It was recently brought to our attention that our paper was missing information regarding when the patient chest computed tomography (CT) scans were obtained and that there were some discrepancies in the clinical metadata, associated with the very large image dataset, that we made publicly available through the China National Center for Bioinformation (http://ncov-ai.big.ac.cn/ download?lang=en). All of the chest CT and clinical metadata used in our prognostic analysis were collected from patients at the time of hospital admission, and we have now added this statement to the STAR Methods section of our paper. We believe that the errors in the clinical metadata were introduced when the chest CT images, clinical metadata, and codes were transferred to the web server, and we have now corrected the errors manually. Although these corrections do not alter any of the conclusions made in the paper, we do apologize for these errors and any confusion that they may have caused.
Quasiparticle excitation energies and optical properties of TiO 2 in the rutile and anatase structures are calculated using many-body perturbation-theory methods. Calculations are performed for a frozen crystal lattice; electron-phonon coupling is not explicitly considered. In the GW method, several approximations are compared and it is found that inclusion of the full frequency dependence as well as explicit treatment of the Ti semicore states are essential for accurate calculation of the quasiparticle energy-band gap. The calculated quasiparticle energies are in good agreement with available photoemission and inverse photoemission experiments. The results of the GW calculations, together with the calculated static screened Coulomb interaction, are utilized in the Bethe-Salpeter equation to calculate the dielectric function ⑀ 2 ͑͒ for both the rutile and anatase structures. The results are in good agreement with experimental observations, particularly the onset of the main absorption features around 4 eV. For comparison to low-temperature optical-absorption measurements that resolve individual excitonic transitions in rutile, the low-lying discrete excitonic energy levels are calculated with electronic screening only. The lowest energy exciton found in the energy gap of rutile has a binding energy of 0.13 eV. In agreement with experiment, it is not dipole allowed but the calculated exciton energy exceeds that measured in absorption experiments by about 0.22 eV and the scale of the exciton binding energy is also too large. The quasiparticle energy alignment of rutile is calculated for nonpolar ͑110͒ surfaces. In the GW approximation, the valence-band maximum is 7.8 eV below the vacuum level, showing a small shift from density-functional theory results.
Abstract-We consider the optimal control of feedback linearizable dynamical systems subject to mixed state and control constraints. In general, a linearizing feedback control does not minimize the cost function. Such problems arise frequently in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. In this paper, we consider a pseudospectral (PS) method to compute optimal controls. We prove that a sequence of solutions to the PS-discretized constrained problem converges to the optimal solution of the continuous-time optimal control problem under mild and numerically verifiable conditions. The spectral coefficients of the state trajectories provide a practical method to verify the convergence of the computed solution. The proposed ideas are illustrated by several numerical examples.
In recent years, many practical nonlinear optimal control problems have been solved by pseudospectral (PS) methods. In particular, the Legendre PS method offers a Covector Mapping Theorem that blurs the distinction between traditional direct and indirect methods for optimal control. In an effort to better understand the PS approach for solving control problems, we present consistency results for nonlinear optimal control problems with mixed state and control constraints. A set of sufficient conditions is proved under which a solution of the discretized optimal control problem converges to the continuous solution. Convergence of the primal variables does not necessarily imply the convergence of the duals. This leads to a clarification of the Covector Mapping Theorem in its relationship to the convergence properties of PS methods and its connections to constraint qualifications. Conditions for the convergence of the duals are described and illustrated. An application of the ideas to the The research was supported in part by NPS, the Secretary of the Air Force, and AFOSR under grant number, F1ATA0-60-6-2G002. optimal attitude control of NPSAT1, a highly nonlinear spacecraft, shows that the method performs well for real-world problems.
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