The aim of this paper is to investigate the well-posedness and stability in set optimization. For the first time, by embedding the original set optimization problem into a family of perturbed problems with the same structure, the notions of generalized $m_{1}$-well-posedness and $m_{1}$-well-setness under perturbation for set optimization problems are introduced. The relationships between them are obtained. Sufficient condition, necessary conditon and many characterizations are given for generalized $m_{1}$-well-posedness under perturbation, respectively. Two new definitions of monotonicity of set-valued mapping are introduced, and from this, the semi-continuity and closedness of $m_{1}$-efficient solution mappings for parametric set optimization problems are studied.

In this paper, we consider the dynamics of a second-order rational difference equation. Through the linear stability analysis, the parameter conditions are gained to guarantee the existence and stability of the equilibrium. By applying the center manifold theorem, the Normal Form theory and the bifurcation theory, we derive the critical parameter values of the fold bifurcation, transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation respectively. In order to identify chaos among regular behaviors, we calculate the maximum Lyapunov exponents and fractal dimensions. These results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamic behaviors, such as period doubling cascade, periodic windows, limit cycles, chaotic behaviors and so on. Interestingly, with the selection of parameters, the dynamic behaviors of the system are completely symmetric.

Malaria is a vector-borne disease caused by a pathogen. In order to study the multiple effects of spatial heterogeneity, vector-bias effect, and seasonality on disease transmission, this study proposes a temporal periodic reaction-diffusion model for malaria transmission. Firstly, the basic reproduction number ($R_{0}$) of the model is introduced. Then it is proved that if $R_{0}\leq1$, the disease-free periodic solution is globally asymptotically stable; while if $R_{0}>1$, the system possesses a globally asymptotically stable positive periodic solution. These proofs utilize the monotone dynamical system theorem, the theory of periodic semiflows, and the chain transitivity theory. Numerical studies on malaria transmission in Maputo Province, Mozambique are carried out to validate the theoretical analysis results. The impact of key parameters in the model is discussed, and it is concluded that neglecting the diffusion of human and mosquito populations and the vector-bias effect underestimates the risk of disease transmission. In addition, the impact of medical resources on disease transmission is analyzed from two aspects: quantity and distribution. It is found that increasing medical resources would reduce the risk of disease transmission. If medical resources are fixed, reducing the variability in the distribution of medical resources would also decrease the risk of disease transmission.

The paper is concerned with the traveling wave solutions of a nonlocal dispersal delayed SEIR model with saturated recovery rate. First, we proved the existence of traveling wave solutions for R₀>1,c> $c^{*}$ by applying the Schauder's fixed point theorem and a limiting argument, and the asymptotic boundary of the traveling wave solutions was obtained by Lyapunov functional. Secondly, we proved the nonexistence of the traveling wave solutions for R₀>1, 0<c<c^* or R₀<1 by two-side Laplace transform. Finally, we discussed how the latent period and the movement of the infective individuals affect the minimal wave speed. Generally speaking, the longer the latent period, the slower the spread of the disease.

At present, research on functional regression models is mainly based on the estimation of mean regression. However, mean regression only studies the influence of covariates on the mean position of response variables in the conditional distribution, and cannot reflect the relationship between the two at the tail of the conditional distribution, which can lead to information leakage and be easily affected by outliers. At the same time, there is currently no relevant research on partially functional linear additive models in the spatial dimension. In fact, economic relationships between variables exhibit more nonlinear characteristics in space, and ignoring this nonlinear relationship in spatial lag models can easily lead to model setting errors. To overcome the above shortcomings, this paper combines parametric models and semi parametric models with functional data to propose a new partially functional linear additive spatial lag quantile. Regression model. Furthermore, A tool variable estimation method for the model was constructed based on functional principal component analysis and B-spline approximation. Under some regular conditions, the consistency and asymptotic normality of the model parameter estimates were given, and the optimal convergence speed of the function estimates was obtained. The large sample nature of these estimates was also proved. The model can reflect spatial dependence and the influence of functional data, as well as capture multiple nonlinear effects caused by covariates, reducing the risk of model error, solving the curse of dimensionality, and having high robustness. Finally, numerical simulations and practical applications show that the proposed model and method are effective.

