THE DIVERGENCE OF LAGRANGE INTERPOLATION IN EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to \x\ at e-qually spaced nodes in [-1.1] diverges everywhere. except at zero and the end-points. In this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [-1.1] still diverges every -where in the interval except at zero and the end-points.

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|~α(2<α<4) AT EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to ｜x｜ at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to ｜x｜^α （2 〈 α 〈 4） on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.

Equidistant combination wavelength screening and step-by-step phase-out method for the near-infrared spectroscopic analysis of serum urea nitrogen

We applied near-infrared(NIR)spectroscopy with chemometrics for the rapid and reagent-fee analysis of serum urea nitrogen(SUN).The modeling is based on the average effect of multiple sample partitions to achieve parameter selection with stability.A multiparameter optimization platform with Norris derivative filter-partial least squares(Norris-PLS)was developed to select the most suitable mode(d=2,s=33,g=15).Using equidistant combination PLS(EC-PLS)with four parameters(initial wavelength I,number of wavelengths N,number of wavelength gaps G and latent variables LV),we performed wavelength screening after eliminating high-absorption wavebands.The optimal EC-PLS parameters were I=1228 nm,N=26,G=16 and LV=12.The root-mean square error(SEP),correlation coefficient(R_(p))for prediction and ratio of performance-to-deviation(RPD)for validation were 1.03 mmol L^(-1),0.992 and 7.6,respectively.We proposed the wavelength step-by-step phase-out PLS(WSP-PLS)to remove redun-dant wavelengths in the top 100 EC-PLS models with improved prediction performance.The combination of 19 wavelengths was identifed as the optimal model for SUN.The SEP,Rp and RPD in validation were 1.01 mmol L^(-1),0.992 and 7.7,respectively.The prediction effect and wavelength complexity were better than those of EC-PIS.Our results showed that NIR spectroscopy combined with the EC-PLS and WSP-PLS methods enabled the high-precision analysis ofSUN.WSP-PLS is a secondary optimization method that can further optimize any wavelength moc odel obtained through other continuous or discrete strategies to establish a simple and better model.

Approximately Conformal,Equivalent and Equidistant Map Projections

If geodetic coordinates from an ellipsoid are included in the equations of a projection for mapping a sphere instead of geographical coordinates,the result will be a projection of the ellipsoid into a plane.This will slightly change the distortion distribution of the initial map projection.The question is to what extent the replacement of geographical with geodetic coordinates will affect this change.In this paper,we deal with conformal,equal-area and equidistant projections of the sphere,which we modify by using geodetic coordinates instead of geographical ones.The result will be an approximately conformal,approximately equal-area and approximately equidistant projection.It is shown that in this case the maximum distortion of the angles in approximately conformal projections will be approximately 23.09′,the maximum distortion of the area in approximately equal-area projections less than 0.7% and the maximum distortion of lengths in approximately equidistant projections less than 0.7%,therefore on the maps imperceptible.

INTEGRAL REPRESENTATION OF SLOWLY GROWING EQUIDISTANT SPLINES

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow a5 O (|x|~5). We present an integral representation of such splines with a distribution kernel. This repre- sentation is related to the Fourier integral of slowly growing functions. The part of the Fourier ex- ponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First. it allows us to con- struct a rich library of splines possessing the property that translations of any such spline form a ba- sis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of growing func- tion. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.

Preliminary Exploration on Properties of the Equidistant Conjugate Points of Higher Dimensional Simplex

In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the property of the equidistant conjugate points of a triangle is generalized to high dimensional simplex.

Lexicographic Constant-Weight Equidistant Codes over the Alphabet of Three, Four and Five Elements

In this paper we consider the problem of finding bounds on the size of lexicographic constant-weight equidistant codes over the alphabet of three, four and five elements with 2 ≤ w 【n ≤ 10. Computer search of lexicographic constant-weight equidistant codes is performed. Tables with bounds on the size of lexicographic constant-weight equidistant codes are presented.

Design of a 10 MeV normal conducting CW proton linac based on equidistant multi-gap CH cavities

Continuous wave （CW） high current proton linacs have wide applications as the front end of high power proton machines. The low energy part of such a linac is the most difficult and there is currently no widely accepted solution. Based on the analysis of the focusing properties of the CW low energy proton linac, a 10 MeV low energy normal conducting proton linac based on equidistant seven-gap Cross-bar H-type （CH） cavities is proposed. The linac is composed of ten 7-gap CH cavities and the transverse focusing is maintained by quadrupole doublets located between the cavities. The total length of the linac is less than 6 meters and the average acceleration gradient is about 1.2 MeV/m. The electromagnetic properties of the cavities are investigated by Microwave Studio. At the nominal acceleration gradient the maximum surface electric field in the cavities is less than 1.3 times the Kilpatrick limit, and the Ohmic loss of each cavity is less than 35 kW. Multi-particle beam dynamics simulations are performed with Tracewin code, and the results show that the beam dynamics of the linac are quite stable, the linac has the capability to accelerate up to 30 mA beam with acceptable dynamics behavior.

