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Avoid confusing reuse of 'x' in spatialDistribution description #3649

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Cleaning up the notation used for spatialDistribution, plus stating that initialPoints and initialValues are parameter expressions (item 3 in #3640 (comment)).

Note that the change about parameter variability is not the one suggested in #3640 (comment), so this still remains to be done.

For the notation part, the most important improvement is to avoid using x as the name for the normalized position in the equations, since this was very confusing when the formal parameter x in the synopsis as well as in the pseudo-code is something else.

@henrikt-ma henrikt-ma requested a review from casella February 7, 2025 08:40
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The other changes look good, but I'm not sure if replacing $x$ by $\xi$ is that beneficial.

On one hand using x both in the differential equation and as input is a bit overloaded, but on the other hand we lose the connection between the x-coordinate as input and the $\xi$, and in particular state that $v=\xi'(t)$, and then describe $x$ as the integral of $v$. Effectively that combines to imply that $x=\xi$, right?

@henrikt-ma henrikt-ma force-pushed the cleanup/spatialDistributionNotation branch from 0ef979e to b288f19 Compare February 24, 2025 22:05
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henrikt-ma commented Feb 24, 2025

On one hand using x both in the differential equation and as input is a bit overloaded, but on the other hand we lose the connection between the x-coordinate as input and the ξ , and in particular state that v = ξ ′ ( t ) , and then describe x as the integral of v . Effectively that combines to imply that x = ξ , right?

Right. The relation $v(t) = \xi'(t)$ is crap, as ξ is an independent variable and not a function of time. This relation is now removed. That x is a function of time, while ξ is not, also highlights how different they are and why using an overloaded symbol gets very confusing.

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