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 $\Frac{R}$.
+ 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}
\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 $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
- of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
- and 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 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}
+ Pair: $\pair{1}{2}$,
+ \end{item}
+ \begin{item}
+ Triple: $\triple{1}{2}{3}$,
+ \end{item}
+ \begin{item}
+ Quadruple: $\quadruple{1}{2}{3}{4}$,
+ \end{item}
+ \begin{item}
+ Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
+ \end{item}
+ \begin{item}
+ Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
+ \end{item}
+ \begin{item}
+ Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
+ \end{item}
+ \end{itemize}
+ %
+ 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 },\\
\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}{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
+ $\transpose{L}$. 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
- $\pseudoinverse{L}$.
+ $\pseudoinverse{L}$. 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.
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
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