Victor Tapia

Conformal fourth-rank gravity, non-vanishing cosmological constant and anisotropy

We show that the recently proposed conformally invariant model for the gravitational field, based on fourth-rank geometry, is equivalent, in vacuum, to general relativity with a non-vanishing cosmological constant coupled to a Weyl vector field. Recent observational evidence suggests a non-vanishing cosmological constant and an essential anisotropy in the large-scale structure of the universe, supporting our model. We also couple the model to matter in the low-energy regime where the conformal invariance has been broken.

Universal field equations for metric-affine theories of gravity

We show that almost all metric-affine theories of gravity yield Einstein field equations with a non-null cosmological constant . Under certain circumstances and for any dimension, it is also possible to incorporate a Weyl vector field and therefore the presence of an anisotropy. The viability of these field equations is discussed in view of recent astrophysical observations.

Invariants and polynomial identities for higher rank matrices

We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors, and in terms of them we construct discriminants and the determinant as the discriminant of order d, where d is the dimension of the matrix. Analogues of the characteristic polynomials and the Cayley–Hamilton theorem are obtained therefrom for higher rank matrices.