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
微元法是分析解决物理问题的常用方法,是一种从部分到整体的思维方法,通过将问题分解为众多微小的“元过程”,运用熟悉的物理规律解决问题,从而使问题简单化。微元法中“元”的寻找是解决问题的关键,对一些特殊的积分区域,如圆柱面,圆柱...微元法是分析解决物理问题的常用方法,是一种从部分到整体的思维方法,通过将问题分解为众多微小的“元过程”,运用熟悉的物理规律解决问题,从而使问题简单化。微元法中“元”的寻找是解决问题的关键,对一些特殊的积分区域,如圆柱面,圆柱体,球体,本文打破常规的微元分割,找到灵活的“微元(——弧棒,柱壳,球壳)”,实现积分运算的简化。The differential element method is a commonly used method for analyzing and solving physical problems. It is a way of thinking that moves from part to whole by breaking down problems into numerous small “element processes” and applying familiar physical laws to solve them, simplifying the problem. The search for “elements” in the differential element method is the key to solving problems. For some special integration regions, such as cylindrical surfaces, cylinders, and spheres, this article breaks away from conventional finite element segmentation and finds flexible “differential elements” (arc rods, cylindrical shells, spherical shells) to simplify integration operations.展开更多
在地震资料反演中,反褶积是一种重要的压缩地震子波、提高薄层纵向分辨率的地震数据处理方法。由于地层为层状结构,反射系数可视作稀疏的脉冲序列,因此地震反褶积可以描述为稀疏求解问题。然而,反褶积问题通常是病态的,需要引入正则化...在地震资料反演中,反褶积是一种重要的压缩地震子波、提高薄层纵向分辨率的地震数据处理方法。由于地层为层状结构,反射系数可视作稀疏的脉冲序列,因此地震反褶积可以描述为稀疏求解问题。然而,反褶积问题通常是病态的,需要引入正则化约束以获得稳定和准确的解。本研究介绍了几种不同的正则化方法,包括L1正则化、L2正则化、Cauchy正则化以及结合L1和L2正则化的方法,给出了它们的数学模型,并着重比较了Cauchy正则化与结合L1和L2正则化的方法。通过简单的一维模型和复杂的Marmousi2 (二维)模型的实验,我们评估了这些正则化方法在稀疏脉冲反褶积中的性能表现。结果表明,结合L1和L2正则化的联合方法在噪声抑制和分辨率提升方面表现优异,能够更准确地恢复地下结构的真实反射特性。本文的研究为选择适当的正则化策略以优化地震数据的反褶积处理提供了理论支持和实用指导。In seismic data inversion, deconvolution is an important seismic data processing method that compresses seismic wavelets and improves the vertical resolution of thin layers. Due to the layered structure of the strata, the reflection coefficient can be regarded as a sparse pulse sequence, so seismic deconvolution can be described as a sparse solution problem. However, deconvolution problems are often pathological and require the introduction of regularization constraints to obtain stable and accurate solutions. This study introduces several different regularization methods, including L1 regularization, L2 regularization, Cauchy regularization, and methods combining L1 and L2 regularization. Their mathematical models are given, and the comparison between Cauchy regularization and methods combining L1 and L2 regularization is emphasized. We evaluated the performance of these regularization methods in sparse pulse deconvolution through experiments using a simple one-dimensional model and a complex Marmousi2 (two-dimensional) model. The results show that the joint method combining L1 and L2 regularization performs well in noise suppression and resolution improvement, and can more accurately restore the true reflection characteristics of underground structures. This study provides theoretical support and practical guidance for selecting appropriate regularization strategies to optimize the deconvolution processing of seismic data.展开更多
非线性Ablowitz-Kaup-Newell-Segur方程是一类应用广泛的非线性偏微分方程。(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程常用于描述孤立波在光纤中传播的物理过程,本文利用复行波变换和扩展的Tanh-函数展开法,获得了(2 + 1)维...非线性Ablowitz-Kaup-Newell-Segur方程是一类应用广泛的非线性偏微分方程。(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程常用于描述孤立波在光纤中传播的物理过程,本文利用复行波变换和扩展的Tanh-函数展开法,获得了(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程的系列新的精确行波解。The Ablowitz-Kaup-Newell-Segur (AKNS) equations, a class of nonlinear partial differential equations, find their utility in a wide array of applications. The space-time fractional (2 + 1)-dimensional AKNS equation, in particular, is capable of describing the physical process of solitary wave propagation in optical fibers. A new class of exact traveling wave solutions of (2 + 1)-dimensional generalized fractional AKNS equation are obtained by employing complex traveling wave transformation and extended Tanh expansion method.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
传统Pearson相关系数计算公式具有不稳健性,离群值的存在会导致计算结果与实际不符。针对此问题,文章给出了一种稳健估计方法。在模拟样本量分别为20、50、100、200,污染率分别为1%、5%、10%情形下,比较传统相关系数值与稳健相关系数值...传统Pearson相关系数计算公式具有不稳健性,离群值的存在会导致计算结果与实际不符。针对此问题,文章给出了一种稳健估计方法。在模拟样本量分别为20、50、100、200,污染率分别为1%、5%、10%情形下,比较传统相关系数值与稳健相关系数值,发现:稳健相关系数公式正确率均显著高于传统相关系数。在实例分析中进一步验证了稳健相关系数的可行性和有效性。文章研究结论可用于含离群值变量的相关系数稳健估计。The traditional Pearson correlation coefficient calculation formula is not robust, and the existence of outliers will cause the calculation results to be inconsistent with reality. To solve this problem, this paper presents a robust estimation method. When the simulated sample size is 20, 50, 100 and 200 respectively, the pollution rate is 1%, 5% and 10% respectively, it is found that the accuracy of the robust correlation coefficient formula is significantly higher than that of the traditional correlation coefficient. The feasibility and effectiveness of a robust correlation coefficient are further verified in the example analysis. The conclusions of this paper can be used for robust estimation of correlation coefficients with outlier variables.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。
文摘微元法是分析解决物理问题的常用方法,是一种从部分到整体的思维方法,通过将问题分解为众多微小的“元过程”,运用熟悉的物理规律解决问题,从而使问题简单化。微元法中“元”的寻找是解决问题的关键,对一些特殊的积分区域,如圆柱面,圆柱体,球体,本文打破常规的微元分割,找到灵活的“微元(——弧棒,柱壳,球壳)”,实现积分运算的简化。The differential element method is a commonly used method for analyzing and solving physical problems. It is a way of thinking that moves from part to whole by breaking down problems into numerous small “element processes” and applying familiar physical laws to solve them, simplifying the problem. The search for “elements” in the differential element method is the key to solving problems. For some special integration regions, such as cylindrical surfaces, cylinders, and spheres, this article breaks away from conventional finite element segmentation and finds flexible “differential elements” (arc rods, cylindrical shells, spherical shells) to simplify integration operations.
