如何解综合梯度题

近几年来,高考数学试卷的布局和编排,没有固守传统的做法,出现了一些变化,即难点分散.例如,2003年的全国高考数学试卷,不再是"一题压轴尾巴高翘",全卷的难度梯次不再严格的由易到难.另一方面增加了要求较高层次能力的梯度题,加强了对学生的数学思想和转化能力的考查.下面结合实践谈谈解综合梯度题的一些策略.

题组引领 梯度推进——例谈题组梯度复习法

“温故而知新”的古训我们早已铭记在心，“温故”即复习之意，数学要学得深透，复习是其中不可或缺的重要环节．复习既担负着将平时相对独立的知识点“串成线、连成片、结成网”的重任，又承载了反馈矫正、优化思维品质的功能．给出了“题组梯度复习法”，即根据复习的内容。精心设计一组逐层递进的题目，将知识点镶嵌于练习之中，沿着学生的认知轨道展开复习，一般包括再现巩固题组、认知集成题组、深化理解题组、反馈演练题组．

小学数学问题串与练习题梯度设计促进学生深度参与学习策略探讨

对于问题串来说,也就是结合教学方案和目标,根据教学内容学生实际情况而设计的各种教学问题,在教学问题的引导下,帮助学生自主学习,从中掌握更多的专业知识,提高学生学习素养。梯度题也是围绕教学目标,按照由浅入深、循序渐进的原则所设计的习题模式。可见,问题串和梯度题的教学理想是相通的。基于此,本文就结合理论与实际,探讨如何通过问题串和梯度题两者结合,进行教学设计,促进其全面发展。

The UNIFORM C^(0)ESTIMATE AND WEIGHTED ESTIMATE OF GENERALIZED CHRISTOFFEL-MINKOWSKI PROBLEMS

In this paper,we consider generalized Christo®el-Minkowski problems as followsσ_(k)(u_(ij)+uδ_(ij))/σ_(l)(u_(ij)+uδ_(ij))=|u^(p-1)f(x),x∈S^(n),where 0≤l≤k≤n,p-1>0 and f is positive,and we establish the weighted gradient estimate and uniform C^(0)estimate for the positive convex even solutions,which is a generalization of Guan-Xia[1]and Guan[2].

Pure bending of simply supported circular plate of transversely isotropic functionally graded material

This paper considers the pure bending problem of simply supported transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate. First, the partial differential equation, which is satisfied by the stress functions for the axisymmetric deformation problem is derived. Then, stress functions are obtained by proper manipulation. The analytical expressions of axial force, bending moment and displacements are then deduced through integration. And then, stress functions are employed to solve problems of transversely isotropic functionally graded circular plate, with the integral constants completely determined from boundary conditions. An elasticity solution for pure bending problem, which coincides with the available solution when degenerated into the elasticity solutions for homogenous circular plate, is thus obtained. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a simply supported circular plate of transversely isotropic functionally graded material (FGM).

Application of a Derivative-Free Method with Projection Skill to Solve an Optimization Problem

Improving numerical forecasting skill in the atmospheric and oceanic sciences by solving optimization problems is an important issue. One such method is to compute the conditional nonlinear optimal perturbation(CNOP), which has been applied widely in predictability studies. In this study, the Differential Evolution(DE) algorithm, which is a derivative-free algorithm and has been applied to obtain CNOPs for exploring the uncertainty of terrestrial ecosystem processes, was employed to obtain the CNOPs for finite-dimensional optimization problems with ball constraint conditions using Burgers' equation. The aim was first to test if the CNOP calculated by the DE algorithm is similar to that computed by traditional optimization algorithms, such as the Spectral Projected Gradient(SPG2) algorithm. The second motive was to supply a possible route through which the CNOP approach can be applied in predictability studies in the atmospheric and oceanic sciences without obtaining a model adjoint system, or for optimization problems with non-differentiable cost functions. A projection skill was first explanted to the DE algorithm to calculate the CNOPs. To validate the algorithm, the SPG2 algorithm was also applied to obtain the CNOPs for the same optimization problems. The results showed that the CNOPs obtained by the DE algorithm were nearly the same as those obtained by the SPG2 algorithm in terms of their spatial distributions and nonlinear evolutions. The implication is that the DE algorithm could be employed to calculate the optimal values of optimization problems, especially for non-differentiable and nonlinear optimization problems associated with the atmospheric and oceanic sciences.

