Shape-Preserving C 2 Functional Interpolation via Parametric Cubics

The paper proposes a method for the construction of a shape preserving C² function interpolating a given set of data. The constructed interpolant is a parametric cubic curve. The shape of the curve can be easily controlled via tension parameters which have an immediate geometric interpretation. The approximation order is investigated and numerical examples are presented.     

On Shape Preserving C2 Hermite Interpolation

We propose a general parametric local approach for functional C ² Hermite shape preserving interpolation. The constructed interpolant is a parametric curve which interpolate values, first and second derivatives of a given function and reproduces the behavior of the data. The method is detailed for parametric curves with piecewise cubic components. For the selected space necessary and sufficient conditions are derived to ensure the convexity of the constructed interpolant. Monotonicity is also studied. The approximation order is investigated for both cases. The use of a parametric curves to interpolate data from a function can be considered a disadvantage of the scheme. However, the simple structure of the used curve greatly reduces such a disadvantage.     

Constructing C3 shape preserving interpolating space curves

We present a method for constructing shape-preserving C ³ interpolants in R ³. The resulting curve is obtained by adding a polynomial perturbation of high degree to a curve which is shape-preserving but not sufficiently smooth. The degree of the perturbed curve is selected in order to maintain the shape properties of the basic curve.     

一种保形C~1三次样条插值方法

本文得到了构造一个保形Ｃ１三次插值样条函数的充要条件，并给出了一种构造保形Ｃ１三次插值样条函数的方法．     

一类含参数均匀三角样条插值曲线

基于一类C3连续的三角样条基函数,首先分别构造了含参数α的C2和C3连续的三角样条插值曲线,然后通过在基函数中引入参数λ,构造了含两个参数α,λ的形状可调控插值曲线,通过α,λ的不同取值,可得到一类有较好保凸和保单调效果的插值曲线,最后用图例验证了理论的有效性和正确性.     

Interpolation Methods for Curve Construction

This paper surveys a wide selection of the interpolation algorithms that are in use in financial markets for construction of curves such as forward curves, basis curves, and most importantly, yield curves. In the case of yield curves the issue of bootstrapping is reviewed and how the interpolation algorithm should be intimately connected to the bootstrap itself is discussed. The criterion for inclusion in this survey is that the method has been implemented by a software vendor (or indeed an inhouse developer) as a viable option for yield curve interpolation. As will be seen, many of these methods suffer from problems: they posit unreasonable expections, or are not even necessarily arbitrage free. Moreover, many methods lead one to derive hedging strategies that are not intuitively reasonable. In the last sections, two new interpolation methods (the monotone convex method and the minimal method) are introduced, which it is believed overcome many of the problems highlighted with the other methods discussed in the earlier sections.     

On Shape-Preserving Additions of Fuzzy Intervals

We prove an open problem suggested by Mesiar (1997, Fuzzy Sets and Systems86, 73-78) concerning continuous t-norm-based additions preserving the LR-fuzzy intervals.     

On discrete hyperbolic tension splines

A hyperbolic tension spline is defined as the solution of a differential multipoint boundary value problem. A discrete hyperbolic tension spline is obtained using the difference analogues of differential operators; its computation does not require exponential functions, even if its continuous extension is still a spline of hyperbolic type. We consider the basic computational aspects and show the main features of this approach.     

Tensioned Quasi-Interpolation Via Geometric Continuity

The paper proposes a method for the construction of C ² quasi-interpolating functions with tension properties. The constructed quasi-interpolant is a parametric cubic curve and its shape can be easily controlled via tension parameters which have an immediate geometric interpretation. Numerical examples are presented.     

