The twistor kinematic-energy model of the space-time and the kinematic-energy tensor as the energy-matter tensor in relativity are considered to demonstrate the possible behavior of gravity as gravitational waves that derive of mass-energy source in the space-time and whose contorted image is the spectrum of the torsion field acting in the space-time. The energy of this field is the energy of their second curvature. Likewise, it is assumed that the curvature energy as spectral curvature in the twistor kinematic frame is the curvature in twistor-spinor framework, which is the mean result of this work. This demonstrates the lawfulness of the torsion as the indicium of the gravitational waves in the space-time. A censorship to detect gravitational waves in the space-time is designed using the curvature energy.

Into the study of quasi-relaxation, in the past researches it has been concluded that the condition of meta-stability in the metallic specimen is given by the plasticity explained by the plastic energy in the process of the quasi-relaxation. It is calculated through quasi-relaxation functional of this energy to obtain a spectra in the space D (σ -ε; t), that induces the existence of functions φ(t), and Ψ(t), related with the fundamental curves of quasi-relaxation given by σ(t), with their poles in, which is got in the maximum of stress given by σ 0 = σ 1 . Also the tensor of plastic deformation that represents the plastic load during the application of specimen machine, cannot be obtained without poles in the space D(σ; t), corresponding the curves calculated into the space D(σ -ε; t), by curves that in the kinetic process of quasi-relaxation are represented by experimental curves in coordinates log σ -t. This situation cannot be eluded, since in this phenomena exist dislocations that go conform fatigue in the nano-crystalline structure of metals. From this point of view, is necessary to obtain a spectral study related to the energy using functions that permits the modeling and compute the states of quasi-relaxation included in the poles in the deformation problem to complete the solutions in the space D(σ -ε; t), and try a new method of solution of the differential equations of the quasi-relaxation analysis. In a nearly future development, the information obtained by this spectral study (by our integral transforms), will be able to give place to the programming through the spectral encoding of the materials in the meta-stability state, which is propitious to a nano-technological transformation of materials, concrete case, some metals.

Abstract:Through the application of the Yoneda embedding in the context of the ∞-categories is obtained a classification of functors with their corresponding extended functors in the geometrical Langlands program. Also is obtained a functor formula that can be considered in the extending of functors to obtaining of generalized Verma modules. In this isomorphism formula are considered the Verma modules as classifying spaces of these functors.