INCREMENTAL AUGMENT ALGORITHM BASED ON REDUCED Q-MATRIX

Reduced Q-matrix （Qr matrix） plays an important role in the rule space model （RSM） and the attribute hierarchy method （AHM）. Based on the attribute hierarchy, a valid/invalid item is defined. The judgment method of the valid/invalid item is developed on the relation between reachability matrix and valid items. And valid items are explained from the perspective of graph theory. An incremental augment algorithm for constructing Qr matrix is proposed based on the idea of incremental forward regression, and its validity is theoretically considered. Results of empirical tests are given in order to compare the performance of the incremental augment algo-rithm and the Tatsuoka algorithm upon the running time. Empirical evidence shows that the algorithm outper-forms the Tatsuoka algorithm, and the analysis of the two algorithms also show linear growth with respect to the number of valid items. Mathematical models with 10 attributes are built for the two algorithms by the linear regression analysis.

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS-THE EXISTENCE AND UNIQUENESS

The concepts of Markov process in random environment, q-matrix in random environment, and q-process in random environment are introduced. The minimal q-process in random environment is constructed and the necessary and sufficient conditions for the uniqueness of q-process in random environment are given.

THE CONSTRUCTION OF DENUMERABLE q-PROCESSES IN RANDOM ENVIRONMENTS SATISFYING (F) OR (B)

This article is a continuation of[9].Based on the discussion of random Kolmogorov forward（backward）equations,for any given q-matrix in random environment, Q（θ）=（q（θ;x,y）,x,y∈X）,an infinite class of q-processes in random environments satisfying the random Kolmogorov forward（backward）equation is constructed.Moreover, under some conditions,all the q-processes in random environments satisfying the random Kolmogorov forward（backward）equation are constructed.

Laplace Transformation and Ergodic Potential Kernel for q-Process in Random Environment

This paper introduces some concepts such as q- process in random environment, Laplace transformation, ergodic potential kernel, error function and some basic lemmas.We study the continuity and Laplace transformation of random transition function. Finally, we give the sufficient condition for the existence of ergodic potential kernel for homogeneous q- processes in random environments.

一类Q过程的存在性问题(1)

本文考虑形如Q=(■)的拟Q-矩阵,其中e1=+∞,且Q_b不要求全稳定。假定存在Q_b过程ψ(λ),记P_(φ(λ))(Q)={(Rλ):(Rλ)为Q过程且_bR(λ)-ψ(λ)}称为由ψ(λ)生成的过程类。本文给出P_(φ(λ))(Q)非空,即由ψ(λ)生成的过程存在的充要条件。

一类Q过程的存在性问题(Ⅱ)

本文考虑形如Q=(■)的拟Q-矩阵,其中el<∞,且Q_b不要求全稳定。假定存在Q_b过程ψ(λ),记Pψ(λ)(Q)={R(λ):R(λ)为Q过程且_bR(λ)=ψ(λ)}称为由ψ(λ)生成的过程类。本文给出P_φ(λ)(Q)非空,即由ψ(λ)生成的过程存在的充要条件。利用这一结果,在Q的非保守量有界时,给出了可和单瞬时F型Q过程的存在准则。

常返Q过程定性理论

对任给的一个全稳定Q—矩阵,本文给出了存在常返Q过程的充要条件以及常返Q过程唯一的充要条件.

Q过程的分类

本文将Q过程重新分类,得到每类Q过程存在、唯一性准则以及一般表达式,我们的结果在Q过程构造及定性理论,特别是在含瞬时态的Q过程的构造及定性理论中有重要应用,关于这些结果,将另文述及.

Q过程构造论中的H-条件

设E为可列集,Q为E×E上全稳定Q-矩阵,在[1]中,侯振挺教授得到“Q过程的唯一性准则”,这个准则的核心是给出了如下条件: ??(1) 其中φ(λ)=(φij(λ),i,j∈E,λ>0)为最小Q过程. 我们称(1)为H-条件.我们发现H-条件在Q过程构造论中起重要的作用.本文就H-条件在Q过程定性理论及瞬时态Q过程构造论中的作用进行一些讨论.

