广义最小二乘估计(Generalized least squares estimation,GLSE)是最佳线性无偏估计,却有计算复杂高和依赖未知信息的局限性,使得普通最小二乘估计(Ordinary least squares estimation,OLSE)经常成为应用的无奈之选。本文探讨该现象背...广义最小二乘估计(Generalized least squares estimation,GLSE)是最佳线性无偏估计,却有计算复杂高和依赖未知信息的局限性,使得普通最小二乘估计(Ordinary least squares estimation,OLSE)经常成为应用的无奈之选。本文探讨该现象背后的三个循序渐进的理论问题:第一,GLSE的退化问题,给出GLSE完全退化为OLSE的充要条件;第二,退化的分类问题,依据设计矩阵和误差协方差阵的结构把退化现象分为三类,并给出典型的退化特例;第三,不完全退化问题,研讨导致效率退化的因素,刻画效率曲线和效率曲面,最后给出效率不低于95%的退化边界。效率退化和边界分析的潜在应用价值主要包括两方面:第一,为进一步优化试验方案提供效率视角和反馈信息;第二,为设计更简洁更可靠的算法提供理论依据。展开更多
在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中...在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中值定理在证明导数极限定理、求函数极限问题、证明不等式以及证明函数单调性方面的应用,以及拉格朗日中值定理的两个推广。希望本文可以对学生学习微积分有所帮助。During the teaching process, we found that the Lagrange Mean Value Theorem is a significant obstacle for students learning calculus. The Lagrange Mean Value Theorem is the core content of the Mean Value Theorem in differential calculus. It is a theoretical tool for studying the relationship between functions and their derivatives and plays a crucial role in calculus, with a wide range of applications. This paper focuses on the application of the Lagrange Mean Value Theorem in proving the derivative limit theorem, solving limit problems of functions, proving inequalities, and proving the monotonicity of functions, as well as two extensions of the Lagrange Mean Value Theorem. It is hoped that this article can be of assistance to students in their study of calculus.展开更多
通过研究非自治传染病SIRS模型解的存在性和建立迭代算法,分析模型的特殊性质,引入特殊性质的函数,将模型转换为积分系统,并构造所需的迭代序列,证明这个序列收敛于模型的解。可以证明,在一定时间内,SIRS模型具有唯一解,解和近似解之间...通过研究非自治传染病SIRS模型解的存在性和建立迭代算法,分析模型的特殊性质,引入特殊性质的函数,将模型转换为积分系统,并构造所需的迭代序列,证明这个序列收敛于模型的解。可以证明,在一定时间内,SIRS模型具有唯一解,解和近似解之间能进行误差估计。该算法克服模型中非线性项可取负值和不满足Lipschitz条件的困难,证明了SIRS模型解存在,且可使用迭代算法求出和进行误差估计。This paper studies the existence and iterative algorithms of solutions to the SIRS model of non- autonomous infectious diseases. By analyzing the special properties of the model and introducing the function with special properties, the SIRS model is changed to an integral system, and the required iterative sequence is constructed. It is proved that this sequence converges to the solution of the model. It is proved that the SIRS model has a unique solution within a certain period of time, and the error estimations between the exact solution and the approximate solution are established. By overcoming the difficulties that the nonlinear terms in the model may take negative values and do not satisfy the Lipschitz condition, it is proved that the solution of the SIRS model can be obtained by an iterative method, and the error estimation can be performed.展开更多
文摘广义最小二乘估计(Generalized least squares estimation,GLSE)是最佳线性无偏估计,却有计算复杂高和依赖未知信息的局限性,使得普通最小二乘估计(Ordinary least squares estimation,OLSE)经常成为应用的无奈之选。本文探讨该现象背后的三个循序渐进的理论问题:第一,GLSE的退化问题,给出GLSE完全退化为OLSE的充要条件;第二,退化的分类问题,依据设计矩阵和误差协方差阵的结构把退化现象分为三类,并给出典型的退化特例;第三,不完全退化问题,研讨导致效率退化的因素,刻画效率曲线和效率曲面,最后给出效率不低于95%的退化边界。效率退化和边界分析的潜在应用价值主要包括两方面:第一,为进一步优化试验方案提供效率视角和反馈信息;第二,为设计更简洁更可靠的算法提供理论依据。
文摘在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中值定理在证明导数极限定理、求函数极限问题、证明不等式以及证明函数单调性方面的应用,以及拉格朗日中值定理的两个推广。希望本文可以对学生学习微积分有所帮助。During the teaching process, we found that the Lagrange Mean Value Theorem is a significant obstacle for students learning calculus. The Lagrange Mean Value Theorem is the core content of the Mean Value Theorem in differential calculus. It is a theoretical tool for studying the relationship between functions and their derivatives and plays a crucial role in calculus, with a wide range of applications. This paper focuses on the application of the Lagrange Mean Value Theorem in proving the derivative limit theorem, solving limit problems of functions, proving inequalities, and proving the monotonicity of functions, as well as two extensions of the Lagrange Mean Value Theorem. It is hoped that this article can be of assistance to students in their study of calculus.
文摘通过研究非自治传染病SIRS模型解的存在性和建立迭代算法,分析模型的特殊性质,引入特殊性质的函数,将模型转换为积分系统,并构造所需的迭代序列,证明这个序列收敛于模型的解。可以证明,在一定时间内,SIRS模型具有唯一解,解和近似解之间能进行误差估计。该算法克服模型中非线性项可取负值和不满足Lipschitz条件的困难,证明了SIRS模型解存在,且可使用迭代算法求出和进行误差估计。This paper studies the existence and iterative algorithms of solutions to the SIRS model of non- autonomous infectious diseases. By analyzing the special properties of the model and introducing the function with special properties, the SIRS model is changed to an integral system, and the required iterative sequence is constructed. It is proved that this sequence converges to the solution of the model. It is proved that the SIRS model has a unique solution within a certain period of time, and the error estimations between the exact solution and the approximate solution are established. By overcoming the difficulties that the nonlinear terms in the model may take negative values and do not satisfy the Lipschitz condition, it is proved that the solution of the SIRS model can be obtained by an iterative method, and the error estimation can be performed.