THE EXACT CONVERGENCE RATE AT ZERO OF LAGRANGE INTERPOLATION POLYNOMIAL TO|x|~α

In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|~a

This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f（z） =｜x｜^α（1〈α〈2） on [-1,1] can diverge everywhere in the interval except at zero and the end-points.

Wavelength Selection for Near-Infrared Spectroscopic Analysis of Teicoplanin

Teicoplanin (TCP) is a multiple drug-resistant lipoglycopeptide antibiotic produced by fermenting Actinoplanes teichomyceticus. In this study, a mixture of TCP with the Tris-HCl buffer (TCP-Tris- HCl) was used to simulate TCP fermentation broth. The reagent-free, rapid and simultaneous quantitative analysis models for TCP and Tris in the TCP-Tris-HCl mixtures were established by near-infrared (NIR) spectroscopy. The equidistant combination partial least squares (EC-PLS) method and the equivalent model sets were proposed, the simplest equivalent model with the smallest number of wavelengths were further selected. The initial wavelength, number of wavelengths, number of wavelength gaps, number of PLS factors were 1520 nm, 28, 5, 5 for TCP and 1084 nm, 13, 6, 4 for Tris, respectively. Compared with the optimal EC-PLS models, the simplest equivalent models adopted fewer wavelengths. Thus, the redundant wavelengths were removed, the models were further simplified. The root-mean-square errors (SEP) and correlation coefficients (RP) for prediction were 0.043 mg·mL-1 and 0.9998 for TCP, and 0.222 mg·mL-1 and 0.9989 for Tris, respectively. The results indicate that NIR method can be applied to highly accurate quantitative analysis for TCP and provide valuable references for further application to TCP fermentation broth.

Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems

The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the boundary layers.The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed.For the central scheme,error estimates are derived in a discrete L^1 norm.They are of second order and decrease together with the perturbation parameterε.The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically.Numerical results showε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.

NUMERICAL METHOD FOR THE SYSTEM OF REACTION-DIFFUSION EQUATIONS WITH A SMALL PARAMETER

This paper deals with the numerical method for the system of reaction-diffusion equations with a small parameter. It is difficult to solve the problems of this kind numerically because of the boundary layer efect Besed on singular perturbed theory and Greens function, we have established the difference scheme that is suited for the solution to the problems. We introduce an idea of feasitbe equidistant degree a here. And this proves that if a>2 the scheme converges in norm with speed uniformly.

Drone for Dynamic Monitoring and Tracking with Intelligent Image Analysis

Traditional monitoring systems that are used in shopping malls or com-munity management,mostly use a remote control to monitor and track specific objects;therefore,it is often impossible to effectively monitor the entire environ-ment.Whenfinding a suspicious person,the tracked object cannot be locked in time for tracking.This research replaces the traditionalfixed-point monitor with the intelligent drone and combines the image processing technology and automatic judgment for the movements of the monitored person.This intelligent system can effectively improve the shortcomings of low efficiency and high cost of the traditional monitor system.In this article,we proposed a TIMT(The Intel-ligent Monitoring and Tracking)algorithm which can make the drone have smart surveillance and tracking capabilities.It combined with Artificial Intelligent(AI)face recognition technology and the OpenPose which is able to monitor the phy-sical movements of multiple people in real time to analyze the meaning of human body movements and to track the monitored intelligently through the remote con-trol interface of the drone.This system is highly agile and could be adjusted immediately to any angle and screen that we monitor.Therefore,the system couldfind abnormal conditions immediately and track and monitor them automatically.That is the system can immediately detect when someone invades the home or community,and the drone can automatically track the intruder to achieve that the two significant shortcomings of the traditional monitor will be improved.Experimental results show that the intelligent monitoring and tracking drone sys-tem has an excellent performance,which not only dramatically reduces the num-ber of monitors and the required equipment but also achieves perfect monitoring and tracking.

