求分段函数在分段点处导数的一种有效方法

为了求解分段函数在分段点处的导数,利用微分学中的相关知识推证定理,并举例说明了如何利用该定理求分段函数在分段点处的导数。     

定义在分形集上函数的H-导数

 分形的生成含有降维与升维两种形式,采用适当的方式总可以对分形集加以定向化,分形的内禀测度是Hausdorff测度。因此,定义在分形集上的函数可以看作是在Hausdorff测度下定义在区间[0, h]。在函数图像也是分形的前提下,依函数的Hausdorff测度的定义,给出了定义在分形集上的函数f(p)关于Hausdorff测度p的导数以及一些性质。     

δ函数导数在电磁学中的一些应用

本文说明理想电偶极子、电四极子的电荷分布可分别用δ函数的一阶、二阶导数来表示．在此基础上，运用δ函数导数的计算公式。可以直观、简便地得出理想电多极子的静电势和辐射场标势的表达式     

基于Buchadhl象差系数和解析偏导数的光学优化设计

本文以基于 Buchadhl 象差系数得到的点列图为目标函数,只需追迹少许光线就可得到近似点列图,大大减少了计算量;同时采用了目标函数对结构参数偏导数解析求导原理并应用于光学自动设计,这样得到的解析偏导数不仅不存在原理误差,同时极大地减少了求得所需的时间;最后给出了使用 DFP-BFGS 优化方法设计一个双高斯物镜的实例.     

Voigt线形函数二阶导数研究

在强展宽条件下, 光谱信号二阶导数相互的交叠影响较小, 是一种潜在的反演光谱信息的手段. 本文研究了光谱Voigt线形函数的二阶导数, 得出了其二阶导数全域积分为0的性质, 计算了二阶导数最小值与偶数高阶导数最大值和最小值的解析结果, 并通过数值计算与曲线拟合得出了其极大值位置与零点位置的比例与洛仑兹-多普勒半宽比的关系, 为强展宽下由光谱二阶导数准确反演光谱信息提供了理论基础. 关键词： Voigt函数 二阶导数最小值 零点位置     

导数在高中物理中的应用

函数对自变量的瞬时变化率, 称为导数. 在日常教学中将导数思想渗透入物理课堂, 能深化对物理概 念和规律的理解, 把握物理知识的内在联系; 在解决实际问题时引入导数, 能有效解决物理量的非均匀变化问题、 极值问题, 有利于提升学生应用数学知识处理物理问题的能力     

升力系数不变时跨音速机翼阻力优化设计

基于共轭方程的优化设计理论,应用三维欧拉方程进行了升力系数不变时跨音速机翼阻力优化设计研究,根据给定的目标函数推导了在物理空间上表述的共轭方程及边界条件,研究了共轭方程的数值求解方法及目标函数对设计变量的敏感性导数求解问题,发展了一种跨音速机翼阻力优化设计方法,应用该设计方法进行了跨音速机翼阻力优化设计研究,优化后机翼表面的激波强度减弱很多,有效减少了波阻.     

量子引力的曲率两点真空相关

以平坦的Minkowski时空为背景，得到了任意坐标系和谐和坐标系中，n维GR引力和高导数引力的引力子自由传播子，求得了四种可能的曲率两点真空相关函数的首项.用微扰计算证明了曲率的两点真空相关函数在GR引力中为零，而在高导数引力中不为零.讨论了高导数引力与GR的引力子传播子、曲率相关函数的关系. 关键词： GR 高导数引力 引力子自由传播子 曲率真空相关函数 平移传播子     

函数的H-导数和分形动力学

 给出了函数的Hausdorff测度与H-导数。从而，为开展自给的分形动力学提供了解析的数学工具。同时，给出了分形动力学的演化方程组并做了初步讨论与分析。     

透平叶栅三维粘性气动反问题的控制理论方法

将基于控制理论的形状优化设计方法应用于粘性可压流动条件下的透平叶栅三维气动反设计,详细推导了三维N-S方程伴随系统的偏微分方程组及其各类边界条件.讨论了伴随系统的解的适定性条件,并由此给出应用N-S方程进行气动优化的目标函数的选取限制.研究了伴随方程的数值求解技术,给出敏感性导数的最终计算式,结合拟牛顿算法发展了三维透平叶栅粘性反问题的气动设计方法.     

