In order to solve the problem that the least squares estimation of the matrix autoregressive model (MAR) is easily affected by outliers or thick-tailed distribution errors and deviates from the true value, this paper improves the loss function. Based on the projection method and iterative least squares method, RE-PROJ and RE-ILS methods are proposed to solve the robust estimation of the MAR model. Both methods use the idea of M estimation in the process of constructing estimators, and the simulation data show that the above two methods can better resist the influence of outliers on parameter estimation. This paper further discusses the BIC criterion for order selection of the MAR(p) model and a robust estimation method with the assumption of rank reduction when the matrix observations are correlated. The actual analysis results show that the robust estimation proposed in this paper has more advantages in fitting and prediction than the least squares estimation in the matrix-valued observation data with many outliers or sharp peaks and thick tails.

BAR (broker adaptive ridge) is a new method to replace $L_0$ penalty regression. Based on a reweighted $L_2$ penalty, BAR penalty combines the advantages of two different penalties, $L_0$ and $L_2$, and avoids the shortcomings of separately using these two penalties. In this paper, the BAR method is extended to linear regression model with robust loss function, and the coordinate descent algorithm is used to estimate the parameters. To characterize the robustness of the proposed method, we provide an influence function for robust BAR estimation. Under appropriate conditions, we theoretically establish the variable selection consistency and Oracle property of BAR estimation, and provide proof. We compared the proposed method with other existing methods through numerical simulation and analysis of real data, further verifying that the new method is more effective in terms of robustness and variable selection performance.

In this paper, we consider the method for optimization problem with $l_1-$norm. This kind of problem is widely used in compressed sensing and other fields. Based on the proposed smoothing function, a new conjugate gradient method to solve this optimization problem with $l_1-$norm is proposed in this paper. The global convergence of the proposed method is analyzed under general conditions, and the related numerical results also show the effectiveness of it.

The information collection capabilities in the era of big data have brought more complex data structures to time series analysis. Matrix-valued time series are common in the fields of macroeconomics, finance, and management, and are manifested as continuous observations of multiple indicators in multiple locations. The matrix autoregressive model is superior to the vector autoregressive model in terms of model expression and prediction due to its bilinear structure and fewer parameters. However, the matrix autoregressive model only contains the prediction structure of the time dimension, and does not have the prediction structure of the space dimension. For this reason, we added the spatial lag regression term containing the spatial weight matrix, and proposed the time-space lag regression model of matrix-valued time series. We assign scale parameters and adjustment parameters to each location and each variable to test whether the spatial prediction effect exists. Since the proposed model has no endogenous problem, this paper uses the partial iterative least squares method to obtain good parameter estimates. We also give the BIC criterion for model order selection and propose a model rank-reduced estimate. In addition, for the case of thick-tail distribution of the residual term, a robust estimation based on Huber loss function is proposed. With the increase of the sample size of the simulated data, the deviation and variance of the estimator tends to decrease gradually. The actual data shows that the proposed model has moderate model complexity and the smallest out-of-sample prediction error.

The modified phase field crystal model is a sixth order nonlinear generalized damped wave equation. Based on the Crank-Nicolson scheme, a second order implicit linearized finite difference scheme is presented by using the method of order reduction. The nonlinear term is approximated by the second order explicit extrapolation. A theoretical analysis is carried out by the energy argument and mathematical induction. The unique solvability and $L^{\infty}$ norm convergence of the numerical scheme are proved rigorously. The convergence order is two in time and space. Numerical results demonstrate that the presented scheme for the modified phase field crystal equation is efficient and can achieve the expected accuracy.

In this paper, a class of HIV time-delayed differential equation model with antiviral drug periodic treatment is established to study the threshold dynamics of~HIV~infection in the host. In order to investigate the influence of viral reverse transcription and budding processes on HIV infection process, we incorporate two time-dependent time delays into the model to characterize viral reverse transcription and budding periods respectively. Firstly, the well posedness of the model, including the global existence of periodic solutions and the dissipation of the system, is studied by using functional differential equation theory; Secondly, the functional expression of the basic reproduction number of the model $R_0$ is derived from the definition of the basic reproduction number for the periodic infectious disease compartment model; Finally, the global dynamic behavior of the system is discussed. More precisely, applying the uniform persistence theory, it is proved that HIV infection and replication in the host are persistent when $R_0>1 $, the infection free periodic solution of the system is globally attractive when $R_0<1$, that is, HIV will eventually be eliminated within the host.

