% 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.
\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 $\unit{R}$.
\end{section}
\begin{section}{Algorithm}
\While{$M$ is not sorted}
\State{Rearrange $M$ randomly}
- \EndWhile
+ \EndWhile{}
\Return{$M$}
\end{algorithmic}
\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:
%
\begin{itemize}
\begin{item}
\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*}
- %
- Finally, we have the four standard types of intervals in $\Rn[1]$,
+ $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. 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 },\\
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
+ $\transpose{L}$. 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-$n$
\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}\\
\end{section}
\begin{section}{Set theory}
- The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
- = 3$, and its powerset is $\powerset{X}$.
+ 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}
\setlength{\glslistdottedwidth}{.3\linewidth}
\setglossarystyle{listdotted}
- \glsaddall
- \printnoidxglossaries
+ \glsaddall{}
+ \printnoidxglossaries{}
\bibliographystyle{mjo}
\bibliography{local-references}
projects / mjotex.git / blobdiff