Height: Difference between revisions
Save the symbol "p" for p-adic valuation or p-limit; ratios are now consistently denoted "q". |
Explain what this "equivalence class" would commonly be |
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<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math> | <math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math> | ||
Exponentiation and logarithm are such functions commonly used for converting a height between the arithmetic and logarithmic scale. | |||
A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true: | A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true: | ||