containing the set $\set{x,y,z}$.
If $R$ has a multiplicative identity (that is, a unit) element,
- then that element is denoted by $\unit{R}$.
+ then that element is denoted by $\unit{R}$. Its additive identity
+ element is $\zero{R}$.
\end{section}
\begin{section}{Algorithm}
\State{Rearrange $M$ randomly}
\EndWhile{}
- \Return{$M$}
+ \State{\Return{$M$}}
\end{algorithmic}
\end{algorithm}
\end{section}
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:
+ d}}$. The tuples go up to seven, for now, and then we give up
+ and use the general construct:
%
\begin{itemize}
\begin{item}
\begin{item}
Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
\end{item}
+ \begin{item}
+ Tuple: $\tuple{1,2,\ldots,8675309}$.
+ \end{item}
\end{itemize}
%
The factorial of the number $10$ is $\factorial{10}$, and the
\begin{section}{Cone}
The dual cone of $K$ is $\dual{K}$. Some familiar symmetric cones
- are $\Rnplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$. If cones
- $K_{1}$ and $K_{2}$ are given, we can define $\posops{K_{1}}$,
- $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$, $\Zof{K_{1}}$,
- $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can also define $x
- \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K} y$, and $x
- \ltcone_{K} y$ with respect to a cone $K$.
+ are $\Rnplus$, $\Rnplusplus$, $\Lnplus$, $\Snplus$, and $\Hnplus$.
+ If cones $K_{1}$ and $K_{2}$ are given, we can define
+ $\posops{K_{1}}$, $\posops[K_{2}]{K_{1}}$, $\Sof{K_{1}}$,
+ $\Zof{K_{1}}$, $\LL{K_{1}}$, and $\lyapunovrank{K_{1}}$. We can
+ also define $x \gecone_{K} y$, $x \gtcone_{K} y$, $x \lecone_{K}
+ y$, and $x \ltcone_{K} y$ with respect to a cone $K$.
\end{section}
\begin{section}{Convex}
\end{itemize}
\end{section}
+ \begin{section}{Hurwitz}
+ Here lies the Hurwitz algebras, like the quaternions
+ $\quaternions$ and octonions $\octonions$.
+ \end{section}
+
\begin{section}{Linear algebra}
The absolute value of $x$ is $\abs{x}$, or its norm is
$\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
instead.
If you want to solve a system of equations, try Cramer's
- rule~\cite{ehrenborg}.
+ rule~\cite{ehrenborg}. Or at least the reduced row-echelon form of
+ the matrix, $\rref{A}$.
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