ON THE GENERALIZED BIRTH AND DEATH PROCESSES(Ⅰ)——THE NUMERAL INTRODUCTION,THE FUNCTIONAL OF INTEGRAL TYPE AND THE DISTRIBUTIONS OF RUNS AND PASSAGE TIMES

This paper makes a painstaking discussion for the generalized birth and death processes, describes some properties of processes in fixed quantity: Such as the uniqueness criterion of processes and the distributions of runs and passage times of the generalized birth     

ON THE GENERALIZED BIRTH AND DEATH PROCESSES (Ⅱ)——THE STAY TIME, LIMIT THEOREM AND ERGODIC PROPERTY

In this paper we follows [1] to discuss the stay time, limit distribution and ergodic property of the generalized birth and death processes. Here we use the notations involved in [1] besides special exposition.     

THE INVARIANCE OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE β AND ORDER α

The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper.From the maximum modulus principle of holomorphic functions,we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallikeness of type β and order α on the unit ball B~n in C~n and on bounded and complete Reinhardt domains.Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirallikeness of type β,strong and almost starlikeness of order α,strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.     

ON THE ACCURACY OF THE QUASICONFORMING AND GENERALIZED CONFORMING FINITE ELEMENTS

It is observed in practice that the numerical accuracy of the two unconventional plate elements, i. e., the nine parameter quasi-conforming and generalized conforming elements, is better than that of the usual Zienkiewicz in compatible cubic element and of a new element proposed recently by Specht, although all these elements have the same asymptotical rate of convergence O(h) in the energy norm. In the paper a careful error analysis for the quasi-conforming and generalized conforming elements is given. It is shown that the consistency error due to nonconformity of the two unconventional elements is of order O(h~2), one order high than that of other conventional nonconforming elements with nine parameters.     

Decomposition and embedment of trajectories after explosion for a birth and death process

Let X = {X(t),t ＜σ} (σ is lifespan) be a birth and death process with explosion whose characteristic triple (Mα,MC,MD) of MX in terms of (α, C, D) and M. This means that a lot of given birth and death processes can be embedded in one and the same birth and death process. If κ∈ E and M = {κ},we decompose X into κX, κ∈ E.     

THE BIRTH AND DEATH PROCESSES WITH ZERO AS THEIR ABSORBING BARRIER

Let E=(0,1,…),Q~b=(q_(ij)),i,j=0,1,…,where q_(i,i-1)=a_i,q_(i,i+1)=b_i,q_(ii)=-(a_i+b_i),q_(ij)=0,when|i-j|>1.a_0=0,b_0=b>0,a_i,b_i>0(i>0).Letting b=0 in Q~b,we get the matrix Q~0.The time homogeneous Markov process X~b={x~b(t,ω),0≤t<σ~b(ω)}(X~0={x~0(t,ω),0≤t<σ~0(ω)}),with Q~b(Q~0,respectively)as its density matrix and with E as its state space,is called Q~b(Q~0,respectively)process in this paper.Q~b and Q~0 processes are all called the birth and death processes,with zero being the reflecting barrier of Q~b processes,the absorbing barrier of Q~0 processes.All the Q~b processes have been constructed by both probability and analytical methods(Wang [2],Yang[1]).In this paper,the Q~0 processes are“imbedded.”into Q~b processes and all the Q~0 processesare directly constructed from the Q~b processes.The main results are:Let b>0 be arbitrarily fixed,then there is a one to one correspondence between the Q~0 processesand the Q~b processes(in the sense of transition probability).The Q~0 process is uni     

MATRIX NORM VERSIONS OF THE KANTOROVICH INEQUALITY AND ITS APPLICATIONS

In this paper we discuss the generalizations of the Kantorovich inequality and obtain some generalized Kantorovich inequalities in the sense of matrix norm. We further illustrate how to use these inequalities to determine the lower bound of relative efficiency of the parameter estimate in linear model.     

THE ERGODICITY FOR BI-IMMIGRATION BIRTH AND DEATH PROCESSES IN RANDOM ENVIRONMENT

The concepts of bi-immigration birth and death density matrix in random environment and bi-immigration birth and death process in random environment are introduced. For any bi-immigration birth and death matrix in random environment Q（θ） with birth rate λ 〈 death rate μ, the following results are proved, （1） there is an unique q-process in random environment, P^-（θ＊（0）;t） = （p^-（θ^＊（0）;t,i,j）,i,j ≥ 0）, which is ergodic, that is, lim t→∞（θ^＊（0）;t,i,j） = π^-（θ^＊（0）;j） ≥0 does not depend on i ≥ 0 and ∑j≥0π （θ＊（0）;j） = 1, （2） there is a bi-immigration birth and death process in random enjvironment （X^＊ = {X^＊,t ≥ 0},ε^＊ = {εt,t ∈ （-∞, ∞）}） with random transition matrix P^-（θ^＊ （0）;t） such that X^＊ is a strictly stationary process.     

