adding equation sheet

Chapters/Genetic_drift_selection.tex

@@ -112,7 +112,7 @@ \section{Stochastic loss of strongly selected alleles}
 \end{equation}
 Solving this we find that
 \begin{equation}
-p_F = 2s.
+p_F = 2s. \label{eqn:prob_fix_strong}
 \end{equation}
 Thus even an allele with a $1\%$ selection coefficient has a $98\%$
 probability of being lost when it is first introduced into the

Chapters/Interaction_selection_mut_mig.tex

@@ -98,7 +98,7 @@ \subsection{Mutation--Selection Balance}
 so eqn.\ \eqref{eqn:mut_sel_bal} is not valid. However, we can make an argument similar to the one above to show
 that, for truly recessive alleles,
 \begin{equation}
-	q_e = \sqrt{\frac{\mu}{s}}.
+	q_e = \sqrt{\frac{\mu}{s}}. \eqref{eqn:recess_mut_sel_bal} 
 \end{equation}
 \graham{Add figure illustrating the freq as a function of h and s}
 \begin{marginfigure}
@@ -370,7 +370,7 @@ \subsection{Migration--selection balance}
 selection model, resulting in a diploid
 migration--selection balance equilibrium frequency of
 \begin{equation}
-q_{e,1} = \frac{m}{hs}
+q_{e,1} = \frac{m}{hs} \label{eqn:mig_sel_eq}
 \end{equation}
 
 \begin{figure}
@@ -477,7 +477,7 @@ \subsection{Migration--selection balance}
 tangent) of $q(x)$ at $x=0$. See Figure \ref{fig:cline_tangent}. Under this definition, the cline width is
 approximately
 \begin{equation}
-  0.6 \sigma/\sqrt{s} ~\textrm{miles},
+  0.6 \sigma/\sqrt{s} ~\textrm{miles}, \label{eqn:cline_width}
 \end{equation}
 note that the units are miles here just because we defined the average
 dispersal distance ($\sigma$) in miles above. Thus the cline will be wider if individuals dispersal further, higher

Chapters/Multi_trait_selection.tex

@@ -426,7 +426,7 @@ \subsection{Hamilton's Rule and the evolution of altruistic and
 % http://darwin-online.org.uk/content/frameset?pageseq=23&itemID=F8.2&viewtype=side
 
 
-%%
+%%  Beaver https://twitter.com/EponymousBreeze/status/1226934486090145799/photo/1
 \paragraph{Other forms of alturism}
 Kin-selection can favour altruism because individuals carrying
 altruistic alleles interact with other \emph{ related} individuals who tend to display altruistic phenotypes and so gain an advantage. However, there are other ways that altruistic behaviours can spread than just through the interactions with kin.

Chapters/One_locus_selection.tex

@@ -111,11 +111,11 @@ \subsection{Haploid selection model}
 In practice, it is often helpful to parametrize the relative fitnesses $w_i$ in a specific way. For example, we may set $w_1 = 1$ and $w_2 = 1 - s$, where $s$ is called the selection coefficient. Using this parametrization, $s$ is simply the difference in relative fitnesses between the two alleles. Equation \eqref{eq:haploid_tau_gen} becomes
 \begin{equation}
 	\label{eq:haploid_tau_gen_expl}
-	p_{t+\tau} = \frac{p_{t}}{p_{t} + q_{t} (1 - s)^{\tau}},
+	p_{\tau} = \frac{p_{0}}{p_0 + q_0 (1 - s)^{\tau}},
 \end{equation}
 as $w_2 / w_1 = 1 - s$. Then, if $s \ll 1$, we can approximate $(1-s)^{\tau}$ in the denominator by $\exp(-s\tau)$ to obtain
 \begin{equation} \label{eq:haploid_logistic growth}
-	p_{t+\tau} \approx \frac{p_t}{p_t + q_t e^{-s\tau}}.
+	p_{\tau} \approx \frac{p_0}{p_0 + q_0 e^{-s\tau}}.
 \end{equation}
 This equation takes the form of a logistic function. That is because
 we are looking at the relative frequencies of two `populations' (of
@@ -849,7 +849,7 @@ \subsection{Heterozygote advantage}
 Using our $s_1$ and $s_2$ parametrization above, we see that the marginal fitnesses of
 the two alleles are equal when
 \begin{equation}
-	p_e = \frac{s_2}{s_1+s_2}
+	p_e = \frac{s_2}{s_1+s_2} \label{eqn:het_ad_eq}
   \end{equation}
       \begin{marginfigure}
 \begin{center}

