Section 6 Some Advanced Ideas
The multi-row displayed mathematics in the proof of the Fundamental Theorem had equations aligned on the equals signs via the & character. Sometimes you don’t want that. Here is an example with some differential equations, with each equation centered and unnumbered,
\begin{gather*}
{\mathcal L}(y')(s) = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\\
{\mathcal L}(y'')(s) = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)\text{.}
\end{gather*}
Just prior to this sentence, in the middle of this paragraph, is an
<idx> and a <notation>, adjacent, but separated by some whitespace in the authored source. That insignificant whitespace will be removed akways, which will be a (slightly) noticeable improvement in the LaTeX output. We test referencing notation here, placed before the sentence-ending period and right after some inline mathematics—for \(\mathbb{Z}_n\) .
LaTeX has a device where you can interrupt a sequence of equations with a small amout of text and preserve the equation alignment on either side. Here are two tests of that device, with aligned equations and non-aligned equations. Study the source to see use and differences. (The math does not make sense.)
Aligned and numbered first.
\begin{align}
{\mathcal L}(y')(s) &= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\tag{6.1}\\
{\mathcal L}(y'')(s) &= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0).\tag{6.2}\\
\end{align}
And so it follows that,
\begin{align}
{\mathcal L}(y')(s)^{++} &= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\tag{6.3}\\
{\mathcal L}(y'')(s)^{5} &= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)\text{.}\tag{6.4}
\end{align}
Now with no numbers and no alignment. We include two cross-references in the
intertext portion for testing.
\begin{gather*}
{\mathcal L}(y')(s) = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\\
{\mathcal L}(y'')(s) = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0).\\
\end{gather*}
First an external reference to
\begin{gather*}
{\mathcal L}(y')(s)^{++} = s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0)\\
{\mathcal L}(y'')(s)^{5} = s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0)\text{.}
\end{gather*}
example.com and internal cross-reference to Corollary 4.1. And so it follows that,Tables can get quite complex. Simple ones are simpler, such as this example of numerical computations for Euler’s method in just a bit.
But first we make a figure with two very simple tables next to each other. This causes the very first instance of
<table> to actually be a “subtable”, which exposes a bug provoked by Emiliano Vega and fixed around 2020-08-06. (So we have to place this early to create the same behavior that exposed the bug.)| \(i\) | \(t_i\) | \(x_i\) | \(y_i\) |
| 0 | 0.00 | 0.0000 | 0.5000 |
| 1 | 0.20 | 0.1000 | 0.4800 |
| 2 | 0.40 | 0.1960 | 0.4560 |
| 3 | 0.60 | 0.2872 | 0.4295 |
| 4 | 0.80 | 0.3731 | 0.4027 |
| 5 | 1.00 | 0.4536 | 0.3783 |
| 6 | 1.20 | 0.5293 | 0.3591 |
| 7 | 1.40 | 0.6011 | 0.3480 |
| 8 | 1.60 | 0.6707 | 0.3474 |
| 9 | 1.80 | 0.7402 | 0.3603 |
| 10 | 2.00 | 0.8123 | 0.3900 |
