Recoverability: Difference between revisions
Wikispaces>genewardsmith **Imported revision 539385306 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 539386598 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-02 13: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-02 13:58:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>539386598</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 10: | Line 10: | ||
W∨2 consists of two segments. The first segment consists of coefficients for basis elements (which are (r-1)-fold wedge products of primes) containing 2 in the product. Since we are examining W∨2, these coefficients are all identically 0. The second segment consists of the initial segment of W itself, being all coefficients for basis elements in W containing 2. Then R = (W∨2)∧J consists of three segments. The first segment consists of coefficients for basis elements with 2 in the product. Because of the zeros in the initial segment of W∨2, these coefficients are identical to the initial segment of W, meaning the coefficients of those basis elements with 2 in the product. There is then a segment where log₂(2) plays a role, and a final segment where only logs of odd primes are involved in the calculation. It follows that Ƹ = W - R also has three segments. The first segment consists of zeros, the second segment corresponds to the second segment of R, and the third segment to the third segment of R. The second segment of Ƹ has coefficients identical to the coefficients of W∧J. | W∨2 consists of two segments. The first segment consists of coefficients for basis elements (which are (r-1)-fold wedge products of primes) containing 2 in the product. Since we are examining W∨2, these coefficients are all identically 0. The second segment consists of the initial segment of W itself, being all coefficients for basis elements in W containing 2. Then R = (W∨2)∧J consists of three segments. The first segment consists of coefficients for basis elements with 2 in the product. Because of the zeros in the initial segment of W∨2, these coefficients are identical to the initial segment of W, meaning the coefficients of those basis elements with 2 in the product. There is then a segment where log₂(2) plays a role, and a final segment where only logs of odd primes are involved in the calculation. It follows that Ƹ = W - R also has three segments. The first segment consists of zeros, the second segment corresponds to the second segment of R, and the third segment to the third segment of R. The second segment of Ƹ has coefficients identical to the coefficients of W∧J. | ||
Assuming as we do throughout this article that multivals are expressed in unweighted coordinates, we may define a norm on them by the L∞ norm, so that ]M[ equals the maximum of the absolute values of the coefficients of the multival M in unweighted coordinates. Then W is recoverable precisely when ]Ƹ[ < 1/2. Since some of the coefficients of Ƹ are identical to the coefficients of W∧J, it follows that ]W∧J[ ≤ ]Ƹ[ . But ]W∧J[ is a measure of relative error, hence if relative error is less than 600 cents/octave | Assuming as we do throughout this article that multivals are expressed in unweighted coordinates, we may define a norm on them by the L∞ norm, so that ]M[ equals the maximum of the absolute values of the coefficients of the multival M in unweighted coordinates. Then W is recoverable precisely when ]Ƹ[ < 1/2. Since some of the coefficients of Ƹ are identical to the coefficients of W∧J, it follows that ]W∧J[ ≤ ]Ƹ[ . But ]W∧J[ is a measure of relative error, hence if W is recoverable, relative error is less than 600 cents/octave. | ||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Recoverability</title></head><body>Suppose W is a p-limit rank r wedgie, and so of dimension n = π(p). Let R be the p-limit rank r wedgie (W∨2)∧J, where J = &lt;1 log₂(3) log₂(5) ... log₂(p)|. Then if rounding the coefficients of R to the nearest integer gives W, W is <em>recoverable</em>. <br /> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Recoverability</title></head><body>Suppose W is a p-limit rank r wedgie, and so of dimension n = π(p). Let R be the p-limit rank r wedgie (W∨2)∧J, where J = &lt;1 log₂(3) log₂(5) ... log₂(p)|. Then if rounding the coefficients of R to the nearest integer gives W, W is <em>recoverable</em>. <br /> | ||
| Line 16: | Line 18: | ||
W∨2 consists of two segments. The first segment consists of coefficients for basis elements (which are (r-1)-fold wedge products of primes) containing 2 in the product. Since we are examining W∨2, these coefficients are all identically 0. The second segment consists of the initial segment of W itself, being all coefficients for basis elements in W containing 2. Then R = (W∨2)∧J consists of three segments. The first segment consists of coefficients for basis elements with 2 in the product. Because of the zeros in the initial segment of W∨2, these coefficients are identical to the initial segment of W, meaning the coefficients of those basis elements with 2 in the product. There is then a segment where log₂(2) plays a role, and a final segment where only logs of odd primes are involved in the calculation. It follows that Ƹ = W - R also has three segments. The first segment consists of zeros, the second segment corresponds to the second segment of R, and the third segment to the third segment of R. The second segment of Ƹ has coefficients identical to the coefficients of W∧J.<br /> | W∨2 consists of two segments. The first segment consists of coefficients for basis elements (which are (r-1)-fold wedge products of primes) containing 2 in the product. Since we are examining W∨2, these coefficients are all identically 0. The second segment consists of the initial segment of W itself, being all coefficients for basis elements in W containing 2. Then R = (W∨2)∧J consists of three segments. The first segment consists of coefficients for basis elements with 2 in the product. Because of the zeros in the initial segment of W∨2, these coefficients are identical to the initial segment of W, meaning the coefficients of those basis elements with 2 in the product. There is then a segment where log₂(2) plays a role, and a final segment where only logs of odd primes are involved in the calculation. It follows that Ƹ = W - R also has three segments. The first segment consists of zeros, the second segment corresponds to the second segment of R, and the third segment to the third segment of R. The second segment of Ƹ has coefficients identical to the coefficients of W∧J.<br /> | ||
<br /> | <br /> | ||
Assuming as we do throughout this article that multivals are expressed in unweighted coordinates, we may define a norm on them by the L∞ norm, so that ]M[ equals the maximum of the absolute values of the coefficients of the multival M in unweighted coordinates. Then W is recoverable precisely when ]Ƹ[ &lt; 1/2. Since some of the coefficients of Ƹ are identical to the coefficients of W∧J, it follows that ]W∧J[ ≤ ]Ƹ[ . But ]W∧J[ is a measure of relative error, hence if relative error is less than 600 cents/octave | Assuming as we do throughout this article that multivals are expressed in unweighted coordinates, we may define a norm on them by the L∞ norm, so that ]M[ equals the maximum of the absolute values of the coefficients of the multival M in unweighted coordinates. Then W is recoverable precisely when ]Ƹ[ &lt; 1/2. Since some of the coefficients of Ƹ are identical to the coefficients of W∧J, it follows that ]W∧J[ ≤ ]Ƹ[ . But ]W∧J[ is a measure of relative error, hence if W is recoverable, relative error is less than 600 cents/octave.</body></html></pre></div> | ||