#
MJOTEX = mjo-algebra.tex mjo-algorithm.tex mjo-arrow.tex mjo-calculus.tex
MJOTEX += mjo-common.tex mjo-complex.tex mjo-cone.tex mjo-convex.tex
-MJOTEX += mjo-eja.tex mjo-font.tex mjo-linear_algebra.tex mjo-listing.tex
-MJOTEX += mjo-proof_by_cases.tex mjo-set.tex mjo-theorem.tex
+MJOTEX += mjo-eja.tex mjo-font.tex mjo-hurwitz.tex mjo-linear_algebra.tex
+MJOTEX += mjo-listing.tex mjo-proof_by_cases.tex mjo-set.tex mjo-theorem.tex
MJOTEX += mjo-theorem-star.tex mjo-topology.tex mjo.bst
# Compile a list of raw source code listings (*.listing) and their
.PHONY: check-sage
check-sage: $(SAGE_LISTING_DSTS)
ifdef SAGE_LISTING_DSTS
- sage -t --timeout=0 --memlimit=0 $^
+ sage -t --timeout=0 $^
endif
# Run a suite of checks.
If $R$ has a multiplicative identity (that is, a unit) element,
then that element is denoted by $\unit{R}$. Its additive identity
- element is $\zero{R}$.
+ 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}
\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}
\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
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}. Or at least the reduced row-echelon form of
\fi
+% The stabilizer subgroup of its first argument that fixes the point
+% given by its second argument.
+\newcommand*{\Stab}[2]{ #1_{#2} }
+
+
+% The affine algebraic variety consisting of the common solutions to
+% every polynomial in its argument, which should be a subset of some
+% polynomial ring.
+\newcommand*{\variety}[1]{ \mathcal{V}\of{{#1}} }
+\ifdefined\newglossaryentry
+ \newglossaryentry{variety}{
+ name={\ensuremath{\variety{I}}},
+ description={variety corresponding to the ideal $I$},
+ sort=v
+ }
+\fi
+
\fi
% A seven-tuple of things.
\newcommand*{\septuple}[7]{ \left({#1},{#2},{#3},{#4},{#5},{#6},{#7}\right) }
+% A free-form tuple of things. Useful for when the exact number is not
+% known, such as when \ldots will be stuck in the middle of the list,
+% and when you don't want to think in column-vector terms, e.g. with
+% elements of an abstract Cartesian product space.
+\newcommand*{\tuple}[1]{ \left({#1}\right) }
+
% The "least common multiple of" function. Takes a nonempty set of
% things that can be multiplied and ordered as its argument. Name
% chosen for synergy with \gcd, which *does* exist already.
% Common cones.
%
-% The nonnegative orthant in the given number of dimensions.
+% The nonnegative and strictly positive orthants in the given number
+% of dimensions.
\newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
+\newcommand*{\Rnplusplus}[1][n]{ \Rn[#1]_{++} }
% The Lorentz ``ice-cream'' cone in the given number of dimensions.
\newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
% a (bilinear) algebra multiplication in any other context.
\newcommand*{\jp}[2]{{#1} \circ {#2}}
+% The "quadratic representation" of the ambient space applied to its
+% argument. We have standardized on the "P" used by Faraut and Korányi
+% rather than the "U" made popular by Jacobson.
+\newcommand*{\quadrepr}[1]{P_{#1}}
+
+% The "left multiplication by" operator. Takes one argument, the thing
+% to multiply on the left by. This has meaning more generally than in
+% an EJA, but an EJA is where I use it.
+\newcommand*{\leftmult}[1]{L_{#1}}
+
+% The ``Jordan automorphism group of'' operator. Using
+% \Aut{} is too ambiguous sometimes.
+\newcommand*{\JAut}[1]{ \operatorname{JAut}\of{{#1}} }
+
\fi
--- /dev/null
+\ifx\havemjohurwitz\undefined
+\def\havemjohurwitz{1}
+
+
+\newcommand*{\quaternions}{\mathbb{H}}
+
+\ifdefined\newglossaryentry
+ \newglossaryentry{quaternions}{
+ name={\ensuremath{\quaternions}},
+ description={the algebra of quaternions},
+ sort=H
+ }
+\fi
+
+
+\newcommand*{\octonions}{\mathbb{O}}
+
+\ifdefined\newglossaryentry
+ \newglossaryentry{octonions}{
+ name={\ensuremath{\octonions}},
+ description={the algebra of octonions},
+ sort=O
+ }
+\fi
+
+
+\fi
\fi
+% The general linear group of square matrices whose size is the first
+% argument and whose entries come from the second argument.
+\newcommand*{\GL}[2]{\operatorname{GL}_{#1}\of{#2}}
+
\fi
% automatically. The bar was chosen over a colon to avoid ambiguity
% with the L : V -> V notation. We can't leverage \set here because \middle
% needs \left and \right present.
-\newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2} \right\rbrace}
+\newcommand*{\setc}[2]{\left\lbrace{#1}\ \middle|\ {#2}\right\rbrace}
% The cardinality of a set. The |X| notation conflicts with the
\input{mjo-convex}
\input{mjo-eja}
\input{mjo-font}
+\input{mjo-hurwitz}
\input{mjo-linear_algebra}
\input{mjo-listing}
\input{mjo-proof_by_cases}
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