SIAM Student Chapter Seminar: Difference between revisions
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|VV911&Zoom | |VV911&Zoom | ||
|Xiaoyu Dong (UMich) | |Xiaoyu Dong (UMich) | ||
|Approximately Hadamard matrices and Riesz bases in frames | |Approximately Hadamard matrices and Riesz bases in frames | ||
|- | |- | ||
|3/22 | |3/22 | ||
Revision as of 23:47, 11 March 2024
- When: Fridays at 1 PM unless noted otherwise
- Where: 9th floor lounge (we will also broadcast the virtual talks on the 9th floor lounge with refreshments)
- Organizers: Yahui Qu, Peiyi Chen, Shi Chen and Zaidan Wu
- Faculty advisers: Jean-Luc Thiffeault, Steve Wright
- To join the SIAM Chapter mailing list: email siam-chapter+join@g-groups.wisc.edu.
- Zoom link: https://uwmadison.zoom.us/j/97976615799?pwd=U2xFSERIcnR6M1Y1czRmTjQ1bTFJQT09
- Passcode: 281031
Spring 2024
| Date | Location | Speaker | Title |
|---|---|---|---|
| 2/2 | VV911 | Thomas Chandler (UW-Madison) | Fluid–body interactions in anisotropic fluids |
| 3/8 | Ingraham 214 | Danyun He (Harvard) | Energy-positive soaring using transient turbulent fluctuations |
| 3/15 | VV911&Zoom | Xiaoyu Dong (UMich) | Approximately Hadamard matrices and Riesz bases in frames |
| 3/22 | VV911&Zoom | Mengjin Dong (UPenn) | TBD |
| 4/5 | VV911 | Sixu Li (UW-Madison) | TBD |
| 4/12 | VV911&Zoom | Anjali Nair (UChicago) | TBD |
| 4/19 | VV911 | Jingyi Li (UW-Madison) | TBD |
| 5/3 | Bella Finkel (UW-Madison) | TBD |
Abstracts
February 2, Thomas Chandler (UW-Madison): Fluid anisotropy, or direction-dependent response to deformation, can be observed in biofluids like mucus or, at a larger scale, self-aligning swarms of active bacteria. A model fluid used to investigate such environments is a nematic liquid crystal. In this talk, we will use complex variables to analytically solve for the interaction between bodies immersed in liquid crystalline environments. This approach allows for the solution of a wide range of problems, opening the door to studying the role of body geometry, liquid crystal anchoring conditions, and deformability. Shape-dependent forces between bodies, surface tractions, and analogues to classical results in fluid dynamics will also be discussed.
March 8, Danyun He (Harvard University): The ability of birds to soar in the atmosphere is a fascinating scientific problem. It relies on an interplay between the physical processes governing atmospheric flows, and the capacity of birds to process cues from their environment and learn complex navigational strategies. Previous models for soaring have primarily taken advantage of thermals of ascending hot air to gain energy. Yet, it remains unclear whether energy loss due to drag can be overcome by extracting work from transient turbulent fluctuations. In this talk, I will present a recent work that we look at the alternative scenario of a glider navigating in an idealized model of a turbulent fluid where no thermals are present. First, I will show the numerical simulations of gliders navigating in a kinematic model that captures the spatio-temporal correlations of atmospheric turbulence. Energy extraction is enabled by an adaptive algorithm based on Monte Carlo tree search that dynamically filters acquired information about the flow to plan future paths. Then, I will demonstrate that for realistic parameter choices, a glider can navigate to gain height and extract energy from flow. Glider paths reflect patterns of foraging, where exploration of the flow is interspersed with bouts of energy extraction through localized spirals. As such, this work broadens our understanding of soaring, and extends the range of scenarios where soaring is known to be possible.
March 15, Xiaoyu Dong (University of Michigan, Ann Arbor): An $n \times n$ matrix with $\pm 1$ entries which acts on $\R^n$ as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools we construct matrices with $\pm 1$ entries which act as approximate scaled isometries in $\R^n$ for all $n \in \N$. More precisely, the matrices we construct have condition numbers bounded by a constant independent of $n$.
Using this construction, we establish a phase transition for the probability that a random frame contains a Riesz basis. Namely, we show that a random frame in $\R^n$ formed by $N$ vectors with independent identically distributed coordinate having a non-degenerate symmetric distribution contains many Riesz bases with high probability provided that $N \ge \exp(Cn)$. On the other hand, we prove that if the entries are subgaussian, then a random frame fails to contain a Riesz basis with probability close to $1$ whenever $N \le \exp(cn)$, where $c<C$ are constants depending on the distribution of the entries.