I’m happy to report that AIP Chaos published our paper on Finite-Time Braiding Exponents (co-authored with Jean-Luc Thiffeault):
MB. and Thiffeault, J.-L. Finite-time braiding exponents. Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 087407 (2015).
In this paper, we use braid group generators to represent the entangling of trajectories of a dynamical system, and compute the Finite-Time Braiding Exponent (FTBE). FTBE should represent a finite-time, finite-information version of the topological entropy of the flow, and we show evidence that this is really the case.
Mean FTBE and top. entropy correlate, and almost-match as number of strands is increased (Fig. 10 from the paper)
In addition to publishing a paper, we have also released the new version of our MATLAB toolbox braidlab (v.3.2) which was used for all computations in the paper. You can see the release notes on our GitHub repository, as well as download source and compiled versions of the toolbox. Please let us know if you’re using braidlab and, especially, submit any problems with it to our issues page.
I am spending this week at my all-time favorite SIAM DS Snowbird meeting. As this is my 5th straight DS meeting, it really feels like coming home. I’ll be speaking about my work with Jean-Luc Thiffeault on Finite-Time Braiding Exponent (arXiv) on Wednesday at 3p (in Ballroom 2). My talk is in the MS92: Topological Fluid and Mass Dynamics, organized by Stefanella Boatto and Mark Stremler. The slides of the presentation are embedded below. Additionally, I am co-organizing a two-session minisymposium (MS111 and MS124) with Jean-Luc Thiffeault and Sanjeeva Balasuriya on Thursday. Come by and see what our speakers have done on control of fluids and things inside fluids.
Update: The video of the talk is available on SIAM website.
Today I gave a short introduction to compressive sensing, following the article:
Bryan, K., and T. Leise. “Making Do with Less: An Introduction to Compressed Sensing.” SIAM Review 55, no. 3 (January 1, 2013): 547–66. doi:10.1137/110837681.
The examples I’ve shown can be found below:
This is a short overview of the Proper Orthogonal and Dynamic/Koopman Mode Decompositions, which are commonly used in analysis of velocity fields of fluid flows. While I worked with the theoretical side of Koopman modes, I never implemented the numerical code myself; I wrote these notes up a I was teaching myself the basics of numerics of these decompositions, and consequently used the notes for two lectures. The notes are based References at the end of the post. Caveat lector: Notes may contain gross oversimplifications — the emphasis was on understanding and not on precision. I welcome your corrections and comments below. (You can always stop by my Van Vleck office if you’re in Madison to discuss any part of this).
Jean-Luc Thiffeault and I have just uploaded our paper on Finite-Time Braiding Exponents (FTBE) to arXiv. In the paper we study how closely topological entropy of a mixing dynamical system can be approximated by sampling only finitely many trajectories from the flow.
Based on numerical results (obtained using braidlab 3.1), we find that FTBEs of finitely-many trajectories converge to topological entropy as number of trajectories is increased. The paper further explores robustness and dependence of FTBEs on time step, number of trajectories, and their length.
The latest full release of braidlab is out! Braidlab is a MATLAB toolbox which incorporates algorithms for analyzing braid groups of punctured disks in both theoretical and applied contexts. It was primarily written by Jean-Luc Thiffeault, but Michael Allshouse and I have contributed code to it as well. Feel free to direct your questions either at JLT or at me.
You can find a good summary of release updates on the main release-3.0 announcement, but here’s my list of favorites:
- We moved the repository to GitHub. This means that you can (and should) use our Issue Tracker to let us know what went wrong or what you would like to see included in future releases. We are also present on MATLAB Central.
- Installation from source should now work on Matlab 2014b without any special configuration. There are two known installation issues which are out of our reach: conflict of “mex” command with a LaTeX command, and lack of GMP libraries on your system. Make sure you read the installation guidelines in the manual first to see how to resolve these (and other) known problems.
- “Data braids” now have a broader support. This type of braids is useful if you are trying to represent physical trajectories, which have an independent variable, e.g., time, attached to them.
- More functions are implemented as MEX C++ code, which means that they ultimately run faster (some of them have even been parallelized!)
To install, go to the release page, scroll down, download the pre-packaged binaries. If you want to build from source, you can either download the source or even clone our git repository to stay up to date with the latest developments.
In all cases: let us know if braidlab works on your end and if you find it useful.
Last night I had an amazing time at Madison Math Circle, hanging out with a bunch of bright school kids, who were undaunted by either diseases or mathematics. Since I had quite a few questions after the lecture, I decided to post an overview so anyone interested can go over it themselves. For those of you who attended and have further questions, or even wish to explore mathematical dynamics on your own, feel free to leave a comment below or e-mail me at firstname.lastname@example.org and I’ll try to answer all your questions and provide some guidance for future.