In recent years, multifractal detrended fluctuation analysis (MF-DFA) has become an important tool for detecting the scale and long correlation of non-stationary time series. With the continuous development of multifractal theory, researchers have widely applied it in physics, chemistry, biology, economy, etc. In this paper, we briefly review various applications of MF-DFA, and present some empirical research using one- and two-dimensional (2D) MF-DFA. 1D MF-DFA is always applied in financial markets, energy markets, heartbeat, and atmospheric science. Furthermore, 2D MF-DFA has been studied in surface science such as image segmentation, medical image classification. In this paper, we use 1D MF-DFA to explore the market efficiency of Korean stock market, and adopt 2D MF-DFA to segment images such as the license plate and hepatic cell image. In addition, we apply the proposed algorithm to segment transmission lines under icing condition, and the proposed method achieves satisfactory segmentation results.

The contamination in water bodies is a big threat to the environment to control the water pollution, a model study is conducted. The pollution model for a system of three lakes that are interconnected by channels is taken into account. Three input models (periodic, exponentially decaying, and linear) were solved by employing a variational iteration technique coupled with various types of multipliers. Identification of exact multipliers for $n{\rm th}$-order differential equations is also presented. All said models were examined mathematically. We noticed that exact Lagrange multipliers in the correction functional bring more accurate and efficient results that have been deliberated graphically. The results obtained via variational iteration method (VIM)-$\lambda_E,$ already published work and the Runge-Kutta method of order four provides an excellent agreement. Additionally, the use of exact Lagrange multipliers with other techniques may be extended to other dynamical problems.

Here, a hierarchical autoregressive spatio-temporal model under the Bayesian framework is proposed to address the simultaneous multi-site ${\rm PM}_{2.5}$ prediction. The true daily average concentration of ${\rm PM}_{2.5}$ is regarded as a potential spatio-temporal process, then the temporal correlation is described by the first-order autoregressive process and the spatial correlation is captured based on the Matérn process, which greatly improves the efficiency in dimension reduction and synchronous prediction. In addition, meteorological factors such as daily maximum temperature, relative humidity and wind speed are used as explanatory variables to improve the prediction accuracy. The combination of Bayesian method and MCMC can realize parameter estimation and prediction process due to the model's hierarchical structure. The empirical analysis of daily ${\rm PM}_{2.5}$ concentration in Beijing shows that the proposed model has good interpolation or prediction performance in both spatial and temporal dimensions.

The operation experiences have shown that the large-scale DC magnetic bias caused by DC grounding electrode can be attributed to the uneven surface potential distribution. Here, the Chebyshev polynomial is used to fit the Hankel transform kernel function in order to solve the surface potential distribution for the complex earth model of wide area depth stratification. The adaptive order fitting method of Chebyshev polynomial for kernel function is obtained via shift operation, coefficient expansion and truncation error determination, which greatly reduces the calculation difficulty of surface potential distribution in a large area caused by DC grounding electrode. Compared with the standard grounding calculation software CDEGS, the proposed Chebyshev polynomial approach achieves less than 1V of earth surface potential deviation in range of 1-100 km when the DC grounding current is 5 000 A. Moreover, the order of the Chebyshev polynomial has influence on the solution results, and it is confirmed that the 20th-order Chebyshev polynomial can meet the accuracy requirements for general DC bias risk assessment. The proposed surface potential assessment method based on shifted Chebyshev polynomial provides a basic technical means for the risk assessment of DC bias, which is helpful to reduce the difficulty of DC bias risk assessment for power grid.

In order to solve the problems of high operating cost and poor service quality of school bus due to the scattered distribution of bus stops in rural areas, multi-objective SBRP (School Bus Routing Problem) models were developed for the mixed-load and non-mixed-load scenarios. In the non-mixed-load scenario, a model of the SBRP was developed to optimize the students' travel cost and school operating cost, while in the mixed-load scenario, another model of the SBRP was developed to consider the input cost and operation cost of the school bus. Several heuristic algorithms were compared, based on which the simulated annealing algorithm was selected to solve the models, and the horizontal comparison of the solution results based on genetic algorithm were determined. Tests were conducted on an international bench mark case and the constructed models were solved by introducing different search operators into the simulated annealing algorithm, then the proposed approach was applied to the optimal design of school bus routes in Wulian county, Rizhao, Shandong province. The results showed that in the non-mixed-load scenario, compared with the original school bus operation mode, the school bus input, mileage and travel cost were reduced by 28.6%, 37.8% and 35.6%, respectively, and students' travel cost was reduced by 4.3% considering the students' perception of school bus service. While in the mixed-load scenario, the proposed approach reduced the school bus input, mileage and travel cost by 37.5%, 42.0% and 35.8%, respectively; due to the complexity of the mixed-load scenario, it is difficult to take the travel cost into account, thus the students' travel cost was increased by 0.5%. The proposed SBRP models were verified to be effective and the simulated annealing approach can optimize service quality and reduce operation cost of rural school bus to a greater extent than the genetic algorithm.