In this paper, we study freely quasiconformal mappings in quasiconvex metric spaces. It is proved that freely quasiconformal mappings and rough quasihyperbolic mappings in quasiconvex metric spaces are equivalent, and the quasisymmetric properties of freely quasiconformal mapping in quasiconvex metric spaces are obtained.

In this paper, we present a new method to test quantum entanglement based on affine mapping. First, we prove the complete positivity of two special affine mappings in two-dimensional Hilbert space. Then, based on this completely positive affine mapping, we give our entanglement test. At last, we illustrate the capability of our test with two examples.

Based on the definition of multipartite quantum discord proposed in [Phys Rev Lett, 124, 110401(2020)], we give the analytic expression for the multipartite geometric quantum discord of one type of three-qubit X states which depend on four real parameters by measurement induction. And we present the level surfaces of the class of three-qubit X states. As an application, we investigate the dynamic behavior of multipartite geometric quantum discord for the three-qubit X states under the phase flip channel.

This paper introduces some work on singular perturbation problems with discontinuous right-hand side, mainly discusses a class of fourth-order Dirichlet boundary value singular perturbation equations with discontinuous right-hand sides. After introducing complex equation form, we construct a formal asymptotic solution with an internal transfer layer by using the boundary layer function method, and give the existence and residual estimation of smooth solutions. Finally, an example is given to verify the correctness of the algorithm.

This paper is concerned with the nonhomogeneous Dirichlet exterior problem for a semilinear hyperbolic differential inequality in the higher dimensional space ($N\geq2$). Using the test function method and contradiction argument, we establish the nonexistence theorem under the nonhomogeneous Dirichlet boundary condition, which depends both on time and space. Meantime, we obtain some existence results.

In this paper, we consider a class of $\phi$-Laplacian Rayleigh equation, where the nonlinear term is non-autonomous and has a singularity at the origin. By applications of Mawhin's continuation theorem and some analysis methods, we prove the existence of periodic solutions to the equation with a strong singularity of repulsive type (or weak and strong singularities of attractive type).

In this paper, we study the properties of normalized solutions for a class of mass critical Kirchhoff type equations by using constrained variational methods, including the existence, nonexistence and concentration behavior of normalized solutions.

This article concerns the following Klein-Gordon-Maxwell system $\begin{equation*} \begin{cases} -\Delta u+u-(2\omega+\phi)\phi u=\lambda Q(x)f(u), & x\in \mathbb{R}^{3},\\ \Delta \phi=(\omega+\phi)u^2, &x\in \mathbb{R}^{3}, \end{cases} \end{equation*}$ where $\omega> 0$ is a constant, $\lambda> 0$ is a parameter, $Q$ is a positive function. When the nonlinear term $f$ is sublinear at infinity, two nontrivial solutions for the system are established via variational methods and three critical points theorem. Furtermore, when $f$ is sublinear only in a neighbourhood of the origin, existence and multiplicity of non-trivial solutions are obtained via variational methods and critical point theorem. Our result completes some recent works concerning the multiplicity of solutions of this system.

This paper explores the maximum Reynolds number that can be simulated by the multiple-relaxation-time lattice Boltzmann method with viscosity counteracting (MRT-VC). Firstly, the accuracy of the model is validated by simulating the classic 2D lid-driven cavity flow. The focus is on the flow fields at Reynolds numbers of 5430 and 7000, analyzing the flow fields, vortex core coordinates, axial velocity, and velocity spectra. Secondly, as the simulated Reynolds number increases, the number of swirling vortices in the flow field gradually increases. The flow exhibits a sequence of stable flow, periodic flow, incomplete chaotic flow, and chaotic flow. The critical transition Reynolds number from stable flow to periodic flow is between 10000 and 12500, from periodic flow to incomplete chaotic flow is between 45000 and 50000, and from incomplete chaotic flow to chaotic flow is between 95000 and 100000.

