X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=examples.tex;h=6e1c5007abf16c19d48f8eadb7c1a03b85a49cab;hp=fbbe8608e002736fef5d43c3e0b49da9e9a2179c;hb=4fd0c97de0370110d6fecab412ec1b5b831611e8;hpb=7a60af0c2fa38f05aecd84b530d3c5b87eaf3de7

diff --git a/examples.tex b/examples.tex
index fbbe860..6e1c500 100644
--- a/examples.tex
+++ b/examples.tex
@@ -28,20 +28,34 @@
   \begin{section}{Arrow}
     The identity operator on $V$ is $\identity{V}$. The composition of
     $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
-    $\inverse{f}$.
+    $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
+    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
     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
-    of things $\pair{1}{2}$ or a triple of them
-    $\triple{1}{2}{3}$. The Cartesian product of two sets $A$ and $B$
-    is $\cartprod{A}{B}$; if we take the product with $C$ as well,
-    then we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$
-    and $W$ is $\directsum{V}{W}$ and the factorial of the number $10$
-    is $\factorial{10}$.
-
+    of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
+    and the factorial of the number $10$ is $\factorial{10}$.
+
+    The Cartesian product of two sets $A$ and $B$ is
+    $\cartprod{A}{B}$; if we take the product with $C$ as well, then
+    we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
+    is $\directsum{V}{W}$. Or three things,
+    $\directsumthree{U}{V}{W}$. How about more things? Like
+    $\directsummany{k=1}{\infty}{V_{k}} \ne
+    \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
+    cartesian products adapt nicely to display equations:
+    %
+    \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
@@ -60,7 +74,20 @@
     \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}
@@ -77,7 +104,9 @@
     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}{Font}
@@ -90,7 +119,9 @@
     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 $\trace{L}$.
+    $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
+    concept is the Moore-Penrose pseudoinverse of $L$, denoted by
+    $\pseudoinverse{L}$.
 
     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
@@ -107,23 +138,24 @@
     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
     writing $\directsumperp{V}{W}$. That operator should survive a
-    display equation, too:
+    display equation, too, and the weight of the circle should match
+    that of the usual direct sum operator.
     %
     \begin{align*}
       Z = \directsumperp{V}{W}\\
-      \oplus\oplusperp\oplus\oplusperp
+      \oplus \oplusperp \oplus \oplusperp
     \end{align*}
     %
     Its form should also survive in different font sizes...
     \Large
     \begin{align*}
       Z = \directsumperp{V}{W}\\
-      \oplus\oplusperp\oplus\oplusperp
+      \oplus \oplusperp \oplus \oplusperp
     \end{align*}
     \Huge
     \begin{align*}
       Z = \directsumperp{V}{W}\\
-      \oplus\oplusperp\oplus\oplusperp
+      \oplus \oplusperp \oplus \oplusperp
     \end{align*}
     \normalsize
   \end{section}
@@ -212,6 +244,10 @@
       fox
     \end{theorem}
 
+    \begin{exercise}
+      jumps
+    \end{exercise}
+
     \begin{definition}
       quod
     \end{definition}
@@ -242,6 +278,10 @@
       fox
     \end{theorem*}
 
+    \begin{exercise*}
+      jumps
+    \end{exercise*}
+
     \begin{definition*}
       quod
     \end{definition*}
@@ -259,5 +299,5 @@
     The interior of a set $X$ is $\interior{X}$. Its closure is
     $\closure{X}$ and its boundary is $\boundary{X}$.
   \end{section}
-  
+
 \end{document}