文摘在地震资料反演中,反褶积是一种重要的压缩地震子波、提高薄层纵向分辨率的地震数据处理方法。由于地层为层状结构,反射系数可视作稀疏的脉冲序列,因此地震反褶积可以描述为稀疏求解问题。然而,反褶积问题通常是病态的,需要引入正则化约束以获得稳定和准确的解。本研究介绍了几种不同的正则化方法,包括L1正则化、L2正则化、Cauchy正则化以及结合L1和L2正则化的方法,给出了它们的数学模型,并着重比较了Cauchy正则化与结合L1和L2正则化的方法。通过简单的一维模型和复杂的Marmousi2 (二维)模型的实验,我们评估了这些正则化方法在稀疏脉冲反褶积中的性能表现。结果表明,结合L1和L2正则化的联合方法在噪声抑制和分辨率提升方面表现优异,能够更准确地恢复地下结构的真实反射特性。本文的研究为选择适当的正则化策略以优化地震数据的反褶积处理提供了理论支持和实用指导。In seismic data inversion, deconvolution is an important seismic data processing method that compresses seismic wavelets and improves the vertical resolution of thin layers. Due to the layered structure of the strata, the reflection coefficient can be regarded as a sparse pulse sequence, so seismic deconvolution can be described as a sparse solution problem. However, deconvolution problems are often pathological and require the introduction of regularization constraints to obtain stable and accurate solutions. This study introduces several different regularization methods, including L1 regularization, L2 regularization, Cauchy regularization, and methods combining L1 and L2 regularization. Their mathematical models are given, and the comparison between Cauchy regularization and methods combining L1 and L2 regularization is emphasized. We evaluated the performance of these regularization methods in sparse pulse deconvolution through experiments using a simple one-dimensional model and a complex Marmousi2 (two-dimensional) model. The results show that the joint method combining L1 and L2 regularization performs well in noise suppression and resolution improvement, and can more accurately restore the true reflection characteristics of underground structures. This study provides theoretical support and practical guidance for selecting appropriate regularization strategies to optimize the deconvolution processing of seismic data.
文摘非线性Ablowitz-Kaup-Newell-Segur方程是一类应用广泛的非线性偏微分方程。(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程常用于描述孤立波在光纤中传播的物理过程,本文利用复行波变换和扩展的Tanh-函数展开法,获得了(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程的系列新的精确行波解。The Ablowitz-Kaup-Newell-Segur (AKNS) equations, a class of nonlinear partial differential equations, find their utility in a wide array of applications. The space-time fractional (2 + 1)-dimensional AKNS equation, in particular, is capable of describing the physical process of solitary wave propagation in optical fibers. A new class of exact traveling wave solutions of (2 + 1)-dimensional generalized fractional AKNS equation are obtained by employing complex traveling wave transformation and extended Tanh expansion method.
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.
文摘传统Pearson相关系数计算公式具有不稳健性,离群值的存在会导致计算结果与实际不符。针对此问题,文章给出了一种稳健估计方法。在模拟样本量分别为20、50、100、200,污染率分别为1%、5%、10%情形下,比较传统相关系数值与稳健相关系数值,发现:稳健相关系数公式正确率均显著高于传统相关系数。在实例分析中进一步验证了稳健相关系数的可行性和有效性。文章研究结论可用于含离群值变量的相关系数稳健估计。The traditional Pearson correlation coefficient calculation formula is not robust, and the existence of outliers will cause the calculation results to be inconsistent with reality. To solve this problem, this paper presents a robust estimation method. When the simulated sample size is 20, 50, 100 and 200 respectively, the pollution rate is 1%, 5% and 10% respectively, it is found that the accuracy of the robust correlation coefficient formula is significantly higher than that of the traditional correlation coefficient. The feasibility and effectiveness of a robust correlation coefficient are further verified in the example analysis. The conclusions of this paper can be used for robust estimation of correlation coefficients with outlier variables.
