分段光滑

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。

Chaikin algorithm is the simplest discrete curve modeling method.In thepaper planar Chaikin algorithm is generalized into sphere,and the practicability ofthe algorithm as well as the continuous differentiable property of the generatingcurve is proved.It is shown that the curve generated by Chaikin algorithm is apiecewise spherical Bezier curve of second kind of order two.Furthermore,generalspherical corner-cutting algorithms are discussed,and it is pointed out that thecurve generated through corner-cutting has"Spherical Variation DiminishingProperty".3Curve interpolation method restricted on a smooth surface based ontransformation.

Chaikin算法是最简单的离散曲线造型方法，文中将Chaikin算法推广到球面，证明了算法的可行性和生成曲线的连续可微性，指出球面Chaikin算法得到的曲线是分段的二次第二类球面Bezier曲线，并给出了用球面Chaikin算法构造球面插值曲线的算法；进一步，文中讨论了一般的球面割角算法，指出由球面割角算法得到的曲线具有"球面变差缩减性质"。3基于变换的约束在光滑曲面上的曲线插值方法。

Test and simulation shows that the method can eliminate the unsmooth shortcoming caused by subsection low order interpolation with its inherent feature, and can meet the vision effect in inner capacity reconstruction. Further more the volume measurement error are reduced after interpolated by cubic spline without increasing any calculation, and the measurement relative error could reach 2‰ required by the system.

实验和仿真结果证明：该方法能够克服直接累加和传统的分段低次线性插值方法由于函数本身的特性，在各次间歇测量点处产生的衔接不光滑性，满足了变径内腔内表面容积重建中的视觉需求；同时在不增加计算量的前提下使得插值后容积的测量精度得到了提高，且满足2‰测量误差的要求。

A class of piecewise smooth maps with two border and three zones is derived to describe the dynamics of a current-programmed Buck converter operating in a discontinuous mode.

本文以不连续导通模式下的电流反馈型Buck变换器为例，推导出两个边界三段形式的分段光滑系统的离散映射模型。

Because of switching actions, power-factor- correction Boost converters are essentially piecewise smooth nonlinear systems, which leads to the occurrence of fast-scale bifurcations.

由于开关的动作，功率因数校正(power-factor-correc- tion，PFC) Boost变换器本质上是分段光滑的非线性系统，从而导致快时标分岔的产生。

The cartoon component is described by piecewise smooth functions (Mumford-Shah model, or M-S model), while the texture is characterized by oscillating functions in G space.

其中结构成分用分段光滑的函数（即Mumford-Shah模型）刻画，纹理部分用振荡函数来描述。

The unified discrete iterative map model of peak current and valley current controlled buck, boost, and buck-boost converters is established, and the unified piecewise smooth iterative map and their corresponding characteristics equations are also derived.

本文建立了峰值电流和谷值电流控制Buck，Boost，及Buck-Boost变换器的统一离散迭代映射模型，并导出了统一的分段光滑迭代映射方程及特征值方程，通过数值仿真得到了占空比变化时的正、逆分岔图和Lyapunov指数谱。

During building 2D model, I use the concepts of point, segment, line and plane, the segment can be formed from points, then line can be formed from segments, the closed curved plane can be obtained by around lines, and finally the 2D model can be built by closed curved plane; I express the un-smooth stratum or fault by linear fitting and express the smooth stratum or default by cubic spline fitting segmentally, in this way I can describe the variation feature of 2D interface properly.

在二维建模中，引用点、段、线、面的概念，即由点形成段，由段形成线，由线围成封闭面，封闭面构建成二维模型；对表示二维地层和断层的界面线，用线性拟合表示非光滑性地层或断层面，用分段三次样条拟合表示光滑性地层或断层面，恰当地表达了二维界面的变化特点。

First,a proposition is presented:for a general kind of biobjective deci-sionmaking problemin real linear topological space satisfyiny someuniversal convexity assumpltions,the Pareto frontier is a piece-wise smoothconvex curve,each piece being either a Ⅰ-curve or a Ⅱ-curve.

对于满足一般凸性假设的实线性拓扑空间上的双目标决策问题，本文提出了"其非劣前沿为分段光滑的凸曲线，每一段要么是工型弧线，要么是直型弧线"的观点，据此详细研究了决策人的偏好与加权和问题之间的关系，提出了4种对话式决策方法，这些方法既适用于一类静态双目标决策问题，又可适用于一类双目标控制问题。

A new analytic model of unsteady oil film forces is used to study the bifurcating and chaotic behavior of rotor systems is obtained.

本文讨论了转子系统的一个新的短轴承非稳态油膜力模型，这是一个分段光滑的模型，通过数值模拟，得到了该系统随转动角速度变化产生的分岔和混沌特性。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。