Application of the form invariance transformations of the scalar cosmological model in inflation theory

In this work, it is shown that the equations of motion of the scalar field for spatially flat, homogeneous, and isotropic space-time Friedmann-Robertson-Walker have a form-invariance symmetry, which is arising from the form invariance transformation. Form invariance transformation is defined by linear function ρ = n 2 ρ in general case. It is shown the method of getting potential and the scalar field for the power law scale factor. The initial model is always stable at exponent of the scale factor α > 1, but stability of the transformation model depends on index n. Slow roll parameters and spectral induces is obtained and at large α they agree with Planck observation data.

Power Invariance Transformation in Power Systems

Many textbooks describe a balanced three-phase circuit by a single-phase equivalent representation. Confusion may arise amongst students regarding per-unit values of line-to-line voltages and phase voltages and, therefore, about the magnitudes of currents and powers. This paper proposes that students must first be taught symmetrical components based on power invariance transformation. A balanced three-phase circuit is to be described only in terms of positive sequence components. In the authors' experience, students understand this approach better and make fewer errors in per-unit calculation than when they use the single-phase equivalent representation.