Right-Permutative Cellular Automata on Topological Markov Chains

In this paper we consider cellular automata $(\mathfrak{G},\Phi)$ with algebraic local rules and such that $\mathfrak{G}$ is a topological Markov chain which has a structure compatible to this local rule. We characterize such cellular automata and study the convergence of the Ces\`aro mean distribution of the iterates of any probability measure with complete connections and summable decay.Comment: 16 pages, 2 figure. A new version with improved redaction of Theorem 6.3(i)) to clearify its consequence

Bifurcation in Weighted Digraphs and Their Applications in Ecology

Merrill (2010) described bifurcation in a Markov chain by examining the eigenvalues of the associated probability matrix. The bifurcation point is that point where the dynamics of the system’s structure changes. He recognized a change in the dynamics of a sample path in a Markov chain when the nature of its eigenvalues changes. We built upon this work and found that not all changes in Markov chain dynamics are accompanied by change in the nature of the eigenvalues. And we introduce other measures that will recognize a change in dynamics. This was applied to solve the problem of evaluating the effectiveness of an ecological corridor. This was also used as a measure to examine bifurcation in metapopulation dynamics.Ovaskeinan and Hanski (2003) gave four definitions of patch value (contribution of a patch to metapopulation dynamics and persistence). One of them denotes a patch value as W_i, the contribution of patch i to colonization in the patch network. It is the left leading eigenvector of matrix B whose entries, b_ij=(p_j c_ij)/(∑▒〖p_k c_ik 〗). This is a Markov chain, where p_i is the probability that patch i is occupied, c_ij is the contribution that occupied patch j makes to the colonization rate of empty patch i. This matrix is in the family of coperiodic cospectral, which will be introduced in this dissertation. Therefore, it could be an effective tool in studying metapopulation dynamics. The goal is to evaluate the effectiveness of corridor introduction on species persistence, richness, and ecosystem dynamics. We focused our application on available data from the Osceola-Ocala black bears in Florida

Large deviations of a class of non-homogeneous Markov chains

Let Sigma = 1, 2, ..., r be a finite set of points. Let Pn = pn( i, j) : i, j [is in] Sigma be an r x r stochastic matrix for n ≥ 1, and p be a distribution on Sigma. Let now Pp= Pp({Pn }) be the (non-homogeneous) Markov measure on the sequence space Sinfinity with Borel sets B(Sinfinity) corresponding to initial distribution p and transition kernels Pn.;We now describe the class of non-homogeneous process focused upon in the article. These are the Markov chains where the transition kernels are asymptotically close to a fixed stochastic matrix. Let p be a distribution and P be a stochastic matrix on Sigma. Define the collection A=A p,P by A={Pp ({Pn}) :lim n→infinityPn =P}. The collection A can be thought of as perturbations of the stationary Markov chain run with P, and is a natural class in which to explore how non-homogeneity enters into the large deviation picture.;Let now f : Sigma → Rd be a d ≥ 1 dimensional function. Let also Pp({P n})[is in]A(p, P) be a perturbed non-homogeneous Markov measure. In terms of the coordinate process, define the additive sums Zn = Zn(f) for n ≥ 1 by Zn=1n i=1nf(Xi). The goal of this paper is to understand the large deviation behavior of the induced distributions of Zn : n ≥ 1 with respect to Pp({P n}) . An immediate question which comes to mind asks whether these large deviations differ from the deviations with respect to the stationary chain run with P. The general answer found in our work is yes and no, and as might be suspected depends on the rate of convergence Pn → P and the structure of the limit matrix P.;More specifically, when P is an irreducible matrix, it turns out that the large deviation of behavior of Zn under Pp({P n}) is exactly that under the stationary chain associated with P no matter the rate of convergence of Pn to P. Therefore, perhaps the most interesting case is when the target matrix P is reducible. In this situation, the large deviations of Zn depend both on the type of reducibilities of P and the convergence rate of Pn to P, and fall roughly into three distinct categories. Namely, when the convergence speed is very fast, the large deviation behavior is the same as for the stationary Markov chain run under P; when the speed is slow, one obtains a trivial large deviation behavior; and finally when the speed is intermediate, a non-trivial behavior is found which differs from stationarity. Moreover, these behaviors are characterized in terms of an explicit rate function which illustrates that among all paths which lead to a deviation those which minimize certain routing and resting costs are selected

Differential interferometry of QSO broad line regions I: improving the reverberation mapping model fits and black hole mass estimates

Reverberation mapping estimates the size and kinematics of broad line regions (BLR) in Quasars and type I AGNs. It yields size-luminosity relation, to make QSOs standard cosmological candles, and mass-luminosity relation to study the evolution of black holes and galaxies. The accuracy of these relations is limited by the unknown geometry of the BLR clouds distribution and velocities. We analyze the independent BLR structure constraints given by super-resolving differential interferometry. We developed a three-dimensional BLR model to compute all differential interferometry and reverberation mapping signals. We extrapolate realistic noises from our successful observations of the QSO 3C273 with AMBER on the VLTI. These signals and noises quantify the differential interferometry capacity to discriminate and measure BLR parameters including angular size, thickness, spatial distribution of clouds, local-to-global and radial-to-rotation velocity ratios, and finally central black hole mass and BLR distance. A Markov Chain Monte Carlo model-fit, of data simulated for various VLTI instruments, gives mass accuracies between 0.06 and 0.13 dex, to be compared to 0.44 dex for reverberation mapping mass-luminosity fits. We evaluate the number of QSOs accessible to measures with current (AMBER), upcoming (GRAVITY) and possible (OASIS with new generation fringe trackers) VLTI instruments. With available technology, the VLTI could resolve more than 60 BLRs, with a luminosity range larger than four decades, sufficient for a good calibration of RM mass-luminosity laws, from an analysis of the variation of BLR parameters with luminosity.Comment: 19 pages, 14 figures, accepted by MNRAS on December 5, 201

Reversibility in Queueing Models

In stochastic models for queues and their networks, random events evolve in time. A process for their backward evolution is referred to as a time reversed process. It is often greatly helpful to view a stochastic model from two different time directions. In particular, if some property is unchanged under time reversal, we may better understand that property. A concept of reversibility is invented for this invariance. Local balance for a stationary Markov chain has been used for a weaker version of the reversibility. However, it is still too strong for queueing applications. We are concerned with a continuous time Markov chain, but dose not assume it has the stationary distribution. We define reversibility in structure as an invariant property of a family of the set of models under certain operation. The member of this set is a pair of transition rate function and its supporting measure, and each set represents dynamics of queueing systems such as arrivals and departures. We use a permutation {\Gamma} of the family menmbers, that is, the sets themselves, to describe the change of the dynamics under time reversal. This reversibility is is called {\Gamma}-reversibility in structure. To apply these definitions, we introduce new classes of models, called reacting systems and self-reacting systems. Using those definitions and models, we give a unified view for queues and their networks which have reversibility in structure, and show how their stationary distributions can be obtained. They include symmetric service, batch movements and state dependent routing.Comment: Submitted for publicatio