The Weierstrauss function is the solution to a differential equation whose order is purely imaginary
The Weierstrauss function is the solution to a differential equation whose order is purely imaginary
© 1992 Michael Clarage
huh?
When I do get sufficient graduate students, some of them will continue with the fractional calculus work. I was reminded of a paper my brother and I wrote in the 90's summarizing some of our work in this area. A scan of the paper is attached at the end of this post.
Traditional calculus has first derivative, second derivative, etc, as well as the integral, second integral, etc. This formalism as we all know kicks ass for describing the natural world. The "operator" is only of integer order: 1, 2, 3, ... This is no problem for Euclidean geometry, Gaussian distributions, and 2-body problems. But to describe things like trees, rivers, lightning, lung branching, that is, fractals, a much better tool is to use fractional order derivatives and integrals. How and why that is so is sketched out in the short paper below. I emphasize the word "sketch".
I was reminded of our work in this area while considering the state of climate science. Climate science is ready for someone to bring in fractal dynamics because we need to study the flow of energy from the Sun-solar wind-magnetosphere-ionosphere-troposphere-lithosphere. When jumping through such a wide range of distance & time scales, Nature chooses fractal geometry, not Euclidean; chooses fractal dynamics, not the smooth evolution of average values. Think about air getting into the body through the lungs. The lungs are not a smooth spherical sack, as they would no doubt be modeled by a climate scientist of our day. Similarly the flow of solar energy to the surface of the Earth is not going to be described by modeling the layers in between as smooth shells, nor considering average flows of average values of average energy.
Consider the electrical energy going into and out of the Earth's poles. When we look at average values over a large scale, we see pictures like this, which is the UV light from aurora.
But when we look closer, each bright band is made up of two parts: one current flow up (red) and a current flow down ( blue ).
And, yes, this structure within structure continues as we look closer. Just like the length of the coastline of England keeps increasing as we use a smaller and smaller ruler. Just like the number of air passages in the lung keeps increasing the more we zoom in. Just like the total length of the blood vessels in the human body keeps increasing as we come in closer to the cellular level. (The last quote I heard was that we have about 25,000 miles of blood vessels - enough to wrap around the Earth.)
My point being that just as we would never understand how the lung conveys Oxygen & Carbon Dioxide if we ignored its fractal geometry, so we will never understand how solar energy & matter is conveyed to the Earth if we ignore the fractal geometry of each layer Sun-solar wind-magnetosphere-ionosphere-troposphere-lithosphere.The experimental first step is to measure the fractal dimension of such energy-transforming structures as the aurora. The next step with the theory is to develop fractal dynamics of energy flow.
Moving back to the title of this post, what about that Weierstrauss function? This is one of those pathological functions dreamed up by 19th century mathematicians exploring the nature of the continuum. Weierstrauss published the function as
where 0<a<1, and b is positive odd integer.
If you agree that you can meaningfully define a continuous function as an infinite sum of other continuous functions, then Prof W's function gives us something that is everywhere continuous but nowhere differentiable. Monstrous. And, 120 years later, as you can read below, the brothers Clarage showed that this monstrous function is the solution to a differential equation whose order is purely imaginary, see "Section 6 Pathological functions."