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-rw-r--r--info/undergradmath/undergradmath.tex260
1 files changed, 137 insertions, 123 deletions
diff --git a/info/undergradmath/undergradmath.tex b/info/undergradmath/undergradmath.tex
index 5d95b59e04..f041bab041 100644
--- a/info/undergradmath/undergradmath.tex
+++ b/info/undergradmath/undergradmath.tex
@@ -1,6 +1,7 @@
\RequirePackage{mmap} % make PDF copy and paste-able
\documentclass[twocolumn]{article}
\usepackage{amsmath, amsthm, amssymb}
+\usepackage{mathrsfs}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
% \usepackage{lmodern} % from http://tex.stackexchange.com/a/115089/121234
@@ -23,7 +24,7 @@
\lstdefinestyle{inline}{basicstyle=\ttfamily}
\usepackage[dvipsnames]{xcolor}
- \definecolor{headcolor}{HTML}{E34234} % vermillion
+ \definecolor{headcolor}{HTML}{004225} % British racing green {E34234} % vermillion
\usepackage{titlesec}
% \titleformat{ command }[ shape ]{ format }{ label }{ sep }{ before-code }[ after-code ]
% \titlespacing*{ command }{ left }{ before-sep }{ after-sep }[ right-sep ]
@@ -42,16 +43,15 @@
\section{Rule One}
Any mathematics at all,
-even a single character, goes in a mathematical setting.
-Thus, for ``the value of \( x \) is \( 7 \)\,'' enter
-`\lstinline[style=inline]!the value of \( x \) is \( 7 \)!'.
+even a single character, gets a mathematical setting.
+Thus, for ``the value of $x$ is $7$\,'' enter
+\lstinline[style=inline]!the value of $x$ is $7$!.
\section{Template}
Your document should contain at least this.
\begin{lstlisting}
\documentclass{article}
-\usepackage{amsmath, amssymb, amsthm}
-\usepackage[utf8]{inputenc}
+\usepackage{mathtools,amssymb,amsthm} % imports amsmath
\begin{document}
--document body here--
@@ -62,19 +62,19 @@ Your document should contain at least this.
\section{Common constructs}
\begin{center}
\begin{tabular}{@{} *{1}{l@{\hspace{1.5em}}} l @{}}
- \(x^2\)\quad\lstinline[style=inline]!x^2!
- &\(\sqrt{2}\), \(\sqrt[n]{3}\)\quad\lstinline[style=inline]!\sqrt{2}!,
+ $x^2$\quad\lstinline[style=inline]!x^2!
+ &$\sqrt{2}$, $\sqrt[n]{3}$\quad\lstinline[style=inline]!\sqrt{2}!,
\lstinline[style=inline]!\sqrt[n]{3}! \\
- \(x_{i,j}\)\quad\lstinline[style=inline]!x_{i,j}!
- &\(\frac{2}{3}\), \(2/3\)\quad\lstinline[style=inline]!\frac{2}{3}!,
+ $x_{i,j}$\quad\lstinline[style=inline]!x_{i,j}!
+ &$\frac{2}{3}$, $2/3$\quad\lstinline[style=inline]!\frac{2}{3}!,
\lstinline[style=inline]!2/3!
\end{tabular}
\end{center}
\section{Calligraphic letters}
-Use as \lstinline[style=inline]!\( \mathcal{A} \)!.
+Use as in \lstinline[style=inline]!$\mathcal{A}$!.
\begin{center}
- \( \mathcal{A}\mathcal{B}\mathcal{C}
+ $ \mathcal{A}\mathcal{B}\mathcal{C}
\mathcal{D}\mathcal{E}\mathcal{F}
\mathcal{G}\mathcal{H}\mathcal{I}
\mathcal{J}\mathcal{K}\mathcal{L}
@@ -82,9 +82,13 @@ Use as \lstinline[style=inline]!\( \mathcal{A} \)!.
\mathcal{P}\mathcal{Q}\mathcal{R}
\mathcal{S}\mathcal{T}\mathcal{U}
\mathcal{V}\mathcal{W}\mathcal{X}
- \mathcal{Y}\mathcal{Z} \)
+ \mathcal{Y}\mathcal{Z} $
\end{center}
-
+Get script letters, such as $\mathscr{P}$
+from \lstinline[style=inline]!$\mathscr{P}$!,
+by putting
+\lstinline[style=inline]!\usepackage{mathrsfs}!
+in the preamble.
