Height: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>2013-08-31 06:53:49 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-08-31 20:57:07 UTC</tt>.

: The original revision id was <tt>~~447782290~~</tt>.

: The original revision id was <tt>447860730</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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# Given any constant C, there are finitely many elements q such that H(q) ≤ C.

# H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.

# H(q) = H(1) iff q = 1.

# H(q) = H(1/q)

# H(q^n) ≥ H(q) for any non-negative integer n

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[[math]]

A **semi-height** is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:

A **semi-height** is a function which does not obey criteria #3 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:

[[math]]

2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q

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A height function H(q) on the positive rationals q should fulfill the following criteria:

<ol><li>Given any constant C, there are finitely many elements q such that H(q) ≤ C.</li><li>H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.</li><li>H(q) = H(1/q)</li><li>H(q^n) ≥ H(q) for any non-negative integer n</li></ol>

<ol><li>Given any constant C, there are finitely many elements q such that H(q) ≤ C.</li><li>H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.</li><li>H(q) = H(1) iff q = 1.</li><li>H(q) = H(1/q)</li><li>H(q^n) ≥ H(q) for any non-negative integer n</li></ol>

If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:

<!-- ws:start:WikiTextMathRule:0:

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--><script type="math/tex">H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</script>

A semi-height is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:

A semi-height is a function which does not obey criteria #3 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:

<!-- ws:start:WikiTextMathRule:1:

[[math]]&lt;br/&gt;

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-08-31 06:53:49 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-08-31 20:57:07 UTC</tt>.<br>
-: The original revision id was <tt>447782290</tt>.<br>
+: The original revision id was <tt>447860730</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 12: / Line 12: @@
 # Given any constant C, there are finitely many elements q such that H(q) ≤ C.
 # H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.
+# H(q) = H(1) iff q = 1.
 # H(q) = H(1/q)
 # H(q^n) ≥ H(q) for any non-negative integer n
@@ Line 20: / Line 21: @@
 [[math]]
-A **semi-height** is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:
+A **semi-height** is a function which does not obey criteria #3 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:
 [[math]]
 ^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q
@@ Line 95: / Line 96: @@
 &lt;br /&gt;
 A height function H(q) on the positive rationals q should fulfill the following criteria:&lt;br /&gt;
-&lt;ol&gt;&lt;li&gt;Given any constant C, there are finitely many elements q such that H(q) ≤ C.&lt;/li&gt;&lt;li&gt;H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.&lt;/li&gt;&lt;li&gt;H(q) = H(1/q)&lt;/li&gt;&lt;li&gt;H(q^n) ≥ H(q) for any non-negative integer n&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
+&lt;ol&gt;&lt;li&gt;Given any constant C, there are finitely many elements q such that H(q) ≤ C.&lt;/li&gt;&lt;li&gt;H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.&lt;/li&gt;&lt;li&gt;H(q) = H(1) iff q = 1.&lt;/li&gt;&lt;li&gt;H(q) = H(1/q)&lt;/li&gt;&lt;li&gt;H(q^n) ≥ H(q) for any non-negative integer n&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
 If we have a function F(x) which is strictly increasing on the positive reals, then F(H(q)) will rank elements in the same order as H(q). We can therefore establish the following equivalence relation:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
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   --&gt;&lt;script type="math/tex"&gt;H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-A &lt;strong&gt;semi-height&lt;/strong&gt; is a function which does not obey criteria #1 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:&lt;br /&gt;
+A &lt;strong&gt;semi-height&lt;/strong&gt; is a function which does not obey criteria #3 above in the strictest sense, so that there is a rational number q ≠ 1 such that H(q) = H(1), resulting in an equivalence relation on its elements. An example would be octave-equivalence, where two ratios p and q are considered equivalent if the following is true:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:1:
 [[math]]&amp;lt;br/&amp;gt;