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 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.

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.

Discovering optimal weights in weighted‑scoring stock‑picking models: a mixture design approach

Certain literature that constructs a multifactor stock selection model adopted a weighted-scoring approach despite its three shortcomings.First,it cannot effectively identify the connection between the weights of stock-picking concepts and portfolio performances.Second,it cannot provide stock-picking concepts’optimal combination of weights.Third,it cannot meet various investor preferences.Thus,this study employs a mixture experimental design to determine the weights of stock-picking concepts,collect portfolio performance data,and construct performance prediction models based on the weights of stock-picking concepts.Furthermore,these performance prediction models and optimization techniques are employed to discover stock-picking concepts’optimal combination of weights that meet investor preferences.The samples consist of stocks listed on the Taiwan stock market.The modeling and testing periods were 1997–2008 and 2009–2015,respectively.Empirical evidence showed(1)that our methodology is robust in predicting performance accurately,(2)that it can identify significant interactions between stock-picking concepts’weights,and(3)that which their optimal combination should be.This combination of weights can form stock portfolios with the best performances that can meet investor preferences.Thus,our methodology can fill the three drawbacks of the classical weighted-scoring approach.

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.

Solving Multivariate Polynomial Matrix Diophantine Equations with Gr?bner Basis Method

Different from previous viewpoints,multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper,that is,regarding the columns of matrices as elements in modules.A necessary and sufficient condition of the existence for the solution of equations is derived.Using powerful features and theoretical foundation of Gr?bner bases for modules,the problem for determining and computing the solution of matrix Diophantine equations can be solved.Meanwhile,the authors make use of the extension on modules for the GVW algorithm that is a signature-based Gr?bner basis algorithm as a powerful tool for the computation of Gr?bner basis for module and the representation coefficients problem directly related to the particular solution of equations.As a consequence,a complete algorithm for solving multivariate polynomial matrix Diophantine equations by the Gr?bner basis method is presented and has been implemented on the computer algebra system Maple.

Some extremal properties of multivariate polynomial splines in the metric L_p(R^d)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the Bd instead of the usual multivariate cardinal interpolation oper-ators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asymptoti-cally optimal for the Kolmogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on Bd in the metric Lp(Bd).

Threshold Signature Scheme with Threshold Verification Based on Multivariate Linear Polynomial

Secret sharing schemes are multi-party protocols related to key establishment. They also facilitate distributed trust or shared control for critical activities (e.g., signing corporate cheques and opening bank vaults), by gating the critical action on cooperation from t(t ∈Z+) of n(n ∈Z+) users. A (t, n) threshold scheme (t < n) is a method by which a trusted party computes secret shares Γi(1 i n) from an initial secret Γ0 and securely distributes Γi to user. Any t or more users who pool their shares may easily recover Γ0, but any group knowing only t-1 or fewer shares may not. By the ElGamal public key cryptophytes and the Schnorr's signature scheme, this paper proposes a new (t,n) threshold signature scheme with (k,m) (k,m ∈Z+) threshold verification based on the multivariate linear polynomial.

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.

Solving sparse non-negative tensor equations： algorithms and applications

We study iterative methods for solving a set of sparse non-negative tensor equations （multivariate polynomial systems） arising from data mining applications such as information retrieval by query search and community discovery in multi-dimensional networks. By making use of sparse and non-negative tensor structure, we develop Jacobi and Gauss-Seidel methods for solving tensor equations. The multiplication of tensors with vectors are required at each iteration of these iterative methods, the cost per iteration depends on the number of non-zeros in the sparse tensors. We show linear convergence of the Jacobi and Gauss-Seidel methods under suitable conditions, and therefore, the set of sparse non-negative tensor equations can be solved very efficiently. Experimental results on information retrieval by query search and community discovery in multi-dimensional networks are presented to illustrate the application of tensor equations and the effectiveness of the proposed methods.