BIFURCATION OF SINGULAR POINTS NEAR A DOUBLE FOLD POINT INZ_2 -SYMMETRIC NONLINEAR EQUATIONS

We consider the bifurcation of singular points near a double fold point in Z2 -symmetric nonlinear equations with two parameters,where the linearization has a two dimensional null space spanned by a symmetric null vector and an ami-symmetric null vector. In particular, we show the existence of a turning point path and a pitchfork point path passing ihrough the double fold point and they are the only singular points nearby. Their nondegeneracy is confirmed. A supporting numerical example is also provided. The main tools for our analysis as well as the compulation are some extended systems.     

Nordsieck Methods with an Off-Step Point

Hybrid methods, incorporating one or more off-step points, are difficult to implement in a variable stepsize situation using the standard representation of input and output data in each step. However, instead of representing this data in terms of solution values and derivative values at a sequence of step points, it is possible to reformulate the method so that it operates on a Nordsieck vector. This has the consequence of reducing stepsize adjustments to nothing more than rescaling the components of the Nordsieck vector. This paper shows how to derive methods in both formulations and considers some implementation details. It is also possible to derive a new type of hybrid method using the Norsieck representation as the starting point and this is also discussed in the paper. The new method is found to have comparable accuracy for corresponding work expended as for standard methods.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.     

A method for minimizing the sum of a convex function and a continuously differentiable function

This paper presents a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function. As in the convex case developed by the author, the algorithm is a descent method which generates successive search directions by solving quadratic programming subproblems. An inexact line search ensures global convergence of the method to stationary points.     

Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation

We present a pathfollowing strategy based on pseudo-arclength parametrization for the solution of parameter-dependent boundary value problems for ordinary differential equations. We formulate criteria which ensure the successful application of this method for the computation of solution branches with turning points for problems with an essential singularity. The advantages of our approach result from the possibility to use efficient mesh selection, and a favorable conditioning even for problems posed on a semi-infinite interval and subsequently transformed to an essentially singular problem. This is demonstrated by a Matlab implementation of the solution method based on an adaptive collocation scheme which is well suited to solve problems of practical relevance. As one example, we compute solution branches for the complex Ginzburg-Landau equation which start from non-monotone ‘multi-bump’ solutions of the nonlinear Schrödinger equation. Following the branches around turning points, real-valued solutions of the nonlinear Schrödinger equation can easily be computed.     

A Simple Multistart Algorithm for Global Optimization

1.IntroductionConsidertheunconstrainedoptimizationproblem:findx*suchthatf(x*)~caf(x),(1)wheref(x)isanonlinearfllnctiondefinedonW"andXCR".Ourobjectiveistofindtheglobalminimizeroff(x)inthefeasibleset.Withoutassuminganyconditionsonf(x)globaloptimizationproblemsareunsolvableinthefollowingsensefnoalgorithmcanbeguaranteedtofindaglobalminimizerofageneralnonlinearfunctionwithinfinitelymanyiterations.Supposethatanalgorithmappliedtoanonlinearfunctionf(x)producesiteratesxlandterminatesafterKiterations.…     

带转点的三阶常微分方程边值问题的渐近解

研究带转点的三阶常微分方程的边值问题,其中f(x;0)在(-a,b)具有多个多重零点。给出边值问题出现共振的必要条件,求得其一致有效渐近解和余项估计。     

The approximation of generalized turning points by projection methods with superconvergence to the critical parameter

Summary A procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between Banach spaces. This characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization. Applications are given to parameter dependent two-point boundary value problems. In particular, collocation at Gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points.     

返航备降航班高风险频发子集搜索模型

返航备降航班是一种保证航空安全的有效措施,但返航备降航班会带来成本损失,所以对返航备降业务规律进行分析,从而重点监控,在确保安全生产的基础上,有效降低返航备降航班的经济损失。我们可以把确定返航备降航班发生的密集时间段描述为一个多属性空间搜索问题,在许多实践领域存在该类问题,比如商业银行对贷款客户的信用评级、绩效考核等。在多属性组成的多维空间中,存在许多正常点和不正常点,通过建立高风险频发子集搜索模型和算法确定不正常点比较密集的多属性范围,使得在该空间范围内不正常点的比例较高并且总的点的数量达到一定的水平。本文以航空公司安全生产为例,其模型结果在实际生产的运用中取得良好效果,在一年内避免30余班次返航备降。     

Numerical solution of second-order nonlinear singularly perturbed boundary-value problems by initial-value methods

Nonlinear singularly perturbed boundary-value problems are considered, with one or two boundary layers but no turning points. The theory of differential inequalities is used to obtain a numerical procedure for quasilinear and semilinear problems. The required solution is approximated by combining the solutions of suitable auxiliary initial-value problems easily deduced from the given problem. From the numerical results, the method seems accurate and solutions to problems with extremely thin layers can be obtained at reasonable cost.This work was supported by CNR, Rome, Italy (Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, Sottoprogetto 1).     

