We develop some estimates under the Ricci flow and use these estimates to study the blowup rates of curvatures at singularities. As applications, we obtain some gap theorems: and must blow up at least at the rate of type-I. Our estimates also imply some gap theorems for shrinking Ricci solitons.

As a critical and fragile rotary supporting component in mechanical equipment, fault diagnosis of rolling bearing has been a hot issue. A rolling bearing fault diagnosis technique based on fined-grained multi-scale symbolic entropy and whale optimization algorithm-multiclass support vector machine (abbreviated as FGMSE-WOA-MSVM) is proposed in this paper. Firstly, the vibration signals are decomposed with fine-grained multi-scale decomposition, and the symbolic entropy of the sub-signals at different analysis scales are extracted and constructed as the multi-dimension fault feature vector. In order to address the problem of sensitive parameters for MSVM model, whale optimization algorithm (abbreviated as WOA) is introduced to optimize the penalty factor and kernel function parameters to construct the optimal WOA-MSVM model. Finally, Instance analysis is carried out with bearing fault dataset from Jiangnan University to verify the parameters influence and the effectiveness on the unbalanced sample set. The results show that compared with different feature vector inputs and learning models such as k-Nearest Neighbor (abbreviated as KNN), Decision Tree (abbreviated as DT), Random Forest (RF), etc., the proposed technique can achieve an accuracy rate of 99.33%, besides, the computation speed is fast and the diagnosis efficiency is high which means its potential value for engineering application.

Rolling bearing is an important rotating support component in mechanical equipment. It is very prone to wear, defects, and other faults, which directly affect the reliable operation of mechanical equipment. Its running condition monitoring and fault diagnosis have always been a matter of concern to engineers and researchers. A rolling bearing fault diagnosis technique based on multi-domain feature and whale optimization algorithm-support vector machine (MDF-WOA-SVM) is proposed. Firstly, recursive analysis is performed on vibration signal and the recursive features are employed as nonlinear recursive feature vector including recursive rate (RR), deterministic rate (DET), recursive entropy (RE), and diagonal average length (DAL). Then, a comprehensive multi-domain feature vector is constructed by combining three time-domain features including root mean square, variance, and peak to peak. Finally, whale optimization algorithm (WOA) is introduced to optimize the penalty factor C and kernel function parameter g to construct the optimal WOA-SVM model. The rolling bearing datasets of Jiangnan University is employed for instance analysis, and the results show that the 10-CV accuracy of the technique proposed is good with an accuracy of 99%. Compared with recursive features or time-domain features, multi-domain features are more accurate and comprehensive in describing characters of the signal. Some popular supervised learning models are also introduced for comparison including K-nearest neighbor (KNN) and decision tree (DT), and the result shows that the proposed method has a higher accuracy and certain advantages.

Owing to the superior mechanical and physical properties, metallic glasses (MGs) have attracted tremendous attention as promising candidates for structural and functional applications. Unfortunately, the ability to form uncontrollable glasses, the poor stability and the unpredicted catastrophic failure stemming from the disordered structure, as the Achilles' heel of MGs, severely restrict their large-scale applications. A number of phenomenological models, such as free volume model, shear transformation zone (STZ) model, flow unit model, etc., have been proposed, intending to relate microstructures to properties of MGs. However, few sophisticated structure-property relationships are established due to a poor understanding of the microstructure of MGs. Recently, heterogeneity is commonly believed to be intrinsic to MGs, and it can be used to establish the structure-property relationship of MGs. In this paper, we review the recent progress of MGs from the angle of heterogeneity, including the static heterogeneities and dynamic heterogeneities. The perspectives of the scientific problems and the challenges of metallic glass researches are also discussed briefly.

In allusion to solve the issue of fault diagnosis for bearing and other rotatory machinery, a technique based on fined-grained multi-scale Kolmogorov entropy and whale optimized multi-class support vector machine (abbreviated as FGMKE-WOA-MSVM) is proposed. Firstly, vibration signals are decomposed by fine-grained multi-scale decomposition, and the Kolmogorov entropy of the sub-signals at different analysis scales is calculated as the multi-dimension feature vector, which quantitatively characterize the complexity of the signal at multi-scales. Aiming at the problem of sensitive parameters selection for multi-class support vector machine model (abbreviated as MSVM), the whale optimization algorithm (abbreviated as WOA) is introduced to optimize the penalty factor and kernel function parameter, and constructing optimal WOA-MSVM model. Finally, an instance analysis is carried out with Jiangnan University bearing datasets to verify the effectiveness and superiority of this technique. The results show that compared with different feature vectors and models such as K nearest neighbors (abbreviated as KNN) and Decision Tree (abbreviated as RF), the proposed technique is superior with fast computation speed and high diagnostic efficiency.