Abstract:
We prove some limit properties of the harmonic mean of a random transition probability for finite Markov chains indexed by a homogeneous tree in a nonhomogeneous Markovian environment with finite state space. In particular, we extend the method to study the tree-indexed processes in deterministic environments to the case of random enviroments. 1. Introduction A tree is a graph which is connected and doesn't contain any circuits. Given any two vertices , let be the unique path connecting and . Define the graph distance to be the number of edges contained in the path . Let be an infinite tree with root . The set of all vertices with distance from the root is called the th generation of , which is denoted by . We denote by the union of the first generations of . For each vertex , there is a unique path from to and for the number of edges on this path. We denote the first predecessor of by . The degree of a vertex is defined to be the number of neighbors of it. If every vertex of the tree has degree , we say it is Cayley’s tree, which is denoted by . Thus, the root vertex has neighbors in the first generation and every other vertex has neighbors in the next generation. For any two vertices and of tree , write if is on the unique path from the root to . We denote by the farthest vertex from satisfying and . We use the notation and denote by the number of vertices of . In the following, we always let denote the Cayley tree . A tree-indexed Markov chain is the particular case of a Markov random field on a tree. Kemeny et al. [1] and Spitzer [2] introduced two special finite tree-indexed Markov chains with finite transition matrix which is assumed to be positive and reversible to its stationary distribution, and these tree-indexed Markov chains ensure that the cylinder probabilities are independent of the direction we travel along a path. In this paper, we omit such assumption and adopt another version of the definition of tree-indexed Markov chains which is put forward by Benjamini and Peres [3]. Yang and Ye[4] extended it to the case of nonhomogeneous Markov chains indexed by infinite Cayley’s tree and we restate it here as follows. Definition 1 (T-indexed nonhomogeneous Markov chains (see [4])). Let be an infinite Cayley tree, a finite state space, and a stochastic process defined on probability space , which takes values in the finite set . Let be a distribution on and a transition probability matrix on . If, for any vertex , then will be called -valued nonhomogeneous Markov chains indexed by infinite Cayley’s tree with initial distribution (1) and

Abstract:
We prove a central limit theorem for th-order nonhomogeneous Markov information source by using the martingale central limit theorem under the condition of convergence of transition probability matrices for nonhomogeneous Markov chain in Cesàro sense. 1. Introduction Let be an arbitrary information source taking values on alphabet set with the joint distribution for . If is an th-order nonhomogeneous Markov information source, then, for , Denote where and are called the m-dimensional initial distribution and the th-order transition probabilities, respectively. Moreover, are called the th-order transition probability matrices. In this case, There are many of practical information sources, such as language and image information, which are often th-order Markov information sources and always nonhomogeneous. So it is very important to study the limit properties for the th-order nonhomogeneous Markov information sources in information theory. Yang and Liu [1] proved the strong law of large numbers and the asymptotic equipartition property with convergence in the sense of a.s. the th-order nonhomogeneous Markov information sources. But the problem about the central limit theorem for the th-order nonhomogeneous Markov information sources is still open. The central limit theorem (CLT) for additive functionals of stationary, ergodic Markov information source has been studied intensively during the last decades [2–9]. Nearly fifty years ago, Dobrushin [10, 11] proved an important central limit theorem for nonhomogeneous Markov information resource in discrete time. After Dobrushin's work, some refinements and extensions of his central limit theorem, some of which are under more stringent assumptions, were proved by Statuljavicius [12] and Sarymsakov [13]. Based on Dobrushin's work, Sethuraman and Varadhan [14] gave shorter and different proof elucidating more the assumptions by using martingale approximation. Those works only consider the case about th-order nonhomogeneous Markov chain. In this paper, we come to study the central limit theorem for th-order nonhomogeneous Markov information sources in Cesàro sense. Let be an th-order nonhomogeneous Markov information source which is taking values in state space with initial distribution of (3) and mth order transition probability matrices (5). Denote We also denote the realizations of by . We denote the th-order transition matrix at step by where . For an arbitrary stochastic square matrix whose elements are , we will set the ergodic -coefficient equal to where . Now we extend this idea to the th-order stochastic

Abstract:
Taking cultural knowledge tests as the case study, this research carries out a series of empirical investigations to verify the moderating effects of item order arranged by difficulty on the relationship between test anxiety and test performance. Groups classified according to test anxiety take tests with two major types of item order: item order arranged according to item bank calibrated item difficulty and item order adjusted according to individual examinee’s perceived item difficulty. The means of those test results are compared between groups to see whether the differences are significant. The investigations obtain the following findings: the higher the test taker’s level of test anxiety, the higher significance of the moderating effects and vice versa; item order adjusted according to individual examinee’s perceived item difficulty may have a more significant moderating effect than item order arranged according to item bank calibrated item difficulty has.

Abstract:
A modified stochastic ratio-dependent Leslie-Gower predator-prey model isformulated and analyzed. For the deterministic model, we focus on the existence of equilibria,local, and global stability; for the stochastic model, by applying Itô formula and constructingLyapunov functions, some qualitative properties are given, such as the existence of global positivesolutions, stochastic boundedness, and the global asymptotic stability. Based on theseresults, we perform a series of numerical simulations and make a comparative analysis of thestability of the model system within deterministic and stochastic environments.

