On the Topology and Isotopic Meshing of Plane Algebraic Curves

This paper presents a symbolic algorithm to compute the topology of a plane curve. This is a full version of the authors' CASC15 paper. The algorithm mainly involves resultant computations and real root isolation for univariate polynomials. Compared to other symbolic methods based on elimination techniques, the novelty of the proposed method is that the authors use a technique of interval polynomials to solve the system f(α, y),?f/?y(α, y)and simultaneously obtain numerous simple roots of f(α, y) = 0 on the α fiber. This significantly improves the efficiency of the lifting step because the authors are no longer required to compute the simple roots of f(α, y) = 0. After the topology is computed, a revised Newton's method is presented to compute an isotopic meshing of the plane algebraic curve. Though the approximation method is numerical, the authors can ensure that the proposed method is a certified one, and the meshing is topologically correct. Several nontrivial examples confirm that the proposed algorithm performs well.     

Visualizing Planar and Space Implicit Real Algebraic Curves with Singularities

Journal of Systems Science and Complexity - This paper presents a new method for visualizing implicit real algebraic curves inside a bounding box in the 2-D or 3-D ambient space based on numerical...     

On the Complexity of Computing the Topology of Real Algebraic Space Curves

Journal of Systems Science and Complexity - This paper presents an algorithm to compute the topology of an algebraic space curve. This is a modified version of the previous algorithm. Furthermore,...     

A Hybrid Procedure for Finding Real Points on a Real Algebraic Set

Journal of Systems Science and Complexity - Motivated by the idea of Shen, et al.’s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy...     

Global Optimization of Polynomials over Real Algebraic Sets

Let f, g_1, ···, g_s be polynomials in R[X_1, ···, X_n]. Based on topological properties of generalized critical values, the authors propose a method to compute the global in?mum f~? of f over an arbitrary given real algebraic set V = {x ∈ R~n| g_1(x) = 0, ···, g_s(x) = 0}, where V is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing f over a basic closed semi-algebraic set S = {x ∈ R~n| g_1(x) ≥ 0, ···, g_s(x) ≥ 0}.     

实时控制计算微分代数系统的代数约束算法

本文结合实际的航天工程背景,针对飞行器轨道约束实时控制模型问题进行了算法研究。首先分析了ＢＤＦ方法用于飞行器轨道约束实时控制模型问题时的缺陷,其后针对实际问题的特点构造了具有三阶收敛的代数约束算法,分析了该算法的数值稳定性,并对潜地式弹道约束实时控制问题及指标（ｉｎｄｅｘ）为２的单摆模型问题进行了实际仿真计算,理论分析以及数值结果表明代数约束算法对指标为２的半显式微分代数系统的实时控制计算是非常有效的。     

一种基于曲率信息的太空帆板空间曲面拟合算法分析

大型空间柔性结构如太阳能帆板,在轨运行期间必须保证很高的运行精度,因此对结构的振动水平和控制要求及其严格。如能实现其振动响应形态的实时感知与重构,将对其振动响应的主动控制具有重要意义。以太阳能帆板为实验结构模型,通过分布植入光纤光栅传感阵列,利用空分复用和波分复用技术精确检测多点的曲率信息,以柔性帆板结构的振动形态感知和可视化为目标,深入探讨了基于曲率信息的插值方法和曲面的拟合算法,并利用计算机软件技术和计算机图像处理技术对算法进行了仿真,实现了空间曲面的拟合和可视化显示。仿真结果表明,所采取的插值方法及拟合算法比较精确地反映了帆板结构的振动形态,为进一步的振动主动控制研究打下了良好的基础。     

Determining Whether a Multivariate Hyperexponential Function is Algebraic

Let F=C（x1,x2,…,xe,xe＋1,…,xm）, where x1, x2,… , xe are differential variables, and xe＋1,…,xm are shift variables. We show that a hyperexponential function, which is algebraic over F,is of form g（x1, x2, …,xm）q（x1,x2,…,xe）^1/lwe＋1^xe＋1…wm^xm, where g∈ F, q ∈ C（x1,x2,…,xe）,t∈Z^＋ and we＋1,…,wm are roots of unity. Furthermore,we present an algorithm for determining whether a hyperexponential function is algebraic over F.     

多步Runge—Kutta方法的代数稳定性

本文获得了多步Runge—是Kutta方法代数稳定的一系列必要充分条件,其中多数结果可视为关于Radau I A、Radau ⅡA及Gauss型Runge—Kutta方法已有结果的推广。     

Algebraic Stability of Multistep Runge-Kutta Methods

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分布时滞细胞神经网络的代数抗饱和补偿设计

针对一类分布时滞神经网络系统提出一种代数抗饱和补偿设计。在不考虑系统输入饱和的前提下,通过系统参数、M矩阵理论、迪尼导数设计出了一种代数判据。在此基础上,当发生输入饱和时,运用舒尔补引理将二次型矩阵不等式(quadratic matrix inequality,QMI)转化成线性矩阵不等式(linear matrix inequality,LMI)。同时给出优化吸引域估计的方法。仿真结果验证了方案的可行性和有效性。     