关于含瞬时态Q过程的几点注记

设E为可列集,Q=(q_(ij):i,j∈E是E×E上的Q矩阵,我们并不要求Q全稳定.考虑如下问题,取△∈E,问Q满足什么条件下,存在E上的非负行向量e和E上的非负向量f,使Q=(q_(ij):i,j∈EU{△})为Q-矩阵.其中q△△=-∞,qi△=f_i,q_(△j)=e_j,q_(ij)=q(i),(i,j∈E),本文给出了上述问题的完整解答.同时,我们给出了上述结果在含瞬时态Q过程方面的三个应用.

全稳定Q过程构造的一个等价条件

给出了 Q过程构造的等价条件 ,证明了 Q过程的构造等价于一个条件方程组的解 ,以此方法求出了所有的 Q过程、F过程、B过程及

非时齐马氏链转移矩阵的性质

对非时齐马氏链转移矩阵的分析性质进行了讨论起因于非时齐马氏链广泛应用于随机控制和奇异摄动 .得到了转移矩阵的一些连续性性质 ,特别是对 Q-矩阵函数的概率意义进行了详细讨论 ,得到了在一状态停留的时间服从一指数分布 ,从一状态跳到另一状态的概率等较完整的结果 ,最后还给出一个 Q-矩阵函数有界的充要条件 .

分支随机Q过程的惟一性

引进了分支随机Q矩阵、分支随机Q过程和随机生成母函数的概念.研究了分支随机转移矩阵与随机生成母函数的关系.本文的主要结果是:在某些合理的条件下,证明了分支随机Q矩阵的随机Q过程的惟一性.

禁止概率在构造论中的应用

本文将禁止概率应用于Q过程构造论,得到了一些新结果,特别是,它可以应用于有瞬时态的情况,同时,我们得到了单瞬时态的一些结果,并得到了拟保守不可和单瞬时Q过程的存在准则。

分枝Q过程的一个特征

设Ｅ＝｛０，１，２，…），Ｑ为Ｅ上的全稳定Ｑ─矩阵，众所周知，如果Ｑ是分枝Ｑ─矩阵，则最小Ｑ过程Ｐ（ｔ）＝（ｐｉｊ（ｔ））满足： 本文证明了Ｅ上任意一个全稳定Ｑ过程Ｐ（ｔ）＝（ｐｉｊ（ｔ））如果满足上式及对某个 则其Ｑ矩阵必为分板Ｑ─矩阵，且该过程就是最小Ｑ过程．

一类常返及遍历Q过程的存在唯一性

设Ｑ为广义Ｋｏｌｍｏｇｏｒｏｖ矩阵，并给出了存在常返Ｑ过程与遍历Ｑ过程及其过程唯一的充要条件．

The Equivalence Forms of Random Kolmogorov Forward(Backward) Equations

The concepts of Markov process in random environment, q-matrix in random environment and q-process in random environment are introduced. Three forms of random Kolmoogrov farward （or backward） equations are introduced and the equivalence of these three forms are also proved. Moreover any conservative q-process in random environment satisfies random Kolmogrov backward equation.

THE STATIONARY DISTRIBUTION OF A CONTINUOUS－TIME RANDOM GRAPH PROCESS WITH INTERACTING EDGES

In previous papers, the stationary distributions of a class of discrete and continuoustime random graph processes with state space consisting of the simple and directed graphs on Nvenices were studied. In this paper, the random graph graph process is extended one impotent stepfurther by allowing interaction of edges. Similarly, We obtha the expressions of the stationarydistributions and prove that the process is ergodic under different editions.

极大Q过程存在准则

本文讨论极大Ｑ过程的存在性问题．我们得到，对任给的Ｑ－矩阵Ｑ，极大Ｑ过程总存在．

REPRESENTATION OF ADDITIVE FUNCTIONALS AND LOCAL TIMES FOR JUMP MARKOV PROCESSES AND THEIR FUNCTIONAL LIMIT THEOREM

The representation of additive functionals and local times for jump Markov processes are obtained. The results of uniformly functional moderate deviation and their applications to birth-death processes are also presented.