三点共线问题赏析

<正>几何问题中,我们常见一类经典问题——三点共线问题.通过学习平面向量知识,我们深刻地体会到:求解三点共线问题,向量的知识和方法非常有用.我们应该学会应用平面向量的有关知识和方法灵活求解几何问题.     

多目标规划求解中修正权系数的方法

我们利用 p级数方法求解多目标规划问题 MOP,并用分层法的思想确定权系数 .求解多目标规划问题 MOP就相当于求解分层的多目标规划问题 L SP.这样 ,我们就可以确定这个函数的目标函数解 ,如果这个解不是满足决策者要求的 Pareto有效解 ,就改变原 MOP问题的权系数。我们就用这个迭代的方法求解多目标规划问题 MOP。     

利用旋转变换作图一例及其推广应用

<正>尺规作图是平面几何的重要内容,掌握好尺规作图有助于我们探索解题思路,有助于加深我们对平面几何的理解与认识.有些作图问题,如果仅仅从基本作图方法考虑,问题解决起来比较困难,但如果我们从旋转变换的角度出发,问题就变得容易思考.下面我们将从一个具体作图问题开始,利用旋转变换解决问题,并将作图方法推广.     

一个求解消失约束数学规划的部分光滑正则化方法（英文）

本文研究带有消失约束的数学规划问题.针对这一问题,我们提出了一种基于伪Huber函数的光滑正则化方法,该方法只对部分消失约束进行光滑化.对于新的光滑问题,我们证明Mangasarian-Fromovitz约束规格在某些情况下是成立的.我们也分析该方法的收敛性质,即,一个光滑正则化问题稳定点序列的聚点是原问题的T-稳定点,并给出光滑正则化问题稳定点序列的聚点是原问题的M-稳定点或S-稳定点的一些充分条件.最后初步的数值结果表明该方法是可行的.     

谈数学解题中的“变更问题法”

数学解题(或证题)中,常遇到一些问题,对问题直接求解(证)较为困难,我们往往将原问題变换为一个新问题,通过新问题的求解(证),达到解决原问题的目的,这种解题方法我们称它为“变更问题法”。“变更问题法”是数学问题中应用极为广泛的解题方法。本文想对“变更问题法”的形式与原则作些探讨。     

“极端分析法”在解题中的妙用

<正>我们知道,极限思想是一种重要的数学思想方法,当面对较复杂问题时,我们也时常采用"极端"的思考方式,以便于得出结论,当然在不同的问题背景下,极端分析的方法也有很多不同的表现形式.在运动变化的问题中,我们往往可以考虑问题在极端情形下的情况,这样的思考方式经常用来猜出结论,如解析几何中定点定值问题,经常可以作为先猜后证的一种重要方法.     

涉及四个三角形的一个几何定理

在平面几何中,我们经常可以见到“在任意三角形的三边上向形外或形内作三个三角形……”这类涉及四个三角形的问题(或者可以化为这种类型的问题),对于这类问题,我们往往是采用三角方法或复数方法来处理的,本文将揭示适于解决这类问题的一个几何定理,其证明方法是纯几何的。     

例谈处理多变元问题的常用方法

<正>在解决数学问题的过程中,经常会遇到变元较多的情况,处理这样的问题会让我们觉得无从下手．是否有一些行之有效的方法呢?下面我们结合实例说明处理多变元问题的常用方法．     

一类非线性外问题的数值解法

本文利用FEM-BEM方法研究平面上一类非线性外问题数值方法, 给出了基于非线性人工边界条件的耦合问题收敛性结果和误差估计.数值算例验证了我们的理论分析结果. 最后, 我们提出求解其耦合问题的一种区域分解算法.     

非凸优化问题的一个对偶结论

本文研究了单约束条件的非凸极小问题的对偶形式,我们的结论是通过变换,可以化成无缝对偶情形,同时我们研究了多约束条件的同类问题的处理方法。     

On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes

T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined on this type of meshes, which provides a solution for the defect of NURBS. In this paper, we generalize the definitions to the three-dimensional (3D) case and discuss a fundamental problem – the dimension of trivariate spline spaces on 3D T-meshes. We focus on a special case where splines are C ^d?1 continuous for degree d. The smoothing cofactor method for trivariate splines is explored for this situation. We obtain a general dimension formula and present lower and upper bounds for the dimension. At last, we introduce a type of 3D T-meshes, where we can give an explicit dimension formula.     

