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The principal goal of this paper is to extend the classical problem of find the values of α ∈ C for which the mappings, eitherare univalent, whenever f belongs to some subclasses of univalent mappings in D, but in the case of harmonic mappings, considering the shear construction introduced by Clunie and Sheil-Small in [3].The second, and third, and fivth authors were partially supported by Fondecyt Grants #1190756 .

In this paper we study the controllability property of invariant control systems on Lie groups. In [1], the authors state: "If there exists a real function strictly increasing on the positive trajectories, then the system cannot be controllable". To develop this idea, the authors define the concept of symplectic vector via the co-adjoint representation. We are interested in finding algebraic conditions to determine the existence of symplectic vectors in nilpotent Lie algebras. In particular, we state a necessary and sufficient condition for controllability in the simply connected nilpotent case.

The aim of this study was to observe the penetration of an aqueous solution into the root canal dentin under sonic activation and ultrasonic activation. Materials and Method: This study consisted of experimental in vitro research. In order to achieve a closed system, the apex of 45 single-rooted teeth was sealed with wax. The step-back technique was manually performed using a K50 apical master file and 3 groups were organized according to the protocol of the final irrigant activation: Group I: non-activated Chinese ink for 30 seconds, Group II: Chinese ink sonically activated with EndoActivator for 30 seconds, and Group III: Chinese ink ultrasonically activated with Varios 350 equipment for 30 seconds. Teeth were sectioned longitudinally, and the samples obtained were observed under a stereomicroscope at 1X magnification in order to be photographed and scanned to calculate the penetration area using the Image J software. The tinted radicular area was evaluated in relation to the total area of the root dentin. The Tukey's post-hoc test and ANOVA were used for the statistical analysis (p<0.05). Results: Group I and II obtained 9.13% and 9.42% penetration respectively, while in group III the highest degree of dye infiltration was achieved (13.9%), being statistically significant (p<0.001).Conclusions: Ultrasonic activation produced a significantly higher penetration of the dye when compared to conventional activation and sonic activation.

This paper shows the results of an experiment applied to 170 students from two Chilean universities who solve a task about reading a graph of an affine function in an online assessment environment where the parameters (coefficients of the graphed affine function) are randomly defined from an ad-hoc algorithm, with automatic correction and automatic feedback. We distinguish two versions: one of them with integer coefficients and the other one with decimal coefficients in the affine function. We observed that the nature of the coefficients impacts the mathematical work used by the students, where we again focus on two of them: by direct estimation from the graph or by calculating the equation of the line. On the other hand, feedback oriented towards the "estimation" strategy influences the mathematical work used by the students, even though a non-negligible group persists in the "calculating" strategy, which is partly explained by the perception of each of the strategies.