In this article, we prove a generalization of Hsiung-Minkowski formula for free boundary hypersurfaces in a ball in space forms. As corollaries, we obtain some Alexandrov-type results.

In this paper, the partial discrete sequence of $SO^\ast(6)/SO(3,\mathbb{C})$ is obtained by local isomorphism of Hermite-type affine symmetric space, and the specific form of the holomorphic discrete sequence generated by the cyclic vector is given.

In this paper, the Ramanujan asymptotic formula of the Gaussian hypergeometric function $_{2}F_{1}$ and its related Ramanujan $R$-function will be generalized to the case of basic hypergeometric series $_{2}\phi_{1}$. On the one hand, we shall present the $q$-Ramanujan asymptotic formula of $_{2}\phi_{1}$ and introduce the $q$-Ramanujan $R$-function; on the other hand, we shall mainly study the $q$-Ramanujan $R$-function, and prove some analytical properties of the $q$-Ramanujan $R$-function including series expansions, complete monotonicity property and monotonicity property with respect to the parameter $q$. As applications, several sharp inequalities for the $q$-Ramanujan $R$-function will be derived.

This paper studies the limit cycle bifurcation problem of a non-smooth differential system with a cuspidal loop under non-smooth perturbation of polynomials of degree $n$. Firstly, the first order Melnikov function $M(h)$ of the perturbed differential system is expressed as a linear combination of several generating integrals with polynomial coefficients, and the independence of coefficients of these polynomials is proved by mathematical induction. Then the lower bounds of the number of limit cycles bifurcating the origin and cuspidal loop are obtained by using the asymptotic expansions of $M(h)$.

This work is concerned with the stabilization and optimal decay rates of a weakly coupled (coupling through displacements) elastic plate system, where the damping (viscous damping, structural damping or Kelvin-Voigt damping) is actuated at only one of the two plates. The optimal polynomial decay rate is derived based on the frequency domain approach and detailed spectral analysis of the system operator. Moreover, the relationship between the optimal decay rates of the system and the order of damping is identified. Besides, an interesting phenomenon is found that the higher the order of indirect damping is actuated, the slower polynomial decay rate of the weakly coupled plate system achieves. Finally, some numerical simulations are presented to verify the theoretical results.

The bounded primitive of a periodic function is periodic, and the bounded primitive of an asymptotically periodic function is not necessarily asymptotically periodic. In this paper, we introduce the concept of slowly periodic functions and prove that the bounded primitive function of an asymptotically periodic function is slowly periodic. Interestingly, slowly periodic functions are just a special class of $\S$-asymptotically periodic functions, which were introduced 15 years ago and extensively studied in recent years. On this basis, a Tauberian theorem for asymptotically periodic functions and two related Tauberian theorems are established. Moreover, we apply our Tauberian theorems to the nonhomogeneous abstract Cauchy problem, and obtain the spectral condition under which the solution of Cauchy problem is $\S$-asymptotically periodic. In our Tauberian theorem for asymptotically periodic functions and its application to abstract Cauchy problem, we completely remove the ergodic assumption in [23] although the conclusions are slightly weaker than asymptotical periodicity. Finally, we construct a concrete Cauchy problem as an example. It is worth mentioning that the inhomogeneous term of this Cauchy problem is asymptotically periodic and its solution is $\S$-asymptotically periodic rather than asymptotically periodic. This demonstrates that $\S$-asymptotically periodic functions are the "natural class'' for solutions to some differential equations.

The paper is concerned with the time-periodic (T-periodic) problem of fractional Burgers equation on the real line. Based on the Galerkin approximates and Fourier expansion, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution for the nonlinear equation are established by constructing a suitable contraction mapping. Furthermore, we show that the unique T-periodic solution is asymptotically stable. In addition, our method can be extended to the classical forced Burgers equation in a bounded region, which improves the known result.

In this paper, we study the existence of normalized solutions for a class of fractional Schrödinger-Poisson equations with coercive potential by using the constrained variational method, which generalizes the results of the relevant literature.

This paper investigates the existence of unique positive solution for a class of fractional boundary value problems involving the $p$-Laplacian operator, a fractional derivative term in the nonlinearity $f$ and nonlinear integral terms in the boundary conditions. By constructing appropriate auxiliary boundary value problems and equivalence classes on cone, and using the theory of cone and the method of sum operators, some sufficient conditions for the existence of unique positive solution are obtained, in addition, a monotone iterative sequence uniformly converging to the unique positive solution is constructed. Finally, an example is given to illustrate the main result.