In this paper, we explain the solution of the non-autonomous stochastic delay lattice equation driven by white noise generating a continuous cocycle. Uniform estimation and tail estimation are performed, and it is proved that the system has a $ \mathcal{D}$-pullback random attractor. We prove the double upper semicontinuity of random attractors for delay system when the delay and a certain parameter converge simultaneously.

We build an estimator of calendar effect in market microstructure noise using high-frequency financial data. The associated asymptotic theory and feasible inference are established. This is, to our knowledge, the first (feasible) asymptotic theory ever built for the estimation of calendar effect in market microstructure noise. This research may lead to a better understanding of market microstructure noise and hence the market mechanism.

The SEIAV infectious disease model containing factors such as incubation period, saturation cure, secondary infection, transmission of free virus was developed by combining the effect of spatial and temporal diffusion and the incidence of Holling-IV in the process of infectious disease transmission. The stability of the disease-free equilibrium for R₀<1 and the persistence of the endemic disease equilibrium point for R₀>1 are obtained, and the global attractiveness of the disease-free equilibrium point of the model for R₀=1 is further discussed. The fitness of the model is proved using operator semigroup theory, and the eigenvalue problem is also constructed to give a general calculation of the basic regeneration number using the property of the existence of principal eigenvalues. Finally, the effect of spatial diffusion on the spread of the infectious disease model is verified by numerical simulations.

The Riemann-Hilbert (RH) problem is performed to study the modified Korteweg-de Vries (mKdV) equation in this work, and we give an effective method to obtain the soliton solution with the rapidly decaying initial value. The important properties of Jost functions and scattering matrix are obtained by the direct scattering to construct a suitable RH problem, and then the relationship between the solution of the RH problem and the potential function is established. In the inverse problem, two cases of scattering data, including simple zeros and double zeros, are considered, and the corresponding RH problem is solved. Then, the general forms of solutions for mKdV equation in two cases are given successfully. Finally, in combination with specific parameters, the multi-soliton solutions image propagation in two cases are given in detail.

Mathematical analysis of pharmacokinetic models plays an important role in drug researches. In this paper, considering a one-compartment pharmacokinetic model with parallel first-order and Hill (n = 2) elimination under different dosage designs with periodic intravenous bolus administrations, we have mathematically studied the steady-state pharmacokinetics. As a result, we have proved that the pharmacokinetic model, represented by an impulsive differential equation, admits a unique steady-state periodic solution. Then we have derived the analytical formulas for two important pharmacokinetic indexes: steady-state drug exposure and steady-state average plasma concentration. Moreover, different to the existing the pharmacokinetic model with parallel first-order and Michaelis elimination pathways, we have been able to discover, both numerically and theoretically, the diversity in the steady-state average plasma concentration for different dose regimens. That is, by increasing the dosing frequency, three circumstances can occur for the steady-state average drug concentration: (i) monotonically decreases and converges to a limit value; (ii) monotonically increases and converges to a limit value; and (iii) decreases first and then increases and eventually converges to a limit value. Finally, we have applied the results to a real drug model of recombinant granulocyte colony-stimulating factor (Filgrastim), for which we have provided the analytical formulas of the steady-state average plasma concentrations using different dose regimens and calculated the steady-state average plasma concentration and minimum plasma concentration.

As a generalization of integer-order differential equations, fractional differential equations have been widely used in science and engineering in recent years, which also have attracted widespread attention from many scholars. In this paper, a novel Crank-Nicolson finite volume method (CN-FVM) is proposed for solving Riesz space-fractional advection-diffusion equations (RSFADEs) with homogeneous Dirichlet boundary conditions. In order to obtain the discrete linear systems arising from RSFADEs, the Crank-Nicolson method is used to discretize the first order time partial derivative, while the finite volume method is adopted to approximate the first order space partial derivative of advection term and the Riesz space fractional partial derivative of the diffusion term. Furthermore, two main theoretical results about stability and convergence of the CN-FVM scheme are also discussed. It is proved that CN-FVM scheme is unconditionally stable and convergent with the accuracy of ${\mathcal O} (h^2+\tau^2)$ in the discrete $L_2$-norm, where $h$ and $\tau$ denote the spatial and temporal step sizes, respectively. Finally, some numerical experiments are presented to confirm the correctness of the theoretical analysis of the proposed scheme.