A Probe Method of Gradient Projection Type

In this paper,a probe method for nonlinear programming wiht equality and inequality is given. Its iterative directions at an arbitrary point x can be obtained through solving a liear system. The terminate conditions and choices of the parameters are given. The global convergence of the method is proved. Further more,some well known gradient projection type algorithms [1-15] and new gradient projection type algorithms from the linear system are given in this paper.

High Slope Stability of Diversion Power System Intake of Jinchuan Hydropower Station

The excavated height of the left bank slope of the diversion power system intake in Jinchuan hydropower station is about 16o m. The stability and safety of the slope during construction and its operation/utilization become one of the most important geological engineering problems. At the same time, it is also crucial to select a safe and economic excavation gradient for the construction. We studied the problem of how to select a safe and economic slope ratio by analyzing the geological condition of the high slope, including the lithology, slope structure, structural surface and their combinations, rock weathering and unloading, hydrology, and the natural gradient. The study results showed that the use of an excavation gradient larger than the gradient observed during site investigation and the gradient recommended in standards and field practice manuals is feasible. Then, we used the finite element method and rigid limit equilibrium method to evaluate the stability of the excavation slope under natural, rainstorm and earthquake conditions. The calculated results showed that the excavated slope only has limited failure, but its stability is greatly satisfactory. The research findings can be useful in excavation and slope stabilization projects.

Mechanical Properties of Graphene within the Framework of Gradient Theory of Adhesion

The gradient model of two-dimensional defectless medium is formulated. A graphene sheet is examined as an example of such two-dimensional medium. The problem statement of a graphene sheet deforming in its plane and the bending problem are examined. It is ascertained that the statement of the first problem is equivalent to the flat problem statement of Toupin gradient theory. The statement of the bending problem is equivalent to the plate bending theory of Timoshenko with certain reserves. The characteristic feature of both statements is the fact that the mechanical properties of the sheet of graphene are not defined by “volumetric” moduli but by adhesive ones which have different physical dimension that coincides with the dimension of the corresponding stiffness of classical and nonclassical plates.

Constraints in Lean Optimization Method Tested on a Real Engineering Problem

Optimizing a vehicle includes testing millions of parameters with hundreds of constraints of the performance. In this article, 162 parameters are optimized with 5 constraints using Lean Optimization combined with Linear Programming. The method converges in this example in about 100 evaluations. This is less than the gradient method needs for its first step.

Optimal design of functionally graded Pm PV/CNT nanocomposite cylindrical tube for purpose of torque transmission

Carbon nanotube(CNT)/polymer nanocomposites have vast application in industry because of their light mass and high strength. In this work, a cylindrical tube which is made up of functionally graded(FG) PmP V/CNT nanocomposite, is optimally designed for the purpose of torque transmission. The main confining parameters of a rotating shaft in torque transmission process are mass of the shaft, critical speed of rotation and critical buckling torque. It is required to solve a multi-objective optimization problem(MOP) to consider these three targets simultaneously in the process of design. The three-objective optimization problem for this case is defined and solved using a hybrid method of FEM and modified non-dominated sorting genetic algorithm(NSGA-II), by coupling two softwares, MATLAB and ABAQUS. Optimization process provides a set of non-dominated optimal design vectors. Then, two methods, nearest to ideal point(NIP) and technique for ordering preferences by similarity to ideal solution(TOPSIS), are employed to choose trade-off optimum design vectors. Optimum parameters that are obtained from this work are compared with the results of previous studies for similar cylindrical tubes made from composite or a hybrid of aluminum and composite that more than 20% improvement is observed in all of the objective functions.

Research on Optimizing Computer Network Structure based on Genetic Algorithm and Modified Convex Optimization Theory

In this paper, we report in-depth analysis and research on the optimizing computer network structure based on genetic algorithm and modified convex optimization theory. Machine learning method has been widely used in the background and one of its core problems is to solve the optimization problem. Unlike traditional batch algorithm, stochastic gradient descent algorithm in each iteration calculation, the optimization of a single sample point only losses could greatly reduce the memory overhead. The experiment illustrates the feasibility of our proposed approach.