\section{Greek}
\begin{center}
@@ -137,19 +141,19 @@ Use as \lstinline[style=inline]!\( \mathcal{A} \)!.
\begin{center}
\begin{tabular}{@{}*{2}{l@{\hspace{1em}}}l@{}}
$\cup$\quad \lstinline[style=inline]!\cup!
- &\(\mathbb{R}\)\quad\lstinline[style=inline]!\mathbb{R}!
+ &$\mathbb{R}$\quad\lstinline[style=inline]!\mathbb{R}!
&$\forall$\quad \lstinline[style=inline]!\forall! \\
$\cap$\quad \lstinline[style=inline]!\cap!
- &\(\mathbb{Z}\)\quad\lstinline[style=inline]!\mathbb{Z}!
+ &$\mathbb{Z}$\quad\lstinline[style=inline]!\mathbb{Z}!
&$\exists$\quad \lstinline[style=inline]!\exists! \\
- \(\subset\)\quad\lstinline[style=inline]!\subset!
- &\(\mathbb{Q}\)\quad\lstinline[style=inline]!\mathbb{Q}!
+ $\subset$\quad\lstinline[style=inline]!\subset!
+ &$\mathbb{Q}$\quad\lstinline[style=inline]!\mathbb{Q}!
&$\neg$\quad \lstinline[style=inline]!\neg! \\
$\subseteq$\quad \lstinline[style=inline]!\subseteq!
- &\(\mathbb{N}\)\quad\lstinline[style=inline]!\mathbb{N}!
+ &$\mathbb{N}$\quad\lstinline[style=inline]!\mathbb{N}!
&$\vee$\quad \lstinline[style=inline]!\vee! \\
$\supset$\quad \lstinline[style=inline]!\supset!
- &\(\mathbb{C}\)\quad\lstinline[style=inline]!\mathbb{C}!
+ &$\mathbb{C}$\quad\lstinline[style=inline]!\mathbb{C}!
&$\wedge$\quad \lstinline[style=inline]!\wedge! \\
$\supseteq$\quad \lstinline[style=inline]!\supseteq!
&$\varnothing$\quad \lstinline[style=inline]!\varnothing!
@@ -157,24 +161,18 @@ Use as \lstinline[style=inline]!\( \mathcal{A} \)!.
$\in$\quad \lstinline[style=inline]!\in!
&$\emptyset$\quad \lstinline[style=inline]!\emptyset!
&$\models$\quad \lstinline[style=inline]!\models! \\
- $\ni$\quad \lstinline[style=inline]!\ni!
+ $\notin$\quad \lstinline[style=inline]!\notin!
&$\aleph$\quad \lstinline[style=inline]!\aleph!
- &$\Rightarrow$\quad \lstinline[style=inline]!\Rightarrow! \\
- $\notin$\quad \lstinline[style=inline]!\notin!
- &$\setminus$\quad \lstinline[style=inline]!\setminus!
- &$\nRightarrow$\quad \lstinline[style=inline]!\nRightarrow! \\
- $\not\in$\quad \lstinline[style=inline]!\not\in!
- &$\equiv$\quad \lstinline[style=inline]!\equiv!
- & \\
+ &$\setminus$\quad \lstinline[style=inline]!\setminus! \\
+ % $\ni$\quad \lstinline[style=inline]!\ni!
\end{tabular}
\end{center}
-Negate an operator, as in \(\not\subset\),
+Negate an operator, as in $\not\subset$,
with \lstinline[style=inline]!\not\subset!.
Get the set complement
-$A^{\mathsf{c}}$ with \lstinline[style=inline]!A^{\mathsf{c}}!,
-get $A^{\complement}$ with \lstinline[style=inline]!A^{\complement}!,
-or
-get $\bar{A}$ with \lstinline[style=inline]!\bar{A}!.
+$A^{\mathsf{c}}$ with \lstinline[style=inline]!A^{\mathsf{c}}!
+(or $A^{\complement}$ with \lstinline[style=inline]!A^{\complement}!,
+or $\overline{A}$ with \lstinline[style=inline]!\overline{A}!).
@@ -199,6 +197,7 @@ If the decorated letter is $i$ or~$j$ then some decorations need
as in \lstinline[style=inline]!\vec{\imath}!.
Some authors use boldface for
vectors: \lstinline[style=inline]!\boldsymbol{x}!.
+% Mention \bm?
