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+\makenoidxglossaries{}
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\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 $\unit{R}$. Its additive identity
+ element is $\zero{R}$. The stabilizer (or isotropy)
+ subgroup of $G$ that fixes $x$ is $\Stab{G}{x}$.
+
+ If $I$ is an ideal, then $\variety{I}$ is the variety that
+ corresponds to it.
\end{section}
\begin{section}{Algorithm}
\While{$M$ is not sorted}
\State{Rearrange $M$ randomly}
- \EndWhile
+ \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
superscript when that superscript would be one: $\Nn[1]$,
$\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
- $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Finally, we have the four standard
- types of intervals in $\Rn[1]$,
+ $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. The symbols $\Fn[1]$, $\Fn[2]$,
+ et cetera, are available for use with a generic field.
+
+ 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 },\\
\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}
\begin{section}{Euclidean Jordan algebras}
The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra
- is $\jp{x}{y}$.
+ $V$ is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is
+ $\JAut{V}$. Two popular operators in an EJA are its quadratic
+ representation and ``left multiplication by'' operator. For a
+ given $x$, they are, respectively, $\quadrepr{x}$ and
+ $\leftmult{x}$.
\end{section}
\begin{section}{Font}
\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
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 $\trace{L}$, 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-$n$
+ 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-$n$
real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
simplify because the ``$n$'' does not indicate the arity of a
- Cartesian product in this case.
+ Cartesian product in this case. A handy way to represent the
+ matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on
+ the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$.
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.
+ instead. If you have matrices instead, then the general linear
+ group of $n$-by-$n$ matrices with entries in $\mathbb{F}$ is
+ $\GL{n}{\mathbb{F}}$.
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
\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}\\
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