不变代数曲线与一类三次系统的中心判定问题

对于一类具有一条抛物线解、两条直线解和中心-焦点型奇点的三次系统,证明它以原点为中心的充要条件是它的第一阶焦点量为零.系统在原点的中心条件是通过不变代数曲线构造积分因子或利用Poincare对称原理得以证明.     

平面点列的自动光顺算法

本文考虑平面点列的光顺问题并将该问题化成最小能量曲线的构成问题，即在原点列和相应允许误差构成的带状区域内构造一条最小能量曲线并给出一种自动算法．整个光顺过程分成两步，第一步利用凸分析原理在原点列的允许变动范围内除去多余拐点；第二步在保凸的前提下构造插值点列的最小能量曲线并通过对最小能量曲线进行修正而达到对原型值点列进行光顺的目的．光顺结果不仅可以得到一光顺点列，同时还得到了一条插值点列的光顺曲线．该方法可以对分布不均匀甚至有较大转角的点列进行光顺，与已有的方法比起来具有光顺能力强光顺范围广的特点．     

对称思想在解题中的应用

对称思想在数学问题中是广泛存在的.近几年的高考中都占有一定的比例.如果能发现或挖掘问题中的对称特征,为解题会带来意想不到的效果. 一、抓住图形的对称特征 例1 在平面直角坐标系中,一个圆心在(a,b)的圆包含原点,设此圆在第一象限及第三象限的面积之和为S1,在第二象限及第四象限的面积之和为S2,求S1-S2的值. 分析如图1,S1=SOAPC SOBD,S2=SODQA SOBMC.由于圆的半径未知及组成S1、S2的四个部分的面积都不便用式子计算,要想用代数计算求S1-S。是很困难的.但是,注意挖掘图     

三次T-Bézier螺线的构造

根据道路轨道路径设计中回旋线的特性,构造了一条起始点曲率为零且曲率单调递增变化的三次T-Bézier螺线.由于该螺线是含有参数的三角多项式参数曲线,所以用其代替诸如回旋线等传统螺线作为道路轨道路径设计中的过度曲线拥有易于计算和便于形状调整的优势.最后,分别利用一对该螺线在两圆弧间构造了满足G^2连续的S型和C型过渡曲线,并给出了详细算法.     

一道最值问题的发散思维

例求曲线C:槡x+槡y=1上的点到原点的距离的最小值.分析一在使用基本不等式求最值时,凑定值是解题的重要一环.本题中虽然有定值"1",但与曲线上的点P(x,y)到原点的距离x2槡+y2所要求的定值无直接的关系.可以考虑用"中间量"x+y来联系槡x+槡y与x2+y2.解法一设点P(x,y)是曲线C上的任意一点,则1=(槡x+槡y)2=x+y+2槡xy,结合基本     

泊松分布高阶原点矩的两种计算方法

考虑到直接用定义计算泊松分布高阶原点矩的复杂性,将组合数学中的第二类Stirling数和二项式定理应用到泊松分布高阶原点矩的计算中,得到了泊松分布高阶原点矩的简单和式与递推表达式,并利用结论计算了泊松分布的前九阶原点矩.     

反比例函数图像的一个优美性质

形如y=k/x(k≠0)的函数叫做反比例函数,它的图像是由两条曲线组成的双曲线.双曲线上的点关于原点成中心对称,关于直线y=±x轴对称.     

一种新型四阶S型轨迹规划算法的研究与应用

提出了一种四阶S型运动轨迹规划的新方法,根据给定参数计算得到最大速度三阶导数作用时间,最大加加速度作用时间和最大加速度作用的时间,通过循环计算的方式得到加速阶段所运动的距离与速度,以加速阶段所达到的速度作为最大速度计算得到最大速度作用的时间.此时,由浮点型转整型数据所产生的误差也可以在其最大速度的匀速阶段得到补偿,最后,减速阶段的数据由S曲线加减速控制中速度曲线的对称性得到.该算法在基于倍福PLC的三动子运料平台中进行了验证,实验表明,提出的S型运动轨迹规划算法在不用过多保存曲线数据的情况下也能获得平滑的速度和加速度,有效地提高了系统的柔性,同时简化了算法的实现,大大地节省了PLC的资源.     

一种新型四阶S型轨迹规划算法的研究与应用北大核心CSCD

提出了一种四阶S型运动轨迹规划的新方法,根据给定参数计算得到最大速度三阶导数作用时间,最大加加速度作用时间和最大加速度作用的时间,通过循环计算的方式得到加速阶段所运动的距离与速度,以加速阶段所达到的速度作为最大速度计算得到最大速度作用的时间.此时,由浮点型转整型数据所产生的误差也可以在其最大速度的匀速阶段得到补偿,最后,减速阶段的数据由S曲线加减速控制中速度曲线的对称性得到.该算法在基于倍福PLC的三动子运料平台中进行了验证,实验表明,提出的S型运动轨迹规划算法在不用过多保存曲线数据的情况下也能获得平滑的速度和加速度,有效地提高了系统的柔性,同时简化了算法的实现,大大地节省了PLC的资源.     

一类Z_2对称五次微分系统的中心条件和极限环分支

本文研究了一类Z2对称五次微分系统的中心条件和小振幅极限环分支.通过前6阶焦点量的计算,获得了原点为中心的充要条件,并证明系统从原点分支出的小振幅极限环的个数至多为6.最后通过构造后继函数,给出系统具有6个围绕原点的小振幅极限环的实例.     

