AN UPPER BOUND OF THE DIRECTIONAL ENTROPY WITH RESPECT TO THE MARKOV MEASURES

2012 ◽  
Vol 22 (11) ◽  
pp. 1250263 ◽  
Author(s):  
HASAN AKIN
Keyword(s):  
Upper Bound ◽  
Stochastic Matrix ◽  
Natural Extension ◽  
Short Paper ◽  
Probability Vector ◽  
Compact Metric Space ◽  
One Dimensional ◽  
Markov Measure ◽  
Shift Map ◽  
Markov Measures

In this short paper, without considering the natural extension we study the directional entropy of a Z2-action Φ generated by an invertible one-dimensional linear cellular automaton [Formula: see text] and [Formula: see text], over the ring Zpk(with p a prime number and k ≥ 2), where gcd (p, λr) = 1 and p ∣ λifor all i ≠ r, and the shift map acting on the compact metric space [Formula: see text]. Without loss of generality, we consider k = 2. We prove that the directional entropy hv(Φ)(v = (s, q) ∈ R) of a Z2-action with respect to a Markov measure μπPover space [Formula: see text] defined by a stochastic matrix P = (aij) and a probability vector π = {π0, π1, …, πp2-1} is bounded above by [Formula: see text].

UPPER BOUND OF THE DIRECTIONAL ENTROPY OF A ℤ2-ACTION

2011 ◽  
Vol 22 (07) ◽  
pp. 711-718 ◽  
Author(s):  
HASAN AKIN
Keyword(s):  
Cellular Automaton ◽  
Metric Space ◽  
Upper Bound ◽  
Natural Extension ◽  
Studia Math ◽  
Short Paper ◽  
Compact Metric Space ◽  
One Dimensional ◽  
Shift Map ◽  
Bernoulli Measures

In this short paper, we study the measure directional entropy of a ℤ2-action generated by an invertible one-dimensional linear cellular automaton and the shift map acting on the compact metric space [Formula: see text]. We obtain an upper bound of the measure directional entropy of the ℤ2-action with respect to arbitrary Bernoulli measures without making use of the natural extension previously used in the paper [M. Courbage and B. Kaminski, Studia Math. 153(3), 285–295 (2002)].

scholarly journals Conservation Laws and Invariant Measures in Surjective Cellular Automata

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Jarkko Kari ◽  
Siamak Taati
Keyword(s):  
Cellular Automata ◽  
Conservation Laws ◽  
Arbitrary Dimension ◽  
Close Link ◽  
One Dimensional ◽  
Full Support ◽  
Markov Measure ◽  
Bernoulli Measure ◽  
International Audience ◽  
Markov Measures

International audience We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.

scholarly journals Chaotic Behavior of One-Dimensional Cellular Automata Rule 24

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zujie Bie ◽  
Qi Han ◽  
Chao Liu ◽  
Junjian Huang ◽  
Lepeng Song ◽  
...  
Keyword(s):  
Cellular Automata ◽  
Characteristic Function ◽  
Symbolic Dynamics ◽  
Bernoulli Shift ◽  
Chaotic Behavior ◽  
Class Ii ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
Shift Map

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 24, which is Bernoulliστ-shift rule and is member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of four rules, whether they possess chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 24 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 24 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Furthermore, we prove that four rules of global equivalenceε52of cellular automata are topologically conjugate. Then, we use diagrams to explain the attractor of rule 24, where characteristic function is used to describe the fact that all points fall into Bernoulli-shift map after two iterations under rule 24.

Error bounds for augmented truncation approximations of Markov chains via the perturbation method

2018 ◽  
Vol 50 (2) ◽  
pp. 645-669 ◽  
Author(s):  
Yuanyuan Liu ◽  
Wendi Li
Keyword(s):  
Markov Chains ◽  
Perturbation Method ◽  
Error Bounds ◽  
Continuous Time ◽  
Stochastic Matrix ◽  
Probability Vector ◽  
The Poisson Equation ◽  
Residual Matrix ◽  
Recurrent Markov Chain ◽  
Positive Recurrent

AbstractLetPbe the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vectorπT, and let(n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n+ 1) x (n+ 1) northwest corner truncation ofParbitrarily, with invariant probability vector(n)πT. We derive computableV-norm bounds on the error betweenπTand(n)πTin terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 asntends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.

Velocity Computation from Measures of Spatiotemporal Gradients at Multiple Orientations

Perception ◽  
1996 ◽  
Vol 25 (1_suppl) ◽  
pp. 77-77 ◽  
Author(s):  
A Johnston ◽  
P W McOwan
Keyword(s):  
Natural Extension ◽  
Image Brightness ◽  
Motion Patterns ◽  
One Dimensional ◽  
Aperture Problem ◽  
Motion Models ◽  
Temporal Domain ◽  
Moving Stimulus ◽  
Zero Values ◽  
Derivatives Of

