Some Identities Involving the High-Order Cauchy Polynomials

In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.

Reliability analysis of structure with random parameters based on multivariate power polynomial expansion

A new method for calculating the failure probabilityof structures with random parameters is proposed based onmultivariate power polynomial expansion, in which te uncertain quantities include material properties, structuralgeometric characteristics and static loads. The structuralresponse is first expressed as a multivariable power polynomialexpansion, of which the coefficients ae then determined by utilizing the higher-order perturbation technique and Galerkinprojection scheme. Then, the final performance function ofthe structure is determined. Due to the explicitness of theperformance function, a multifold integral of the structuralfailure probability can be calculated directly by the Monte Carlo simulation, which only requires a smal amount ofcomputation time. Two numerical examples ae presented toillustate te accuracy ad efficiency of te proposed metiod. It is shown that compaed with the widely used first-orderreliability method （ FORM） and second-order reliabilitymethod （ SORM）, te results of the proposed method are closer to that of the direct Monte Carlo metiod,and it requires much less computational time.

A sludge volume index (SVI) model based on the multivariate local quadratic polynomial regression method

In this study, a multivariate local quadratic polynomial regression(MLQPR) method is proposed to design a model for the sludge volume index(SVI). In MLQPR, a quadratic polynomial regression function is established to describe the relationship between SVI and the relative variables, and the important terms of the quadratic polynomial regression function are determined by the significant test of the corresponding coefficients. Moreover, a local estimation method is introduced to adjust the weights of the quadratic polynomial regression function to improve the model accuracy. Finally, the proposed method is applied to predict the SVI values in a real wastewater treatment process(WWTP). The experimental results demonstrate that the proposed MLQPR method has faster testing speed and more accurate results than some existing methods.

A NEW METHOD FOR THE CONSTRUCTIONOF MULTIVARIATE MINIMALINTERPOLATION POLYNOMIAL

The extended Hermite interpolation problem on segment points set over n-dimensional Euclidean space is considered. Based on the algorithm to compute the Gr?bner basis of Ideal given by dual basis a new method to construct minimal multivariate polynomial which satisfies the interpolation conditions is given.

A Novel Multivariate Polynomial Approximation Factorization of Big Data

In actual engineering, processing of big data sometimes requires building of mass physical models, while processing of physical model requires relevant math model, thus producing mass multivariate polynomials, the effective reduction of which is a difficult problem at present. A novel algorithm is proposed to achieve the approximation factorization of complex coefficient multivariate polynomial in light of characteristics of multivariate polynomials. At first, the multivariate polynomial is reduced to be the binary polynomial, then the approximation factorization of binary polynomial can produce irreducible duality factor, at last, the irreducible duality factor is restored to the irreducible multiple factor. As a unit root is cyclic, selecting the unit root as the reduced factor can ensure the coefficient does not expand in a reduction process. Chinese remainder theorem is adopted in the corresponding reduction process, which brought down the calculation complexity. The algorithm is based on approximation factorization of binary polynomial and calculation of approximation Greatest Common Divisor, GCD. The algorithm can solve the reduction of multivariate polynomials in massive math models, which can obtain effectively null point of multivariate polynomials, providing a new approach for further analysis and explanation of physical models. The experiment result shows that the irreducible factors from this method get close to the real factors with high efficiency.

HIGH-ORDER RUNGE-KUTTA DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR 2-D RESONATOR PROBLEM

The Runge-Kutta discontinuous Galerkin finite element method （RK-DGFEM） is introduced to solve the classical resonator problem in the time domain. DGFEM uses unstructured grid discretization in the space domain and it is explicit in the time domain. Consequently it is a best mixture of FEM and finite volume method （FVM）. RK-DGFEM can obtain local high-order accuracy by using high-order polynomial basis. Numerical experiments of transverse magnetic （TM） wave propagation in a 2-D resonator are performed. A high-order Lagrange polynomial basis is adopted. Numerical results agree well with analytical solution. And different order Lagrange interpolation polynomial basis impacts on simulation result accuracy are discussed. Computational results indicate that the accuracy is evidently improved when the order of interpolation basis is increased. Finally, L^2 errors of different order polynomial basis in RK-DGFEM are presented. Computational results show that L^2 error declines exponentially as the order of basis increases.

On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices

A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented.The key idea is to establish a relationship between a matrix and any of its full row rank submatrices.Based on the new result,the authors propose an algorithm for factorizing matrices and have implemented it on the computer algebra system Maple.Two examples are given to illustrate the effectiveness of the algorithm,and experimental data shows that the algorithm is efficient.

