Abstract:
The purpose of this short notice is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of partial differential equations and Lie pseudo groups to engineering (elasticity theory, electromagnetism, coupling phenomena) and mathematical (gauge theory, general relativity) along the following scheme: 1) Lie groups of transformations may be considered as Lie pseudo groups of transformations but no action type method can be used as parameters never appear any longer. 2) The work of Cartan is superseded by the use of the canonical Spencer sequence while the work of Vessiot is superseded by the use of the canonical Janet sequence but the link between these two sequences and thus these two works is not known today. 3)Using duality theory, the formal adjoint of the Spencer operator for the conformal group of transformations of space-time provides the Cosserat equations, the Maxwell equations and the Weyl equations on equal footing but such a result leads to deep contradictions. Accordingly, the results thus obtained prove that the foundations of engineering and mathematical physics must be revisited within the framework of jet theory though striking it may look like for established theories.

Abstract:
The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli matrices or covariant differentiation. We think of our 3-dimensional space as an elastic continuum and assume that material points of this continuum can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points of the space continuum are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We choose a particular potential energy which is conformally invariant and then incorporate time into our action in the standard Newtonian way, by subtracting kinetic energy. The main result of our paper is the theorem stating that in the stationary setting our model is equivalent to a pair of Weyl equations. The crucial element of the proof is the observation that our Lagrangian admits a factorization.

Abstract:
The theory of the Clausius' virial maximum to explain the Fundamental Plane (FP) proposed by Secco (2000, 2001,2005) is based on the existence of a maximum in the Clausius' Virial (CV) potential energy of a early type galaxy (ETG) stellar component when it is completely embedded inside a dark matter (DM) halo. At the first order approximation the theory was developed by modeling the two-components with two cored power-law density profiles. An higher level of approximation is now taken into account by developing the same theory when the stellar component is modeled by a King-model with a cut-off. Even if the DM halo density remains a cored power law the inner component is now more realistic for the ETGs. The new formulation allows us to understand more deeply what is the dynamical reason of the FP tilt and in general how the CV theory may really be the engine to produce the FP main features. The degeneracy of FP in respect to the initial density perturbation spectrum may be now full understood in a CDM cosmological scenario. A possible way to compare the FPs predicted by the theory with those obtained by observations is also exemplified.

Abstract:
In a gravitational virialized bound system built up of two components, one of which is embedded in the other, the Clausius' virial energy of one subcomponent is not, in general, equal to its total potential energy, as occurs in a single system without external forces. This is the main reason for the presence, in the case of two non-coinciding concentric spheroidal subsystems, of a minimum (in absolute value) in the Clausius' virial of the inner component B, when it assumes a special configuration characterized by a value of its semi-major axis we have named "tidal radius". The physical meaning, connected with its appearance, is to introduce a scale length on the gravity field of the inner subsystem, which is induced from the outer one. Its relevance in the galaxy dynamics has been stressed by demonstrating that some of the main features of the Fundamental Plane may follow as consequence of its existence. More physical insight into the dynamics of a two component system may be got by looking at the location of this scale length inside the plots of the potential energies of each subsystem and of the whole system and by also taking into account the trend of the anti-symmetric residual-energy, that is the difference between the tidal and the interaction-energy of each component. Some thermodynamical arguments related to the inner component are also added to prove as special is the "tidal radius configuration". Moreover the role of the divergency at the center of the two subsystems in obtaining this scale length is considered. For the sake of simplicity the analysis has been performed in the case of a frozen external component even if this constraint does not appear to be too relevant in order to preserve the main results.

Abstract:
I discuss generalized Maxwell and Weyl equations. They may lead to dynamics which are different from those accepted at the present time. For instance, the photon may have non-transverse components and the neutrino may be not in the chiral states.

Abstract:
Topological mechanical structures exhibit robust properties protected by topological invariants. In this letter, we study a family of deformed square lattices that display topologically protected zero-energy bulk modes analogous to the massless fermion modes of Weyl semimetals. Our findings apply to sufficiently complex lattices satisfying the Maxwell criterion of equal numbers of constraints and degrees of freedom. We demonstrate that such systems exhibit pairs of oppositely charged Weyl points, corresponding to zero-frequency bulk modes, that can appear at the origin of the Brillouin zone and move away to the zone edge (or return to the origin) where they annihilate. We prove that the existence of these Weyl points leads to a wavenumber-dependent count of topological mechanical states at free surfaces and domain walls.

Abstract:
We show that for a metastable system there exists a theoretical possibility of a violation of the Clausius inequality without a violation of the second law. Possibilities of experimental detection of this hypothetical violation are pointed out.

Abstract:
The classical unified theory of Weyl is revisited. The possibility of stable extended electron model in the Einstein-Weyl space is suggested.

Abstract:
A new family of nonlinear partial differential equations is presented. They represent a generalization of the hyperbolic Ernst equations for an Einstein-Mawxell-Weyl field in general relativity. A B\"acklund transformation for the system of equations under consideration is given, and their direct relation to the complete Boussinesq hierarchy of soliton equations is illustrated.

Abstract:
Static solutions of the electro-gravitational field equations exhibiting a functional relationship between the electric and gravitational potentials are studied. General results for these metrics are presented which extend previous work of Majumdar. In particular, it is shown that for any solution of the field equations exhibiting such a Weyl-type relationship, there exists a relationship between the matter density, the electric field density and the charge density. It is also found that the Majumdar condition can hold for a bounded perfect fluid only if the matter pressure vanishes (that is, charged dust). By restricting to spherically symmetric distributions of charged matter, a number of exact solutions are presented in closed form which generalise the Schwarzschild interior solution. Some of these solutions exhibit functional relations between the electric and gravitational potentials different to the quadratic one of Weyl. All the non-dust solutions are well-behaved and, by matching them to the Reissner-Nordstr\"{o}m solution, all of the constants of integration are identified in terms of the total mass, total charge and radius of the source. This is done in detail for a number of specific examples. These are also shown to satisfy the weak and strong energy conditions and many other regularity and energy conditions that may be required of any physically reasonable matter distribution.