二次函数

Based on the equilibrium equation of an arch element the cubic nonlinear dynamic model with second order is established and the nonlinear differential dynamic system without constant term is obtained through a transfer process of T=U+b. Hence three fixed points and the homoclinic orbits are found. The critical condition of the chaotic vibration of the elastic arch is given by using the method of Melnikov function. The numerical results of the stationary and chaotic motion are given and discussed.

首先利用拱的单元平衡方程建立了拱的二阶三次非线性动力学模型；然后通过变换使方程转化为不含常数项的非线性微分动力系统，并由此得到无扰动系统的3个不动点与同宿轨道；再用Melnikov函数法给出了发生混沌的临界条件；最后给出了该系统出现定常运动和混沌运动的数值结果。

However, the smoothness and convergence of these methods are not satisfactory. Because cubic spline function can be a piecewise, and its second order derivative is continuous, it is smooth and convergent.

但以上算法的光滑性和收敛性不好，而三次样条函数具有连续的二阶导数，且可采用分段函数的形式，具有很好的光滑性和收敛性。

According to different characteristics of CMM error distribution, single error measurement model is built by cubic spline principle and double cubic spline function.

根据三种类型三坐标测量机误差分布的特征，提出了利用三次样条原理和双三次样条函数建立测量机一维和二维相关性误差模型。

A contact problem of decagonal two?

通过引入位移函数和应用Fourier分析与对偶积分方程理论圆满解决了在一个平底冲头作用下十次二维准晶材料的接触问题，得到了此材料接触问题应力与位移的解析表达式。

On the basis of the power spectra of earthquake records, the parameters of the power spectral density function of the twice filtered Gaussian white noise process are determined by adopting the least squares procedure of a nonlinear function.

根据地震记录的功率谱，采用非线性函数的最小二乘法，确定了两次过滤Gauss白噪声过程的功率谱密度函数的参数。

It's the first time that from the angle of computational mathematics combined with the structure model of parallel machine, the parallel degree of the relevant algorithms in orthopaedics clinic engineering modeling system, such as piecewise cubic Lagrange interpolation formula, piecewise cubic Hermite's interpolation formula, cubic spline, Chebyshev multinomial fitting algorithms, least square method and contour curve multinomial algorithms, were discussed.

论文首次从计算数学的角度并结合并行机的结构模式出发，讨论了在骨科临床工程造型系统中可能涉及的如分段三次拉格郎日插值公式、分段三次埃尔米特插值公式、三次样条函数、切比雪夫多项式拟合算法，最小二乘法以及轮廓曲线多项式求值算法的并行度。

First of all,we have given some of the basic concepts of differential equations, described the constant coefficient linear ordinary differential equation solution, for a class of second-order variable coefficient linear ordinary differential equation initial value problem, an approximate solution, the method is first unknown function of a definition for N sub-interval, and then in between each district within a constant coefficient ordinary differential equations similar to the replacement, the solution has been the problem as similar to the original analytical solution, and then gives a detailed second-order change order coefficient of linear homogeneous ordinary differential equation solution examples, the examples of the approximate method proposed in this paper is valid.

首先给出了微分方程的一些基本概念，讲述了常系数线性常微分方程的解法，针对一类二阶变系数线性常微分方程初值问题，提出了一个近似解法，本方法是先对未知函数的一个定义区间作N等分，然后在每一个小区间内用一个常系数常微分方程近似替换，所得到的解作为原问题的近似解析解，随后详细给出了一个求二阶变系数齐次线性常微分方程的解的实例，该实例说明本文提出的近似方法是有效的。

Compared with deterministic systems, the infinitesimal generator for the system is a second-order differential operation. In this dissertation, we designed the quartic control Lyapunov function and systematically design the state-feedback and output feedback adaptive controller using integrator backstepping methodology. The closed-loop system is asymptotically stable in the large.

与确定性系统相比，随机系统的Lyapunov函数的微分运算包含二阶导，所以设计适当形式的控制Lyapunov函数十分关键，本文设计了四次型的控制Lyapunov函数，并应用积分反推技术，给出了状态反馈及输出反馈自适应控制器和鲁棒自适应控制器的设计方法，证明了闭环系统概率意义下的渐近稳定性和有界稳定性。

In Chapter 4 we discuss the solvable theorem of adaptive inverse optimal control problems, and proceed with controller designs of global asymptotic stability in probability, adaptive inverse optimal stabilization in probability and output-feedback adaptive inverse optimal stabilization in probability for strict-feedback stochastic nonlinear continuous systems with additive standard Wiener noises and constant unknown parameters using It〓's differentiation rule and an adaptive backstepping algorithm. Control laws and adaptive laws can be obtained at one time by this design scheme. Many simulations have been performed to validate the properties of the proposed adaptive control scheme.

在第4章中，针对具有标准Wiener噪声扰动和未知定常参数的不确定随机非线性系统，提出并证明了自适应逆最优控制问题可解定理，构造了适当形式的四次型随机控制Lvapunov函数，基于It〓微分规则和自适应Backstepping算法，系统地设计了全局依概率渐近稳定控制器、自适应逆最优控制器、输出反馈逆最优控制器以及在设计中如何处理二阶Hessian矩阵函数的方法，这种方法可同时获得控制律和自适应律，通过实例仿真，表明该控制算法是有效性的。

Based on the analysis and summary of basic ideas and methods,this thesis proposes two new estimation of error of higher order approximation.

方法二是在等距分布的条件下，基于三次样条插值函数的三弯距算法，通过利用节点处弯距的加权和近似计算区间中点处的二阶导数。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。