[math-fun] How does this function manage to oscillate on the unit interval?
https://www.wolframcloud.com/objects/d4fa673b-0423-42fb-b7b5-0f70a7c14c6e Sidarth Ghoshal has exactly determined the mean value and amplitude of an O(10^-6) ripple in a function whose definition appears nonoscillatory. It graphs like the error in a Remez minimax approximation, even unto the ripples Chebychummying at the endpoints. But unboundedly. Worse, Sid says there are O(10^-13) ripples on the O(10^-6) ripples. Whether or not he discovers their formula, I consider Sid's present formula a breakthrough. —rwg
EEK! I carelessly let ries substitute Out[107]= Sqrt[(2 Csch[(2 \[Pi]^2)/Log[2]])/Log[2]] -> 2 E^(-π^2/Log[2])/Sqrt[Log[2]] In[109]:= N[Subtract @@ %%, 9] Out[109]= 1.44664592*10^-31 Quad precision fraud! Fixed at https://www.wolframcloud.com/objects/637b5c6b-f016-43e1-8d1e-935506bb4d24 A few minutes ago, yesterday's Cloud link below was 404ing. Now it's back. How long are these objects supposed to last? —rwg On Thu, Mar 21, 2019 at 3:28 AM Bill Gosper <billgosper@gmail.com> wrote:
https://www.wolframcloud.com/objects/d4fa673b-0423-42fb-b7b5-0f70a7c14c6e Sidarth Ghoshal has exactly determined the mean value and amplitude of an O(10^-6) ripple in a function whose definition appears nonoscillatory. It graphs like the error in a Remez minimax approximation, even unto the ripples Chebychummying at the endpoints. But unboundedly. Worse, Sid says there are O(10^-13) ripples on the O(10^-6) ripples. Whether or not he discovers their formula, I consider Sid's present formula a breakthrough. —rwg
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Bill Gosper