In this paper, we first establish some second main theorems for algebraic curves from a compact Riemann surface into a complex projective subvariety of the complex projective space, which is ramified over hypersurfaces in subgeneral position. Then we use it to study the ramification for the generalized Gauss map of complete regular minimal surfaces in $\mathbb{R}^{m}$ with finite total curvature.

In this paper, the normal theorems of meromorphic functions involving discrete values are studied by using the theory of Ahlfors covering surfaces. Firstly, the discrete values with one leaf of meromorphic functions are defined, then the inequalities about islands are investigated and two precise inequalities about islands are obtained. Finally, the inequalities are used to study the discrete values and the normal family of meromorphic functions, then a normal theorem involving a monophyletic island and a normal theorem involving discrete values of one leaf are obtained. All these theorems promote the famous Ahlfors' five islands theorem and five single valued theorem of Nevanlinna.

In this paper, we study commutators of weighted composition operators with linear fractional non-automorphisms on Hardy space of the unit ball. First, we obtain the formula of commutators of weighted composition operators. Then, we characterize compactness of commutators according to two special situations of linear fractional maps. Finally, we obtain that commutators are compact when linear fractional maps are parabolic and commutators are not compact when linear fractional maps are hyperbolic.

Let $\mu$ be a normal function on $[0, 1)$. In this paper, the dependencies among the three conditions for which the multiplier operator is bounded on the normal weight Zygmund space ${\cal Z}_{\mu}(B)$ in the unit ball $B$ of ${{\bf C}}^{n}$ are discussed.

In this paper, we construct a shape functional for p-torsional rigidity problems and prove that the optimal shape of this shape functional is a ball. Using a method of the shape derivative, we give an alternative proof of the overdetermined problem for p-torsional rigidity.

Space-time fractional quantum mechanics, described by Schrödinger equation with Caputo derivative and Riesz derivative, is a generalization of quantum mechanics and can depict more extensive quantum phenomena. This paper studies the one-dimensional space-time fractional Schrödinger equation for a particle in the single and double δ-potential well, and gives the wave functions and energy levels of the particle. In addition, the space-time fractional quantum mechanical path integrals kernels of a particle in the δ-potential well are established by using integral transformation, and the corresponding Fox's H-function forms are derived, and the relation between space-time fractional Schrodinger equation and path integrals is constructed. It provides more possibilities to study space-time fractional quantum mechanics from the perspective of path integrals.

In this paper, we study indefinite Sturm-Liouville operators with discontinuity at interior point. The analyticity characteristics of the eigencurves is discussed and the sufficient conditions for the existence and exact number and evaluations on the upper bounds for non-real eigenvalues are obtained. Then two examples are given.

This paper deals with the weighted temporal-spatial estimates and strong solvability of the Navier-Stokes equation in ${\mathbb R}_{+}^{n}$. With the aid of Ukai's representation of the Stokes semigroup, and weighted inequalities for the fractional integral operators, $L^{r}$-$L^{q}$ estimates with mixed spatial weights are made for the Stokes flow. Then by means of Hardy's inequality, and interpolation method for the weak $L^{s}$ space, existence of the integral solution in $L^{b}(0, T;L^{q}({\mathbb R}_{+}^{n}))$ with temporal and spatial weights for the Navier-Stoke equation, where the initial velocity $u_{0}$ belongs to $L^{s}({\mathbb R}_{+}^{n})$ with the weight $w^{s-n}$ for some $n\leq s<\infty$ is established. This solution is proved to be the regular one provided $n=3$, $n\leq s\leq4$, and $u_{0}$ also lies in $L_{\sigma}^{2}({\mathbb R}_{+}^{n})$. Considering that $L_{w^{s-n}}^{s}({\mathbb R}_{+}^{n})$ does not coincide with $L^{s}({\mathbb R}_{+}^{n})$ whenever $s>n$, results obtained here can be viewed as useful supplements to the literatures.

In this paper, we study the nematic liquid crystals system under the simplified Ginzburg-Landau model, which is probably the simplest mathematical model that one can derive, without destroying the basic nonlinear structure [1]. We get the local existence and uniquness of the Serrin's type of solutions provided the initial data $u_{0}\in L^{p}\cap H, $ $d_{0}\in W^{1, p}, p\geq n$. According to the Serrin's regularity criteria for the incompressible liquid crystals system [2], we actually prove the local existence of smooth solutions to liquid crystals system for big data and global existence of smooth solutions for small data.

In this paper, we investigate a nonlinear viscoelastic equation with a time-varying delay effect and velocity-dependent material density. Under suitable assumptions on the relaxation function and time-varying delay effect, we prove the global existence of weak solutions and general decay of the energy by using Faedo-Galerkin method and the perturbed energy method respectively. This result improves earlier ones in the literature, such as Refs.[1, 48-50].

