in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

Let k be an algebraic number field. Let α be a root of a polynomial f ∈ k[x] which is solvable by radicals. Let L be the splitting field of α over k. Let n be a natural number divisible by the discriminant of the maximal abelian subextension of L, as well as the exponent of G(L/k), the Galois group of L over k. We show that an optimal nested radical with roots of unity for α can be effectively ...

Abstract It is an interesting and difficult topic to determine the structure of a finite group by number elements maximal order. This related Thompson’s conjecture, that is, if two groups have same order type one them solvable, then other solvable. In this article, we continue work prove G G which has 4 <m...

Using an isomorphism between Hilbert spaces L and l 2 we consider Hamiltonians which have tridiagonal matrix representations (Jacobi matrices) in a discrete basis and an eigenvalue problem is reduced to solving a three term difference equation. Technique of intertwining operators is applied to creating new families of exactly solvable Jacobi matrices. It is shown that any thus obtained Jacobi m...

It is proved that the free boundary problem for two phase viscous compressible and incompressible fluids in spaces $$ {W}_p^{2+l,1+l/2} (QT) with p > 2, l ∈ (1/p, 2/p), solvable locally time.

The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is induced from a character φ of U with solvable factor group U/ker(φ). Using the classification of finite simple groups, we prove that these groups are also s...

Solvable Graphs (also known as Reachable Graphs) are types of graphs that any arrangement of a specified number of agents located on the graph’s vertices can be reached from any initial arrangement through agents’ moves along the graph’s edges, while avoiding deadlocks (interceptions). In this paper, the properties of Solvable Graphs are investigated, and a new concept in multi agent moti...

First examples of quasi-exactly solvable models describing spin-orbital interaction are constructed. In contrast with other examples of matrix quasi-exactly solvable models discussed in the literature up to now, our models admit infinite (but still incomplete) sets of exact (algebraic) solutions. The hamiltonians of these models are hermitian operators of the form H = −∆ +V1(r)+ (s · l)V2(r)+ (...