Multiple Semiclassical Standing Waves for Fractional Nonlinear Schr\"{o}dinger Equations

Via a Lyapunov-Schmidt reduction, we obtain multiple semiclassical solutions to a class of fractional nonlinear Schr\"odinger equations. Precisely, we consider \begin{equation*} \varepsilon^{2s}(-\Delta)^{s}u+u+V(x)u=|u|^{p-1}u,\quad u\in H^s(\mathbf R^n), \end{equation*} where $04-4s$, $12s$) and $1<p<\infty$ (if $n\le 2s$), $V(x)$ is a non-negative potential function. If $V$ is a sufficiently smooth bounded function with a non-degenerate compact critical manifold $M$, then, when $\varepsilon$ is sufficiently small, there exist at least $l(M)$ semiclassical solutions, where $l(M)$ is the cup length of $M$.Comment: Some error corrected, several references adde

Deep neural network approximations for the stable manifolds of the Hamilton-Jacobi equations

As the Riccati equation for control of linear systems, the Hamilton-Jacobi-Bellman (HJB) equations play a fundamental role for optimal control of nonlinear systems. For infinite-horizon optimal control, the stabilizing solution of HJB equation can be represented by the stable manifold of the associated Hamiltonian system. In this paper, we study the neural network (NN) semiglobal approximation of the stable manifold. The main contribution includes two aspects: firstly, from the mathematical point of view, we rigorously prove that if an approximation is sufficiently close to the exact stable manifold of the HJB equation, then the corresponding control derived from this approximation is near optimal. Secondly, we propose a deep learning method to approximate the stable manifolds, and then numerically compute optimal feedback controls. The algorithm is devised from geometric features of the stable manifold, and relies on adaptive data generation by finding trajectories randomly in the stable manifold. The trajectories are found by solving two-point boundary value problems (BVP) locally near the equilibrium and extending the local solution by initial value problems (IVP) for the associated Hamiltonian system. A number of samples are chosen on each trajectory. Some adaptive samples are selected near the points with large errors after the previous round of training. Our algorithm is causality-free basically, hence it has a potential to apply to various high-dimensional nonlinear systems. We illustrate the effectiveness of our method by stabilizing the Reaction Wheel Pendulums.Comment: The algorithm is modified. The main point is that the trajectories on stable manifold are found by a combination of two-point BVP near the equilibrium and initial value problem far away from the equilibrium. The algorithm becomes more effectiv

Z-ligustilide reduces cisplatin-induced nephrotoxicity via activation of NRF2/HO-1 signaling pathways

Purpose: To investigate the effect of Z-ligustilide (Z-lig) on cisplatin-induced nephrotoxicity and examine whether NRF2 signaling mediates the underlying mechanism of action.Methods: Human proximal tubular epithelial cells (HK-2) were pretreated with 20 or 100 μM Z-lig for 2 h, followed by 10 μM cisplatin treatment for 24 h. Cell viability was measured using (3-(4,5- dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay. A commercial kit was used todetermine lactate dehydrogenase (LDH) release. Apoptosis was determined by flow cytometry while Western blotting was used to evaluate protein levels. Levels of malondialdehyde (MDA), superoxide dismutase (SOD), glutathione (GSH), and glutathione peroxidase (GSH-Px) were assessed by enzymelinked immunosorbent assay (ELISA).Results: Cisplatin decreased HK-2 cell viability and increased LDH release, while Z-lig increased cell viability and decreased LDH release in a dose-dependent manner (p &lt; 0.05). Moreover, Z-lig reduced cisplatin-induced apoptosis (p &lt; 0.01), and alleviated cellular oxidative stress caused by cisplatin (p &lt; 0.05). Furthermore, Z-lig activated NRF2/HO-1 signaling in cells treated with cisplatin (p &lt; 0.05).Conclusion: Z-lig reduces cisplatin-induced nephrotoxicity via activation of NRF2/HO-1 signaling. Thus, Z-lig is a potential drug for the treatment of nephrotoxicity caused by cisplatin. Keywords: Z-ligustilide, Cisplatin, Nephrotoxicity, Oxidative stress, Apoptosis, Nuclear factor erythroid 2-related factor 2, Heme oxygenase-

Energy-saving Potential Study on Telecommunication Base Station Free Cooling With a Thermosyphon Heat Exchanger in China

The energy consumption of air conditioning for a telecommunication base station (TBS) in China can be significantly decreased when an air to air thermosyphon heat exchanger is used for free cooling in low atmosphere temperature days. The model for the typical TBS in China was established and the air conditioning energy consumption was analyzed for the cities with climatic data in China. Taking the sandwich steel panel envelope with the thickness of 50 mm for an example, the energy consumption of an air conditioner combined with a thermosyphon heat exchanger in a TBS and the energy-saving potential for using ambient energy were calculated, and were shown on the map of China by different colors according to the energy-saving rates. The static payback periods were also presented. The results demonstrate that the operation mode of the combined system of an air conditioner and a thermosyphon heat exchanger for controlling the indoor temperature can bring enormous energysaving potential for a TBS in different Chinese cities except the southern areas