
are close to the nicest things ever:
they turn out to be given by a Taylor series
(i.e.\ are basically polynomials).
This means we'll be able to prove unreasonably nice results
about holomorphic functions $\CC \to \CC$, like
\begin{itemize}
	\ii they are determined by just a few inputs,
	\ii their contour integrals are all zero,
	\ii they can't be bounded unless they are constant,
	\ii \dots.
\end{itemize}
We then introduce \buzzword{meromorphic functions},
which are like quotients of holomorphic functions,
and find that we can detect their zeros by simply drawing
loops in the plane and integrating over them:
the famous \buzzword{residue theorem} appears.
(In the practice problems, you will see this even gives
us a way to evaluate real integrals that can't be evaluated otherwise.)

\pitch{part:measure}
Measure theory is the upgraded version of integration.
The Riemann integration is for a lot of purposes not really sufficient;
for example, if $f$ is the function equals $1$ at rational numbers
but $0$ at irrational numbers,
we would hope that $\int_0^1 f(x) \; dx = 0$,
but the Riemann integral is not capable of handling this function $f$.

The \buzzword{Lebesgue integral} will handle these mistakes
by assigning a \emph{measure} to a generic space $\Omega$,
making it into a \buzzword{measure space}.
This will let us develop a richer theory of integration
where the above integral \emph{does} work out to zero
because the ``rational numbers have measure zero''.
Even the development of the measure will be an achievement,
because it means we've developed a rigorous, complete way
of talking about what notions like area and volume mean ---
on any space, not just $\RR^n$!
So for example the Lebesgue integral will let us
integrate functions over any \buzzword{measure space}.

\pitch{part:prob}
Using the tools of measure theory, we'll be able to start
giving rigorous definitions of \buzzword{probability}, too.
We'll see that a \buzzword{random variable} is actually
a function from a measure space of worlds to $\RR$,
giving us a rigorous way to talk about its probabilities.
We can then start actually stating results like
the \buzzword{law of large numbers} and \buzzword{central limit theorem}
in ways that make them both easy to state and straightforward to prove.

\pitch{part:diffgeo}
Multivariable calculus is often confusing
because of all the partial derivatives.
But we'll find out that, armed with our good understanding
of linear algebra, that we're really looking at a \buzzword{total derivative}:
at every point of a function $f \colon \RR^n \to \RR$
we can associate a \emph{linear map} $Df$ which
captures in one object the notion of partial derivatives.
Set up this way, we'll get to see versions of \buzzword{differential forms}
and \buzzword{Stokes' theorem},
and we finally will know what the notation $dx$ really means.
In the end, we'll say a little bit about manifolds in general.
\end{itemize}

\section{Algebraic number theory}
\begin{itemize}
\pitch{part:algnt1}
Why is $3+\sqrt5$ the conjugate of $3-\sqrt5$?
How come the norm $\norm{a+b\sqrt5} = a^2-5b^2$ used in Pell equations
just happens to be multiplicative?
Why is it we can do factoring into primes in $\ZZ[i]$
but not in $\ZZ[\sqrt{-5}]$?
All these questions and more will be answered in this part,
when we learn about \buzzword{number fields},
a generalization of $\QQ$ and $\ZZ$ to things like $\QQ(\sqrt5)$
and $\ZZ[\sqrt{5}]$.
We'll find out that we have unique factorization into prime ideals,
that there is a real \emph{multiplicative norm} in play here,
and so on.
We'll also see that Pell's equation falls out of this theory.

\pitch{part:algnt2}
All the big buzz-words come out now:
\buzzword{Galois groups}, the \buzzword{Frobenius}, and friends.
We'll see quadratic reciprocity is just a shadow of
the behavior of the Frobenius element,
and meet the \buzzword{Chebotarev density theorem},
which generalizes greatly the Dirichlet theorem on the infinitude
of primes which are $a \pmod n$.
Towards the end, we'll also state \buzzword{Artin reciprocity},
one of the great results of \buzzword{class field theory},
and how it generalizes quadratic reciprocity and cubic reciprocity.
\end{itemize}

\section{Algebraic topology}
\begin{itemize}
\pitch{part:algtop1}
What's the difference between an annulus and disk?
Well, one of them has a ``hole'' in it,
but if we are just given intrinsic topological spaces
it's hard to make this notion precise.
The \buzzword{fundamental group} $\pi_1(X)$
and more general \buzzword{homotopy group}
will make this precise --- we'll find a way to define an abelian group
$\pi_1(X)$ for every topological space $X$ which captures the idea
there is a hole in the space, by throwing lassos into the space
and seeing if we can reel them in.

