Model of Markov Chains in Space-Time Random Environments

A general framework of stochastic model for a Markov chain in a space-time random environment is introduced, here the environment ξ^＊：={ξ1,x∈N,x∈ X}is a random field. We study the dependence relations between the environment and the original chain, especially the ＂feedback＂. Some equivalence theorems and law of large numbers are obtained.

THE DIMENSIONS OF THE RANGE OF RANDOM WALKS IN TIME-RANDOM ENVIRONMENTS

Suppose {Xn} is a random walk in time-random environment with state space Z^d, ｜Xn｜ approaches infinity, then under some reasonable conditions of stability, the upper bound of the discrete Packing dimension of the range of {Xn} is any stability index α. Moreover, if the environment is stationary, a similar result for the lower bound of the discrete Hausdorff dimension is derived. Thus, the range is a fractal set for almost every environment.

EXTINCTION OF POPULATION-SIZE-DEPENDENT BRANCHING CHAINS IN RANDOM ENVIRONMENTS

We consider a population-size-dependent branching chain in a general random environment.We give suffcident conditions for certain extinction and for non-certain extinction.The chain exhibits different asymptotic according to supk,θmk,θ1, mk,θn→1 as k →∞, n→∞, infk,θmk,θ1.

The Moments of First Entrance Time Distributions of Markov Chains in Random Environments

After defining generating functions,this paper discusses their properties,and then provides a sufFcient and necessary condition for a finite property of the moments of first entrance time distributions of Markov chains in random environments by generating functions.Finally,the paper obtains relevant conclusions of the moments of first entrance time distributions.

Traditional herbal medicine in preventing recurrence after resection of small hepatocellular carcinoma: a multicenter randomized controlled trial

BACKGROUND： Disease recurrence is a main challenge in treatment of hepatocellular carcinoma （HCC）. There is no generally accepted method for preventing recurrence of HCC after resection. OBJECTIVE： To compare the efficacy of a traditional herbal medicine （THM） regimen and transarterial chemoembolization （TACE） in preventing recurrence in post-resection patients with small HCC. DESIGN, SETTING, PARTICIPANTS AND INTERVENTIONS： This is a multicenter, open- label, randomized, controlled study, which was undertaken in five centers of China. A total of 379 patients who met the eligibility criteria and underwent randomization were enrolled in this trial. One hundred and eighty-eight patients were assigned to the THM group and received Cinobufacini injection and Jiedu Granule, and the other 191 patients were assigned to the TACE group and received one single course of TACE. MAIN OUTCOME MEASURES： Primary outcome measures were the annual recurrence rate and the time to recurrence. Incidence of adverse events was regarded as the secondary outcome measure. RESULTS： Among the 364 patients who were included in the intention-to-treat analysis, 67 patients of the THM group and 87 of the TACE group had recurrence, with a hazard ratio of 0.695 （P = 0.048）. Median recurrence-free survival of the patients in the THM and TACE groups was 46.89 and 34.49 months, respectively. Recurrence rates at 1, 2 and 3 years were 17.7%, 33.0% and 43.5% for the THM group, and 28.8%, 42.5% and 54.0% for the TACE group, respectively （P = 0.026）. Multivariate analysis indicated that the THM regimen had a big advantage for prolonging the recurrence-free survival. Adverse events were mild and abnormality of laboratory indices of the two groups were similar. CONCLUSION： In comparison with TACE therapy, the THM regimen was associated with diminished risk of recurrence of small-sized HCC after resection, with comparable adverse events. TRIAL REGISTRTION IDENTIFIER： This trial was registered in the Chinese Clinical Trial Registry with the identifier ChiCTR-TRC-07000033.

ON MARKOV CHAINS IN SPACE-TIME RANDOM ENVIRONMENTS

In Section 1, the authors establish the models of two kinds of Markov chains in space-time random environments (MCSTRE and MCSTRE(+)) with abstract state space. In Section 2, the authors construct a MCSTRE and a MCSTRE(+) by an initial distribution Φ and a random Markov kernel (RMK) p(γ). In Section 3, the authors es-tablish several equivalence theorems on MCSTRE and MCSTRE(+). Finally, the authors give two very important examples of MCMSTRE, the random walk in spce-time random environment and the Markov br...

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 MULTITYPE CANONICAL MARKOV BRANCHING CHAINS IN RANDOM ENVIRONMENTS

The investigation for branching processes has a long history by their strong physics background, but only a few authors have investigated the branching processes in random environments. First of all, the author introduces the concepts of the multitype canonical Markov branching chain in random environment （CMBCRE） and multitype Markov branching chain in random environment （MBCRE） and proved that CMBCRE must be MBCRE, and any MBCRE must be equivalent to another CMBCRE in distribution. The main results of this article are the construction of CMBCRE and some of its probability properties.

