Orthogonal Chebyshev Polynomials

The Strong Chebyshev Distribution and Orthogonal Laurent Polynomials

The paper compares three methods of coding nominal variables in regression analysis: coding of each level as a separate variable, coding with binary code, numbering of factor levels. Although these methods have existed for a long time and even have a theoretical justification (except for encoding with binary code), there were no recommendations and comparisons for their practical application. The features of the application of each method and the existing limitations are analyzed. In the article, there are considered two examples that provide a detailed comparison of these three methods. Comparative analysis has been carried out in the following areas: the presence of restrictions in use; statistical properties of plans; labour intensity and difficulty of obtaining mathematical models and the final result of their building; convenience of semantic analysis and use. Additionally, there have been made comparisons with models based on Chebyshev orthogonal polynomials. It has been established that different methods of coding nominal variables, when used correctly, lead to regression models that are approximately identical in their properties. Moreover, the method of encoding each level as a separate variable is possible only if there are experiments in which there is no nominal variable as an influence effect. The binary coding method is inconvenient to use with a large number of levels of variation of the nominal variable and inconvenient to analyze. When coding by level numbering, it is necessary that the average response values, according to the dispersion diagram of this factor, are sorted by value in accordance with the assigned numbers. With this encoding method, a natural number of factors is preserved. Sharply distinguishable best results are achieved with this coding method using Chebyshev orthogonal polynomials. The highest accuracy and uniformity of approximation are ensured.

Chebyshev collocation method for static intrinsic equations of geometrically exact beams

Fingerprint image segmentation heavily influences the performance of automated fingerprint identification systems. We propose a segmentation technique based on the topographic properties of the image surface. The digital image is treated as a noisy sampling of the underlying continuous surface which is approximated by orthogonal Chebyshev polynomials. Ridge and valley regions are discriminated by the sign of the maximum normal curvature of this continuous surface. Different approximation accuracies are used along the orthogonal and parallel directions to the ridge orientation. This determines a lower sensitivity to small irregularities along the ridge direction then along the orthogonal direction, resulting in a high quality segmentation.

Abstract. Buckling of nonuniform carbon nanotubes are studied with the axial load taken as a combination of concentrated and axially distributed loads. Constitutive modelling of the nanotubes is implemented via nonlocal continuum mechanics. Problem solutions are obtained by employing a weak formulation of the problem and the Rayleigh-Ritz method which is implemented by using orthogonal Chebyshev polynomials. The accuracy of the method of solution is verified against available results. Solutions are obtained for the cases of uniformly distributed and triangularly distributed axial loads. Contour plots are given to assess the effect of nonuniform cross-sections and the small-scale parameter on the buckling load for a combination of simply supported, clamped and free boundary conditions.

Via simulating the contribution of spin-orbit interaction to nuclear masses by orthogonal Chebyshev polynomials, the mass excesses are calculated for a group of atomic nuclei characterized by the isospins of Tz = 51/2, 26, 53/2, 55/2, 28, and 57/2 and by the presence of restored Wigner’s spin-isospin SU(4) symmetry. The root-mean-square deviation of the results of these calculations from their experimental counterparts is σ = 140 keV. The potential of the proposed method is discussed.

Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

If a finite Markov chain (discrete time, discrete states) has a number of absorbing states, one of these will eventually be reached. In this paper are given theoretical formulae for the probability distribution, its generating function and moments of the time taken to first reach an absorbing state, and these formulae are applied to an example taken from genetics. While first passage time problems and their solutions are known for a wide variety of Markov chain processes (e.g., [2], [7], [4]), the theory seems not to have been used in population genetics. Suppose a genetic population consists of a constant number of individuals and the state of the population is defined by the numbers of the various genotypes existing at a given time. Then if mutation is absent, all individuals will eventually become of the same genotype because of random influences such as births, deaths, mating, selection, chromosome breakages and recombinations. The population behavior may in some circumstances be approximated by a Markov chain with absorbing states. In Section 2, two alternative approaches are given for the theoretical determination of absorption time properties, using well known techniques. In Section 3, the consequences of the theoretical results are investigated for a particular population model introduced by Moran [9], [10], and explicit expressions for the distribution of the gene fixation time are obtained in terms of Chebyshev's orthogonal polynomials. The derivation requires finding the pre- and post-eigenvectors of the matrix of transition probabilities, and an incidental by-product is the proof of certain identities for the orthogonal polynomials. The material presented in Section 2 and Section 3 is obtained by exact methods. In Section 4, the Fokker-Planck diffusion equation is used to obtain approximate results, and these are compared with those of the exact theory to ascertain the accuracy of the diffusion approximation.

The objective of this paper is to reduce the computational effort in reservoir flooding optimization problems by a combination of different optimization parametrization methods and model order reduction techniques. We compare three different parametrization methods that reduce the cardinality of the original infinite set of control-decision variables to a finite set. The three methods include a traditional piece-wise constant (PWC) approximation, a polynomial approximation by Chebyshev orthogonal polynomials and a piece-wise polynomial approximation by cubic Spline interpolation. The Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD-DEIM) accomplishes the reduced order modeling (ROM).. We compare a gradient-free global stochastic search approach and a gradient-based local search approach. We used Particle Swarm Optimization (PSO) as a gradient-free algorithm and Interior-Point Optimization (IPOPT) with L-BFGS method as a gradient-based algorithm. First, we compare the performances of the three parametrization methods solved by each optimizer, using fine scale simulations for an increasing level of parametrization refinement. Then, in the second part of this paper, we combine the parametrization methods with the reduced modeling workflow. For a given level of parametrization refinement, we compare the performance of each parametrization method coupled with POD-DEIM, and solved by each optimizer. In this part, we introduce an online training procedure, where the first optimization iteration is used to construct the snapshot matrix. The results demonstrate how refining the control approximation with more decision variables per well lead to better NPV values, but with a higher computational cost. The best NPV was achieved using the highest refining level with Chebyshev polynomial approximation. Both polynomial and piece-wise polynomial approximations served as better training sets for POD-DEIM leading to a more accurate and fast reduced model. With the strategy proposed, POD-DEIM showed the best optimization accuracy for Chebyshev polynomial with the gradient-free optimizer, thus permitting the use of the model reduction methodology for global-stochastic search methods. However, the gradient-based approach seems to consistently outperform the gradient-free approach in terms of NPV and number of iterations for the cases shown.