Modified Smoothed Particle Hydrodynamics (MSPH) basis functions for meshless methods,and their application to axisymmetric Taylor impact test

The Modified Smoothed Particle Hydrodynamics (MSPH) method proposed earlier by the authors and applied to the analysis of transient two-dimensional (2-D) heat conduction, 1-D transient simple shearing deformations of a thermoviscoplastic material, 1-D wave propagation in a functionally graded plate, and 2-D elastodynamic crack propagation is extended to the analysis of axisymmetric deformations of a thermoviscoplastic material. In the MSPH method, different shape functions are used to find kernel estimates of the function, and of its first and second derivatives. It differs from the classical finite element method in which derivatives of a function are usually obtained by differentiating the shape function used to approximate the function. It is shown that results computed with the MSPH method for the Noh problem agree well with its analytical solution. The MSPH basis functions can be used in any meshless method to numerically solve either static or dynamic problems. The method is then applied to analyze transient deformations of a cylindrical rod impacting at normal incidence a rigid smooth stationary flat plate. The computed solution is found to agree very well with those obtained by analyzing axisymmetric and 3-D transient deformations of the rod with the commercial code LS-DYNA. The final length of the deformed rod, the final radius of the impacted face, and the final length of the relatively undeformed portion of the rod for twelve test configurations computed with the MSPH method are also found to agree well with their corresponding experimental values.     

Parallelization of analytical Hartree—Fock and density functional theory Hessian calculations. Part I: parallelization of coupled-perturbed Hartree—Fock equations

Solving the coupled-perturbed Hartree-Fock (CPHF) equations is the most time consuming part in the analytical computation of second derivatives of the molecular energy with respect to the nuclei. This paper describes a unique parallelization approach for solving the CPHF equations. The computational load is divided by the nuclear perturbations and distributed evenly among the computing nodes. The parallel algorithm is scalable with respect to the size of the molecule, i.e. the larger the molecule, the greater the parallel speedup. The memory storage requirements are also distributed among the processors, with little communication among the processors. The method is implemented in the Q-Chem software package and its performance is discussed. This work represents the first step in a research project to parallelize analytical frequency calculations at Hartree-Fock and density functional theory levels.     

Computation of collective excitations and elastic constants of expanded trigonal tellurium

The potential function derived from experimental structure factor data gives a repulsive minimum and also an attractive minimum. These two minima correspond to the first and second nearest-neighbour distances. Similar results have been obtained through the use of Percus-Yevick (PY) and hyper-netted chain (HNC) equations. The Hafner molecular dynamics results also show a similar trend. The first, second and third derivatives of the potential function have been obtained from the expression of the potential function. The first and second derivatives have been used to compute the phonon frequencies using Bhatia and Singh's (BS) method. The phonon frequencies obtained from Hubbard and Beeby's (HB) method show similar results. The elastic constants have also been calculated and the values obtained for C₁₁ by different methods are in good agreement. The Grüneisen constant has been calculated by two different methods which are in very good agreement with each other. From the Grüneisen constant the pressure derivative of the diffusion coefficient has been obtained and has been compared with that of Swalin's equation. The results are in excellent agreement with each other.     

Analytical Expressions of the Various Derivatives of the Output Power of the Semiconductor Optical Amplifiers with Respect to the Input Power

Based on the physical fact that the nonlinear distortions induced by the travelingwave semiconductor optical amplifiers in the subcarrier multiplexing CATV transmission system can be calculated by expressing the output power P o of the amplifiers with the Taylor series of the input optical power P i and specifying the various order derivatives of P o with respect to P i , the solutions to the rate equations of the amplifiers have been derived, and the analytical expressions of the various order derivatives of P o with respect to P i have been obtained and simulated numerically by selecting typical amplifier parameters.     

