In this paper, we extend the results of Whitt, W. (1971) on the weak convergence of a queueing system with a single server for PR priority discipline. We investigate the weak convergence of queueing length and load processes in a queueing system with a single server for HL priority discipline in various cases and for PRE priority discipline in the case p₂ > 1.

We discuss a class of prey-predator population models with time lag msing the method of the functional differential equations. Under some conditions of the parameters Under (i.e. some ecological environment conditions), the existence of a nonconstant periodic solution around the equilibrium is proved. That is, the ecological balance betweea the specaea can be retained.

Starting from the discrete Sturm-Liouville problem we study the discrete Gelfand-Levitan equations, i.e. the discrete version of G-L linear and nonlinear integral equations, as well as the discrete inverse scattering problems. Our study of various discrete problems provides an easy way to solving the G-L problems numerically.

We first obtain the existence theorem of periodic solution of the nonautonomous periodic system

by using the Leray-Schauder degree theory. Then we transfer the existence problem of periodic solution of the periodic system

into that of the fixed point of set-valued mapping, and obtain a general existence theorem of periodic solution and its corollary by using Kakutani's theorem. Finally, examples of application of the theorem are cited.

In this paper, Babuska-Brezzi's condition is shown to be also necessary for the existence and uniqueness of the saddle point problem.

In this paper the pursuit for linear differential games is discussed and some conditions for completing the pursuit are given.

The estiu}rtes of the coeditional L-functional

are constructed by using the kernel fuaetion method. The asymptotic normality of these estimates is established. Moreover, the bootstrapping statistics of the estimates are also given and their asymptotic properties discussed.

This paper deals with the continuous dependence on domains of the optimal control for a system governed by a parabolic partial differential equation. Generally, for most physical operating systems, because of interior or exterior reason, small disturbance of the domain often appears. So problems of this type are clearly of significant practical interests. By using the Sobolev space theory under the perturbation of domains, it is shown that a small change of the domain will imply a small change of corresponding optimal control in a sense. This is obviously an expected well-posed property.

The first boundary value problem for the nonlinear multidimensional coupled systems of the hyperbolic type, the pseudo-hyperbolic type, the parabolic type and pseudo-parabolic type of higher order are considered. The existence, uniqueness and regularities of the generalized global solution for the first boundary value problem of the nonlinear coupled evolutional systerns are proved by the Galerkin method and energy estimates.

Consider the model Y=Xβ+ε,where β and ε arep-dimensional and ndime nsional unobservable random vectors respectively, p (or σ²V);X,A,V are known matrices; α∈R^k and σ>0 are parameters, The nece-ssary and sufficient conditions for an estimate LY(LY+α) of Sα+Qβ to be admissible among the class of homogeneous linear estimates (linear estimates) under the quadratic or matrix loss function are gained. When the sufficient conditions for LY+α to be admissible within the class that contains all the esrimates for Sα+Qβ with the quadratic loss function are detived; some necessary conditions for LY+α to be admissible are also given.

The conjecture proposed by Geoffroy [3] is proved to be true for triangle-free graphs and false for graphs comaining triangles. A modified form of this conjecture is provided.

In this paper we consider first-order nonlinear differential equations with retarded and advanced arguments. We give some new conditions for oscillations of every solution of these equations and generalize and improve the main results in [3, 4].

First,the mimmax property of ∞-MINQE(U) is studied.Then,the definition of biased Bayesian quadratic estimation is introduced and discussed.Finally,the optimal solutions of these estimators are shown to reduce to the solutions of a system of linear equations.

The number of rooted simple planar triangulations with the face number given was first determined by W.T.Tutte^[3] by a recursive formula.However,in this note,a simpler recursive formula is found by the method the author used for 3-nets^[2].Further,an explicit formula,seemingly the first ever is also obtained.

In the paper we discuss the problem of how todetermine the distribution,the expectation and the variance of time statistics on PERT,which was proposed by Hua Loo-keng.Our answer is that the determination of the distribution is a result of the choice in Pearson's distributed system,and that in the determination of both expectation and variance,one of them must be assumed and the other is derived from the approximate formula of linear interpolation with Tchebycheff knods.

A sufficient and necessary condition is found for the existence of a Ляпунов function,constructed by Li Senlin in 1964,for constant coefficient linear equations.Moreover a simple construction of this Ляпунов function is presented in the form of determinant.

By constructing appropriate comparison functions on different areas,Filippov's theorem is extended to the more generalized system
.

In this paper we discuss bifurcations in 2-jet system with universal unfolding and 3-jet system with π-rotation perserving unfolding of codimension two generation systems which are under weak periodic perturbations.We find that under some parameter conditions,the two systems both have various harmonic or ultraharmonic bifurcations and smale horseshoes.Its dynamic behavior are complex.

Let X_n=(X_n⁽¹⁾,X_n^(m)),n=1,2,…,be a m-dimensional stationary sequence

,i=1,…,m,M_n=(M_n⁽¹⁾,…,M_n⁽¹⁾) is called the maximum of X₁,…,X_n,In this paper,we define the mixing conditions for {X_n},and under these conditions,discuss the asymptotic distribution for M_n.The results are similar to those in the i.i.d.case.

