Below are two surprising mathematical results which I found interesting recently. They kind of demonstrate some really fascinating ‘arbitrarily complex’ behaviours of very simple objects, and are primarily useful for philosophical grounding in terms of understanding tools used in science, such as data fitting and mathematical modelling. Continue reading
This has been an incredibly busy and hectic year, but overall an extraordinarily rewarding one. I will catalogue some of what I did here in case anyone is interested, and point out some of the things that will hopefully happen early next year. I do not know how often I plan to blog going forward, but I think having at the very least one reminiscence a year about things that I have been involved in is useful.
As with the previous post, I want to mention some other work I did related to my DPhil course. This project was about implementing an algorithm to numerically compute conformal maps between simply connected domains and the unit disk. By the Riemann Mapping Theorem we know that for every such domain, a map must exist, but finding it analytically is rarely easy if possible at all. A good discussion of these ideas can be found, for instance, in Terence Tao’s lecture notes on the subject. Given a domain, there are several ways to numerically construct a map which approximates the conformal map guaranteed by this Theorem, such as by approximating the domain by polygons and using Schwarz-Christoffel maps to map it to the unit disk.
I implemented the Geodesic or Zipper algorithm due to Don Marshall. The full code (in Matlab) can be found here, along with a few graphical tools to explore it. It has some precision problems for domains with “sharp” boundaries for the inverse map (that is, mapping the unit disk to a given domain) but overall I’m pretty happy with the results. I did not include all of the approximations one could to make the algorithm both more accurate and more efficient, but I may come back to this later and do that, as well as demonstrate some of the applications of these maps. If you’re interested, I would also check out this Thesis on the topic which I found to be useful. It also contains Python code for this algorithm as well as another approach using sphere packing. Below I will include a few examples of what this code can generate, but I encourage you to download it and play with it yourself. As always, comments and questions are appreciated! Continue reading
“…the equations we deal with are probably more complicated than even most physical scientists are accustomed to. This is because the phenomena we are attempting to describe are generally more complex than most physical systems, although it may reflect our own ineptness in perceiving their underlying simplicity.” -James Murray, Mathematical Biology
Due to the complexity of modelling biological phenomena, analytical and numerical approaches are often used together to give a more thorough understanding of a model than either could provide alone. This is particularly true in the area of mathematical physiology, where there has been a dual development of increasingly sophisticated models, as well as analytical and numerical tools to analyze these models. In this post I want to describe some visualizations of simple models in this area, as well as reference some useful tools to explore these simulations further. These visualizations complement the lecture notes for a corresponding course that I tutored last year, which can be found here for the moment, and a much older version of the notes can be found here. Continue reading
In writing up several recent papers I have spent some time reading about Reaction-Diffusion systems and the kinds of behaviour they can have. I will likely blog about these kinds of equations more often, as well as applications of them to chemical, biological, and other systems. Today I want to just record some interesting observations from the literature. This will be about the existence of large time asymptotic solutions to these equations. Roughly speaking, these must be either uniform or non-uniform steady states, or more exotic time-periodic solutions such as oscillations, spatiotemporal waves, and uniform or spatial chaos. Other interesting transient behaviours can occur, such as wave propagation, but here I’ll focus entirely on the long time dynamics, and specifically on the existence of stable equilibria. For a broad overview of these models in some detail see . Continue reading
I want to review some aspects of dynamical systems theory for a class of dissipative systems which are particularly simple. These are systems that posses a Lyapunov function. Dissipative systems are mathematical objects that evolve in time, but remain in some bounded region of state-space for all times. Below I will define the concept of a Lyapunov function, how it relates to the long-time dynamics of differential equations, and finally develop some intuition for why the (scalar) Reaction-Diffusion equation is relatively simple in terms of dynamics, while Reaction-Diffusion systems can have much richer structure. Continue reading
Today I want to talk about dynamical systems on networks. A good review article can be found here. In this post I want to focus on the interplay between the topology of the underlying network, and the asymptotic dynamics of the system. I will focus on two aspects of this area that I have found particularly interesting. Continue reading