Current models of speed and direction of motion which use measures of spatiotemporal gradients can suffer from ill-conditioning. This problem arises either because local measures of the derivatives of image brightness take zero values or because the motion equations cannot be solved for one-dimensional (1-D) signals in two-dimensional (2-D) images—the aperture problem. One way around this predicament is to select image points or introduce constants to deal with ill-conditioned calculations. Here we describe an analytic method that combines measures of speed in a range of directions to provide a well-conditioned measure of velocity at all points in the moving stimulus. This approach is a natural extension of a one-dimensional model which has been successful in predicting perceived motion in a variety of 1-D spatiotemporal motion patterns (Johnston, McOwan and Buxton 1992 Proceedings of the Royal Society of London, Series B250 297 – 306). Speed is computed with the use of biologically plausible filters that are derivatives of Gaussians in the spatial domain and log Gaussians in the temporal domain. Measures of speed and inverse speed are computed for a range of orientations consistent with the number of direction columns in MT/V5. The pattern of velocities measured over this set of orientations is then used to recover the speed and direction of motion of the stimulus. The model can correctly compute the velocity of moving 1-D patterns, such as gratings, patterns that prove a problem for many current 2-D motion models as they form degenerate cases, as well as the motion of rigid 2-D patterns.

scholarly journals Lyapunov optimization for non-generic one-dimensional expanding Markov maps

2019 ◽  
Vol 40 (9) ◽  
pp. 2571-2592 ◽  
Author(s):  
MAO SHINODA ◽  
HIROKI TAKAHASI
Keyword(s):  
Periodic Orbit ◽  
Finite Number ◽  
Dense Subset ◽  
Equilibrium States ◽  
Positive Entropy ◽  
Hölder Continuous ◽  
Perturbation Theorem ◽  
One Dimensional ◽  
Lyapunov Optimization ◽  
Shift Map

For a non-generic, yet dense subset of$C^{1}$expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new$C^{1}$perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

scholarly journals The Extrapolation-Accelerated Multilevel Aggregation Method in PageRank Computation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bing-Yuan Pu ◽  
Ting-Zhu Huang ◽  
Chun Wen ◽  
Yi-Qin Lin
Keyword(s):  
Stochastic Matrix ◽  
Numerical Results ◽  
Extrapolation Method ◽  
Stationary Probability ◽  
Probability Vector ◽  
Aggregation Method ◽  
Vector Extrapolation ◽  
Stationary Probability Vector

An accelerated multilevel aggregation method is presented for calculating the stationary probability vector of an irreducible stochastic matrix in PageRank computation, where the vector extrapolation method is its accelerator. We show how to periodically combine the extrapolation method together with the multilevel aggregation method on the finest level for speeding up the PageRank computation. Detailed numerical results are given to illustrate the behavior of this method, and comparisons with the typical methods are also made.

scholarly journals Close-packed structures with finite-range interaction: computational mechanics of layer pair interaction

2017 ◽  
Vol 73 (4) ◽  
pp. 357-369 ◽  
Author(s):  
Edwin Rodriguez-Horta ◽  
Ernesto Estevez-Rams ◽  
Reinhard Neder ◽  
Raimundo Lora-Serrano
Keyword(s):  
Phase Diagram ◽  
Computational Mechanics ◽  
Stochastic Matrix ◽  
Processing System ◽  
Pair Interaction ◽  
Finite Range ◽  
Range Interaction ◽  
One Dimensional ◽  
Finite State ◽  
General Method

The stacking problem is approached by computational mechanics, using an Ising next-nearest-neighbour model. Computational mechanics allows one to treat the stacking arrangement as an information processing system in the light of a symbol-generating process. A general method for solving the stochastic matrix of the random Gibbs field is presented and then applied to the problem at hand. The corresponding phase diagram is then discussed in terms of the underlying ∊-machine, or optimal finite-state machine. The occurrence of higher-order polytypes at the borders of the phase diagram is also analysed. The applicability of the model to real systems such as ZnS and cobalt is discussed. The method derived is directly generalizable to any one-dimensional model with finite-range interaction.

scholarly journals Reducing the 0-1 Knapsack Problem with a Single Continuous Variable to the Standard 0-1 Knapsack Problem

2012 ◽  
Vol 3 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Marcel Büther ◽  
Dirk Briskorn
Keyword(s):  
Knapsack Problem ◽  
Upper Bound ◽  
Original Problem ◽  
Natural Extension ◽  
Continuous Variable ◽  
Computational Effort ◽  
Objective Value ◽  
The One

The 0-1 knapsack problem with a single continuous variable (KPC) is a natural extension of the binary knapsack problem (KP), where the capacity is not any longer fixed but can be extended which is expressed by a continuous variable. This variable might be unbounded or restricted by a lower or upper bound, respectively. This paper concerns techniques in order to reduce several variants of KPC to KP which enables the authors to employ approaches for KP. The authors propose both, an equivalent reformulation and a heuristic one bringing along less computational effort. The authors show that the heuristic reformulation can be customized in order to provide solutions having an objective value arbitrarily close to the one of the original problem.

Geometric bounds on iterative approximations for nearly completely decomposable Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 521-529 ◽  
Author(s):  
Guy Louchard ◽  
Guy Latouche
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Perturbation Analysis ◽  
Stochastic Matrix ◽  
Stationary Probability ◽  
Probability Vector ◽  
Finite Markov Chain ◽  
Stationary Probability Vector

We consider a finite Markov chain with nearly-completely decomposable stochastic matrix. We determine bounds for the error, when the stationary probability vector is approximated via a perturbation analysis.

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