X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=examples.tex;h=2d05431e2ceb2fcdcdf3314980f3348929a69ea8;hp=babc886abe26b31ef223cb783e2fa43d4d848d79;hb=HEAD;hpb=cab35094298c00d10035e926b64ff69df33393a5

diff --git a/examples.tex b/examples.tex
index babc886..383cef2 100644
--- a/examples.tex
+++ b/examples.tex
@@ -39,7 +39,11 @@
 
     If $R$ has a multiplicative identity (that is, a unit) element,
     then that element is denoted by $\unit{R}$. Its additive identity
-    element is $\zero{R}$.
+    element is $\zero{R}$. The stabilizer (or isotropy)
+    subgroup of $G$ that fixes $x$ is $\Stab{G}{x}$.
+
+    If $I$ is an ideal, then $\variety{I}$ is the variety that
+    corresponds to it.
   \end{section}
 
   \begin{section}{Algorithm}
@@ -57,7 +61,7 @@
           \State{Rearrange $M$ randomly}
         \EndWhile{}
 
-        \Return{$M$}
+        \State{\Return{$M$}}
       \end{algorithmic}
     \end{algorithm}
   \end{section}
@@ -79,7 +83,8 @@
     The function $f$ applied to $x$ is $f\of{x}$, and the restriction
     of $f$ to a subset $X$ of its domain is $\restrict{f}{X}$. We can
     group terms like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c -
-        d}}$. The tuples go up to seven, for now:
+        d}}$. The tuples go up to seven, for now, and then we give up
+    and use the general construct:
     %
     \begin{itemize}
       \begin{item}
@@ -100,6 +105,9 @@
       \begin{item}
         Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
       \end{item}
+      \begin{item}
+        Tuple: $\tuple{1,2,\ldots,8675309}$.
+      \end{item}
     \end{itemize}
     %
     The factorial of the number $10$ is $\factorial{10}$, and the
@@ -139,12 +147,12 @@
 
   \begin{section}{Cone}
     The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
-    are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.  If cones
-    $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
-    $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
-    $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
-    \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
-    \ltcone_{K} y$ with respect to a cone $K$.
+    are $\Rnplus$, $\Rnplusplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.
+    If cones $K_{1}$ and $K_{2}$ are given, we can define
+    $\posops{K_{1}}$, $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$,
+    $\Zof{K_{1}}$, $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can
+    also define $x \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K}
+    y$, and $x \ltcone_{K} y$ with respect to a cone $K$.
   \end{section}
 
   \begin{section}{Convex}
@@ -158,7 +166,11 @@
 
   \begin{section}{Euclidean Jordan algebras}
     The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
-    is $\jp{x}{y}$.
+    $V$ is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is
+    $\JAut{V}$. Two popular operators in an EJA are its quadratic
+    representation and ``left multiplication by'' operator. For a
+    given $x$, they are, respectively, $\quadrepr{x}$ and
+    $\leftmult{x}$.
   \end{section}
 
   \begin{section}{Font}
@@ -172,6 +184,11 @@
     \end{itemize}
   \end{section}
 
+  \begin{section}{Hurwitz}
+    Here lies the Hurwitz algebras, like the quaternions
+    $\quaternions$ and octonions $\octonions$.
+  \end{section}
+
   \begin{section}{Linear algebra}
     The absolute value of $x$ is $\abs{x}$, or its norm is
     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
@@ -200,7 +217,9 @@
 
     The set of all bounded linear operators from $V$ to $W$ is
     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
-    instead.
+    instead. If you have matrices instead, then the general linear
+    group of $n$-by-$n$ matrices with entries in $\mathbb{F}$ is
+    $\GL{n}{\mathbb{F}}$.
 
     If you want to solve a system of equations, try Cramer's
     rule~\cite{ehrenborg}. Or at least the reduced row-echelon form of