There are almost no simple linear dynamic systems in nature, and most of them exist in the form of non-conservative nonlinear dynamic systems. Non-standard Lagrange functions can be used for dynamic modeling of non-conservative nonlinear problems. The fractional model is also a good choice for studying complex dynamics and physical behavior. Therefore, this paper studies the Noether symmetry and conserved quantity of non-standard Lagrange systems under generalized fractional operators. Firstly, the Lagrange equation of non-standard Lagrangian system under generalized operator is established. Then, based on the invariance of Hamilton action under infinitesimal transformation, the Noether theorem of non-standard Lagrangian system under generalized fractional operator is established, and the symmetry and corresponding conserved quantity of the system are given. Under certain conditions, the Noether conserved quantities of non-standard Lagrangian systems under generalized fractional operators can be reduced to the Noether conserved quantities of non-standard Lagrangian systems of integer order. Finally, examples are given to illustrate the specific application of the obtained results.

The structural stability for the Brinkman-Forchheimer fluid interfacing with a Darcy fluid in a bounded region in $\mathbb{R}^3$was studied. We assumed that the velocity of fluid was slow and it was governed by the Brinkman-Forchheimer equations in $\Omega_1$, while in $\Omega_2$, we supposed that the saturated flow satisfies the Darcy equations. With the aid of the fourth norm estimates for the temperatures and the Sobolev inequality, we formulated an energy expression, and the expression satisfies a differential inequality. By integrating, we were able to demonstrate the continuous dependence result for the Brinkman coefficient.

Preview control can improve control system performance by utilizing future desired ox external disturbance information, repetitive control introduces human learning mechanism into the control system. Therefore, more and more applications are seen in various real engineering fields, attracting a wide spread attention of researchers. For a class of uncertain system with time-varying delay, the problem of preview repetitive control (PRC) is proposed under the assumption that the reference signals are previewed and periodic. First, the repetitive controller is introduced and the error system in the preview control theory is adopted, and an two-dimensional (2D) augmented error system is constructed. This leads to the problem of preview repetitive controller’s design being transformed into a output feedback control problem of the augmented error system. Then, for the 2D augmented error system, the output equation is modified to fuse the previewed future information and the repetitive controller while considering output feedback. Based on Lyapunov method and LMI technique, the conditions of asymptotic stability of the closed-loop system and the design method of the preview repetitive controller are given. Finally, the effectiveness of the proposed method is illustrated by two simulation experiments.

The Oscillation of a class of second order nonlinear neutral delay differential equation with damping term is studied in this paper. By using the generalized Riccati transformation technique and some special techniques, some new oscillation criteria for the equation are obtained in the noncanonical. The oscillation results in several recent literatures are improved, generalized and unified. Finally, some examples demonstrate the applicability of the results of this paper.

Due to the influence of observation noise, the identification test for volatility function of diffusion model will fail in high frequency environment. In this paper, we used local-averaging method to denoise the observed data. Based on the smoothing values, we combined the conditional moment and the nonparametric kernel estimation method to construct U statistics to identify the volatility function of the diffusion model. The test statistic converges to the standard normal distribution under the condition that the form of the volatility function is specified correctly. The Monte Carlo simulation results show that this statistic has more reasonable test size and stronger test power than the existing test methods. Using the constructed test statistic to identify the volatility function of the logarithmic price data of bank of China stock, more reasonable test results are obtained.

For the aim of exploring the influencing factors of periodic outbreaks of spruce budworm, we proposed a delayed diffusive spruce budworm model with Holling II predation function under the homogeneous Neumann boundary condition. As is known to all, the evolution of many populations is relate with not only the present situation, but also with the past situation. This implies that it is essential to take over the effect of delay on reflecting the phenomena. For this aim, choosing the delay as bifurcating parameter, we investigated the stability of the positive equilibrium and the existence of Hopf bifurcation deduced by single delay or two delays by using the characteristic equation and mathematical analysis skills. Finally, through performing numerical simulations, we obtained the stable periodic solutions of this model, which provided the theoretical basis for periodic outbreaks of spruce budworm. Moreover, the numerical results indicated that the critical value of Hopf bifurcation deduced by two delays is smaller than one of Hopf bifurcation deduced by single delay.

