From: Michael Orlitzky Date: Sat, 13 Apr 2024 21:23:09 +0000 (-0400) Subject: mjo-algebra.tex: fix glossary sorting of \variety X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=commitdiff_plain;h=HEAD;hp=cab35094298c00d10035e926b64ff69df33393a5 mjo-algebra.tex: fix glossary sorting of \variety --- diff --git a/GNUmakefile b/GNUmakefile index 73d8988..3669ac2 100644 --- a/GNUmakefile +++ b/GNUmakefile @@ -31,8 +31,8 @@ BIBS = local-references.bib # 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 @@ -197,7 +197,7 @@ check-undefined: $(PN).log .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. diff --git a/examples.tex b/examples.tex index babc886..383cef2 100644 --- a/examples.tex +++ b/examples.tex @@ -39,7 +39,11 @@ 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} @@ -57,7 +61,7 @@ \State{Rearrange $M$ randomly} \EndWhile{} - \Return{$M$} + \State{\Return{$M$}} \end{algorithmic} \end{algorithm} \end{section} @@ -79,7 +83,8 @@ 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} @@ -100,6 +105,9 @@ \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 @@ -139,12 +147,12 @@ \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} @@ -158,7 +166,11 @@ \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} @@ -172,6 +184,11 @@ \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 @@ -200,7 +217,9 @@ 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 diff --git a/mjo-algebra.tex b/mjo-algebra.tex index 25b96d9..3bb51c1 100644 --- a/mjo-algebra.tex +++ b/mjo-algebra.tex @@ -90,4 +90,21 @@ \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 diff --git a/mjo-common.tex b/mjo-common.tex index 6b357ab..ccb22da 100644 --- a/mjo-common.tex +++ b/mjo-common.tex @@ -39,6 +39,12 @@ % 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. diff --git a/mjo-cone.tex b/mjo-cone.tex index 78e8741..00a1309 100644 --- a/mjo-cone.tex +++ b/mjo-cone.tex @@ -23,8 +23,10 @@ % 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}}_{+} } diff --git a/mjo-eja.tex b/mjo-eja.tex index b6974be..efa68de 100644 --- a/mjo-eja.tex +++ b/mjo-eja.tex @@ -22,5 +22,19 @@ % 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 diff --git a/mjo-hurwitz.tex b/mjo-hurwitz.tex new file mode 100644 index 0000000..9dca960 --- /dev/null +++ b/mjo-hurwitz.tex @@ -0,0 +1,27 @@ +\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 diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex index 7f2484a..5f25544 100644 --- a/mjo-linear_algebra.tex +++ b/mjo-linear_algebra.tex @@ -174,4 +174,8 @@ \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 diff --git a/mjo-set.tex b/mjo-set.tex index 4177f5a..66cd56b 100644 --- a/mjo-set.tex +++ b/mjo-set.tex @@ -22,7 +22,7 @@ % 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 diff --git a/mjotex.sty b/mjotex.sty index f85f779..90db53c 100644 --- a/mjotex.sty +++ b/mjotex.sty @@ -8,6 +8,7 @@ \input{mjo-convex} \input{mjo-eja} \input{mjo-font} +\input{mjo-hurwitz} \input{mjo-linear_algebra} \input{mjo-listing} \input{mjo-proof_by_cases}