In this paper, we adopt the MF-DCCA (Multifractal Detrended Cross-Correlation Analysis) method to study the nonlinear correlation between the returns of financial stock markets and investors' sentiment index (SI). The return series of Shanghai Securities Composite Index (SSEC) of China, Shenzhen Securities Component Index (SZI) of China, Nikkei 225 Index (N225) of Japan, and Standard & Poor's 500 Index (S&P500) of the United States are adopted. Firstly, we preliminarily analyze the correlation between SSEC and SI through the Pearson correlation coefficient. In addition, by MF-DCCA, we observe a power-law correlation between investors' sentiment index and SSEC stock market returns, with a significant multifractal correlation. Besides, SI series and SSEC return series have positive persistence. We compare the differences in multifractal cross-correlation between SI and stock return sequences in different markets. We found that the values of SZI-SI in terms of cross-correlation persistence and cross-correlation strength are relatively close to those of SSEC-SI, while the H_xy(2), ∆H_xy, and ∆α_xy of N225-SI and S&P500 are much smaller than those of SSEC-SI and SZI-SI. This reason is related to the fact that the investors' sentiment index originated from the Shanghai Composite Index Tieba. The SI is obtained through natural language processing method. Finally, we study the rolling of H_xy(2) and ∆α_xy. Results indicate that the macroeconomic environment may cause fluctuations in two sequences of H_xy(2) and ∆α_xy.

In this article, we propose an efficient and accurate adaptive time-stepping numerical method for the Black-Scholes (BS) equations. The numerical scheme used is the finite difference method (FDM). The proposed adaptive time-stepping computational scheme is based on the maximum norm of the discrete Laplacian values of option values on a discrete domain. Most numerical solvers for the BS equations require a small time step when there are large variations in the solutions. To resolve this problem, we propose an adaptive time-stepping algorithm that uses a small time step size when the maximum norm of the discrete Laplacian values on a discrete domain is large; otherwise, a larger time step size is used to speed up the computation. To demonstrate the high performance of the proposed adaptive time-stepping methodology, we conduct several computational experiments. The numerical tests confirm that the proposed adaptive time-stepping method improves both the efficiency and accuracy of computations for the BS equations.

We consider the bifocusing method (BFM) for a fast identification of small objects in microwave imaging. In many researches, it was very hard to measure the scattering parameter data if the location of the transmitter and the receiver is the same. Due to this reason, the imaging function of BFM has mainly been designed by converting unknown measurement data into the zero constant; this approach has yielded reliable imaging results, but the theoretical reason for this conversion has not been investigated yet. In this study, we converted unknown measurement data to a fixed constant and applied the BFM to retrieve small objects. To demonstrate the effect of the converted constant, we show that the imaging function of the BFM can be represented in terms of an infinite series of the Bessel functions of an integer order, antenna setting, material properties, and applied constant. Based on the theoretical result, we concluded that converting unknown measurement data to constant zero guarantees good imaging results, including the unique determination of the objects. Simulation results obtained with synthetic and real data support the theoretical result.

The Reynolds-averaged Navier-Stokes (RANS) simulations are commonly used in industrial applications due to their computational efficiency. However, the linear eddy viscosity model (LEVM) used in RANS often fails to accurately capture the anisotropy of Reynolds stress in complex flow conditions. To enhance RANS predictive accuracy, data-driven closure models, such as Tensor Basis Neural Network (TBNN) and Tensor Basis Random Forest (TBRF), have been proposed. However existing models, including TBNN and TBRF, have limitations in capturing the nonlocal patterns of turbulence models, resulting in irregular and unsmooth predictions. Convolutional neural networks (CNNs) are considered as an alternative approach, but their reliance on discretization poses challenges when dealing with arbitrarily designed meshes in RANS simulations. In this study, we propose a nonlinear convolutional neural operator as the RANS closure model. Our model satisfies Galilean invariance, can learn nonlocal physics, and recovers high-resolution physics even when trained on undersampled grids. The model outperforms existing TBNN and TBRF models, successfully predicting smooth fields of Reynolds stress in flows with adverse pressure gradients, separations, and streamline curvature, where existing models struggle or fail to provide accurate predictions.