In the current paper we improve the recent results established in [2] concerning the traveling wave solutions for a Holling-Tanner predator-prey system. It is shown that there is a $c^*>0$ such that for every $c>c^*$, this system has a traveling wave solution $(u(\xi), v(\xi))$ with speed $c$ connecting the constant steady states $(1, 0)$ and $(\frac{1}{1+\beta}, \frac{1}{1+\beta})$ under the technical assumptions $\limsup\limits_{\xi\rightarrow+\infty}u(\xi) < 1$ and $\liminf\limits_{\xi\rightarrow+\infty}v(\xi)>0$. Here we do not assume these assumptions and obtain the existence of traveling waves for every $c>c^*$ by some analysis techniques. Moreover, we deal with the open problem in [2] and complete the study of traveling waves with the critical wave speed $c^*$ by the approximating method. We also point out that both the nonlocal dispersal and coupling of the system in the model bring some difficulties in the study of traveling wave solutions.

In this paper, we consider a fourth order evolution equation involving a singular nonlinear term $\frac{\lambda}{(1-u)^{2}}$ in a bounded domain $\Omega \subset \mathbb{R}^{n}$. This equation arises in the modeling of microelectromechanical systems. We first investigate the well-posedness of a fourth order parabolic equation which has been studied in [1], where the authors, by the semigroup argument, obtained the well-posedness of this equation for $n\leq2$. Instead of semigroup method, we use the Faedo-Galerkin technique to construct a unique solution of the fourth order parabolic equation for $n\leq7$, which completes the result of [1].

In this paper, we study the global existence and regularity of the 2D generalized tropical climate model, which has the standard Laplacian term Δv in the first baroclinic mode and partial dissipation in the barotropic mode and the temperature equation.

In this paper, we consider the following critical Schrödinger-Poisson system \begin{eqnarray*} \left\{ {\begin{array}{*{20}{l}}{\begin{array}{*{20}{l}}{ - \Delta u + \lambda V{\rm{(}}x{\rm{)}}u + \phi u = \mu |u{|^{p - 2}}u + |u{|^4}u{\rm{, }}\; \; \; }\\{ - \Delta \phi = {u^2}, \; \; \; \; \; \; \; }\end{array}\begin{array}{*{20}{c}}{x \in {\mathbb{R}^3},}\\{x \in {\mathbb{R}^3},}\end{array}}\end{array}} \right. \end{eqnarray*} where $\lambda, \mu$ are two positive parameters, $p\in(4, 6)$ and $V$ satisfies some potential well conditions. By using the variational arguments, we prove the existence of ground state solutions for $\lambda$ large enough and $\mu>0$, and their asymptotical behavior as $\lambda\to\infty$. Moreover, by using Lusternik-Schnirelmann theory, we obtain the existence of multiple solutions if $\lambda$ is large and $\mu$ is small.

In this paper, we study the existence standing wave solutions for a nonlinear Schrödinger equation coupled with a neutral scalar field and a gauge field. We establish the existence result for the case that the exponent p of nonlinear term is greater than 2.

This paper is devoted to study a class of semilinear elliptic equation with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of positive solutions to this problem is obtain.

This paper investigates the stability in the sense of Hyers-Ulam-Rassias for a class of generalized fractional differential systems by the generalized Laplace transform method. Several examples are given to illustrate the theoretical results.

In this paper, we investigate the $n$-dimensional $(n\geq2)$ Magnetohydrodynamics-Boussinesq system with fractional diffusion. When the nonnegative constants $\alpha, \beta$ and $\gamma$ satisfy $\alpha\geq\frac{1}{2}+\frac{n}{4}, \ \alpha+\beta\geq 1+\frac{n}{2}$ and $\alpha+\gamma\geq\frac{n}{2}$, by using the energy methods, we obtain the global existence and uniqueness of solution for the system, which generalizes the existing result.

In this paper, we consider the initial boundary value problem of a viscoelastic equation with logarithmic nonlinearity. Under some suitable conditions, we obtain the existence of global weak solutions. Otherwise, we get that the solution does not blow up in any finite time. This is different from the situation of the viscoelastic equation with a polynomial nonlinearity, in which case the solution blows up in finite time.

Using the functional generalized variational iteration method, a class of nonlinear disturbed dynamic system was considered. First introduce solitary solution to a corresponding typical system. And then a set of functional generalized variation constructed, and Lagrange multiplier functions were solved. Finally, the generalized variational iteration was received. Thus, the asymptotic travelling wave solution to the original nonlinear disturbed generalized dynamic system was obtained

In this paper, the oscillation of second order delay dynamic equations with super-linear neutral terms on time scales is studied. By using Riccati transformation and Bernoulli inequality techniques, several new oscillation theorems for the equation are obtained. The corresponding results in the existing literature are generalized and improved, some of which are new even for differential equations. Finally, some examples are given to verify the validity of the theorems.

In this paper, we consider the asymptotic behavior of solutions to the linear spatially inhomogeneous Boltzmann equation for hard potentials in the torus. We obtain an optimal rate of exponential convergence towards equilibrium in a Lebesgue space with polynomial weight $L_{v}^{1} L_{x}^{2}\left(\langle v\rangle^{k}\right)$. This model is analyzed from a spectral point of view and from the point of view of semigroups. Our strategy is taking advantage of the spectral gap estimate in the Hilbert space with inverse Gaussian weight, the factorization argument and the enlargement method.

