79 research outputs found
On a zero speed sensitive cellular automaton
Using an unusual, yet natural invariant measure we show that there exists a
sensitive cellular automaton whose perturbations propagate at asymptotically
null speed for almost all configurations. More specifically, we prove that
Lyapunov Exponents measuring pointwise or average linear speeds of the faster
perturbations are equal to zero. We show that this implies the nullity of the
measurable entropy. The measure m we consider gives the m-expansiveness
property to the automaton. It is constructed with respect to a factor dynamical
system based on simple "counter dynamics". As a counterpart, we prove that in
the case of positively expansive automata, the perturbations move at positive
linear speed over all the configurations
Cellular automata and Lyapunov exponents
In this article we give a new definition of some analog of Lyapunov exponents
for cellular automata . Then for a shift ergodic and cellular automaton
invariant probability measure we establish an inequality between the entropy of
the automaton, the entropy of the shift and the Lyapunov exponent
Space-time directional Lyapunov exponents for cellular automata
Space-time directional Lyapunov exponents are introduced. They describe the
maximal velocity of propagation to the right or to the left of fronts of
perturbations in a frame moving with a given velocity. The continuity of these
exponents as function of the velocity and an inequality relating them to the
directional entropy is proved
Invariant Measures and Decay of Correlations for a Class of Ergodic Probabilistic Cellular Automata
We give new sufficient ergodicity conditions for two-state probabilistic
cellular automata (PCA) of any dimension and any radius. The proof of this
result is based on an extended version of the duality concept. Under these
assumptions, in the one dimensional case, we study some properties of the
unique invariant measure and show that it is shift-mixing. Also, the decay of
correlation is studied in detail. In this sense, the extended concept of
duality gives exponential decay of correlation and allows to compute
explicitily all the constants involved
On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems
For a given subspace, the Rayleigh-Ritz method projects the large quadratic
eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar
to the Rayleigh-Ritz method for the linear eigenvalue problem, the
Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP
with respect to the projection subspace. We analyze the convergence of the
method when the angle between the subspace and the desired eigenvector
converges to zero. We prove that there is a Ritz value that converges to the
desired eigenvalue unconditionally but the Ritz vector converges conditionally
and may fail to converge. To remedy the drawback of possible non-convergence of
the Ritz vector, we propose a refined Ritz vector that is mathematically
different from the Ritz vector and is proved to converge unconditionally. We
construct examples to illustrate our theory.Comment: 20 page
Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems
The final publication is available at Springer via http://dx.doi.org/ 10.1007/s10543-016-0601-5.We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs of with M, C, K symmetric matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization , where A, B are symmetric matrices but the pair (A, B) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices M, C, K as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.This work was supported by the Spanish Ministry of Economy and Competitiveness under Grant TIN2013-41049-P. Carmen Campos was supported by the Spanish Ministry of Education, Culture and Sport through an FPU Grant with reference AP2012-0608.Campos, C.; Román Moltó, JE. (2016). Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems. BIT Numerical Mathematics. 56(4):1213-1236. https://doi.org/10.1007/s10543-016-0601-5S12131236564Bai, Z., Su, Y.: SOAR: a second-order Arnoldi method for the solution of the quadratic eigenvalue problem. SIAM J. Matrix Anal. 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Towards generalized measures grasping CA dynamics
In this paper we conceive Lyapunov exponents, measuring the rate of separation between two initially close configurations, and Jacobians, expressing the sensitivity of a CA's transition function to its inputs, for cellular automata (CA) based upon irregular tessellations of the n-dimensional Euclidean space. Further, we establish a relationship between both that enables us to derive a mean-field approximation of the upper bound of an irregular CA's maximum Lyapunov exponent. The soundness and usability of these measures is illustrated for a family of 2-state irregular totalistic CA
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