In this paper, a multi-fidelity Monte Carlo (Multi-fidelity Monte Carlo, MFMC) estimation method based on data-driven low-fidelity models is proposed for the multi-parameter uncertainty quantitative analysis of the numerical solution of the Advection-Diffusion-Reaction equation. In our method, a high-fidelity model of numerical solution is first obtained according to the finite element method. Then, two types of low-fidelity models of DEIM and POD-DEIM are given respectively based on the POD reducing dimension method of the finite element discrete linear equations and the DEIM interpolation method for parameter space. Finally, through numerical experiments, the mean value and sensitivity of the multi-parameter uncertainty are analyzed for the ADR equation. The results show that, compared with the standard Monte Carlo method, the MFMC estimation method based on the data-driven low-fidelity model can effectively reduce the computational cost and the relative mean square error.

In this paper, we investigate a partially observation insider trading model with different market structures and different risk preferences agents. With the help of conditional expectation, game theory and projection theorem, the related market characteristics are given, and then we explicate the corresponding economic significance under equilibrium conditions. It shows that: (1) The higher the observation precision by market makers, the market is basically strong efficient; on the contrary, it is counterproductive, and even the insider can get higher expected profits when she/he releases less private information. (2) Under Cournot game, the risk averse doesn't willing to trade, so the private information releases relatively slow. In Stackelberg game, the leader is conservative, while the follower is impulsive. In short, it is beneficial for risk neutral trader under Cournot game, and it is useful to follower under Stackelberg game. (3) When both insiders adopt a constant strategy, with the decreasing of market makers' observation accuracy, both the market liquidity and the residual information are increasing, however the price pressure of partially observation decrease slowly.

In the realm of financial data analysis, where data often exhibits volatility clustering, heteroscedasticity, and asymmetry, capturing these inherent features authentically requires the adoption of conditional heteroscedasticity models within a skewed distribution framework. This study addresses the Bayesian statistical diagnostic challenges associated with skew-normal GARCH models. The research begins by employing the Griddy-Gibbs algorithm for effective parameter estimation within the skew-normal distribution embedded in the GARCH model. The investigation considers three distinct sources of perturbations: disruptions from prior assumptions, anomalies within the data, and variations in the model itself. To facilitate comprehensive statistical diagnosis, the study leverages three objective functions: Bayes factor, Kullback-Leibler divergence, and posterior mean, enhancing the precision of the diagnostic process. Empirical validation is achieved through rigorous numerical simulations, conclusively establishing the method's efficacy and resilience. This is further supported by an empirical application involving GARCH modeling for Chevron Stock. By utilizing the skew-normal distribution to encapsulate weekly logarithmic returns, the study empirically underscores the distinct advantages of Bayesian local influence analysis, demonstrating superior results.

In this paper, we undertake a comprehensive study the orbital stability of standing waves for the inhomogeneous nonlinear Schrödinger equation $$ \begin{cases} i\partial_t\psi+\Delta \psi+|\psi|^{p}\psi+|x|^{-b}|\psi|^q\psi=0, &\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \\ \psi(0,x) = \psi_0 (x), &\quad x\in\mathbb{R}^N, \end{cases} $$ where $N\geq3$, $\psi:\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{C}$, $\frac{4-2b}{N}<q<\frac{4-2b}{N-2}$, $0<b<2$. In the case of $0<p<\frac{4}{N}$, the energy functional corresponding to this equation has a local minimizing structure. Therefore, we introduce a local minimizing problem. By studying the compactness of the minimizing sequence of this minimizing problem, we prove the existence of the minimizer of this minimizing problem, and finally obtain that the set of minimizer is orbitally stable.

