Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

Stability and Hopf bifurcation analysis of a predator-prey model with time delayed incomplete trophic transfer

A new nonlinear predator-prey model with incomplete trophic transfer is introduced. In this model,we assume that the rate of the trophic absorption of the predator is less than the rate of the conversion of consumed prey to predator in the Ivlev-type functional responses. The existence and uniqueness of the positive equilibrium of the model and the stability of the equilibrium of the model are studied under various conditions.Hopf bifurcation analysis of the delayed model is provided.     

拟线性可化约的双曲型方程组的奇性分析

In this paper we investigate the formation of singularities of hyperbolic systems. Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.     

Time to most recent common ancestor for stationary continuous state branching processes with immigration

Motivated by sample path decomposition of the stationary continuous state branching process with immigration, a general population model is considered using the idea of immortal individual. We compute the joint distribution of the random variables： the time to the most recent common ancestor （MRCA）, the size of the current population, and the size of the population just before MRCA. We obtain the bottleneck effect as well. The distribution of the number of the oldest families is also established. These generalize the results obtained by Y. T. Chen and J. F. Delmas.     

THE AVERAGE ABUNDANCE FUNCTION WITH MUTATION OF THE MULTI-PLAYER SNOWDRIFT EVOLUTIONARY GAME MODEL

This article explores the characteristics of the average abundance function with mutation on the basis of the multi-player snowdrift evolutionary game model by analytical analysis and numerical simulation.The specific field of this research concerns the approximate expressions of the average abundance function with mutation on the basis of different levels of selection intensity and an analysis of the results of numerical simulation on the basis of the intuitive expression of the average abundance function.In addition,the biological background of this research lies in research on the effects of mutation,which is regarded as a biological concept and a disturbance to game behavior on the average abundance function.The mutation will make the evolutionary result get closer to the neutral drift state.It can be deduced that this affection is not only related to mutation,but also related to selection intensity and the gap between payoff and aspiration level.The main research findings contain four aspects.First,we have deduced the concrete expression of the expected payoff function.The asymptotic property and change trend of the expected payoff function has been basically obtained.In addition,the intuitive expression of the average abundance function with mutation has been obtained by taking the detailed balance condition as the point of penetration.It can be deduced that the effect of mutation is to make the average abundance function get close to 1/2.In addition,this affection is related to selection intensity and the gap.Secondly,the first-order Taylor expansion of the average abundance function has been deduced for when selection intensity is sufficiently small.The expression of the average abundance function with mutation can be simplified from a composite function to a linear function because of this Taylor expansion.This finding will play a significant role when analyzing the results of the numerical simulation.Thirdly,we have obtained the approximate expressions of the average abundance function corresponding to small and large selection intensity.The significance of the above approximate analysis lies in that we have grasped the basic characteristics of the effect of mutation.The effect is slight and can be neglected when mutation is very small.In addition,the effect begins to increase when mutation rises,and this effect will become more remarkable with the increase of selection intensity.Fourthly,we have explored the influences of parameters on the average abundance function with mutation through numerical simulation.In addition,the corresponding results have been explained on the basis of the expected payoff function.It can be deduced that the influences of parameters on the average abundance function with mutation will be slim when selection intensity is small.Moreover,the corresponding explanation is related to the first-order Taylor expansion.Furthermore,the influences will become notable when selection intensity is large.     

The Formual of Generalized Center－Weak Saddle

In his paper,a problem of center-weak focus of a general system of degree n is transformed into a problem of generalized center-weak saddle.It offers:(1) the calculating formula of the saddle values of the former 4 orders in the general system of degree three;(2) some calculating formulae of the saddle values in the general homogeneous system of degree n. The problem of center-focus(saddle)plays an important role in the qualitative theory of ordinary differential equations.However,since the calculations are difficult,the results have not been satisfactory.On the basis of the author‘s work,Northeast.Math.J.,13(4)(1997),487-494,the paper offers some calculating formulae of the saddle values.     

基于Legendre多项式的数值微分: 加权$L^2$空间中的收敛性分析

We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-Legendre series of f~δ. Instead of the observation L~2 space, we perform the reconstruction of the derivative in a weighted L~2 space. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.     

集合$\tau$-标准部分的闭性

In this paper,the closeness of the τ-standard part of a set is discussed.Some related propositions of the τ-neighborhood system of a set are given.And then some related conclusions of the τ-monad of a set and the τ-standard part of a set are presented.And based on it,the necessary and sufficient conditions of the enlarged model and the saturated model are showed.Finally,some sufficient conditions that the τ-standard part of a set is closed are proved in the enlarged model and the saturated model.     

The Closeness of the τ-Standard Part of a Set

In this paper,the closeness of the τ-standard part of a set is discussed.Some related propositions of the τ-neighborhood system of a set are given.And then some related conclusions of the τ-monad of a set and the τ-standard part of a set are presented.And based on it,the necessary and sufficient conditions of the enlarged model and the saturated model are showed.Finally,some sufficient conditions that the τ-standard part of a set is closed are proved in the enlarged model and the saturated model.     

Eigenvalue problem of a class of fourth-order Hamiltonian operators

The eigenvalue problem of a class of fourth-order Hamiltonian operators is studied. We first obtain the geometric multiplicity, the algebraic index and the algebraic multiplicity of each eigenvalue of the Hamiltonian operators. Then, some necessary and sufficient conditions for the completeness of the eigen or root vector system of the Hamiltonian operators are given, which is characterized by that of the vector system consisting of the first components of all eigenvectors. Moreover, the results are applied to the plate bending problem.