Extension of Fermat’s last theorem in Minkowski natural spaces

Minkowski natural (N + 1)-dimensional spaces constitute the framework where the extension of Fermat’s last theorem is discussed. Based on empirical experience obtained via computational results, some hints about the extension of Fermat’s theorem from (2 + 1)-dimensional Minkowski spaces to (N + 1)-dimensional ones. Previous experience permits to conjecture that the theorem can be extended in (3 + 1) spaces, new results allow to do the same in (4 + 1) spaces, with an anomaly present here but difficult to find in higher dimensions. In (N + 1) dimensions with $$N > 4$$ N > 4 there appears an increased difficulty to find Fermat vectors, there is discussed a possible source of such an obstacle, separately of the combinatorial explosion associated to the generation of natural vectors of high dimension.

Dynamical C*-algebras and Kinetic Perturbations

The framework of dynamical C*-algebras for scalar fields in Minkowski space, based on local scattering operators, is extended to theories with locally perturbed kinetic terms. These terms encode information about the underlying spacetime metric, so the causality relations between the scattering operators have to be adjusted accordingly. It is shown that the extended algebra describes scalar quantum fields, propagating in locally deformed Minkowski spaces. Concrete representations of the abstract scattering operators, inducing this motion, are known to exist on Fock space. The proof that these representers also satisfy the generalized causality relations requires, however, novel arguments of a cohomological nature. They imply that Fock space representations of the extended dynamical C*-algebra exist, involving linear as well as kinetic and pointlike quadratic perturbations of the field.

Mappings that preserve helices in the n-dimensional Minkowski spaces

We introduce two types of mappings that preserve nonnull helices in Minkowski spaces. The first type constructs helices in the [Formula: see text]-dimensional Minkowski space from helices in the same Minkowski space. The second type constructs helices in the [Formula: see text]-dimensional Minkowski space from helices in the [Formula: see text]-dimensional Minkowski space. Furthermore, we study invariants of these mappings and present examples.