The metric dimension problem (MDP) for graphs is a class of combinatorial optimization problems widely used in machine navigation, chemistry, network discovery and other fields. In order to solve this problem, a general optimization model with combinatorial properties is established in this paper, the resolving relationships between vertices and vertex pairs are described theoretically, the resolving graph and resolving probability distribution of graphs are proposed, and then the resolving probability distributions of some special graph classes are determined. An approximate metric dimension computation framework based on reinforcement learning and graph convolutional neural network is established by using machine learning sampling method. Numerical experiments show the effectiveness of graph machine learning algorithm to solve the graph metric dimension problem.

The adapted power generalized Weibull (APGW) model, which is generalized from the Weibull distribution, is a flexible parametric model and includes many popular survival models as special cases. In this paper, we extend the APGW model to the modeling of recurrent event data. A maximum likelihood estimating approach is proposed and the asymptotic properties are established. A model selection method is provided via variable selection through the SCAD penalty likelihood approach. The performance of both parameter estimation and model selection under finite samples is examined by simulation studies. To the end, the proposed method is applied to a bladder cancer recurrent event data for illustration.

During the process of androgen deprivation therapy for prostate cancer (PCa), stochastic effects in the tumor microenvironment can cause treatment to fall short of the desired effect on tumor clearance. In this paper, the Lévy noise is introduced to describe the random changes of tumor cell number, and a system of stochastic differential equations (SDEs) driven by Lévy noise is established and analyzed. Firstly, through appropriate Lyapunov functions and the solution formula of SDEs driven by Lévy noise, the existence and uniqueness of the global positive solution are proved. Then, employing the theories and methods of stochastic dynamical systems, such as Lévy-Itô's formula, the comparison theorem of SDEs driven by Lévy noise, the exponential martingale inequality and the Borel-Cantelli lemma, the stochastic dynamics of extinction, non-persistence in the mean, weak persistence in the mean and stochastic permanence of PCa cells are studied. Finally, numerical simulation further verifies the theoretical results. Combined with theoretical analysis and numerical simulation, it can be found that the higher the intensity of Lévy noise, the easier the PCa cells are removed.

This paper deals with the uniqueness of a finite-order meormophic solution $f(z)$ of some delay differential equation $$ f(z+1)-f(z-1)+a(z)\frac{f'(z)}{f(z)}=b(z) $$ sharing $0, 1, \infty$ CM with a meromorphic function $g(z)$, where $a(z), b(z)$ are nonzero rational functions, then either $f(z)\equiv g(z)$ holds, either $f(z)=Ce^{ik\pi z}$ and $f(z)g(z)\equiv1$ holds, where $C$ is a nonzero constant, $k$ is a nonzero integer.

Quantum Bernoulli noises are the family of annihilation and creation operators on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. Stochastic Schrödinger equation is a classical stochastic differential equation describing the evolution of open quantum systems in continuous quantum measurement. In this paper, the diffusion of linear stochastic Schrödinger equation with quantum Bernoulli noise method is studied. This model is expected to play a role in describing the evolution of open quantum systems interacting with QBN. Firstly, we prove some technical conclusions in Bernoulli functional space. In particular, we get spectral decomposition of counting operators. And then, with these theorems as main tools, inspired by the ideas of Pellegrini and Mora, we establish a theorem about the existence and uniqueness of a regular solution to the associated stochastic Schrödinger equation, which gives particular versions of previous results via QBN. Finally, we prove that the diffusion case can be obtained by the limit of the discrete process obtained by the measurement of non-diagonal observables, and some further results are obtained.

In this paper, under the assumption that the regularity domain $\Pi(T_{0})$ of the minimal operator $T_{0}$ on two intervals is non-empty, we give an analytic characterization of the $J$-self-adjoint extension domains for singular higher-order $J$-symmetric differential operators on two intervals by employing the direct sum theory and the solutions to the equation $\tau y=\lambda y$. This result includes the case when the endpoints are regular or singular and the deficiency index is arbitrary.

