We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties SL(n)/P , including the quantum Pieri and quantum Giambelli formulas and theâ€¦ (More)

1. In the present study the chromosome numbers of about 120 species representing 25 genera of the family Malvaceae have been described. The following chromosome numbers were found: 5, 7, 8, 9, 10,â€¦ (More)

1.Gossypium Davidsonii (n = 13), a wild species from California, has smaller chromosomes thanG. Sturtii (n = 13), a wild species from Australia. This difference in chromosome size is maintained inâ€¦ (More)

We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. Ourâ€¦ (More)

We prove a positive combinatorial formula for the equivariant class of an orbit closure in the space of representations of an arbitrary quiver of type A. Our formula expresses this class as a sum ofâ€¦ (More)

We introduce a mutation algorithm for puzzles that is a threedirection analogue of the classical jeu de taquin algorithm for semistandard tableaux. We apply this algorithm to prove our conjecturedâ€¦ (More)