Matrix Schur complement is an important part of matrix theory and its application, which has a wide application background. Strictly double diagonally dominant matrices are a very important class of special matrices, which are closely related to fluid mechanics calculation, material simulation and design, electromagnetic field calculation and so on. The study of strictly double diagonally dominant matrices mainly focuses on two aspects: eigenvalue localization of Schur complement of strictly double diagonally dominant matrices; Infinite norm estimation of inverse of Schur complement of strictly double diagonally dominant matrices. First, a new lower bound estimation of the double diagonally dominant degree of Schur complement of strictly double diagonally dominant matrices is given. Then, the new eigenvalue inclusion set of Schur complement of strictly double diagonally dominant matrices and the new upper bound of infinite norm for the inverse of Schur complement of strictly diagonally dominant matrices are obtained by using the obtained estimations. Numerical examples show that the results obtained in this paper improve some existing results.

A new projection algorithm is proposed for the common elements of variational inequality problems and fixed point problems. Under appropriate conditions, it is proved that the iterative sequence generated by the algorithm strongly converges to the common element of the solution set for pseudomonotone variational inequality problem and the common fixed point set of two semi-contractive mappings by using viscous approximation method and projection operator skills. Finally, numerical experiments illustrate the effectiveness of the algorithm and shou that the performance of the proposed algorithm is better then existing alrorithm.

An analytical and numerical algorithm for solving the displacement response of fractional oscillator system under cosine excitation is presented. The analytical method means that steady-state response and transient response solutions of the system can be obtained by the average method. The total displacement response solution is the sum of the steady-state solution and transient solutions. In the numerical method, the Grunwald-Letnikov definition of fractional derivative is used to discretize the fractional differential term in the system, so as to reduce the order of the original system. Considering the general periodic excitation, the approximate response solution of the system can be obtained by using the Fourier series expansion method and the linear system superposition principle. Finally, the effectiveness and feasibility of the proposed method are verified by numerical simulation. The effects of fractional order, linear damping coefficient and fractional derivative coefficient on steady-state response amplitude and total displacement response of the system are analyzed.

Based on the inner-outer iteration sequence for solving the PageRank vector, a relaxed inner-outer iteration method is obtained by introducing a relaxed factor. Combining the multi-splitting iteration framework with two different relaxed factors, a relaxed multi-splitting iteration method for solving the PageRank vector is proposed, and its convergence property is analyzed. Furthermore, by using the relaxed inner-outer iteration format, a preconditioned matrix for accelerating the projection subspace methods is constructed, the spectral distribution is theoretically investigated, and choice criteria of the parameters in the relaxed multi-splitting iteration method and preconditioner are provided. Several numerical examples validate the effectiveness of the relaxed multi-splitting iteration method and preconditioner, the relaxed multi-splitting iteration method is more efficient compared to the multi-splitting iteration method with appropriate relaxed factors.

The problem of variable selection is considered in partially linear varying coefficient spatial autoregressive model. By combining profile quasi-maximum likelihood method and a class of non-convex penalty function, a variable selection method is proposed to simultaneously select important explanatory variables in parametric component of the partially linear varying coefficient spatial autoregressive model and estimate the corresponding nonzero parameters. Extensive simulation studies show that the proposed variable selection method is of satisfactory finite sample performance. Especially, the proposed variable selection method is quite robust to degree of sparseness of spatial weight matrix, intensity of spatial dependence, degree of complexity of coefficient function and non-normality of error distribution, and even works well in the case where correlation among explanatory variables is strong or number of unimportant explanatory variables is large provided that appropriate penalty function is used and sample size is moderately large. As an illustrative example, the proposed variable selection method is applied to analyze the popular Boston housing price data.

