Generalized fairing algorithm of parametric cubic splines

Kjellander has reported an algorithm for fairing uniform parametric cubic splines. Poliakoff extended Kjellander’s algorithm to non-uniform case. However, they merely changed the bad point’s position, and neglected the smoothing of tangent at bad point. In this paper, we present a fairing algorithm that both changed point’s position and its corresponding tangent vector. The new algorithm possesses the minimum property of energy. We also proved Poliakoff’s fairing algorithm is a deduction of our fairing algorithm. Several fairing examples are given in this paper.

DEFICIENT CUBIC SPLINES WITH AVERAGE SLOPE MATCHING

We obtain a deficient cubic spline function which matches the functions with certain area matching oner a greater mesh intervals, and also provides a greater flexibility in replacing area matching as interpolation. We also study their convergence properties to the interpolating functions.

AN APPLICATION OF THE EXPONENTIAL CUBIC SPLINES TO NUMERICAL SOLUTION OF A SELF-ADJOINT PERTURBATION PROBLEM

We present a class of the second order optimal splines difference schemes derived from ex- ponential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estimate and give illustrative numerical example.

Several Ways to Calculate the Universal Gravitational Constant <i>G</i>Theoretically and Cubic Splines to Verify Its Measured Value

In 1686, Newton discovered the laws of gravitation [1] and predicted the universal gravitational constant . In 1798, with a torsion balance, Cavendish [2] measured . Due to the low intensity of gravitation, it is difficult to obtain reliable results because they are disturbed by surrounding masses and environmental phenomena. Modern physics is unable to link G with other constants. However, in a 2019 article [3], with a new cosmological model, we showed that G seams related to other constants, and we obtained a theoretical value of . Here, we want to show that our theoretical value of G is the right one by interpreting measurements of G with the help of a new technique using cubic splines. We make the hypothesis that most G measurements are affected by an unknown systematic error which creates two main groups of data. We obtain a measured value of . Knowing that our theoretical value of G is in agreement with the measured value, we want to establish a direct link between G and as many other constants as possible to show, with 33 equations, that G is probably linked with most constants in the universe. These equations may be useful for astrophysicists who work in this domain. Since we have been able to link G with Hubble parameter H0 (which evolve since its reverse gives the apparent age of the universe), we deduce that G is likely not truly constant. It’s value probably slowly varies in time and space. However, at our location in the universe and for a relatively short period, this parameter may seem constant.

Numerical solution of Poisson equation with wavelet bases of Hermite cubic splines on the interval

A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.

AN INTERPOLATION METHOD BY BIVARIATE CUBIC SPLINES WITH C^2 -JOIN ON TYPE-II TRIANGULARS AT A RECTANGULAR DOMAIN

In this paper, an interpolating method for bivariate cubic splines with C2-join on type-II triangular at a rectangular domain is given, and the approximation degree, inter- polating existence and uniqueness of the cubic splines are studied.

On the Transfinite interpolation and Approximation by a Class of Periodic Bivariate Cubic Splines on Type-Ⅱ Triangulation

In this paper, we discuss the transfinite interpolation and approxiulation by a class of periodic bivariate cubic Splines on type-Ⅱ triangulated partition △（2）mn. the existence, uniqueness and the expression of interpolation periodic bivariate splines are given. And at last, we estimate their approximation order.

Explicit Gaussian Quadrature Rules for C1 Cubic Splines with Non-uniform Knot Sequences

This paper provides the explicit and optimal quadrature rules for the cubic C1 spline space,which is the extension of the results in Ait-Haddou et al.(J Comput Appl Math 290:543–552,2015)for less restricted non-uniform knot values.The rules are optimal in the sense that there exist no other quadrature rules with fewer quadrature points to exactly integrate the functions in the given spline space.The explicit means that the quadrature nodes and weights are derived via an explicit recursive formula.Numerical experiments and the error estimations of the quadrature rules are also presented in the end.

Theoretical Modeling and Surface Roughness Prediction of Microtextured Surfaces in Ultrasonic Vibration-Assisted Milling

