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Section 6 Some Advanced Ideas

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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 output. We test referencing notation here, placed before the sentence-ending period and right after some inline mathematics—for \(\mathbb{Z}_n\) .

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 example.com and internal cross-reference to Corollary 4.1. And so it follows that,

\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*}

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.)

One
(a) First
Two
(b) Second
Figure 6.1. Buggy sub-tables
Table 6.2. Euler's approximation for Duffing's Equation with \(h = 0.2\)
\(i\) \(t_i\) \(x_i\) \(y_i\)
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