Inverse Passage Research Articles

The Parabola: Section of a Cone or Locus of Points of a Plane? Tips for Teaching of Geometry from Some Writings by Mydorge and Wallis

This article proposes a possible path devoted to upper secondary school and early university students, as well as training teachers, with the aim to build a conscious approach to the learning/teaching of the conics, which uses, for an educational purpose, the close relationship between conics as loci of points of a plane and conics as sections of a cone. In this path, we will refer to some elements taken from the history of mathematics relating to a particular conic: the parabola. These elements could help students to discover and realize the transition from a parabola considered as a curve in a plane to the same parabola considered on a cone of which it is a section, as well as the inverse passage, and to grasp the profound link between two presentations of the same geometric object. Both steps will be carried out through constructions made with the use of the GeoGebra dynamic geometry software. In addition, it will be highlighted how the construction of conics by points has allowed the creation of lenses and mirrors, which represents a practical application of geometry very relevant to physics and astronomy. Such a practical approach could help students to overcome the difficulty in understanding conics by making the argument less abstract. Moreover, this path could build up an environment in which teachers and students could explore some semiotic registers and their changes, through which Mathematics expresses itself. In the final part, an educational experiment of the path that was proposed will be shown to the students of the Master’s degree course of “mathematics education” at the Department of Mathematics and Computer Science of the University of Calabria. The results of this experiment are described in detail and seem to confirm that the twofold view of the parabola as a section of a cone and as locus of points of a plane helps the students in understanding its meaning in both theoretical and applicative fields.

Passage through resonance of a statically unbalanced rotor with an imperfect autobalancing device

Nonstationary modes of direct and inverse passage through resonance of unbalanced rotor equipped with an imperfectly mounted autobalancing device (ABB) are investigated. Two mathematical models are considered. The first one is the passage of the critical region with constant angular acceleration, and the second one is the rotor motion under the influence of a constant external torque. The study of the first model revealed that the presence of ABB significant negatively influences on the process of direct passage through resonance, and the maximum amplitude of whirling in this case can exceed the maximum amplitude in the steady state case. At the same time, during inverse passage through resonance, the influence of ABB is positive, and the maximum amplitude of whirling is much less than in the case of the direct passage through resonance. The investigation of the influence of viscous damping in the ABB revealed that too small damping influences negatively on the transient process dynamics, delaying the time of the rotor resonant vibrations until the moment of their stop. The study of the processes of acceleration and rundown of the rotor according to the second model confirms the main conclusions obtained from the first model.

Mesoscopic transport at finite frequencies.

We derive a set of equations that describe electrical transport in mesoscopic systems, at finite frequencies. The electron-electron interactions are treated at the level of a self-consistent field. It is shown that the static limit corresponds to frequencies \ensuremath{\omega} smaller than the inverse passage time ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}^{\mathrm{\ensuremath{-}}1}$ through the system. In this limit, the conductance is not affected by the complicated self-consistent field due to the current flow, in agreement with earlier results. However, for frequencies greater than ${\mathrm{\ensuremath{\tau}}}_{\mathit{L}}^{\mathrm{\ensuremath{-}}1}$ the transport equations for current, charge, and potential become strongly coupled and it is no longer possible to ignore the self-consistent field in the calculation of conductance.