X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=0cf8afa51a90a84b624fee9a7f4232ae7ff9a52d;hb=6dfb93ac68463f1f47a009e2d1672c1c78f1e847;hp=dde74d0f6997418a8ab283ee1f6fb53452f6a6bf;hpb=8f3a3f952b0e5bd692b9b41d1417c877b0e0425e;p=mjotex.git diff --git a/examples.tex b/examples.tex index dde74d0..0cf8afa 100644 --- a/examples.tex +++ b/examples.tex @@ -72,10 +72,10 @@ \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} @@ -100,38 +100,21 @@ % 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 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 },\\ @@ -171,7 +154,14 @@ \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} @@ -180,10 +170,15 @@ 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 - $\rank{L}$. + $\rank{L}$. 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 @@ -249,11 +244,6 @@ 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} @@ -300,6 +290,35 @@ \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