In this paper, the blow-up properties of solutions for a class of fractional diffusion equations with time dependent coefficients is studied. By means of the potential well theory, Nehari manifold, concave conex method, and various differential inequalities, the finite time blow-up of the solutions under subcritical initial energy level, negative initial energy level and any positive initial energy level is discussed.And the upper bound of blow-up time is obtained.In particular, due to the energy functional and the depth of the potential well are related to the time-dependent coefficient $f(t)$, in the case of sub-critical initial energy level, the upper bound of blow-up time will change with $f(t)$.

In this paper, we investigate the evolution of the initial discontinuity for the generalized Gardner equation through the Whitham modulation theory, which the generalized Gardner equation can describe the transcritical flow of stratified fluids over topography. Firstly, we derive the linear harmonic wave, soliton and nonlinear trigonometric wave in different limiting cases via the periodic waves represented by the Jacobi elliptic functions. Then we obtain the Whitham characteristic velocities and modulation system based on the Riemann invariants by the finite-gap integration method. Since the modulation system of the generalized Gardner equation is neither strictly elliptic nor hyperbolic type, which makes the dynamical evolution behavior more varied in different regions compared to the KdV equation. Furthermore, we perform a complete classification for all wave structures in the cases of positive and negative cubic nonlinear terms, including the dispersive shock wave, rarefaction wave, trigonometric dispersive shock wave, solibore and their combinations. In addition, the correctness of the results is verified by numerical simulations, and the numerical solutions are in good agreement with the analytical solutions. Finally, the influences of the coefficients of the linear and nonlinear terms on the step initial value problem under certain conditions are analyzed.

In this paper, a new algorithm for solving the second Volterra type integral equation is proposed by combining the least square method with the reproducing kernel method. By constructing the multi-scale orthogonal basis of the reproducing kernel space, the solution expression of the model is obtained. To reduce the amount of computation and simplify the calculation process, this paper transforms the model into linear algebraic equation by using least square method and obtains the approximate solution of $\varepsilon$. In addition, to verify the rigor of the algorithm, the uniform convergence and stability of the algorithm are proved in detail, and the error estimation is discussed and analyzed. The feasibility and applicability of the proposed algorithm are verified by numerical examples. Compared with some known methods, the results obtained in this paper are more accurate.

PNP Equations are a class of nonlinear partial differential equations coupled from Poisson and Nernst planck equations, and the efficiency of its Gummel iteration, a commonly used linearization iteration method, is largely affected by the relaxation parameter. The Gaussian Process Regression (GPR) method in machine learning, due to its small training size and the fact that it does not need to provide a functional relationship, is applied in that paper to predict the preferred relaxation parameters for the Gummel iteration and accelerate the convergence of the iteration. Firstly GPR method with predictable relaxation parameters is designed for the Gummel iteration of the PNP equation. Secondly, the Box-Cox transformation method is utilized to preprocess the data of Gummel iteration to improve the accuracy of the GPR method. Finally, based on the GPR method and Box-Cox transformation algorithm, a new Gummel iteration algorithm for the PNP equation is proposed. Numerical experiments show that the new Gummel iterative algorithm is more efficient in solving and has the same convergence order compared to the classical Gummel iterative algorithm.

Let $\{X_{i,n},1\leq i\leq n\}$ be a centered stationary Gaussian triangular array with unit variance. Assuming the correlation $ \rho_{j,n}=E\left( X_{i,n}X_{i+j,n}\right)$ satisfies the conditions in [14], this paper is interested in the joint behavior of the point process of clusters and the partial sum of the Gaussian triangular array. It is shown that the point process of clusters converges in distribution to a Poisson process and is asymptotically independent with the partial sums.

In the paper, the symplectic-preserving schemes are presented for fractional Klein-Gordon-Schrödinger equations. First, we give the infinite-dimensional Hamilton with fractional Laplacian operator and conservation laws, and change the above quantum mechanical equations into Hamilton system. We apply the central finite difference schemes to discrete Klein-Gordon-Schrödinger in space, and yield a large Hamilton ordinary differential system. Second, we use the midpoint rule in time to Hamiltonian ordinary differential system, and obtain a symplectic approximation of the these equations. Moreover, we analyze the conservation of the numerical scheme. Finally, we give numerical experiments to show the verify the efficiency of the conservative finite difference scheme.

