多项式,即由递推公式To（x）=1,T1（x）=x,Tn+1（x）=2xTn（x）-Tn-1（x） n=1,2,…所决定的多项式 Tn（x）,有许多奇妙而有趣的性质,并在计算方法、函数逼近等现代教学研究领域中有着极其广泛的应用。正因此,人们在研究多项式 y（x）=s...多项式,即由递推公式To（x）=1,T1（x）=x,Tn+1（x）=2xTn（x）-Tn-1（x） n=1,2,…所决定的多项式 Tn（x）,有许多奇妙而有趣的性质,并在计算方法、函数逼近等现代教学研究领域中有着极其广泛的应用。正因此,人们在研究多项式 y（x）=sum from l=0 to m aixm-i 或在作函数级数的数值计算等方面,往往要把一般多项式形式用 Tn（x）（n=0,1,…）线性表出,以便于研究或减少运算次数。本文主要讨论一般多项式怎群用 Tn（x）（n=0,1,…）线性表出(关于能够线性表出性,展开更多
In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].
The correct answer of Pal's interpolation polynomial problem has been given in this paper ,especially, the explict form of this polynomial has been obtained.
By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial....By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.展开更多
We calculate Wigner function, tomogram of the pair coherent state by using its Sehmidt decomposition in the coherent state representation. It turns out that the Wigner function can be seen as the quantum entanglement ...We calculate Wigner function, tomogram of the pair coherent state by using its Sehmidt decomposition in the coherent state representation. It turns out that the Wigner function can be seen as the quantum entanglement （QE） between two two-variable Hermite polynomials （TVHP） and the tomogram is further simplified as QE of two single-variable Hermite polynomials. The Husimi function of pair coherent state is also calculated.展开更多
Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried ...Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.展开更多
Let P(t) be a product of(possibly repeated) linear factors over Q and K/Q an abelian extension. Under a strict condition, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the ...Let P(t) be a product of(possibly repeated) linear factors over Q and K/Q an abelian extension. Under a strict condition, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for any smooth proper model of the variety over Q defined by P(t) = NK/Q(x).展开更多
The conformal transformations with respect to the metric defining the orthogonal Lie algebra o（n, C） give rise to a one-parameter （c） family of inhomogeneous first-order differential operator representations of th...The conformal transformations with respect to the metric defining the orthogonal Lie algebra o（n, C） give rise to a one-parameter （c） family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o（n ＋ 2, C）. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a^→∈ C^n, we prove that the space forms an irreducible o（n ＋ 2, C）-module for any c ∈ C if a^→ is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o（n＋2, C） on the polynomial algebra in n variables. Moreover, we prove that l forms an infinite-dimensional irreducible weight o（n ＋2, C）-module with finite-dimensional weight subspaces if c Z/2.展开更多
文摘多项式,即由递推公式To（x）=1,T1（x）=x,Tn+1（x）=2xTn（x）-Tn-1（x） n=1,2,…所决定的多项式 Tn（x）,有许多奇妙而有趣的性质,并在计算方法、函数逼近等现代教学研究领域中有着极其广泛的应用。正因此,人们在研究多项式 y（x）=sum from l=0 to m aixm-i 或在作函数级数的数值计算等方面,往往要把一般多项式形式用 Tn（x）（n=0,1,…）线性表出,以便于研究或减少运算次数。本文主要讨论一般多项式怎群用 Tn（x）（n=0,1,…）线性表出(关于能够线性表出性,
文摘In this paper, we discuss the positive definite problem of a binary quartic form and obtain a necessary and sufficient condition. In addition we give two examples to show that there are some errors in the paper [1].
文摘The correct answer of Pal's interpolation polynomial problem has been given in this paper ,especially, the explict form of this polynomial has been obtained.
文摘By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10775097 and 10874174the Research Foundation of the Education Department of Jiangxi Province
文摘We calculate Wigner function, tomogram of the pair coherent state by using its Sehmidt decomposition in the coherent state representation. It turns out that the Wigner function can be seen as the quantum entanglement （QE） between two two-variable Hermite polynomials （TVHP） and the tomogram is further simplified as QE of two single-variable Hermite polynomials. The Husimi function of pair coherent state is also calculated.
文摘Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.
文摘Let P(t) be a product of(possibly repeated) linear factors over Q and K/Q an abelian extension. Under a strict condition, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for any smooth proper model of the variety over Q defined by P(t) = NK/Q(x).
基金supported by National Natural Science Foundation of China(Grant Nos.11171324 and 11321101)
文摘The conformal transformations with respect to the metric defining the orthogonal Lie algebra o（n, C） give rise to a one-parameter （c） family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o（n ＋ 2, C）. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a^→∈ C^n, we prove that the space forms an irreducible o（n ＋ 2, C）-module for any c ∈ C if a^→ is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o（n＋2, C） on the polynomial algebra in n variables. Moreover, we prove that l forms an infinite-dimensional irreducible weight o（n ＋2, C）-module with finite-dimensional weight subspaces if c Z/2.
