In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △2u-a△u+bu=Λρu,inΩRn,u|Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper boun...In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △2u-a△u+bu=Λρu,inΩRn,u|Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper bounds of the (k+1)-th eigenvalueΛ_k+1 in terms of the first k eigenvalues.Moreover,these results contain some results for the biharmonic operator.展开更多
Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators (BEQPO's)I we present the analytic expression of arbitrary m...Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators (BEQPO's)I we present the analytic expression of arbitrary matrix elements for BEQPO's. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO's.展开更多
In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the ...In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.展开更多
Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with s...Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.展开更多
We study integral spline operators of order k. exact on polynomials of degree 2m. with 0≤2m<k, having the form T_(k,t)^((m))f=∑ i∈J [∫_lf(x)C_(l,k)^(x)dx]N_IK, where {N_(l,k),i∈J} is the classical Bspline bas...We study integral spline operators of order k. exact on polynomials of degree 2m. with 0≤2m<k, having the form T_(k,t)^((m))f=∑ i∈J [∫_lf(x)C_(l,k)^(x)dx]N_IK, where {N_(l,k),i∈J} is the classical Bspline basis associated with the sequence t of knots on the interval I and C_(l,k)~is a linear combination of B-splines {N_(l+l,k),-m≤j≤m}. We prove a general theorem of eristence and uniqueness. Then we study the L^D -norms of these operators and error bounds for smooth furlctions f. We then obtain partial results about the L~∞--boundedness of T_(k,t)^((m)), independently of the pertition t. We also give the complete description of these operators in the case of a uniform partition of the real line.展开更多
In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechani...In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.展开更多
By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive...By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.展开更多
This paper generalizes the basic principle of multiplier-enlargement approach to approximating any nonbounded continuous functions with positive linear operators, and as an example, Bernstein polynomial operators are ...This paper generalizes the basic principle of multiplier-enlargement approach to approximating any nonbounded continuous functions with positive linear operators, and as an example, Bernstein polynomial operators are analysed and studied. This paper gives a certain theorem as a general rule to approximate any nonbounded continuous functions.展开更多
Extending the results of [4] in the univariate case, in this paper we prove that the bivariate interpolation polynomials of Hermite-Fejér based on the Chebyshev nodes of the first kind, those of Lagrange based o...Extending the results of [4] in the univariate case, in this paper we prove that the bivariate interpolation polynomials of Hermite-Fejér based on the Chebyshev nodes of the first kind, those of Lagrange based on the Chebyshev nodes of second kind and ±1, and those of bivariate Shepard operators, have the property of partial preservation of global smoothness, with respect to various bivariate moduli of continuity.展开更多
By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-seri...By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min(m,n)∑l=0^min(m,n)n!m!(-1)^l/l!(n-1)!(m-l)!/Hn-l(x)y^m-l≡ n,m(x,y).By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators(IWOP) we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.展开更多
By virtue of the technique of integration within an ordered product (IWOP) of operators and the bipartite entangled state representation, we derive some new identities about operator Hermite polynomials in both the si...By virtue of the technique of integration within an ordered product (IWOP) of operators and the bipartite entangled state representation, we derive some new identities about operator Hermite polynomials in both the single-and two-variable cases. We also find a binomial-like theorem between the single-variable Hermite polynomials and the two-variable Hermite polynomials. Application of these identities in deriving new integration formulas, but without really doing the integration in the usual sense, is demonstrated.展开更多
In this paper, we study the convergence order of a new polynominal operator H n(f;x,r) through Grnwald polynomial operator appoximating f(x)∈C j [-1,1] ,j≤r. The result of paper [1] is improved.
We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix.The efficiency of the presented approach is demon...We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix.The efficiency of the presented approach is demonstrated by solving some differential equations.Also,this technique is combined with the standard Laplace Homotopy Per-turbation Method.The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions.展开更多
The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The RoperSuffridge operator on the unit ball B^n in C^n is modified....The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The RoperSuffridge operator on the unit ball B^n in C^n is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [Φ_(G,γ)(f)](z) preserves the properties of S_Ω~*(A, B), as well as strong and almost spirallikeness of type β and order α on B^n. Thus, the mappings in S_Ω~*(A, B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.展开更多
基金supported by the National Natural Science Foundation of China(11001130)the NUST Research Funding(2010ZYTS064)supported by China Postdoctoral Science Foundation(20080430351)
文摘In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △2u-a△u+bu=Λρu,inΩRn,u|Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper bounds of the (k+1)-th eigenvalueΛ_k+1 in terms of the first k eigenvalues.Moreover,these results contain some results for the biharmonic operator.