The rationality of the reward function is crucial for enhancing the performance of the Deep Reinforcement Learning algorithms. In portfolio management, this study identifies and solves two major flaws in existing reward functions: first, overemphasis on short-term market fluctuations and neglect of long-term trends; second, the equivalent rewards or punishments for actions that result in gains or losses, which is not in line with the investor's loss aversion psychology. To this end, drawing on the loss aversion theory in behavioral finance, this paper innovatively proposes a multi-step loss aversion (MSLA) reward function, which more accurately captures the behavioral patterns of investors in trading and constructs an online portfolio management strategy based on the MSLA. The study selects three representative indices from the A-share market to build corresponding portfolios and conducts several backtesting experiments on historical data from 2019 to 2023. The experimental results demonstrate that the MSLA reward function significantly improves the overall performance of the portfolio strategy, outperforming other existing algorithms in terms of cumulative returns, Sharpe ratio, and maximum drawdown. Furthermore, the proposed strategy is not only applicable to portfolios composed of stocks with different market capitalizations, but also maintains robust performance in rising, falling, and volatile market conditions, fully illustrating its effectiveness and practicality in portfolio management.

A multi time delay HIV infection model with adaptive immune responses is proposed, in which both the virus-to-cell infection and the cell-to-cell transmission are considered. The existence of five equilibria and five basic reproduction numbers are calculated. By using the Lyapunov functionals, the sufficient conditions on the global stability of five equilibria are established. Using a time delay $\tau_3$ as a bifurcation parameter, we show that $\tau_3$ may destabilize two equilibria $\widetilde{E}_2$ and $\widetilde{E}_4$ leading to Hopf bifurcation. The results indicate that $\tau_3$ can lead to periodic oscillations in viral load and immune responses, which can reduce the risk of infection. Finally, numerical simulations are carried out to illustrate the corresponding theoretical results, and reveal the effects of different delay parameters on the stability of the equilibria $\widetilde{E}_2$ and $\widetilde{E}_4$.

AIDS is an extremely harmful infectious disease since HIV virus destroy the immune system of people. Homosexuality is one of the important ways to spread HIV. Considering the transmission characteristics of men who have sex with men (MSM) and the existence of random factors in the environment, a stochastic HIV/AIDS epidemic dynamic model based on MSM is established. Using the basic theory of stochastic differential equation, the dynamic behaviors of the system are analyzed. First, for any given positive initial value, it is proved that there is a unique and global positive solution. From a biological point of view, this conclusion must hold, which ensure it is value of studying such a system. When a virus dynamical model is investigated, the extinction and persistence of virus require the immediate attention. Therefore, by using It$\hat{\rm o}$'s formula and constructing some special functions, the sufficient conditions of disease extinction are given. Moreover, the existence of a unique positive ergodic stationary distribution of the system is discussed, which implies the prevalence of disease. Finally, the obtained theoretical results are verified by numerical simulation, and the influences of random disturbance on the dynamical behavior of the system are analyzed.

Predator-prey reaction-diffusion model with refuge effect is an important type of population dynamics model, and its pattern formation provides key information for studying the spatiotemporal distribution of population. A Leslie-Gower predator-prey reaction-diffusion model with both refuge and nonlocal predation effects is proposed. Firstly, the conditions for local stability and Turing instability of the positive equilibrium point are derived through linear stability analysis and Turing instability analysis, respectively. Secondly, the evolution of the patterns of the prey with nonlocal predation and refuge effects was demonstrated through numerical simulations. Finally, by analyzing the relationship between the spatial mean density of predator and prey populations and the changes in nonlocal prey and refuge effect parameters, it is shown that the spatial mean density of prey increases, while the number of predator decreases with the enhancement of nonlocal predation effects, indicating that nonlocal effects promote the growth of prey density while inhibiting the growth of predator populations; The spatial mean density of predator and prey increases simultaneously with the enhancement of refuge effect, indicating that strong refuge effect has a protective effect on prey and can promote the persistence and coexistence of predator and prey. The result helps people better understand the law of nonlocal effects and refuge effects on the pattern formation of predator and prey from a mathematical perspective.

