On weakly separable polynomials and weakly quasi-separable polynomials over rings

Separable extensions of noncommutative rings have already been studied extensively. Recently, N. Hamaguchi and A. Nakajima introduced the notions of weakly separable extensions and weakly quasiseparable extensions. They studied weakly separable polynomials and weakly quasi-separable polynomials in the case that the coefficient ring is commutative. The purpose of this paper is to give some improvements and generalizations of Hamaguchi and Nakajima's results. We shall characterize a weakly separable polynomial f(X) over a commutative ring by using its derivative f′(X) and its discriminant δ(f(X)). Further, we shall try to give necessary and sufficient conditions for weakly separable polynomials in skew polynomial rings in the case that the coefficient ring is noncommutative

Kadanoff-Baym approach to the thermal resonant leptogenesis

Using the non-equilibrium Green function method (Kadanoff-Baym equations) in the expanding universe, we investigate evolution of the lepton number asymmetry when the right-handed (RH) neutrinos have almost degenerate masses $|M_i^2-M_j^2| \ll M_i^2$. The resonantly enhanced $CP$-violating parameter $\varepsilon_i$ associated with the decay of the RH neutrino $N_i$ is obtained. It is proportional to an enhancement factor $(M_i^2-M_j^2) M_i \Gamma_j/ ((M_i^2-M_j^2)^2 +R_{ij}^2)$ with the regulator $R_{ij}=M_i \Gamma_i+M_j \Gamma_j$. The result is consistent with the previous result obtained by Garny et al., in a constant background with an out-of-equilibrium initial state. We discuss the origin of such a regulator, and why it is not like $R_{ij}=M_i \Gamma_i-M_j \Gamma_j$.Comment: 51 pages + appendices (46 pages), 5 figures; typos corrected, references adde

On Weakly Separable Polynomials in skew polynomial rings

The notion of weakly separable extensions was introduced by N. Hamaguchi and A. Nakajima as a generalization of separable extensions. The purpose of this article is to give a characterization of weakly separable polynomials in skew polynomial rings. Moreover, we shall show the relation between separability and weak separability in skew polynomial rings of derivation type

On tildeDtilde{D}-separable polynomials in skew polynomial rings of derivation type (Algebraic system, Logic, Language and Related Areas in Computer Sciences II)

The notion of ($tilde{p}$, $tilde{D}$)-separable polynomials in skew polynomial rings was introduced by S. Ikehata, and X. Lou gave a characterization of $tilde{p}$-separable polynomials in skew polynomial rings of automorphism type. In this paper, we shall give a new characterization of $tilde{D}$-separable polynomials in skew polynomial rings of derivation type

On weakly (tilderho,tildeD)(tilde{rho}, tilde{D})-separable polynomials in skew polynomial rings (Algebras, logics, languages and related areas)

Separable polynomials in skew polynomial rings were studied extensively by Y. Miyashita, T. Nagahara, S. Ikehata, and G. S eto. In particular, Ikehata gave the characteri ation of (overline{rho}, overline{D})-separable polynomials in skew polynomial rings. In this article, we shall introduce the notion of weakly (overline{rho}, tilde{D})-separable polynomials in skew polynomial rings, and we shall give a characteri ation of the (overline{p}, overline{D})-separability and that of the weak (overline{rho}, overline{D})-separability

Note on (tilderhotilde{rho}, tildeDtilde{D})-separable polynomials in skew polynomial rings (Logic, Language, Algebraic system and Related Areas in Computer Science)

The notion of ($tilde{rho}$, $tilde{D}$)-separable polynomials in skew polynomial rings was introduced by S. Ikehata. In this paper, we shall give a new characterization of ($tilde{rho}$, $tilde{D}$)-separable polynomials in skew polynomial rings which shows the difference between separable systems and ($tilde{rho}$, $tilde{D}$)-separable systems

紫外線スペクトルスロープβを用いた赤方偏移4に存在する星形成銀河の性質への制限

Tohoku University千葉柾司課