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.

Traditional mean regression models have been widely applied in forecasting; however, they often fail to capture the tail behaviors of data, especially in the presence of skewness and heavy-tailed distributions. Expectile regression, as an extension of the mean regression model, provides a more flexible framework that adapts to different data distributions and quantiles, offering a more detailed perspective on predictability. This paper proposes a unified predictability test for expectile predictive regression models, accounting for high persistence and conditional heteroscedasticity in financial time series. The asymptotic distribution of the test statistic is derived, and the method is robust against different persistences of the predictor. The empirical analysis re-examines the predictability of monthly returns on the S&P 500 index using 11 macroeconomic indicators, revealing significant variations in predictive power across different expectiles. This study highlights the effectiveness of expectile regression in capturing the complexities of financial data and improving predictive accuracy under challenging conditions.

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.

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.

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 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.

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.

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 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 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.

As a new research direction in the field of artificial intelligence, deep learning has received widespread attention in recent years and has made significant progress in many application areas. Conjugate gradient method, as an effective optimization method, achieves excellent numerical performance by iteratively approximating the optimal solution. Compared to other methods, the conjugate gradient method does not require the computation of the Hessian matrix, thereby greatly reducing the computational and storage requirements. Therefore, this paper aims to investigate the application of the conjugate gradient method in deep learning and proposes a new conjugate gradient method, demonstrating its sufficient descent property and trust region characteristics. In addition, we introduce the stochastic subspace algorithm and an improved version of it with variance reduction techniques, providing detailed steps for the new algorithm to facilitate a better understanding of its purpose and significance. Through theoretical analysis, we prove that the new algorithm exhibits good convergence properties and high iteration efficiency, with a complexity of $O(\epsilon^{-\frac{1}{1-\beta}})$. Furthermore, experimental results demonstrate the favorable numerical performance of this method.

This paper focuses on the existence and stability of Nash equilibria for population games with trapezoidal fuzzy payoffs. Firstly, a new order relation and a new distance formula between trapezoidal fuzzy numbers are defined, then some properties of trapezoidal fuzzy numbers and trapezoidal fuzzy payoff functions similar to the deterministic case are obtained. Secondly, the existence of Nash equilibria for such games is proved by Kakutani fixed point theorem. Finally, most of such games are proved to be essential on the meaning of Baire category by Fort theorem.

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.

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.

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 this paper, we propose and explore a modified Leslie-Gower with nonlinear harvesting in prey. Through an examination of the existence and stability of all possible equilibria, we find the system may exhibit complex bifurcation phenomena. Using Sotomayor's theorem, the rigorous proof of the occurrence of saddle-node bifurcation is derived. To investigate the stability of the limit cycle of Hopf bifurcation, the Lyapunov coefficient is calculated, and a numerical example is conducted to illustrate this visually. By computing a universal unfolding near the cusp, we show that the system experiences a codimension 2 Bogdanov-Takens bifurcation and provide its bifurcation diagram. At the same time, the dynamic behavior of the model is demonstrated in detail by numerical simulation. Our findings enhance the understanding of Leslie-type predator-prey dynamics.

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 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.

At present, there have been many achievements in the study of Turing instability of reaction-diffusion models, but the study of Turing pattern formation and evolution process of reaction-diffusion system pattern formation is still in the early stage, especially under the drive of cross diffusion. Therefore, the Turing pattern dynamics of a class of Oregonator reaction-diffusion models with cross-diffusion terms are analyzed. First, the conditions of Turing instability induced by cross-diffusion term are obtained when self-diffusion term drives the system to be stable. Secondly, the effect of the cross-diffusion term of the reactants on the pattern formation and evolution process of the system is studied, and whether the cross-diffusion term can change the Turing unstable state caused by the self-diffusion term and whether the different cross-diffusion coefficients can affect the stability rate of the system is discussed. Finally, the simulation results show that the cross-diffusion term plays a significant role in Turing instability and pattern evolution.

In this paper, the asymptotic behavior of the solutions to the beam equation with rotational inertia and strong damping: $\varepsilon(t)(1+(-\Delta) ^{\alpha})\partial^{2}_tu+\Delta^2 u-\gamma\Delta\partial_tu+f(u)=g(x),$ where $\alpha\in[0,1)$ is discussed. When the growth exponent of nonlinear terms satisfies $1\leqslant p< p^{*}=\frac{N+2}{N-4},$ $N\geqslant5,$ firstly, by using the Faedo-Galerkin approximation method and the asymptotic regular estimate technique, the well-posedness and regularity of solutions are established; secondly, the asymptotic compactness of the solution process is proved via the method of contraction function; finally, the existence of a time-dependent global attractor is obtained in the time-dependent space $\mathcal{H}_{t}^{\alpha}$.