In this paper, we consider a one-dimensional Schrödinger equation with Dirichlet boundary condition, where Neumann control is suffered from bounded disturbance. On the one hand, we design sliding mode control and analyze stability property of the closed-loop system. Firstly, we transform the system by an invertible transformation. And then we prove the system is well-posed. Secondly, we design the sliding mode surface. And we show the system on the sliding mode surface is exponentially stable. Thirdly, we construct sliding mode control, and prove the system will reach the sliding mode surface in a finite time interval which shows stability of the closed-loop system. On the other hand, we construct high gain estimators and design active disturbance rejection control, and then analyze asymptotical stability of the closed-loop system. Finally, we simulate states of the closed-loop systems under both sliding mode control and active disturbance rejection control respectively. Simulation results show the two controllers are effective to make the original system stable.

In a general unbounded domain, we study the large time behavior for the initial boundary value problem of a three-dimensional incompressible viscous magneto-hydrodynamic system. Using the theory of polishing operators, we first establish an approximate solution sequence; secondly, using spectral decomposition method and analytic semigroup theory, we give a new unified estimate for all the nonlinear terms in the equation system. Combining the energy estimation method and weak convergence theory, the existence of the global weak solution is ultimately proved, and long time decay rate is also give. In addition, it reveals that the algebraic decay property of the weak solution is generally dominated by its linear part (i.e., the semigroup solution of the Stokes equation).

Uniform designs are a kind of space-filling designs that are widely used in scientific experiments and industrial production due to their runs flexibility. Discrepancies are used to measure the overall uniformity of designs, but in practical problems, it is often necessary to consider the uniformity of the low-dimensional projection designs. In this paper, the conclusions of Wang et al. (2021) are generalized. Firstly, the uniformity pattern of any $q$-level design based on the reproducing kernel function is defined, and the minimum projection uniformity criterion is provided to select the designs with minimum low-dimensional projection uniformity. Secondly, the analytical relationship between the uniformity pattern and the generalized word length pattern of $q$-level designs is built, and an improved lower bound of the uniformity pattern is obtained, which can be a basis to measure the uniformity of projection designs. Finally, some numerical examples are given to verify the theoretical results.

The convergence and the order of the convergence for the truncated EM numerical solution of the stochastic delay differential equations are researched by using the result that numerical scheme which satisfies stochastic C stability and stochastic B consistency is strongly convergent. Under the local Lipschitz condition and Khasminskii condition and monotonicity condition, the truncated EM scheme for the stochastic delay differential equations is strongly convergent with the order 1/2.

This paper establishes an existence and uniquness result and a comparison theorem for solutions to backward stochastic differential equation driven by $G$-Brownian motion, where $p>1$, the generators $f$ and $g$ satisfy the $p$-order weak monotonicity condition in $y$, and Lipschitz condition in $z$, the terminal condition $\xi$ satisfies the $p'$-order integrable condition, and $p'>p$.

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In this paper, a maximum likelihood estimator (MLE) of the parameter of the SBB distribution and its properties are respectively studied under simple random sampling (SRS) and ranked set sampling(RSS). Both theoretical and numerical results demonstrate that the MLE under RSS is asymptotically more effective than the MLE under SRS. Additionally, we investigate the asymptotic efficiency of the MLE under imperfect ranked set sampling (IRSS), taking into account the potential presence of ranked errors. Numerical results show that the asymptotic efficiency is influenced by the ranked judgement, but the MLE under IRSS is at least as effective as the MLE under SRS.

We study dynamical systems which have bounded complexity with respect to the Bowen metric $d_{n}$. It is shown that any topological dynamical system $G\curvearrowright X$ for actions of countable group $G$ is equicontinuous if and only if $X$ has bounded topological complexity with respect to $\{d_{n}\}_{n=1}^{\infty}$. Meanwhile, it is shown that for any topological dynamical system $G\curvearrowright X$ for actions of countable group $G$ and a Borel probability measure $\mu$ on $X$, $\mu$ has bounded measure-theoretic complexity with respect to $\{d_{n}\}_{n=1}^{\infty}$ if and only if $G\curvearrowright X$ is $\mu$-equicontinuous. These generalize some results of Huang, Li, Thouvenot, Xu and Ye.

