分数应用题教学要重视学法指导

分数应用题是小学阶段数学教学的重点和难点,如何正确引导学生灵活合理地分析、解答分数应用题,成为众多教师教学研究的热点。通过多年的教学实践与研究,笔者总结出数形结合法、等量关系式法、口算法、乘除转换法等几种方法。多种方法的应用,适应了不同学生的需求,使难点得到了突破,学生的思维得到拓展,学生创新意识得到增强,取得了事半功倍的教学效果。

ADRC FRACTIONAL ORDER PID CONTROLLER DESIGN OF HYPERSONIC FLIGHT VEHICLE

Active disturbance rejection controller（ADRC）uses tracking-differentiator（TD）to solve the contradiction between the overshoot and the rapid nature.Fractional order proportion integral derivative（PID）controller improves the control quality and expands the stable region of the system parameters.ADRC fractional order（ADRFO）PID controller is designed by combining ADRC with the fractional order PID and applied to reentry attitude control of hypersonic vehicle.Simulation results show that ADRFO PID controller has better control effect and greater stable region for the strong nonlinear model of hypersonic flight vehicle under the influence of external disturbance,and has stronger robustness against the perturbation in system parameters.

一题多解,人脑胜电脑(续)

姜克华老师撰写的《一题多解,人脑胜电脑》一文于本刊今年第2期(5月号)发表后,在读者中引起较大反响,许多读者来信来电询问有关问题,认为“一题多解”有利于启智研习,尤对激发中小学生数学课的学习兴趣大有助益。姜老于是以年逾八十之高龄冒着八月酷暑又写了“续文”以飨广大读者。

Exact Solutions of Fractional-Order Biological Population Model

In this paper, the Adomian＇s decomposition method （ADM） is presented for finding the exact solutions of a more general biological population models. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided.

Analytic Study on a Plane Stress Tensile Crack in a Power-Law Hardening Material

Aim To construct an analytic solution for the asymptotic field near a tensile cracktip of power-law hardening material under Plane stress condition. Methods Constructing funtion method was used. Results The exact asymptotic field was found. Conclusion The exact analytic solution for the problem is available.

An Algebraic Method for Constructing Exact Solutions to Difference-Differential Equations

In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.

Error Analysis on Corrector Formula for Rectangular Rule

This paper presents truncation errors among Corrector Formula for left Rectangular rule and Corrector Formula for middle Rectangular rule respectively. It also displays an analysis on convergence order of compound corrector formulas for rectangular rule. Examples of numerical calculation have validated theoretical analysis.

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.

Analytical Hopf Bifurcation and Stability Analysis of T System

Complex dynamics are studied in the T system, a three-dimensional autonomous nonlinear system. In particular, we perform an extended Hopf bifurcation analysis of the system. The periodic orbit immediately following the Hopf bifurcation is constructed analytically for the T system using the method of multiple scales, and the stability of such orbits is analyzed. Such analytical results complement the numerical results present in the literature. The analytical results in the post-bifurcation regime are verified and extended via numerical simulations, as well as by the use of standard power spectra, autocorrelation functions, and fractal dimensions diagnostics. We find that the T system exhibits interesting behaviors in many parameter regimes.

Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized Nonlinear Wave Equations

Using the generalized conditional symmetry approach, we obtain a number of new generalized (1+1)-dimensional nonlinear wave equations that admit derivative-dependent functional separable solutions.

Algebraic Method for Constructing Exact Discrete Soliton Solutions of Toda and mKdV Lattices

In this paper, with the aid of symbolic computation, we present a new method for constructing soliton solutions to nonlinear differentiM-difference equations. And we successfully solve Toda and mKdV lattice.

Study on fast calibration of a ULA's gain and phase error

Array calibration is important in engineering practice. In this paper, fast calibration methods for a ULA's gain and phase errors both in far and near fields are proposed. In the far field, using a single sound source without exact orientation, this method horizontally rotates the array exactly once, performs eigen value decomposition for the covariance matrix of received data, then computes the gain and phase error according to the formulas. In the near field, using the same single sound source, it is necessary to rotate the array horizontally at most three times, build equations according to geometric relations, then solve them. Using the formula proposed in this paper, spherical waves are modified into plane waves. Then eigen values decomposition is performed. These two calibration methods were shown to be valid by simulation and are fast, accurate and easy to use. Finally, an analysis of factors influencing estimation precision is given.

