ORTHOGONAL POLYNOMIALS AND DETERMINANT FORMULAS OF FUNCTION-VALUED PADE-TYPE APPROXIMATION USING FOR SOLUTION OF INTEGRAL EQUATIONS

To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.

AN ALGEBRAIC APPROACH TO DEGENERATE APPELL POLYNOMIALS AND THEIR HYBRID FORMS VIA DETERMINANTS

It is remarkable that studying degenerate versions of polynomials from algebraic point of view is not limited to only special polynomials but can also be extended to their hybrid polynomials.Indeed for the first time,a closed determinant expression for the degenerate Appell polynomials is derived.The determinant forms for the degenerate Bernoulli and Euler polynomials are also investigated.A new class of the degenerate Hermite-Appell polynomials is investigated and some novel identities for these polynomials are established.The degenerate Hermite-Bernoulli and degenerate Hermite-Euler polynomials are considered as special cases of the degenerate Hermite-Appell polynomials.Further,by using Mathematica,we draw graphs of degenerate Hermite-Bernoulli polynomials for different values of indices.The zeros of these polynomials are also explored and their distribution is presented.

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

Key Management Using Chebyshev Polynomials for Mobile Ad Hoc Networks

A dedicated key server cannot be instituted to manage keys for MANETs since they are dynamic and unstable. The Lagrange's polynomial and curve fitting are being used to implement hierarchical key management for Mobile Ad hoc Networks(MANETs). The polynomial interpolation by Lagrange and curve fitting requires high computational efforts for higher order polynomials and moreover they are susceptible to Runge's phenomenon. The Chebyshev polynomials are secure, accurate, and stable and there is no limit to the degree of the polynomials. The distributed key management is a big challenge in these time varying networks. In this work, the Chebyshev polynomials are used to perform key management and tested in various conditions. The secret key shares generation, symmetric key construction and key distribution by using Chebyshev polynomials are the main elements of this projected work. The significance property of Chebyshev polynomials is its recursive nature. The mobile nodes usually have less computational power and less memory, the key management by using Chebyshev polynomials reduces the burden of mobile nodes to implement the overall system.

Chebyshev Polynomials for Solving a Class of Singular Integral Equations

This paper is devoted to studying the approximate solution of singular integral equations by means of Chebyshev polynomials. Some examples are presented to illustrate the method.

Chebyshev Polynomials with Applications to Two-Dimensional Operators

A new application of Chebyshev polynomials of second kind Un(x) to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows to reduce powers and smooth functions of them to superpositions of the first N-1 powers of the considered operator in N-dimensional case. The method leads in two-dimensional case first to the recurrence relations for Chebyshev polynomials and due to initial conditions to the application of Chebyshev polynomials of second kind Un(x). Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind Un(x) comprising the known Generating function as special cases is constructed by means of a derived identity for operator functions f(A) of a general two-dimensional operator A. The basic results are Formulas (9.5) and (9.6) which are then specialized for different examples of functions f(x). The generalization of the theory for three-dimensional operators is started to attack and a partial problem connected with the eigenvalue problem and the Hamilton-Cayley identity is solved in an Appendix. A physical application of Chebyshev polynomials to a problem of relativistic kinematics of a uniformly accelerated system is solved. All operator calculations are made in coordinate-invariant form.

Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

DENSITY OF MARKOV SYSTEMS AND ZEROS OF CHEBYSHEV POLYNOMIALS

We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.

Lucas Symbolic Formulae and Generating Functions for Chebyshev Polynomials

This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functions of them by operator calculus built from the derivative and the positional operators.

Analysis of stochastic bifurcation and chaos in stochastic Duffing-van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.

Precise integration methods based on the Chebyshev polynomial of the first kind

This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.

Application of Chebyshev Polynomial to simulated modeling

Chebyshev polynomial is widely used in many fields, and used usually as function approximation in numerical calculation. In this paper, Chebyshev polynomial expression of the propeller properties across four quadrants is given at first, then the expression of Chebyshev polynomial is transformed to ordinary polynomial for the need of simulation of propeller dynamics. On the basis of it, the dynamical models of propeller across four quadrants are given. The simulation results show the efficiency of mathematical model.

Finite Indeterminacy of Homogenous Polynomial Germs under Some Subgroups R_(I_r) of R

Using the finite determinacy relation with the regular sequence in the Ring Theory and the complete intersection in Analytic Geometry, the finite indeterminacy of homogeneous polynomial germs under some subgroups R1（r） of R in both real and complex case is proven by the homogeneity of the polynomial germs. It results in the finite determinacy of homogeneous polynomial germs needn＇t be discussed respectively.

