X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=cd615de16cbab3fc8381aac6aeb4039d945f5601;hb=123eb0365b7a38288f0cd80a0c70b203b8ff33fd;hp=cb7d28a1f41d3a17c02635b370c3cb0fbe36a54f;hpb=37b342a8a6fada8b0fbe828bdf97f346b538f5f4;p=mjotex.git

diff --git a/examples.tex b/examples.tex
index cb7d28a..cd615de 100644
--- a/examples.tex
+++ b/examples.tex
@@ -29,7 +29,13 @@
     If $R$ is a \index{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 $\Frac{R}$.
+    domain, then its fraction field is $\Frac{R}$. If $x,y,z \in R$,
+    then $\ideal{\set{x,y,z}}$ is the ideal generated by
+    $\set{x,y,z}$, which is defined to be the smallest ideal in $R$
+    containing that set. Likewise, if we are in an algebra
+    $\mathcal{A}$ and if $x,y,z \in \mathcal{A}$, then
+    $\alg{\set{x,y,z}}$ is the smallest subalgebra of $\mathcal{A}$
+    containing the set $\set{x,y,z}$.
   \end{section}
 
   \begin{section}{Algorithm}
@@ -106,6 +112,7 @@
     \begin{equation*}
       \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
     \end{equation*}
+    %
     Here are a few common tuple spaces that should not have a
     superscript when that superscript would be one: $\Nn[1]$,
     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
@@ -124,8 +131,9 @@
     \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 $\Rn[1]$,
+    %
+    The powerset of $X$ displays nicely, as $\powerset{X}$. 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 },\\
@@ -165,7 +173,14 @@
   \end{section}
 
   \begin{section}{Font}
-    We can write things like Carathéodory and Güler and $\mathbb{R}$.
+    We can write things like Carathéodory and Güler and
+    $\mathbb{R}$. The PostScript Zapf Chancery font is also available
+    in both upper- and lower-case:
+    %
+    \begin{itemize}
+      \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
+      \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
+    \end{itemize}
   \end{section}
 
   \begin{section}{Linear algebra}
@@ -176,7 +191,8 @@
     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
     $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
-    $\pseudoinverse{L}$.
+    $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
+    $\rank{L}$.
 
     The span of a set $X$ is $\spanof{X}$, and its codimension is
     $\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The