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| 1 | Strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree显示文摘In this paper,we study the strong law of large numbers and Shannon-McMillan (S-M) theorem for Markov chains indexed by an infinite tree with uniformly bounded degree.The results generalize the analogous results on a homogeneous tree. | HUANG HuiLin~1 YANG WeiGuo~(2+) 1 Department of Mathematics,Shanghai Jiaotong University,Shanghai 200240,China 2 Faculty of Science,Jiangsu University,Zhenjiang 212013,China | 2008 | Science China Mathematics2008，51，2： | 23 |
| 2 | Estimation of Spectral Gap for Markov Chains | Chen Mufa Department of Mathematics Beijing Normal University Beijing, 100875 China | 1996 | Acta Mathematica Sinica,English Series1996，12，4： | 14 |
| 3 | Population-size-dependent branching processes in Markovian random environments显示文摘A branching model {Z n} n≥0is considered where the offspring distribution of the population’s evolution is not only dependent on the population size, but also controlled by a Markovian environmental process {ξ n} n≥0. For this model, asymptotic behaviour is studied such as limn→∞Z n and limn→∞Z n/m n in the case that the mean m k, θof the offspring distribution converges to m>1 as the population size k grows to ∞. In the case that {ξ n} n≥0is an irreducible positive recurrent Markov chain, certain extinction (i.e. P(Z n=0 for some n)=1) and noncertain extinction (i.e. P(Z n=0 for some n)<1) are studied. | WANG Hanxing 1 and DAI Yonglong 2 1. Department of Mathematics, Shanghai University, Shanghai 201800, China 2. Department of Mathematics, Zhongshan University, Guangzhou 510275, China | 1998 | Chinese Science Bulletin1998，43，8： | 5 |
| 4 | Coordinating Exploration and Exploitation To Construct Genetic Algorithms显示文摘A new genetic algorithm is proposed based on the careful coordination of the exploration in the solution space of the given problem and the exploitation of the information from the previous search. In the new algorithm architecture, the population in each generation consists of three sub-populations: a preserved part, a reproduced part, and a randomized part. Two parameters are incorporated into the algorithm to efficiently control the percentage of each sub-population to achieve good balance between the exploration and exploitation processes during the optimization. By modeling the algorithm as a homogeneous finite Markov chain, the new genetic algorithm is shown to converge towards the global optimum of the problem at hand. Experiments were designed to test the algorithm using the Rastrigin function, the Griewangk function, and the Schaffer function. Data analyses using the average success ratio, the average objective calculating number, the average first passage time to solution, and the standard deviation of the first passage time were compared with those of the canonical genetic algorithm, the elitist genetic algorithm, and the steady genetic algorithm. The results show strong evidence that our algorithm is superior in performance in terms of economy, robustness and efficiency. | 江瑞 罗予频 胡东成 司徒国业 | 2002 | Tsinghua Science and Technology2002，7，6： | 3 |
| 5 | STRONG LAW OF LARGE NUMBERS AND SHANNON-MCMILLAN THEOREM FOR MARKOV CHAINS FIELD ON CAYLEY TREE显示文摘This paper studies the strong law of large numbers and the Shannom-McMillan theorem for Markov chains field on Cayley tree. The authors first prove the strong law of large number on the frequencies of states and orderd couples of states for Markov chains field on Cayley tree. Then they prove the Shannon-McMillan theorem with a.e. convergence for Markov chains field on Cayley tree. In the proof, a new technique in the study the strong limit theorem in probability theory is applied. | 杨卫国 刘文 | 2001 | Acta Mathematica Scientia2001，21，4： | 2 |
| 6 | Coupling, spectral gap and related topics (Ⅱ)显示文摘THIS is the second one of a series of three reviews. The ideas introduced in the last review are used to study the estimate of spectral gap and four classes of typical eigenvalue problems on manifolds. The comparison with the known optimal estimates is given, some new progress is reported and some open problems are proposed. | CHEN Muf aDepartment of Mathematics, Beijing Normal University, Beijing 100875, China | 1997 | Chinese Science Bulletin1997，42，17： | 2 |