Current preventive treatment for recurrence after curative hepatectomy for liver metastases of colorectal carcinoma: A literature review of randomized control trials

To review the preventive approaches for recurrence after curative resection of hepatic metastases from colorectal carcinoma, we have summarized all available publications reporting randomized control trials (RCTs) covered in PubMed. The treatment approaches presented above include adjuvant intrahepatic arterial infusion chemotherapy, systemic chemotherapy, neoadjuvant chemotherapy, and immunotherapy. Although no standard treatment has been established, several approaches present promising results, which are both effective and tolerable in post-hepatectomy patients. Intrahepatic arterial infusion chemotherapy should be regarded as effective and tolerable and it increases overall survival (OS) and disease free survival (DFS) of patients, while 5-fluorouracil-based systemic chemotherapy has not shown any significant survival benefit. Fortunately chemotherapy combined with hepatic arterial infusion and intravenous infusion has shown OS and DFS benefit in many researches. Few neoadjuvant RCT studies have been conducted to evaluate its effect on prolonging survivals although many retrospective studies and case reports are published in which unresectable colorectal liver metastases are downstaged and made resectable with neoadjuvant chemotherapy. Liver resection supplemented with immunotherapy is associated with optimal results; however, it is also questioned by others. In conclusion, several adjuvant approaches have been studied for their efficacy on recurrence after hepatectomy for liver metastases from colorectal cancer (CRC), but multi-centric RCT is still needed for further evaluation on their efficacy and systemic or local toxicities. In addition, new adjuvant treatment should be investigated to provide more effective and tolerable methods for the patients with resectable hepatic metastases from CRC.

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.

MOMENTS OF CONTINUOUS-STATE BRANCHING PROCESSES IN LéVY RANDOM ENVIRONMENTS

For continuous-state branching processes in Lévy random environments, the recursion of n-moments and the equivalent condition for the existence of general f-moments are established, where f is a positive continuous function satisfying some standard conditions.

The Laplace Functional and Moments for Markov Branching Chains in Random Environments

The concepts of random Markov matrix, Markov branching chain in randomenvironment (MBCRE) and Laplace functional of Markov branching chain in random environment (LFMBCRE)are introduced. The properties of LFMBCRE and the explicit formulas of momentsof MBCRE are given.

MOMENTS AND LARGE DEVIATIONS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION IN RANDOM ENVIRONMENTS

Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.

The Classification of States for Markov Chain in Random Environments and m-irreducible

First of all, we introduces the concept of m-irreducible of Markov chain in random environment. Then under the condition of m-irreducible, the relationship of recurrent and positive recurrent between two states is studied. We also give several conditions that can imply a state is recurrent and positive recurrent. And then the period of a state is discussed and we obtained that under the condition of m-irreducible, for any two states in x, they have the same period.

On the Extension of the Three-Term Recurrence Relation to Probabilities Distributions without Moments

In this paper, we extend the three-term recurrence relation for orthogonal polynomials associated with a probability distribution having a finite moment of all orders to a class of orthogonal functions associated with an infinitely divisible probability distribution µ?having a finite moments of order less or equal to four. An explicit expression of these functions will be given in term of the Lévy-Khintchine function of the measure?µ.

Recurrence and invariant measure of Markov chains in double-infinite random environments

The concepts of π-irreduciblity, recurrence and transience are introduced into the research field of Markov chains in random environments.That a π-irreducible chain must be either recurrent or transient is proved, a criterion is shown for recurrent Markov chains in double-infinite random environments, the existence of invariant measure of π-irreducible chains in double-infinite environments is discussed,and then Orey's open-questions are partially answered.

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on ? with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.

A HybridManufacturing ProcessMonitoringMethod Using Stacked Gated Recurrent Unit and Random Forest

This study proposed a new real-time manufacturing process monitoring method to monitor and detect process shifts in manufacturing operations.Since real-time production process monitoring is critical in today’s smart manufacturing.The more robust the monitoring model,the more reliable a process is to be under control.In the past,many researchers have developed real-time monitoring methods to detect process shifts early.However,thesemethods have limitations in detecting process shifts as quickly as possible and handling various data volumes and varieties.In this paper,a robust monitoring model combining Gated Recurrent Unit(GRU)and Random Forest(RF)with Real-Time Contrast(RTC)called GRU-RF-RTC was proposed to detect process shifts rapidly.The effectiveness of the proposed GRU-RF-RTC model is first evaluated using multivariate normal and nonnormal distribution datasets.Then,to prove the applicability of the proposed model in a realmanufacturing setting,the model was evaluated using real-world normal and non-normal problems.The results demonstrate that the proposed GRU-RF-RTC outperforms other methods in detecting process shifts quickly with the lowest average out-of-control run length(ARL1)in all synthesis and real-world problems under normal and non-normal cases.The experiment results on real-world problems highlight the significance of the proposed GRU-RF-RTC model in modern manufacturing process monitoring applications.The result reveals that the proposed method improves the shift detection capability by 42.14%in normal and 43.64%in gamma distribution problems.

Bisexual Galton-Watson Branching Processes in Random Environments

In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.