On the unimprovability of full-memory strategies in the risk minimization problem

We continue the study of approximation properties of local exponential splines on a uniform grid with step h > 0 corresponding to a linear differential operator L with constant coefficients and real pairwise different roots of the characteristic polynomial (such splines were constructed by E.V. Strelkova and V.T. Shevaldin). We find order estimates as h → 0 for the error of approximation of certain Sobolev classes of functions by splines of the described type that are exact on the kernel of the operator L.     

Multivariate Bernoulli splines and the periodic interpolation problem

Periodic spline interpolation in Euclidian spaceR d is studied using translates of multivariate Bernoulli splines introduced in [25]. The interpolating polynomial spline functions are characterized by a minimal norm property among all interpolants in a Hilbert space of Sobolev type. The results follow from a relation between multivariate Bernoulli splines and the reproducing kernel of this Hilbert space. They apply to scattered data interpolation as well as to interpolation on a uniform grid. For bivariate three-directional Bernoulli splines the approximation order of the interpolants on a refined uniform mesh is computed.     

Remarkable Haar spaces of multivariate piecewise polynomials

Some families of Haar spaces in \(\mathbb {R}^{d},~ d\ge 1,\) whose basis functions are d-variate piecewise polynomials, are highlighted. The starting point is a sequence of univariate piecewise polynomials, called Lobachevsky splines, arised in probability theory and asymptotically related to the normal density function. Then, it is shown that d-variate Lobachevsky splines can be expressed as products of Lobachevsky splines. All these splines have simple analytic expressions and subsets of them are suitable for scattered data interpolation, allowing efficient computation and plain error analysis.     

Complex Wavelets and Framelets from Pseudo Splines

In this paper, we introduce complex pseudo splines that are derived from pseudo splines of type I. First, we show that the shifts of every complex pseudo spline are linearly independent. Therefore we can construct a biorthogonal wavelet system. Next, we investigate the Riesz basis property of the corresponding wavelet system generated by complex pseudo splines. The regularity of the complex pseudo splines will be analyzed. Furthermore, by using complex pseudo splines, we will construct symmetric or antisymmetric complex tight framelets with desired approximation order.     

Reconstruction matrices and calibration relations for minimal splines

On a nonuniform gird, we consider twice continuously differentiable splines of third order and obtain calibration relations expressing the coordinate splines on the original grid as a linear combination of splines of the same type on a refined grid. We also obtain calibration relations representing the coordinate splines on an enlarged grid as a linear combination of splines of the same type on the original grid. We derive the reconstruction matrices on an interval and on a segment for the space of third order splines associated with infinite and finite nonuniform grids respectively. Bibliography: 10 titles.     

On Lebesgue constants of local parabolic splines

Lebesgue constants (the norms of linear operators from C to C) are calculated exactly for local parabolic splines with an arbitrary arrangement of knots, which were constructed by the second author in 2005, and for N.P. Korneichuk’s local parabolic splines, which are exact on quadratic functions. Both constants are smaller than the constants for interpolating parabolic splines.     

A local Lagrange interpolation method based on cubic splines on Freudenthal partitions

A trivariate Lagrange interpolation method based on cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.

     

Lösung von Differentialgleichungen mit Splinefunktionen. Eine Störungstheorie

Summary In this paper non-linear splines (depending onn+1 parameters) are used to patch up the solution of an initial value problem in intervals of stepsizeh. The elements of the solution are fixed byq smoothness conditions andd conditions derived from the differential equation in an appropriate setup. The feasibility of the method can be connected to that of the polynomial spline method by a perturbation type argument. Thus the question of convergence forh0 is closely connected to the linear (polynomial) case.A new elementary prove is given for divergence of the polynomial splines ifq is larger thand+1, as was done by Mülthei [4] with other techniques.A byproduct is an extention of the famous result for polynomial interpolation by Runge on equidistant grids that interpolation of a given function by splines of too high smoothness can cause divergence forh0.

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 72 entstanden     

Approximation by Piecewise Constant Functions on a Square

In this paper we consider several algorithms for approximating functions defined on the unit square I = [0,1]² and ranging in $\mathbb{R}^2 $ . We use functions of zeroth-order Lagrange spline type as the approximation apparatus. They differ from the standard Lagrange splines on the plane by the rule for choosing grid lines according to which the spline is constructed; namely, a set of one-dimensional splines is used instead of a family of parallel lines determining the interpolation nodes.