| 7 | Coupling, spectral gap and related topics (Ⅲ)显示文摘THIS is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the convergence in total variation, the algebraic convergence and the infinite-dimensional case. The perturbation of spectral gap and the logarithmic Sobolev constant under a linear transform is given (Theorem 5). A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7). | CHEN MufaDepartment of Mathematics, Beijing Normal University, Beijing 100875, China | 1997 | Chinese Science Bulletin1997，42，18： | 2 |
| 8 | Coupling, spectral gap and related topics (Ⅰ)显示文摘THIS is the first of a series of three reviews. They are partially surveys on three aspects: (ⅰ) explaining the main ideas of our recent application of the coupling method to the estimation of spectral gap, (ⅱ) introducing some more recent progress on the study on some related topics,(ⅲ) collecting some open problems for the further study. | CHEN MufaDepartment of Mathematics, Beijing Normal University, Beijing 100875, China | 1997 | Chinese Science Bulletin1997，42，16： | 1 |
| 9 | CHAOS IN TOPOLOGICAL MARKOV CHAINS显示文摘CHAOSINTOPOLOGICALMARKOVCHAINS￥CAOLiangyue;WUShijing;ZHAOHanzhang(InstituteofSystemsScience,AcademiaSinica,Beijing100080,Chin... | CAO Liangyue WU Shijing ZHAO Hanzhang(Institute of Systems Science, Academia Sinica, Beijing 100080, China) | 1994 | Systems Science and Mathematical Sciences1994，7，2： | 0 |
| 10 | Using Markov chains to predict the natural progression of diabetic retinopathy显示文摘AIM: To study the natural progression of diabetic retinopathy in patients with type 2 diabetes.METHODS: This was an observational study of 153 cases with type 2 diabetes from 2010 to 2013. The state of patient was noted at end of each year and transition matrices were developed to model movement between years. Patients who progressed to severe non-proliferative diabetic retinopathy(NPDR) were treated.Markov Chains and Chi-square test were used for statistical analysis.RESULTS: We modelled the transition of 153 patients from NPDR to blindness on an annual basis. At the end of year 3, we compared results from the Markov model versus actual data. The results from Chi-square test confirmed that there was statistically no significant difference(P =0.70) which provided assurance that the model was robust to estimate mean sojourn times. The key finding was that a patient entering the system in mild NPDR state is expected to stay in that state for 5y followed by 1.07 y in moderate NPDR, be in the severe NPDR state for 1.33 y before moving into PDR for roughly8 y. It is therefore expected that such a patient entering the model in a state of mild NPDR will enter blindness after 15.29 y.CONCLUSION: Patients stay for long time periods in mild NPDR before transitioning into moderate NPDR.However, they move rapidly from moderate NPDR to proliferative diabetic retinopathy(PDR) and stay in that state for long periods before transitioning into blindness. | Priyanka Srikanth | 2015 | International Journal of Ophthalmology(English edition)2015，8，1： | 0 |
| 11 | BLOCKWISE PERTURBATION THEORY FOR 2×2 BLOCK MARKOV CHAINS显示文摘Let P be a transition matrix of a Markov Chain and be of the form The stationary distribution πT is partitioned conformally in the form (π1T,π2T). This paper establish the relative error bound in πiT (i =1, 2) when each block Pij get a small relative perturbation. | Jun-gong Xue Wei-guo Gao (Department of Mathematics, Fudan University, Shanghai, 200433, China) | 2000 | Journal of Computational Mathematics2000，18，3： | 0 |
| 12 | A Note on Markov Chains in Stationary Random Environments显示文摘1IntroductionAstandardMarkovchainisastochasticproceswithaconstanttransitionprobabilitymatrixP,wherePisaonesteptransitionprob... | Wang Hanxing(College of Sciences, Shanghai University) Fang Dafan(Yueyang Normal College) | 1999 | Advances in Manufacturing1999，，2： | 0 |
| 13 | SOME LIMIT PROPERTIES AND THE GENERALIZED AEP THEOREM FOR NONHOMOGENEOUS MARKOV CHAINS显示文摘Let(ξ_n)_(n=0)~∞ be a Markov chain with the state space X = {1, 2, ···, b},(g_n(x, y))_(n=1)~∞ be functions defined on X × X, and F_(m_n,b_n)(ω) =1 /b_n sum from k=m_n+1 to m_n+b_n g_k(ξ_(k-1), ξ_k).In this paper the limit properties of F_(m_n,b_n)(ω) and the generalized relative entropy density f_(m_n,b_n)(ω) =-(1/b_n) log p(ξ_(m_n,m_n+b_n)) are discussed, and some theorems on a.s. convergence for(ξ_n)_n=0~∞ and the generalized Shannon-McMillan(AEP) theorem on finite nonhomogeneous Markov chains are obtained. | Ping Hu Zhongzhi Wang | 2018 | Annals of Applied Mathematics2018，34，3： | 0 |
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