docs(gh-993): fix description of formula for DirichletElement

[Qazalbash]

closes #993

[Qazalbash]

@ColmTalbot

[ColmTalbot]

Thanks for putting this together, just a few small comments.

It looks like one of the comments has been duplicated, sorry about that!

Also, congratulations on getting #1000!

[ColmTalbot]

It looks like it's generally rendering well, except for the issue with $S_n$

[Qazalbash]

It looks like it's generally rendering well, except for the issue with $S_n$

Can you point to the issue in $S_n$?

[ColmTalbot]

I think it should be

$$S_n = \sum_{0}^{n - 1} x_i$$

This satisfies $S_0 = 0$ and $S_1 = x_0$.

[ColmTalbot]

Thanks for continuing to work on this! I think I'm not being clear. For the definition of $S_n$, there is no distinction between $n < N - 1$ and $n = N - 1$, in all cases it is

$$ S_n = \sum_{0}^{n - 1} x_i \quad ; \quad n < N $$

Unless there's something else I'm missing.

[Qazalbash]

It's because of the $S_0=0$ case. The current definition explains all; otherwise, there is a chance of confusion.

[ColmTalbot]

We currently have

$$ S_n := \begin{cases} 0 & n = N - 1 \\ \displaystyle\sum_{i=0}^{n-1}x_i & n < N - 1 \end{cases} $$

The top case here is not correct. Also, it doesn't mention $S_0$, which is also covered since $\sum_{0}^{0} x_i = 0$ trivially.

[ColmTalbot]

This looks good to me now!

[mj-will]

The equation looks good to me but I have questions on the examples.

[Qazalbash]

@mj-will, sorry for the delay. You can see the fix in the recent commit.

[mj-will]

@mj-will, sorry for the delay. You can see the fix in the recent commit.

Thanks @Qazalbash, did you have a chance to check the first part of my comment here (for the N=1 case)?

[Qazalbash]

@mj-will, I tried to accommodate in the current expressions, but it is hard to fit, because the $N=1$ should lie in the first condition, but it is forced into the latter one. A jugaad would be to write $N=1$ case separately.