% 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 $\unit{R}$. Its additive identity
+ element is $\zero{R}$.
\end{section}
\begin{section}{Algorithm}
\While{$M$ is not sorted}
\State{Rearrange $M$ randomly}
- \EndWhile
+ \EndWhile{}
\Return{$M$}
\end{algorithmic}
\end{item}
\end{itemize}
%
- The factorial of the number $10$ is $\factorial{10}$.
+ 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
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: $\Nn[2]$, $\Zn[2]$,
- $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Finally, we have the four standard
- types of intervals in $\Rn[1]$,
+ $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. 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 $\Rn[1]$,
%
\begin{align*}
\intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
\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
- $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
+ $\transpose{L}$. 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.
+ 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
instead.
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
\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}\\
\setlength{\glslistdottedwidth}{.3\linewidth}
\setglossarystyle{listdotted}
- \glsaddall
- \printnoidxglossaries
+ \glsaddall{}
+ \printnoidxglossaries{}
\bibliographystyle{mjo}
\bibliography{local-references}
projects / mjotex.git / blobdiff