Quantum cosmology of the Friedmann-Lemaître-Robertson-Walker model with cosmological constant in the Ho$\check{\rm r}$ava-Lifshitz type gravity is studied in the phase space by means of the Wigner function. The modification of the usual general relativity description by the Ho$\check{\rm r}$ava-Lifshitz type gravity induces a new scenario for the origin of the Universe with an embryonic era where the Universe can exist classically before the tunneling process takes place and which gives rise to the current evolution of the Universe. The Wigner functions corresponding to the Hartle-Hawking, Vilenkin and Linde boundary conditions are obtained by means of numerical calculations. In particular three cases were studied for the potential of the Wheeler-DeWitt equation: tunneling barrier with and without embryonic era and when the potential barrier is not present. The quantum behavior of these three cases are analyzed using the Wigner function for the three boundary conditions considered.

The four-body decays $D^+ \to K^-\pi^+e^+\nu_e$ ($D_{e4}^+$) and $D^0\to \overline{K^0}\pi^-e^+\nu_e$ ($D^0_{e4}$) are studied in a model where the momentum-dependence of the hadronic matrix elements are described in terms of $K^*(892)$ and $D^*(2010)$ pole contributions. From fits to the recent data of the BESIII collaboration we find that the $D^*$-pole can mimic the effect of the $S$-wave contribution of the $K\pi$ system, which has been suggested as an observational evidence for the light scalar strange resonance. Implications for the determination of the $|V_{cs}|$ quark mixing matrix element are discussed.

The paper is devoted to a discussion of general properties of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. First we propose a general method of a recursive construction of the CKM matrix for any number of generations. This allows to set up a parameterization with desired properties. As an application we generalize the Wolfenstein parameterization to the case of 4 generations and obtain restrictions on the CKM suppression of the fourth generation. Motivated by the rephasing invariance of the CKM observables we next consider the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show, that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In the Main Theorem we show that all rephasing invariant monomials can be expressed as a product of at most 5 factors: 4 of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements.