\xmpheader 8/{More mathematics}
The absolute value of $X$, $|x|$, is define by:
$$|x| = \cases{x, & if $x\ge 0$; \cr
-x,&otherwise.\cr}$$
Now for some numbered equations. It is the case that for $k \ge 0$:
$$x^{k^2} =\overbrace{x\>x\>\cdots\> x}^{k^2\rm\ times} \eqno(1)$$
Here's an example that shows some spacing controls, with a number on the left:
$$[u]\![v][w]\,[x]\>[y]\;[z] \leqno(2a)$$
The amount of space between the items in brackets gradually increases from left to right.
(We've made the space between the first two items be {\it less\/} than the natural space.)
It is sometimes the case that
$$\leqalignno{u'_1 + tu''_2 &= u'_2 + tu_1''&(2b)\cr
\hat\imath &\ne \hat\jmath&(2c)\cr
\vec{\vphantom{b}a}&\approx \vec b\cr}$$
% The \vphantom is an invisible rule as tall as a `b'.
The result is of order $O(n \log\log n)$. Thus
$$\sum_{i=1}^{n} x_i = x_1+x_2+\cdots+x_n = {\rm Sum}(x_1,x_2,\cdots,x_n). \eqno(3)$$
and $$dx\,dy = r\, dr\, d\theta\!. \eqno(4)$$
The set of all $q$ such that $q\le 0$ is written as:
$$ \{\,q\mid q\le0\, \}$$
Thus
$$\forall x\exists y\; P(x,y) \Rightarrow \exists x\exists y\;P(x,y)$$
where
$$P(x,y) \buildrel \rm def \over \equiv \hbox{\rm any predicate in $x$ and $y$}.$$