By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater tha...By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater than the existing ways.These products are very useful in obtaining some new differential relations and useful mathematical integral formulas.Further,we derive the normally ordered form of the operator(fQ+gP)^-n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate-momentum representation.In addition,general mutual transformation rules of the normal and anti-normal orderings,which have good universality,are derived and hence the anti-normal ordering of(fQ+gP)^-n is also obtained.Finally,the application of some new identities is given.展开更多
By extending the usual Wigner operator to the s-parameterized one as 1/4π2 integral (dyduexp [iu(q-Q)+iy(p-P)+is/2yu]) from n=- ∞ to ∞ with s beng a,real parameter,we propose a generalized Weyl quantization scheme ...By extending the usual Wigner operator to the s-parameterized one as 1/4π2 integral (dyduexp [iu(q-Q)+iy(p-P)+is/2yu]) from n=- ∞ to ∞ with s beng a,real parameter,we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule.This rule recovers P-Q ordering,Q-P ordering,and Weyl ordering of operators in s = 1,1,0 respectively.Hence it differs from the Cahill-Glaubers' ordering rule which unifies normal ordering,antinormal ordering,and Weyl ordering.We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P.The formula that can rearrange a given operator into its new s-parameterized ordering is presented.展开更多
Normal form theory is, a very effective method when we study degeneratebifurcations of nonlinear dynamical systems. In this paper by using adjoint operatormethod, normal forms of order 3 and 4 for nonlinear dynamical ...Normal form theory is, a very effective method when we study degeneratebifurcations of nonlinear dynamical systems. In this paper by using adjoint operatormethod, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotentlinear part and Z2-asymmetry are computed. According to normal forms obtained,universal unfoldings for some degenerate bifurcation cases of codimension 3 and simpleglobal characterizations, are studied.展开更多
By virtue of the technique of integration within an ordered product of operators we present a new approach to obtain operators' normal ordering. We first put operators into their Weyl ordering through the Weyl-Wig...By virtue of the technique of integration within an ordered product of operators we present a new approach to obtain operators' normal ordering. We first put operators into their Weyl ordering through the Weyl-Wigner quantization scheme, and then we convert the Weyl ordered operators into normal ordering by virtue of the normally ordered form of the Wigner operator.展开更多
For the first time,we derive the compact forms of normalization factors for photon-added(-subtracted) two-mode squeezed thermal states by using the P-representation and the integration within an ordered product of ope...For the first time,we derive the compact forms of normalization factors for photon-added(-subtracted) two-mode squeezed thermal states by using the P-representation and the integration within an ordered product of operators(IWOP) technique.It is found that these two factors are related to the Jacobi polynomials.In addition,some new relationships for Jacobi polynomials are presented.展开更多
We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., V(x1, x2, x3)|x1, x2, x3 x1, x2, x3| d3 x = V(X1, X2, X3) = e-λ2/4: V(X1, X2, X3) :,where V(x1, x2, x3) ...We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., V(x1, x2, x3)|x1, x2, x3 x1, x2, x3| d3 x = V(X1, X2, X3) = e-λ2/4: V(X1, X2, X3) :,where V(x1, x2, x3) is the solution to the Helmholtz equation ?2V + λ2V = 0, the symbol : : denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac's radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.展开更多
In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and fo...In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.展开更多
The present work is based on the third-order partial differential equation (PDE) of acoustics of viscoelastic solids for the quasi-equilibrium (QE) component of the average normal stress. This PDE includes the stress-...The present work is based on the third-order partial differential equation (PDE) of acoustics of viscoelastic solids for the quasi-equilibrium (QE) component of the average normal stress. This PDE includes the stress-relaxation time (SRT) for the material and is applicable at any value of the SRT. The notion of a smart deicing system (SDS) for blade shells (BSs) of a wind turbine is specified. The work considers the stress in a BS as the one caused by the operational load on the BS. The work develops key design issues of a prospective ice-detection system (IDS) able to supply an array of the heating elements of an SDS with the element-individual spatiotemporal data and procedures for identification of the material parameters of atmospheric-ice (AI) layer accreted on the outer surfaces of the BSs. Both the SDS and IDS flexibly allow for complex, curvilinear and space-time-varying shapes of BSs. The proposed IDS presumes monitoring of the QE components of the normal stresses in BSs. The IDS is supposed to include an array of pressure-sensing resistors, also known as force-sensing resistors (FSRs), and communication hardware, as well as the parameter-identification software package (PISP), which provides the identification on the basis of the aforementioned PDE and the data measured by the FSRs. The IDS does not have hardware components located outside the outer surfaces of, or implanted in, BSs. The FSR array and communication hardware are reliable, and both cost- and energy-efficient. The present work extends methods of structural-health/operational-load monitoring (SH/OL-M) with measurements of the operational-load-caused stress in closed solid shells and, if the prospective PISP is used, endows the methods with identification of material parameters of the shells. The identification algorithms that can underlie the PISP are computationally efficient and suitable for implementation in the real-time mode. The identification model and algorithms can deal with not only the single-layer systems such as the BS layer without the AI layer or two-layer systems but also multi-layer systems. The outcomes can be applied to not only BSs of wind turbines but also non-QE closed single- or multi-layer deformable solid shells of various engineering systems (e.g., the shells of driver or passenger compartments of ships, cars, busses, airplanes, and other vehicles). The proposed monitoring of the normal-stress QE component in the mentioned shells extends the methods of SH/OL-M. The topic for the nearest research is a better adjustment of the settings for the FSR-based measurement of the mentioned components and a calibration of the parameter-identification model and algorithms, as well as the resulting improvement of the PISP.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.11347026)the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2016AM03 and ZR2017MA011)the Natural Science Foundation of Heze University,China(Grant Nos.XY17KJ09 and XY18PY13).