PSEUDO-DIVISION ALGORITHM FOR MATRIX MULTIVARIABLE POLYNOMIAL AND ITS APPLICATION

Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equations into infinite dimensional Hamiltonianian system and its concrete form are obtained. Then by combining this method with Wu's method, a new method of constructing general solution of a class of mechanical equations is got, which several examples show very effective.

A Multivariate Public Key Cryptographic Scheme

This paper presents a multivariate public key cryptographic scheme over a finite field with odd prime characteristic.The idea of embedding and layering is manifested in its construction.The security of the scheme is analyzed in detail,and this paper indicates that the scheme can withstand the up to date differential cryptanalysis.We give heuristic arguments to show that this scheme resists all known attacks.

Spectral Solutions of Linear and Nonlinear BVPs Using Certain Jacobi Polynomials Generalizing Third- and Fourth-Kinds of Chebyshev Polynomials

This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems.For this purpose,we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials.These two classes generalize the two celebrated non-symmetric classes of polynomials,namely,Chebyshev polynomials of third-and fourth-kinds.The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials.The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved.Furthermore,and based on the first-order derivatives formula of certain Jacobi polynomials,the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method.Convergence analysis of the proposed expansions is investigated.Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

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.

HIGHER-ORDER MULTIVARIABLE EULER'S POLYNOMIALAND HIGHER-ORDER MULTIVARIABLEBERNOULLI'S POLYNOMIAL

In this paper, the definitons of both higher-order multivariable Euler's numbersand polynomial. higher-order multivariable Bernoulli's numbers and polynomial aregiven and some of their important properties are expounded. As a result, themathematical relationship between higher-order multivariable Euler's polynomial(numbers) and higher-order higher -order Bernoulli's polynomial (numbers) are thusobtained.

On the Average Errors of Multivariate Lagrange Interpolation

In this paper, we discuss the average errors of multivariate Lagrange interpolation based on the Chebyshev nodes of the first kind. The average errors of the interpolation sequence are determined on the multivariate Wiener space.

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

On the High-Order Quasi Exactly Solvable Differential Equations

In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.

Smith Form of Triangular Multivariate Polynomial Matrix

The Smith form of a matrix plays an important role in the equivalence of matrix.It is known that some multivariate polynomial matrices are not equivalent to their Smith forms.In this paper,the authors investigate mainly the Smith forms of multivariate polynomial triangular matrices and testify two upper multivariate polynomial triangular matrices are equivalent to their Smith forms respectively.

A New Algorithm for Computing the Extended Hensel Construction of Multivariate Polynomials

This paper presents a new algorithm for computing the extended Hensel construction(EHC) of multivariate polynomials in main variable x and sub-variables u1, u2, ···, um over a number field K. This algorithm first constructs a set by using the resultant of two initial coprime factors w.r.t. x, and then obtains the Hensel factors by comparing the coefficients of xi on both sides of an equation. Since the Hensel factors are polynomials of the main variable with coefficients in fraction field K(u1, u2, ···, um), the computation cost of handling rational functions can be high. Therefore,the authors use a method which multiplies resultant and removes the denominators of the rational functions. Unlike previously-developed algorithms that use interpolation functions or Grobner basis, the algorithm relies little on polynomial division, and avoids multiplying by different factors when removing the denominators of Hensel factors. All algorithms are implemented using Magma, a computational algebra system and experiments indicate that our algorithm is more efficient.

Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation

The problem of computing the greatest common divisor（GCD） of multivariate polynomials, as one of the most important tasks of computer algebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement,robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel＇s method and Ben-Or/Tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.

On Rank Factorizations and Factor Prime Factorizations for Multivariate Polynomial Matrices

In this paper,rank factorizations and factor left prime factorizations are studied.The authors prove that any polynomial matrix with full row rank has factor left prime factorizations.And for a class of polynomial matrices,the authors give an algorithm to decide whether they have rank factorizations or factor left prime factorizations and compute these factorizations if they exist.

基于模算术系数解析的稀疏插值算法

稀疏多元多项式插值是利用多项式的稀疏结构及其给定的插值点信息重构黑盒函数的一种有效策略,被广泛应用于科学和工程领域。传统的基于Prony方法的稀疏插值算法,其复杂度与多项式项数和次数相关,遇到大规模问题时由于执行多个高阶代数运算而效率较低。提出一种新的求解稀疏多元多项式插值问题的算法,核心操作是利用模算术解析单变元多项式的系数,避免了传统方法必需的高阶方程组求解、高次方程求根等。该算法设定一变元为主元,将黑盒多元多项式视为该主元的单变元多项式,通过解析主元的系数多项式在不同插值点处的函数值,进而重构这些系数多项式以恢复整个多元多项式。理论分析和数值实验表明了算法的有效性和可行性。