Unified characteristic numbers and solutions of equations for birth and death processes with barriers

The state 0 of a birth and death process with state space E = {0, 1, 2,....} is a barrier which can be classified into four kinds： reflection, absorption, leaping reflection, quasi-leaping reflection. For the first, second and fourth barriers, the characteristic numbers of different forms have been introduced. In this paper unified characteristic numbers for birth and death processes with barriers were introduced, the related equations were solved and the solutions were expressed by unified characteristic numbers. This paper concerns work solving probability construction problem of birth and death processes with leaping reflection barrier and quasi-leaping reflection barrier.     

ON THE EXISTENCE AND UNIQUENESS THEOREMS OF SOLUTIONS FOR THE SYSTEMS OF MIXED MONOTONE OPERATOR EQUATIONS WITH APPLICATIONS

The existence, uniqueness and non-symmetric iterative approximation of solutions for a class of systems of mixed monotone operator equations are discussed. As an appplication, we utilize the results presented in this paper to study the existence and uniqueness problems of common solutions for a class of systems of functional equations arising in dynamic programming of multistage decision processes and a class of systems of nonlinear integral equations. The results obtained in this paper not only answer an open question suggested in [3] but also generalize the corresponding results of.     

POTENTIALITY AND REVERSIBILITY FOR THE N-SPIN-FLIP PROCESSES AND THE N-GENERALIZED SPIN-FLIP-PROCESSES

The Markov processes with infinite particle system are one of the new branches in probability, recently it is developed rapidly. It relates not only to the statistical physics of balanced state, but also to the statistical physics of the unbalanced state and other subjectst In this paper, two kinds of Markov processes with infinite     

ON THE CENTER OF GENERALIZED LIENARD SYSTEM

This paper discusses the local and global center of the generalized Lienard system, gives two theorems which extend and improve the relative results of [1]-[3]. Moreover the lemmas in this paper are also useful to the study for the oscillation, stability and the existence of limit cycles of the generalized Lienard system.     

ON THE ESTIMATION OF GENERALIZED LEBESGUE CONSTANT AND MODULUS OF GENERALIZED SINGULAR QUADRATURE FORMULAS AND ITS APPLICATION

This article, first gives the estimaties of two modulus, namely, generalized Lebesgue constant and modulus of generalized singular integral quadrature formulas, then applies them to obtain the error bounds of the operator BLmp to the operator B.     

ON THE DUALITY OF GENERALIZED GEOMETRIC PROGRAMMING

It is found that generalized geometric programming (GGP) is in fact a special case of generalized convex programming. By selecting a suitable bifunetion and calculating its adjoint function, the dual form of the standard GGP problem is derived. Some duality theorems are also obtained with this point of view. The method used is simpler and more general than what appeared in the literature.     

ASYMPTOTIC BEHAVIOR OF POSITIVE RADIAL SOLUTIONS FOR THE GENERALIZED EMDEN-FOWLER EQUATION WITH APPLICATION

In this paper, we consider a class of semilinear parabolic differential equation where nonlinear term has local bounded coefficients. Under some assumptions, we get the existence and uniqueness of solution. Terminate to the time variable, we obtain the so called generalized Emden-Fowler equation and the asymptotic behavior of positive radial solutions have been given in all dimensions. At the end of this paper, we give its application to critical branching Brownian motion (also called measure-valued branching processes).     

SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.     

ASYMPTOTIC SOLUTIONS OF THE REDUCED WAVE EQUATION IN THE VICINITY OF A HIGHER CAUSTIC

In this paper the asymptotic solutions of the reduced wave equation (1) in the vicinity of a higher caustic have been found out by means of asymptotic expansions of the generalized Airy function, and the order (or the singularity inder) and the phase's increment of solutions on the higher caustic have been obtained, too.     

THE STATIONARY DISTRIBUTIONS AND THE STEADY PROBABILITY CURRENTS OF ONEDIMENSIONAL DIFFUSION PROCESSES UNDER NON-LOCAL BOUNDARY CONDITIONS

In the present paper we study the limit distributions of one-dimensional diffusion processes under non-local boundary conditions. We obtain the existence conditions for the non-trivial stationary distributions, get their concrete formulas in all cases according to boundary classification, and give the results of some probability explanations. One-dimensional diffusion processes, which provide a simple mathematical model for studying irreversible steady systems, show that the basic characteristic of an irreversible stationary state is the occurrence of the probability current, i. e. the circulation phenomenon.     

POTENTIALITY AND REVERSIBILITY FOR THE N-SPIN-FLIP PROCESSES AND THE N-GENERALIZED SPIN-FLIP-PROCESSES

§1 Introduction The Markov processes with infinite particle system are one of the new branches in probability, recently it is developed rapidly. It relates not only to the statistical physics of balanced state, but also to the statistical physics of the unbalanced state and other subjects.     

DECOMPOSITION, REPRESENTATION AND EXTENSION OF THE HARMONIC FUNCTIONS

In this paper, by using the probabilistic method, we obtained a decomposition theorem for the harmonic functions, generalized and strengthened some theorems: Doob's theorem on the limit behaviors of the superharmonic functions, Bocher's theorem on the representation of the harmonic functions and the extension theorem of the harmonic functions etc.