Robust user equilibrium model based on cumulative prospect theory under distribution-free travel time

The assumption widely used in the user equilibrium model for stochastic network was that the probability distributions of the travel time were known explicitly by travelers. However, this distribution may be unavailable in reality. By relaxing the restrictive assumption, a robust user equilibrium model based on cumulative prospect theory under distribution-free travel time was presented. In the absence of the cumulative distribution function of the travel time, the exact cumulative prospect value(CPV) for each route cannot be obtained. However, the upper and lower bounds on the CPV can be calculated by probability inequalities.Travelers were assumed to choose the routes with the best worst-case CPVs. The proposed model was formulated as a variational inequality problem and solved via a heuristic solution algorithm. A numerical example was also provided to illustrate the application of the proposed model and the efficiency of the solution algorithm.

Decomposable Algorithm for Large Dynamic Programming

An approach about large dynamic programming based on discrete linear system with a quadratic index function is proposed by importing two Lagrange multipliers.

Approximate Solutions of Perturbed Nonlinear Schroedinger Equations

By applying Lou＇s direct perturbation method to perturbed nonlinear Schroedinger equation and the critical nonlinear SchrSdinger equation with a small dispersion, their approximate analytical solutions including the zero-order and the first-order solutions are obtained. Based on these approximate solutions, the analytical forms of parameters of solitons are expressed and the effects of perturbations on solitons are briefly analyzed at the same time. In addition, the perturbed nonlinear Schroedinger equations is directly simulated by split-step Fourier method to check the validity of the direct perturbation method. It turns out that the analytical results given by the direct perturbation method are well supported by numerical calculations.

Several Properties of p-w-hyponormal Operators

In this paper, let T be a bounded linear operator on a complex Hilbert H. We give and prove that every p-w-hyponormal operator has Bishop's property(β) and spectral properties; Quasi-similar p-w-hyponormal operators have equal spectra and equal essential spectra. Finally, for p-w-hyponormal operators, we give a kind of proof of its normality by use of properties of partial isometry.

On the Acceleration Problem of q-Bernstein Polynomials

In this paper, we investigate not only the acceleration problem of the q-Bernstein polynomials Bn（f, q; x） to B∞ （f, q; x） but also the convergence of their iterated Boolean sum. Using the methods of exact estimate and theories of modulus of smoothness, we get the respective estimates of the convergence rate, which suggest that q-Bernstein polynomials have the similar answer with the classical Bernstein polynomials to these two problems.

Solving Nonlinear Wave Equations by Elliptic Equation

The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.

A NEW PUBLIC-KEY ENCRYPTION SCHEME BASED ON LUCAS SEQUENCE

Investigated the properties of LUCas sequence(LUC), the paper proposed a new variant of (probabilistic) public-key encryption scheme. Security analysis of the proposed encryption schemes shows that its one-wayness is equivalent to partial LUC discrete logarithm problem in ZN, and for the proposed probabilistic encryption scheme, its semantic security is equivalent to decisional LUC Diffie-Hellman problem in ZN. At last, the efficiency of the proposed schemes is briefly analyzed.

Nonfactorizable B→xcoK Decay and QCD Factorization

We study the unexpectedly large rate for the factorization-forbidden decay B →xcoK within the QCD factorization approach. We use a non-zero gluon mass to regularize the infrared divergences in vertex corrections. The end-point singularities arising from spectator corrections are regularized and carefully estimated by the off-shellness of quarks. We find that the contributions arising from the vertex and leading-twist spectator corrections are numerically small, and the twist-3 spectator contribution with chiral enhancement and linear end-point singularity becomes dominant. With reasonable choices for the parameters, the branching ratio for B → XcoK decay is estimated to be in the range （2 ～ 4）× 10^-4, which is compatible with the Belle and BaBar data.