The Schrödinger equation is one of the most basic equations in quantum mechanics. In this paper, we study the convergence of symmetric discretization models for the nonlinear Schrödinger equation in dark solitons’ motion and verify the theoretical results through numerical experiments. Via the second-order symmetric difference, we can obtain two popular space-symmetric discretization models of the nonlinear Schrödinger equation in dark solitons’ motion: the direct-discrete model and the Ablowitz–Ladik model. Furthermore, applying the midpoint scheme with symmetry to the space discretization models, we obtain two time–space discretization models: the Crank–Nicolson method and the new difference method. Secondly, we demonstrate that the solutions of the two space-symmetric discretization models converge to the solution of the nonlinear Schrödinger equation. Additionally, we prove that the convergence order of the two time–space discretization models is O(h2+τ2) in discrete L2-norm error estimates. Finally, we present some numerical experiments to verify the theoretical results and show that our numerical experiments agree well with the proven theoretical results.

In this paper, we study efficiency of numerical simulation for integrable and nonintegrable discrete Non-linear Schrodinger equations (NLSE). We first discretize the NLSE into two classical spatial models, nonintegrable direct discrete model and integrable Ablowitz-Ladik model. By some simple transformations and Doubarx trans-formation, we obtain two integrable models from Ablowitz-Ladik model. Then, five different kinds of schemes can be applied to simulate four models in bright and dark cases for comparing the performance in preserving the conserved quantities' approximations of NLSE. The numerical experiments indicate that Gauss symplectic method is more efficient than nonsymplectic schemes and splitting schemes when simulating the same model. Both intergrable models and nonintergrable model have their own advantages in preserving the conserved quantities' approximations. For the three integrable models, Ablowitz-Ladik Model and the model which has a general symplectic structure have similar simulation effects, and the model owing a cononical symplectic structure has low efficiency because the complicated Doubarx transformations make the model difficult to solve. Moreover, symplectic scheme and symmetric scheme have overwhelming superiorities over nonsymplectic schemes in preserving the invariants of Hamiltonian system.

In this paper, symplectic schemes and symmetric schemes are presented to simulate Nonlinear Schrödinger Equation (NLSE) in case of dark soliton motion. Firstly, by Ablowitz–Ladik model (A–L model), the NLSE is discretized into a non-canonical Hamiltonian system. Then, different kinds of coordinate transformations can be used to standardize the non-canonical Hamiltonian system. Therefore, the symplectic schemes and symmetric schemes can be employed to simulate the solitons motion and test the preservation of the invariants of the A–L model and the conserved quantities approximations of the original NLSE. The numerical experiments show that symplectic schemes and symmetric schemes have similar simulation effect, and own significant superiority over non-symplectic and non-symmetric schemes in long-term tracking the motion of solitons, preserving the invariants and the approximations of conserved quantities. Moreover, it is obvious that coordinate transformations with more symmetry have a better simulation effect.

A multi-model modeling method in near infrared spectra analysis is introduced. The basic idea of the modeling method is relatively simple. The method is easy to be mastered, and the modeling effect is good. The authors have been engaged in the research of this method in recent years. With the contents of some chemical components of acacia, populus tomentosa and populus euramericana as the research objects, some quantitative analysis models of near infrared spectroscopy were established by using multi-model method, and some understanding to the characteristics of this modeling method is obtained. This method is expected to get more development. It is possible that the method will be used to determine the contents of some chemical components in other materials.

This paper focuses on the research about selecting the best college sports coaches during the last century. Firstly, we build a reasonable evaluation index system, taking basketball for instance, we choose 9 evaluation indexes which are the coaching years, rate of winning, times of attending major competitions, etc. Because of too many indexes, it brings difficulty in building judgment matrix. Therefore, using the Principal Component Analysis, we get not only a fewer of indexes which almost explain the original indexes, but also the corresponding weight coefficients. Furthermore, based on the results, we give the evaluation scores of basketball coaches, and then rank them from top to bottom. AnalyticalHierarchyProcess and Comprehensive Evaluation are integrated to test the ranking result. Finally, this evaluation model is applied to football and hockey fields to select the best coaches.

