The adiabatic theorem describes the time evolution of the pure state and gives an adiabatic approximate solution to the Schödinger equation by choosing a single eigenstate of the Hamiltonian as the initial state. In quantum systems, states are divided into pure states (unite vectors) and mixed states (density matrices, i.e., positive operators with trace one). Accordingly, mixed states have their own corresponding time evolution, which is described by the von Neumann equation. In this paper, we discuss the quantitative conditions for the time evolution of mixed states in terms of the von Neumann equation. First, we introduce the definitions for uniformly slowly evolving and δ-uniformly slowly evolving with respect to mixed states, then we present a necessary and sufficient condition for the Hamiltonian of the system to be uniformly slowly evolving and we obtain some upper bounds for the adiabatic approximate error. Lastly, we illustrate our results in an example.

本文研究一类具有分段常数变量的三维食饵-捕食者系统的稳定性和分支行为,该系统由一个捕食者和两个食饵构成,其中一个食饵可由捕食者对另一个食饵的捕食行为中获益.首先通过计算得到三维食饵-捕食者系统对应的差分模型,其次通过选择合适的参数讨论边界和正平衡点的存在性,进而利用线性稳定性理论讨论平衡点局部渐近稳定的充分条件.将两个食饵种群的出生率以及最大环境容纳量作为分支参数,使用分支理论研究差分模型在平衡点处产生翻转分支、Neimark-Sacker分支、折-翻转分支和1:2共振分支的充分条件.最后通过数值模拟验证了理论分析的正确性.

A differential-algebraic biological economic system with time delay and Holling type Ⅲ functional response is considered,into which are incorporated a constant prey refuge and prey harvesting.A sufficient condition for the existence of a positive equilibrium is discussed and then an ordinary differential equation is transformed from the system by homeomorphic transformation.The time delay is considered as a bifurcation parameter,and the stability and Hopf bifurcation of the system were analyzed based on the characteristic root method and the bifurcation theorem.It is found that the Hopfbifurcation will occur when the bifurcation parameter is via an exceptive value.Numerical simulations illustrate the effectiveness of our results.

The qualitative analysis of a predator–prey model with rapid evolution and piecewise constant arguments is investigated in this work. The discrete model, which determines the dynamical behavior of the corresponding differential model, is achieved by calculation. First, the sufficient conditions for the existence and local stability of the equilibriums are concluded from the linearized stability theorem and latent root method. Second, the global stability of the equilibriums is discussed through the Poincaré–Bendixson theorem. Furthermore, it is proved that the system has at most one limit cycle. Third, by using the bifurcation theory it is found that the model can undergo the saddle-node bifurcation; the flip bifurcation; and the Neimark–Sacker bifurcation. From the qualitative analysis it can be found that the exponential growth rate and the ratio between the fast and slow timescales have profound influence on the dynamic behavior of the model. Finally, numerical examples carry out to justify the main results in this work.

In this paper,the stability and bifurcation behavior of a prey-predator with epidemic and piecewise constant arguments are researched.Firstly,through calculation the discrete solution of the model is achieved,which has the same dynamical behavior.Next,applying the linearized stability theorem,some sufficient conditions for the local asymptotic stability of equilibrium are achieved.Secondly,by choosing the intrinsic rate of increase as the bifurcation parameter,it is shown that the discrete model can undergo Fold bifurcation,Flip bifurcation and Neimark-Sacker bifurcation through using the bifurcation theory.Finally,the illustration of the analytic results and the complex dynamical behaviors of the model are shown from numerical simulations.

研究一类带有分段常数变量和避难所的天敌-害虫模型的稳定性和分支行为.首先通过计算转化得到天敌-害虫模型对应的差分模型,利用线性稳定性理论讨论了正平衡态局部渐近稳定的充分条件.其次以害虫种群的内禀增长率或逃脱率为分支参数,利用分支理论研究了模型正平衡态处产生翻转分支周期解和Neimark-Sacker分支周期解的充分条件;并且使用正规形理论和中心流形定理构造了判断分支周期解稳定性的阈值.最后数值模拟验证了理论分析的正确性,并展示了该模型复杂的动力学行为.

The stability and bifurcation behavior of a population model with piecewise con-stant arguments are investigated in this paper. The discrete model determining the dynamical behavior of corresponding differential model is achieved by calculation. Firstly, the sufficient conditions for the local asymptotic stability of the steady state are achieved in three aspects based on the linearized stability analysis. Secondly, by choosing the parameter r as the bifurcation parameter and using the bifurcation theory, we find that the discrete equation undergoes a flip bifurcation at an excep-tive value of the parameter r. Finally, numerical examples are carried out to justify the main results in this work.

本文研究一类带有分段常数变量的Lorenz系统的稳定性和分支行为.首先通过计算转化得到Lorenz系统对应的差分系统,利用线性稳定性理论讨论平衡点局部渐近稳定的充要条件.其次选择差分系统三个参数的一个参数为分支参数,利用分支理论研究平衡点处产生Neimark-Sacker分支不变闭曲线的充要条件,并使用分支理论给出判断分支不变闭曲线的稳定性的阈值.最后数值模拟验证了理论分析的正确性.

A delayed stage-structured single-species model with disease in infancy and vaccination is investigated.Using limit system theorem and constructing Liapunov function,the sufficient conditions for the global asymptotically stability of the positive equilibrium point and the infection-free equilibrium point are obtained.The results show that in the certain immunization coverage rate,the disease will eventually become extinct when the birth rate of the species is located in an interval,and when the birth rate is large than a threshold,the disease will eventually become an endemic.

A delayed prey-predator system with disease in prey and predator was investigated. First the su?cient conditions for the existence of the positive equilibrium point and the infection-free equilibrium point were acquired, and then the nonlinear characteristic equation was acquired using the method of characteristic roots. Next the su?cient conditions that the equilibrium points are local asymptotically stability were obtained. Furthermore the Hopf bifurcation behavior of the equilibrium point time was discussed. At last some numerical simulations were carried out to support the theoretical analysis of the research.

An SI model with epidemic in the time delays predator and prey is proposed.Using the method of latent root the equilibrium points are obtained.Through analysis,the result is achieved that the predator and the prey with epidemic will be extinct when the infection percentage of susceptible is smaller than a threshold.The predator will be extinct when the infection percentage of the susceptible prey is larger than a threshold and the conversion coefficient of the susceptible predator is smaller than a threshold.The boundary equilibrium is locally asymptotically stable when the time delay of prey is suitable small,while a loss of stability by a Hopf bifurcation can occur as the delay increases.The time delay of predator has no in- fluence on the stability of the equilibrium.

An impulsive delayed SI model with disturbance and a nonlinear in-cidence was formulated and analyzed. By introducing three thresholds, the sufficient conditions for eradication and permanence of the disease are obtained. Furthermore the global attractive of the infection-free periodic solution and permanence of the model are both influenced by time delay, the disease will disappear if the ratio of the inaximum to minimum of the pulse vaccination rate is lager than some value. The main feature of this paper is that multi-delays and variable coefficients are introduced into the SI model. Numerical results show that the system has complex dynamics including the infection-free periodic solution and periodic oscillations.

A nonautonomous predator-prey difference model with functional response and time delays is investigated. By using some Lemmas and continuation theorem of coincidence degree theory, a set of sufficient conditions are established for the existence of difference system. Our result is substantiated through numerical example.

In this paper the representation of Heat equation and the fairing energy is used in constructing blending surfaces which is link to two surfaces.The blending surfaces which satisfy the demand of C0 or C1-continuation boundary condition and fairing can be achieved by PDF,and the detailed approaches are researched.The infection of the initial surfaces and the parameter of PDE to blending surfaces are discussed.

A food web model consisting of two competing preys and one predator with impulsive,time delay and general diffusion functions was proposed.The sufficient criteria were established for the existence of positive periodic solutions and the approach was based on the coincidence degree and the related continuation theorem as well as some prior estimates.Meanwhile,some numerical simulations were carried out to support the theoretical analysis of the research.

全国大学生数学建模竞赛,是教育部面向在校大学生的群众性科技活动之一,在各个院校普及的深度和广度越来越大。如何选取三个优秀的学生组成一个配合默契、能够发挥各自优势的参赛队伍是一个非常重要的问题。通过实践,我们总结出了一套行之有效的选拔方法。

In order to get an approximation with better effect of parameterization of Bézier curves,we proposed a method for arc-length parameterization and the corresponding algorithms by square approximation for the discrete even de-parameterization of the curves. This method is simple and easy to implement,and the property of the approximation has no change compared with the original curve. A quantitative criterion for estimating the effect of parameterization is also built to quantitatively characterize the parameterization effect of the algorithms. As a result,the nearly arc-length parameterized curve has a smaller relative deviation using either the algorithm with point constraint at endpoints or the algorithm with point constraint plus the first derivative constraint at endpoints. Experiments show that after re-parameterization with our algorithms,the relative deviation will have at least a 20% reduction.

The constructing method of blending surfaces by heat equation with fluctuant boundary condition is researched.In different initial condition and fluctuant boundary condition,the representation of heat equation is used,such that the modeling of blending surfaces by heat equation is brought forward.The equation can be resolved by the segregation variable method or the numerical method,then the blending surfaces which satisfy C0 or C1 continuation can be achieved by the image space of the equation.The experimental results show that the model is effective and feasible.

In this paper,Heat equation and the fairing energy are used in constructing blending surfaces.With initial condition and fixed boundary condition,the Heat equation can be established and resolved.Then the fairing energy of surfaces in the image space of the equation about t is acquired,the function of fairing energy about t is regarded as the fairing ruler.Through computing,the point of the minimal value can be get,such that the blending surfaces which satisfy the demand of continuation and fairing can be achieved.The experimental results show that the model is effective and feasible.

In computer-aided design,transformations among different forms of curves and surfaces are often required to carry out operations of degree-reduction of curves and surfaces and data exchanging between different geometric modeling systems. The errors of these trans-formations would depend on the condition numbers of the corresponding transformation matrices. For this reason,we studied some properties of Jacobi-Bernstein basis transformation matrices related to their condition numbers,and by computing the infinite norms of the transformation matrices and their inverse matrices,we obtained explicit upper bounds to these condition numbers. An example of applications of these condition numbers in CAGD was also provided.