Is it possible to share files between iCloud Music Libraries?

WITH RECURSIVE cte AS (
  SELECT id, domain_name, valid 
  FROM domains
  WHERE parent_id IS NULL
UNION ALL
  SELECT domains.id, domains.domain_name, domains.valid 
  FROM domains
  JOIN cte ON domains.parent_id = cte.id
  WHERE NOT cte.valid -- stop recursion when valid node reached
)

SELECT id, domain_name
FROM cte
WHERE valid

fiddle


You can find details on the Maass waveform calculation in SAGE in this talk by Frederik Strömberg. I guess you'll have to ask Dr. Strömberg for the precise parameters used in the waveform you mention.


Let's just do it for a simple example. By $\vec{c}$ I imagine you mean the location of the particle relative to some origin, so $\vec{c}=\vec{r}$. Later on for simplicity we'll suppose further the particle is located on the x-axis (but it is important to do this only after differentiating as we will see).

We'll also suppose we are rotating around the z axis so that $\hat{n}=\hat{z}$.

Then we have \begin{equation} \{\vec{c} , \vec{l}\cdot\vec{n}\}=\{\vec{r},xp_y - y p_x\} =\{x\hat{x}+y\hat{y}+z\hat{z},xp_y - y p_x\}= -y \hat{x}+x\hat{y}. \end{equation}

Now that we have differentiated (meaning, evaluated the brackets) we can set $y=0$ and $x=R$ (that is, we can suppose our particle started on the $x$-axis at the position $R$). Then \begin{equation} \{R \hat{x},\vec{l}\cdot\hat{z}\}=R \hat{y}=\hat{z}\times(R\hat{x}) \end{equation} which is consistent with your formula.

Incidentally, you might be worried that I started off by setting $\vec{c}=\vec{r}$. I think in the framework you are working in--particle mechanics--the vectors should all start from the same origin. If you want to start taking poisson brackets of vectors with different origins, I think you really need to generalize this discussion to field theory (which will complicate the story a bit because in addition to rotating the direction of the vector you need to rotate the origin, so you will end up with an additional term). So I think that may be what you have in mind but that is a more complicated story.