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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 c...
Answer
#3: Post edited
by
Monica Cellio
·
2023-08-02T20:02:08Z (over 1 year ago)
- 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](https://math.codidact.com/posts/288658).
- ----
- I'm exploring a function that takes a non-negative integer vector $$ \theta\in {\mathbb{N}_0}^4$$ and returns the modulus of each adjacent vector components, wrapping around:
- $$
- s:{\mathbb{N}_0}^4 \rightarrow {\mathbb{N}_0}^4 \\
- $$
- $$
- \begin{align*}
- & s(\boldsymbol{\theta}) =
- s\left(\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \right) =
- \begin{bmatrix}
- |y-x| \\
- |z-y| \\
- |w-z| \\
- |x-w|
- \end{bmatrix}
- \end{align*}
- $$
- Example:
- $$
- \begin{align*}
- & s\left(\begin{bmatrix} 6 \\ 2 \\ 15 \\ 7 \end{bmatrix} \right) =
- \begin{bmatrix}
- |2-6| \\
- |15-2| \\
- |7-15| \\
- |6-7|
- \end{bmatrix} =
- \begin{bmatrix}
- |-4| \\
- |+13| \\
- |-8| \\
- |-1|
- \end{bmatrix} =
- \begin{bmatrix}
- 4 \\
- 13 \\
- 8 \\
- 1
- \end{bmatrix}
- \end{align*}
- $$
- Basic properties of $s$ that I can prove:
- * $ s \text{ is not linear} $
- * $ s(\boldsymbol{0}) = \boldsymbol{0} $
- * $ s(\lambda\boldsymbol{\theta}) = |\lambda|s(\boldsymbol{\theta}) $
- * Let $\boldsymbol\theta' = \boldsymbol\theta - \theta_{min}$, where $\theta_{min}$ is the smallest component of $\boldsymbol\theta$. Then $s(\boldsymbol\theta)=s(\boldsymbol\theta') $
- 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 $\boldsymbol{\theta}\in {\mathbb{N}_0}^4$
- As a purely recreational exercise, my goal is to prove or disprove that this holds *for all* $\boldsymbol{\theta}$:
- $ \forall_{\boldsymbol{\theta} \in {\mathbb{N}_0}^4} \exists_{n\ge 0} \space \space \overbrace{s \circ \ s \circ \dots s}^{n} (\boldsymbol{\theta}) = \boldsymbol{0} $
Now my questions:1. Is $s$ a common function in some domain or application?1. Which research keyworks would you recommend to find more about $s$ (or similar function)?Search expressions like "adjacent element distance" and variations thereof didn't help much. These often lead to pages more on the implementation/coding side of the problem.This is purely a recreational problem. There is no real application that I know of - but I would like to know. I'm not a professional mathematician. My background is computer science.Thank you
- 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](https://math.codidact.com/posts/288658).
- ----
- I'm exploring a function that takes a non-negative integer vector $$ \theta\in {\mathbb{N}_0}^4$$ and returns the modulus of each adjacent vector components, wrapping around:
- $$
- s:{\mathbb{N}_0}^4 \rightarrow {\mathbb{N}_0}^4 \\
- $$
- $$
- \begin{align*}
- & s(\boldsymbol{\theta}) =
- s\left(\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \right) =
- \begin{bmatrix}
- |y-x| \\
- |z-y| \\
- |w-z| \\
- |x-w|
- \end{bmatrix}
- \end{align*}
- $$
- Example:
- $$
- \begin{align*}
- & s\left(\begin{bmatrix} 6 \\ 2 \\ 15 \\ 7 \end{bmatrix} \right) =
- \begin{bmatrix}
- |2-6| \\
- |15-2| \\
- |7-15| \\
- |6-7|
- \end{bmatrix} =
- \begin{bmatrix}
- |-4| \\
- |+13| \\
- |-8| \\
- |-1|
- \end{bmatrix} =
- \begin{bmatrix}
- 4 \\
- 13 \\
- 8 \\
- 1
- \end{bmatrix}
- \end{align*}
- $$
- Basic properties of $s$ that I can prove:
- * $ s \text{ is not linear} $
- * $ s(\boldsymbol{0}) = \boldsymbol{0} $
- * $ s(\lambda\boldsymbol{\theta}) = |\lambda|s(\boldsymbol{\theta}) $
- * Let $\boldsymbol\theta' = \boldsymbol\theta - \theta_{min}$, where $\theta_{min}$ is the smallest component of $\boldsymbol\theta$. Then $s(\boldsymbol\theta)=s(\boldsymbol\theta') $
- 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 $\boldsymbol{\theta}\in {\mathbb{N}_0}^4$
- As a purely recreational exercise, my goal is to prove or disprove that this holds *for all* $\boldsymbol{\theta}$:
- $ \forall_{\boldsymbol{\theta} \in {\mathbb{N}_0}^4} \exists_{n\ge 0} \space \space \overbrace{s \circ \ s \circ \dots s}^{n} (\boldsymbol{\theta}) = \boldsymbol{0} $
- Now my questions: (...)