Multispherical Euclidean distance matrices

In this paper we introduce new necessary and sufficient conditions for an Euclidean distance matrix to be multispherical. The class of multispherical distance matrices studied in this paper contains not only most of the matrices studied by Hayden et al. (1996) 2, but also many other multispherical structures that do not satisfy the conditions in Hayden et al. (1996) 2.We also study the information provided by the origin of coordinates when it is placed at the center of the spheres and the origin representation property is satisfied. These vectors associated with the origin of coordinates generate a number of supporting hyperplanes for a family of multispherical matrices and also describe part of the null space of the corresponding distance matrices.     

Wavefront evolution of a plane wave passing through an area with heterogeneities

A two-dimensional eikonal equation with the right-hand side tending to unity as the distance from the origin of coordinates tends to infinity is considered. Some formulas describing the wavefront in such a medium are obtained.     

Octahedral Projections of a Point onto a Polyhedron

In computational methods and mathematical modeling, it is often required to find vectors of a linear manifold or a polyhedron that are closest to a given point. The “closeness” can be understood in different ways. In particular, the distances generated by octahedral, Euclidean, and Hölder norms can be used. In these norms, weight coefficients can also be introduced and varied. This paper presents the results on the properties of a set of octahedral projections of the origin of coordinates onto a polyhedron. In particular, it is established that any Euclidean and Hölder projection can be obtained as an octahedral projection due to the choice of weights in the octahedral norm. It is proven that the set of octahedral projections of the origin of coordinates onto a polyhedron coincides with the set of Pareto-optimal solutions of the multicriterion problem of minimizing the absolute values of all components.     

Tiling Polygons with Lattice Triangles

Given a simple polygon with rational coordinates having one vertex at the origin and an adjacent vertex on the x-axis, we look at the problem of the location of the vertices for a tiling of the polygon using lattice triangles (i.e., triangles which are congruent to a triangle with the coordinates of the vertices being integer). We show that the coordinates of the vertices in any tiling are rationals with the possible denominators odd numbers dependent on the cotangents of the angles in the triangles.     

Spherical rarefaction flow in the neighbourhood of a reflection point of a “boundary” characteristic

The structure of the one-dimensional steady spherically symmetric rarefaction flow of an ideal (inviscid and non-heat-conducting) gas in the neighbourhood of a reflection point of a “boundary” C⁻-characteristic is investigated in principal order. The “boundary” C⁻-characteristic separates the gas at rest from the flow due to the outward motion of a piston which confines the gas. In the rt plane, where r is the distance from the centre of symmetry and t is the time, the reflection point, which coincides with the point of arrival on the t axis of the boundary characteristic, coincides with the origin of coordinates. The initial velocity of the piston may be zero (for positive acceleration) or finite. When two symmetrical plane pistons advance, the “derived” derivatives of all the flow parameters on the C⁻-characteristic at the origin of coordinates, which in this case lies on the plane of symmetry, are finite. When a cylindrical and spherical piston advance, the derived derivative of the pressure (velocity) of the gas on the C⁻-characteristic at the origin of coordinates becomes minus (plus)-infinity although without intersecting characteristics of the same family [1–4].     

样条扇形单元

本文在样条分片插值及样条矩形单元的基础之上进而讨论极坐标中二次及三次样条分片插值及样条(圆环)扇形单元.以用于求解圆(环)域与(圆环)扇形域上的各类问题.圆环扇形单元(r≠0)是样条分片插值在极坐标中简单的推广应用,但扇形单元则不然.本文根据扇形单元在r=0处的特殊性对各位移插值函数作了合理的处理,使得该单元即体现了r=0处的几何特性又可以消除该处应变、应力的奇异性.文中给出了用样条(圆环)扇形单元求解平面问题及薄板弯曲问题的数值算例用以说明该单元的效能.     

Construction of a stabilizing control and solution to a problem about the center and the focus for differential systems with a polynomial part on the right side

The stationary differential systems with polynomial right sides are considered. Necessary and sufficient conditions are formulated when a given domain is a domain of asymptotic stability and the origin of coordinates is either the focus or the center. The problem of construction of a stabilizing control in a form of polynomial is studied.     

Construction of a stabilizing control and solution to a problem about the center and the focus for differential systems with a polynomial part on the right side

The stationary differential systems with polynomial right sides are considered. Necessary and sufficient conditions are formulated when a given domain is a domain of asymptotic stability and the origin of coordinates is either the focus or the center. The problem of construction of a stabilizing control in a form of polynomial is studied.     

分数外微分在三维空间中的应用

首先介绍了分数微积分和分数微分形式。讨论了在原点处对曲线坐标的分数外微分变换,并且获得了从三维卡氏坐标到球面坐标和柱面坐标的两个分数微分变换。特别地,当v=m=1时,这两个分数微分变换约化的结果与通过外微积分获得的结果是一致的。     

Addendum to the paper “on certain exact solutions of the fourier equation for regions varying with time” : PMM vol. 37, n6, 1973, pp. 1145–1146

The exact solutions given in [1] are generalized to the case of cylindrical and spherical sectors rotating about the azimuth relative to the coordinate origin either at a uniform rate or with uniform acceleration (or deceleration). The class of equations of motion of the boundaries of the half-space (in the Cartesian coordinates) which lead to exact solutions of the Fourier equation defined in these domains, is enlarged.