In this paper, we investigate an overdetermined problem involving a fourth order elliptic operator defined within convex cones. The primary objective is to establish the radial symmetry of solutions under specified boundary conditions. Our approach entails the construction of a $P$-function and the application of the maximum principle, leading to a proof that any smooth solution in a bounded sector-like domain with a mean-convex boundary portion necessitates the domain being a spherical sector—the intersection of the cone with a ball. A major contribution is overcoming the challenge of deriving precise boundary estimates for the $P$-function on the cone, a setting with more intricate geometry than classical bounded domains. We also present the solution's explicit form and the relation between the Neumann data and the sphere's radius, thereby extending several classical rigidity results to the context of convex cones.

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.

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.

With the rapid development of high technology, the influx of high dimensional data brings new challenges to the existing statistical methods and theories. Huber regression is a statistical analysis method that uses regression analysis in mathematical statistics to determine the interdependent quantitative relationship between two or more variables. Existing methods in Huber regression treat all the predictors equally with the same priori, we take advantage of the graphical structure among predictors to improve the performance of parameter estimation, model selection and prediction in sparse Huber regression. In order to overcome the difficulty of solving Huber regression with graphic structure, we propose an alternating direction method of multipliers (ADMM) algorithm with a linearization technique. The simulation and empirical results show that the Huber regression method combining graphical structure among predictors is superior to the adaptive Lasso penalty Huber regression without graphical structure in estimation accuracy and prediction performance.

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.

This paper studies the optimal reinsurance decision-making problem between an insurer and $n$ reinsurers based on competition under the influence of inside information of claims. The insurer and $n$ reinsurers joint share claims, the competition between the insurer and reinsurers is quantified by relative performance. Inside information of claims refers to partial information about future claims, which is modeled by filtration expansion theory. The insurer's aim is to maximize his expected relative surplus while minimizing the variance of his relative surplus at the time of reinsurance termination. By using stochastic control and stochastic analysis theory, we establish the Hamilton-Jacobi-Bellman (HJB) equation and verification theorem. By solving the HJB equation and constructing Lagrange function, we obtain the explicit solutions for the optimal reinsurance strategy and the corresponding optimal value function. Finally, the influence of key model features such as inside information of claim, competition and the number of reinsurers on the optimal reinsurance strategy is examined by numerical experiments, and the insurance and economic significance behind the influence is also analyzed.

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.

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.

In the era of big data, fields such as biomedicine and finance face significant challenges in modeling complex relationships among variables. Accurate identification of key interaction effects is essential for improving predictive accuracy and uncovering underlying mechanisms. Nevertheless, high-dimensional data contain numerous covariates, and the number of interaction terms rises sharply. Incorporating all candidate terms into models will inevitably result in heavy computational burden and serious overfitting. Accordingly, efficient screening of influential main and interaction effects has become an urgent research issue. This paper proposes a model-free variable screening approach based on the Hilbert——Schmidt Independence Criterion (HSIC) and two-step screening strategy for ultrahigh-dimensional right-censored survival data. The method can simultaneously select significant main and interaction effects and accommodate ultrahigh dimensionality. Numerical simulations and real data analyses verify that the proposed method possesses satisfactory screening accuracy and desirable robustness under various circumstances.

The fuzzy mappings are led into a class of coupled map lattices related to chaotic cryptographic algorithm. It is proved that the $\mathcal{P}_1$-chaos of fuzzy coupled systems means that the initial value mappings also have the same chaotic properties. Where $\mathcal{P}_1$-chaos includes $({{\mathcal{F}}_{1}},{{\mathcal{F}}_{2}})$-chaos, Li-Yorke chaos, distributional chaos, spatio-temporal chaos, densely $\delta $-chaos, densely chaos, Ruelle-Takens chaos and Kato chaos. In particular, by limiting the initial value mappings to the diagonal of the space, a sufficient condition for the fuzzy system to has $\mathcal{P}_2$-chaos is obtained. Where $\mathcal{P}_2$-chaos is one of the followings: initial value sensitive dependence, Li-Yorke sensitive, densely Li -Yorke sensitive, infinite sensitive, synthetically sensitive, cofinitely sensitive, $({{\mathcal{F}}_{1}},{{\mathcal{F}}_{2}})$-sensitive, $\mathcal{F}$-sensitive, transitive, exact, or accessible.

Let $G=(V,E)$ be a graph. Let $k$ and $d$ be positive integers. If we can color these vertices with $k$ colors such that at most $d$ neighbors of $v$ receive the same color as $v$, then $G$ is called to be $(k,d)^{*}$-colorable. A list assignment of $G$ is a function $L$ that assigns a color list $L(v)$ to each vertex $v\in V(G)$. An $(L,d)^{*}$-coloring of $G$ is a mapping $\pi$ that assigns a color $\pi(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most $d$ neighbors of $v$ receive the color $\pi(v)$. If there exists an $(L,d)^{*}$-coloring for every list assignment $L$ with $|L(v)|\ge k$ for all $v\in V(G)$, then $G$ is called to be $(k,d)^{*}$-choosable. In this paper, we prove every planar graph $G$ without adjacent $i$-cycles and $7$-cycles is $(3,1)^{*}$-choosable, for all $i\in\{3,4\}$.

