数学建模素养发展:从“咬文嚼字”开始——一道例题的教学感悟

数学建模是对现实问题进行数学抽象,用数学语言表述问题,用数学方法构建模型解决问题的素养.对于初中数学教学而言,让学生从数学问题中抽象出数量关系,建构出方程、不等式、函数等模型来解决问题,是本学段中发展数学建模素养的基本途径.     

初中数学建模对高中数学教学的意义

数学建模是一个创造性的思维过程,数学建模的教学内容、教学方法、教学原则都围绕着一个培养创新人才的主题而进行,目的是学生真正学到"有用的数学",懂得数学是人类文化的重要组成部分,数学与人类生活有密切的联系.它与培养学生的创造性思维是相辅相成、辩证统一的.在初中数学教学中构建学生建模意识十分重要,是实现初中阶段数学课程目标的策略要求,又对后续高中数学的学习有着重要的意义.一、初高中数学建模知识内涵与思想方法的传承与发展初中数学建模常用到6类模型:方程(组)模型、不等式(组)模型、函数模型、几何或三角     

基于信息技术的数学建模教学研究

运用信息技术进行模型求解是数学教育的重要内容.“三门问题”通俗易懂却反直觉,通过分析“三门问题”建模过程,提炼出建模的关键在于构造出抽奖盒子、策略1和策略2三个主要模拟函数,基于分析设计了“三门问题”数学建模项目式学习的教学设计.研究的意义在于一方面是对高中数学教材数学建模主题案例较少的补充,另一方面也为教师运用信息技术进行数学问题仿真模拟提供一定的参考.     

具有类反斜线回滞执行器的非线性系统的自适应控制

针对类反斜线回滞非线性系统,设计了自适应控制器,保持系统稳定并实现对参考信号的任意精度跟踪.在控制器设计中,通过构造足够光滑的非线性函数作为符号函数的逼近,从而解决了传统自适应控制器设计中由于符号函数的引入而导致控制器不连续的问题,避免了因此而产生的抖震现象.在系统模型中充分考虑未建模动态,使模型更具一般性.最后应用MATLAB软件进行仿真实验,结果进一步验证了该控制器的有效性.     

基于广义函数空间的不连续梁振动分析

首先运用广义函数建立了轴向力作用下含任意不连续点的弹性基础Euler(欧拉)梁的自由振动的统一微分方程.不连续点的影响由广义函数(Dirac delta函数)引入梁的振动方程.微分方程运用Laplace变换方法求解;与传统方法不同的是,该文方法求得的模态函数为整个不连续梁的一般解.由于模态函数的统一化以及连续条件的退化,特征值的求解得到了极大地简化.最后,以梁-质量块模型和轴向力作用下弹性基础裂纹梁模型为例验证了该文方法的正确性与有效性.     

基于因子分析的学生数学建模能力结构研究

通过编制测试卷对深圳市龙岗区17327名初中生进行数学建模能力测试.根据测试数据,利用探索性因素分析,探索初中生数学建模能力的整体结构和组成因素,对于探索出的数学建模能力结构模型分别利用一阶和二阶验证性因素分析验证其合理存在性.研究表明,我国初中生数学建模能力包含模型建立能力,模型阐释能力,模型计算能力和模型评估能力四个组成因素,每个组成要素包含若干数学基本技能,该能力模型可以与现有数学建模能力理论形成三角互证.最后,根据初中生的数学建模能力结构模型,提出三个相应的教学策略,以期帮助学生提高数学建模能力.     

数学建模中的创新意识培养

针对数学建模中模型建立与求解以及模型的评价与推广这两个环节,提出了"旁观者"与"当局者"这两个角色,并指出优秀数学建模参与者应该具备"从旁观者到当局者"以及"从当局者再回到旁观者"的双向角色转换能力.同时,还从形式和内容两个方面对数学建模优秀论文的共性进行了分析,并联系数学建模的两个环节,指出了优秀论文和数学建模过程中两个环节这两者与角色转换的紧密关系.最后,对如何培训学生成为一个优秀的数学建模参与者提供了一些参考和建议.     

例谈函数应用问题中的“误区”

函数的应用问题主要是指将实际问题转化为函数问题,就是“数学建模”,它是解决数学应用题的重要方法．在建模时常会因出现“忽视从实际出发”、“理解不全面”、“与事实不符”等几种解题误区,下面就函数应用问题中的这几个误区进行举例分析：     

数学建模思想在高中数学教学中的应用探析

数学建模是高中数学的六大核心素养之一.在高中数学教学中如何应用数学建模思想是亟待加强研究的重要课题.本文从函数、几何与代数、概率与统计等几个方面探索了数学建模思想实施的方法与步骤.     

最小正基直接搜索共轭梯度法

1引言直接搜索算法是求解最优化问题的一类重要算法,它特别适用于目标函数的梯度信息无法得到或很难计算的情形.直接搜索算法大致可以分为两大类,一类是包含单纯形搜索[19]和模式搜索[12,22]等的直接搜索算法,这类算法不使用泰勒展开等方法对目标函数进行建模;另一类算法则依赖于对目标函数的建模,常被称为以模型为基础的方法,有     

Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication

This paper investigates drive-response synchronization of chaotic systems with discontinuous right-hand side. Firstly, a general model is proposed to describe most of known discontinuous chaotic system with or without time-varying delay. An uniform impulsive controller with multiple unknown time-varying delays is designed such that the response system can be globally exponentially synchronized with the drive system. By utilizing a new lemma on impulsive differential inequality and the Lyapunov functional method, several synchronization criteria are obtained through rigorous mathematical proofs. Results of this paper are universal and can be applied to continuous chaotic systems. Moreover, numerical examples including discontinuous chaotic Chen system, memristor-based Chua’s circuit, and neural networks with discontinuous activations are given to verify the effectiveness of the theoretical results. Application of the obtained results to secure communication is also demonstrated in this paper.     