The mean residual life function is an important tool for describing the distribution of survival data. This paper investigates variable selection methods for the mean residual life model with right-censored survival data. Focusing on the proportional mean residual life model, we propose a method based on penalized estimating functions, which enables simultaneous variable selection and parameter estimation. It is shown that, with a suitable choice of penalty function and tuning parameter, the resulting estimator is $\sqrt{n}$-consistent and possesses the oracle property. Furthermore, we develop an implementation algorithm based on local quadratic approximation and a BIC-type selection criterion. Simulation studies demonstrate that the proposed method performs well in variable selection and parameter estimation. Finally, the proposed method is applied to the Mayo Clinic primary biliary cirrhosis dataset.

In this paper, a (2+1)-dimensional Yu-Toda-Sasa-Fukuyama (YTSF) equation is investigated, which can model the interfacial wave in a two-layer liquid or elastic quasiplane wave in a lattice. Lie group method is a powerful and fundamental tool in studying the properties of differential equations and obtaining the invariant solutions. Using Lie symmetry approach, infinitesimal generators, symmetry groups and invariant solutions of this equation are presented, and the optimal system is given with adjoint representation. By means of the optimal system, some symmetry reductions to partial differential equations (PDEs) are obtained and some similarity solutions are provided. With Lagrangian, it is shown that the YTSF equation is nonlinearly self-adjoint. Furthermore, based on Lie point symmetries and nonlinear self-adjointness, the conservation laws for YTSF equation are derived, then an infinite number of conservation laws can be constructed through choosing different parameter functions.

Let $\left \{ X_{n},n\ge 1 \right \}$ be a sequence of independent and identically distributed random variables. Let $N\left (n\right)$ be a sequence of positive integer random variables. In this paper, we obtain the joint limit distribution of the extremes $M_{N\left(n \right)}=\{X_{1}, X_{2},\cdots,X_{N(n)}\}$ and the partial sums $S_{N\left (n \right)}=\sum\limits_{i=1}^{N(n)}X_{i}$. The results are also extended to the case of the extreme order statistics and the partial sums. In the end, the strongly mixing cases are also considered.

We consider an initial boundary value problem for hyperbolic compressible Navier-Stokes equations on a half line. After transforming the system into Lagrangian coordinate, the resulting system possesses a structure with uniform characteristic boundary. We first construct an approximate system with non-characteristic boundary, and get a uniform global smooth solutions by basic energy methods. Then, by passing to a limit and using compactness argument, we obtain a global solution of the original problem. Moreover, as the relaxation parameter goes to zero, we show that the solutions of relaxed system converge globally to that of classical compressible Navier-Stokes system.

In this paper, the properties of the analytic solution of the nonhomogeneous linear complex differential equation
f^(k)+B_k-1(z) f^(k-1)+…+B₁(z) f'+B₀(z) f=B_k(z)
is discussed by combining the theory of analytic function space and complex differential equation. Firstly, the condition of coefficients belonging to the weighted Bergman space$(A_\omega^p)$ is obtained. Secondly, the inverse problem is discussed, that is, the coefficients belong to the weighted Bergman space $\left(A_{\omega_{[k p]}}^p\right)$ when all the solutions belong to the weighted Bergman space $(A_\omega^p)$. Finally, the properties of weighted Bergman space $(A_{2(\rho+2)}^p)$ for solutions of second-order homogeneous differential equations are discussed.

In non-cylindrically symmetric media, by investigating the Maxwell equations with Kerr-like nonlinear terms, a new semilinear elliptic equation is derived. Then, using the Hilbert-Schmidt theory, the spectrum of the operator $L$ is given, where the eigenvalue $0$ has infinite multiplicity. Since the kernel space of the operator $L$ is infinite-dimensional, the energy functional of this semilinear elliptic equation is strongly indefinite. Therefore, we construct an appropriate Sobolev space and prove the existence of a ground state solution for the equation by means of the variational method. In addition, if the nonlinear term is even, the energy functional has an unbounded sequence of critical values.

