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1. Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in $$\dot{H}^{s} \times \dot{H}^{s - 1}$$, $$s> \frac{1}{2}$$

   https://doi.org/10.1093/imrn/rnx323  

   Dodson, Benjamin (February 2018, International Mathematics Research Notices)

   Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $$\dot{H}^{1/2} \times \dot{H}^{-1/2}$$. We show that if the initial data is radial and lies in $$\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$$ for some $$s> \frac{1}{2}$$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11]. 

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2. Asymptotic Network Independence and Step-Size for a Distributed Subgradient Method

   Alex Olshevsky (January 2022, Journal of machine learning research)

   We consider whether distributed subgradient methods can achieve a linear speedup over a centralized subgradient method. While it might be hoped that distributed network of n nodes that can compute n times more subgradients in parallel compared to a single node might, as a result, be n times faster, existing bounds for distributed optimization methods are often consistent with a slowdown rather than speedup compared to a single node. We show that a distributed subgradient method has this “linear speedup” property when using a class of square-summable-but-not-summable step-sizes which include 1/t^β when β ∈ (1/2,1); for such step-sizes, we show that after a transient period whose size depends on the spectral gap of the network, the method achieves a performance guarantee that does not depend on the network or the number of nodes. We also show that the same method can fail to have this “asymptotic network independence” property under the optimally decaying step-size 1/t^{1/2} and, as a consequence, can fail to provide a linear speedup compared to a single node with 1/t^{1/2} step-size. 

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3. Dataset: Simulating Hindered Grain Boundary Diffusion Using the Smoothed Boundary Method

   https://doi.org/10.13011/m3-0mxz-xd55  

   Hanson, Erik; Andrews, W Beck; Powers, Max; Jenkins, Kaila G; Thornton, Katsuyo (January 2024, Materials Commons)

   Data associated with the article "Simulating Hindered Grain Boundary Diffusion Using the Smoothed Boundary Method" [1]. 1. Hanson, E., Andrews, W. B., Powers, M., Jenkins, K. G., & Thornton, K. (2024). Simulating hindered grain boundary diffusion using the smoothed boundary method. Modelling and Simulation in Materials Science and Engineering, 32(5), 055027. 

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4. Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Discontinuity Interface

   https://doi.org/10.1137/23M1568107  

   Cai, Zhiqiang; Choi, Junpyo; Liu, Min (August 2024, SIAM Journal on Scientific Computing)

   We studied the least-squares ReLU neural network (LSNN) method for solving a linear advection-reaction equation with discontinuous solution in [Z. Cai et al., J. Comput. Phys., 443 (2021), 110514]. The method is based on a least-squares formulation and uses a new class of approximating functions: ReLU neural network (NN) functions. A critical and additional component of the LSNN method, differing from other NN-based methods, is the introduction of a properly designed and physics preserved discrete differential operator. In this paper, we study the LSNN method for problems with discontinuity interfaces. First, we show that ReLU NN functions with depth \(\lceil \log\_2(d+1)\rceil+1\) can approximate any \(d\)-dimensional step function on a discontinuity interface generated by a vector field as streamlines with any prescribed accuracy. By decomposing the solution into continuous and discontinuous parts, we prove theoretically that the discretization error of the LSNN method using ReLU NN functions with depth \(\lceil \log\_2(d+1)\rceil+1\) is mainly determined by the continuous part of the solution provided that the solution jump is constant. Numerical results for both two- and three-dimensional test problems with various discontinuity interfaces show that the LSNN method with enough layers is accurate and does not exhibit the common Gibbs phenomena along discontinuity interfaces. 

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5. 1/f Noise Mitigation in an Opto-Mechanical Sensor with a Fabry–Pérot Interferometer

   https://doi.org/10.3390/s24061969  

   Nelson, Andrea M; Sanjuan, Jose; Guzmán, Felipe (March 2024, Sensors)

   Low-frequency and 1/f noise are common measurement limitations that arise in a variety of physical processes. Mitigation methods for these noises are dependent on their source. Here, we present a method for removing 1/f noise of optical origin using a micro-cavity Fabry–Pérot (FP) interferometer. A mechanical modulation of the FP cavity length was applied to a previously studied opto-mechanical sensor. It effectively mimics an up-conversion of the laser frequency, shifting signals to a region where lower white-noise sources dominate and 1/f noise is not present. Demodulation of this signal shifts the results back to the desired frequency range of observation with the reduced noise floor of the higher frequencies. This method was found to improve sensitivities by nearly two orders of magnitude at 1 Hz and eliminated 1/f noise in the range from 1 Hz to 4 kHz. A mathematical model for low-finesse FP cavities is presented to support these results. This study suggests a relatively simple and efficient method for 1/f noise suppression and improving the device sensitivity of systems with an FP interferometer readout. 

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