In this note we study the local behaviour of the multi-variate Bernstein polynomials B on the d-dimen- sional simplex . For function f admitting derivatives of sufficient high order in we derive the complete asymptoti...In this note we study the local behaviour of the multi-variate Bernstein polynomials B on the d-dimen- sional simplex . For function f admitting derivatives of sufficient high order in we derive the complete asymptotic expansion of Bnf as n tends to infinity. All the coefficients of n-k that only depend on f and x are calculated explicitly . It turns out that combinatorial numbers play an important role . Our results generalize recent formulae due to R, Zhany in a way.展开更多
The extended Hermite inter polation problem on segment points set over n-dimensinonal Euclidean space is considered. Based on the algorithm to compute the Grobner basis of Ideal given by dual basis a new method to con...The extended Hermite inter polation problem on segment points set over n-dimensinonal Euclidean space is considered. Based on the algorithm to compute the Grobner basis of Ideal given by dual basis a new method to construct minimal multivariate polynomial which satis fies the interpolation conditions is given.展开更多
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 stoc...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.展开更多
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 main...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.展开更多
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 nece...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.展开更多
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 ...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).展开更多
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)...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.展开更多
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 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.展开更多
文摘In this note we study the local behaviour of the multi-variate Bernstein polynomials B on the d-dimen- sional simplex . For function f admitting derivatives of sufficient high order in we derive the complete asymptotic expansion of Bnf as n tends to infinity. All the coefficients of n-k that only depend on f and x are calculated explicitly . It turns out that combinatorial numbers play an important role . Our results generalize recent formulae due to R, Zhany in a way.
文摘The extended Hermite inter polation problem on segment points set over n-dimensinonal Euclidean space is considered. Based on the algorithm to compute the Grobner basis of Ideal given by dual basis a new method to construct minimal multivariate polynomial which satis fies the interpolation conditions is given.
文摘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.
基金supported by the National Natural Science Foundation of China under Grant Nos.11971161 and 11871207。
文摘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.
基金supported by the National Natural Science Foundation of China under Grant No.12001030the CAS Key Project QYZDJ-SSW-SYS022the National Key Research and Development Project2020YFA0712300。
文摘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.
基金Scientific Research Foundation for Returned Overseas Chinese Scholars of the Ministry of Education of China.
文摘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).
基金the National Natural Science Foundation of China (No. 10671051)the Natural Science Foundation of Zhejiang Province (No. Y6110782)the Key Laboratory Foundation of Hangzhou(No. 20100331T11)
文摘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.
基金supported by the National Natural Science Foundation of China under Grant Nos.12171469,12001030 and 12201210the National Key Research and Development Program under Grant No.2020YFA0712300the Fundamental Research Funds for the Central Universities under Grant No.2682022CX048。
文摘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.