We prove a restriction of an analogue of the Robinson-Schensted-Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means that the pair of their right keys is such that one key is bounded by the Schützenberger evacuation of the other. This bijection is then used to obtain an expansion formula of the non-symmetric Cauchy kernel, over staircases or truncated staircases, in the basis of Demazure characters of type A, and the basis of Demazure atoms. The expansion implies Lascoux expansion formula, when specialised to staircases or truncated staircases, and make explicit, in the latter, the Young tableaux in the Demazure crystal by interpreting Demazure operators via elementary bubble sorting operators acting on weak compositions.
In this paper, we study a system of partial differential equations defined in a moving domain. This system is defined by a heat equation and a diffusion equation for a concentration of non-Fickian type whose diffusion coefficient depends on the temperature, completed with suitable initial and boundary conditions. The non-Fickian mass flux is established considering the viscoelastic properties of the medium where the strain depends on the temperature and on the concentration. The initial boundary value problem (IBVP) analyzed can be used to describe the drying of viscoelastic materials where the internal structure offers a resistance to the movement of the moisture molecules and a consequent delay in the moisture removal. Due to heat transference into the materials and moisture removal, shrinkage of the medium occurs. The stability of the IBVP defined in a moving domain is analyzed and its qualitative behavior is numerically studied.
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