A method for determining the equilibrium states of dynamic systems

A heuristic method is presented for determining the equilibrium states of motion of dynamic systems, in particular, spacecraft. The method can also be applied to the solution of sets of linear or nonlinear algebraic equations. A positive-semidefinite functional is formed to convert the problem to that of finding those minimum points where the functional vanishes. The process is initiated within a selecteddomain of interest by random search; convergence to a minimum is obtained by a modified Davidon's deflected gradient technique. To render this approach feasible in the presence of constraints, the functional is modified to include penalty terms which cause the functional to approach infinity at the constraint boundaries. Close approximations to solutions near the constraint boundaries are found by applying Carroll's approach in successively reducing the weighting factors of the penalty terms. After finding a minimum, the local domain around this point is eliminated by adding to the functional an interior constraint term, representing the surface under a hypersphere centered at the minimum point. The domain of consideration now becomes the subdomain formed by subtracting the space contained within this hypersphere from the previous domain of interest. Minima are now sought within the remaining space, as before.This paper is derived from research performed by the author while employed by TRW Systems Group, Redondo Beach, California.The author acknowledges the helpful suggestions of Dr. G. Bekey, University of Southern California, and those of Mr. E. A. Quast, Dr. M. P. Scher, and Dr. R. J. Wiley, Dynamics Department, TRW Systems Group, Redondo Beach, California.     

Three weak solutions for elliptic Dirichlet problems

An existence result of three non-zero solutions for non-autonomous elliptic Dirichlet problems, under suitable assumptions on the nonlinear term, is presented. The approach is based on a recent three critical points theorem for differentiable functionals.     

Preservation of equilibria for symplectic methods applied to Hamiltonian systems

In this paper, the linear stability of symplectic methods for Hamiltonian systems is studied. In par- ticular, three classes of symplectic methods are considered： symplectic Runge-Kutta （SRK） methods, symplectic partitioned Runge-Kutta （SPRK） methods and the composition methods based on SRK or SPRK methods. It is shown that the SRK methods and their compositions preserve the ellipticity of equilibrium points uncondi- tionally, whereas the SPRK methods and their compositions have some restrictions on the time-step.     

一种基于二部图的试题知识点标注方法

针对在线教育中试题知识点自动标注问题,本文提出了一种基于二部图的试题知识点标注方法。首先,为了合理划分知识点粒度,本文构建了知识点知识图谱来融合知识点;其次,基于教材等语料抽取知识点与特征词二部图和试题与特征词二部图,并利用TF-IDF公式计算知识点与特征词、试题与特征词之间的边权,以构建试题-知识点二部图模型;再次,提出词频加权的相似性度量方法计算试题和知识点之间的相似度,将相似度最高的知识点作为试题知识点标签。最后,以某在线教育平台提供的高中历史试题为实验数据集进行数值实验,实验结果表明该方法的效果显著优于朴素贝叶斯、K最近邻、随机森林以及支持向量机。     

Polynomial function and derivative approximation of Sinc data

Sinc methods consist of a family of one dimensional approximation procedures for approximating nearly every operation of calculus. These approximation procedures are obtainable via operations on Sinc interpolation formulas. Nearly all of these approximations–except that of differentiation–yield exceptional accuracy. The exception: when differentiating a Sinc interpolation formula that gives an approximation over an interval with a finite end-point. In such cases, we obtain poor accuracy in the neighborhood of the finite end-point. In this paper we derive novel polynomial-like procedures for differentiating a function that is known at Sinc points, to obtain an approximation of the derivative of the function that is uniformly accurate on the whole interval, finite or infinite, in the case when the function itself has a derivative on the closed interval.     

A new method for constructing invariant subspaces

The new idea that is used in this article for producing non-trivial (closed) invariant subspaces of (bounded linear) operators on reflexive Banach spaces, is the use of fixed points of set-valued functions. The advantage of this new method is that it is reasonable to expect that the famous method of Lomonosov for producing invariant subspaces using fixed points of functions, can be viewed as a special case of the use of fixed points of set-valued functions. Further uses of this new idea and open questions are suggested at the end of the article.     

Rainflow analysis in Coastal Engineering using switching second order Markov models

The paper deals with the use of Markov and switching Markov chain models of turning points to reproduce random sets of sea states. The advantages of these models are emphasized and compared with existing models based on wave height records, indicating that long and short range and period cycles are included, while the wave height records ignore this important information from the point of view of damage accumulation. Existing models for first order Markov processes are extended to the case of second order processes and closed formulas are given to derive the rainflow matrices of these processes. Finally, one illustrative example of application is given.     

Singular Points in Electromechanical Systems and Their Determination

An approach to determine turning points in electromechanical systems is presented and applied to the determination of pull-in parameters. The pull-in phenomenon is an inherent instability in devices using electrostatic actuation where an electrostatic force works against an elastic restoring force. The paper presents a relation between multi-valued characteristics describing MEMS devices and weakly singular tracking problems. The technique of augmented systems that characterize turning points is applied to the analytical and numerical determination of pull-in parameters. The method is discussed using a computer algebra system and a VHDL-AMS simulation engine.