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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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 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.

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.

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.

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.

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 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.

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.

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.

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.

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 $\mathcal{H}_{t}^{\alpha}$.

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 $i\in\{3,4\}$.

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 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.

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.

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.