Entering \lstinline[style=inline]!\overline{x+y}! produces
$\overline{x+y}$,
@@ -213,8 +212,8 @@ Comment on an expression as here
\section{Dots}
Use low dots in a list
$\{0,1,2,\,\ldots\}$, entered as \lstinline[style=inline]!\{0,1,2,\,\ldots\}!.
-(If you use \lstinline[style=inline]!\ldots! in plain text as
-\lstinline[style=inline]!London, Paris, \ldots{}\,.! note
+(If you use \lstinline[style=inline]!\ldots! in plain text as with
+\lstinline[style=inline]!London, Paris, \ldots{}\,.! then note
the thinspace \lstinline[style=inline]!\,! before the period.)
Use centered dots in a sum or product
$1+\cdots+100$, entered as \lstinline[style=inline]!1+\cdots+100!.
@@ -315,35 +314,13 @@ These get the same treatment.
&$\checkmark$\quad \lstinline[style=inline]!\checkmark! \\
\end{tabular}
\end{center}
-Enter \lstinline[style=inline]!a|b! for the divides relation \( a|b \).
-Use \lstinline[style=inline]!\mid! as in
-\lstinline[style=inline]!\{a\in S\mid\text{\(a=0\) or \(a\) is odd}\}!
-for the set \( \{a\in S \mid\text{\(a=0\) or \(a\) is odd}\} \).
-
-\section{Variable-sized operators}
-The summation
-\(\sum_{j=0}^3j^2\)\quad\lstinline[style=inline]!\sum_{j=0}^3 j^2!
-and the integral $\int_{x=0}^3x^2\,dx$
-\lstinline[style=inline]!\int_{x=0}^3 x^2\,dx!
-expand when displayed.
-\begin{equation*}
- \sum_{j=0}^3j^2
- \qquad
- \int_{x=0}^3x^2\,dx
-\end{equation*}
-These do the same.
-\begin{center}
- \begin{tabular}{@{}*{2}{l@{\hspace{.35in}}}l@{}}
- $\int$\quad\lstinline[style=inline]!\int!
- &$\iiint$\quad\lstinline[style=inline]!\iiint!
- &$\bigcup$\quad\lstinline[style=inline]!\bigcup! \\
- $\iint$\quad\lstinline[style=inline]!\iint!
- &$\oint$\quad\lstinline[style=inline]!\oint!
- &$\bigcap$\quad\lstinline[style=inline]!\bigcap! \\
- % &$\bigvee$\quad\lstinline[style=inline]!\bigvee!
- % &$\bigwedge$\quad\lstinline[style=inline]!\bigwedge! \\
- \end{tabular}
-\end{center}
+Use \lstinline[style=inline]!a\mid b! for the divides relation,
+\mbox{$a\mid b$},
+and \lstinline[style=inline]!a\nmid b! for the negation, \mbox{$a\nmid b$}.
+Also use \lstinline[style=inline]!\mid! to get
+set builder notation \mbox{$\{a\in S \mid\text{$a$ is odd}\}$},
+with
+\lstinline[style=inline]!\{a\in S\mid\text{$a$ is odd}\}!.
\section{Arrows}
\begin{center}
@@ -369,6 +346,31 @@ The right arrows in the first column have matching left arrows,
such as \lstinline[style=inline]!\nleftarrow!,
and there are some other matches for down arrows, etc.
+\section{Variable-sized operators}
+The summation
+$\sum_{j=0}^3j^2$\quad\lstinline[style=inline]!\sum_{j=0}^3 j^2!
+and the integral $\int_{x=0}^3x^2\,dx$
+\lstinline[style=inline]!\int_{x=0}^3 x^2\,dx!
+expand when displayed.
+\begin{equation*}
+ \sum_{j=0}^3j^2
+ \qquad
+ \int_{x=0}^3x^2\,dx
+\end{equation*}
+These do the same.
+\begin{center}
+ \begin{tabular}{@{}*{2}{l@{\hspace{.35in}}}l@{}}
+ $\int$\quad\lstinline[style=inline]!\int!
+ &$\iiint$\quad\lstinline[style=inline]!\iiint!
+ &$\bigcup$\quad\lstinline[style=inline]!\bigcup! \\
+ $\iint$\quad\lstinline[style=inline]!\iint!
+ &$\oint$\quad\lstinline[style=inline]!\oint!
+ &$\bigcap$\quad\lstinline[style=inline]!\bigcap! \\
+ % &$\bigvee$\quad\lstinline[style=inline]!\bigvee!
