Bivariate fibonacci like p-polynomials

In this article, we study the bivariate Fibonacci and Lucas p-polynomials (p ? 0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal-Lucas polynomials, Fibonacci and Lucas p-polynomials, Fibonacci and Lucas p-numbers, Pell and Pell-Lucas p-numbers and Chebyshev polynomials of the first and second kind, are obtained. Afterwards, we obtain some properties of the bivariate Fibonacci and Lucas p-polynomials.     

Powell’s optimal identification of material constants of thin-walled box girders based on Fibonacci series search method

A dynamic Bayesian error function of material constants of the structure is developed for thin-walled curve box girders. Combined with the automatic search scheme with an optimal step length for the one-dimensional Fibonacci series, Powell’s optimization theory is used to perform the stochastic identification of material constants  of the thin-walled curve box. Then, the steps in the parameter identification are presented. Powell’s identification procedure for material constants of the thin-walled curve box is compiled, in which the mechanical analysis of the thin-walled curve box is completed based on the finite curve strip element (FCSE) method. Some classical examples show that Powell’s identification is numerically stable and convergent, indicating that the present method and the compiled procedure are correct and reliable. During the parameter iterative processes, Powell’s theory is irrelevant with the calculation of the FCSE partial differentiation, which proves the high computation efficiency of the studied methods. The stochastic performances of the system parameters and responses are simultaneously considered in the dynamic Bayesian error function. The one-dimensional optimization problem of the optimal step length is solved by adopting the Fibonacci series search method without the need of determining the region, in which the optimized step length lies.     

A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem

In this article, we combine mixed finite element method, multiscale discretization, and Rayleigh quotient iteration to propose a new adaptive algorithm based on residual type a posterior error estimates for the Stokes eigenvalue problem. Both reliability and efficiency of the error indicator are proved. The efficiency of the algorithm is also investigated using Chen's Innovation Finite Element Method (iFEM) package. Numerical results are satisfying.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 31–53, 2015     

Tuned preconditioners for inexact two‐sided inverse and Rayleigh quotient iteration

Convergence results are provided for inexact two‐sided inverse and Rayleigh quotient iteration, which extend the previously established results to the generalized non‐Hermitian eigenproblem and inexact solves with a decreasing solve tolerance. Moreover, the simultaneous solution of the forward and adjoint problem arising in two‐sided methods is considered, and the successful tuning strategy for preconditioners is extended to two‐sided methods, creating a novel way of preconditioning two‐sided algorithms. Furthermore, it is shown that inexact two‐sided Rayleigh quotient iteration and the inexact two‐sided Jacobi‐Davidson method (without subspace expansion) applied to the generalized preconditioned eigenvalue problem are equivalent when a certain number of steps of a Petrov–Galerkin–Krylov method is used and when this specific tuning strategy is applied. Copyright © 2014 John Wiley & Sons, Ltd.     

Origin of the log-periodic oscillations in the quantum dynamics of electrons in quasiperiodic systems

Recently, the occurrence of log-periodic oscillations in the quantum dynamics of electrons was reported for the one-dimensional Fibonacci quasicrystal by Lifshitz and Even-Dar Mandel. We apply a real-space renormalization group approach to show that these log-periodic oscillations are related to the underlying quasiperiodic structure of the Fibonacci quasicrystal. We find that they originate from the superposition of bonding and antibonding states associated to strongly coupled atoms in the chains, and that they show a hierarchical structure closely related to the atomic configurations.     

Pell graphs

In this paper, we introduce the Pell graphs, a new family of graphs similar to the Fibonacci cubes. They are defined on certain ternary strings (Pell strings) and turn out to be subgraphs of Fibonacci cubes of odd index. Moreover, as well as ordinary hypercubes and Fibonacci cubes, Pell graphs have several interesting structural and enumerative properties. Here, we determine some of them. Specifically, we obtain a canonical decomposition giving a recursive structure, some basic properties (bipartiteness and existence of maximal matchings), some metric properties (radius, diameter, center, periphery, medianicity), some properties on subhypercubes (cube coefficients and polynomials, cube indices, decomposition in subhypercubes), and, finally, the distribution of the degrees.     

A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FE ELASTIC STABILITY ANALYSIS OF EULER BEAMS1)

本文将有限元p型超收敛算法应用于欧拉梁弹性稳定分析。该法基于有限元解答中失稳载荷和失稳模态结点位移的超收敛特性,建立了单元上失稳模态近似满足的线性常微分方程边值问题,在每个单元上,对该边值问题采用一个高次元进行求解,获得失稳模态的超收敛解,再将失稳模态的超收敛解代入瑞利商的解析表达式,最终获得失稳载荷的超收敛解。该法思路简明,通过少量计算即可显著提高失稳载荷和失稳模态的精度与收敛阶。数值算例表明,该法高效、可靠,值得进一步研究和推广到各类杆系结构。     

空间型中紧致无边子流形上的第一特征值

得到了两个关于空间形式中紧致无边子流形的广义位置向量场和其上Laplace算子第一特征值λ_1的积分不等式。并由此首先给出了λ_1与其上界间的间隔估计,其次得到了此紧致无边子流形等距浸入在空间形式的测地超球面或等距于测地超球面的充分条件,推广了Deshmukh[6]在欧氏空间中的相应结论。     

Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation

Gradient-type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD-id) is one of such methods. The convergence behavior of the PSD-id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non-asymptotic estimates indicate a superlinear convergence of the PSD-id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD-id using a restricted formulation of the PSD-id. More importantly, we extend the new convergence analysis of the PSD-id to a practically preferred block version of the PSD-id, or BPSD-id, and show the cluster robustness of the BPSD-id. Numerical examples are provided to validate the theoretical estimates.