Arc Length Estimation and the Convergence of Polynomial Curve Interpolation

When fitting parametric polynomial curves to sequences of points or derivatives we have to choose suitable parameter values at the interpolation points. This paper investigates the effect of the parameterization on the approximation order of the interpolation. We show that chord length parameter values yield full approximation order when the polynomial degree is at most three. We obtain full approximation order for arbitrary degree by developing an algorithm which generates more and more accurate approximations to arc length: the lengths of the segments of an interpolant of one degree provide parameter intervals for interpolants of degree two higher. The algorithm can also be used to estimate the length of a curve and its arc-length derivatives. AMS subject classification (2000) 65D05, 65D10     

Piecewise linear interpolants to Lagrange and Hermite convex scattered data

This paper concerns two fundamental interpolants to convex bivariate scattered data. The first,u, is the supremum over all convex Lagrange interpolants and is piecewise linear on a triangulation. The other,l, is the infimum over all convex Hermite interpolants and is piecewise linear on a tessellation. We discuss the existence, uniqueness, and numerical computation ofu andl and the associated triangulation and tessellation. We also describe how to generate convex Hermite data from convex Lagrange data.Research partially supported by the EU Project FAIRSHAPE, CHRX-CT94-0522. The first author was also partially supported by DGICYT PB93-0310 Research Grant.     

C~2 CONTINUOUS NEARLY ARC-LENGTH PARAMETERIZED CURVE

An efficient method for C~2 nearly arc-length parameterized curve is presented. An idea of approximation for the arc-length function of parametric curve which interpolates CAD data points is discussed. The parameterization is implemented by using parameter transformation. Finally, two numerical examples are given..     

任意三角形单元上的三次Lagrange型插值逼近

本文推广了文［２］中的结果，对于任意三角形单元的三次Ｌａｇｒａｎｇｅ型插值多项式给出了原函数ｕ与被插函数Ｕ之间的误差估计     

Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差

在L_q-范数逼近的意义下,确定了基于Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差的弱渐近阶.从我们的结果可以看出,当2≤q<∞,1≤p<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列的p-平均误差弱等价于相应的最佳逼近多项式列的p-平均误差.在信息基计算复杂性的意义下,如果可允许信息泛函为计算函数在固定点的值,那么当1≤p,q<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差弱等价于相应的最小非自适应p-平均信息半径.     

Preconditioned Iterative Methods for Scattered Data Interpolation

The scattered data interpolation problem in two space dimensions is formulated as a partial differential equation with interpolating side conditions. The system is discretized by the Morley finite element space. The focus of this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis functions for the equations corresponding to the interpolation conditions. The effect of the preconditioners are documented by numerical experiments.     

The State of Elliptic Curve Cryptography

Since the introduction of public-key cryptography by Diffie and Hellman in 1976, the potential for the use of the discrete logarithm problem in public-key cryptosystems has been recognized. Although the discrete logarithm problem as first employed by Diffie and Hellman was defined explicitly as the problem of finding logarithms with respect to a generator in the multiplicative group of the integers modulo a prime, this idea can be extended to arbitrary groups and, in particular, to elliptic curve groups. The resulting public-key systems provide relatively small block size, high speed, and high security. This paper surveys the development of elliptic curve cryptosystems from their inception in 1985 by Koblitz and Miller to present day implementations.     

草地蝗虫密度的克立格插值分析——以青海湖西侧铁卜加样区为例

以青海湖西侧铁卜加样区草地蝗虫密度的空间插值为例 ,对比分析了趋势面分析、普通克立格分析和泛克立格分析三种方法 .结果表明 ,1指数模型能够较好地模型化克立格分析中的半方差变异函数 ;2草地蝗虫的空间相关距离为 1 3 5米左右 ;3泛克立格分析插值效果最好     

Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差

在Lq-范数逼近的意义下，确定了基于Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差的弱渐近阶．从我们的结果可以看出，当2≤q〈∞，1≤p〈∞时，基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite—Fejér插值多项式列的p-平均误差弱等价于相应的最佳逼近多项式列的p-平均误差．在信息基计算复杂性的意义下，如果可允许信息泛函为计算函数在固定点的值，那么当1≤p，q〈∞时，基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差弱等价于相应的最小非自适应p-平均信息半径．     

三维插值对称正交小波的构造

高维小波分析是分析和处理多维数字信号的有力工具.基于任意的三维正交尺度函数及相应的正交小波,提出一种构造三维插值对称尺度函数和对称小波的方法,并建立了多维信号采样定理,这一点在信号处理中具有很好的应用价值.最后给出了数值算例.