The asymptotic spread properties are investigated for an SIRS reaction-diffusion infectious disease model simulating the disease propagation in temporally periodic environment by using the theory of asymptotic speed of spread. Different from two-dimensional SI type systems, the R component in current model cannot be decoupled from the entire system. This means to overcome the difficulties caused by the coupling of high-dimension and non-autonomy to establish the existence of asymptotic speed of spread for current high-dimensional system. Firstly, the asymptotic spread characteristics of I component is obtained in disease-free region by using the abstract theory of asymptotic spreading speed of monotone systems and comparison principle. Further，the asymptotic spread property of R component in disease-free region is verified through the use of the entire solution and the maximum principle. Secondly, the propagation properties of the invaded region are analyzed by applying comparison principle and uniformly persistent idea to I and R equations, respectively. As a result, the value of asymptotic spreading speed by which we divide the two regions is obtained. In addition, numerical experiments are tested for more specific propagation dynamics of the invaded area in temporally periodic environment.

The Painlevé equation is a typical nonlinear differential equation and has a wide range of applications. The complex method combines complex analysis and differential equation theory, and is an effective method for solving meromorphic solutions of nonlinear differential equations. The complex method is used to prove that meromorphic solutions of generalized algebraic differential equations related to the third Painlevé equation belong to class $W$, and meromorphic solutions of the mentioned equation were obtained. Through appropriate transformations and application of the obtained results, class $W$ meromorphic exact solutions for the modified double sine Gordon equation and modified Kortweg de Vries (mKdV) equation were derived, and non class $W$ meromorphic exact solutions for the relevant nonlinear partial differential equations were obtained. It is well known that based on the suitable transformations and these results, it is convenient to seek meromorphic exact solutions of other related mathematical and physical equations. In future work, further research can be conducted on the meromorphic solutions and applications of generalized algebraic differential equations related to other categories of the Painlevé equation.

The traditional graph-based fault diagnosis framework usually uses a certain structural relationship of high-dimensional dataset to construct a similarity graph to reveal the geometric structure between samples, resulting in the loss of other structural information of the dataset, and it is impossible to accurately extract the low-dimensional features that characterize the running state of the bearing. A new graph-based unsupervised feature extraction method is proposed, which considers both the global and local structures of high-dimensional dataset in the process of constructing graphs, which is called global-local graph embedding. The method first constructs an undirected graph by using the global structure information of the dataset. Then, by constructing local structure information and assigning corresponding weights to the edges in the undirected graph, a global-local graph joint representation convex optimization problem is obtained, and the similarity between samples is evaluated according to the obtained weights. Finally, the low-dimensional embedding result is calculated by keeping the similarity between samples unchanged in the low-dimensional space. Compared with the single graph structure representation, our constructed global-local joint graph takes full advantage of the global and local structural information inherent in high-dimensional dataset. In addition, the essential features of high-dimensional bearing data can be effectively extracted by maintaining the similar performance between samples. Experimental results show that our proposed feature extraction method based on global-local graph embedding has obvious advantages over existing methods.

Spodoptera frugiperda (SF) is a pest that invaded China since 2019. It mainly eats corn, wheat and other crops, which has caused a great threat to agricultural production and has been highly valued by agricultural department. The study aims at clarifying the population mechanism of SF and its distributions in China. First, a new differential system is developed to simulate the SF evolution process among different age stages and then stability theory is employed to calculate the population reproduction number and analyze the conditions of population extinction and persistent existence. Finally, based on the temperature-based model parameter, the effectiveness of temperature on SF survival is quantified, and the distribution patterns in Chinese provinces are estimated. The results indicate that temperature has a significant effect on the growth and migration of SF, in which SF could not survive when temperature is smaller than 19.6$^{\circ}$C or larger than 35.7$^{\circ}$C. The suitable temperature for SF survival is 24$\sim$28$^{\circ}$C, and when temperature is around 24.3$^{\circ}$C, SF develops fastest. SF is more sensitive to lower or higher temperature. It is further found that the best approach to control SF is reducing spawning rate and accelerating the death rate of adult SF.