This paper studies the controllability and the existence of time optimal control for a class of nonlinear parabolic equations, which has drift-diffusion term with nonlinear terms $h(u,v)$ and $g(u,v)$. It applies the $L^{p}$-$L^{q}$ estimate of semigroups and the recently developed maximal regularity theory to study the regularity and the cost estimate of control functions. Moreover, this paper establishes the local exact controllability of the nonlinear control system by utilizing the Kakutani's fixed point theorem. Therefore, it is applied to the existence of time optimal controls.

In this paper, we study a class of degenerative quasilinear elliptic equations with critical exponent. Using a change of variable, we reformulate the equation into a semilinear one, and prove the existence of solutions by employing Lions concentration compactness principle and linking theorem based on cone.

In this paper, we consider a class of Gierer-Meinhardt activation inhibition model with time delay diffusion under homogeneous Neumann boundary conditions. Firstly, the local asymptotic stability of the positive equilibrium of the model is obtained by using spectral theory; second, choosing the time delay as the bifurcation parameter, the existence of the Hopf bifurcation of the model is analyzed; next, the formulae to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by applying the center manifold theorem and normal form theory for partial differential equation are derived; finally, numerical simulations are also carried out to simulate the Hopf bifurcation that the model go through near the critical point.

In this paper, we study the well-poseness of a non-autonomous delayed incompressible non-Newtonian fluid on $2D$ unbounded domains. With a minimal regularity of the force, we prove the existence of solutions by the method of combining the technique of domain decomposition with the Garlerin approximation. Then we use the energy method to prove the uniqueness and the stability of solutions.

Based on the Saint-Venant equations, the star-shaped open channel network with bottom slope and bottom friction is studied in this paper. The system consists of $n$ subsystems: $n-1$ inlet channels and one outlet channel. It is assumed that both online output measurement and input control are located on the boundary. The feedback control law with a linear combination of position and delayed position is established based on the restriction of flow relation before and after the gate. Considering that the delay term can be characterized by the first-order hyperbolic partial differential equation, the closed-loop system is rewritten into the form of PDE-PDE infinite-dimensional coupling system by means of Riemannian coordinate transformation and linearization. By constructing the weighted Lyapunov function, the exponential stability of the system under the $L^2$-norm is discussed, and the sufficient conditions for the control parameters and time-delay value are given. Furthermore, it is proved that the online output measurement is regulated to the specified reference signal. Finally, numerical simulation is carried out by using Matlab to prove the rationality of the parameters conditions and the validity of the conclusion.

For $n\geq1$, let $p_n>1$ and $D_n=\{0,a_n,b_n\}\subset \mathbb{Z}$, where $0

$\mu:=\delta_{p_1^{-1}\{0,a_1,b_1\}} \ast \delta_{p_1^{-1}p_2^{-1}\{0,a_2,b_2\}} \ast \cdots \ast \delta_{p_1^{-1}p_2^{-1}\cdots p_n^{-1}\{0,a_n,b_n\}} \ast \cdots$

which is generated by the sequence of integers $\{p_n\}_{n=1}^\infty$ and the sequence of number sets $\{D_n\}_{n=1}^\infty$. The author shows that when all digit sets are uniformly bounded, $\mu$ is a spectral measure if and only if the numbers of factors 3 in the sequence $\{\frac{p_1p_2\cdots p_n}{3{\rm gcd}(a_n,b_n)}\}_{n=1}^\infty$ are different from each other and $\{\frac{a_n}{{\rm gcd}(a_n,b_n)},\frac{b_n}{{\rm gcd}(a_n,b_n)}\}\equiv\{1,-1\}$ (mod 3) for all $n\geq1$.

In this paper, we study Finsler warped product metrics. We obtain the differential equations that characterize Landsberg Finsler warped product metrics. By solving these equations, we obtain the expression of these metrics. Furthermore, we construct a class of Finsler warped product metrics $F$ with the following properties: $(1)$ $F$ is a Landsberg metric; $(2)$ $F$ is not a Berwald metric; $(3)$ $F$ has zero flag curvature (or Ricci curvature).

In this paper, we investigates optimal dividend and reinsurance policies for an insurer with a random time horizon. The goal of the insurer is to maximize the value of the insurance company when the random time or the ruin time arrives. This value consists of three parts: the dividends up to the random time or the ruin time, the surplus at the random time or the ruin time and the company's brand value. We identify the insurer's joint optimal strategies using stochastic control methods. The results reveal that managers should consider no reinsurance if and only if the brand value or the surplus is too high, less reinsurance is bought when the surplus increases, and dividends are always distributed using the barrier strategy.

The non-stationarity of time series is closely related to its application in many fields. How to measure the non-stationarity of time series is an important research topic. A complete non-stationarity measurement framework has been constructed, in which the criterion of stable set is the core. In this paper, based on the law of iterated logarithm of independent identically distributed random variables, a new criterion of stable set is proposed. Compared with the existing criterion, the convergence criterion is more strict and requires fewer parameters when the level of significance is the same. This method improves the resolution of non-stationary measurement when NS is small. For the parameter calibration of deterministic signals, the new method can get more accurate results.