基金The authors would like to thank Prof. Y.D. Zhang for selfless helps and valuable discussions.
文摘Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators (BEQPO's)I we present the analytic expression of arbitrary matrix elements for BEQPO's. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO's.
基金This work was supported by Junta de Andalucia. Grupo de investigacion Matematica Aplioada. Codao 1107
文摘In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175113)
文摘Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.
文摘We study integral spline operators of order k. exact on polynomials of degree 2m. with 0≤2m<k, having the form T_(k,t)^((m))f=∑ i∈J [∫_lf(x)C_(l,k)^(x)dx]N_IK, where {N_(l,k),i∈J} is the classical Bspline basis associated with the sequence t of knots on the interval I and C_(l,k)~is a linear combination of B-splines {N_(l+l,k),-m≤j≤m}. We prove a general theorem of eristence and uniqueness. Then we study the L^D -norms of these operators and error bounds for smooth furlctions f. We then obtain partial results about the L~∞--boundedness of T_(k,t)^((m)), independently of the pertition t. We also give the complete description of these operators in the case of a uniform partition of the real line.
基金Project supported by the National Natural Science Foundation of China(Grant No.11775208)the Foundation for Young Talents at the College of Anhui Province,China(Grant Nos.gxyq2021210 and gxyq2019077)the Natural Science Foundation of the Anhui Higher Education Institutions of China(Grant Nos.KJ2020A0638 and 2022AH051586)。
文摘In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.
文摘By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.
文摘This paper generalizes the basic principle of multiplier-enlargement approach to approximating any nonbounded continuous functions with positive linear operators, and as an example, Bernstein polynomial operators are analysed and studied. This paper gives a certain theorem as a general rule to approximate any nonbounded continuous functions.
文摘Extending the results of [4] in the univariate case, in this paper we prove that the bivariate interpolation polynomials of Hermite-Fejér based on the Chebyshev nodes of the first kind, those of Lagrange based on the Chebyshev nodes of second kind and ±1, and those of bivariate Shepard operators, have the property of partial preservation of global smoothness, with respect to various bivariate moduli of continuity.
基金supported by the Natural Science Fund of Education Department of Anhui Province,China(Grant No.KJ2016A590)the Talent Foundation of Hefei University,China(Grant No.15RC11)the National Natural Science Foundation of China(Grant Nos.11247009 and 11574295)
文摘By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min(m,n)∑l=0^min(m,n)n!m!(-1)^l/l!(n-1)!(m-l)!/Hn-l(x)y^m-l≡ n,m(x,y).By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators(IWOP) we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.
基金supported by the National Natural Science Foundation of China (Grant Nos.10775097,11074190 and 10947017/A05)the specialized research fund for the Doctorial Progress of Higher Education of China (Grant No.20070358009)
文摘By virtue of the technique of integration within an ordered product (IWOP) of operators and the bipartite entangled state representation, we derive some new identities about operator Hermite polynomials in both the single-and two-variable cases. We also find a binomial-like theorem between the single-variable Hermite polynomials and the two-variable Hermite polynomials. Application of these identities in deriving new integration formulas, but without really doing the integration in the usual sense, is demonstrated.
文摘In this paper, we study the convergence order of a new polynominal operator H n(f;x,r) through Grnwald polynomial operator appoximating f(x)∈C j [-1,1] ,j≤r. The result of paper [1] is improved.
文摘We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix.The efficiency of the presented approach is demonstrated by solving some differential equations.Also,this technique is combined with the standard Laplace Homotopy Per-turbation Method.The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions.
基金supported by the National Natural Science Foundation of China(Nos.11271359,11471098)the Joint Funds of the National Natural Science Foundation of China(No.U1204618)the Science and Technology Research Projects of Henan Provincial Education Department(Nos.14B110015,14B110016)
文摘The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The RoperSuffridge operator on the unit ball B^n in C^n is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [Φ_(G,γ)(f)](z) preserves the properties of S_Ω~*(A, B), as well as strong and almost spirallikeness of type β and order α on B^n. Thus, the mappings in S_Ω~*(A, B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.