The F5 Algorithm in Buchberger's Style

The famous F5 algorithm for computing \gr basis was presented by Faug\`ere in 2002. The original version of F5 is given in programming codes, so it is a bit difficult to understand. In this paper, the F5 algorithm is simplified as F5B in a Buchberger's style such that it is easy to understand and implement. In order to describe F5B, we introduce F5-reduction, which keeps the signature of labeled polynomials unchanged after reduction. The equivalence between F5 and F5B is also shown. At last, some versions of the F5 algorithm are illustrated

Double symbolic joint entropy in nonlinear dynamic complexity analysis

Symbolizations, the base of symbolic dynamic analysis, are classified as global static and local dynamic approaches which are combined by joint entropy in our works for nonlinear dynamic complexity analysis. Two global static methods, symbolic transformations of Wessel N. symbolic entropy and base-scale entropy, and two local ones, namely symbolizations of permutation and differential entropy, constitute four double symbolic joint entropies that have accurate complexity detections in chaotic models, logistic and Henon map series. In nonlinear dynamical analysis of different kinds of heart rate variability, heartbeats of healthy young have higher complexity than those of the healthy elderly, and congestive heart failure (CHF) patients are lowest in heartbeats' joint entropy values. Each individual symbolic entropy is improved by double symbolic joint entropy among which the combination of base-scale and differential symbolizations have best complexity analysis. Test results prove that double symbolic joint entropy is feasible in nonlinear dynamic complexity analysis.Comment: 7 pages, 4 figure