Edgeworth expansions for random walks on covering graphs with groups of polynomial volume growths are obtained under a few natural assumptions. The coefficients appearing in this expansion depends not only geometric features the underlying but also modified harmonic embedding graph into certain nilpotent Lie group. Moreover, we apply rate convergence Trotter's approximation theorem to establish...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

Abstract An asymmetric three-layered laminate with prescribed stresses along the faces is considered. The outer layers are assumed to be much stiffer than inner one. focus on long-wave low-frequency anti-plane shear. Asymptotic analysis of original dispersion relation reveals a harmonic supporting slow quasi-static (or static at limit) decay near cut-off wave propagation. In spite asymmetry pro...

The harmonic polytope and the bipermutahedron are two related polytopes that arose in Lagrangian geometry of matroids. We study bipermutahedron. show it is a simple whose faces bijection with vertex-labeled edge-labeled multigraphs no isolated vertices; generating function for its \(f\)-vector evaluation three variable Rogers-Ramanujan function. introduce biEulerian polynomial, which counts bip...