MOS substitution: Difference between revisions
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=== If the template is a primitive MOS, and for some perfect generators <math>p_T, p_F, \ \left|p_T\right|_\mathbf{X} = \left|p_F\right|,</math> then MOS substitution yields almost parallelograms in the lattice === | === If the template is a primitive MOS, and for some perfect generators <math>p_T, p_F, \ \left|p_T\right|_\mathbf{X} = \left|p_F\right|,</math> then MOS substitution yields almost parallelograms in the lattice === | ||
With the additional assumption that the number of X's in a perfect generator ''p''<sub>''T''</sub> of the template MOS be a generator class of the filling MOS, the generator sequence yields ''q'' parallel chains ''C''<sub>1</sub>, | With the additional assumption that the number of X's in a perfect generator ''p''<sub>''T''</sub> of the template MOS be a generator class of the filling MOS, the generator sequence yields ''q'' parallel chains ''C''<sub>1</sub>, | ||
..., ''C''<sub>''q''</sub> of the aggregate generator. The offset between ''C''<sub>''i''</sub> and ''C''<sub>''i''+1</sub> is equal to subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>), where ''p''<sub>''T''</sub> and ''p''<sub>''F''</sub> are perfect generators (of appropriate lengths) of the template and filling MOSes, respectively. The aggregate generator is subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>'' | ..., ''C''<sub>''q''</sub> of the aggregate generator. The offset between ''C''<sub>''i''</sub> and ''C''<sub>''i''+1</sub> is equal to subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>), where ''p''<sub>''T''</sub> and ''p''<sub>''F''</sub> are perfect generators (of appropriate lengths) of the template and filling MOSes, respectively. The aggregate generator is subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>''r''</sup>), where ''G'' is the period of the filling MOS. | ||
Hence in the GS, | Hence in the GS, | ||
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* <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a prefix of the last row. | * <math>\{\mathbf{a} + i\mathbf{v} + (m-1)\mathbf{w}\}_{i=0}^{b}</math> is a prefix of the last row. | ||
In the above case, {{nowrap| ''n'' {{=}} ''q'' | '''v''' {{=}} subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>) | and '''w''' {{=}} subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>'' | In the above case, {{nowrap| ''n'' {{=}} ''q'' | '''v''' {{=}} subst(''p''<sub>''T''</sub>, '''X''', ''p''<sub>''F''</sub>) | and '''w''' {{=}} subst((''p''<sub>''T''</sub>)<sup>''q''</sup>, '''X''', ''G''<sup>''r''</sup>) (the aggregate generator)}}. | ||
The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram. | The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram. | ||