Investigation of the Appropriate Partial Level Density Formula for Pre-Equilibrium Nuclear Exciton Model

Ericson formula represents the first formula, which was suggested to describe the partial level density (PLD) formula in pre-equilibrium region of the nuclear reactions. Then a number of corrections were added to this formula in order to make it more suitable to physical meaning. In this paper, there are two aims to be done: the first aim is to study the correspondence between one and two-components formulae in Ericson, Pauli, and pairing corrections;the second aim is to compare and study the results of Comprehensive formula, which contents with all corrections, with Ericson, Pauli, and pairing formulae. The Comprehensive formula was suggested to simulate the reality. To achieve these aims the 56Fe and 90Zr nuclei were chosen and the results showed that the difference between one and two-components formulae was too small which can be neglected. Furthermore, the results strongly recommended that for cross section calculations of the nuclear reaction, one must use Comprehensive formula rather than Pauli formula.

Construction of Supersaturated Design with Large Number of Factors by the Complementary Design Method

Supersaturated designs （SSDs） have been widely used in factor screening experiments. The present paper aims to prove that the maximal balanced designs are a kind of special optimal SSDs under the E（fNOD） criterion. We also propose a new method, called the complementary design method, for constructing E（fNoD） optimal SSDs. The basic principle of this method is that for any existing E（fNOD） optimal SSD whose E（fNOD） value reaches its lower bound, its complementary design in the corresponding maximal balanced design is also E（fNOD） optimal. This method applies to both symmetrical and asymmetrical （mixed-level） cases. It provides a convenient and efl：icient way to construct many new designs with relatively large numbers of factors. Some newly constructed designs are given as examples.

Structure Optimization of Air Filter Based on Parametric Sensitivity

As the supporting supplier of the main engine plant, the general air filter manufacturers have insufficient technical reserves. The structural optimization of air filter is often based on the bench experiment, which has high implementation cost and poor performance. In view of this, taking computational fluid dynamics(CFD) as the basic technical means, an optimization design method based on parametric sensitivity combined with equidistant search was proposed. Specifically, the sensitivity of local structure parameters to pressure loss was analyzed by taking local structure of air filter as the object. According to the sensitivity, the method of equidistant search was used to optimize the parameters in order, so as to achieve the goal of overall optimization. After optimization, the pressure loss decreased by 45.13% and the effect was remarkable.

Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

For the approximation in L_(p)-norm,we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots.For p=1,∞,we obtain its values.By these results we know that for the Sobolev classes,the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.At the same time,the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.

Design of Twin-Screw Compressor Rotor Tooth Profile with Meshing Clearance Based on Graphic Method and Alpha Shape Algorithm

Rotor clearance is necessary for the safe operation of twin-screw compressors,and it has a major impact on the performance of twin-screw compressors.The purpose of this study was to obtain a rotor tooth profile with reasonable meshing clearance on the rotor end surface,so that the clearance on the rotor contact line would be uniform and the rotor could be smoothly meshed.Under ideal conditions,the rotor of a screw compressor should have no clearance or interference.However,owing to assembly errors,thermal compression,stress deformation,and other factors,a rotor without backlash modification will inevitably produce interference during operation.A new design method based on the Alpha shape solution was proposed to achieve an efficient and high-precision design of the clearance of the twin-screw rotor profile.This method avoids the complex analytical calculations in the traditional envelope principle.The best approximation of the points on the rotor conjugate motion sweeping surface in the points is illuminated using a specific color.The sweeping surface of the screw rotor single-tooth profile is roughly scanned to capture the base point set of the sweeping surface boundary points.The chord length and tilt angle of each interval are calculated using the value of the base point set to adjust the position,phase,and magnification of each interval sweeping surface.Finally,the data point set is converted to the same coordinate system to generate the conjugated rotor profile.An example was used to verify the feasibility and adaptability of this method.Based on the equidistant profile method,the clearance between male and female rotors of a screw compressor was obtained under actual operation conditions.Therefore,this study provides a basis for the meshing clearance design in the machining of twin-screw compressor rotors.

Ordering and Pricing Decisions of a Retailer in the Presence of Multiple-Time Ordering

This paper studies ordering and pricing issues of a retailer who faces a linear-sensitive demand.The definition of a reasonable retail price is put forward.After that,a deterministic model for trading off sales revenue and total cost is constructed,in which the retailer determines the optimal ordering schedule and the optimal retail price simultaneously for the sake of maximizing its total profit.It is shown that the acquired retail prices are all reasonable.During the process of derivation,both ordering time points and the optimal retail price are expressed as functions of ordering times.By solving a quadratic programming model with an undetermined parameter,we demonstrate that the optimal ordering time points of the retailer are equidistant time points on the given selling periods.Further,the sensitivity of each parameter is analyzed and some meaningful conclusions are drawn.Finally,the ordering times is obtained by analyzing the derivatives of the profit function.