In this paper, the general nonlinear moisture migration equations are studied with a mixed finite element method. The superconvergence of the equations is proved by use of bilinear element $Q_{11}$ and zero order Raviart-Thomas element ($Q_{10}\times Q_{01}$). With the help of the interpolation operators of the above two elements and mean-value technique, the superconvergence results of order $O(h^2)$ are obtained for the semi-discrete scheme of the equations. For the linearized Crank-Nicolson (C-N) fully-discrete scheme, the superconvergence results of order $O(h^2+\tau^2)$ are also derived, here $h$ is the subdivision parameter, $\tau$ is the time step. This method shows that if the linearization problem has superconvergence, the corresponding nonlinear problem has the same superconvergence. Finally, a numerical example is provided to illustrate the correctness of the theoretical analysis and the feasibility of the proposed method.

A 3-rainbow dominating function of a graph $G$ is a mapping $f$ from the vertex set $V$ of $G$ to the power set of the set $\{1,2,3\}$ such that any vertex $v$ with $f(v)=\varnothing$ satisfies that $\bigcup\limits_{u\in N(v)}f(u)=\{1,2,3\}$, where $N(v)$ is the neighborhood of $v$. The weight of a 3-rainbow dominating function $f$ of $G$ is $\sum\limits_{v\in V}|f(v)|.$ If $f$ is a 3-rainbow dominating function of a graph $G$ and its complement, then $f$ is called a global 3-rainbow dominating function of $G$. The global 3-rainbow domination number of $G$ is the minimum weight of a global 3-rainbow dominating function of $G$. By analyzing the structure of graphs, using the method of categorical discussion, we completely characterize the graphs with global 3-rainbow domination number equal to the number of vertices.

In this paper, we mainly introduce a topological definition of quasi-shadowing property with respect to a leaf-set (or a foliation) on a dynamical system, which is equivalent to the general definition of quasi-shadowing property with respect to a foliation on Riemannian manifolds. At the same time, we discuss the relationship between the quasi-shadowing property with respect to a leaf-set (or a foliation) and an inverse limit of a sequence of shifts of finite type on the compact Hausdorff totally disconnected spaces and Riemannian manifolds respectively. In addition, we give a sufficient condition for the factor maps to preserve quasi-shadowing property.

In this paper, we study the dynamics of a predator-prey system with delay of fear effect. First we discuss the positivity, boundedness and permanence of the solution, then base on center manifold theorem and normal form, we study Hopf bifurcation caused by delay of fear effect. Furthermore, we discuss the global existence of bifurcated periodic solutions. At last, some simulations are given to support our results.

In order to handle the coexistence of spatial heterogeneity and spatial correlation in spatial data, This paper considers a class of error spatial autocorrelation ~GWR~ models, This model is considered an effective fusion of classical GWR model and spatial error model.~the existing estimation methods of the GWR model cannot obtain a consistent estimate of the parameters, due to the endogeneity problem caused by the spatial lag of the error term in the model. Therefore this article propose a new estimation method for the model by combines the ideas of local linear estimation and profile least square estimation metod.Discussed the asymptotic properties of estimators, evaluated the performance of the proposed method under limited samples through data simulation, and compared our method with existing methods.The data simulation results indicates that the method proposed in this paper, it can provide more accurate estimates of model parameters and coefficient functions by contrast . Finally, an empirical case is presented to demonstrate the practicality of the proposed method.

We consider the existence of weak solutions to a class of elliptic equations with nonstandard growth condition and zero order term. The main tools are the weak convergence method for PDEs and Young measure method. The perturbed problem is treated as a starting point. In view of the assumptions on the zero order term and the integrability of the source term, we pick up some appropriate test functions, which are vital to some necessary a priori estimates and the limit process. Thanks to the Young measure method, the weak limit of the nonlinear term is identified. The existence of a weak solution is obtained by passage of the limit.

This work is concerned with a stochastic evolutionary equation with a multiplicative noise. Verifying the convergence from the solution of the stochastic evolutionary equation to one of its Wong-Zakai approximation and applying the exponential martingale argument, the Kallianpur-Striebel formula and Itô formula, we prove that the nonlinear filter generated by the stochastic evolutionary equation converges to one generated by its Wong-Zakai approximation in the observation system with colored noise.

Our primary objective of this article is to study a class of nonlinear telegraph systems with a parameter. Using index theory of fixed points for compact operators, some new criteria for the existence, multiplicity and nonexistence of positive doubly periodic solutions are established in terms of different values of parameter.