math_background/Equation_sheet.tex

@@ -0,0 +1,44 @@
+% \usepackage{multicol} %%For eqn sheet https://tex.stackexchange.com/questions/152093/troubles-with-a-two-column-document-using-tufte-handout
+
+
+%\d% ocumentclass[12pt,twocolumn]{article}
+% % \usepackage{nicefrac}
+% %  \usepackage{amsmath}
+% %   \usepackage{amsfonts}
+% %   \usepackage{amssymb}
+% %   \usepackage[dvipsnames]{xcolor}
+% % \newcounter{question}[section]   %%modified from https://www.sharelatex.com/learn/Counters
+% % %\newenvironment{question}[1][]{\refstepcounter{question}\par   \begin{tcolorbox}
+% %  %   \medskip \textbf{Question~\thequestion. #1}\rmfamily}{\medskip} \end{tcolorbox}
+% % \newcommand{\E}{\mathbb{E}}
+% % \renewcommand{\P}{\mathbb{P}}
+% % \newcommand{\half}{\tfrac{1}{2}}
+
+%\newcommand{\wbar}{\overline{w}}
+% New commands added by Simon:
+%\newcommand{\fis}{F_{\mathrm{IS}}}
+%\newcommand{\fit}{F_{\mathrm{IT}}}
+%\newcommand{\fst}{F_{\mathrm{ST}}}
+%\newcommand{\Wbar}{\overline{W}}
+
+%\newenvironment{question}[1][]{\refstepcounter{question}\par\medskip
+ %  \textbf{Question~\thequestion. #1} \rmfamily}{\medskip} 
+
+
+%\begin{document}
+\section*{Equation Sheet}
+\begin{table*}
+% \begin{tabular}{llll}
+ % \csvreader[separator=semicolon]{math_background/Equation_sheet.txt} 
+%\end{tabular}
+   \csvautobooktabular[separator=semicolon]{math_background/Equation_sheet.txt}  % can control this more https://mirror.hmc.edu/ctan/macros/latex/contrib/csvsimple/csvsimple.pdf
+  % https://tex.stackexchange.com/questions/324777/one-of-the-fields-in-the-csv-read-by-csvreader-has-a-lot-of-comma
+\end{table*}  %, table head=\hline, table foot=\hline
+
+%\newpage 
+%\begin{multicols}{2}
+
+%\end{multicols} 
+
+
+%\end{document}