Amazingly, the fundamental group $\pi_1(X)$ will, under mild conditions,
tell you about ways to cover $X$ with a so-called
\buzzword{covering projection}.
One picture is that one can wrap a real line $\RR$ into a helix shape
and then project it down into the circle $S^1$.
This will turn out to correspond to the fact that $\pi_1(S^1) = \ZZ$
which has only one subgroup.
More generally the subgroups of $\pi_1(X)$ will be in
bijection with ways to cover the space $X$!

\pitch{part:cats}
What do fields, groups, manifolds, metric spaces, measure spaces,
modules, representations, rings, topological spaces, vector spaces,
all have in common?
Answer: they are all ``objects with additional structure'',
with maps between them.

The notion of \buzzword{category} will appropriately generalize all of them.
We'll see that all sorts of constructions and ideas
can be abstracted into the framework of a category,
in which we \emph{only} think about objects and arrows between them,
without probing too hard into the details of what those objects are.
This results in drawing many \buzzword{commutative diagrams}.

For example, any way of taking an objection in one category
and getting another one (for example $\pi_1$ as above,
from the category of spaces into the category of groups)
will probably be a \buzzword{functor}.
We'll unify $G \times H$, $X \times Y$, $R \times S$,
and anything with the $\times$ symbol into the notion of a product,
and then even more generally into a \buzzword{limit}.
Towards the end, we talk about \buzzword{abelian categories}
and talk about the famous
\buzzword{snake lemma}, \buzzword{five lemma}, and so on.

\pitch{part:algtop2}
Using the language of category theory,
we then resume our adventures in algebraic topology,
in which we define the \buzzword{homology groups}
which give a different way of noticing holes in a space,
in a way that is longer to define but easier to compute in practice.
We'll then reverse the construction to get so-called
\buzzword{cohomology rings} instead,
which give us an even finer invariant for telling spaces apart.
\end{itemize}

\section{Algebraic geometry}
\begin{itemize}
\pitch{part:ag1}
We begin with a classical study of classical \buzzword{complex varieties}:
the study of intersections of polynomial equations over $\CC$.
This will naturally lead us into the geometry of rings,
giving ways to draw pictures of ideals,
and motivating \buzzword{Hilbert's nullstellensatz}.
The \buzzword{Zariski topology} will show its face,
and then we'll play with \buzzword{projective varities}
and \buzzword{quasi-projective varieties},
with a bonus detour into \buzzword{Bezout's theorem}.
All this prepares us for our journey into schemes.

\pitch{part:ag2}
We now get serious and delve into Grothendiek's definition of
an \buzzword{affine scheme}:
a generalization of our classical varieties
that lets us start with any ring $A$
and construct a space $\Spec A$ on it.
We'll equip it with its own Zariski topology
and then a sheaf of functions on it,
making it into a \buzzword{locally ringed space};
we will find that the sheaf can be understood
effectively in terms of \buzzword{localization} on it.
We'll find that the language of commutative algebra provides
elegant generalizations of what's going on geometrically:
prime ideals correspond to irreducible closed subsets,
radical ideals correspond to closed subsets,
maximal ideals correspond to closed points, and so on.
We'll draw lots of pictures of spaces and examples to accompany this.

\pitch{part:ag3}
Not yet written! Wait for v2.
\end{itemize}

\section{Set theory}
\begin{itemize}
\pitch{part:st1}
Why is \buzzword{Russell's paradox} such a big deal
and how is it resolved?
What is this \buzzword{Zorn's lemma}
that everyone keeps talking about?
In this part we'll learn the answers to these questions
by giving a real description of the \buzzword{Zermelo-Frankel}
axioms, and the \buzzword{axiom of choice},
delving into the details of how math is built axiomatically
at the very bottom foundations.
We'll meet the \buzzword{ordinal numbers} and \buzzword{cardinal numbers}
and learn how to do \buzzword{transfinite induction} with them.

\pitch{part:st2}
The \buzzword{continuum hypothesis}
states that there are no cardinalities
between the size of the natural numbers and the size of the real numbers.
It was shown to be \emph{independent} of the axioms ---
one cannot prove or disprove it.
How could a result like that possibly be proved?
Using our understanding of the ZF axioms,
we'll develop a bit of \buzzword{model theory}
and then use \buzzword{forcing} in order to show
how to construct entire models of the universe
in which the continuum hypothesis is true or false.
\end{itemize}