The involute-evolute counterparts in 3-space are defined in this paper. Based on this definition, the existence and relationship of null involute-evolutes derived from pseudo null curves in Minkowski 3-space are studied. Meanwhile, the null evolutes are expressed by the structure function of pseudo null curves and the detailed structure of the null evolutes derived from pseudo null helices is explored. Last but not least, several practical examples and corresponding graphs are given.

The fracture problem of multi-branch fast propagation crack in one -dimensional hexagonal piezoelectric quasicrystals is studied, the analytical expressions of stress, field intensity factor and energy release rate of fast propagation crack with multi-branch under electric non-permeability are given by using the complex function method, the influence of the deflection angle and the relative size of the crack on the field intensity factor of the fast propagation crack and the energy release rate on the propagation velocity are analyzed. The results show that the field intensity factor at the crack tip decreases with the increase of the deflection angle, and the field intensity factor at the crack tip decreases with the increase of the relative size of the crack The energy release rate increases with the increase of the propagation velocity of the crack, the field intensity factor at the crack tip decreases with the increase of the propagation velocity of the crack, and the energy release rate increases with the increase of the propagation length of the main crack, the field intensity factor at the crack tip decreases and the energy release rate increases with the crack propagation velocity.

This paper studies a system of coupled Choquard equations with a weakly attractive potential. By employing methods such as comparison theory and min-max principles, we prove that the system admits positive radial ground state normalized solutions when the coupling constant is sufficiently large.

Considering of the cross-sectional dependence, heterogeneity and the possible outliers, this article applies spatial autoregressive model based on the quantile regression, these are the advantages: (1) able to show overall properties of the conditional distribution, (2) able to describe spatial effect at different quantile level; (3) more robust so that could be adapted to more general structure of spatial errors. According to endogeneity due to the spatial lag and differentiability of the objective function, this article uses instrument variables to handle the issue of endogeneity, establishes smoothed moment condition in order to make the objective function differentiable, then chooses the optimal bandwidth to estimate the parameter. Through mathematical proof, large sample properties of consistency, asymptotic normality and efficiency to the estimators are explicitly shown. Through simulation, this method exhibits faster speed and better performance in finite sample. Finally, based on the spatial quantile regression model, the heterogeneity and spatial aggregation effects of rural labor transfer on the incidence of rural poverty are studied.

In the study of differential equations and dynamic systems, the investigation of singular differential equations has more important scientific significance and widely application, which has attracted great attention and exploration by many scholars. This paper considers the existence of periodic solutions for a singular $\phi$-Laplacian generalized Liénard equations, where the nonlinear term exhibits singularity at the origin and is non-autonomous. By applying Manásevich-Mawhin continuation theorem and some analytical methods, we prove the existence of periodic solutions for this equation under conditions of strong and weak singularities of attractive type, as well as strong and weak singularities of repulsive type.

In this paper, we mainly study the self-adjointness, Green's function and the dependence of eigenvalues of a class of Sturm-Liouville operator with eigenparameter dependent internal point conditions. Firstly, a linear operator $T$ related to the problem is defined in an appropriate Hilbert space, then the problem to be studied is transformed into the research of the operator $T$ in this space, the operator $T$ is proved to be self-adjoint and its Green's function is obtained. In particular, on the basis of self-adjointness, we show that the eigenvalues not only continuously but also differentiably dependent on each parameter of the problem, and the corresponding differential expressions are given. Meanwhile, the monotonicity of eigenvalues with respect to some parameters of the problem is also discussed.