Sound absorption coefficient optimization of gradient sintered metal fiber felts

An optimization method for sound absorption of gradient(multi-layered) sintered metal fiber felts is presented. The theoretical model based on dynamic flow resistivity is selected to calculate the sound absorption coefficient of the sintered metal fiber felts since it only requires three key morphological parameters: fiber diameter, porosity and layer thickness. The model predictions agree well with experimental measurements. Objective functions and constraint conditions are then set up to optimize separately the distribution of porosity, fiber diameter, and simultaneous porosity and fiber diameter in the metal fiber. The optimization problem for either a sole frequency or a pre-specified frequency range is solved using a genetic algorithm method. Acoustic performance comparison between optimized and non-optimized metal fibers is presented to confirm the effectiveness of the optimization method. Gradient sintered metal fiber felts hold great potential for noise control applications particularly when stringent restriction is placed on the total volume and/or weight of sound absorbing material allowed to use.

An improved nonlinear conjugate gradient method with an optimal property

Conjugate gradient methods have played a special role in solving large scale nonlinear problems. Recently, the author and Dai proposed an efficient nonlinear conjugate gradient method called CGOPT, through seeking the conjugate gradient direction closest to the direction of the scaled memoryless BFGS method. In this paper, we make use of two types of modified secant equations to improve CGOPT method. Under some assumptions, the improved methods are showed to be globally convergent. Numerical results are also reported.

Aerodynamic design optimization by using a continuous adjoint method

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows.General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived.With the adjoint method,the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function,regardless of the number of design parameters.An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method.Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil,wing,and wing-body configuration,and the aerodynamic performance improvement of turbine and compressor blade rows.The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.

Inverse Heat Transfer Problem of Thermal Contact Conductance Estimation in Periodically Contacting Surfaces

The problems involving periodic contacting surfaces have different practical applications. An inverse heat conductionproblem for estimating the periodic Thermal Contact Conductance (TCC) between one-dimensional, constantproperty contacting solids has been investigated with conjugate gradient method (CGM) of function estimation.This method converges very rapidly and is not so sensitive to the measurement errors. The advantage of thepresent method is that no a priori information is needed on the variation of the unknown quantities, since the solutionautomatically determines the functional form over the specified domain. A simple, straight forward techniqueis utilized to solve the direct, sensitivity and adjoint problems, in order to overcome the difficulties associatedwith numerical methods. Two general classes of results, the results obtained by applying inexact simulatedmeasured data and the results obtained by using data taken from an actual experiment are presented. In addition,extrapolation method is applied to obtain actual results. Generally, the present method effectively improves theexact TCC when exact and inexact simulated measurements input to the analysis. Furthermore, the results obtainedwith CGM and the extrapolation results are in agreement and the little deviations can be negligible.

Inverse Coefficient Problems for Elliptic Hemivariational Inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.

Linear response eigenvalue problem solved by extended locally optimal preconditioned conjugate gradient methods

The locally optimal block preconditioned 4-d conjugate gradient method(LOBP4dC G) for the linear response eigenvalue problem was proposed by Bai and Li(2013) and later was extended to the generalized linear response eigenvalue problem by Bai and Li(2014). We put forward two improvements to the method: A shifting deflation technique and an idea of extending the search subspace. The deflation technique is able to deflate away converged eigenpairs from future computation, and the idea of extending the search subspace increases convergence rate per iterative step. The resulting algorithm is called the extended LOBP4 dC G(ELOBP4dC G).Numerical results of the ELOBP4 dC G strongly demonstrate the capability of deflation technique and effectiveness the search space extension for solving linear response eigenvalue problems arising from linear response analysis of two molecule systems.

A NEW DESCENT MEMORY GRADIENT METHOD AND ITS GLOBAL CONVERGENCE

In this article, a new descent memory gradient method without restarts is proposed for solving large scale unconstrained optimization problems. The method has the following attractive properties： 1） The search direction is always a sufficiently descent direction at every iteration without the line search used; 2） The search direction always satisfies the angle property, which is independent of the convexity of the objective function. Under mild conditions, the authors prove that the proposed method has global convergence, and its convergence rate is also investigated. The numerical results show that the new descent memory method is efficient for the given test problems.

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.