Purpose. Modification of the numerical calculation method of transients in linear electric circuits, as well as the development of a schematic model of the method, leading to ease the calculation and to increase the speed of calculation on a computer..Methodology. Polynomials approximation of functions, numeral methods of solution of integro-differential equations, matrix methods, computer programming and methods of electric circuits theory are used.Findings The improved method of transient calculation is worked out as the modification of the known one. It allows to reduce the computer modeling time of dynamic electric processes in linear electric circuits. The example of the application of the proposed method is shown. Based on the developed method, the computer program was compiled for calculating the transient in the model circuit. This example showed the reduction in CPU time of 45% compared with the use of known methodsOriginality. The approximation of the functions of currents with time by series in the Chebyshev orthogonal polynomials is used. Chebyshev polynomials possess the property of the approximation error uniformity of functions in the entire range of the argument, which distinguishes them from a number of other orthogonal functions. It is proposed to use a polynomial approximation not of the solution functions themselves, but of its derivative. This significantly reduces the approximation error of both the solution function and the derivative of the solution, which appears in the calculation method.. It is shown how, in the proposed method, integro-differential equations of state are transformed into linear algebraic equations. The schematic model of the proposed numerical calculation method is developed. The schematic model allows you to move from an electrical circuit in which processes are described by integro-differential equations to an equivalent circuit with images of currents. Images of true currents in the equivalent circuit are interpreted as direct currents. It is shown that the Kirchhoff laws are valid for current images in the equivalent circuit, which leads to the calculation of a system of algebraic equations. Such a circuit model creates convenience for an engineer performing transient calculations.Practical value. The developed method opens up the possibility of using the entire diverse apparatus of the theory of electric circuits for working with current images. Based on this, a universal software package is developed for calculating transients in electrical circuits of arbitrary

In a Banach space E, we consider the Neumann problem for a second-order homogeneous equation with a constant operator coefficient densely defined in E. Under a weakened strong positivity condition on the operator coefficient (we use the approach due to V.I. Gorbachuk and A.I. Knyazyuk to the study of a-positive operators), we present a criterion for the well-posed solvability of the problem. We obtain a representation of the solution via Chebyshev operator polynomials of the first and second kind.

In this paper, we use maize yield and meteorological data during the period of 1980-2015 in the Yilan County, Heilongjiang Province to study the impact of meteorological factors (rainfall, sunshine hours, temperature) on maize yield. Based on the computer simulation, Fisher integral model and Chebyshev orthogonal polynomials are used to analyze the coefficients between meteorological factors (temperature, precipitation and sunshine hours) and the maize yield so as to establish a mathematics model between the two. Through the coefficients between meteorological factors and maize yield during different periods, it is discovered that meteorological factors have a great effect on maize yield. By fitting degree of maize yield meteorological model, the maize yield can be predicted in an accurate way. It provides a theoretical basis for the information management of maize production in Heilongjiang Province based on the computer simulation. It is of great significance in providing scientific guidance for the sustainable development of Heilongjiang's corn industry.

ABSTRACTA formulation for static and dynamic vibration analysis of composite laminated curved beams is presented by using an efficient domain decomposition method. Multi-segment partitioning strategy is used to accommodate the accurate requirements of responses. The system potential functional is introduced to satisfy the continuity constraints on the segment interfaces. The Chebyshev orthogonal polynomials are used as displacement admissible functions. The theoretical results are verified by comparing with those previously published in literature, and the ones obtained by using the program ABAQUS. Based on the verification, the parametric study of composite laminated curved beam is carried out.

Latest aircraft design advances have started to recognize the important significance of defining multiple types of uncertainty. An important issue faced in the previous aeroelastic theory is how to deal with uncertain parameters in gust and structure. We describe the governing equation of the structural response of elastic wing in atmosphere due to gust excitation. The uncertain parameters describing the gust model and the wing structure are modeled as interval sets before the unified treatment of the uncertainties. A collocation interval analysis method( CIAM) for gust response based on the first Chebyshev orthogonal polynomials,interval collocation scheme and finite element method is proposed. The formula of CIAM is derived. The method does not require the sensitivities of the objective function with respect to uncertain variables and the assumption of narrow interval is also not needed. The proposed method can be used to solve gust response problems with uncertainties. Numerical example demonstrates that CIAM gives tighter gust response bound including exact response by comparing its results with Taylor interval analysis method( TIAM),which illustrates the efficiency and significant engineering value of the proposed method.

Summary Using Chebyshev orthogonal polynomial and optimizing algorithm, we develop a new asymmetric Chebyshev travel time formula and compare it with high-precision taylor travel time formulas. Based on this, considering the bend-way effect, we apply the new Chebyshev travel time formula to calculate incident angles, and achieve more precise ADCIG from the original seismic gathers. The method is testified by both synthetic data and real marine data. It is verified that using our method to obtain ADCIG is more precise and consume less calculation time.