We propose to deploy limits that arise from different tests of the Pauli Exclusion Principle in order: i) to provide theories of quantum gravity with an experimental guidance; ii) to distinguish among the plethora of possible models the ones that are already ruled out by current data; iii) to direct future attempts to be in accordance with experimental constraints. We firstly review experimental bounds on nuclear processes forbidden by the Pauli Exclusion Principle, which have been derived by several experimental collaborations making use of different detector materials. Distinct features of the experimental devices entail sensitivities on the constraints hitherto achieved that may differ one another by several orders of magnitude. We show that with choices of these limits, renown examples of flat noncommutative space-time instantiations of quantum gravity can be heavily constrained, and eventually ruled out. We devote particular attention to the analysis of theκ-Minkowski and θ-Minkowski noncommutative spacetimes. These are deeply connected to some scenarios in string theory, loop quantum gravity and noncommutative geometry. We emphasize that the severe constraints on these quantum spacetimes, although cannot rule out theories of top-down quantum gravity to whom are connected in various way, provide a powerful limitations of those models that it will make sense to focus on in the future.

Discrete Asymmetries for Neutrino Oscillations are contaminated by Matter Effects. Their Physics is discussed in terms of a basis of 3 independent components in correspondence with invariance under CPT (genuine), Time Reversal T (matter) and Charge-Parity CP (interference). They have definite and different parities under the baseline L -theory independent-, the imaginary part of the flavor mixing, the matter potential and the hierarchy of the neutrino spectrum. In terms of the Standard Model of neutrino oscillations, these definite model-independent parities manifest in terms of definite parities in the standard parameters, which can be studied analytically thanks to the smallness of some of these quantities. The two fake components vanish at the same MAGIC ENERGY E, with the connection L/E = 1420 Km/GeV near the second oscillation maximum, where the genuine component is also maximal. The experimental CPV Asymmetry for the appearance nu_mu --> nu_e can separate the genuine (CPT-invariant) and the fake matter-induced (T-invariant) components by either L-dependence (HKK) or E-dependence (DUNE).

For a singular hypersurface of arbitrary type in quadratic gravity motion equations were obtained using only the least action principle. The coefficients in the motion equations are zeroed with a combination corresponding to the Gauss-Bonnet term. Therefore it does not create neither double layers nor thin shells. It has been demonstrated that there is no “external pressure” for any type of null singular hypersurface. For spherically symmetric lightlike singular hypersurfaces additionally the “external flux” is equal to zero and the system of motion equations is reduced to one which is expressed through the invariants of spherical geometry along with the Lichnerowicz conditions. In this case there are no double layers only thin shells. Spherically symmetric null thin shells were investigated for spherically symmetric solutions of conformal gravity as an application, in particular, for various vacua and Vaidya-type solutions. By virtue of Lichnerowicz conditions scalar curvature R₂ of the two-dimensional non-spherical part of the metric without conformal factor must be continuous on the null shell. Therefore the following combinations for two vacua are possible: matching a vacuum with a constant R₂ and a vacuum with a variable R₂, matching two vacuums with a variable R₂, matching two vacua with a coinciding constant R₂. In the first case, the hypersurface is an analogue of the double horizon for the vacuum with variable R₂; in other cases, junction is possible only if the metrics coincide up to a conformal factor. With the addition of the Vaidya-type solution new possible matchings appear: Vaidya-type metric with vacuum with variable R₂ and two Vaidya-type metrics. In the first version, the null shell is actually the singular part of the Vaidya-type solution, in the second, they must coincide up to the conformal factor. Moreover the null shell does not emit in both cases.

From the assumption that the slow roll parameter \epsilon has a Lorentzian form as a function of the
e-folds number N, a successful model of a quintessential inflation is obtained. The form corresponds
to the vacuum energy both in the inflationary and in the dark energy epochs. The form satisfies
the condition to climb from small values of \epsilon to 1 at the end of the inflationary epoch. At the
late universe \epsilon becomes small again and this leads to the Dark Energy epoch. The observables
that the models predicts fits with the latest Planck data: r ∼ 10−3 , ns ≈ 0.965. Naturally a large dimensionless factor that exponentially amplifies the inflationary scale and exponentially suppresses the dark energy scale appears, producing a sort of cosmological see saw mechanism. We find the
corresponding scalar Quintessential Inflationary potential with two flat regions - one inflationary
and one as a dark energy with slow roll behavior. Moreover, a reheating mechanism is suggested with numerical estimation for the homogeneous evolution of the universe. The suggested mechanism is consistent with the BBN bound.

The dihedral group represents a group of symmetries of a regular polygon. Moreover, it is generated by rotation and axial symmetries. Due to this structure, in a recently published paper, we analyzed the HX-groups associated with them and we computed the commutativity degree of the associated HX-groups. Since there is an interesting connection between the HX-groups and the complete hypergroups- both of them can be constructed using the structure of a group, in this paper, we aim to determine the relationships between the lattice of the dihedral group and the lattice of the associated HX-groups, as well as that of the associated complete hypergroups. Furthermore, we will present some conditions to describe the modular and the distributive lattices, using elements from hypercompositional algebra. This has the advantage that the interaction called hyperproduct or hyperoperation between two elements of the considered set is not anymore just an element, as in the classical algebra, but a subset of the support set.