Nuno Franco

Symbolic dynamics and renormalization of non-autonomous k periodic dynamical systems

The purpose of this paper was to introduce the symbolic formalism based on kneading theory, which allows us to study the renormalization of non-autonomous periodic dynamical systems.

Families of Hyperbolic Lorenz Knots

Lorenz knots and links have been an area of research for over thirty years. Their study combines several fields of mathematics, including topology, geometry, and dynamical systems. The introduction of the Lorenz template permitted a dimension reduction, the definition of the one-dimensional Lorenz map, and the use of symbolic dynamics to code its orbits. Meanwhile, Thurston’s geometrization theorem implies that all Lorenz knots can be classified as torus knots, satellites, or hyperbolic knots. Based on a list of hyperbolic knots presented by Birman and Kofman, we define three families of Lorenz knots, compute their hyperbolic volume using the topology and geometry software SnapPy and use the results to present conjectures concerning their hyperbolicity.

Conjugacy Problem for Subgroups with Applications to Artin Groups and Braid Type Group

Let G be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in G of any element of G. Let H be a subgroup of G. We give some conditions on H, under which we provide a solution to the conjugacy problem in H. We apply our results to some Artin groups and braid type groups. In particular, we give solutions to the explicit conjugacy problem in the Artin groups of type and .

Zeta-functions of renormalizable sub-Lorenz templates

Abstract We describe the Williams zeta-functions and the twist zeta-functions of sub-Lorenz templates generated by renormalizable Lorenz maps, in terms of the corresponding zeta-functions of the sub-Lorenz templates generated by the renormalized map and by the map that determines the renormalization type.

Effectiveness of In-office Hydrogen Peroxide With Two Different Protocols: A Two-center Randomized Clinical Trial

OBJECTIVESnThe aim of this study was to compare the bleaching efficacy and tooth sensitivity (TS) of a 38% hydrogen peroxide bleaching agent used for in-office bleaching, applied under different time protocols: a 40-minute application or two 20-minute applications.nnnMETHODS AND MATERIALSnForty-four patients from Brazil and Colombia, with right superior canines darker than C2, were selected for this multicenter, single-blind, randomized trial. The teeth were bleached in two sessions, with a one-week interval between them, in a split-mouth design. The bleaching agent was applied in two 20-minute (2×20) applications or one 40-minute (1×40) application in each session according to the manufacturers instructions. The color changes were evaluated by using subjective (Vita Classical and Vita Bleachedguide) and objective (Easyshade Spectrophotometer) methods at baseline and 30 days after the second session. Tooth sensitivity was recorded up to 48 hours with a 0-10 visual analog scale. Also, the pH values during the application of bleaching were recorded. Color change in shade guide units and ΔE were analyzed by using the Student t-test (α=0.05). The absolute risk and intensity of TS were evaluated with the McNemar test, the Wilcoxon signed-rank test, and the Friedman test, respectively (α= 0.05).nnnRESULTSnSignificant whitening was observed in both groups after 30 days of clinical evaluation. The use of a 40-minute application did not significantly influence the absolute risk of TS (68%, 95% confidence interval [CI] = 53-80) as well as the intensity of TS compared with the acid bleaching gel (absolute risk of 82%, 95% CI = 68-91). The pH values did not differ significantly between groups and at the different assessment periods ( p=0.42).nnnCONCLUSIONnThe use of a 40-minute in-office bleaching agent gel application produced the same whitening degree and TS that the two 20-minute bleaching agent applications did. The former preferably should be applied because one 40-minute application does not require gel refreshing.

Farey neighbors and hyperbolic Lorenz knots

Based on symbolic dynamics of Lorenz maps, we prove that, pro- vided one conjecture due to Morton is true, then Lorenz knots asso- ciated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type (X,Y) * S, where the sequences X and Y are Farey neighbors verifying some conditions, are hyperbolic.

Computation of Genus and Braid Index for Renormalizable Lorenz Links

We present some recent results concerning the structure of renormalizable Lorenz links. Then we use these results to derive formulae for the computation of knot and link invariants. Finally we analyze the complexity of the algorithms obtained and compare it with the complexity of the algorithms derived from the definitions, obtaining a reduction from exponential complexity, in the classic algorithms, to linear in our algorithms.