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1. Critical local well‐posedness for the fully nonlinear Peskin problem

   https://doi.org/10.1002/cpa.22139  

   Cameron, Stephen; Strain, Robert M (February 2024, Communications on Pure and Applied Mathematics)

   Abstract We study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space . We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure. 

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2. An efficient partial-differential-equation-based method to compute pressure boundary conditions in regional geodynamic models

   https://doi.org/10.5194/se-13-1107-2022  

   Jourdon, Anthony; May, Dave A. (January 2022, Solid Earth)

   Abstract. Modelling the pressure in the Earth's interior is a common problem in Earth sciences. In this study we propose a method based on the conservation of the momentum of a fluid by using a hydrostatic scenario or a uniformly moving fluid to approximate the pressure. This results in a partial differential equation (PDE) that can be solved using classical numerical methods. In hydrostatic cases, the computed pressure is the lithostatic pressure. In non-hydrostatic cases, we show that this PDE-based approach better approximates the total pressure than the classical 1D depth-integrated approach. To illustrate the performance of this PDE-based formulation we present several hydrostatic and non-hydrostatic 2D models in which we compute the lithostatic pressure or an approximation of the total pressure, respectively. Moreover, we also present a 3D rift model that uses that approximated pressure as a time-dependent boundary condition to simulate far-field normal stresses. This model shows a high degree of non-cylindrical deformation, resulting from the stress boundary condition, that is accommodated by strike-slip shear zones. We compare the result of this numerical model with a traditional rift model employing free-slip boundary conditions to demonstrate the first-order implications of considering “open” boundary conditions in 3D thermo-mechanical rift models. 

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3. Steinberg slices and group-valued moment maps

   https://doi.org/10.1016/j.aim.2022.108344  

   Bălibanu, Ana (June 2022, Advances in Mathematics)

   We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification of this space by taking the closure of each centralizer fiber in the wonderful compactification. By realizing this relative compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic. 

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4. The “Notch” in Unstable Layers and the Stability Minimum in the Tropics

   https://doi.org/10.1175/JCLI-D-24-0135.1  

   Geller, Marvin_A; Wang, Ling (December 2024, Journal of Climate)

   Abstract In a previous paper, we identified a “notch” in unstable layers at Koror (7.3°N, 134.5°E), where there was a relative deficiency in thin unstable layers and a corresponding relative excess in thicker layers, at altitudes centered at 12 km. We hypothesized that this feature was associated with the previously identified stability minimum in the tropics at that same altitude. In this paper, we extend our studies of this notch and its association with the tropical stability minimum by examining other stations in the deep tropics and also some stations at higher latitudes within the tropics. We find that this notch feature is found at all the other radiosonde stations in the deep tropics that we examined. We also find that the annual variations in unstable layer occurrences at stations at higher latitudes within the tropics show variations consistent with our hypothesis that this notch is associated with the region of minimum stability in the tropics at altitudes centered around 12 km, in that the annual variation in this notch feature is consistent with the annual variation of minimum stability in this region. Two factors contribute to the notch feature. One is that the data quality control procedure of the analysis rejects many thin layers due to the small trend-to-noise ratio in the region of minimum stability. The other is that the cloud-top outflow, which was previously identified with the stability minimum, advects thicker unstable layers throughout the deep tropics at the altitudes of the notch. Significance StatementPrevious papers have separately identified a stability minimum in the tropics and a “notch” feature in the thicknesses of unstable atmospheric layers where there are less thin unstable layers and a corresponding excess of thicker unstable layers, both at altitudes around 12 km. We previously hypothesized that these two features were associated with one another. In this paper, we examine this notch feature and the minimum in atmospheric stability at both deep tropical radiosonde stations and stations located at higher latitudes in the tropics, and we find that the annual variation of this notch feature is consistent with the latitudinal migration of the latitudes of the stability minimum. Turbulence associated with this notch feature might be significant for aircraft operations. 

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5. Neural Networks with Sparse Activation Induced by Large Bias: Tighter Analysis with Bias-Generalized NTK

   Yang, Hongru; Jiang, Ziyu; Zhang, Ruizhe; Liang, Yingbin; Wang, Zhangyang (November 2024, Journal of machine learning research)

   We study training one-hidden-layer ReLU networks in the neural tangent kernel (NTK) regime, where the networks' biases are initialized to some constant rather than zero. We prove that under such initialization, the neural network will have sparse activation throughout the entire training process, which enables fast training procedures via some sophisticated computational methods. With such initialization, we show that the neural networks possess a different limiting kernel which we call bias-generalized NTK, and we study various properties of the neural networks with this new kernel. We first characterize the gradient descent dynamics. In particular, we show that the network in this case can achieve as fast convergence as the dense network, as opposed to the previous work suggesting that the sparse networks converge slower. In addition, our result improves the previous required width to ensure convergence. Secondly, we study the networks' generalization: we show a width-sparsity dependence, which yields a sparsity-dependent Rademacher complexity and generalization bound. To our knowledge, this is the first sparsity-dependent generalization result via Rademacher complexity. Lastly, we study the smallest eigenvalue of this new kernel. We identify a data-dependent region where we can derive a much sharper lower bound on the NTK's smallest eigenvalue than the worst-case bound previously known. This can lead to improvement in the generalization bound. 

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