The gravity model can effectively fuse multiple information of nodes, which make up for the problem of incomplete node information considered by traditional node importance evaluation methods. However, the existing gravity model related methods consider a single factor when defining node mass, and ignore the important role of neighbor topology in measuring node importance. To solve the above problems, a gravity model based on neighborhood hierarchy distribution is proposed for node importance evaluation. Firstly, the neighborhood of nodes and position information are fused to represent the mass of objects in the gravity model. Secondly, the gravity coefficient is defined according to the similarity of the topological structure of the node and neighborhood. Finally, the importance of nodes is measured by the interaction between nodes and neighbor nodes within a given scope. The simulation on six real network datasets shows that the proposed method performs better than other gravity model-related ones in both monotonicity and accuracy.

In this paper, we consider a common solution of three problems in real Hilbert spaces including the fixed points problem for asymptotically nonexpansive mapping, a system of variational inequalities and the split equilibrium problem. Under some suitable conditions imposed on the sequence of parameters, we prove that the sequence generated by the modified viscosity approximation method converges strongly to a common element of the solution set of these three kinds of problems. The results obtained in this article extend and improve the corresponding results of the relevant literature.

In this paper, the Karush-Kuhn-Tucker (KKT) and the weakly complementary approximate Karush-Kuhn-Tucker (W-CAKKT) optimality conditions for an LU-solution of the interval-valued optimization problem with inequality and equality constraints (IVOP) are investigated, where the interval-valued objective function is weakly continuously differentiable. Firstly, under suitable constraint qualification, it is proved that the KKT condition is necessary for an LU-solution of (IVOP). Secondly, the W-CAKKT condition is introduced. Absence of any constraint qualification, it is proved that the W-CAKKT condition is necessary for an LU-solution of (IVOP). In addition, under the assumption of convexity, the W-CAKKT condition is sufficient for an LU-solution of (IVOP). Moreover, when some constraint qualification is satisfied, it is shown that the W-CAKKT necessary condition is better than the KKT necessary condition. The results presented in this paper generalize known results from scalar optimization problems to interval-valued optimization problems.

This paper presents an epidemic model with asymptomatic infection and isolation in the context of population transmission of a Corona Virus Disease 2019 (COVID-19), we analyze the basic reproduction number of the model, the final epidemic size, the existence and uniqueness and solvability of the solution for the implicit final size equation. On this basis, we consider two possible control strategies and analyze the existence of optimal control by using the Filippov-Cesari existence theorem and Pontryagin extreme principle. Base on the historical data of COVID-19 infection in Zhejiang Province, the model parameters are estimated using the Markov Chain Monte Carlo method. The numerical simulation results show that the control strategy can reduce the peak isolation rate by 33.92% and final epidemic size by 76.54%. This suggests that reducing transmission rates and vaccinating susceptible individuals are still effective means of controlling the development of COVID-19 outbreaks, and provides recommendations for controlling COVID-19 outbreaks and responding to emerging infectious diseases.

A diffusive predator-prey model with Beddington-DeAngelis function response and harvesting is studied. Firstly, the sufficient conditions for the existence of positive solutions are obtained by the fixed point index theory, and the multiplicity of positive solutions is discussed by the bifurcation theorem. Then, by virtue of the combination of the perturbation theory for linear operators, degree theory and stability theory, the uniqueness and stability of positive solution are investigated when the interaction among predators is large. In addition, we analyze the extinction and permanence of the two species by means of the comparison principle of parabolic systems. Finally, we make some numerical simulations to validate and complement the theoretical results. If the maximal growth rate of the prey is large and low density mortality of the predator is small, the findings suggest that the model has only one unique asymptotically stable positive solution provided that the effect of the interaction among predators is sufficiently large and the harvesting rate of the predator lies in a certain range; and has at least two positive solutions when the harvesting rate of the predator belongs to another range.

Based on the take-out operation model of the store, this paper constructed and analyzed a fluid model driven by the M/M/1/N queueing system with set-up time, optional service and two types of mixed vacations. Firstly, the driving system is described and the infinite small generators of the two-dimensional Markov process are decomposed into a blocky Jacobian matrix form. Then, the stationary probability distribution of the driving system is obtained by using matrix-geometric method. Secondly, based on the net input rate structure of the fluid model, the differential and difference equations satisfied by the fluid level in steady-state conditions are obtained using the probability analysis method. Then, the expected buffer content and the probability of the empty buffer under steady-state conditions are obtained by using Laplace transform(LT) and Laplace-Stieltjes transform(LST) methods. Finally, the influence of parameters changing on the performance indicators and cost function are illustrated in numerical analysis.