Boundary differentiability for inhomogeneous infinity Laplace equations

We study the boundary regularity of the solutions to inhomogeneous infinity Laplace equations. We prove that if $u\in C(\bar{\Omega})$ is a viscosity solution to $\Delta_{\infty}u:=\sum_{i,j=1}^n u_{x_i}u_{x_j}u_{x_ix_j}=f$ with $f\in C(\Omega)\cap L^{\infty}(\Omega)$ and for $x_0\in \partial\Omega$ both $\partial\Omega$ and $g:=u|_{\partial\Omega}$ are differentiable at $x_0$, then u is differentiable at $x_0$

Physiological Noise Filtering in Functional Near-Infrared Spectroscopy Signals Using Wavelet Transform and Long-Short Term Memory Networks

Activated channels of functional near-infrared spectroscopy are typically identified using the desired hemodynamic response function (dHRF) generated by a trial period. However, this approach is not possible for an unknown trial period. In this paper, an innovative method not using the dHRF is proposed, which extracts fluctuating signals during the resting state using maximal overlap discrete wavelet transform, identifies low-frequency wavelets corresponding to physiological noise, trains them using long-short term memory networks, and predicts/subtracts them during the task session. The motivation for prediction is to maintain the phase information of physiological noise at the start time of a task, which is possible because the signal is extended from the resting state to the task session. This technique decomposes the resting state data into nine wavelets and uses the fifth to ninth wavelets for learning and prediction. In the eighth wavelet, the prediction error difference between the with and without dHRF from the 15-s prediction window appeared to be the largest. Considering the difficulty in removing physiological noise when the activation period is near the physiological noise, the proposed method can be an alternative solution when the conventional method is not applicable. In passive brain-computer interfaces, estimating the brain signal starting time is necessary