Ternary parallelogram scales are MOS substitution: Difference between revisions
m Removing from Category:Math using Cat-a-lot |
Sintel bot (talk | contribs) Fix <math> environments for MathJax v4 (align->aligned, etc.) |
||
| Line 37: | Line 37: | ||
For any integers ''i''<sub>0</sub> and ''j''<sub>0</sub>, if {{nowrap|φ((''i''<sub>0</sub>, ''j''<sub>0</sub>)) {{=}} ''a'',}} then for any (''i'', ''j'') in [0 : ''m''] × [0 : ''n''], we have | For any integers ''i''<sub>0</sub> and ''j''<sub>0</sub>, if {{nowrap|φ((''i''<sub>0</sub>, ''j''<sub>0</sub>)) {{=}} ''a'',}} then for any (''i'', ''j'') in [0 : ''m''] × [0 : ''n''], we have | ||
<math>\begin{ | <math>\begin{aligned}\varphi((i_0 + i, j_0 + j)) &= \varphi((i_0, j_0)) + \varphi((i, j)) | ||
\\ &= a + \varphi((i, j)),\end{ | \\ &= a + \varphi((i, j)),\end{aligned}</math> | ||
as φ is a homomorphism. Hence corresponding elements of any two ''m'' × ''n'' windows get the same {{nowrap|ℤ/''mn''ℤ}} labeling under φ modulo a shift. | as φ is a homomorphism. Hence corresponding elements of any two ''m'' × ''n'' windows get the same {{nowrap|ℤ/''mn''ℤ}} labeling under φ modulo a shift. | ||