Harmonic entropy: 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:<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-05 02:52:41 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-05 03:24:15 UTC</tt>.<br>

: The original revision id was <tt>~~515616600~~</tt>.<br>

: The original revision id was <tt>515617058</tt>.<br>

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

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Other spreading functions have also been explored, such as the use of the "Vos function" of a·exp(b|x|) rather than the Gaussian distribution.

Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming it is possible to suitably modify the HE equation so that the curve converges in the limit. ~~The pictures above use these~~ approximations for ~~the widths of each rational rather than~~ actually ~~computing mediants and~~ integrating over the ~~domains as stated~~ above.

Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming that it's possible to suitably modify the HE equation so that the curve converges in the limit.

These approximations have sometimes been used as quick computational substitutes for actually integrating the spreading function over each domain, so long as the resulting probabilities are normalized so that the sum is 1. The pictures above use these approximations.

=Harmonic Rényi Entropy=

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Other spreading functions have also been explored, such as the use of the &quot;Vos function&quot; of a·exp(b|x|) rather than the Gaussian distribution.<br />

<br />

Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming it is possible to suitably modify the HE equation so that the curve converges in the limit. ~~The pictures above use these~~ approximations for ~~the widths of each rational rather than~~ actually ~~computing mediants and~~ integrating over the ~~domains as stated~~ above.<br />

Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming that it's possible to suitably modify the HE equation so that the curve converges in the limit.<br />

<br />

These approximations have sometimes been used as quick computational substitutes for actually integrating the spreading function over each domain, so long as the resulting probabilities are normalized so that the sum is 1. The pictures above use these approximations.<br />

<br />

<h1 id="toc3"><a name="Harmonic Rényi Entropy"></a>Harmonic Rényi Entropy</h1>

@@ 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>2014-07-05 02:52:41 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-05 03:24:15 UTC</tt>.<br>
-: The original revision id was <tt>515616600</tt>.<br>
+: The original revision id was <tt>515617058</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 56: / Line 56: @@
 Other spreading functions have also been explored, such as the use of the "Vos function" of a·exp(b|x|) rather than the Gaussian distribution.
-Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming it is possible to suitably modify the HE equation so that the curve converges in the limit. The pictures above use these approximations for the widths of each rational rather than actually computing mediants and integrating over the domains as stated above.
+Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming that it's possible to suitably modify the HE equation so that the curve converges in the limit.
+These approximations have sometimes been used as quick computational substitutes for actually integrating the spreading function over each domain, so long as the resulting probabilities are normalized so that the sum is 1. The pictures above use these approximations.
 =Harmonic Rényi Entropy=
@@ Line 150: / Line 152: @@
 Other spreading functions have also been explored, such as the use of the &amp;quot;Vos function&amp;quot; of a·exp(b|x|) rather than the Gaussian distribution.&lt;br /&gt;
 &lt;br /&gt;
-Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming it is possible to suitably modify the HE equation so that the curve converges in the limit. The pictures above use these approximations for the widths of each rational rather than actually computing mediants and integrating over the domains as stated above.&lt;br /&gt;
+Finally, it should be noted that the use of Tenney height in bounding the rationals seems to lead to domain widths that are proportional to 1/sqrt(nd) for each rational, and that the use of Weil-bounded subsets of the rationals leads to domain widths that are proportional to 1/max(n,d) for each rational. It is a conjecture of Erlich's that this is true in the limit as N→∞, assuming that it's possible to suitably modify the HE equation so that the curve converges in the limit.&lt;br /&gt;
+&lt;br /&gt;
+These approximations have sometimes been used as quick computational substitutes for actually integrating the spreading function over each domain, so long as the resulting probabilities are normalized so that the sum is 1. The pictures above use these approximations.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Harmonic Rényi Entropy"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Harmonic Rényi Entropy&lt;/h1&gt;