An estimation problem of fundamental interest is that of phase (or angular) synchronization, in which the goal is to recover a collection of phases (or angles) using noisy measurements of relative phases (or angle offsets). It is known that in the Gaussian noise setting, the maximum likelihood estimator (MLE) has an expected squared ℓ2-estimation error that is on the same order as the Cramér-Rao lower bound. Moreover, even though the MLE is an optimal solution to a non-convex quadratic optimization problem, it can be found with high probability using semidefinite programming (SDP), provided that the noise power is not too large. In this paper, we study the estimation and convergence performance of a recentlyproposed low-complexity alternative to the SDP-based approach, namely, the generalized power method (GPM). Our contribution is twofold. First, we bound the rate at which the estimation error decreases in each iteration of the GPM and use this bound to show that all iterates-not just the MLE-achieve an estimation error that is on the same order as the Cramér-Rao bound. Our result holds under the least restrictive assumption on the noise power and gives the best provable bound on the estimation error known to date. It also implies that one can terminate the GPM at any iteration and still obtain an estimator that has a theoretical guarantee on its estimation error. Second, we show that under the same assumption on the noise power as that for the SDP-based method, the GPM will converge to the MLE at a linear rate with high probability. This answers a question raised in [3] and shows that the GPM is competitive in terms of both theoretical guarantees and numerical efficiency with the SDP-based method. At the heart of our convergence rate analysis is

A fundamental class of matrix optimization problems that arise in many areas of science and engineering is that of quadratic optimization with orthogonality constraints. Such problems can be solved using line-search methods on the Stiefel manifold, which are known to converge globally under mild conditions. To determine the convergence rate of these methods, we give an explicit estimate of the exponent in a Lojasiewicz inequality for the (non-convex) set of critical points of the aforementioned class of problems. By combining such an estimate with known arguments, we are able to establish the linear convergence of a large class of line-search methods. A key step in our proof is to establish a local error bound for the set of critical points, which may be of independent interest.

Twenty-four secondary metabolites, including 16 isoflavonoids, 7 astragalasides, and 1 benzoquinone, have been isolated from the roots of Astragalus membranaceus (Astragali radix). Among these isolated isoflavonoids, (-)-methylinissolin 3-O-β-d-(6'-acetyl)-glucoside (1), (-)-methylinissolin 3-O-β-d-{6'-[(E)-but-2-enoyl]}-glucoside (2), and calycosin 7-O-β-d-(6''-acetyl)-glucoside (3) have been identified as new compounds on the basis of spectroscopic analysis; (-)-methylinissolin 3-O-β-d-glucoside (4) was isolated from the natural products for the first time. The nitric oxide (NO) production inhibitory activity of the major compounds has been assessed in lipopolysaccharide (LPS)-stimulated RAW 264.7 cells. To identify A. membranaceus, a fingerprint method was developed by using a high-performance liquid chromatography-evaporative light scattering detector (HPLC-ELSD) method. Furthermore, characteristic peaks for the 11 major compounds in the chromatogram were unambiguously confirmed.

they were then compared with rats that received a normal diet (ND). RESULTSHyperglycaemia and dyslipidaemia were induced in HFD + STZ rats, suggesting that T2D was established. The rats with T2D had associated ED, as both nonadrenergic noncholinergic-mediated corporal relaxation and increased ICP by cavernous nerve stimulation were significantly attenuated compared to the ND group. Western blot analysis revealed diabetes-associated lower expression of endothelial and neuronal nitric oxide synthase (e and nNOS), and cGMPdependent protein kinase (PKG)-1 α / β expression in penile tissue than in the ND group. Contrary to the proteins that regulate corporal relaxation, there were relatively high levels of RhoA/Rho kinase receptor 1 (ROCK1) and ET-A receptor (ETAR) in T2D rats. However, the expressed level of phosphodiesterase-5 and insulin-like growth factor binding protein 3 was not altered significantly in response to T2D. CONCLUSIONDecreased expression of certain proteins that mediate the relaxant mechanism, associated with increased expression of certain proteins that mediate contractile mechanisms, might be important in the development of T2D-associated ED. In particular, down-regulated eNOS/nNOS/ PKG1 as well as up-regulated ETAR/RhoA/ ROCK1 might participate in the aetiology of ED in T2D. KEYWORDS

The rapid drop of frequency under the disturbance is a major threat to the safe and stable operation of a microgrid (MG) system. Emergency load shedding is the main measure to prevent continuous frequency drop and power outage. The existing load shedding strategies have poor adaptability to deal with the problem of MG load shedding under different disturbance situations, and it is difficult to ensure the safe and stable operation of an MG in different operating environments. To address this problem, this paper proposes a data-driven load shedding strategy. First, considering the importance of the load and the frequency recovery time of the system, a load shedding contribution indicator is designed that takes into account the load frequency adjustment effect and the load shedding priority. This contribution indicator is introduced as a load shedding criterion into the reward value function of dueling deep Q learning. Second, considering the suddenness and uncertainty of emergency load shedding, a MG emergency load shedding strategy (ELSS) based on dueling deep Q-learning is proposed. On this basis, the dueling deep Q learning algorithm is used to obtain the load shedding decision with the maximum cumulative reward. Finally, taking the MG load shedding cases in two different scenarios as examples, a simulation study is carried out on a modified IEEE-25 bus MG. The simulation results show that, compared with the model-driven implicit enumeration strategy (IES), the proposed ELSS has superiority in maintaining stable power supply for important loads and reducing load shedding decision-making time and frequency fluctuations.