Analytical Expressions of the Various Derivatives of the Output Power of the Semiconductor Optical Amplifiers with Respect to the Input Power

Based on the physical fact that the nonlinear distortions induced by the travelingwave semiconductor optical amplifiers in the subcarrier multiplexing CATV transmission system can be calculated by expressing the output power P o of the amplifiers with the Taylor series of the input optical power P i and specifying the various order derivatives of P o with respect to P i , the solutions to the rate equations of the amplifiers have been derived, and the analytical expressions of the various order derivatives of P o with respect to P i have been obtained and simulated numerically by selecting typical amplifier parameters.     

一种改进的光学层析图像重建方法

Alexander D．Klose将联合差分方法用于光学层析图像重建的梯度计算中,但给出的对光学参量的求导算法有局限,他的算法只能实现对边界点光学参量的导数计算,而无法实现对内部点光学参量导数的计算,会导致图像重建失败。在联合差分算法的基础上,研究了针对内部点光学参量的求导方法,给出了一种基于树形结构的对内部点光学参量求导的策略。具体实现时,为了降低计算复杂度,采用近似梯度计算方法。算法的仿真实验结果表明：该方法可以有效地实现对内部点光学参量的导数计算,提出的近似计算方法可降低梯度计算复杂度,提高运算速度,并可得到良好的图像重建质量。     

Comparison between a thin matrix decomposition method and the rigorous coupled wave theory applied to volume diffraction gratings

Diffraction gratings have been an interesting field during the last decades. A great deal of research has been done in order to understand how light propagates inside diffraction gratings. Since its formulation, the Rigorous Coupled Wave Theory (RCWT) has been applied to different periodic structures providing a basis to check the validity of approximate theories, such as Kogelnik's coupled wave theory or the coupled wave theory neglecting second derivatives. In this work we will compare the results of the diffraction efficiencies of the first and second orders at second on-Bragg replay angular condition, obtained by using an analytical formula provided by Alferness using an approximate thin-matrix decomposition method (TMDM) with the results obtained by using the RCWT. Good agreement between the results obtained using TMDM and the RCWT were found.     

膜系光谱系数对膜层参数的一阶和二阶偏导数的计算模型

为了寻找一种膜系深入分析与设计的有力辅助计算工具,从光学薄膜光谱系数的矩阵计算理论出发,基于对矩阵求迹运算的巧妙运用,从数学上建立了准确计算膜系光谱系数关于膜层几何厚度、实际折射率和消光系数等膜层参数的一阶和二阶偏导数的解析模型。这一偏导数计算模型物理上与矩阵理论具有一致的通用性和普适性,适用于任何各向同性的均匀膜系统。在数学上也严格成立,数值编程计算不存在差分近似,精度达到了计算机浮点运算的极限精度。而且算法运算费时少,计算速度快,取得了高度准确性和便于实时化的双重优越性能,非常有利于应用于膜系设计领域来提高膜层数较大时的设计速度和效率。同时,这一模型能非常便捷和准确地给出膜系许多实用的导数信息,对膜系分析、测量和膜厚监控等领域中的相关应用具有重要的参考意义。     

An analytical solution to Li/Li+ insertion into a porous electrode

The dynamic faradic properties of the lithium ion batteries are primarily determined by the process of lithium ion insertion into a porous electrode. In this paper, we present an analytical result of the intercalating process of Li/Li⁺ into a spherical particle of graphite or cobalt oxide immersed in a conductive electrolyte. Using the finite integral transform method, an exact solution to the concentration profile was obtained for arbitrary linear initial and boundary conditions. To avoid analytical difficulties with respect to the boundary conditions of second kind, the method of pseudo-steady-state is applied. The final solution uniformly converges, and can be used for accurate and fast dynamic modelling and simulation.