X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=examples.tex;h=2d05431e2ceb2fcdcdf3314980f3348929a69ea8;hp=e1f375fdc68d219d144010732030850d31cd5869;hb=HEAD;hpb=66f91bd13a303f46822639420a20b5bffbe7a963 diff --git a/examples.tex b/examples.tex index e1f375f..383cef2 100644 --- a/examples.tex +++ b/examples.tex @@ -13,7 +13,7 @@ % We have to load this after hyperref, so that links work, but before % mjotex so that mjotex knows to define its glossary entries. \usepackage[nonumberlist]{glossaries} -\makenoidxglossaries +\makenoidxglossaries{} % If you want an index, we can do that too. You'll need to define % the "INDICES" variable in the GNUmakefile, though. @@ -26,16 +26,24 @@ \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} @@ -51,9 +59,9 @@ \While{$M$ is not sorted} \State{Rearrange $M$ randomly} - \EndWhile + \EndWhile{} - \Return{$M$} + \State{\Return{$M$}} \end{algorithmic} \end{algorithm} \end{section} @@ -72,10 +80,11 @@ \end{section} \begin{section}{Common} - The function $f$ applied to $x$ is $f\of{x}$. We can group terms - like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a - set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go - up to seven, for now: + 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, and then we give up + and use the general construct: % \begin{itemize} \begin{item} @@ -96,40 +105,30 @@ \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 $\factorial{10}$. - - The Cartesian product of two sets $A$ and $B$ is - $\cartprod{A}{B}$; if we take the product with $C$ as well, then - we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$ - is $\directsum{V}{W}$. Or three things, - $\directsumthree{U}{V}{W}$. How about more things? Like - $\directsummany{k=1}{\infty}{V_{k}} \ne - \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and - cartesian products adapt nicely to display equations: + The factorial of the number $10$ is $\factorial{10}$, and the + least common multiple of $4$ and $6$ is $\lcm{\set{4,6}} = + 12$. + + The direct sum of $V$ and $W$ is $\directsum{V}{W}$. Or three + things, $\directsumthree{U}{V}{W}$. How about more things? Like + $\directsummany{k=1}{\infty}{V_{k}}$. Those direct sums + adapt nicely to display equations: % \begin{equation*} - \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}. + \directsummany{k=1}{\infty}{V_{k}} \ne \emptyset. \end{equation*} + % Here are a few common tuple spaces that should not have a 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: $\Nn[2]$, $\Zn[2]$, - $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. - - We also have a few basic set operations, for example the union of - two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of - course with union comes intersection: $\intersect{A}{B}$, - $\intersectthree{A}{B}{C}$. We can also take an arbitrary - (indexed) union and intersections of things, like - $\unionmany{k=1}{\infty}{A_{k}}$ or - $\intersectmany{k=1}{\infty}{B_{k}}$. The best part about those - is that they do the right thing in a display equation: - % - \begin{equation*} - \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}} - \end{equation*} + $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. 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 $\Rn[1]$, % @@ -148,12 +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} @@ -167,7 +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} @@ -181,16 +184,28 @@ \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 their tensor product is $\tp{x}{y}$. The Kronecker product of 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 $\trace{L}$. Another matrix-specific - concept is the Moore-Penrose pseudoinverse of $L$, denoted by - $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is - $\rank{L}$. + $\transpose{L}$. 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-$n$ + 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. 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 @@ -202,10 +217,13 @@ 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 @@ -218,7 +236,7 @@ \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}\\ @@ -256,11 +274,6 @@ system to test them. \end{section} - \begin{section}{Miscellaneous} - The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} - = 3$. - \end{section} - \begin{section}{Proof by cases} \begin{proposition} @@ -307,6 +320,35 @@ \renewcommand{\baselinestretch}{1} \end{section} + \begin{section}{Set theory} + Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The + cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} = + 3$, and its powerset is $\powerset{X}$. + + We also have a few basic set operations, for example the union of + two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of + course with union comes intersection: $\intersect{A}{B}$, + $\intersectthree{A}{B}{C}$. The Cartesian product of two sets $A$ + and $B$ is there too: $\cartprod{A}{B}$. If we take the product + with $C$ as well, then we obtain $\cartprodthree{A}{B}{C}$. + + We can also take an arbitrary (indexed) union, intersection, or + Cartesian product of things, like + $\unionmany{k=1}{\infty}{A_{k}}$, + $\intersectmany{k=1}{\infty}{B_{k}}$, or + $\cartprodmany{k=1}{\infty}{C_{k}}$. The best part about those is + that they do the right thing in a display equation: + % + \begin{equation*} + \unionmany{k=1}{\infty}{A_{k}} + \ne + \intersectmany{k=1}{\infty}{B_{k}} + \ne + \cartprodmany{k=1}{\infty}{C_{k}}. + \end{equation*} + % + \end{section} + \begin{section}{Theorems} \begin{corollary} The @@ -382,8 +424,8 @@ \setlength{\glslistdottedwidth}{.3\linewidth} \setglossarystyle{listdotted} - \glsaddall - \printnoidxglossaries + \glsaddall{} + \printnoidxglossaries{} \bibliographystyle{mjo} \bibliography{local-references}