Abstract:
We extend the classical SIRS epidemic model incorporating media coverage from a deterministic framework to a stochastic differential equation (SDE) and focus on how environmental fluctuations of the contact coefficient affect the extinction of the disease. We give the conditions of existence of unique positive solution and the stochastic extinction of the SDE model and discuss the exponential -stability and global stability of the SDE model. One of the most interesting findings is that if the intensity of noise is large, then the disease is prone to extinction, which can provide us with some useful control strategies to regulate disease dynamics. 1. Introduction Recent years, a number of mathematical models have been formulated to describe the impact of media coverage on the dynamics of infectious diseases [1–10]. Mass media (television, radio, newspapers, billboards, and booklets) has been used as a way of delivering preventive health messages as it has the potential to influence people’s behavior and deter them from risky behavior or from taking precautionary measures in relation to a disease outbreak [7, 11, 12]. Hence, media coverage has an enormous impact on the spread and control of infectious diseases [2, 3, 9]. On the other hand, for human disease, the nature of epidemic growth and spread is inherently random due to the unpredictability of person-to-person contacts [13], and population is subject to a continuous spectrum of disturbances [14, 15]. In epidemic dynamics, stochastic differential equation (SDE) models could be the more appropriate way of modeling epidemics in many circumstances and many realistic stochastic epidemic models can be derived based on their deterministic formulations [16–28]. In [10], Liu investigated an SIRS epidemic model incorporating media coverage with random perturbation. He assumed that stochastic perturbations were of white noise type, which were directly proportional to distance susceptible , infectious , and recover from values of endemic equilibrium point , influence on the , , , respectively. In fact, besides the possible equilibrium approach in [10], there are different possible approaches to introduce random effects in the epidemic models affected by environmental white noise from biological significance and mathematical perspective [28–30]. Some scholars [17, 28, 30, 31] demonstrated that one or more system parameter(s) can be perturbed stochastically with white noise term to derive environmentally perturbed system. In [10], the author proved that the endemic equilibrium of the stochastic model is

Abstract:
Autism spectrum disorder is a neuro-developmental disorder characterized by abnormalities of neural synchronization. In this study, functional near infrared spectroscopy (fNIRS) is used to study the difference in functional connectivity in left and right inferior frontal cortices (IFC) and temporal cortices (TC) between autistic and typically developing children between 8-11 years of age. 10 autistic children and 10 typical ones were recruited in our study for 8-min resting state measurement. Results show that the overall interhemispheric correlation of HbO was significantly lower in autistic children than in the controls. In particular, reduced connectivity was found to be most significant in TC area of autism. Autistic children lose the symmetry in the patterns of correlation maps. These results suggest the feasibility of using the fNIRS method to assess abnormal functional connectivity of the autistic brain and its potential application in autism diagnosis.

Abstract:
Introduction: Osteoarthritis is the most common form of arthritis. It is a slowly progressive joint disease typically seen in middle-age to elderly people. Intra-articular injection of hyaluronic acid is a well-documented treatment for knee osteoarthritis. Celebrex ？ (celecoxib) is a novel nonsteroidal anti-inflammatory drug, which could help to reduce inflammation and to reduce pain. The aim of this study was to evaluate the effects of intra-articular injection of celecoxib in a rabbit osteoarthritis model. Methods: Thirty New Zealand white rabbits underwent unilateral knee joint surgery using the Hulth technique. Six weeks post-surgery, the animals were randomly divided into three groups, and each group was respectively given weekly intra-articular injections with Celebrex ？, hyaluronic acid and saline. On the sixth week, the results were assessed in rabbit models by gross observation, histological evaluation, and expression of IL-1β, TNF-α, MMP-3. Results: In the group given Celebrex ？ and hyaluronic acid, the pathological changes in the rabbit articular cartilage improved significantly, much more than in the saline group. The statistically significant suppression of IL-1β, TNF-α, MMP-3 was shown in the Celebrex group. No significant differences were detected between two treatment groups. Conclusions: Intra-articular injection of celecoxib is beneficial for knee osteoarthritis. It might repair and protect early osteoarthritis cartilage by delaying cartilage degeneration and impairing the function of inflammatory mediators, therefore, intra-articular injection of celecoxib can be used as an alternative to the current treatment of osteoarthritis.

Abstract:
Ghrelin, a 28-amino acid peptide, is mainly secreted by the stomach. Ghrelin has been shown to have neuroprotective effects. However, whether ghrelin protects the spinal cord from ischemia/reperfusion (I/R) injury is unknown. To investigate this, 60 rats were randomly divided into three different groups: the sham group ( n = 20), the vehicle group ( n = 20), and the Ghrelin group (100 μg/kg, n = 20). Rats were sacrificed 12, 24, 48 and 72 h after ischemia. After the evaluation of neurologic function (48 h), the spinal cords were immediately removed for the determination of myeloperoxidase (MPO) activity (12–72 h). Apoptosis was quantitatively measured using the terminal transferase UTP nick end-labeling (TUNEL) method (24 h). The expression of bax and bcl-2 were evaluated by Western blot analysis (1 h), and GHSR-1a mRNA expression was detected using reverse transcriptase polymerase chain reaction (24 h). The neurological motor function was evaluated by ‘Tarlov’s score’. The neurologic outcomes in the ghrelin-group were significantly better than those in the vehicle group ( p < 0.05). Serum tumor necrosis factor (TNF-α) levels were assessed in the peripheral venous blood. Ghrelin decreased the serum TNF-α levels and ameliorated the down regulation of spinal cord MPO activity. The expression of ghrelin receptors (GHSR-1a) in the rat spinal cord was decreased by I/R injury and increased by ghrelin. Ghrelin reduced the TUNEL-positive rate. Greater bcl-2, HSP27, HSP70, and attenuated bax expression were observed in the ghrelin-treated rats. Our results suggest that ghrelin administration may inhibit spinal I/R injury. Moreover, the improvement of neurologic function in rats was increased after the ghrelin treatment.