The problems of identifiability of mixtures are considered.Two limit laws for mixtures are obtainrd.Some results on the joint asymptotic distributions of the upper and lower extremes are obtained by applying these limit laws to the asymptotic theory of extreme.

The necessary and sufficient algebraic criteria for second order 2-delay differential-difference equations

to be stable for all delays are obtained.

In this paper we discuss the following general quadratic programming problem: min{f(x)=1/2x^TGx+c^Tx|A^Tx≥b},An exchange type null space algorithm for this problem is given.Some computational results indicate that our algorithm is superior to both Gill and Murray's in [3] and Fletcher's in [2] when the number of active constraints at the solution is large.Our algorithm can also be applied to linear programming problems.

We consider the equation

where R is a polynomial with quadratic and cubic terms of x(t) and x(t-1).We prove that if α>0,then this equation has no solution with period 1;if α=0,then under certain conditions it has a solution with period 1.By using Hale's theory,under more general conditions we prove that the equation has a periodic solution (of course the period may not be 1).

Given a general nonlinear system

We tackle the problems of its parallel decomposition and cascade decomposition.We first discuss the relation between the strong accessibility Lie algebra and a set of its ideals.Then necessary and sufficient conditions of the parallel decomposition are derived from it.Similary,we obtain necessary and sufficient conditions of the cascade decomposition.All the results may be considered as a generalization of those of[1].Finally we discuss the area on which the decompositions exist,and the algorithm.An example is included to illustrate the deeomposition procedure.

In this paper a Lyapunov function concerning partial stability for linear systems with constant coefficients is constructed,and the Lyapunov functions concerning partial stability for a kind of linear systems with time-varying coefficients and a kind of non-linear and nonautonomous systems are given.These results are applied to partial stability of the large-scale systems whose isolated subsystems are those systems as mentioned above.This makes it feasible to determine partial stability.

For the purpose of estimating the population distribution by sample statistical data,two new types of theoretical distributions,the Γ-type distribution and the B-type distribution,are suggested in this paper,which include several parameters and an arbitrary function is the expressions and contain most of the widely used distributions as special cases.The new distributions greatly expand the range of the theoretical distributions in use and are of wide adaptability.A new method for determining the population distribution according to the sample distribution is also given,which can be used to determine the parameters and the undetermined function in the distribution expression.This new method is superior to the traditional method using probability papers or type curve match.Finally,an example is given to show how to use the new method.

This paper is devoted to studying the following initial-boundary value ptoblekn

vrhich comes from the mathematical model for nonlinear vibrations of finite rods with linear viscoelasticity.The existence,uniqueness and smoothness properties of the solution of this problem are obtained.

In this paper,we discusse the rth-mean and strong consistency of the least squares atimators of linear regressions with dependent errors,i.e,pairwise m-dependent,generalized Gaussion,multiplicative and *-miming sequences,and generalize the main theorem in [1].some sufficient conditions on the rth-mean and a.s.ceusistency are given.

In this paper,we ohtain the following results.When the origin is a node,a focus (spirals),or a saddle of a second order linear system with ordinary coefficient,the high dimeasion term will not change the type,stability and oscillation of the singalar point.But it may turn the center into a focus.

We investigate the perturbed quadratic system with two symmetric centers.By computing defection functions of the autonomous perturbation system,global and local bifurcations can be completely listed in the parameter space.For the time periodic perturbation system,we study the subharmonic bifurcation and chaotic behavior.

A Lax pair for the modified Boussinesq equation and the Darboux transformation associated with a liven eigenvalue problem are derived in this paper.The htiura traosformation of this equation and the Boussinesq equation is obtained.The Darboux and Miura transformations are used to give the solutions of the two equations.

In this paper,we study two classes of factor incident graphs of feedback shift register.one is the incident graph of the product of ta}oreciprocal primitive polynomials over F₂ with degree n,from which we construct 2²ⁿ(2^n-2-1)2n stage and 2²ⁿ(2ⁿ-1)^2n-12n+1 stage feedback functions of an M sequence.The other is the incident graph of the power of a primitive polynomial with degreen,from which we construct 2^2n-1-1(2ⁿ-1)2n stage feedback functions of an M sequence.

In this paper,we discuss the exponent of third order differential equations with unbounded piecewise-constant matrix,and give the upper and lower bounds of the exponent of these equations and the existence of the solutions whose exponent is equal to the upper or lower bound.Finally,we also study the exponent of its perturbation equations.

Under some conditions on the functional perturbation (t,y_t),we study the asymptotic relationship between solutions of the perturbed equation

and of the linear ordinary equation

We also discuss the same problem of equations

and
x⁽ⁿ⁾(t)+a_n-1(t)x^(n-1)(t)+…+a₀(t)x(t)=0.(4)
For equations (2) and (4),we assume that the general solutions and their asymptotic behavior are known.An integral inequality of Bellman-Gronwall-Bihari type is used.Our conclusions generalize the Theorem 7 in [1] and Theorems 1,2 in[2].

The existence,uniqueness and stability of the discontinuous solution of impulsive flow in a hydraulic pipe due to impact are discussed.The possibility of occurance of a shock wave and its behavior are analyzed marhematically.Moreover,the connection condition and the stability condition for a discontinuous solution are given.