X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=1d79079ee0ba4c1f5fe09422a8e0f363da81e139;hb=7dfe6fdee9d819258393544917c871f6b85a3eb8;hp=f922d655715645da0e00309d1fb7973f7f4b3c51;hpb=4febc3ad82bc8ac73e660c484e105835feb1ed84;p=mjotex.git

diff --git a/examples.tex b/examples.tex
index f922d65..1d79079 100644
--- a/examples.tex
+++ b/examples.tex
@@ -13,7 +13,7 @@
 % We have to load this after hyperref, so that links work, but before
 % mjotex so that mjotex knows to define its glossary entries.
 \usepackage[nonumberlist]{glossaries}
-\makenoidxglossaries
+\makenoidxglossaries{}
 
 % If you want an index, we can do that too. You'll need to define
 % the "INDICES" variable in the GNUmakefile, though.
@@ -26,16 +26,19 @@
 \begin{document}
 
   \begin{section}{Algebra}
-    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 $\Frac{R}$. 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
+    If $R$ is a commutative ring\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
+    $\Frac{R}$. 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 $\set{x,y,z}$.
+
+    If $R$ has a multiplicative identity (that is, a unit) element,
+    then that element is denoted by $\unit{R}$.
   \end{section}
 
   \begin{section}{Algorithm}
@@ -51,7 +54,7 @@
 
         \While{$M$ is not sorted}
           \State{Rearrange $M$ randomly}
-        \EndWhile
+        \EndWhile{}
 
         \Return{$M$}
       \end{algorithmic}
@@ -72,10 +75,10 @@
   \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
-    set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
-    up to seven, for now:
+    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:
     %
     \begin{itemize}
       \begin{item}
@@ -98,7 +101,9 @@
       \end{item}
     \end{itemize}
     %
-    The factorial of the number $10$ is $\factorial{10}$.
+    The factorial of the number $10$ is $\factorial{10}$, and the
+    least common multiple of $4$ and $6$ is $\lcm{\set{4,6}} =
+    12$.
 
     The direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three
     things, $\directsumthree{U}{V}{W}$. How about more things? Like
@@ -170,7 +175,8 @@
     their tensor product is $\tp{x}{y}$. The Kronecker product of
     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
-    $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
+    $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
+    set of its eigenvalues---is $\spectrum{L}$. Another matrix-specific
     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
     $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
     $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
@@ -205,7 +211,7 @@
       \oplus \oplusperp \oplus \oplusperp
     \end{align*}
     %
-    Its form should also survive in different font sizes...
+    Its form should also survive in different font sizes\ldots
     \Large
     \begin{align*}
       Z = \directsumperp{V}{W}\\
@@ -290,8 +296,9 @@
   \end{section}
 
   \begin{section}{Set theory}
-    The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
-    = 3$, and its powerset is $\powerset{X}$.
+    Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
+    cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
+    3$, and its powerset is $\powerset{X}$.
 
     We also have a few basic set operations, for example the union of
     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
@@ -392,8 +399,8 @@
 
   \setlength{\glslistdottedwidth}{.3\linewidth}
   \setglossarystyle{listdotted}
-  \glsaddall
-  \printnoidxglossaries
+  \glsaddall{}
+  \printnoidxglossaries{}
 
   \bibliographystyle{mjo}
   \bibliography{local-references}