In this paper, some new methods of computation for zero point index are given by using the theory of cones. As applications, 
the existence of positive solutions for the second order periodic boundary value problem is considered.

In this paper, for normal, AR(1) model, when Ro xixi+1=0, and , it is proved that the maximum likelihood estimation exists but is not uqique. This result is different from the global solution of maximum likelihood estimation for R1 and R2. The analytical expression and the mathematical properties of the maximum likelihood estimation R1 are studied as well.

In this paper, we study nonparametric estimation of regression quantiles by inverting a weighted Naraya-Watson (WNW) estimator of conditional distribution, which was first used by Hall, Wolff and Yao (1999). %and establish %the asymptotic normality and consistency of the WNW conditional distribution estimator %and $WNW$ conditional quantile estimator. Furthermore, We propose a new test for homoskedasticity of variance in nonparametric regression model based on local empirical likelihood and WNW conditional quantile estimator. Under null hypothesis, the test statistic is asymptotically chi-square distuibution with degree of freedom $m-1$. A simulation study is carried out to illustrate the performance of the new test.

By the comparison of transition matrices of two nonhomogeneous Markov chains, we discuss the relations of ergodicity for the two chains and obtain some sufficient conditions for a nonhomogeneous Markov chain to be strongly ergodic. The relations of uniformstrong ergodicity and ulilform weall ergodicity for a nonhomogeneous Markov chain are analysedand some sufficient conditions of uniform strong ergodicity for a nonhomogeneous Markov chainare obtained.

In this paper, we will give an overview on the status of the research on the nonlinear output regulation problem with the emphasis on the establishment of a nonlinear version of the internal model principle and its role in shaping a general framework for handling the nonlinear output regulation problem.

This paper discusses the consistency of the expected utility theory with the mean-variance rule in location-scale family, and broadens the application of mean-variance rule from the normal distribution to location-scale family and proves the Mossin theorem by the obtained result. This not only enlarges the application range and can consolidate the foundation of CAPM.

In this paper, we discuss isomorphic factorization for tensor product of two divisible graphs and prove the conditions for the tensor product graphs to be factorized isomorphically.

This paper deals with global structure of the following periodic boundary value problem of third order differential equation $u^{\prime\prime\prime}+\r^3u =\ld f(t, u)$, $0< t < 2\pi$, with $u^{(i)}(0)= u^{(i)}(2\pi)$, $i=0, 1, 2$, where $\r\in (0, \f{1}{\s})$ is a constant, $ \ld\in R^+=[0, +\i)$ is a parameter, and $f$ is singular at $t=0$, $t=2\pi$ and $u=0$. Also, $f$ is sublinear at $\i$.

In this paper the sufficient condition and necessary condition are given for relative compactness of a class of abstract continuous function groups. By applying them, we prove the existence of solutions of terminal value problems for nonlinear first order differential equation and solutions for Fredholm integro-equation in Banach spaces.

For Bernstein type operators, we prove that the operators are
unbounded in usual weighted norm. After introducing a new
weighted norm, we study the properties of approximation with Jacobi weight and obtain the direct and inverse theorems of weighted approximation.
From these theorems the equivalent theorem of characterization follows.

Polynomial factorizations are basic problems in symbolic computation. Polynomial factorization algorithms appeared in the 1960's are considered to be the origin of the field of symbolic computation. At present, polynomial factorization algorithms are well established and implemented in symbolic computation software such as MAPLE. But factorization algorithms over successive algebraic extension fields are still under investigation. The basic factorization algorithm over algebraic extension fields is Trager's algorithm. Algorithms for a single algebraic extension field based
on Hensel lifting are given by Weinberger et al. However, in order to compute the irreducible ascending chain in Wu's method, polynomial factorizations over successive algebraic extension fields are needed. Wu, Hu, and Wang independently put forward factorization algorithms over successive algebraic extension fields based on methods of equation solving. Similar to the Trager's algorithm, Wang and Lin proposed another algorithm reducing the problem to the
factorization over the rational number field. In their approach, Wu's triangularization algorithm is used, and hence the termination of the algorithm depends on the computation of Wu's method. Zhi applied the lifting technique to the factorization over successive algebraic extension fields. A direct algorithm on factorization over successive algebraic extension fields is given in this paper,
extending Trager's algorithm to factorization over successive algebraic extension fields. The proposed algorithm only uses resultant computation and factorization over the rational number field.

In this paper, a new concept of the Z-P-S space is introduced, based on Menger probabilistic linear normed space. Some problems of the intrinsic value and the intrinsic element of operator A are investigated in the Z-P-S space($E,F,\Delta $). Four sufficient conditions for which the compact continuous operator A has an intrinsic value $\lambda $ which is larger than 1 and has an intrinsic element corresponding with $\lambda $ on $\partial D$ are established.