一种新型椭圆曲线密码系统协处理器的VLSI设计

研究了椭圆曲线运算的方法及椭圆曲线密码系统协处理器VLSI实现问题。基于一种新的投影坐标系,给出了求解椭圆曲线点乘运算的点加与倍点算法,提出了一种新的优化存储单元调度方案。根据研究的算法,得到了一种新型的协处理器VLSI结构。仿真结果表明,在50MHz时钟下,点乘运算速度为平均155次/秒。     

基于虚拟样机技术的齿轮啮合动力学仿真研究

齿轮传动是机械装备中重要的传动装置。运用Pro/E建立齿轮的三维实体模型,结合MATLAB编程生成齿形线,基于ADAMS建立了齿轮啮合传动模型和虚拟样机模型。针对传统计算方法难以获取全齿面接触力的局限性,将Hertz接触理论嵌入仿真模型,运用曲线接触实现了齿轮离散齿及连续齿啮合的动态实时仿真。基于虚拟样机仿真研究了齿轮偏心、轴偏心、轮齿磨损等缺陷对啮合力的影响,为深入研究齿轮传动系统动态特性提供了理论参考依据。     

多尺度条件下制造系统的监控器设计——极大代数方法

针对在时间多尺度条件下制造系统同时具有离散和连续事件,经典的极大代数方法已经不能描述连续事件问题,提出了一种基于混杂时间Petri网的建模方法,通过建立一个新型的双子代数系统,把混杂时间Petri网模型转化为极大代数模型.在极大代数模型基础上,借鉴自动机理论和格理论,通过定义保并映射和保交映射等概念,提出了一种监控器设计方法.这样我们可以继续在极大代数框架下研究时间多尺度条件下制造系统的建模和控制.一个简单的示例说明了极大代数方法的有效性.     

Rational Solutions of High-Order Algebraic Ordinary Differential Equations

This paper considers algebraic ordinary differential equations(AODEs) and study their polynomial and rational solutions. The authors first prove a sufficient condition for the existence of a bound on the degree of the possible polynomial solutions to an AODE. An AODE satisfying this condition is called noncritical. Then the authors prove that some common classes of low-order AODEs are noncritical. For rational solutions, the authors determine a class of AODEs, which are called maximally comparable, such that the possible poles of any rational solutions are recognizable from their coefficients. This generalizes the well-known fact that any pole of rational solutions to a linear ODE is contained in the set of zeros of its leading coefficient. Finally, the authors develop an algorithm to compute all rational solutions of certain maximally comparable AODEs, which is applicable to 78.54% of the AODEs in Kamke's collection of standard differential equations.     

基于谐波传动啮合仿真的齿形干涉研究

根据谐波齿轮传动基本理论,建立谐波传动局部啮合仿真的计算模型.已知原始曲线采用四力作用式模型计算得出,柔轮齿形曲线采用渐开线曲线,根据包络法计算出刚轮齿廓,并求解柔轮齿廓的运动轨迹.在此基础上,完成谐波传动全图和局部啮合仿真,并对谐波传动的特性进行论证.最后通过局部啮合图形对谐波传动过程进行干涉研究.结果表明:谐波传动采用不等移距变位,可以很好地控制轮齿啮合侧隙和啮合深度.柔轮变位系数越大,侧隙越大;柔轮径向变形量与模数比越大,侧隙越小.     

粗糙集与拓扑空间的子集

引入拓扑空间的子集关于子基的内部和闭包的定义,并获得了它们的一些基本性质;讨论它们与粗糙集理论中的下近似集和上近似集之间的关系.这些结果可以看作是粗糙集的理论基础.     

Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let f(t,y,y’)=∑_i=0ⁿ a_i(t,y)y’ⁱ=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y’)=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y’ but also on the coefficients of f viewed as the polynomial in t,y,y’.In this paper,the authors show that if f s...     

椭圆与抛物线及双曲线位置关系的代数条件

在计算机动画、计算机图形学、计算机辅助设计、机器人等领域中，经常需要检测多个实体间的位置关系。我们基于广义特征多项式的方法，给出了判断平面上椭圆与抛物线，椭圆与双曲线所有位置关系（分离、外切、相交、内切、内含等）的代数条件。这些代数条件表示为由椭圆与抛物线，椭圆与双曲线确定的广义特征多项式的根的分布。判别方法简单实用。     

线性系统同时极点配置的代数几何方法

利用代数几何方法,研究两个线性系统状态反馈和输出反馈同时极点配置问题。通过讨论代数几何中的有理映射是否为到上的,来判别线性系统的特征多项式的系数可否几乎任意配置,从而推导出两个线性系统状态反馈和输出反馈同时极点配置的充分条件。将此结论应用到同时镇定问题上,得到了两个线性系统同时镇定的充分条件,并证明了如果两个线性系统存在复反馈同时配置极点,则一定存在实反馈同时配置极点。