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I'm trying to understand what is meant by the -(-w etc. in the equation below. Can someone enlighten me?

Source: Section 2.2.3 Case Amplification equation

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  • $\begingroup$ as it's written, it seems as though the piece-wise definition isn't necessary; $-(-w)=w$ ... $\endgroup$ – Zubin Mukerjee Apr 2 '14 at 21:14
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This is a misprint and should read $$ -(-w_{a,i})^\rho, $$ or, equivalently, since this applies when $w_{a,i}\lt0$, $$ -|w_{a,i}|^\rho. $$

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  • $\begingroup$ Well, after the parentheses are rearranged, whether or not the term is negative depends on the parity of $\rho$. EDIT: oops $\endgroup$ – Zubin Mukerjee Apr 2 '14 at 21:15
  • $\begingroup$ No, "after the parentheses are rearranged", $w_{a,i}'$ has the sign of $w_{a,i}$, for every $\rho$. $\endgroup$ – Did Apr 2 '14 at 21:16
  • $\begingroup$ Makes a lot more sense now! +1. $\endgroup$ – Zubin Mukerjee Apr 2 '14 at 21:17
  • $\begingroup$ @ZubinMukerjee Here is an idea for you: read my answer slowly, then ponder it for a while, and then, if you still have a problem, post a comment. Deal? $\endgroup$ – Did Apr 2 '14 at 21:18
  • $\begingroup$ Yeah, sorry about that! Will do in the future :) $\endgroup$ – Zubin Mukerjee Apr 2 '14 at 21:19
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The author seems to be assuming that the unary minus has a higher precedence than exponentiation; i.e. that $-x^y$ means $(-x)^y$ rather than $-(x^y)$.

Without any parentheses $--x^y$ contains a pointless double negative which happens before exponentiation if unary minus has a higher precedence, or after exponentiation if it has a lower precedence.

Therefore, at least one set of parentheses is necessary to express the idea that negation is applied, then the exponent, and then another negation. An opening parenthesis, moreover, must come between the two negation signs, that much is clear. The only question is where to put the matching closing one.

If you believe exponentiation has a higher precedence, then it can be partially parenthesized as $-(-x)^y$. Precedence then resolves the remaining ambiguity in favor of doing the exponentiation first, then the outer negation.

If you believe that exponentiation has a lower precedence than negation, then you must bring the exponent into the parentheses like this: $-(-x^y)$, which has the same structure as the formula in question. This is probably what your author intends, because otherwise if exponentiation is assumed to have high precedence, then this placement of parentheses achieves nothing, leaving a useless double negation.

If you don't know the precedence, you must parenthesize it fully, to eliminate all reliance on precedence: $-((-x)^y)$.

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