This article proposes a hypothesis testing method to detect serial correlation and ARCH effect in high-dimensional time series based on $L_{2}$ norm and Spearman's correlation. In this article, We study the asymptotic behavior of our test statistic and provide a bootstrap-based approach to generate critical values, we prove our test can control Type-I errors. Our test is dimensional-free, which means it is independent of the dimension of the data, hence our test can be used for high dimensional time series data. Our test does not require tail properties of data, hence it can be used for heavy-tailed time series. The simulation results indicate that our new test performs well in both empirical sizes and powers and outperforms other tests. The practical usefulness of our test is illustrated via simulation and a real data analysis.

Based on the spatial heterogeneity, the asymptomatic hosts and the multiplicity of pathogen transmission routes, a model of reactive diffusion host-pathogen with asymptomatic hosts and multiple infection routes is proposed, which is discussed the existence and uniqueness of the global positive solution of the model by using semigroup theory. Further, according to the spectral radius method of the next generation operator, the basic reproduction number $\mathcal{R}_{0}$ of the model is given, and the extinction and persistence of the disease are described. That is, if $\mathcal{R}_0<1$, the disease-free steady state is globally asymptotically stable; while if $\mathcal{R}_0>1$, the disease is uniformly persistent and the model admits at least one endemic steady state. In addition, the global asymptotic stability of the disease-free and endemic equilibrium states of the model in a spatially homogeneous environment is obtained by constructing suitable Lyapunov functions. Finally, some numerical simulations are conducted to explain the main theoretical results and to explore the influence of diffusion rates on the distribution of infected hosts.

Under widely orthant dependent samples, two kinds of risk measure are considered. The Bahadur representation and strong consistency of the quantile estimator for VaR are discussed. And the strong consistency and its rate of the CVaR estimator are established, by the suitable choice of some constants, their rates are near $O(n^{-\frac{1}{2}})$. In order to better illustrate performances of the VaR and CVaR estimators, we conduct numerical simulations under some WOD sequences by ARMA(1,1) and MA(1) models, and discover that the estimators are high performance by tables of the exact and estimated values of VaR and CVaR, and their curve figures.

In the big data environment, the traditional centralized method is no longer feasible due to the problems of computer storage, computing power, security and privacy. In the context of massive data sets, due to the limitations of data collection, transmission and management, complete data sets are usually difficult to obtain. Statistical inference must rely on the inference results of local data sets to reduce the important problems in the research of estimation bias based massive data regression methods. Therefore, this paper constructs a distributed estimation framework for general parameters based on moment distributed estimation, which improves the computing and storage capabilities and solves the problem of depolarization in distributed environments. Firstly, the parameter to be estimated is expressed as a function of multiple moments. Secondly, the unbiased estimation of each moment parameter is obtained through a simple average distributed framework. Finally, the distributed unbiased estimation of the moment parameter is brought into the aggregate function to obtain the final estimator of the parameter to be estimated The theoretical results show that under general conditions, the probability convergence bound of the estimator is the same order as the centralized estimator and the estimator has asymptotic normality The simulation results show that the distributed estimator and the centralized estimator have similar statistical efficiency and better computational efficiency.

The Kelvin-Voigt fluid passing through a semi infinite cylinder is considered, in which the Kelvin-Voigt fluid satisfies the homogeneous boundary condition on the side of the cylinder. Using the methods of energy estimation and a priori estimations, the exponential decay property of strain energy with distance from the finite end of the cylinder is proved. This type of result can be regarded as the "distance effect" of Saint-Venant principle.

Based on the expected utility maximization criterion, this paper studies the reinsurance and investment problem. There is an insurer and a reinsurer in the market. By using extrapolation deviation method, the insurance model under correlated claims is obtained. The joint interests of the insurer and the reinsurer are reflected by maximizing the weighted of their respective wealth. In the framework of ambiguity aversion, the robust stochastic optimization problem is established. By using stochastic control and stochastic dynamic programming theory to solve the robust stochastic optimization problem, the explicit solution of the robust optimal reinsurance and investment strategy is obtained. Finally, the effects of model parameters on the robust optimal reinsurance and investment strategy are explained by numerical experiments, and the practical guiding significance of the research results is pointed out.

