解析变换

By the deep analysis on winding eddy current loss mechanism of the transformer in forward-flyback converter, a new idea is proposed that the magnetic field in winding window of the transformer can be considered as the combination of the two components, that is, transformer field component and inductor field component. The orthogonal characteristics of the two magnetic field components in CCM are further revealed.

通过深入研究正反激变换器中的变压器线圈涡流损耗机制，指出其线圈窗口磁场可分离为变压器特性的磁场和电感器特性的磁场，证明了连续导通模式时上述两种特性磁场为正交的，进而提出一新型线圈损耗解析建模方法，新方法可以有效提高模型精度。

With aperture expanded as the sum of finite complex Gaussian terms, a more practical approximate analytical solution of fractional Fourier transform of Gaussian beam in an apertured paraboloid refracting system is obtained and also numerical investigation is presented.

通过将孔径函数展开为有限项复高斯函数之和，得到了高斯光束在受孔径约束的抛物面折射系统中分数傅里叶变换较符合实际的近似解析表达式，并进行了数值模拟。

The quasi-analytical expressions of the relative concentrations at the mobile region and the immobile region are deduced respectively by using hypergeometric equation, hypergeometric function and the method of Laplace transform under the initial concentration zero and the first or second boundary condition of a semi-infinite one-dimensional space.

并分别在初始浓度为零，半无限一维空间内第一类边界条件和第二类边界条件下，利用超几何方程、超几何函数和Laplace变换法推导出了可动区和不动区溶质相对浓度的准解析表达式。

The analytical solution of the beam response was obtained by applying the Fourier integral transformation and the residue theorem, followed by a comprehensive parametric study to investigate the effects of the subsoil depth, the material stiffness and the load speed on the beam response.

运用Fourier积分变换和留数定理对动力控制方程进行求解，得到了平面应变条件下覆盖于基岩的均质弹性层和非均质弹性层地基上梁的动力响应解析解，并进一步分析了荷载运动速度、地基厚度及地基非均质性等对梁变形的影响规律。

The mixed boundary value problem is simplified to a pair dual integral equations using Laplace and Fourier integral transform method.

利用拉普拉斯和傅立叶变换将混合边界值问题简化为对偶积分方程，并得到裂纹尖端应力场的解析解答。

And a model of layer—disperising dual medium is established (2) Some new theories and methods, such as Fourier dual integral transformation and half numeral value analytic method and so on, are used to establish the calculating formula in this paper.

2本文在理论推导上，应用了数学、力学和数学物理方程方面的新理论和新方法，如：二重富里叶积分变换、板壳理论、分离变量法以及半解析数值方法等，使本论文在理论推导过程上，增加了理论的严密性和逻辑性。

Then, the differential equations are solved by the Fourier expanding and Hankel integral transform method. Integral solutions of soil skeleton displacements and pore pressure as well as the total stresses for poroelastic media are obtained. Furthermore, a systematic study on Lamb's problems in transversely isotropic saturated half-space is performed. Integral solutions for surface radial, vertical and tangentical displacements are obtained both in the case of drained surface and in the case of undrained surface excited by vertical and tangentical harmonic resources respectively. Numerical results show the obvious difference between the model of isotropic saturated poroelastic media and that of transversely isotropic saturated poroelastic media.

其次，基于Biot波动理论，在圆柱坐标系下求解了横观各向同性饱和土的Biot波动方程：通过引入位移函数，在圆柱坐标系下将横观各向同性饱和土的Biot波动方程转化为两个解耦的6阶和2阶控制方程，然后根据方位角的Fourier展开和径向Hankel变换，求解波动方程，得到以土骨架位移和孔隙水压力为基本未知量的积分形式一般解，并用一般解给出了饱和土总应力分量的表达式；再以基本解为基础，系统地研究了横观各向同性饱和半空间体的Lamb问题，考虑表面排水或不排水两种情况后，首次得到横观各向同性饱和半空间体在表面竖向和水平谐振力作用的下径向、竖向和周向位移的解析解。

Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations, which are then reduced to a Fredholm integral equation of second kind by the Copson-Sih method, and anti-plane displacement, electric potential and stress are obtained.

通过Fourier积分变换，将混合边值问题转化为对偶积分方程，并利用Copson-Sih方法将对偶积分方程转化为第二类Fredholm积分方程进行求解，给出了反平面位移、电势及应力分量的解析表达式。

The evolvement and mechanism of border-collision bifurcations and chaos were analyzed after the collision border line was studied. Results show the limitation of duty cycle is the fundamental cause of non-smooth bifurcations.

分析边界碰撞分岔和混沌演化过程及其产生机理，推导出碰撞边界线的解析表达式，指出发生边界碰撞的根本原因在于开关变换器占空比的有界性。

Concretely, firstly with Timoshenko beams, the fundamental formulae of solutions of dynamical equations, boundary conditions and initial conditions have been transformed by Laplace transformation. Secondly, in the Laplace regions , we gain the analytical solutions of impact forces at the impact contact points, shear and maximum inverse stress of beams. Then, by using Durbin numerical inverse method, we gain the numerical solutions of Timoshenko beams under various restriction conditions. Moreover, the graphs about impact forces at the impact contact points, shear and maximum inverse stress of beams can be obtained too.

把Timoshenko梁力学模型动力学方程的基本解形式、边界条件以及初始条件进行Laplace变换，在Laplace区域得到了冲击接触处的冲击力、整个梁的最大弯曲应力和剪力的解析解，然后利用Durbin数值反演的方法得到了各种约束条件下的Timoshenko梁模型时间-空间的数值解，得到了冲击接触处的冲击力、梁各处的最大弯曲应力和剪力的曲线图。

We will see more and more activist government policies that distinguish economic activities according to the nationality of the actors.

我们将看到越来越多激进的政府政策，这些政策根据企业的国籍来区分经济活动。

If we can prove the independence of an axiomatic system, then we can guarantee the conciseness of this system, ie.

如果能够证明某个一阶谓词系统的独立性，那么就能够保证该系统的建立是精炼的，即没有任何一条公理或推理规则是多余的。