A UNIDADE IMAGINÁRIA: AS IMAGENS CONCEITUAIS EVOCADAS POR ALUNOS DE UM CURSO DE LICENCIATURA EM CIÊNCIAS / IMAGINARY UNIT: THE CONCEPT IMAGE EVOKED BY STUDENTS OF A BACHELOR'S DEGREE IN SCIENCE

O objetivo deste artigo foi realizar um levantamento da imagem conceitual de alunos de um curso de Licenciatura em Ciências com relação à unidade imaginária e verificar se e em medida esta imagem conceitual pode ser incrementada por meio de uma intervenção didática. A unidade imaginária é um obstáculo para a aprendizagem dos números complexos, o que demanda um levantamento da imagem conceitual evocada pelos alunos diante de atividades envolvendo os números complexos. Na parte metodológica foi elaborada e aplicada uma intervenção didática, adaptada a partir das três fases iniciais da Dialética Ferramenta-Objeto, de Douady (1986), a alunos de uma universidade pública do estado de São Paulo. Os resultados revelaram certo incremento na imagem conceitual dos alunos, promovida pela mudança do quadro aritmético para o quadro geométrico advindo pela aplicação da intervenção didática.

Time and Reversal in Birtwistle's Punch and Judy

We examine the occurrence of peripeteia in Harrison Birtwistle's 1967 opera Punch and Judy, as manifest in a reversal of cyclic time. Specifically, we extend a metaphorical association between the passage of cyclic time in the opera and discrete rotation in the complex plane generated by the imaginary unit i. Such a rotation moves alternately between the real and the imaginary axes, as scenes in the opera pass correspondingly through sacred and profane orientations. The instance of peripeteia results in a counter rotation, a dramaturgical inversion. To bring this reversal into the metaphor, we extend it from its situation in the complex plane to one in the space of Hamilton's quaternions, wherein such negation is obtained through the product of upper-level imaginary units. The scene that contains the reversal and that which consists of the opera's comic resolution epitomize the drama and occupy the highest level of dramatic structure.

Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).

On the q-deformed circular imaginary unit and hyperbolic imaginary unit: q-deformed rotation in two dimension and q-deformed special relativity in 1 + 1 dimension

In this paper, the [Formula: see text]-deformed circular unit and hyperbolic imaginary unit are studied. With a help of these units, the invariant [Formula: see text]-deformed length is defined. As applications, the [Formula: see text]-deformed rotation in two dimension and [Formula: see text]-deformed special relativity in [Formula: see text] dimension are also investigated.

SHADOW DARK MATTER AS A MANIFESTATION OF i ↔ -i SYMMETRY IN PRE-QUANTUM TRACE DYNAMICS

We propose an alternate version of the "shadow world" hypothesis for the origin of dark matter. Instead of postulating that the shadow world is a "mirror" world under parity or charge-parity reflection, we suggest that the existence of a shadow world arises from the fact that i or –i can be the imaginary unit in complex quantum mechanics. This mechanism is a natural consequence of "trace dynamics" pre-quantum theory, from which quantum mechanics over the complex number field emerges as a statistical mechanical approximation. Because the pre-quantum dynamics does not pick out a preferred imaginary unit, the emergent quantum dynamics contains two sectors, one based on i and the other on –i, with both sectors coupled to gravity.

Forming groups with 4 × 4 matrices

The three Pauli matrices are normally given [1] as the 2 × 2 matrices:where ‘i’ is the usual complex number imaginary unit.These matrices obey the relations a2 = I = b2 = c2(where I is the 2 × 2 identity matrix), as well as the anticommutation relations:Within the quantities ia,ib and ic,i is a scalar multiplier of the 2 × 2 Pauli matrices and, of course, commutes with each of a, b, c.

On Valuations, Places and Graded Rings Associated to ∗-Orderings

We study natural ∗-valuations, ∗-places and graded ∗-rings associated with ∗-ordered rings. We prove that the natural ∗-valuation is always quasi-Ore and is even quasi-commutative (i.e., the corresponding graded ∗-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded ∗-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimprič regarding ∗-orderability of quantum groups.