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1. Differentiability of the Arrival Time

   https://doi.org/10.1002/cpa.21635  

   Colding, Tobias Holck; Minicozzi, William P (December 2016, Communications on Pure and Applied Mathematics)

   Abstract For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional 2 Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method, a~priori not even differentiable but only satisfying the equation in the viscosity sense . Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black‐Scholes option pricing. From variations of the Sard and Łojasiewicz theorems, we relate differentiability to whether singularities all occur at only finitely many times for flows.© 2016 Wiley Periodicals, Inc. 

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2. Time-dependent Dynamics of the Corona

   https://doi.org/10.3847/2041-8213/ad00bd  

   Mason, Emily I.; Lionello, Roberto; Downs, Cooper; Linker, Jon A.; Caplan, Ronald M.; DeRosa, Marc L. (December 2023, The Astrophysical Journal Letters)

   Abstract We present in this Letter the first global comparison between traditional line-tied steady-state magnetohydrodynamic models and a new, fully time-dependent thermodynamic magnetohydrodynamic simulation of the global corona. To approximate surface magnetic field distributions and magnitudes around solar minimum, we use the Lockheed Evolving Surface-Flux Assimilation Model to obtain input maps that incorporate flux emergence and surface flows over a full solar rotation, including differential rotation and meridional flows. Each time step evolves the previous state of the plasma with a new magnetic field input boundary condition, mimicking photospheric driving on the Sun. We find that this method produces a qualitatively different corona compared to steady-state models. The magnetic energy levels are higher in the time-dependent model, and coronal holes evolve more along the following edge than they do in steady-state models. Coronal changes, as illustrated with forward-modeled emission maps, evolve on longer timescales with time-dependent driving. We discuss implications for active and quiet Sun scenarios, solar wind formation, and widely used steady-state assumptions like potential field source surface calculations. 

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3. Trainable Time Warping: Aligning Time-series in the Continuous-time Domain

   https://doi.org/10.1109/icassp.2019.8682322  

   Khorram, Soheil; McInnis, Melvin G; Mower Provost, Emily (May 2019, International Conference on Acoustics, Speech, and Signal Processing (ICASSP))

   DTW calculates the similarity or alignment between two signals, subject to temporal warping. However, its computational complexity grows exponentially with the number of time-series. Although there have been algorithms developed that are linear in the number of time-series, they are generally quadratic in time-series length. The exception is generalized time warping (GTW), which has linear computational cost. Yet, it can only identify simple time warping functions. There is a need for a new fast, high-quality multisequence alignment algorithm. We introduce trainable time warping (TTW), whose complexity is linear in both the number and the length of time-series. TTW performs alignment in the continuoustime domain using a sinc convolutional kernel and a gradient-based optimization technique. We compare TTW and GTW on S5 UCR datasets in time-series averaging and classification. TTW outperforms GTW on 67.1% of the datasets for the averaging tasks, and 61.2% of the datasets for the classification tasks. 

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4. Field research in the time of the pandemic

   Inouye, David W. (December 2020, Mountain Views Chronicle)

   null (Ed.)

   A short article about the travails of field research during a pandemic 

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5. On the Cover Time of the Emerging Giant

   https://doi.org/10.1137/21M1441468  

   Frieze, Alan M.; Pegden, Wesley; Tkocz, Tomasz (September 2022, SIAM Journal on Discrete Mathematics)