In this paper we consider an $M/G/1$ repairable queueing system with Bernoulli maintenance and admission control policy, in which whenever the system becomes empty, the manager shuts down the system immediately with probability $p(0\leq p\leq1)$ and arranges the repairman to maintenance it with a random time length, and at most $M(\geq1)$ customers are allowed to enter into the system during its maintenance period. Otherwise, the manager does not shut down the system with probability $(1-p)$ and the customers who arrive at the system will be served at once until the system becomes empty again. Applying the total probability decomposition, renewal process theory and Laplace transform tool, we study the transient distribution of queue size at any time $t$ in detail,and obtain the expressions of the Laplace transform of the transient queue-length distribution. Furthermore, the explicit recursive formulas of the steady-state distribution of the queue size at any time $t$ are derived by using L'Hospital's rule. Meanwhile, the expressions for the probability generating function of the steady-state queue-length distribution and the expected queue size are presented. Finally, we employ the renewal reward theory to derive the expression of the long-run expected cost per unit time of the system. Moreover, numerical examples are presented to discuss the one-dimensional optimal control strategy for minimizing the expected cost of the system as well as the two-dimensional optimal control strategy.

In this paper, a new nonmonotone modified L-M algorithm for solving nonlinear equations is proposed by combining the nonmonotone line search technique with the modified Levenberg-Marquardt algorithm (L-M algorithm). In each iteration of the new algorithm, a modified step is introduced, and the gradient norm of the value function is used to update the L-M parameters. If the trial step is not accepted, the nonmonotone line search technique is used to obtain the new iteration point. Under certain assumptions, the global convergence and local convergence of the algorithm are proved. Numerical experiment results show that the algorithm is feasible and effective.

Ecological researches show that there exist dominance ranks of individuals in many species. Moreover, natural populations are actually subject to seasonal fluctuations which makes their habitats often undergoes some periodic changes. Motivated by these considerations, in this paper, we investigate the optimal harvesting problem for a hierarchical age-structured population system in a periodic environment. Here the objective functional represents the net economic benefit yielded from harvesting. Firstly, by means of frozen coefficients and fixed point theory we show that the state system is well posed if the reproducing number is less than one. Meanwhile, it is shown that the population density depends continuously on control parameters. Similarly, we show that the adjoint system is also well posed. Then, the optimality conditions given by the feedback forms of state variable and adjoint variable are obtained by using the adjoint system and tangent-normal cone techniques. The existence of optimal harvesting policy is verified via Ekeland's variational principle and fixed point reasoning. Finally, we use numerical simulations to verify the main results and find other dynamic properties of the system. The results in this paper generalize and improve the previous related results.

In this paper, we extend the network oligopoly with finitely many markets to the model with infinitely many markets, and propose the network oligopoly model with continuous-variable markets. Under the fully noncooperative hypothesis, we first prove the existence of Cournot-Nash equilibria. Furthermore, we assume that there exists a partition of the set of firms. By defining the coalitional cost function and coalitional profit function of every coalition, we establish a noncooperative game basing the coalition structure, and prove the existence of Nash equilibria. New network oligopoly models and Nash equilibrium existence theorems are our main contributions.

This paper is concerned with the time decay rates of strong solutions for the one-dimensional isentropic compressible micropolar fluid model with density-dependent viscosity. The pressure $p(\rho)=\rho^\gamma$ and the viscosity coefficient $\mu(\rho)=\rho^\alpha$ for some parameters $\alpha,\gamma\in\mathbb{R}$ are considered. By using a priori assumption and some refined energy estimates, we show that the global existence and large-time behavior of strong solutions with large initial data for the Cauchy problem under the perturbation of the constant state. Furthermore, by using the anti-derivative and time weighted energy method, the algebraic time decay rates for the specific volume $v(t,x)$ and the velocity $u(t,x)$ are also established.

In this paper, the exact periodic traveling wave solutions of modified short pulse (mSP) equation are systematically discussed. With the aid of new variables, the mSP equation is converted to an ordinary differential equation of first order. By means of elliptic integrals and Jacobi elliptic functions, this equation is solved and the periodic traveling solutions of the mSP equation are constructed. Through exploring the limits of these periodic solutions, the 1-cuspon solution and a new solution are obtained.

In this paper, we consider the non-collapsing property on a class of planar convex curves deformed by non-linear curvature flow.Firstly, we give the definition of the inner non-collapsing property and the outer non-collapsing property on planar curves, then by defining a function $Z$, we prove that the inner non-collapsing property on planar curves is equivalent to the non-negativity of $Z$ and the outer non-collapsing property on planar curves is equivalent to the non-positivity of $Z$. Finally, we use maximum principle to prove the non-collapsing property on the planar convex curve shortening flow is maintained under some conditions.