+ % &$\bigwedge$\quad\lstinline[style=inline]!\bigwedge! \\
+ \end{tabular}
+\end{center}
+
\section{Fences}
\begin{center}
\begin{tabular}{@{}*{2}{l@{\hspace{1.5em}}}l@{}}
@@ -382,8 +384,18 @@ and there are some other matches for down arrows, etc.
&$\lceil\;\rceil$\quad\lstinline[style=inline]!\lceil\rceil!
\end{tabular}
\end{center}
-They will grow with the enclosed formula
-using \lstinline[style=inline]!\left! and \lstinline[style=inline]!\right!.
+Fix the size with
+\lstinline[style=inline]!\big!,
+\lstinline[style=inline]!\Big!,
+\lstinline[style=inline]!\bigg!, or
+\lstinline[style=inline]!\Bigg!.
+\begin{center}
+ $\displaystyle\Big[\sum_{k=0}^n e^{k^2}\Big]$\quad
+ \lstinline[style=inline]!\Big[\sum_{k=0}^n e^{k^2}\Big]!
+\end{center}
+To have them grow with the enclosed formula,
+use \lstinline[style=inline]!\left! and \lstinline[style=inline]!\right!
+(although sometimes \lstinline[style=inline]!\big!, etc., are necessary).
\begin{center}
$\displaystyle\left\langle i,2^{2^i}\right\rangle$\quad
\lstinline[style=inline]!\left\langle i,2^{2^i}\right\rangle!
@@ -392,22 +404,13 @@ Every
\lstinline[style=inline]!\left! must match a
\lstinline[style=inline]!\right!
and they must end on the same line in the output.
-For a one-sided fence put a period
+For a one-sided fence, put a
\lstinline[style=inline]!\left.! or
-\lstinline[style=inline]!\right.! on the other side.%
+\lstinline[style=inline]!\right.! on the other side.
\begin{center}
$\displaystyle\left.\frac{df}{dx}\right|_{x_0}$\quad
\lstinline[style=inline]!\left.\frac{df}{dx}\right|_{x_0}!
\end{center}
-Fix the size with
-\lstinline[style=inline]!\big!,
-\lstinline[style=inline]!\Big!,
-\lstinline[style=inline]!\bigg!, or
-\lstinline[style=inline]!\Bigg!.
-\begin{center}
- $\displaystyle\Big[\sum_{k=0}^n e^{k^2}\Big]$\quad
- \lstinline[style=inline]!\Big[\sum_{k=0}^n e^{k^2}\Big]!
-\end{center}
\section{Arrays, Matrices}
@@ -441,7 +444,7 @@ Definition by cases is an array with two columns.
\begin{center}
$\displaystyle
f_n=\begin{cases}
- a &\text{if \(n=0\)} \\
+ a &\text{if $n=0$} \\
r\cdot f_{n-1} &\text{else}
\end{cases}
$
@@ -450,14 +453,15 @@ Definition by cases is an array with two columns.
\begin{lstlisting}[xleftmargin=-1em]
f_n=
\begin{cases}
- a &\text{if \(n=0\)} \\
+ a &\text{if $n=0$} \\
r\cdot f_{n-1} &\text{else}
\end{cases}
\end{lstlisting}
\end{minipage}
\end{center}
-A matrix is another array variant.
-With this abbreviation you need not specify column alignments.
+A matrix is an array with fences.
+With a \lstinline[style=inline]!pmatrix! environment,
+you need not specify column alignments.
\begin{center}
$\displaystyle
\begin{pmatrix}
@@ -479,39 +483,49 @@ For the determinant use \lstinline[style=inline]!|A|! inline
and \lstinline[style=inline]!vmatrix! in display.
\section{Spacing in mathematics}
-\begin{center}
- \begin{tabular}{@{}*{1}{l@{\hspace{2.25em}}}l@{}}
- \(\rightarrow\,\leftarrow\)\quad\lstinline[style=inline]!\,!
- &\(\rightarrow\quad\leftarrow\)\quad\lstinline[style=inline]!\quad! \\
- \(\rightarrow\:\leftarrow\)\quad\lstinline[style=inline]!\:!
- &\(\rightarrow\qquad\leftarrow\)\quad\lstinline[style=inline]!\qquad! \\
- \(\rightarrow\;\leftarrow\)\quad\lstinline[style=inline]!\;!