Textured surfaces with certain micro/nano structures have been proven to possess some advanced functions,such as reducing friction,improving wear and increasing wettability.Accurate prediction of micro/nano surface textures is of great significance for the design,fabrication and application of functional textured surfaces.In this paper,based on the kinematic analysis of cutter teeth,the discretization of ultrasonic machining process,transformation method of coordinate systems and the cubic spline data interpolation,an integrated theoretical model was established to characterize the distribution and geometric features of micro textures on the surfaces machined by different types of ultrasonic vibration-assisted milling(UVAM).Based on the theoretical model,the effect of key process parameters(vibration directions,vibration dimensions,cutting parameters and vibration parameters)on tool trajectories and microtextured surface morphology in UVAM is investigated.Besides,the effect of phase difference on the elliptical shape in 2D/3D ultrasonic elliptical vibration-assisted milling(UEVAM)was analyzed.Compared to conventional numerical models,the method of the cubic spline data interpolation is applied to the simulation of microtextured surface morphology in UVAM,which is more suitable for characterizing the morphological features of microtextured surfaces than traditional methods due to the presence of numerous micro textures.The prediction of surface roughness indicates that the magnitude of ultrasonic amplitude in z-direction should be strictly limited in 1D rotary UVAM,2D and 3D UEVAM due to the unfavorable effect of axial ultrasonic vibration on the surface quality.This study can provide theoretical guidance for the design and fabrication of microtextured surfaces in UVAM.

The burden of upper motor neuron involvement is correlated with the bilateral limb involvement interval in patients with amyotrophic lateral sclerosis:a retrospective observational study

Amyotrophic lateral sclerosis is a rare neurodegenerative disease characterized by the involvement of both upper and lower motor neurons.Early bilateral limb involvement significantly affects patients'daily lives and may lead them to be confined to bed.However,the effect of upper and lower motor neuron impairment and other risk factors on bilateral limb involvement is unclear.To address this issue,we retrospectively collected data from 586 amyotrophic lateral sclerosis patients with limb onset diagnosed at Peking University Third Hospital between January 2020 and May 2022.A univariate analysis revealed no significant differences in the time intervals of spread in different directions between individuals with upper motor neuron-dominant amyotrophic lateral sclerosis and those with classic amyotrophic lateral sclerosis.We used causal directed acyclic graphs for risk factor determination and Cox proportional hazards models to investigate the association between the duration of bilateral limb involvement and clinical baseline characteristics in amyotrophic lateral sclerosis patients.Multiple factor analyses revealed that higher upper motor neuron scores(hazard ratio[HR]=1.05,95%confidence interval[CI]=1.01–1.09,P=0.018),onset in the left limb(HR=0.72,95%CI=0.58–0.89,P=0.002),and a horizontal pattern of progression(HR=0.46,95%CI=0.37–0.58,P<0.001)were risk factors for a shorter interval until bilateral limb involvement.The results demonstrated that a greater degree of upper motor neuron involvement might cause contralateral limb involvement to progress more quickly in limb-onset amyotrophic lateral sclerosis patients.These findings may improve the management of amyotrophic lateral sclerosis patients with limb onset and the prediction of patient prognosis.

Dynamic GM（1,1） Model Based on Cubic Spline for Electricity Consumption Prediction in Smart Grid

Electricity demand forecasting plays an important role in smart grid expansion planning.In this paper,we present a dynamic GM(1,1) model based on grey system theory and cubic spline function interpolation principle.Using piecewise polynomial interpolation thought,this model can dynamically predict the general trend of time series data.Combined with low-order polynomial,the cubic spline interpolation has smaller error,avoids the Runge phenomenon of high-order polynomial,and has better approximation effect.Meanwhile,prediction is implemented with the newest information according to the rolling and feedback mechanism and fluctuating error is controlled well to improve prediction accuracy in time-varying environment.Case study using the living electricity consumption data of Jiangsu province in 2008 is presented to demonstrate the effectiveness of the proposed model.

A Finite Element Cable Model and Its Applications Based on the Cubic Spline Curve

For accurate prediction of the deformation of cable in the towed system, a new finite element model is presented that provides a representation of both the bending and torsional effects. In this paper, the cubic spline interpolation function is applied as the trial solution. By using a weighted residual approach, the discretized motion equations for the new finite element model are developed. The model is calculated with the computation program complier by Matlab. Several numerical examples are presented to illustrate the numerical schemes. The results of numerical simulation are stable and valid, and consistent with the mechanical properties of the cable. The model can be applied to kinematics analysis and the design of ocean cable, such as mooring lines, towing, and ROV umbilical cables.

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the short- coming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a gl-cubic Hermite in- terpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global G2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an al- ternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15（1） （2007）, pp. 41-53].

Denoising of hyperspectral imagery by cubic smoothing spline in the wavelet domain

The acquired hyperspectral images （HSIs） are inherently attected by noise wlm Dano-varylng level, which cannot be removed easily by current approaches. In this study, a new denoising method is proposed for removing such kind of noise by smoothing spectral signals in the transformed multi- scale domain. Specifically, the proposed method includes three procedures： 1 ） applying a discrete wavelet transform （DWT） to each band; 2） performing cubic spline smoothing on each noisy coeffi- cient vector along the spectral axis; 3 ） reconstructing each band by an inverse DWT. In order to adapt to the band-varying noise statistics of HSIs, the noise covariance is estimated to control the smoothing degree at different spectra｜ positions. Generalized cross validation （GCV） is employed to choose the smoothing parameter during the optimization. The experimental results on simulated and real HSIs demonstrate that the proposed method can be well adapted to band-varying noise statistics of noisy HSIs and also can well preserve the spectral and spatial features.

