The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex...The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex variable are non-integers.This kind of systems can be approximated by high-order integer-order systems,and can be analyzed and designed by the sophisticated integer-order systems methodology.A new closed-form algorithm for fractional-order linear differential equations is proposed based on the definitions of fractional-order derivatives,and the effectiveness of the algorithm is illustrated through examples.展开更多
In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</...In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has exactly two solutions (x, y, z) in non-negative integer i.e., {(3, 0, 3),(4, 1, 5)} but 5<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has no non-negative integer solution.展开更多
A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primit...A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.展开更多
A(n,k)=sum from m=1 to k sum r=1 to m sum j=0 to [k/m]-1 (tm,r,j (k)×nj×s(r,m)×ζmnr,ζm=e2πi/m,s(r,m)={1,gcd(r,m)=1 0,其他)为丢番图方程sum i=1 to k (ixi=n)的非负整数解的个数.虽然用解线性方程组的方法...A(n,k)=sum from m=1 to k sum r=1 to m sum j=0 to [k/m]-1 (tm,r,j (k)×nj×s(r,m)×ζmnr,ζm=e2πi/m,s(r,m)={1,gcd(r,m)=1 0,其他)为丢番图方程sum i=1 to k (ixi=n)的非负整数解的个数.虽然用解线性方程组的方法可求得A(n,k)的所有系数,然而,该求解过程却非常耗时.本文利用方程(1-x)(1-x2)...(1-xk)=0的相异根的幂可能存在的相等关系,即取适当的正整数g使某些相异根的g次幂相等来实现同类项系数的合并以降低方程的维数,达到提高方程求解速度的目的.展开更多
设A(n,k)为丢番图方程sum from t=1 to k(ixi)=n的非负整数解的个数,本文利用A(n.k)精确公式一般形式非常方便地求出了A (n,4)、A(n,5)、A(n,6)、A(n,7)的精确公式,从而实质上给出了无序分拆数P(n,4)、P(n,5)、P(n,6)、P(n,7)的精确公式...设A(n,k)为丢番图方程sum from t=1 to k(ixi)=n的非负整数解的个数,本文利用A(n.k)精确公式一般形式非常方便地求出了A (n,4)、A(n,5)、A(n,6)、A(n,7)的精确公式,从而实质上给出了无序分拆数P(n,4)、P(n,5)、P(n,6)、P(n,7)的精确公式.此方法比过去使用的方法要方便且不需要复杂的解题技巧.展开更多
基金supported by the National Natural Science Foundation of China (Grant No.60475036).
文摘The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex variable are non-integers.This kind of systems can be approximated by high-order integer-order systems,and can be analyzed and designed by the sophisticated integer-order systems methodology.A new closed-form algorithm for fractional-order linear differential equations is proposed based on the definitions of fractional-order derivatives,and the effectiveness of the algorithm is illustrated through examples.
文摘In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has exactly two solutions (x, y, z) in non-negative integer i.e., {(3, 0, 3),(4, 1, 5)} but 5<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has no non-negative integer solution.
文摘A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.
文摘A(n,k)=sum from m=1 to k sum r=1 to m sum j=0 to [k/m]-1 (tm,r,j (k)×nj×s(r,m)×ζmnr,ζm=e2πi/m,s(r,m)={1,gcd(r,m)=1 0,其他)为丢番图方程sum i=1 to k (ixi=n)的非负整数解的个数.虽然用解线性方程组的方法可求得A(n,k)的所有系数,然而,该求解过程却非常耗时.本文利用方程(1-x)(1-x2)...(1-xk)=0的相异根的幂可能存在的相等关系,即取适当的正整数g使某些相异根的g次幂相等来实现同类项系数的合并以降低方程的维数,达到提高方程求解速度的目的.
文摘设A(n,k)为丢番图方程sum from t=1 to k(ixi)=n的非负整数解的个数,本文利用A(n.k)精确公式一般形式非常方便地求出了A (n,4)、A(n,5)、A(n,6)、A(n,7)的精确公式,从而实质上给出了无序分拆数P(n,4)、P(n,5)、P(n,6)、P(n,7)的精确公式.此方法比过去使用的方法要方便且不需要复杂的解题技巧.