This article mainly studies the improvement problem of integral inequality and applies it to the stability study of additive time-varying delay systems: Firstly, the existing Wirtinger inequality is proven from the perspectives of monotonicity and adding positive terms using the method of constructing parameter functions. Secondly, we propose extended inequalities, including the extended Wirtinger inequality and the extended cross-convex inequality which is based on third-order matrices. Thirdly, utilizing these two extended inequalities, we provide criteria for determining the asymptotic stability of additive time-varying delay systems in the form of Linear Matrix Inequalities (LMIs). Finally, numerical examples are presented to demonstrate the effectiveness and superiority of the proposed method.

To study the effects of individual diffusion and spatial heterogeneity on the transmission of syphilis, we construct a heterogeneous spatial reaction diffusion model of syphilis. Firstly, the well posed problem of the model is studied, including the global existence of the solution, the dissipativity of the system and the existence of the attractor for the semiflow; Secondly, based on the definition of the next generation regeneration operator, we derive the functional expression of the basic regeneration number $R_0$; Thirdly, we discussed the dynamical behaviors of the solution regarding the threshold-$R_0 $, specifically, when $R_0>1$, the disease-free steady state is globally stable, when $R_0>1 $, the system is uniformly persistent. In special cases, we also prove the existence, uniqueness, and global stability of the positive equilibrium of the system. Finally, the theoretical results were validated and the influence of spatial factors on the transmission of syphilis was analyzed through numerical simulation. Our numerical results indicate that: (1) strengthening the treatment of early latent syphilis carriers can effectively reduce the risk of syphilis transmission among population; (2) Ignoring spatial heterogeneity will underestimate the epidemic trend of syphilis. In addition, the impact of individual diffusion rate on the transmission of syphilis cannot be ignored.

In the paper, a stochastic predator-prey model with Ornstein-Uhlenbeck process, fear effect, Crowley-Martin type and Leslie-Gower type functional responses is considered. Firstly, by constructing suitable Lyapunov functions, we prove that the existence and uniqueness of the global solution, and the sufficient conditions for the exponential extinction and the existence of stationary distribution are obtained. Secondly, we get access to the specific expression of probability density function via dealing with the corresponding Fokker-Planck equation. Finally, our theoretical results are verified by three numerical examples. The results show that the intensity of volatility and the reversion speed of stochastic disturbance will affect the survival of species.

A stochastic predator-prey system with fear effect and Holling-type III functional response is studied. Firstly, the existence and uniqueness of global positive solutions, mean boundedness and stochastic ultimate boundedness of the system for any given positive initial value are proved. Secondly, the sufficient conditions for extinction and persistence in mean of the prey and predator populations are obtained, and Lyapunov function is constructed to prove the existence of stationary distribution and ergodicity. Finally, the results are verified by numerical simulations.

This paper considers the SIS stochastic epidemic model on the simplex complex under the influence of noise, the difference in stochastic stability and stochastic bifurcation of models with 1 simplex contagion strength ($\lambda$) or 2 simplex contagion strength ($\lambda_{\triangle}$) perturbed by noise is compared. The results show that the threshold of the stochastic model infectious disease extinction with probability 1 after the noise acts on the low-order term where $\lambda$ is located is related to the noise intensity, and vice versa when it acts on the high-order term where $\lambda_{\triangle}$ is located has no effect; increasing $\lambda_{\triangle}$ causes the point near 1 of the steady-state probability density function appearing peak, larger the pear value or nearing the 1 point, that is, increasing $\lambda_{\triangle}$ promotes the outbreak of the disease.

This paper incorporates quadratic transaction costs in the optimal risk control and investment problem for an insurer. Moreover, suppose that the insurance and financial markets are correlated. Under the dynamic mean-variance criterion, by solving a system of extended HJB equations, the equilibrium risk control and investment strategies and the corresponding value function are derived in terms of the solution to a system of matrix Riccati equations. Finally, the effects of transaction costs level and the market correlation coefficient on the equilibrium strategy and the efficient frontier are analyzed by some numerical examples. It turns out that the growth rate of investment slows down as the transaction costs level or the correlation coefficient increases, and the increase of transaction costs level will lead to the decrease of the efficient frontier.