X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=cdf2359d4fe22b68738d76ccca052d8b8d719b89;hb=e6bb4f9ae2d3785b331388703b8793e0409d30af;hp=26db4b1a03c81a218f354a6eaa23c038de592352;hpb=2719020929ee56f190a9e0e91083fb70ee086c9c;p=mjotex.git

diff --git a/examples.tex b/examples.tex
index 26db4b1..cdf2359 100644
--- a/examples.tex
+++ b/examples.tex
@@ -5,6 +5,13 @@
 
 \begin{document}
 
+  \begin{section}{Algebra}
+    If $R$ is a commutative ring, then $\polyring{R}{X,Y,Z}$ is a
+    multivariate polynomial ring with indeterminates $X$, $Y$, and
+    $Z$, and coefficients in $R$. If $R$ is a moreover an integral
+    domain, then its fraction field is $\Frac{R}$.
+  \end{section}
+
   \begin{section}{Algorithm}
     An example of an algorithm (bogosort) environment.
 
@@ -32,6 +39,11 @@
     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
   \end{section}
 
+  \begin{section}{Calculus}
+    The gradient of $f : \Rn \rightarrow \Rn[1]$ is $\gradient{f} :
+    \Rn \rightarrow \Rn$.
+  \end{section}
+
   \begin{section}{Common}
     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
@@ -69,6 +81,20 @@
     \begin{equation*}
       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
     \end{equation*}
+
+    Finally, we have the four standard types of intervals in $\Rn[1]$,
+    %
+    \begin{align*}
+      \intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
+      \intervaloc{a}{b} &= \setc{ x \in \Rn[1]}{ a < x \le b },\\
+      \intervalco{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x < b }, \text{ and }\\
+      \intervalcc{a}{b} &= \setc{ x \in \Rn[1]}{ a \le x \le b }.
+    \end{align*}
+  \end{section}
+
+  \begin{section}{Complex}
+    We sometimes want to conjugate complex numbers like
+    $\compconj{a+bi} = a - bi$.
   \end{section}
 
   \begin{section}{Cone}
@@ -85,7 +111,14 @@
     The conic hull of a set $X$ is $\cone{X}$; its affine hull is
     $\aff{X}$, and its convex hull is $\conv{X}$. If $K$ is a cone,
     then its lineality space is $\linspace{K}$, its lineality is
-    $\lin{K}$, and its extreme directions are $\Ext{K}$.
+    $\lin{K}$, and its extreme directions are $\Ext{K}$. The fact that
+    $F$ is a face of $K$ is denoted by $F \faceof K$; if $F$ is a
+    proper face, then we write $F \properfaceof K$.
+  \end{section}
+
+  \begin{section}{Euclidean Jordan algebras}
+    The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
+    is $\jp{x}{y}$.
   \end{section}
 
   \begin{section}{Font}
@@ -140,7 +173,7 @@
   \end{section}
 
   \begin{section}{Listing}
-    Here's an interactive sage prompt:
+    Here's an interactive SageMath prompt:
 
     \begin{tcblisting}{listing only,
                        colback=codebg,
@@ -153,6 +186,14 @@
     [0 0], [0 0], [1 0], [0 1]
     ]
     \end{tcblisting}
+
+    However, the smart way to display a SageMath listing is to load it
+    from an external file (under the ``listings'' subdirectory):
+
+    \sagelisting{example}
+
+    Keeping the listings in separate files makes it easy for the build
+    system to test them.
   \end{section}
 
   \begin{section}{Miscellaneous}
@@ -223,6 +264,10 @@
       fox
     \end{theorem}
 
+    \begin{exercise}
+      jumps
+    \end{exercise}
+
     \begin{definition}
       quod
     \end{definition}
@@ -253,6 +298,10 @@
       fox
     \end{theorem*}
 
+    \begin{exercise*}
+      jumps
+    \end{exercise*}
+
     \begin{definition*}
       quod
     \end{definition*}