General existence principles for Stieltjes differential equations with applications to mathematical biology

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.     

Designing the shape of a large scientific balloon

Large scientific balloons are used to carry out research in the upper atmosphere. By Archimedes' principle, a balloon in equilibrium at a fixed altitude must displace an amount of air equal to its weight. The system that we model here includes the balloon film, reinforcing caps, load tapes and payload. We consider two mathematical models for the design shape of a balloon. One is a design in the shape of an ellipsoid-on-cone. The equilibrium condition gives rise to a cubic equation whose solution gives the unique shape for a given set of design parameters. A second model is the natural-shape design, a system of nonlinear ordinary differential equations that is commonly used for modeling the shape of large scientific balloons. In both models, caps are included as an added thickness, thus the resulting film weight density is discontinuous. The discontinuities in the natural-shape model are handled with a parallel shooting method. We present numerical solutions for various design parameters and consider issues related to the efficient design of a balloon.     

2010年全国研究生数模竞赛D题的评阅综述

2010年全国研究生数学建模竞赛D题的素材取自一项科研项目.将本题建为何种模型取决于建模人的数学基础和对本题描述运动过程的认识深度.评述介绍了三种建模思路:刚体平面运动模型,最优控制模型和微分方程模型,其中微分方程模型是在阅卷过程中从考生答卷中发现的,从答卷中发现了不少有共性的毛病,反应出国内大学和研究生数学教育中的弊病,值得有识之士们重视.     

An optimal switching approach toward cost‐effective control of a stand‐alone photovoltaic panel system under stochastic environment

The operation of a stand‐alone photovoltaic (PV) system ultimately aims for the optimization of its energy storage. We present a mathematical model for cost‐effective control of a stand‐alone system based on a PV panel equipped with an angle adjustment device. The model is based on viscosity solutions to partial differential equations, which serve as a new and mathematically rigorous tool for modeling, analyzing, and controlling PV systems. We formulate a stochastic optimal switching problem of the panel angle, which is here a binary variable to be dynamically controlled under stochastic weather condition. The stochasticity comes from cloud cover dynamics, which is modeled with a nonlinear stochastic differential equation. In finding the optimal control policy of the panel angle, switching the angle is subject to impulsive cost and reduces to solving a system of Hamilton‐Jacobi‐Bellman quasi‐variational inequalities (HJBQVIs). We show that the stochastic differential equation is well posed and that the HJBQVIs admit a unique viscosity solution. In addition, a finite‐difference scheme is proposed for the numerical discretization of HJBQVIs. A demonstrative computational example of the HJBQVIs, with emphasis on a stand‐alone experimental system, is finally presented with practical implications for its cost‐effective operation.     

A Soft-Computing Algorithm-based Cognitive Observer for an Elastic Robotic Arm

This paper proposes the use of Support Vector Machine (SVM) algorithm for modeling and states estimation of an elastic robotic arm. Due to the complexity of the elastic robotic arm, an accurate mathematical model and large number of sensors are needed to achieve accurate estimation. Initially, it is assumed that all system states are measurable for a short period of time. Additionally, the system output and input signals are being continuously measured. The modeling module builds two models, the internal model which will capture the dynamics of the elastic robotic arm and the data-driven observer model which will estimate the system states. The internal model and data-driven observer are obtained based on the experimental measurements using SVM algorithm in contrast to classical methods that uses the mathematical system model to drive an observer. The internal model is able to make multi-steps ahead predictions of all system states; therefore it can be used to generate a suitable control strategy. Once the models are ready, the main sensors can be removed or turned off. The proposed modeling module will eliminate the need for mathematical modeling and reduces the number of permanent sensors needed. If the main sensors are removed completely, the hardware price can be radically reduced. The simulation result demonstrates the efficiency and high performance of the modeling module. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized solutions to a linear discontinuous differential equation

In this paper we construct generalized solutions (weak solutions) to a linear discontinuous differential equation which appears as mathematical model in mechanics and for which usual methods are not quite adequate. The result is applied to the equation of an elastic beam on a Winkler type of foundation.     

Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set‐valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non‐autonomous biological and mathematical model with a discontinuous right‐hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

考虑不同商业目标和公平关切的动态博弈模型及复杂性研究

建立双寡头零售商具有不同商业目标和公平关切的动态价格博弈模型,着重分析了基于零售商考虑不同商业目标和公平关切下的价格博弈模型的复杂性。数值模拟了不同参数数值组合条件下的价格动态博弈过程,通过系统稳定域,分岔,李雅普诺夫指数,混沌吸引子等对模型进行了复杂性分析,发现零售商考虑公平关切会使自身稳定域减小;公平关切的水平越高,系统越容易进入混沌状态。同时研究了价格调整速度对零售商利润的影响,结果发现当价格调整速度过大时,系统会进入混沌状态,利润值波动剧烈且平均利润随着价格调整速度的增大而减小。最后选择控制因子对系统混沌进行了控制,该研究对零售商价格决策有着很好的借鉴意义。     

Indifference pricing and hedging in a multiple-priors model with trading constraints

This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.