Factorization of Cyclotomic Polynomials with Quadratic Radicals in the Coefficients

In this article we continue the consideration of geometrical constructions of regular n-gons for odd n by rhombic bicompasses and ruler used in [1] for the construction of the regular heptagon (n=7). We discuss the possible factorization of the cyclotomic polynomial in polynomial factors which contain not higher than quadratic radicals in the coefficients whereas usually the factorization of the cyclotomic polynomials is considered in products of irreducible factors with integer coefficients. In considering the regular heptagon we find a modified variant of its construction by rhombic bicompasses and ruler. In detail, supported by figures, we investigate the case of the regular tridecagon (n=13) which in addition to n=7 is the only candidate with low n (the next to this is n=769 ) for which such a construction by rhombic bicompasses and ruler seems to be possible. Besides the coordinate origin we find here two points to fix for the possible application of two bicompasses (or even four with the addition of the complex conjugate points to be fixed). With only one bicompass one has in addition the problem of the trisection of an angle which can be solved by a neusis construction that, however, is not in the spirit of constructions by compass and ruler and is difficult to realize during the action of bicompasses. As discussed it seems that to finish the construction by bicompasses the correlated action of two rhombic bicompasses must be applied in this case which avoids the disadvantages of the neusis construction. Single rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points on one circle determines the positions of all the other points on the second circle in unique way. The known case n=17 embedded in our method is discussed in detail.

Probabilistic density function estimation of geotechnical shear strength parameters using the second Chebyshev orthogonal polynomial

A method to estimate the probabilistic density function （PDF） of shear strength parameters was proposed. The second Chebyshev orthogonal polynomial（SCOP） combined with sample moments （the origin moments） was used to approximate the PDF of parameters. X^2 test was adopted to verify the availability of the method. It is distribution-free because no classical theoretical distributions were assumed in advance and the inference result provides a universal form of probability density curves. Six most commonly-used theoretical distributions named normal, lognormal, extreme value Ⅰ , gama, beta and Weibull distributions were used to verify SCOP method. An example from the observed data of cohesion c of a kind of silt clay was presented for illustrative purpose. The results show that the acceptance levels in SCOP are all smaller than those in the classical finite comparative method and the SCOP function is more accurate and effective in the reliability analysis of geotechnical engineering.

Multivariate Vandermonde Determinants and General Birkhoff Interpolation

In this paper, we consider the Straight Line Type Node Configuration C （SLTNCC） in multivariate polynomial interpolation as the result of different kinds of transformations of lines （such as parallel translations, rotations）. Corresponding to these transformations we define different kinds of interpolation problems for the SLTNCC. The expression of the confluent multivariate Vandermonde determinant of the coefficient matrix for each of these interpolation problems is obtained, and from this expression we conclude the related interpolation problem is unisolvent. Also, we give a kind of generalization of the SLTNCC in Section 5. As well, we obtain an expression of the interpolating polynomial for a kind of interpolation problem discussed in this paper.

Using Chebyshev polynomial interpolation to improve the computational efficiency of gravity models near an irregularly-shaped asteroid

In asteroid rendezvous missions, the dynamical environment near an asteroid’s surface should be made clear prior to launch of the mission. However, most asteroids have irregular shapes,which lower the efficiency of calculating their gravitational field by adopting the traditional polyhedral method. In this work, we propose a method to partition the space near an asteroid adaptively along three spherical coordinates and use Chebyshev polynomial interpolation to represent the gravitational acceleration in each cell. Moreover, we compare four different interpolation schemes to obtain the best precision with identical initial parameters. An error-adaptive octree division is combined to improve the interpolation precision near the surface. As an example, we take the typical irregularly-shaped nearEarth asteroid 4179 Toutatis to demonstrate the advantage of this method; as a result, we show that the efficiency can be increased by hundreds to thousands of times with our method. Our results indicate that this method can be applicable to other irregularly-shaped asteroids and can greatly improve the evaluation efficiency.

基于Chebyshev多项式和区间重叠率的对接圆柱壳结构区间模型修正

采用基于Chebyshev多项式和区间重叠率的区间模型修正方法对对接圆柱壳结构进行了模型修正研究.首先,通过模态试验获取了结构固有频率,运用核密度估计法得到了固有频率的区间范围;然后,采用薄层单元法模拟螺栓连接部件,建立了对接圆柱壳结构的有限元模型,分析了结构的模态特性;之后,采用确定性模型修正方法修正了圆柱壳和法兰的材料参数;最后,选择基于Chebyshev多项式和区间重叠率的区间模型修正方法对薄层单元中的不确定性参数进行了修正.研究结果表明:修正后参数区间上界和下界的误差满足工程精度要求,修正后输出空间与试验输出空间吻合良好,验证了所用区间模型修正方法的可行性和有效性.

Pitfalls in Identity Based Encryption Using Extended Chebyshev Polynomial

Chebyshev polynomials are used as a reservoir for generating intricate classes of symmetrical and chaotic pattems, and have been used in a vast anaount of applications. Using extended Chebyshev polynomial over finite field Ze, Algehawi and Samsudin presented recently an Identity Based Encryption （IBE） scheme. In this paper, we showed their proposal is not as secure as they chimed. More specifically, we presented a concrete attack on the scheme of Algehawi and Samsudin, which indicated the scheme cannot be consolidated as a real altemative of IBE schemes since one can exploit the semi group property （bilinearity） of extended Chebyshev polynomials over Zp to implement the attack without any difficulty.

Numerical solution to the Falkner-Skan equation:a novel numerical approach through the new rational α-polynomials

The new rationalα-polynomials are used to solve the Falkner-Skan equation.These polynomials are equipped with an auxiliary parameter.The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients.To find the unknown coefficients and the auxiliary parameter contained in the polynomials,the collocation method with Chebyshev-Gauss points is used.The numerical examples show the efficiency of this method.