math_background/Equation_sheet.txt

@@ -3,7 +3,7 @@ Equation	;	ref.	;	Equation	;	ref.
  $F_{ij}= 0 \times r_0 + (\nicefrac{1}{4}) r_1  + (\nicefrac{1}{2}) r_2$	;	\eqref{eqn:coeffkinship_step}	;	$(1-F) p^2 + F p (1-F) 2pq   (1-F) q^2 + F q$	;	\eqref{table:GeneralizedHWE}
 	;		;		;	
 F statistics	;	a	;	b	;	c
-$\fst =1-\frac{H_S}{H_T}$	;	\eqref{eqn:FST}.	;	$\fit =1-\frac{H_I}{H_T},~~\fis =1-\frac{H_I}{H_S}$	;	\eqref{eqn:FIT}, ~ \eqref{eqn:FIS}
+$\fst =1-\frac{H_S}{H_T}$	;	\eqref{eqn:FST}.	;	"$\fit =1-\frac{H_I}{H_T},~~\fis =1-\frac{H_I}{H_S}$"	;	"\eqref{eqn:FIT}, ~ \eqref{eqn:FIS}"
 	;		;		;	
 Relationship among F statistics	;		;	Linkage disequilibrium (LD)	;	~
 $(1-\fit) =(1-\fis)(1-\fst)$	;	\eqref{eqn:F_relationships}	;	$D = p_{AB} - p_Ap_B $	;	\eqref{eqn:LD_def}
@@ -12,16 +12,28 @@ Decay of LD	;		;	Decay of Heterozygosity	;	~
  $D_t=  (1-r)^t D_0 \approx D_0 e^{-rt}$	;	\eqref{eqn_LD_decay}	;	  $H_t = \left(1-\frac{1}{2N_e} \right)^tH_0 \approx H_0 e^{-\nicefrac{t}{2N_e}}$	;	\eqref{eqn:loss_het_discrete}
 	;		;		;	
 Equilibrium level of neutral polymorphism	;	\eqref{eqn:hetero}	;	Coalescent time and time to MRCA	;	~
-  $H = \frac{4N_e\mu}{1+4N_e\mu} \approx 4N_e\mu  $	;	~	;	$\E[T_k] = \frac{2 N_e}{ {k \choose 2} },~~~~\E[T_{MRCA}] =4N_e(1-1/n) $	;	~
+  $H = \frac{4N_e\mu}{1+4N_e\mu} \approx 4N_e\mu  $	;	~	;	"$\E[T_k] = \frac{2 N_e}{ {k \choose 2} },~~~~\E[T_{MRCA}] =4N_e(1-1/n) $"	;	~
 	;		;		;	
-Pairwise diversity \& number of segregating sites	;	~	;	Expectation of $\dNdS$	;	\eqref{eqn:dNDS_C_B}
-$\E[\pi] = 4N_e\mu ,~~~~ \E[S] = 4N_e\mu \sum_{k=n}^2 \frac{1}{k-1} $	;		;	$\dNdS = (1-C-B) +  2 N B f_B $	;	~
+Number pairwise diffs. \& segregating sites	;	~	;	Expectation of $\dNdS$	;	\eqref{eqn:dNDS_C_B}
+"$\E[\pi] = 4N_e\mu ,~~~~ \E[S] = 4N_e\mu \sum_{k=n}^2 \frac{1}{k-1} $"	;		;	$\dNdS = (1-C-B) +  2 N B f_B $	;	~
 	;		;		;	
-Model-based $\fst$ expectations.	;	~	;	Phenotypic covariance between relatives ($i$ \& $j$)	;	~
-  $\fst = \frac{ T}{ T + 4N_e }, ~~~~F_{IM} = \frac{1}{1 + 4N_I m} $	;	\eqref{eqn:FST_split},  \eqref{eqn:FIM}	;	$Cov(X_1,X_2)  = 2F_{1,2} V_A + r_2 V_D~~~\textrm{if } V_D>0$	;	~
+Model-based $\fst$ expectations.	;	~	;	Phenotypic covar. between relatives ($i$ \& $j$)	;	~
+"  $\fst = \frac{ T}{ T + 4N_e }, ~~~~F_{IM} = \frac{1}{1 + 4N_I m} $"	;	"\eqref{eqn:FST_split},  \eqref{eqn:FIM}"	;	"$Cov(X_1,X_2)  = 2F_{1,2} V_A + r_2 V_D~~~\textrm{if } V_D>0$"	;	~
 	;		;		;	
-Cross trait ($1$ \& $2$) covariance between relatives	;	~	;	Breeder's equation	;	~
