A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Til...A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.展开更多
The family of cubic Thue equation which depend on two parameters ｜ x^3 ＋ mx^2 y-（m＋3） xy^2＋y^3｜=k is studied. Using rational approximation, we give a smaller upper bound of the solution of the equation, that is...The family of cubic Thue equation which depend on two parameters ｜ x^3 ＋ mx^2 y-（m＋3） xy^2＋y^3｜=k is studied. Using rational approximation, we give a smaller upper bound of the solution of the equation, that is quite better than the present result. Moreover, we study two inequalities ｜ x^3 ＋ mx^2y-（m ＋ 3） xy^2＋y^3 ｜ =k≤2m＋3 and ｜x^3 ＋mx^2y- （m＋3）xy^2 ＋ y^3｜ = k≤ （2m＋3）^2 separately. Our result of upper bound make it easy to solve those inequalities by simple method of continuous fraction expansion.展开更多
Fractional-order differentiator is a principal component of the fractional-order controller.Discretization of fractional-order differentiator is essential to implement the fractionalorder controller digitally.Discreti...Fractional-order differentiator is a principal component of the fractional-order controller.Discretization of fractional-order differentiator is essential to implement the fractionalorder controller digitally.Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the Z-domain as evident from the existing literature.In this paper,a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional Z-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency.The discretization of fractional-order differentiator is accomplished in two stages.In the first stage,the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage,the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method.The proposed method has been compared with two well-known direct discretization methods taken from the existing literature.Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.展开更多
Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 ...Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].展开更多
文摘A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.
基金Supported by the National Natural ScienceFoundation of China (2001AA141010)
文摘The family of cubic Thue equation which depend on two parameters ｜ x^3 ＋ mx^2 y-（m＋3） xy^2＋y^3｜=k is studied. Using rational approximation, we give a smaller upper bound of the solution of the equation, that is quite better than the present result. Moreover, we study two inequalities ｜ x^3 ＋ mx^2y-（m ＋ 3） xy^2＋y^3 ｜ =k≤2m＋3 and ｜x^3 ＋mx^2y- （m＋3）xy^2 ＋ y^3｜ = k≤ （2m＋3）^2 separately. Our result of upper bound make it easy to solve those inequalities by simple method of continuous fraction expansion.
文摘Fractional-order differentiator is a principal component of the fractional-order controller.Discretization of fractional-order differentiator is essential to implement the fractionalorder controller digitally.Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the Z-domain as evident from the existing literature.In this paper,a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional Z-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency.The discretization of fractional-order differentiator is accomplished in two stages.In the first stage,the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage,the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method.The proposed method has been compared with two well-known direct discretization methods taken from the existing literature.Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.
文摘Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].