In this paper, by using Lyapunov functional, we prove the global attractivity of the endemic equilibrium for a nonlocal delayed and diffusive SVIR model when $\mathcal{R}_{0}>1$, which cover and improve some known results.

Conjugate gradient method is an important algorithm to solve a class of large-scale optimization problems, and it has the advantages of simple calculation and fast convergence. This method satisfies the sufficient descent condition without relying on any line search method, and it has global convergence under the modified Armijo line search. Numerical results of experiments show that the method is effective.

The discontinuous Galerkin finite element approximation schemes for the Extended Fisher-Kolmogorov (EFK) equation are studied by using the Wilson element. Without using the technique of postprocessing technique, the convergence results with order $O(h^{2})/O(h^{2}+\tau)$ for the primitive solution $u$ and intermediate variable $v=-\triangle u$ are obtained for the semi-discrete and linearized Euler fully discrete approximation schemes respectively through a new splitting technique for the nonlinear terms. The above results are just one order higher than the usual error estimates of the Wilson element. Here, $h$ and $\tau$ are parameters of the subdivision in space and time step, respectively.

In this paper, we introduce a new inertial contraction and projection algorithm for pseudomonotone variational inequality problems. We prove the strong convergence theorem without the knowledge of the Lipschitz constant of the mapping. Finally, we give some numerical experiments to show the efficiency of the algorithm.

This paper deals with problems of a stochastic chemostat model with general response function and wall growth. We show the conditions for the microorganism to be extinct. On the other hand, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the model which means the microorganism can become persistent. Finally, example and numerical simulations are introduced to illustrate the analytical results.

Generalized estimating equation (GEE) is widely adopted in analyzing longitudinal (clustered) data with discrete or nonnegative responses. In this paper, we prove the existence, weak consistency and asymptotic normality of generalized estimating equations estimator with adaptive designs under some mild regular conditions. The accuracy of the asymptotic approximation is examined via numerical simulations. Our results extend the elegant work of Xie and Yang (Ann Statist, 2003, 31: 310-347) and Balan and Schiopu-Kratina (Ann Statist, 2005, 33: 522-541).

In this paper, we investigate the global dynamics of a glucose-insulin model and its corresponding stochastic differential equation version. For the deterministic model, we show that there exists a unique equilibrium point, which is globally asymptotically stable for all parameter values. For the stochastic model, we show that the system admits unique positive global solution starting from the positive initial value and derive the stochastic permanence of the solutions of the stochastic system. In addition, by using Hasminskiis methods, we prove that there exists a unique stationary distribution and it has ergodicity. Finally, numerical simulations are carried out to support our theoretical results. It is found that: (ⅰ) the difficulty of the prediction of the peak size of the plasma glucose concentration always increases with the increase of the intensity of environmental fluctuations; (ⅱ) environmental fluctuations can result in the irregular oscillating of the plasma glucose concentration and plasma insulin concentration. Moreover, the volatility of the plasma glucose concentration and plasma insulin concentration always increase with the increase of the intensity of environmental fluctuations.

In this paper, a kind of SVEIR measles epidemic model with age structure is established. Firstly, the model is transformed into Volterra integral equation and the well-possdness of solutions of the model is obtained, including non-negativity, boundedness, asymptotic smoothness, etc. Then the equilibria and the basic reproduction number ${{\cal R}}_{0}$ of the model is derived, and it is proved that the epidemic is uniformly persistent when ${{\cal R}}_{0}>1$. Further by analyzing the characteristic equations and selecting suitable Lyapunov functions, we get the model only has the disease-free equilibrium that is globally asymptotically stable if ${{\cal R}}_{0}<1$; if ${{\cal R}}_{0}>1$, the disease-free equilibrium is unstable, the endemic disease equilibrium exist and is globally asymptotically stable. These main theoretical results are applied in the analysis of the trend in data on measles infectious diseases across the country.

In this paper, we propose and investigate a tumor-immune system interaction model with antigenicity. The existence of equilibria of the model is determined, and the local dynamics of each feasible equilibrium is analyzed. The global dynamics of the model is obtained by excluding the existence of periodic solutions. It is found that, under certain conditions, the saddle-node bifurcation and the bi-stability of strong equilibrium with tumor and equilibrium without tumor may occur for the model, which imply that the growth and development of the tumor will depend on its initial state. The obtained theoretical analysis results are verified by numerical simulations.

This paper combines theoretical derivation and numerical simulation to study the dynamics of a delayed reaction-diffusion predator-prey model with fear effect. First, the existence and stability of the positive equilibrium point of the system are studied. Secondly, the Hopf bifurcation problem of the system is studied through linear stability analysis. The results show that the fear effect affects the Hopf bifurcation point, and then affects the stability interval of the system. Finally, the theoretical results are verified by numerical simulations, and the nonlinear relationship between the fear effect and the stability interval is found, that is, as the fear effect continues to increase, the system will change from a stable state to an unstable state, and then to a stable state.