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By investigating Newton\'s Problem of the Solid of Least Resistance,John Bernoulli\'s Brachystochrone Problem and Jacob Bernoulli\'s Isoperimetric Problem, the physics and Calculus background of the calculus of variations is illustrated,the fundamental thought together with the patterns of solutions are summarized,and the ideas and the interaction of the pioneers,such as the Bernoullis and B.

通过考察牛顿最小阻力体问题、约翰·伯努利最速降线问题和雅可布·伯努利等周问题等问题的提出过程及解决方法，深入探究了变分法诞生的深刻背景，提炼和概括出了伯努利兄弟和泰勒等先驱者解决变分问题的基本思想和求解模式，追溯了其微积分渊源，并分析了三位先驱(来源：ABCbc论文64网www.abclunwen.com)者之间的变分法思想传承及相互影响。3。

The problem of sub-Riemannian geodesics is a Lagrange problem with constraint. How to describe the constraint condition "γ'∈ D, a. e." is a difficult problem.

次黎曼测地线问题是变分学中的一个有约束的Lagrange问题，但在变分的过程中有个难点―如何推出约束条件"γ'∈D，在上几乎处处成立"的解析式。

There is no difficulty for us to solve N-K problem for submerged bodies by the coupled element method, but for wavemaking problems of the bodies piercing free surface, we get a difficult problem of line integral on the free surface when we solve the boundary value problem in the external region by boundary element method.

对于深潜物体的N—K问题，用耦合元方法求解不存在根本性的困难，但对于水面船波兴波问题，采用边界元方法来处理外域中的边值问题而出现了所谓的自由面线积分问题。

This paper surveys a new exact method which uses exact dynamic stiffnesses and therefore results in a nonlinear eigenvalue problem in contrast to the traditional finite element method which reduces the problem into a linear algebraic eigenvalue problem by using approximate shape functions.

与常规有限元方法采用近似形函数将原问题化为线性特征值问题不同，本文的精确方法从杆件精确的形函数出发获得精确的动力刚度，将原问题化为非线性特征值问题。

Inverse eigenvalue problem; Constrained matrix equation problem; Procrustes problem; Matrix norm; Optimal approximation solution

矩阵逆特征值问题；约束矩阵方程问题； Procrustes问题；矩阵范数；最佳逼近解

The basic idea was to discrete the structure and find the load function according to the energy criterion of buckling and the principle of random displacement method. As the critical load is the minimum of the load function, a problem of frame buckling can be transformed into an optimization problem. Therefore, the critical load can be found through solving the optimization problem.

先将结构离散为一个有限自由度系统；然后根据有限自由度系统平衡稳定性的能量准则和随机位移法的基本原理，建立符合能量准则的载荷函数，该函数不为零的最小值即为临界载荷，由此将稳定问题转化为无约束的多维优化问题；最后应用遗传算法求出相应于最优解的目标函数值。

The quasi-phsical method makes the original problem an optimization problem in mathematics. There is often the possibility of going to a local minimum of object function when solving the optimization problem mathematically.

所谓拟物就是主动地向自然界学习解决问题的方法，即寻找与原始数学问题等价的物理世界并观察这个世界中物质运动的生动形象，然后从中得出启发并逐步形式化为算法以求解问题。

In the algorithm, radial-basis function neural network was utilized as forward model, and genetic algorithm was used to solve the optimization problem in the inverse problem, which can avoid the local minimum solution possibly encountered in the iterative inverse algorithm based on gradient descent and consequently achieve the global optimized solution of the inverse problem.

在该算法中，径向基函数神经网络用作前向模型，遗传算法用于求解逆问题中的优化问题，其优点是能够避免基于梯度下降法的迭代逆算法中可能遇到的局部最小问题，并能得到逆问题的全局最优解。

Based on the fundamental principles of the improved method mentioned above, multi-functional computation software package, which can be used to analyze effectively the interference effects of random flaws in plane problem, boundary problem and space problem, is compiled and a number of problems involved in the interference effects of different kinds of flaws such as random equal length through-cracks, random branched throughcracks, random unequal length through-cracks, random holes to through-cracks, unequal length edge cracks, equal length and distance multiple edge cracks, random notches with different aspect ratio, random notch to edge crack, non-coplanar surface cracks with different aspect ratio, staggered surface cracks, surface crack to embedded crack and embedded cracks near free boundary are analyzed and computed by the package.

依据上述改进法的基本原理，本文编制了能有效地分析平面问题，边界问题及空间问题的任意缺陷群相互干涉效应的多功能计算软件包。据此，本文分析计算了各类缺陷群。诸如，任意等长裂纹群。任意折裂纹群；任意不等长裂纹群，任意孔与裂纹群，不等长边裂纹群。等长等间距多裂纹群，任意形状比缺口群，任意缺口与边裂纹，非共面不同形状比表面裂纹群，交错型表面裂纹群，表面裂纹与埋藏裂纹群及自由边界附近埋藏裂纹群等的相互干涉效应问题。

Delay constrained multicast problem is an important part of multipoint routing optimization problem and has been proved to be a NP-Complete problem.

有时延约束的组播问题是通信网络多点路由优化问题中的重要部分，已被证明是NP complete问题。

Analog output can display output results.

可以模拟输出显示输出效果。

It implied that RAPD technique pro vided a new alternative for checking irradiation induced effects.

对扩增产物进行数据分析，结果表明RAPD技术能够检测出辐射后唐菖蒲基因组变异情况，可用于对唐菖蒲辐射后基因变异的分析。