Generator-offset property: Difference between revisions
m →Proposition 3 (Properties of even GO scales): The proof in Theorem 1 did not apply to all even GO scales, just the ones that were SGA. |
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By modular arithmetic we have ''rk'' mod ''n'' = ''k''/2 iff ''r'' = ceil(''n''/2) mod ''n''. (Since gcd(2, ''n'') = 1, 2 is multiplicatively invertible mod ''n'', and we can multiply both sides by 2 to check this.) This proves that the offset, which must be reached after ceil(''n''/2) generator steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in ceil(''n''/2) steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.) | By modular arithmetic we have ''rk'' mod ''n'' = ''k''/2 iff ''r'' = ceil(''n''/2) mod ''n''. (Since gcd(2, ''n'') = 1, 2 is multiplicatively invertible mod ''n'', and we can multiply both sides by 2 to check this.) This proves that the offset, which must be reached after ceil(''n''/2) generator steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in ceil(''n''/2) steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.) | ||
=== Proposition 3 (Properties of even GO scales) === | <!--=== Proposition 3 (Properties of even GO scales) === | ||
A GO scale of even size have the following properties: | A GO scale of even size have the following properties: | ||
# It is a union of two mosses of size ''n''/2 generated by g | # It is a union of two mosses of size ''n''/2 generated by g | ||
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Note that x<sup>''r''</sup>yx<sup>''r''</sup>z (generator x, offset x<sup>''r''</sup>y) is a counterexample to the claim that the offset must be an odd number of steps. | Note that x<sup>''r''</sup>yx<sup>''r''</sup>z (generator x, offset x<sup>''r''</sup>y) is a counterexample to the claim that the offset must be an odd number of steps. | ||
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=== Theorem 4 (Classification of PWF scales) === | === Theorem 4 (Classification of PWF scales) === | ||