Simply modified GKL density classifiers that reach consensus faster

The two-state Gacs-Kurdyumov-Levin (GKL) cellular automaton has been a staple model in the study of complex systems due to its ability to classify binary arrays of symbols according to their initial density. We show that a class of modified GKL models over extended neighborhoods, but still involving only three cells at a time, achieves comparable density classification performance but in some cases reach consensus more than twice as fast. Our results suggest the time to consensus (relative to the length of the CA) as a complementary measure of density classification performance.Comment: Short note, 3 pages, 1 table, 2 composite figures, 18 reference

Sensitivity to noise and ergodicity of an assembly line of cellular automata that classifies density

We investigate the sensitivity of the composite cellular automaton of H. Fuk\'{s} [Phys. Rev. E 55, R2081 (1997)] to noise and assess the density classification performance of the resulting probabilistic cellular automaton (PCA) numerically. We conclude that the composite PCA performs the density classification task reliably only up to very small levels of noise. In particular, it cannot outperform the noisy Gacs-Kurdyumov-Levin automaton, an imperfect classifier, for any level of noise. While the original composite CA is nonergodic, analyses of relaxation times indicate that its noisy version is an ergodic automaton, with the relaxation times decaying algebraically over an extended range of parameters with an exponent very close (possibly equal) to the mean-field value.Comment: Typeset in REVTeX 4.1, 5 pages, 5 figures, 2 tables, 1 appendix. Version v2 corresponds to the published version of the manuscrip

Electromagnetic surface wave propagation in a metallic wire and the Lambert WW function

We revisit the solution due to Sommerfeld of a problem in classical electrodynamics, namely, that of the propagation of an electromagnetic axially symmetric surface wave (a low-attenuation single TM$_{01}$ mode) in a cylindrical metallic wire, and his iterative method to solve the transcendental equation that appears in the determination of the propagation wave number from the boundary conditions. We present an elementary analysis of the convergence of Sommerfeld's iterative solution of the approximate problem and compare it with both the numerical solution of the exact transcendental equation and the solution of the approximate problem by means of the Lambert $W$ function.Comment: REVTeX double column, 9 pages, 3 figures, minor differences between v3 and published version; "Editor's Pick" for June 2019 edition of AJ

Numerical evidence against a conjecture on the cover time of planar graphs

We investigate a conjecture on the cover times of planar graphs by means of large Monte Carlo simulations. The conjecture states that the cover time $\tau(G_{N})$ of a planar graph $G_{N}$ of $N$ vertices and maximal degree $d$ is lower bounded by $\tau(G_{N}) \geq C_{d} N(\ln N)^2$ with $C_{d} = (d/4\pi) \tan (\pi/d)$, with equality holding for some geometries. We tested this conjecture on the regular honeycomb ($d=3$), regular square ($d=4$), regular elongated triangular ($d=5$), and regular triangular ($d=6$) lattices, as well as on the nonregular Union Jack lattice ($d_{\rm min}=4$, $d_{\rm max}=8$). Indeed, the Monte Carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices, with an interesting issue in the case of the Union Jack lattice. The data for the honeycomb lattice, however, violates the bound with the conjectured constant. The empirical probability distribution function of the cover time for the square lattice is also briefly presented, since very little is known about cover time probability distribution functions in general.Comment: Typeset in RevTEX 4.1, 4 pages, 3 figure