Rank-3 scale theorems: Difference between revisions
→Definitions: Billiard scale: formatting |
|||
| Line 32: | Line 32: | ||
==== Definitions: Billiard scale ==== | ==== Definitions: Billiard scale ==== | ||
Let | Let | ||
* ''w'' be a scale word with signature ''a''<sub>1</sub>X<sub>1</sub>, ..., ''a''<sub>''r''</sub>X<sub>''r''</sub> (i.e. ''w'' is a scale word with ''a''<sub>''i''</sub>-many X<sub>i</sub> steps); | * ''w'' be a scale word with signature ''a''<sub>1</sub>X<sub>1</sub>, ..., ''a''<sub>''r''</sub>X<sub>''r''</sub> (i.e. ''w'' is a scale word with ''a''<sub>''i''</sub>-many X<sub>''i''</sub> steps); | ||
* ''n'' = ''a''<sub>1</sub> + ... + ''a''<sub>''r''</sub> be the length of ''w''; | * ''n'' = ''a''<sub>1</sub> + ... + ''a''<sub>''r''</sub> be the length of ''w''; | ||
* ''L'' be a line of the form ''L''(''t'') = (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + '''v'''<sub>0</sub>, where '''v'''<sub>0</sub> is a constant vector in '''R'''<sup>''r''</sup>. | * ''L'' be a line of the form ''L''(''t'') = (''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub>)''t'' + '''v'''<sub>0</sub>, where '''v'''<sub>0</sub> is a constant vector in '''R'''<sup>''r''</sup>. | ||