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88 comments on General Dispersive Discovery and The Laplace Transform

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yy2asky on October 2, 2008 - 11:22pm Permalink | Subthread | Parent | Comments top

So with an inbetween search speed we would get something between the two?

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WebHubbleTelescope on October 3, 2008 - 1:14am Permalink | Subthread | Parent | Comments top

Yes, you certainly can vary the speed along the path, and this will get you something in between the formulas to the left and right.

For example, here are a couple of sets of graphs demonstrating exponential growth and power-law growth over different container distributions. The first curve shown is the classic Logistic Hubbert curve.


As you can see, the power-law growth starts off faster than exponential but then slows down relative to the exponential. This gives you a much broader peak as it takes longer to sweep through the container volume along the critical interval.

Also what I find interesting is that the tails of the Hubbert peak have nothing to do with the underlying distribution of potential finds, but everything to do with the dispersion of search rates.

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yy2asky on October 3, 2008 - 10:14am Permalink | Subthread | Parent | Comments top

"but everything to do with the dispersion of search rates"
Applying the map to the territory, does it suggest that the various dispersed searchers in the various dispersed geologically and constrained politically basins will broaden the peak but not so greatly affect the shape or area under the tail?

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WebHubbleTelescope on October 3, 2008 - 10:25pm Permalink | Subthread | Parent | Comments top

yes. First, if you look at all those curves, the area underneath each curve is identical. (This is not always obvious on a semi-logarithmic plot, but trust me)

The speed in the search will effect the profile like squeezing a water-filled balloon. If you slow it down one place, it will pop up in another part of the curve

The speed also has an effect on the tail but apparently only in its slope. Once the fast dispersions sweep through their search space container and the slow ones start to approach the boundaries of the container, the actual distribution of discoveries ceases to become important. It essentially amounts to picking up stragglers from all over the search space. This is basically an entropy argument, as all the details in the original distribution (seam, uniform, gamma, etc) get washed out with dispersion.

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88 comments on General Dispersive Discovery and The Laplace Transform

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