Harmonic entropy: Difference between revisions
→Equivalence of exp-UHE and UHE for a \leq 2: fix dropped exponent |
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We can expand the above into a Taylor series as follows: | We can expand the above into a Taylor series as follows: | ||
<math>\displaystyle \text{UHE}_a(c) = \frac{1}{1-a} \left(\log(U(0)) + \frac{\tilde{U}(c)}{U(0)} - \frac{\tilde{U}(c)^2}{U(0)^2} + \frac{\tilde{U}(c)^3}{U(0)^3} - ...\right)</math> | <math>\displaystyle \text{UHE}_a(c) = \frac{1}{1-a} \left(\log(U(0)) + \frac{\tilde{U}(c)}{U(0)} - \frac{\tilde{U}(c)^2}{2 U(0)^2} + \frac{\tilde{U}(c)^3}{3 U(0)^3} - ...\right)</math> | ||
Now, suppose we only care about the behavior of this function up to a constant vertical shift and scaling. Then we can drop the <math>\frac{1}{1-a}</math> term, since it's just a constant scaling, and also get rid of the <math>\log(U(0))</math> term, which simply subtracts a constant offset. This leaves us with | Now, suppose we only care about the behavior of this function up to a constant vertical shift and scaling. Then we can drop the <math>\frac{1}{1-a}</math> term, since it's just a constant scaling, and also get rid of the <math>\log(U(0))</math> term, which simply subtracts a constant offset. This leaves us with | ||
<math>\displaystyle \text{UHE}_a(c) \sim \left(\frac{\tilde{U}(c)}{U(0)} - \frac{\tilde{U}(c)^2}{U(0)^2} + \frac{\tilde{U}(c)^3}{U(0)^3} - ...\right)</math> | <math>\displaystyle \text{UHE}_a(c) \sim \left(\frac{\tilde{U}(c)}{U(0)} - \frac{\tilde{U}(c)^2}{2 U(0)^2} + \frac{\tilde{U}(c)^3}{3 U(0)^3} - ...\right)</math> | ||
where <math>\sim</math> denotes the two sides are now "equivalent" up to a constant shifting and scaling. | where <math>\sim</math> denotes the two sides are now "equivalent" up to a constant shifting and scaling. | ||
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Finally, we can go one step further and multiply the above by the constant <math>U(0)</math>. Doing so, we get | Finally, we can go one step further and multiply the above by the constant <math>U(0)</math>. Doing so, we get | ||
<math>\displaystyle \text{UHE}_a(c) \sim \left(\tilde{U}(c) - \frac{\tilde{U}(c)^2}{U(0)} + \frac{\tilde{U}(c)^3}{U(0)} - ...\right)</math> | <math>\displaystyle \text{UHE}_a(c) \sim \left(\tilde{U}(c) - \frac{\tilde{U}(c)^2}{2 U(0)} + \frac{\tilde{U}(c)^3}{3 U(0)^2} - ...\right)</math> | ||
And we now have a function that is equivalent to our original, up to a constant shift and scaling, but which is fairly easy to analyze asymptotically. | And we now have a function that is equivalent to our original, up to a constant shift and scaling, but which is fairly easy to analyze asymptotically. | ||
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Note again that this does not hold for <math>a \gt 2</math>, where the graph does display a very large difference between UHE and exp-UHE. | Note again that this does not hold for <math>a \gt 2</math>, where the graph does display a very large difference between UHE and exp-UHE. | ||
==Why not Normalized HE?== | ==Why not Normalized HE?== | ||