分段连续函数

Based on general orthogonal polynomials and general hybrid orthogonal functions, a new linear, continuous and bounded operator has been proposed in the dissertation. The definitions and properties of PGOPO are given, a various operational rules of the operator have been set up in a systematic way, the convergence of approximate solutions has been discussed. After introducing PGOPO and discussing its main properties and operational rules, a systematic methodology for solving various control problems via PGOPO has been developed.

本文在广义正交多项式及广义混合正交函数的基础上，提出了一种新的线性、连续、有界算子—分段广义正交多项式算子，探讨了这种算子的基本性质，并系统地建立了与该算子有关的各种运算规则及收敛性结论，给出了它在控制理论中的各种应用。

Firstly, we prove that if we select nodes properly, and let the distance between these nodes sufficiently small, cubic polynomial spline can approximate a continuously differentiable function, which with continuous derivatives of second order and discontinuity of the first kind in the derivatives of fourth order, to any desirable accuracy. This result lays a theoretical foundation for representing trajectories by splines and double orthogonal basis.

一是证明了适当选取节点并使节点距充分小时，三次多项式样条可以逼近具有二次连续导数的分段四次连续可微函数到足够的精度，这一结论为用样条和双正交基表示弹道打下了理论基础。

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。

On the basis of method of derivative on the left and the right of separable function,the derivability of separable function on demarcation point was simply distinguished,and derivative continuity on demarcation point discussed.

运用研究分段函数的导函数的左、右极限的方法，简便地判明分段函数在分界点处的可导性；同时讨论了分界点处导数的连续

The identity and difference in the expressional patterns between distribution functions of discrete random variables and distribution functions of continuous random variables are elaborated,and the main expressional patterns of distribution functions of continuous random variables are summarized,and the derivability of distribution functions of continuous random variables is discussed.

0引言在概率论中，连续型随机变量分布函数的表达形式及其可导性是一难点。其一，离散型随机变量的分布函数有较明显的性质和统一的表达形式，而连续型随机变量，由于其概率密度函数一般为分段函数，根据其分段区间的差异，表达形式差异较大，不容易总结其规律，本文主要比较连续型随机

The guidance system is built according to the relative relationship in three dimensions. For longitudinal plane, by introducing the subsidiary smooth function, the attraction domain of sliding mode are analyzed and determined, and the stability of the guidance system is proven, considering from the coupling relations of two planes in three dimensions.

根据末端三维弹目相对运动学关系，建立了三维制导系统的数学模型，并以纵向平面模型和变结构控制中到达条件为基础，引入分段光滑连续函数，以三维平面间的耦合关系为出发点，从两个方面分析并确定了过载受限变结构制导律的滑动模态吸引区，证明其制导系统的稳定性。

Theorem(Stokes's Theorem) Let be a piecewise smooth oriented curve in three-space, be a piecewise smooth oriented surface with boundary curve .The surface and the curve are related by a right-hand relationship, i.e., if the fingers of the right hand are cupped in the direction of ,then the thumbpoints in the direction of orientation of .,, have continuous first partial derivatives on and ,then

定理 设为分段光滑的空间有向闭曲线，是以为边界的分片光滑的有向曲面，的正向与的侧符合右手规则(即如果右手大拇指表示曲面的方向，则弯曲四指表示曲线的方向)，函数，，在曲面上具有一阶连续偏导数，则有下述公式成立

Using normal distribution transform probability model to represent a laser range scan.Not need features corresponded,but transform the discrete scan date to the piecewise continuous and differentiable probability density function,and using Newton's algorithm to optimize scan matching,so can build the 2D global map.

利用正态分布转换概率模型来表示激光扫描，不需利用特征对应，而是通过将扫描得到的离散数据点转换成分段连续可微的概率密度函数，并利用牛顿优化算法来进行扫描匹配，从而建立2维全局地图。

The contour is second-order continuous piecewise cubic but not a linear function, and a storage structure of contour information was also studied.

研究结果表明：内插结果是具有整体二阶连续的分段三次函数而不是线性函数；该内插方法与常规方法相比，存贮结构节省60%~70%的存贮空间，并且更适用于基于等高线的各种地学分析。

A merge-split algorithm based on study property of neural net and sum aggregate is presented. Its merit is that its merge is disconnected a little and freedom satisfies the adaptability, and it is realized during learning rather than stipulating in advance. Furthermore, the method of fitting edge of uncontinuous image with neural net is given. The method can carry on the fit at self-learning, using its learning function fully under the circumstances without the mathematical function form of edge.

提出了一种基于神经网络学习特性和集合论的边缘分段算法(Merge-Split)，优点在于它的分段间断点、自由度满足自适应性，能在学习过程中实现而不必事先规定；进一步提出神经网络拟合不连续的图像边缘的方法，在无边缘数学函数形式的情况下，充分利用其学习函数的任意逼近和自学习性进行拟合，实践证明该方法可以克服以往边缘拟合方法的不足。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新