From 07b10e9bcd59b16af1cd6b143f07f0a3cd56d33d Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 3 Mar 2021 22:38:29 -0500 Subject: [PATCH 01/12] mjo-hurwitz.tex: new file for Hurwitz algebras (quaternions and octonions). --- GNUmakefile | 4 ++-- examples.tex | 5 +++++ mjo-hurwitz.tex | 27 +++++++++++++++++++++++++++ mjotex.sty | 1 + 4 files changed, 35 insertions(+), 2 deletions(-) create mode 100644 mjo-hurwitz.tex diff --git a/GNUmakefile b/GNUmakefile index 73d8988..61c9eca 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 diff --git a/examples.tex b/examples.tex index babc886..2d05431 100644 --- a/examples.tex +++ b/examples.tex @@ -172,6 +172,11 @@ \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 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/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} -- 2.43.2 From 42b7c70a17afda63b29fa9f61be142c127ef1db6 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 6 Dec 2021 07:13:18 -0500 Subject: [PATCH 02/12] GNUmakefile: drop obsolete --memlimit argument to sage -t. --- GNUmakefile | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/GNUmakefile b/GNUmakefile index 61c9eca..3669ac2 100644 --- a/GNUmakefile +++ b/GNUmakefile @@ -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. -- 2.43.2 From 47e0baf2d737176fe6f9ab4e45cfd202c92fb72e Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 31 Jan 2022 15:33:04 -0500 Subject: [PATCH 03/12] examples.tex: fix algorithm \Return usage. --- examples.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/examples.tex b/examples.tex index 2d05431..f23b9e9 100644 --- a/examples.tex +++ b/examples.tex @@ -57,7 +57,7 @@ \State{Rearrange $M$ randomly} \EndWhile{} - \Return{$M$} + \State{\Return{$M$}} \end{algorithmic} \end{algorithm} \end{section} -- 2.43.2 From a883c37180e9a8aaa9d1b2f611c93b6be7dc48be Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Tue, 27 Dec 2022 16:26:01 -0500 Subject: [PATCH 04/12] mjo-set.tex: consistent whitespace in \setc{}. --- mjo-set.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) 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 -- 2.43.2 From 4d76b8a320e183fc1fcae6f0c55d7af96806a00b Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 20 Jan 2023 18:30:02 -0500 Subject: [PATCH 05/12] mjo-cone.tex: add the strictly positive orthant, \Rnplusplus. --- examples.tex | 12 ++++++------ mjo-cone.tex | 4 +++- 2 files changed, 9 insertions(+), 7 deletions(-) diff --git a/examples.tex b/examples.tex index f23b9e9..e70e0e1 100644 --- a/examples.tex +++ b/examples.tex @@ -139,12 +139,12 @@ \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} 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}}_{+} } -- 2.43.2 From 1efe9c09be9e93ed5dbb2ae60d787517eb0609ca Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 12 Feb 2023 20:12:44 -0500 Subject: [PATCH 06/12] mjo-common: add \tuple. --- examples.tex | 6 +++++- mjo-common.tex | 6 ++++++ 2 files changed, 11 insertions(+), 1 deletion(-) diff --git a/examples.tex b/examples.tex index e70e0e1..5232635 100644 --- a/examples.tex +++ b/examples.tex @@ -79,7 +79,8 @@ 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 +101,9 @@ \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 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. -- 2.43.2 From 906dd6be4d6f832228c3d97c9c755fa4fc8d43bf Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 3 Mar 2023 13:52:05 -0500 Subject: [PATCH 07/12] mjo-eja.tex: add the Jordan-automorphism group operator, \JAut{}. --- examples.tex | 4 ++-- mjo-eja.tex | 5 +++++ 2 files changed, 7 insertions(+), 2 deletions(-) diff --git a/examples.tex b/examples.tex index 5232635..d59975e 100644 --- a/examples.tex +++ b/examples.tex @@ -161,8 +161,8 @@ \end{section} \begin{section}{Euclidean Jordan algebras} - The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra - is $\jp{x}{y}$. + The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra $V$ + is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is $\JAut{V}$. \end{section} \begin{section}{Font} diff --git a/mjo-eja.tex b/mjo-eja.tex index b6974be..8b4efbf 100644 --- a/mjo-eja.tex +++ b/mjo-eja.tex @@ -23,4 +23,9 @@ \newcommand*{\jp}[2]{{#1} \circ {#2}} +% The ``Jordan automorphism group of'' operator. Using +% \Aut{} is too ambiguous sometimes. +\newcommand*{\JAut}[1]{ \operatorname{JAut}\of{{#1}} } + + \fi -- 2.43.2 From 5b800f8b14e519dc313fbcd2db2553080587d9c8 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 15 Mar 2023 18:33:02 -0400 Subject: [PATCH 08/12] mjo-eja.tex: add \quadrepr and \leftmult. --- examples.tex | 8 ++++++-- mjo-eja.tex | 9 +++++++++ 2 files changed, 15 insertions(+), 2 deletions(-) diff --git a/examples.tex b/examples.tex index d59975e..740dd02 100644 --- a/examples.tex +++ b/examples.tex @@ -161,8 +161,12 @@ \end{section} \begin{section}{Euclidean Jordan algebras} - The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra $V$ - is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is $\JAut{V}$. + The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra + $V$ 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} diff --git a/mjo-eja.tex b/mjo-eja.tex index 8b4efbf..efa68de 100644 --- a/mjo-eja.tex +++ b/mjo-eja.tex @@ -22,6 +22,15 @@ % 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. -- 2.43.2 From c8cd642ae6f1dd3455d2ddc0f535508aedc07aaf Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Thu, 30 Mar 2023 17:51:41 -0400 Subject: [PATCH 09/12] mjo-algebra: add \Stab command for stabilizer subgroups. --- examples.tex | 3 ++- mjo-algebra.tex | 4 ++++ 2 files changed, 6 insertions(+), 1 deletion(-) diff --git a/examples.tex b/examples.tex index 740dd02..d8e2c32 100644 --- a/examples.tex +++ b/examples.tex @@ -39,7 +39,8 @@ 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 $\Stab{G}{x}$. \end{section} \begin{section}{Algorithm} diff --git a/mjo-algebra.tex b/mjo-algebra.tex index 25b96d9..1ab8880 100644 --- a/mjo-algebra.tex +++ b/mjo-algebra.tex @@ -90,4 +90,8 @@ \fi +% The stabilizer subgroup of its first argument that fixes the point +% given by its second argument. +\newcommand*{\Stab}[2]{ #1_{#2} } + \fi -- 2.43.2 From af9f65d664b95a3090d29a109e9efb1d9d08be5b Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Thu, 30 Mar 2023 17:56:52 -0400 Subject: [PATCH 10/12] mjo-linear_algebra: add \GL command for general linear groups. --- examples.tex | 4 +++- mjo-linear_algebra.tex | 4 ++++ 2 files changed, 7 insertions(+), 1 deletion(-) diff --git a/examples.tex b/examples.tex index d8e2c32..6b60b00 100644 --- a/examples.tex +++ b/examples.tex @@ -214,7 +214,9 @@ 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-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 -- 2.43.2 From fc43a7cd831de2042a728269e791ab82b65962ba Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sat, 13 Apr 2024 17:10:40 -0400 Subject: [PATCH 11/12] mjo-algebra.tex,examples.tex: add the \variety command --- examples.tex | 3 +++ mjo-algebra.tex | 13 +++++++++++++ 2 files changed, 16 insertions(+) diff --git a/examples.tex b/examples.tex index 6b60b00..383cef2 100644 --- a/examples.tex +++ b/examples.tex @@ -41,6 +41,9 @@ 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} diff --git a/mjo-algebra.tex b/mjo-algebra.tex index 1ab8880..42ab2aa 100644 --- a/mjo-algebra.tex +++ b/mjo-algebra.tex @@ -94,4 +94,17 @@ % 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=p + } +\fi + \fi -- 2.43.2 From 1e7cb9d3126565735b49bfc76f9f34d14199a65c Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sat, 13 Apr 2024 17:23:09 -0400 Subject: [PATCH 12/12] mjo-algebra.tex: fix glossary sorting of \variety --- mjo-algebra.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/mjo-algebra.tex b/mjo-algebra.tex index 42ab2aa..3bb51c1 100644 --- a/mjo-algebra.tex +++ b/mjo-algebra.tex @@ -103,7 +103,7 @@ \newglossaryentry{variety}{ name={\ensuremath{\variety{I}}}, description={variety corresponding to the ideal $I$}, - sort=p + sort=v } \fi -- 2.43.2