A Robust predictive control algorithm for a class of nonlinear systems subject to input and state constraints is proposed based on invariant set switch.The trajectories of the nonlinear system are embedded within those of a bank of Linear Parameter Varying uncertain systems via a so called Piece-Wise Embedding scheme at first. Then, the equilibrium surface of the nonlinear system is obtained by caculation, and for each LPV embedding model a polyhedral invariant set corresponding to the equilibrium point is constructed. Then finite invariant sets overlapped each other are obtained. These invariant sets cover the equilibrium surface such that the closed-loop stability of the nonlinear system can be guaranteed by switching the control laws according to the current state. Compared with the traditional nonlinear predictive control, this algorithm is computationally efficient because most of the problems are solved offline, so that the online computational burden can be greatly reduced.

This paper gives the focus values of each order at $O(0,0)$ for a class of accompanying system of the quadratic differential system I, and proves that the system with $\de lmn=0 $ has at most one limit cycle surrounding $O$, which means that the system has at most one limit cycle surrounding weak focus. Finally the geometric characteristics of all singular points are given.

In this paper, the authors consider a class of multivariate exponential distributions with survival function
$$
\overline{F}(x_1,x_2,\cdots,x_n)=P\{X_1>x_1,\cdots,X_n>x_n\}=\exp\left\{{-\left[
{\sum\limits_{i=1}^n{\left(\frac{x_i}{\theta_i}\right)^{\fen{1}{\delta}}}}
\right]^\delta}\right\},
$$
where $0The density function is given and its properties are discussed. The estimator
$\hat{\delta}$ of the parameter $\delta$ is proposed, its consistence
and asymptotic normality are established, and the asymptotic variance $\sigma_\delta^2$ is derived. Some simulation results are provided.

The paper first gives a comprehensive comparison of the frame of the Active Disturbance Rejection Control (AQDRC) with other methods for controlling uncertain systems, which includes the adaptive control, robust control, disturbance observers based methods, etc. Then the process of theoretical analysis for ADRC, mainly concluding the convergence of the tracking differentiator (TD), the capabilities of the extended state observer (ESO) for estimating
uncertainties and the performance of the ADRC based closed-loop system is reviewed. Finally, ADRC's contributions to the research of controlling uncertain systems are concluded.

In this paper, for multivariable systems with unmodeled dynamics, the problem of robust direct model reference adaptive control using high frequency gain matrix $K_{P}=LDU$ factorization is studied, and the stability and robustness of the closed-loop system are analyzed rigorously.

In this paper, the problem of local state feedback stabilization of a class of nonlinear systems is considered. Based on the center manifold theory, a sufficient condition of asymptotical stabilization of a class of nonlinear systems is given, and the control law, which stabilizes the nonlinear systems, is designed. Using the method of Lyapunov function with homogenous derivative, we investigate specially the problem of the stabilization for a class of planar nonlinear systems and a class of nonlinear systems with zero eigenvalue of multiplicity 2, respectively. Some sufficient conditions for stabilization are established, and the control laws are constructed. Some examples are given which show the validity of the results.

In this paper, a population dynamic model of virus infection is established and considered. It is found that an orbitally asymptotically stable periodic solution exists in that model. Thus, explain the fluctuant phenomena of virus population that was observed in chronically infected individuals.

In this paper, the notion of logarithmic likelihood ratio, as a measure of the deviation of a sequence of integer-valued random variables from an independent random sequence with geometric distribution, is introduced. By restricting the logarith

This note concerns two papers by Guo Yuanming[1,2]. By introducing the concept of solar point, and by proving that each closed weak quasi-convex set must be convex in normed linear spaces, we point out that the main theorems contained in [1, 2]

The problem of observer design for a class of nonlinear system is considered. Based on input-output linearization approach, the observer design for the nonlinear system is developed. The nonlinear system presented in this paper is of characteristic of multi-input multi-output. Under appropriate conditions, it is proved that the proposed observer assured that the observation error converges to zero asymptotically. A simulation result shows the validity of the results.

In this paper, stabilization and control design for a class of nonlinear systems with unknown constant parameters are investigated. For the class of nonlinear systems,new design method of parameters estimator and adaptive backstepping controller is presented. The Lyapunov function is constructed, and a new sufficient condition that guarantees the closed-loop system is globally asymptotically stable is presented. Finally, an example shows the validity of the result.

Let $D$ be a digraph and $S$ a subset of $V(D)$. {Pushing $S$} in $D$
means reversing the orientation of all arcs with exactly one end in $S$.
Klostermeyer proved that it is $NP-$complete to decide if a given
digraph $D$ can be made strongly connected by pushing vertices.
In this paper, we show that, for any bipartite tournament $D$ with the bipartition
$(X,Y)$ of $V(D)$, if $3\leq |X|\leq |Y|\leq 2^{|X|-1}-1$, then $D$ can be
made strongly connected by pushing vertices, and this result is best possible.

Observing the development of the control theory one sees that there are basically two diff-erent approaches:Model Analysis and Direct Control Approach.In the past three decades themodel analysis approach has been the dominating one,while the classical regulation theory andthe guidance theory,etc.are of Direct Control Approach.There is no clear line between the linearity and non-linearity in control systems.The pur-pose of the control is,more or less,a kind of time-varying process of optimization;it is not aglobal system optimization.The value of feedback is not necessarily a function of state variables;it may be some time-varying quantity.To solve the problems of robustness and adaptability of control systems,it is necessary toget rid of the mathematical model of the systems,and to find the control law,which does notdepend on particular expressions of the system.

