We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any ...

First, we recall some notations and definitions which will be used throughout the paper (cf. [12]). A double sequence x = (xjk) of real or complex numbers is said to be bounded if ‖x‖∞ = supj,k |xjk| < ∞. We denote the space of all bounded double sequences by L. A double sequence x = (xjk) is said to converge to the limit L in Pringsheims sense (shortly, p-convergent to L) if for every ε > 0 th...

soil water retention curve is one of the crucial input parameters in any modeling study on water flow and solute transport in soils. direct measurement of soil water retention curve is time-consuming and costly. alternatively, pedotransfer functions (ptfs) may be used to indirectly estimate the soil water retention curve from easily measured soil properties. many equations for the soil water re...

Given a double-well potential F , a Z-periodic function H, small and with zero average, and ε > 0, we find a large R, a small δ and a function Hε which is ε-close to H for which the following two problems have solutions: 1. Find a set Eε,R whose boundary is uniformly close to ∂BR and has mean curvature equal to −Hε at any point, 2. Find u = uε,R,δ solving −δ∆u+ F (u) δ + c0 2 Hε = 0, such that ...

We prove density estimates for level sets of minimizers of the energy ε‖u‖Hs(Ω) + ∫ Ω W (u) dx, with s ∈ (0, 1), where ‖u‖Hs(Ω) denotes the total contribution from Ω in the Hs norm of u, and W is a double-well potential. As a consequence we obtain, as ε → 0+, the uniform convergence of the level sets of u to either a Hs-nonlocal minimal surface if s ∈ (0, 1 2 ), or to a classical minimal surfac...

We consider a mass conserved Allen-Cahn equation ut = ∆u + ε−2(f(u)− ελ(t)) in a bounded domain with no flux boundary condition, where ελ(t) is the average of f(u(·, t)) and −f is the derivative of a double equal well potential. Given a smooth hypersurface γ0 contained in the domain, we show that the solution uε with appropriate initial data approaches, as ε ց 0, to a limit which takes only two...

I shortly review the present status of the theoretical estimates of ε/ε . I consider a few aspects of the theoretical calculations which may be relevant in understanding the present experimental results. I discuss the role of higher order chiral corrections and in general of non-factorizable contributions for the explanation of the ∆I = 1/2 selection rule in kaon decays and ε/ε . Lacking reliab...