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differential algebra
相关语句
  微分代数
    AN AUTOMATIC DIFFERENTIATION TECHNIQUE: DIFFERENTIAL ALGEBRA AND ITS APPLICATION IN CHARGED PARTICLE OPTICS
    一种自动微分技术:微分代数及其在带电粒子光学中的应用
短句来源
    Application of Differential Algebra to Calculation of High Order Aberration of Realistic Electron Lenses
    利用微分代数方法计算实际电子透镜的高阶像差
短句来源
    Differential algebra is a new mathematical method and a promising tool in charged particle optics.
    微分代数是带电粒子光学中新近出现的一个很有前途的数学工具。
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  differential algebra
The present method depends on a result from differential algebra andp-adic analysis.
      
The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on .
      
In the approach based on the global covariance with respect to the stability subgroup U(1), we show that the gauge-covariant exterior differential algebra on SUq(2) is uniquely determined.
      
Each system of partial differential equations generates an ideal in some differential algebra.
      
We consider tensors with coefficients in a commutative differential algebra A.
      
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Differential algebra is a new mathematical method and a promising tool in charged particle optics. As a kind of automatic differentiation techniques, it is based on the nonstandard analysis and normal power series theory. It presents a straightforward method for the study of arbitrary order property of nonlinear systems, with accuracy limited only by the computer. The differential algebraic method is widely applicable, and it is an effective and elegant tool for the tracking, analysis and correction of...

Differential algebra is a new mathematical method and a promising tool in charged particle optics. As a kind of automatic differentiation techniques, it is based on the nonstandard analysis and normal power series theory. It presents a straightforward method for the study of arbitrary order property of nonlinear systems, with accuracy limited only by the computer. The differential algebraic method is widely applicable, and it is an effective and elegant tool for the tracking, analysis and correction of high_order aberrations, sensitivity analysis and optimization of structure design, etc., of the charged particle optical systems. The principle and virtues of differential algebra are summarized. The application of differential algebra method in charged particle optics and its implementation in various computer languages are analysed. Some new research subjects in this field are discussed.

微分代数是带电粒子光学中新近出现的一个很有前途的数学工具。作为一种自动微分技术,它结合了非标准分析、简正级数理论等的思想。微分代数提供了一种可达到机器精度的研究非线性系统任意阶性质的极为简便的方法,故应用领域非常广泛,对带电粒子光学中的轨迹追踪、高阶像差分析与校正、灵敏度分析与结构优化设计等都是一个很有效的工具。综述了微分代数的基本原理和独特性,分析了它在带电粒子光学中的应用,以及在不同计算机语言中的软件实现,提出了今后的一些发展方向。

Differential algebra has been successfully modified and applied to numerical calculation and simulation of high order aberrations of a set of electron lenses with the electric and/or magnetic field distributions,which are given as discrete numerical arrays,highly close to realistic and well recognized in finite element method(FEM) and finite differential method(FDM).The software package developed works well in engineering calculation and design of the realistic lenses.Our simulated results of the...

Differential algebra has been successfully modified and applied to numerical calculation and simulation of high order aberrations of a set of electron lenses with the electric and/or magnetic field distributions,which are given as discrete numerical arrays,highly close to realistic and well recognized in finite element method(FEM) and finite differential method(FDM).The software package developed works well in engineering calculation and design of the realistic lenses.Our simulated results of the 3~(rd) and 5~(th) order aberration show a rather high accuracy,which is only limited by mechanical precision and arithmetic errors,and which is good enough for engineering purposes.

微分代数是计算机数值分析领域中的一个非常有效的新方法,它以非标准分析理论为基础,可以方便地实现任意高阶微分的运算。利用微分代数扩展数的映射关系,可以非常有效地处理一些困难的非线性动力学问题[1],包括电子光学的高阶像差问题。但是,已有的研究限于电子光学系统的电磁场具有解析表达式的情形。本文根据微分代数的基本原理,将其用于实际电子光学聚焦系统高阶像差的数值模拟和计算,计算了电子透镜的五阶几何像差。这些电子透镜的电场及磁场的分布,是根据实用的FEM或FDM等数值方法的计算结果,以数组形式给出的。因而这一计算方法可以应用于工程的设计计算问题。编写了相关的计算软件,该软件的界面可以方便地应用于实际的电子透镜系统的高阶像差计算。对于一个电磁透镜例的计算结果表明,这一计算方法具有足够高的精确度,其精度仅受限于机器精度和算法误差,完全可以满足工程计算的需要。

 
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