Non-overshooting and Non-undershooting Cubic Spline Interpolation for Empirical Mode Decomposition

To suppress the overshoots and undershoots in the envelope fitting for empirical mode decomposition （EMD）, an alternative cubic spline interpolation method without overshooting and undershooting is proposed. On the basis of the derived slope constraints of knots of a non-overshooting and non-undershooting cubic interpolant, together with ＂not-a-knot＂ conditions the cubic spline interpolants are constructed by replacing the requirement for equal second order derivatives at every knot with Brodlie＇ s derivative formula. Analysis and simulation experiments show that this approach can effectively avoid generating new extrema, shifting or exaggerating the existing ones in a signal, and thus significantly improve the decomposition performance of EMD.

Improvement of Orbit Prediction Algorithm for Spacecraft Through Simplified Precession-Nutation Model Using Cubic Spline Interpolation Method

For the on-orbit flight missions,the model of orbit prediction is critical for the tasks with high accuracy requirement and limited computing resources of spacecraft.The precession-nutation model,as the main part of extended orbit prediction,affects the efficiency and accuracy of on-board operation.In this paper,the previous research about the conversion between the Geocentric Celestial Reference System and International Terrestrial Reference System is briefly summarized,and a practical concise precession-nutation model is proposed for coordinate transformation computation based on Celestial Intermediate Pole(CIP).The idea that simplifying the CIP-based model with interpolation method is driven by characteristics of precession-nutation parameters changing with time.A cubic spline interpolation algorithm is applied to obtain the required CIP coordinates and Celestial Intermediate Origin locator.The complete precession nutation model containing more than 4000 parameters is simplified to the calculation of a cubic polynomial,which greatly reduces the computational load.In addition,for evaluating the actual performance,an orbit propagator is built with the proposed simplified precession-nutationmodel.Compared with the orbit prediction results obtained by the truncated series of IAU2000/2006 precession-nutation model,the simplified precession-nutation model with cubic spline interpolation can significantly improve the accuracy of orbit prediction,which implicates great practical application value in further on-orbit missions of spacecraft.

Combining Cubic Spline Interpolation and Fast Fourier Transform to Extend Measuring Range of Reflectometry

The reflectometry is a common method used to measure the thickness of thin films. Using a conventional method,its measurable range is limited due to the low resolution of the current spectrometer embedded in the reflectometer.We present a simple method, using cubic spline interpolation to resample the spectrum with a high resolution,to extend the measurable transparent film thickness. A large measuring range up to 385 m in optical thickness is achieved with the commonly used system. The numerical calculation and experimental results demonstrate that using the FFT method combined with cubic spline interpolation resampling in reflectrometry, a simple,easy-to-operate, economic measuring system can be achieved with high measuring accuracy and replicability.

OPTIMAL ERROR BOUNDS FOR THE CUBIC SPLINE INTERPOLATION OF LOWER SMOOTH FUNCTIONS(1)

In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu- bic spline interpolation of lower smooth functions is obtained.

Stability Analysis of Spatial Cubic Spline Geometric Nonlinear Beam Element Considering the Second-Order Effect

To analyze the stability problem of spatial beam structure more accurately, a spatial cubic spline geometric nonlinear beam dement was proposed considering the seeond-order effect. The deformation field was built with cubic spline function, and its curvature degree of freedom （DOF） was eliminated by static condensation method. Then we got the geometric nonlinear stiffness matrix of the new spatial two.node Euler-Bernouili beam dement. Several examples proved calculation accuracy of the critical load by meshing a bar to one element using the method of this paper was equivalent to mesh a bar to 3 or 4 traditional nonlinear beam dements.

A polynomial smooth epsilon-support vector regression based on cubic spline interpolation

Regression analysis is often formulated as an optimization problem with squared loss functions. Facing the challenge of the selection of the proper function class with polynomial smooth techniques applied to support vector regression models, this study takes cubic spline interpolation to generate a new polynomial smooth function ｜×｜ε^ 2, in g-insensitive support vector regression. Theoretical analysis shows that Sε^2 -function is better than pε^2 -function in properties, and the approximation accuracy of the proposed smoothing function is two order higher than that of classical pε^2 -function. The experimental data shows the efficiency of the new approach.