We introduce a new notion of measure-theoretic invariance pressure for control systems and present an inverse variational principle for this invariance pressure. Moreover, we obtain two characterizations of this invariance pressure for nonsingular measures: Bowen measure-theoretic invariance pressure and measure-theoretic feedback pressure.

In this paper, a mathematical model for a solid spherically symmetric vascular tumor growth with nutrient periodic supply is studied. The external radius of the tumor $R(t)$ changes with time, so the model is a free boundary problem. The cells inside the tumor obtain nutrient $\sigma(r,t)$ through blood vessels, and the tumor attracts blood vessels at a rate proportional to $\alpha(t)$. Thus, the boundary value condition \begin{equation*} \sigma_r(R(t),t)+\alpha(t)(\sigma(R(t),t)-\psi(t))=0 \end{equation*} holds on the boundary, where the function $\psi(t)$ is the concentration of nutrient externally supplied to the tumor. Considering that the nutrients provided by the outside world are often periodic, the research in this paper assumes that $\psi(t)$ is a periodic function. $\alpha(t)$ is a uniformly bounded function with a positive lower bound. The purpose of this study is to investigate the impact of periodic nutrient supply on the growth of vascularized tumors. Sufficient and necessary conditions for the global stability of zero steady state (i.e., tumor free equilibrium) are provided. Under the condition that the zero steady state is unstable, if $\lim\limits_{t\rightarrow\infty}(\alpha(t)-\bar{\alpha}(t))=0,$ where $\bar{\alpha}(t)$ is a periodic function, by using the Brouwer fixed-point theorem, we prove that there exists a unique periodic solution which is the global attractor of all solutions of the problem. The results are illustrated by computer simulations.

Research on dividend problems has always been a core focus in the field of risk theory. In real situations, insurance companies do not fully know the specific distribution of claims and can only obtain information related to claims before a specific time. Therefore, It would be more meaningful to study the statistical estimation of dividend functions using the data. This paper studied the expected present value of dividend payments before ruin of the perturbed compound Poisson model under the threshold dividend strategy. Based on the available observation data of claims and dividends, the Fourier cosine series expansion method (COS method) was used to obtain the statistical estimator of the expected present value of dividend payments before ruin and analyzed its convergence speed in a large sample environment. Finally, numerical results were given to further prove the effectiveness of the estimation method.

The bilinear time series model, as an important class of nonlinear models, has been widely applied in fields such as control theory and econometrics, particularly for modeling seismic data and other scenarios exhibiting abrupt volatility characteristics. Compared with traditional linear models, this class of models can more effectively capture occasional explosive features in time series data. This paper focuses on a class of bilinear time series models with time-functional variance (TFV) noise, establishing asymptotic theory for the generalized autoregressive conditional heteroskedasticity-type maximum likelihood estimator (GMLE) based on sieve estimation. Under the condition of finite fourth moments for error terms, we prove that the generalized maximum likelihood estimator is consistent and asymptotically normally distributed. Furthermore, we conduct numerical simulations to evaluate the finite-sample performance of the sieve-based GMLE.

In this paper, a maximum likelihood estimator (MLE) of the parameter of the Epanechnikov-exponential distribution and its properties are respectively studied under simple random sampling(SRS) and balanced ranked set sampling(RSS). Both theoretical and numerical results demonstrate that the MLE under balanced RSS is asymptotically more effective than the MLE under SRS. Additionally, we investigate the asymptotic efficiency of the MLE under imperfect balanced RSS, taking into account the potential presence of ranked errors. Both theoretical and numerical results show that the MLE under imperfect balanced RSS is at least as effective as the MLE under SRS.

Solving Volterra calculus equations has various applications in many aspects of science. The delay weakly singular kernel makes it difficult for most existing numerical simulations to deal with. Therefore developing an efficient and accurate solver is a challenge. In this paper, the approximation by Jacobi spectral method is constructed for the delay Volterra calculus equation with weakly singular kernel. The error analysis is also provided to justify the high-order accuracy of convergence for the error of approximate solution and the error of approximate derivative. We get the conclusion that both kinds of errors decay exponentially in both $L^{\infty}$ norm and $L^{2}_{\omega^{-r,-r}}$ norm. In the last section, numerical tests are displayed to confirm the reliability of the Jacobi spectral analysis.

We consider the source term identification problem of space-time fractional diffusion equation. The ill-posedness of the problem is analyzed. The regularized solution of the inverse source problem is obtained by using the iterative generalized quasi-reversibility regularization method, and the error estimates between the regularized solution and the exact solution are given under the prior and posterior regularization parameter selection rules. Finally, numerical results show the effectiveness and stability of the method.