The Classical Central Limit Theorem

In this chapter, the first approach is made to establishing the convergence of scaled random sums, considering independent sequences. The classic Lindeberg–Lévy, Khinchine, Lindeberg–Feller, and Liapunov theorems are proved. The main focus is on the treatment of heterogeneous summands, applying the Lindeberg condition, and extensions are given to allow trending (growing or shrinking) variances. The final sections review cases of the central limit theorem under non-standard conditions and α‎-stable convergence.

Decoupled elastic least-squares reverse time migration and its application in tunnel geological forward-prospecting

Accurate seismic imaging can ensure safe and efficient tunnel construction under complex geological conditions. As a high-precision migration method, reverse time migration (RTM) has been introduced into tunnel seismic forward-prospecting. However, the resolution of traditional RTM imaging results may not meet the requirements in a complex tunnel environment, which affects the interpretation of tunnel seismic forward-prospecting results. In this study, we propose a least-squares RTM method based on the decoupled elastic wave equation in tunnels. The Born forward modeling operator and its exact adjoint migration imaging operator are derived to ensure a stable convergence of the conjugate gradient method. Moreover, a pseudo-Hessian based preconditioning operator is adopted to accelerate the convergence. Numerical examples are provided to verify the efficiency of the proposed scheme. A field test in a traffic tunnel construction site is performed to show the good application effect of the decoupled elastic least-squares RTM in practical situations.

Low-Order Scaling Quasiparticle Self-Consistent GW for Molecules

Low-order scaling GW implementations for molecules are usually restricted to approximations with diagonal self-energy. Here, we present an all-electron implementation of quasiparticle self-consistent GW for molecular systems. We use an efficient algorithm for the evaluation of the self-energy in imaginary time, from which a static non-local exchange-correlation potential is calculated via analytical continuation. By using a direct inversion of iterative subspace method, fast and stable convergence is achieved for almost all molecules in the GW100 database. Exceptions are systems which are associated with a breakdown of the single quasiparticle picture in the valence region. The implementation is proven to be starting point independent and good agreement of QP energies with other codes is observed. We demonstrate the computational efficiency of the new implementation by calculating the quasiparticle spectrum of a DNA oligomer with 1,220 electrons using a basis of 6,300 atomic orbitals in less than 4 days on a single compute node with 16 cores. We use then our implementation to study the dependence of quasiparticle energies of DNA oligomers consisting of adenine-thymine pairs on the oligomer size. The first ionization potential in vacuum decreases by nearly 1 electron volt and the electron affinity increases by 0.4 eV going from the smallest to the largest considered oligomer. This shows that the DNA environment stabilizes the hole/electron resulting from photoexcitation/photoattachment. Upon inclusion of the aqueous environment via a polarizable continuum model, the differences between the ionization potentials reduce to 130 meV, demonstrating that the solvent effectively compensates for the stabilizing effect of the DNA environment. The electron affinities of the different oligomers are almost identical in the aqueous environment.

Research on Pepper External Quality Detection Based on Transfer Learning Integrated with Convolutional Neural Network

A pepper quality detection and classification model based on transfer learning combined with convolutional neural network is proposed as a solution for low efficiency of manual pepper sorting at the current stage. The pepper dataset was amplified with data pre-processing methods including rotation, luminance switch, and contrast ratio switch. To improve training speed and precision, a network model was optimized with a fine-tuned VGG 16 model in this research, transfer learning was applied after parameter optimization, and comparative analysis was performed by combining ResNet50, MobileNet V2, and GoogLeNet models. It turned out that the VGG 16 model output anticipation precision was 98.14%, and the prediction loss rate was 0.0669 when the dropout was settled as 0.3, learning rate settled as 0.000001, batch normalization added, and ReLU as activation function. Comparing with other finetune models and network models, this model was of better anticipation performance, as well as faster and more stable convergence rate, which embodied the best performance. Considering the basis of transfer learning and integration with strong generalization and fitting capacity of the VGG 16 finetune model, it is feasible to apply this model to the external quality classification of pepper, thus offering technical reference for further realizing the automatic classification of pepper quality.

Stable Convergence Theorems for Products of Uniformly Continuous Mappings in Metric Spaces

We study the behavior of inexact products of uniformly continuous self-mappings of a complete metric space that is uniformly continuous and bounded on bounded sets. It is shown that previously established convergence theorems for products of non-expansive mappings continue to hold even under the presence of computational errors.

Identification of eccentricity of a motorized spindle-tool system with random parameters

In order to improve the efficiency of identifying parameters using the maximum likelihood method and to avoid the sensitivity of initial values, a proposed method that combines the micro-genetic algorithm with the advance and retreat method is presented in order to identify the eccentricity of the spindle-tool system with random input and output parameters, which obey a certain probability distribution. Eccentricity without prior information is determined through an iterative procedure. The initial value starts from zero, and the interval is determined by the advance and retreat method. Then, the optimal value is searched in the corresponding interval, utilizing the micro-genetic algorithm. The initial value and interval at each of iterations are changed to ensure a fast and stable convergence. Eventually, a numerical example with three kinds of random deviations verifies the feasibility and validity of the proposed method.

Optimum speed controller structure utilizing the MCA approach

In this paper, an optimal speed controller for dc motor is considered using a PID controller and tuned its parameters of gain to offer an optimal solution by using a modified camel algorithm MCA approach. The proposed MCA scheme was applied to solve the difficulty of getting the optimum gains of PID parameters. The MCA has good evolutionary speed with the simple construction of optimization depend on camel searching performance. The characteristics of the MCA algorithm were confirmed by optimizing the gains parameters of proportional, integral, derivative PID controller. The performance of PID-MCA is comparing with a classic PID controller enhanced with GA genetic algorithm optimization method to tune the gain parameters of the speed controller system. It was shown that the utilize of optimization processes indicated better performance for the MCA procedure in term of speed of execution and the size of memory compared with the GA method by applying computer simulations analysis. The proposed scheme has an efficient feature that includes the ease of implementation, good efficiency of computational performances with stable convergence characteristics. The results indicated that the proposed MCA scheme is a useful tool for search ability, produced efficient outcomes compared with the GA optimized method when applied in the proposed system.

A Bilevel Programming Location Approach to Regional Waste Electric and Electronic Equipment Collection Centers: A Study in China

As one of the largest markets of electronic and electric equipment, China has not completely established the formal recycling system of WEEE compared with the developed countries. As a result, China is facing the huge challenge of resource waste and water/soil environmental pollution. In this paper, according to the current regulations on WEEE recycling and disposal issued by Chinese government, the business model of the Chinese WEEE recycling system was designed, and a bilevel programming-based model was proposed to help the disposal factories to establish the regional efficient and economical WEEE recycling network. This model addressed the optimization of bilateral benefits of disposal factories and the third-party recycling agencies/stations. An experiment based on a regional WEEE recycling business data was solved by the NSGA algorithm to validate the proposed model. With the result, the proposed model was compared with the related studies on Chinese WEEE recycling network planning. With the comparison and the analysis on the experiment result, it was found that the proposed model had considerably stable convergence and optimization performance, which proved that this model can be regarded as a useful tool to the planning of the Chinese regional WEEE recycling network. In the last part, the future improvement of the proposed model was also discussed.