The authors investigate the global behavior of the solutions of the difference equation xn＋1=axn-1xn-k/bxn-p＋cxn-q,n=0,1,…where the initial conditions x-r, x-r＋1, x-r＋2,… , x0 are arbitrary positive real numbers...The authors investigate the global behavior of the solutions of the difference equation xn＋1=axn-1xn-k/bxn-p＋cxn-q,n=0,1,…where the initial conditions x-r, x-r＋1, x-r＋2,… , x0 are arbitrary positive real numbers, r = max{l, k,p, q） is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper.展开更多
We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs...We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs of the tree. We define two operations, the sum and the product of structure polynomials, giving a graph interpretation of them. Then we introduce an equivalence relation between polynomials, using the action of the full automorphism group of the tree, and we count equivalence classes of subgraphs modulo this equivalence. We also prove that this action gives rise to symmetric Gelfand pairs. Finally, when the regularity degree of the tree is a prime p, we regard each level of the tree as a finite dimensional vector space over the finite field Fp, and we are able to completely characterize structure polynomials corresponding to subgraphs whose leaf set is a vector subspace.展开更多
Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form rig, mN are positive integers while coeffic...Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form rig, mN are positive integers while coefficients bk ∈ C. As in the case of even degree polynomial potentials with integer powers, all the integrals in the expansion can be evaluated analytically in terms of F functions. With the help of two examples, we demonstrate the usefulness of these expansions in getting analytic insight into the quantum systems having rational power polynomial potentials.展开更多
文摘The authors investigate the global behavior of the solutions of the difference equation xn＋1=axn-1xn-k/bxn-p＋cxn-q,n=0,1,…where the initial conditions x-r, x-r＋1, x-r＋2,… , x0 are arbitrary positive real numbers, r = max{l, k,p, q） is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper.
文摘We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs of the tree. We define two operations, the sum and the product of structure polynomials, giving a graph interpretation of them. Then we introduce an equivalence relation between polynomials, using the action of the full automorphism group of the tree, and we count equivalence classes of subgraphs modulo this equivalence. We also prove that this action gives rise to symmetric Gelfand pairs. Finally, when the regularity degree of the tree is a prime p, we regard each level of the tree as a finite dimensional vector space over the finite field Fp, and we are able to completely characterize structure polynomials corresponding to subgraphs whose leaf set is a vector subspace.
文摘Asymptotic energy expansion method is extended for polynomial potentials having rational powers. New types of recurrence relations are derived for the potentials of the form rig, mN are positive integers while coefficients bk ∈ C. As in the case of even degree polynomial potentials with integer powers, all the integrals in the expansion can be evaluated analytically in terms of F functions. With the help of two examples, we demonstrate the usefulness of these expansions in getting analytic insight into the quantum systems having rational power polynomial potentials.