Duality theory is an important branch of vector optimization theory, which plays an important role in establishing the optimality conditions and solving vector optimization problems, and is widely used in fields such as game theory and economic equilibrium problems. In this paper, the conjugate duality and Lagrangian duality for generalized vector optimization are investigated. Firstly, under the order relationship induced by convex cones, a new conjugate mapping is introduced by the weak supremal of sets, and an example is provided to illustrate its economic significance. And the conjugate duality for generalized vector optimization is defined by using a perturbation mapping. The weak duality, strong duality and inverse duality theorems are obtained, and an example is provided to illustrate the strong duality theorem. Secondly, a new Lagrangian mapping is introduced, with which a Lagrangian duality for generalized vector optimization is introduced. The objective value of the original problem is characterized by a Lagrangian mapping, and the Lagrangian duality theory is established. Finally, a kind of saddle point is defined, and the saddle point theorem is obtained. The corresponding results in the references are generalized.

Population game theory, as a popular research direction in recent years, has been applied to road traffic networks. Due to uncertain factors such as incomplete information, incomplete rationality, or uncertain environments, this paper aims to incorporate uncertain parameters into population games, investigate the existence of cooperative NS equilibrium, and apply it to the problem of multi-vehicle cooperative path planning. Before that, we first investigate the existence of cooperative NS equilibrium in normal form games with uncertain parameters, and use Zhao's (1992) hybrid equilibrium idea to prove the existence of this cooperative NS equilibrium. Then, we provide corresponding numerical examples for analysis. Actually, Cooperative NS equilibrium is a hybrid equilibrium between cooperative equilibrium $\alpha$-core and noncooperative NS equilibrium, which aligns more closely with the real economic environment and holds significant research importance.

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In this article, we consider maximum likelihood estimation (MLE) of the parameter of the Ailamujia distribution and its properties under ranked set sampling (RSS). Both theoretical and numerical results demonstrate that the MLE of RSS is more effective than that of simple random sample (SRS). Considering the possible effects of ranking errors, this article further considers the asymptotic efficiency of MLE of the parameter under the imperfect ranked set sampling (IRSS). Both theoretical and numerical results demonstrate that MLE under IRSS is at least as effective as MLE under SRS.

On the basis of Type-I censoring test scheme, a new censoring test scheme, namely generalized Type-I censoring, is proposed. Under generalized Type-I censored samples, the maximum likelihood estimates and approximate confidence intervals of the unknown parameters are studied for Burr XII distribution. When the scale parameter is known, the prior distribution of shape parameter is taken as the Gamma distribution, and the Bayesian estimates of shape parameter and reliability are obtained under three types of loss functions. When the model parameters are all unknown, the Bayesian estimates of unknown parameters and reliability are obtained using the Lindley’s approximation method under the squared loss function. The remaining useful life of censored components is predicted using classical and Bayesian methods, including point prediction and interval prediction, and using classical methods to predict future failure times. Calculate the average lengths of the approximate confidence intervals through stochastic simulation and compare the accuracy of classical estimation and Bayesian estimation, from the mean square error point of view, Bayesian estimation is better than maximum likelihood estimation. Finally, a real data set is analyzed.

Based on the conformal multisymplectic theory of Hamilton system, a conformal high-order compact structure-preserving algorithm for a class of damped eKdV equations is studied. Firstly, by introducing intermediate variables, the eKdV equation is transformed into a conformal multi-symplectic Hamilton system that satisfies the local conservation laws, and the conformal multi-symplectic Hamilton system is split into a conservation subsystem and a dissipation subsystem by using the Strang splitting method. Furthermore, the sixth-order compact difference method is used in the spatial direction, and the implicit midpoint method is used in the temporal direction to discretize the Hamilton system to obtain a conformal high-order compact poly-symplectic scheme. Under the periodic boundary conditions, the discrete scheme satisfies the global conformal symplectic conservation law and the mass conservation law. Finally, in the numerical example, the conformal sixth-order compact multi-symplectic algorithm is compared with the sixth-order compact difference method, which demonstrate the effectiveness of the proposed scheme and its capability for long-time numerical simulations.