In this paper, we explore the mechanism and properties of the alternating direction method (ADMM) in the meaning of variational inequality and algorithmic unified framework. The augmented Lagrangian method (ALM) is a proximal point algorithm (PPA) of the multipliers, while ADMM is derived from the relaxation of ALM. Using the variational inequalities as a tool we briefly proved the convergence of ADMM, indicated that although ADMM is relaxed from ALM, it maintains the convergence properties of the ALM for the essential variables. The article also clarifies that ADMM is a specific algorithm within the algorithmic unified framework which is called Splitting and Contraction Method for convex optimization. Finally, the article demonstrates some results on the iterative complexity in terms of ergodic and point-wise for the algorithms within the unified framework.

We provide a numerical method for solving the immersed interface problem.  The immersed interface model is governed with incompressible Navier-Stokes equations with a surface stress tensor corresponding to the interface motions. We first introduce the scalar auxiliary variables (SAV) method to reformulate the immersed interface model as an equivalent system and then propose two second-order discrete schemes for the new system. The program calculates two unknown surface tensions, two velocity fields, and two pressure fields at each time step, as well as solving a nonlinear algebraic equation using a scalar number. It is also proven that both SAV schemes provide unconditional energy stability. A rigorous error analysis of the SAV-CN full discrete scheme is carried out and a couple of numerical experiments were presented to demonstrate the accuracy and effectiveness of the algorithm.

In order to study the mechanism of chaotic rotation of circular truncated continuous water wheel, the mathematical model of circular truncated continuous waterwheel is constructed as a set of coupled nonlinear partial differential equations by using the law of conservation of mass, the theorem of moment of momentum and the element method. The simplified mathematical model of conical continuous waterwheel is obtained by reducing the coupled nonlinear partial differential equations by Fourier expansion, and the rationality of the simplified mathematical model is analyzed. The mechanism of chaotic rotation of circular truncated continuous waterwheel is explained by studying the dynamic behavior of simplified mathematical model. The corresponding numerical simulation of the uniform rotation, periodic inversion and chaotic rotation of circular truncated continuous waterwheel is carried out.

In this paper, we consider the incubation period of organisms, and study the stochastic optimal control problem for a class of delayed competitive population with size structure and fractional Brownian motion in a polluted environment. The existence and uniqueness of the system solution is proved by using the fixed point theorem, and the necessary and sufficient conditions for the optimal control are obtained by means of the adjoint equation, Hamiltonian function and It\^{o} formula. The conclusion is an extension of the competitive population with general size structure, which lays a theoretical foundation for practical application.

In the field of image processing, image denoising is a hot spot of the current research. The total variation (TV) image denoising algorithm can maintain the edge information of the image well, but it will produce a serious ``staircase" effect in the smooth area of the image. To avoid the above problem, an improved TV model that is essentially a time-dependent Hamilton-Jacobi equation is proposed. In order to suppress the ``staircase" effect and retain the edge information of the image, a high order numerical method coupled with an edge detection method for solving the model is designed. In the areas near the edges, a third-order WCNS (Weighted compact nonlinear scheme) is used to retain the image edge information. In the smooth areas, a third-order WCNS-P (Perturbed WCNS) with perturbed terms is used to preserve image details. Experimental results show that the designed denoising algorithm has better convergence and can achieve the  better denoising effect, compared to several classical denoising algorithms.

To address the computational bottlenecks of PageRank problems in large-scale sparse network analysis, particularly under high damping factors, this paper proposes a hybrid solution strategy integrating the thick restarted Arnoldi algorithm with the Power-Minus-Splitting Iteration (PMSI) technique, termed the Arnoldi-PMSI iteration method. To overcome the significant convergence slowdown of the traditional power method as the damping factor approaches 1, an alternating iteration framework based on parameter control is designed. In terms of implementation, the algorithm first utilizes the thick restarted Arnoldi process to rapidly generate high-quality approximate solutions within the Krylov subspace. When the residual precision fails to meet the target, the method automatically switches to the PMSI iteration stage to further reduce the error, feeding the resulting vector back into the Arnoldi process as a new initial vector, thereby achieving a complementary advantage of both acceleration mechanisms. The convergence of this hybrid algorithm is rigorously proved under reasonable mathematical assumptions. Numerical experiments conducted on multiple real-world web link matrices demonstrate that, compared with the classical power method, the thick restarted Arnoldi-power method, and the thick restarted Arnoldi-MSI method, the Arnoldi-PMSI method exhibits significant advantages in key metrics, including the number of iteration steps, matrix-vector multiplications, and CPU time. This study not only effectively improves the computational efficiency of solving large-scale PageRank problems but also provides a robust numerical tool for complex network analysis.

To respect the decision-makers' own expression habits in the decision-making process，a method is proposed in this paper for achieving consensus in group decision-making based on multi-granularity personalized individual semantics. First, the expert's multi-granularity language sets are converted into a single-granularity language set and an optimal additive consistency optimization model is proposed to calculate the numerical scale corresponding to each expert linguistic variable.  Second, calculate the subjective and objective weights of experts based on trust networks and minimizing distance between collective and individual semantics, respectively. Then, a consensus feedback adjustment strategy is proposed, which takes different measures according to the experts' different adjustment willingness. Finally,  feasibility and effectiveness of the proposed method are verified by an example of selecting the best emergency alternative and comparison analysis.

