Recoverability
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Original Wikitext content:
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 = <1 log₂(3) log₂(5) ... log₂(p)|. Then if rounding the coefficients of R to the nearest integer gives W, W is //recoverable//. 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 ]Ƹ[ ≤ ]W∧J[ and so if ]W∧J[ < 1/2 then W is recoverable. But ]W∧J[ is a measure of relative error, hence for small enough relative error, W is recoverable.
Original HTML content:
<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 = <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 /> <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 /> 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 ]Ƹ[ ≤ ]W∧J[ and so if ]W∧J[ < 1/2 then W is recoverable. But ]W∧J[ is a measure of relative error, hence for small enough relative error, W is recoverable.</body></html>