#2: Post edited
by
Monica Cellio
·
2023-08-02T20:00:20Z (over 1 year ago)
- I need some Mathjax to test with, so I'm copying from [this post from Mathematics Codidact](https://math.codidact.com/posts/288658).
- ----
- I'm exploring a function that takes a non-negative integer vector $$ \theta\in {\mathbb{N}_0}^4$$ and returns the modulus of each adjacent vector components, wrapping around:
- $$
- s:{\mathbb{N}_0}^4 \rightarrow {\mathbb{N}_0}^4 \\
- $$
- $$
- \begin{align*}
- & s(\boldsymbol{\theta}) =
- s\left(\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \right) =
- \begin{bmatrix}
- |y-x| \\
- |z-y| \\
- |w-z| \\
- |x-w|
- \end{bmatrix}
- \end{align*}
- $$
- Example:
- $$
- \begin{align*}
- & s\left(\begin{bmatrix} 6 \\ 2 \\ 15 \\ 7 \end{bmatrix} \right) =
- \begin{bmatrix}
- |2-6| \\
- |15-2| \\
- |7-15| \\
- |6-7|
- \end{bmatrix} =
- \begin{bmatrix}
- |-4| \\
- |+13| \\
- |-8| \\
- |-1|
- \end{bmatrix} =
- \begin{bmatrix}
- 4 \\
- 13 \\
- 8 \\
- 1
- \end{bmatrix}
- \end{align*}
- $$
- Basic properties of $s$ that I can prove:
- * $ s \text{ is not linear} $
- * $ s(\boldsymbol{0}) = \boldsymbol{0} $
- * $ s(\lambda\boldsymbol{\theta}) = |\lambda|s(\boldsymbol{\theta}) $
- * Let $\boldsymbol\theta' = \boldsymbol\theta - \theta_{min}$, where $\theta_{min}$ is the smallest component of $\boldsymbol\theta$. Then $s(\boldsymbol\theta)=s(\boldsymbol\theta') $
- 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 $\boldsymbol{\theta}\in {\mathbb{N}_0}^4$
- As a purely recreational exercise, my goal is to prove or disprove that this holds *for all* $\boldsymbol{\theta}$:
- $ \forall_{\boldsymbol{\theta} \in {\mathbb{N}_0}^4} \exists_{n\ge 0} \space \space \overbrace{s \circ \ s \circ \dots s}^{n} (\boldsymbol{\theta}) = \boldsymbol{0} $
- Now my questions:
- 1. Is $s$ a common function in some domain or application?
- 1. Which research keyworks would you recommend to find more about $s$ (or similar function)?
- Search expressions like "adjacent element distance" and variations thereof didn't help much. These often lead to pages more on the implementation/coding side of the problem.
- This is purely a recreational problem. There is no real application that I know of - but I would like to know. I'm not a professional mathematician. My background is computer science.
- Thank you
- 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](https://math.codidact.com/posts/288658).