- &\(\rightarrow\!\leftarrow\)\quad\lstinline[style=inline]+\!+
- \end{tabular}
-\end{center}
-The left column spaces are in ratio \(3\mathbin{:}4\mathbin{:}5\).
-The last in the right column is a negative space,
-opposite to \lstinline[style=inline]!\,!.
+Improve $\sqrt{2}x$ to $\sqrt{2}\,x$ with a thin space,
+as in \lstinline[style=inline]!\sqrt{2}\,x!.
+Slightly wider are \lstinline[style=inline]!\:!
+and \lstinline[style=inline]!\;!
+(the three are in ratio $3\mathbin{:}4\mathbin{:}5$).
+Get the improvement of $n/\!\log n$ instead of $n/\log n$
+by using a negative thin space,
+as in \lstinline[style=inline]+n/\!\log n+.
+Bigger spaces are: \lstinline[style=inline]!\quad! for
+$\rightarrow\quad\leftarrow$, and \lstinline[style=inline]!\qquad! for
+$\rightarrow\qquad\leftarrow$,
+which are useful between parts of a display.
+% \begin{center}
+% \begin{tabular}{@{}*{1}{l@{\hspace{2.25em}}}l@{}}
+% $\rightarrow\mkern3mu\leftarrow$\quad\lstinline[style=inline]!\,!
+% &$\rightarrow\quad\leftarrow$\quad\lstinline[style=inline]!\quad! \\
+% $\rightarrow\mkern4mu\leftarrow$\quad\lstinline[style=inline]!\:!
+% &$\rightarrow\qquad\leftarrow$\quad\lstinline[style=inline]!\qquad! \\
+% $\rightarrow\mkern5mu\leftarrow$\quad\lstinline[style=inline]!\;!
+% &$\rightarrow\!\leftarrow$\quad\lstinline[style=inline]+\!+
+% \end{tabular}
+% \end{center}
Get arbitrary space as in
-% \(\rightarrow\hspace*{0.5cm}\leftarrow\)
-\lstinline[style=inline]!\hspace{0.5cm}!.
+% $\rightarrow\hspace*{0.5cm}\leftarrow$
+\lstinline[style=inline]!\hspace*{0.5cm}!.
\section{Displayed equations}
-Put equations on a separate line with the
-\lstinline[style=inline]!equation*! environment.
+The
+\lstinline[style=inline]!equation*! environment
+puts an equation on a separate line.
\begin{center}
\vspace*{-\topsep}
\begin{minipage}{0.25\linewidth}\vspace*{-\abovedisplayskip}
\begin{equation*}
- S=k\log W
+ S=k\cdot\lg W
\end{equation*}
\end{minipage}
\quad
\begin{minipage}{0.65\linewidth}
\begin{lstlisting}[xleftmargin=0ex]
\begin{equation*}
- S=k\log W
+ S=k\cdot\lg W
\end{equation*}
\end{lstlisting}
\end{minipage}
@@ -535,7 +549,7 @@ You can break into multiple lines.
\end{lstlisting}
\end{minipage}
\end{center}
-Align using the \lstinline[style=inline]!align*! environment
+Align equations using \lstinline[style=inline]!align*!
\vspace*{-\topsep}
\begin{center}
\begin{minipage}[c]{0.25\linewidth}\vspace*{-\abovedisplayskip}
@@ -554,7 +568,7 @@ Align using the \lstinline[style=inline]!align*! environment
\end{lstlisting}
\end{minipage}
\end{center}
-(you can have an empty left or right side of the alignment).
+(the left or right side of an alignment can be empty).
For each environment, get a numbered version by
dropping the asterisk from the name.
@@ -563,19 +577,19 @@ dropping the asterisk from the name.
The last three here are display style.
\begin{center} \small \def\arraystretch{2}
\vspace*{-\topsep}\vspace*{-0.25cm}
- \begin{tabular}{@{}r@{\hspace{1em}}l@{}}
- \( f\colon\mathbb{R}\to\mathbb{R} \)
- &\lstinline[style=inline]!f\colon\mathbb{R}\to\mathbb{R}! \\
- \( 9.8~\text{m}/\text{s}^2 \)
- &\lstinline[style=inline]!9.8~\text{m}/\text{s}^2! \\
+ \begin{tabular}{@{}l@{}}
+ $ f\colon\mathbb{R}\to\mathbb{R} $
+ \quad\lstinline[style=inline]!f\colon\mathbb{R}\to\mathbb{R}! \\
+ $ 9.8~\text{m}/\text{s}^2 $
+ \quad\lstinline[style=inline]!9.8~\text{m}/\text{s}^2! \\
$\displaystyle
\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
$
- &\lstinline[style=inline]!\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}! \\
+ \quad\lstinline[style=inline]!\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}! \\
$\displaystyle
\int x^2\,dx=x^3/3+C
$
- &\lstinline[style=inline]!\int x^2\,dx=x^3/3+C! \\
+ \quad\lstinline[style=inline]!\int x^2\,dx=x^3/3+C! \\
% $\displaystyle
% \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}
% $
@@ -585,41 +599,41 @@ The last three here are display style.