Scalable vocabulary tree (SVT) is a data compression structure which gains scalable visual vocabularies from hierarchical k-means clustering of local image features. Due to both high robustness in data retrieval and great potentials to process huge data, it has become one of the state-of-the-art methods building on the bag-of-features. However, such bag-of-words representations mainly suffer from two limitations. The paper gives a performance research of re-ranking in sub-image retrieval using SVT which is built from local Speed Up Robust Features descriptors. Firstly, the paper gives a study on retrieval performance using different single layers of the tree, which tells it divides data too coarsely for low layers with a small quantity of leaf nodes, while too finely for the 6-th layer with too many leaf nodes. Then using the best selected layer, the authors give a comparative analysis with popular advanced re-ranking strategies in the existing literatures. Finally, the authors propose a weighted score method that calculates matching score from dominating layers. The experimental results prove that the weighted score method achieves almost optimal retrieval performance when using SVT for data representations. Meanwhile, it almost doesn't increase any computational complexity, and can be implemented easily.

首先对2012年5月北京林业大学举办的第一届数学建模竞赛的概况进行了介绍,并对数学建模竞赛的B题进行了详细的点评.其次阐述了这次数学建模竞赛所展现的特点,即参赛学生具有扎实的基本功;最优选择下关系的确定是影响竞赛成绩的关键;参赛学生对试题使用了多样求解方法;部分学生误解题意,最终导致了错误的解决方案.针对这些特点,提出将数学建模纳入大学数学教学,并应增加数学问题的计算机实现环节,利用数学建模可以多角度讲述大学数学重要的知识点.

The research is aimed to accomplish an automatic visual inspection system for circular objects using cameras and image processing technique. The main concern of the paper is on the pattern analysis of circular objects based on image analysis. First, the authors propose a computational method using global symmetry to locate objects before the further inspection process. It aims at designing a symmetry measure based on distance weight, phase weight and intensity weight. Such measure can be used to locate centers of circles, even for those with weak contrast under uncertain complex backgrounds. Then based on the measure, similarities of arbitrary circular objects are given and tested. The experimental results for the proposed approach are promising.

In this paper, an efficient numerical procedure is presented to implement the Gaussian Runge–Kutta (GRK) methods (also called Gauss methods). The GRK technique first discretizes each marching step of the initial value problem using collocation formulations based on Gaussian quadrature. As is well known, it preserves the geometric structures of Hamiltonian systems. Existing analysis shows that the GRK discretization with s nodes is of order 2s, A-stable, B-stable, symplectic and symmetric, and hence "optimal" for solving initial value problems of general ordinary differential equations (ODEs). However, as the unknowns at different collocation points are coupled in the discretized system, direct solution of the resulting algebraic equations is in general inefficient. Instead, we use the Krylov deferred correction (KDC) method in which the spectral deferred correction (SDC) scheme is applied as a preconditioner to decouple the original system, and the resulting preconditioned nonlinear system is solved efficiently using Newton–Krylov schemes such as Newton–GMRES method. The KDC accelerated GRK methods have been applied to several Hamiltonian systems and preliminary numerical results are presented to show the accuracy, stability, and efficiency features of these methods for different accuracy requirements in short- and long-time simulations.

We review the two different approaches for symplecticity of linear multi-step methods (LMSM) by Eirola and Sanz-Serna, Ge and Feng, and by Feng and Tang, Hairer and Leone, respectively, and give a numerical example between these two approaches. We prove that in the conjugate relation G3λτ∘G1τ=G2τ∘G3λτ with G1τ and G3τ being LMSMs, if G2τ is symplectic, then the B-series error expansions of G1τ, G2τ and G3τ of the form Gτ(Z)=∑i=0+∞(τi/i!)Z[i]+τs+1A1+τs+2A2+τs+3A3+τs+4A4+O(τs+5) are equal to those of trapezoid, mid-point and Euler forward schemes up to a parameter θ (completely the same when θ=1), respectively, this also partially solves a problem due to Hairer. In particular we indicate that the second-order symmetric leap-frog scheme Z2=Z0+2τJ-1∇H(Z1) cannot be conjugate-symplectic via another LMSM.

Many techniques for locating circular objects in images have been presented. However, it's still a challenge to extract circles for images coincidently with weak contrast in strong complex backgrounds. In this study, we propose a computational method using global symmetry, which can robustly locate circles in images. Furthermore, it can independently inspect circular objects with simple patterns. Our method aims at designing a symmetry measure based on distance weight, phase weight and intensity weight. Then based on the measure, a systemic approach is given to inspect arbitrary circular objects under uncertain situations in virtue of an invariant content-based descriptor. The experimental results show the presented approach to be promising.