The naturalness of maximal mixing between myon-and tau-neutri-nos is investigated. A spontaneously broken nonabelian generation symmetry can explain a small parameter which governs the deviation from maximal mixing. In many cases all three neutrino masses are almost degenerate. Maximal ν µ − ν τ-mixing would indicate that the leading contribution to the light neutrino masses arises from the exp...

Let G=(G j ) j≥0 be a strictly increasing linear recurrent sequence of integers with G 0 =1 having characteristic polynomial X d -a 1 d-1 -⋯-a X-a . It is well known that each positive integer ν can uniquely represented by the so-called greedy expansion ν=ε (ν)G +⋯+ε ℓ for ℓ∈ℕ satisfying ≤ν<G ℓ+1 Here digits are defined recursively in way 0≤ν-ε -⋯-ε <G holds 0≤j≤ℓ. In present paper we study sum...

Let M be a d-dimensional connected compact Riemannian manifold with boundary ∂M, let V∈C2(M) such that μ(dx):=eV(x)dx is probability measure, and Xt the diffusion process generated by L:=Δ+∇V τ:=inf⁡{t≥0:Xt∈∂M}. Consider conditional empirical measure μtν:=Eν(1t∫0tδXsds|t<τ) for initial distribution ν ν(∂M)<1. Thenlimt→∞⁡{tW2(μtν,μ0)}2=1{μ(ϕ0)ν(ϕ0)}2∑m=1∞{ν(ϕ0)μ(ϕm)+μ(ϕ0)ν(ϕm)}2(λm−λ0)3, where ν...

If massive neutrinos are Majorana particles, then the lepton number should be violated in nature and neutrino-antineutrino oscillations να↔ν‾β (for α,β=e,μ,τ) will definitely take place. In present paper, we study properties of CP violation with non-unitary leptonic flavor mixing matrix, which is actually a natural prediction canonical seesaw model due to between light heavy neutrinos. The osci...

Let G be a graph with vertex set V(G) and edge E(G). A invariant for is number related to the structure of which under symmetry G. The Sombor reduced indices are two new invariants defined as SO(G)=∑uv∈E(G)dG(u)2+dG(v)2 SOred(G)=∑uv∈E(G)dG(u)−12+dG(v)−12, respectively, where dG(v) degree v in We denote by Hn,ν constructed from star Sn adding ν edge(s), 0≤ν≤n−2, between fixed pendent other verti...

Calcium carbonate nano-particles were synthesized by a reverse micro-emulsion method at room temperature with Tween® 80 and Span® 80 as co-surfactants. The nano-particles synthesized were surface modified by stearic acid. An important operating variable in the Span 80-Tween 80/toluene/water reverse micro-emulsion system, the ω-value (water/surfactant molar ratio) was i...