In this paper, we use a quite new method---reflective function, establish Poincare
mapping for the nonautonomous nonlinear systems, and set up the necessary and
sufficient conditions of existence of the periodic solution and stability for these systems.

In this paper,we discuss the wellposedness of the time-dependent population system and the optimal control on the fertility rate by using the functional analysis methods and Liyapunov function methods.The population system is wellposed and the optimal fertility rate is obtained.

A method of LMI, a kind of Lyapunov-Krasovskii functional, which is based on ``descriptor form", and a new type of memory state feedback controller are used to solve the problem of $H^{\infty}$ controller, designing new adaptive strategies for two unknown constant: input delay and up bound for norm of the uncertain nonlinear part. This type of strategy isn't limited in the estimate value as that of the old adaptive controllers and manages to avoid calculating ``adjusting constant", which guarantees the limitation as the old adaptive
controllers, by reflecting difference between the estimate value and the value being close to the practical. The sufficient condition of its solution is given with a numerical example.

The symplectic transvection of symplectic transformation over local rings is studied. The decomposition of symplectic transformation about symplectic transvection over local rings by theory of defective number and residue number is discussed.

Computational methods based on a linearized implicit scheme are proposed for generalized Korteweg-de Vries(KDV) equation. The methods are applied with minor modifications to the generalized Kadomtsew-Petviashvili(KP) equation. An important advantage gained from the linearized implicit methods is unconditional stability. Numerical results portraying a single-line-soliton solution and the interaction of two-line solition are reported for the 5-order KDV and 6-order KP equations.

In this paper, we gave some new sufficient conditions of existence and uniqueness for certain nonlinear differential systems (E$_{1})$, (E$_{2}$) and (E$_{3})$. Many known results of the existence and uniqueness of limit cycles for nonlinear differentinal systems are generalized and their conditions are improved greatly.

A class of non-selfadjoint operators is discussed in this paper. The necessary and sufficient conditions for an operator with purely discrete spectrum to be u-scalar are given. And some examples are given to illustrate the difference between a u-scalar operator and a spectral operator of scalar type.

A nonlinear robust controller is proposed for a class of single input single output time-delay nonlinear systems with system uncertainty. Based on an iterative procedure known as backstepping, the Lyapunov-Krasovskii functions are
constructed at each step. By magnifying inequation at each step, a robust controller can be acquired, and the corresponding closed loop system is stable.
The simulation result of two stage CSTR shows that the controller proposed in this paper has well control behavior.

First,Wu-characteristic sets and Janet bases for linear homogeneous partial differential equations(LHPDES) are introduced, and then using the concept of reduced bases for LHPDES, we draw a conclusion that both canonical Wu-characteristic set and canonical-autoreduced Janet bases for LHPDES are the reduced bases, and equality between them is obtained according to the uniqueness of the reduced bases.

It was shown by R Nevanlinna that k(λ) is positive, when f ranges over all meromorphic functions of positive non-integral order A. Here k(λ)Let f be a meromorphic function of finite order A. Ozawa provedwith a positive constant d satisfying??????? We shown that (*) hold still when d satisfies In this paper, me prove a more exact and general result: Let f be a meromorphic function of finite order A. Then, for any positive integer n,δ(a,f′~（（n））)≤2－dκ(λ) with a positive constant d=d(n, λ) satisfying（2n（n+1））/（4n~2+7N+2）≤d≤（4n（n+1）/（4n~2+6n+1+16n~4+56n~3+60n~2+20n+1~（1/2））.

In this paper, the synthetic function for the time optimal control of the second-order discretetime system is obtained by the Isochronic Region method. Based on this function, the discrete form of tracking-differentiator is constructed. Numerical simulation shows that this discrete form of tracking-differentiator can quickly trck an input signal without overshoot and chattering and can produce a good differential signal.

Recently, Wei and other authors proposed a class of new quasi-Newton equations $B_{k+1} s_k=y_k^*=y_k+A_k s_k,$ where\ $A_k$\ is some matrix, and based on these, they gave two kinds of modified Broyden's familes(MBC). In this paper, generalized Wolfe linesearches procedures are used, which are combined with the modified Broyden's family. Under some suitable conditions, we prove the global and superlinear convergence property of the preconvex part of the modified Broyden's family.

Some new oscillation and nonoscillation criteria for second order nonlinear neutral delay differential equations are established. Illustrative examples are given. Our theorems improve and generalize several known results.

The paper studies the closedness, semicontinuity and cone semicontinuity of cone efficient point sets and cone weakly efficient point sets of the perturbed attainable objective sets when both the objective maps and constraint maps are continuous and the constraint cone is semicontinuous in topological vector spaces.On this basis,we obtain the closedness and semicontinuity of cone efficient solution sets and cone weakly efficient solution sets for multiobjective programming problem under constraint cone perturbations.