Would it be possible to enable special features in the Incubator Q&A?

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Ok, this was weird. The setting was on, I pasted in some Mathjax, it didn't render, I turned it off and on, and now it's rendering.

Experiment below:

I need some Mathjax to test with, so I'm copying from this post from Mathematics Codidact.

I'm exploring a function that takes a non-negative integer vector $θ \in {N_{0}}^{4}$ and returns the modulus of each adjacent vector components, wrapping around:

s : {N_{0}}^{4} \to {N_{0}}^{4}

\begin{aligned} s (θ) = s ([\begin{array}{c} x \\ y \\ z \\ w \end{array}]) = [\begin{array}{c} | y - x | \\ | z - y | \\ | w - z | \\ | x - w | \end{array}] \end{aligned}

Example:

\begin{aligned} s ([\begin{array}{c} 6 \\ 2 \\ 15 \\ 7 \end{array}]) = [\begin{array}{c} | 2 - 6 | \\ | 15 - 2 | \\ | 7 - 15 | \\ | 6 - 7 | \end{array}] = [\begin{array}{c} | - 4 | \\ | + 13 | \\ | - 8 | \\ | - 1 | \end{array}] = [\begin{array}{c} 4 \\ 13 \\ 8 \\ 1 \end{array}] \end{aligned}

Basic properties of $s$ that I can prove:

$s is not linear$
$s (0) = 0$
$s (λ θ) = | λ | s (θ)$
Let $θ^{'} = θ - θ_{m i n}$ , where $θ_{m i n}$ is the smallest component of $θ$ . Then $s (θ) = s (θ^{'})$

In my experiments, I observed that the repeated application of $s$ converged to the zero vector for all sample inputs I tried: millions of unique $θ \in {N_{0}}^{4}$

As a purely recreational exercise, my goal is to prove or disprove that this holds for all $θ$ :

$\forall_{θ \in {N_{0}}^{4}} \exists_{n \geq 0} \overset{n}{\overset{⏞}{s \circ s \circ \dots s}} (θ) = 0$

Now my questions: (...)

posted over 1 year ago

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1y ago

Monica Cellio‭ staff

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Would it be possible to enable special features in the Incubator Q&A?

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