+\boldsymbol{j}\frac{d}{dy}
+\boldsymbol{k}\frac{d}{dz}
$
- &\lstinline[style=inline]!\nabla=\boldsymbol{i}\frac{d}{dx}+!\,\(\cdots\)\\
+ \quad\lstinline[style=inline]!\nabla=\boldsymbol{i}\frac{d}{dx}+!\,$\cdots$\\
\end{tabular}
\end{center}
\section{Discrete mathematics examples}
There are four modulo forms:
-\(m\bmod n\) is from \lstinline[style=inline]!m\bmod n!,
-and \(a\equiv b\pmod m\) is from \lstinline[style=inline]!a\equiv b\pmod m!,
-and \(a\equiv b\mod m\) is from \lstinline[style=inline]!a\equiv b\mod m!,
-and \(a\equiv b\pod m\) is from \lstinline[style=inline]!a\equiv b\pod m!.
+$m\bmod n$ is from \lstinline[style=inline]!m\bmod n!,
+and $a\equiv b\pmod m$ is from \lstinline[style=inline]!a\equiv b\pmod m!,
+and $a\equiv b\mod m$ is from \lstinline[style=inline]!a\equiv b\mod m!,
+and $a\equiv b\pod m$ is from \lstinline[style=inline]!a\equiv b\pod m!.
For combinations the binomial symbol $\binom{n}{k}$ is from
\lstinline[style=inline]!\binom{n}{k}!.
This resizes to be bigger in a display
(to require the display version use \lstinline[style=inline]!\dbinom{n}{k}!
-and for the inline version use \lstinline[style=inline]!\tbinom{n}{k}!).
+and require the inline version with \lstinline[style=inline]!\tbinom{n}{k}!).
For permutations use
-\( n^{\underline{r}} \) from \lstinline[style=inline]!n^{\underline{r}}!
-(some authors use \( P(n,r) \),
-or \( {}_nP_r \) from \lstinline[style=inline]!{}_nP_r!).
+$ n^{\underline{r}} $ from \lstinline[style=inline]!n^{\underline{r}}!
+(some authors use $ P(n,r) $,
+or $ {}_nP_r $ from \lstinline[style=inline]!{}_nP_r!).
\section{Statistics examples}
\begin{center} \small\def\arraystretch{1.5}
\vspace*{-\topsep}\vspace*{-0.15cm}
- \begin{tabular}{@{}r@{\hspace{1em}}l@{}}
+ \begin{tabular}{@{}l@{}}
$
- \sigma^2=\sqrt{\sum (x_i-\mu)^2/N}
+ \sigma^2=\sqrt{\,\sum (x_i-\mu)^2/N}
$
- &\lstinline[style=inline]!\sigma^2=\sqrt{\sum (x_i-\mu)^2/N}! \\
+ \quad\lstinline[style=inline]!\sigma^2=\sqrt{\,\sum (x_i-\mu)^2/N}! \\
$
E(X)=\mu_X=\sum (x_i-P(x_i))
$
- &\lstinline[style=inline]! E(X)=\mu_X=\sum (x_i-P(x_i))! \\
+ \quad\lstinline[style=inline]! E(X)=\mu_X=\sum (x_i-P(x_i))! \\
\end{tabular}
\end{center}
The probability density of the normal distribution
@@ -644,6 +658,6 @@ and De\TeX ify at \url{detexify.kirelabs.org/classify.html}.
\noindent\parbox{\columnwidth}{\small%
{\color{headcolor}\hrulefill\\
\vspace*{0ex}
-Jim Hef{}feron, Saint Michael's College, VT USA\hfill 2017-Jan-10
+Jim Hef{}feron, Saint Michael's College, VT USA\hfill 2020-Dec-30
}}
\end{document}