\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
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
course with union comes intersection: $\intersect{A}{B}$,
- $\intersectthree{A}{B}{C}$.
+ $\intersectthree{A}{B}{C}$. We can also take an arbitrary
+ (indexed) union and intersections of things, like
+ $\unionmany{k=1}{\infty}{A_{k}}$ or
+ $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those
+ is that they do the right thing in a display equation:
+ %
+ \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}
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}
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
The set of all bounded linear operators from $V$ to $W$ is
$\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
instead.
+
+ 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, 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
+ \end{align*}
+ %
+ Its form should also survive in different font sizes...
+ \Large
+ \begin{align*}
+ Z = \directsumperp{V}{W}\\
+ \oplus \oplusperp \oplus \oplusperp
+ \end{align*}
+ \Huge
+ \begin{align*}
+ Z = \directsumperp{V}{W}\\
+ \oplus \oplusperp \oplus \oplusperp
+ \end{align*}
+ \normalsize
\end{section}
\begin{section}{Listing}
fox
\end{theorem}
+ \begin{exercise}
+ jumps
+ \end{exercise}
+
\begin{definition}
quod
\end{definition}
fox
\end{theorem*}
+ \begin{exercise*}
+ jumps
+ \end{exercise*}
+
\begin{definition*}
quod
\end{definition*}
The interior of a set $X$ is $\interior{X}$. Its closure is
$\closure{X}$ and its boundary is $\boundary{X}$.
\end{section}
-
+
\end{document}
projects / mjotex.git / blobdiff