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\usepackage[nonumberlist]{glossaries}
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+\makenoidxglossaries{}
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% 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.
+ 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}
\While{$M$ is not sorted}
\State{Rearrange $M$ randomly}
- \EndWhile
+ \EndWhile{}
- \Return{$M$}
+ \State{\Return{$M$}}
\end{algorithmic}
\end{algorithm}
\end{section}
\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}
\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 $\Rn[1]$,
%
\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}
\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}
- We can write things like Carathéodory and Güler and $\mathbb{R}$.
+ We can write things like Carathéodory and Güler and
+ $\mathbb{R}$. The PostScript Zapf Chancery font is also available
+ in both upper- and lower-case:
+ %
+ \begin{itemize}
+ \begin{item}$\mathpzc{abcdefghijklmnopqrstuvwxyz}$\end{item}
+ \begin{item}$\mathpzc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$\end{item}
+ \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}
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
- $\pseudoinverse{L}$.
+ $\transpose{L}$. 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
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
\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}\\
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}
\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
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projects / mjotex.git / blobdiff