-$Cov(X_{1,i},X_{2,j}) = 2 F_{i,j} V_{A,1,2}$	;	~	;	$R = h^2 S = V_A \beta = \frac{V_A}{\wbar} \frac{\partial \wbar}{\partial \bar{x}} $	;	\eqref{breeders_eqn}, \eqref{eqn:R_beta}, \eqref{eqn:pheno_fitness_landscape} 
+Cross trait ($1$ \& $2$) covar. between relatives	;	~	;	Breeder's equation	;	~
+"$Cov(X_{1,i},X_{2,j}) = 2 F_{i,j} V_{A,1,2}$"	;	~	;	$R = h^2 S = V_A \beta = \frac{V_A}{\wbar} \frac{\partial \wbar}{\partial \bar{x}} $	;	"\eqref{breeders_eqn}, \eqref{eqn:R_beta}, \eqref{eqn:pheno_fitness_landscape} "
 	;		;		;	
 Multi-variate breeders equation	;	~	;	Hamilton's Rule 	;	~
-$\bf{R} = \bf{G} \bf{V}^{-1} \bf{S} = \bf{G} \boldsymbol{ \beta} $	;	\ref{eqn:MV_breeders_eqn}	;	$ 2 F_{i,j} B > C$	;	\eqref{eqn:Hamiltons_rule}
+$\bf{R} = \bf{G} \bf{V}^{-1} \bf{S} = \bf{G} \boldsymbol{ \beta} $	;	\ref{eqn:MV_breeders_eqn}	;	"$ 2 F_{i,j} B > C$"	;	\eqref{eqn:Hamiltons_rule}
+	;		;		;	
+Frequency next generation (haploid \& diploid). 	;		;	Frequency change	;	
+"$p_{t+1}  = \frac{w_1 }{\wbar} p_t , ~p_{t+1} = \frac{w_{11} p_t^2 + w_{12} p_tq_t}{\wbar} $"	;	"\eqref{eq:deltap_haploid}, \eqref{deltap_dip1}"	;	$\Delta p_t=\frac{ (\wbar_1-\wbar_2)}{\wbar} p_t q_t =\frac{1}{2} \frac{p_tq_t}{\wbar} \frac{d \wbar}{dp}$	;	" \eqref{deltap_dip2}, \eqref{deltap_dip3}"
+	;		;		;	
+Haploid cumulative change {\tiny (use $\nicefrac{s}{2}$ for diploid case)}	;		;	Heterozygote advantage equilibrium	;	
+"$p_{\tau} \approx \frac{p_0}{p_0 + q_0 e^{-s\tau}},~~~\tau \approx \frac{1}{s} \log \left(\frac{p_{\tau} q_0}{q_{\tau} p_0}\right) $"	;	\eqref{eq:haploid_logistic growth}~\eqref{eq:estTauExplSimpl}	;	$p_e = \frac{s_2}{s_1+s_2}$	;	\eqref{eqn:het_ad_eq}
+	;		;		;	
+Diploid mutation-selection equilibrium	;		;	Migration-selection equil. \& cline width.	;	
+$q_e = \sqrt{\frac{\mu}{s}}~~(\textrm{if }h=0)$	;	"\eqref{eqn:mut_sel_bal}, \eqref{eqn:recess_mut_sel_bal} "	;	"$q_{e,1} = \frac{m}{hs},~~~~0.6 \sigma/\sqrt{s} $"	;	"\eqref{eqn:mig_sel_eq},~\eqref{eqn:cline_width}"
+	;		;		;	
+Selected prob. fixation  (haploid \& diploid)	;		;	Prob. fixation for weakly selected alleles ($h=\nicefrac{1}{2}$)	;	
+"$p_F \left(\nicefrac{1}{2N} \right) = 2s, ~~~P_F \left(\nicefrac{1}{2N} \right) \approx 2 h s,\text{ \small , ~$Ns \gg 1$} $"	;	 \eqref{eqn:prob_fix_strong} \& \eqref{eqn:diploid_escape}	;	"$P_F \left(\frac{1}{2N} \right) = \frac{1-e^{-s }}{1-e^{-2Ns}} \text{ {\small ,~ $s<0$ for deleterious allele.} } $"	;	\eqref{eqn:new_mut_prob_fixed}