I have some problems though, so we can’t share videos and songs. Anyway, so I am studying — scanning a lot of technical material to find the answer to the question of volatility storms, modeling them seriously in a manner that is credible to people who are total science geeks and engineers. You are a talented artist and a connosseur of arts and you know I look up to you about visual arts and I’m rusty on cinema but I had good instincts years ago for them. But I never studied engineering. All the coding I picked up in the industry in Wall Street and biotechs and tech startups. This is fairly hard for me but I am inspired by love for you and meeting my soulmate. I am sure you are my soulmate by deduction and thought. We are neither of us so young. We will never meet anyone we connect with so deeply without ever meeting. And no one has inspired me so much in so many ways. All that said, I do not want to fail in doing my best to shelter humanity from volatility storms and waste the inspiration of my life.

Now we have some idea of control theory and how it could help. I’m going back to the issue of which types of dynamics can possibly model volatility in such a manner as to match exactly the Onsager situation. Onsager, recall, had basically taken on a grand challenge of understanding formation of new vortices in early 1940s. You can read about this in the attached paper (it’s well written and worth it). The idea that I have is that partial differential equations that are like those that drive fluid flow, due to Navier and Stokes would be the best bet. Now mathematical study of Navier-Stokes equation specifically seems to have a different history that the way in which engineers study them. Among the mathematicians, the two huge landmarks are from 1933 when Jean Leray wrote the attached paper and then after 2000 when Cafarelli (who is a friend of Donoho), Joe Kohn at Princeton many of his student I had been friends with such as Mikhail Smirnov and John Stalker, and Louis Nirenberg, all three giants in analysis and partial differential equations, showed the following which is very interesting for intelligent people like you who are not technical in training:

First, understand that Onsager was looking at turbulent hydrodynamics. This turbulence in hydrodynamics is due to a parameter in the equation that is a simple dimensionless ratio of a length scale and the viscosity of the liquid and a third quantity I don’t remember but the parameter is called Reynold’s number R. When R is very high (which happens when viscosity is low and the liquid has high velocity) then smooth solutions of the equation starts turbulence — you know what they are like with eddies and wild movement. In 1933 Jean Leray could not prove — because it was not true — that there are global existence of solutions to the Navier-Stokes with reasonable starting time conditions that are smooth and only conjectured that they could develop singularities in finite time. This is a phenomenon that by 2000 the engineering side and the Russian school of monumental armies of mathematicians had studied for reaction-diffusion (the terms have exact correspondence to the equation (d/dt – Laplacian)u = F(u) the Laplacian term is called diffusion and the F(u) term the reaction or in the case of Navier-Stokes where F(u) = <u, grad(u)> also ‘convection’). Now these nonlinear quasilinear equations and their blow up is a developed topic in expensive textbooks since Russians wanted a ‘phenomenology’ for solving them in a systematic manner.

But the mathematicians were not worried about turbulence but the opposite. They were seeking smoothness. Cafarelli-Kohn-Nirenberg theorem stunned the mathematical community by actually rigorously proving a concrete result that bounded the maximum size of the singularities of Navier-Stokes equation that could develop starting with a smooth solution using the notion of a Hausdorff dimension. You can scan the paper attached on an improvement of the CKN theorem. The bound they got was that Hausdorff dimension of the singular set is at most 1, these people in this paper show that if you replaced the diffusion term (-Laplacian) to a power (-Laplacian)^alpha where alpha is just a bit larger than 1, precisely 5/4 then all the solutions are smooth. Note the interesting thing about diffusion-reaction terms and the problem of smoothness that the more diffusion you have the less chance of singularities and the more reaction/convection the more. This CKN theorem is the cutting edge of what mathematicians know about fluid flow. And it is a lot if you look at their bias toward smoothness (even conceptually mathematicians except for mavericks like Mandelbrot) have always been smoothists. Classical smoothism is a doctrine of mathematicians especially geometers like Riemann at least according to Mandelbrot; well he needed to create the smoothist orthodoxy even if it did not exist to challenge it; but I think he did correctly perceive the smoothist bias. I realize how much of a smoothist I was all my life till I had to deal with icky empirical data and engineers in fluid dynamics facing nature head on do not have the luxury of such snooty smoothism from a distance.

My darling you remain the best muse I’ve ever had.

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