The classical Liouvile integrability means that there exist n independent first integrals in involution for 2n-dimensional phase space. However, in the infinite-dimensional case, an infinite number of independent first integrals in involution don’t indicate that the system is solvable. How many first integrals do we need in order to make the system solvable? To answer the question, we obtain an...

In many fields of physics and chemistry, explicit analytical solutions of the fundamental dynamical equations are much valuable for a general understanding of phenomena, e. g. the role played by physical parameters. Two typical examples in quantum mechanics are the exact solutions of the Schrödinger equation (SE) for a hydrogen atom (Coulombic) and for a harmonic oscillator [1 – 3]. The Mie-typ...

Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree l (l = 1, 2, . . .) eigenpolynomial. These polynomials...

‎In this paper we prove that a finite group $G$ having at most three‎ ‎conjugacy classes of non-normal non-abelian proper subgroups is‎ ‎always solvable except for $Gcong{rm{A_5}}$‎, ‎which extends Theorem 3.3‎ ‎in [Some sufficient conditions on the number of‎ ‎non-abelian subgroups of a finite group to be solvable‎, ‎Acta Math‎. ‎Sinica (English Series) 27 (2011) 891--896.]‎. ‎Moreover‎, ‎we s...

We prove that every digraph of circumference l has DAG-width at most l and this is best possible. As a consequence of our result we deduce that the k-linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in [2]. We also prove that the weak k-linkage problem (where we ask for arc-disjoint paths) is polynomial...

In this note we prove a group theoretic statement about expressing certain characters of a finite solvable group as a sum of monomial characters. This is used to prove holomorphy of certain products of Artin L-functions which can be thought of as a variant of the Dedekind Conjecture. This variant is then used to improve, in the solvable case, a certain inequality due to R. Foote and K. Murty wh...

A new predictor-corrector algorithm is proposed for solving P ()-matrix linear complementarity problems. If the problem is solvable, then the algorithm converges from an arbitrary positive starting point (x 0 ; s 0). The computational complexity of the algorithm depends on the quality of the starting point. If the starting point is feasible or close to being feasible, it has O((1+) p n== 0 L)-i...

‎let $s$ be a subset of a finite group $g$‎. ‎the bi-cayley graph ${rm bcay}(g,s)$ of $g$ with respect to $s$ is an undirected graph with vertex set $gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin g‎, ‎ sin s}$‎. ‎a bi-cayley graph ${rm bcay}(g,s)$ is called a bci-graph if for any bi-cayley graph ${rm bcay}(g,t)$‎, ‎whenever ${rm bcay}(g,s)cong {rm bcay}(g,t)$ we have $t=gs^alpha$ for some $...

‎let $g$ be a finite group‎. ‎we say that the derived covering number of $g$ is finite if and only if there exists a positive integer $n$ such that $c^n=g'$ for all non-central conjugacy classes $c$ of $g$‎. ‎in this paper we characterize solvable groups $g$ in which the derived covering number is finite‎.‎