Damn it Grant. Why do responses to you not go to the list by default? ;-)
Grant Holland wrote at 05/17/2013 02:41 PM: > Looks like to me that your p(h) function's sensitivity to human > population size is well-considered. If I understand your parameter > constants h_o and h_f correctly, then I believe the exponent of e in > both of your cases is a positive integer. I believe this means that your > p(h) is monotonically decreasing in both cases. Not quite. The first one is a normal S curve. The second mode is inverted. I don't know if I can add attachments. So, try this: first mode: https://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427eolc4anlkqf second mode: https://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427elo9c75852c So, together, the bimodal function should look like a mesa. > So, the next thing is to consider the acceleration of p(h) - its second > derivative. This means that we are interested in its convexity. I > suspect that it is always convex for positive h. If so, then its > acceleration is always positive. Of course, a more analytical approach > to taking these derivatives is called for. { (e^(h+h_o))/(e^(h+h_o)+1)^2 d/dh = { { -(e^(h+h_o))/(e^(h+h_o)+1)^2 (The sign on h_o doesn't really matter, I suppose.) So, the curvature is positive for the first mode and negative for the second. The 2nd derivative will have the same sign as the 1st derivative, I think, which means the convexity flips at h_o. > So, assuming that the population h is always increasing with time - > probably a reasonable case, then p(t) is also convex. This implies, if I > am correct, that your production function is always accelerating. Is > this correct? Given the above, no. It goes through a high acceleration period near h_o, but much less h << h_o and switches to mode 2 at h >> h_o. > Do these considerations reflect your thinking about technology growth? Well, as I said before, I don't think it's accurate. But I do think my "mesa" function might generally capture what people like Steve _perceive_. I actually think that technology doesn't grow any faster or slower on any variable. But I can see how one might _think_ it does. E.g. with Geoff West's concept of more innovation in higher densities. > On 5/17/13 2:35 PM, glen e. p. ropella wrote: >> But absent the time to put that together, I'll go with something like: >> >> { 1/(1+e^-(h-h_o)), h near h_o >> p(h) = { >> { 1/(1+e^(h+h_f)), h >> h_o >> >> where h is the population of humans and h_o is some >> tech-accelerating-maximum population of humans. h_o becomes some sort >> of "optimal clique size". h_f is some sort of failure size larger >> than h_o. -- glen e. p. ropella, 971-255-2847, http://tempusdictum.com We are drowning in information, while starving for wisdom. -- E.O. Wilson ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