- ----
- I'm exploring a function that takes a non-negative integer vector $$ \theta\in {\mathbb{N}_0}^4$$ and returns the modulus of each adjacent vector components, wrapping around:
- $$
- s:{\mathbb{N}_0}^4 \rightarrow {\mathbb{N}_0}^4 \\
- $$
- $$
- \begin{align*}
- & s(\boldsymbol{\theta}) =
- s\left(\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \right) =
- \begin{bmatrix}
- |y-x| \\
- |z-y| \\
- |w-z| \\
- |x-w|
- \end{bmatrix}
- \end{align*}
- $$
- Example:
- $$
- \begin{align*}
- & s\left(\begin{bmatrix} 6 \\ 2 \\ 15 \\ 7 \end{bmatrix} \right) =
- \begin{bmatrix}
- |2-6| \\
- |15-2| \\
- |7-15| \\
- |6-7|
- \end{bmatrix} =
- \begin{bmatrix}
- |-4| \\
- |+13| \\
- |-8| \\
- |-1|
- \end{bmatrix} =
- \begin{bmatrix}
- 4 \\
- 13 \\
- 8 \\
- 1
- \end{bmatrix}
- \end{align*}
- $$
- Basic properties of $s$ that I can prove:
- * $ s \text{ is not linear} $
- * $ s(\boldsymbol{0}) = \boldsymbol{0} $
- * $ s(\lambda\boldsymbol{\theta}) = |\lambda|s(\boldsymbol{\theta}) $
- * Let $\boldsymbol\theta' = \boldsymbol\theta - \theta_{min}$, where $\theta_{min}$ is the smallest component of $\boldsymbol\theta$. Then $s(\boldsymbol\theta)=s(\boldsymbol\theta') $
- 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 $\boldsymbol{\theta}\in {\mathbb{N}_0}^4$
- As a purely recreational exercise, my goal is to prove or disprove that this holds *for all* $\boldsymbol{\theta}$:
- $ \forall_{\boldsymbol{\theta} \in {\mathbb{N}_0}^4} \exists_{n\ge 0} \space \space \overbrace{s \circ \ s \circ \dots s}^{n} (\boldsymbol{\theta}) = \boldsymbol{0} $
- Now my questions:
- 1. Is $s$ a common function in some domain or application?
- 1. Which research keyworks would you recommend to find more about $s$ (or similar function)?
- Search expressions like "adjacent element distance" and variations thereof didn't help much. These often lead to pages more on the implementation/coding side of the problem.
- This is purely a recreational problem. There is no real application that I know of - but I would like to know. I'm not a professional mathematician. My background is computer science.
- Thank you
#1: Initial revision
by
Monica Cellio
·
2023-08-02T19:57:56Z (over 1 year ago)
I need some Mathjax to test with, so I'm copying from [this post from Mathematics Codidact](https://math.codidact.com/posts/288658).
----
I'm exploring a function that takes a non-negative integer vector $$ \theta\in {\mathbb{N}_0}^4$$ and returns the modulus of each adjacent vector components, wrapping around:
$$
s:{\mathbb{N}_0}^4 \rightarrow {\mathbb{N}_0}^4 \\
$$
$$
\begin{align*}
& s(\boldsymbol{\theta}) =
s\left(\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \right) =
\begin{bmatrix}
|y-x| \\
|z-y| \\
|w-z| \\
|x-w|
\end{bmatrix}
\end{align*}
$$
Example:
$$
\begin{align*}
& s\left(\begin{bmatrix} 6 \\ 2 \\ 15 \\ 7 \end{bmatrix} \right) =
\begin{bmatrix}
|2-6| \\
|15-2| \\
|7-15| \\
|6-7|
\end{bmatrix} =
\begin{bmatrix}
|-4| \\
|+13| \\
|-8| \\
|-1|
\end{bmatrix} =
\begin{bmatrix}
4 \\
13 \\
8 \\
1
\end{bmatrix}
\end{align*}
$$
Basic properties of $s$ that I can prove:
* $ s \text{ is not linear} $
* $ s(\boldsymbol{0}) = \boldsymbol{0} $
* $ s(\lambda\boldsymbol{\theta}) = |\lambda|s(\boldsymbol{\theta}) $
* Let $\boldsymbol\theta' = \boldsymbol\theta - \theta_{min}$, where $\theta_{min}$ is the smallest component of $\boldsymbol\theta$. Then $s(\boldsymbol\theta)=s(\boldsymbol\theta') $
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 $\boldsymbol{\theta}\in {\mathbb{N}_0}^4$
As a purely recreational exercise, my goal is to prove or disprove that this holds *for all* $\boldsymbol{\theta}$:
$ \forall_{\boldsymbol{\theta} \in {\mathbb{N}_0}^4} \exists_{n\ge 0} \space \space \overbrace{s \circ \ s \circ \dots s}^{n} (\boldsymbol{\theta}) = \boldsymbol{0} $
Now my questions:
1. Is $s$ a common function in some domain or application?
1. Which research keyworks would you recommend to find more about $s$ (or similar function)?
Search expressions like "adjacent element distance" and variations thereof didn't help much. These often lead to pages more on the implementation/coding side of the problem.
This is purely a recreational problem. There is no real application that I know of - but I would like to know. I'm not a professional mathematician. My background is computer science.
Thank you