Aiming at the problem of fuzzy uncertainty in the process of complex network games, this paper introduces the interval theory to describe the relationship between nodes in complex network. For a class of affine quadratic two-person deterministic discrete-time zero-sum dynamic game open-loop process, the interval matrix is used to represent the interaction between player nodes and non-player nodes in the network. The optimal parameter solution of the total cost function is obtained through a parameterization method, and the optimal trajectory and network optimal strategy of the network nodes are obtained after being substituted into the model. Finally, a simulation example is given to verify the superiority of the model and method in prediction accuracy and breadth as well as statistical inference.

Addressing the critical challenges of excessive packaging material varieties and low space utilization in e-commerce warehousing, this study develops a packaging material size optimization model under multi-constraint conditions based on three-dimensional bin packing data from a major e-commerce platform. Considering constraints including material specifications, item loading sequences, placement orientations, and loading point distributions, a three-dimensional bin packing mathematical model is established. An integrated comprehensive heuristic algorithm is designed to generate initial packing solutions, and a material size optimization algorithm based on K-means clustering theory is proposed to achieve systematic simplification of material specifications through item stacking simulation and cluster analysis. Experimental results demonstrate that the proposed method successfully optimizes and integrates 29 original material specifications into 10 standardized configurations, reducing material varieties by 65.5%, improving average loading cost by 0.977, and enhancing algorithm execution efficiency by 5.4%. Compared with existing material configuration schemes, the optimized material system exhibits significant performance in resource conservation and environmental friendliness, with notable improvements in loading efficiency and space utilization.

This paper studies the influence of supply chain enterprises' emissions reduction investment spillover effect on supply chain pricing strategy under the carbon trading mechanism, and analyzes it under three scenarios: retailers do not invest in emission reduction, single retailer invests in emission reduction, and double retailers invest in emission reduction The impact of carbon trading policy and emission reduction investment spillover effects on the wholesale price, market price and profit of supply chain products. The research shows that under the three scenarios, the spillover effect of emission reduction investment will increase the profit of retailers, the price of carbon trading is proportional to the wholesale price of retailers, and the spillover effect of investment in emission reduction is proportional to the wholesale price of retailers. This trend will gradually slow down with the increase of carbon trading price, and the wholesale price will even drop sharply under the double retailer investment. Under the scenario of single retailer's emission reduction investment, the carbon trading price and emission reduction investment overflow. The increase of the effect will increase the market price of non-investors, and at the same time, the spillover effect of emission reduction investment will increase. Moreover, the difference between the wholesale price of investors and non-investors will become smaller, and the increase of carbon trading prices will make investors and investors. The difference in the wholesale price of non-investors shrinks first and then gradually widens.

This paper considers a parallel repairable system with two identical components where each component has c failure states. Assume that the operating time of each component is exponentially distributed random variable, and the repair time of failed component follow general distribution, we employing the Markov renewal process theory, the Laplace transform and the Laplace-Stieltjes transform to derive the distribution and the mean time to first system failure, the system availability, and the rate of failures of the system. Further, the expressions of these reliability measures are obtained. Finally, some special cases of the model are discussed. Finally, numerical results are reported to illustrate the influence of system parameters on the system reliability measures.

Research on joint modeling of recurrent and failure events with covariate functions was conducted to address data analysis scenarios where both event types coexist with complex covariate effects, as well as the inherent distribution dependency issues in existing methodologies. Conventional frailty models typically assume multiplicative or additive covariate effects, which fail to effectively handle complex time-varying and nonlinear influences. To overcome these limitations, a novel semi-parametric joint model is proposed. This framework introduces predefined link functions to allow covariates to affect event intensities in flexible forms, and uses unobservable frailty variables to characterize the correlation between the two types of events. The proposed estimation method does not rely on specific distributional assumptions for the recurrent event process, nor does it require specifying the distribution form of the frailty variable. Statistical inference is performed solely based on the rate function structure of counting processes and martingale estimation equations. The effectiveness of the proposed method is verified through Monte Carlo simulations. The results show that the parameter estimates exhibit favorable unbiasedness with moderate sample sizes. Furthermore, the method still maintains robust estimation performance even when the recurrent event process deviates from the Poisson assumption or the frailty variable is perturbed. Empirical analysis of hospitalization and mortality data from colorectal cancer patients confirms the practical value of this method in medical follow-up studies. It can accurately identify significant factors affecting the risks of recurrent hospitalization and death, thus providing a more reliable basis for clinical decision-making.

The traditional Brownian motion and the fractional Brownian motion cannot simultaneously characterize the random walk and long-memory properties of financial assets. Most existing pricing models for equity warrant bond generally ignore the equity dilution effect arising from warrant exercise, and also suffer from the problems that firm value volatility is unobservable and cannot be accurately matched with stock volatility. Therefore, under the arbitrage-free market assumption, this paper adopts a mixed fractional Brownian motion to describe the firm value process, incorporates the equity dilution effect, and establishes a pricing model via the martingale method. It derives the correlation between volatilities and obtains the analytical pricing formula, and further presents two degenerate models. Finally, empirical tests are conducted on three domestic equity warrant bonds. The results show that the pricing model under mixed fractional Brownian motion achieves better fitting performance, which can provide a theoretical basis for bond pricing and risk management.