## 5.321. period_vectors

Origin
Constraint

$\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}\left(\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳},\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝙲𝚃𝚁𝚂}\right)$

Types
 $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝙲𝚃𝚁}$ $\mathrm{𝚊𝚝𝚘𝚖}$
Arguments
 $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚎𝚌}-\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}\right)$ $\mathrm{𝙲𝚃𝚁𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚌𝚝𝚛}-\mathrm{𝙲𝚃𝚁}\right)$
Restrictions
 $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|\ge 1$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁},\mathrm{𝚟𝚊𝚛}\right)$ $\mathrm{𝙲𝚃𝚁}\in \left[=,\ne ,<,\ge ,>,\le \right]$ $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}\ge 1$ $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}\le |\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$ $\mathrm{𝚜𝚊𝚖𝚎}_\mathrm{𝚜𝚒𝚣𝚎}$$\left(\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂},\mathrm{𝚟𝚎𝚌}\right)$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝙲𝚃𝚁𝚂},\mathrm{𝚌𝚝𝚛}\right)$ $|\mathrm{𝙲𝚃𝚁𝚂}|=|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|$
Purpose

Let us note ${\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{0},{\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{1},\cdots ,{\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{n-1}$ the vectors of the $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ collection, and $d$ the number of components of each vector (all vectors have the same size). $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}$ is the period of the sequence of vectors ${\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{0},{\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{1},\cdots ,{\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{n-1}$ according to constraints $\mathrm{𝙲𝚃𝚁𝚂}$. This means that $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}$ is the smallest natural number such that $\forall i\in \left[0,n-\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}-1\right],\forall j\in \left[0,d-1\right]:{\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{i}.\mathrm{𝚟𝚎𝚌}\left[j\right]\mathrm{𝙲𝚃𝚁𝚂}\left[j\right]{\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}}_{i+\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}}.\mathrm{𝚟𝚎𝚌}\left[j\right]$.

Example
$\left(\begin{array}{c}3,〈\begin{array}{c}\mathrm{𝚟𝚎𝚌}-〈1,0〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈1,5〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈4,4〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈1,0〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈1,5〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈4,4〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈1,0〉,\hfill \\ \mathrm{𝚟𝚎𝚌}-〈1,5〉\hfill \end{array}〉,\hfill \\ 〈=,=〉\hfill \end{array}\right)$

The $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}_\mathrm{𝚟𝚎𝚌𝚝𝚘𝚛𝚜}$ constraint holds since its first argument $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}=3$ is equal (i.e., since $\mathrm{𝙲𝚃𝚁𝚂}$ is set to $〈=,=〉$) to the period of the sequence $\mathrm{𝚟𝚎𝚌}-〈1,0〉$, $\mathrm{𝚟𝚎𝚌}-〈1,5〉$, $\mathrm{𝚟𝚎𝚌}-〈4,4〉$, $\mathrm{𝚟𝚎𝚌}-〈1,0〉$, $\mathrm{𝚟𝚎𝚌}-〈1,5〉$, $\mathrm{𝚟𝚎𝚌}-〈4,4〉$, $\mathrm{𝚟𝚎𝚌}-〈1,0〉$, $\mathrm{𝚟𝚎𝚌}-〈1,5〉$.

Typical
 $\mathrm{𝙲𝚃𝚁}\in \left[=\right]$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁}|>1$ $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}>1$ $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}<|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|$ $|\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}|>2$
Symmetry

Items of $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ can be reversed.

Arg. properties
• Functional dependency: $\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}$ determined by $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$ and $\mathrm{𝙲𝚃𝚁𝚂}$.

• Prefix-contractible wrt. $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$.

• Suffix-contractible wrt. $\mathrm{𝚅𝙴𝙲𝚃𝙾𝚁𝚂}$.

specialisation: $\mathrm{𝚙𝚎𝚛𝚒𝚘𝚍}$ (vector replaced by $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$).