文摘By using the parameter differential method of operators,we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings,which is more ecumenical,simpler,and neater than the existing ways.These products are very useful in obtaining some new differential relations and useful mathematical integral formulas.Further,we derive the normally ordered form of the operator(fQ+gP)^-n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate-momentum representation.In addition,general mutual transformation rules of the normal and anti-normal orderings,which have good universality,are derived and hence the anti-normal ordering of(fQ+gP)^-n is also obtained.Finally,the application of some new identities is given.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11175113 and 11147009)the Natural Science Foundation of Shandong Province of China (Grant No. ZR2010AQ027)the Program of Higher Educational Science and Technology of Shandong Province,China (Grant No. J10LA15)
文摘By extending the usual Wigner operator to the s-parameterized one as 1/4π2 integral (dyduexp [iu(q-Q)+iy(p-P)+is/2yu]) from n=- ∞ to ∞ with s beng a,real parameter,we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule.This rule recovers P-Q ordering,Q-P ordering,and Weyl ordering of operators in s = 1,1,0 respectively.Hence it differs from the Cahill-Glaubers' ordering rule which unifies normal ordering,antinormal ordering,and Weyl ordering.We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P.The formula that can rearrange a given operator into its new s-parameterized ordering is presented.
文摘Normal form theory is, a very effective method when we study degeneratebifurcations of nonlinear dynamical systems. In this paper by using adjoint operatormethod, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotentlinear part and Z2-asymmetry are computed. According to normal forms obtained,universal unfoldings for some degenerate bifurcation cases of codimension 3 and simpleglobal characterizations, are studied.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10775097 and 10947017/A05)the Specialized Research Fund for the Doctorial Progress of Higher Education of China (GrantNo. 20070358009)
文摘By virtue of the technique of integration within an ordered product of operators we present a new approach to obtain operators' normal ordering. We first put operators into their Weyl ordering through the Weyl-Wigner quantization scheme, and then we convert the Weyl ordered operators into normal ordering by virtue of the normally ordered form of the Wigner operator.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11264018 and 60978009)the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91121023)+1 种基金the National Basic Research Project of China (Grant No. 2011CBA00200)the Young Talents Foundation of Jiangxi Normal University,China
文摘For the first time,we derive the compact forms of normalization factors for photon-added(-subtracted) two-mode squeezed thermal states by using the P-representation and the integration within an ordered product of operators(IWOP) technique.It is found that these two factors are related to the Jacobi polynomials.In addition,some new relationships for Jacobi polynomials are presented.
基金supported by the National Natural Science Foundation of China(Grant No.11175113)
文摘We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., V(x1, x2, x3)|x1, x2, x3 x1, x2, x3| d3 x = V(X1, X2, X3) = e-λ2/4: V(X1, X2, X3) :,where V(x1, x2, x3) is the solution to the Helmholtz equation ?2V + λ2V = 0, the symbol : : denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac's radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.
文摘In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.
文摘The present work is based on the third-order partial differential equation (PDE) of acoustics of viscoelastic solids for the quasi-equilibrium (QE) component of the average normal stress. This PDE includes the stress-relaxation time (SRT) for the material and is applicable at any value of the SRT. The notion of a smart deicing system (SDS) for blade shells (BSs) of a wind turbine is specified. The work considers the stress in a BS as the one caused by the operational load on the BS. The work develops key design issues of a prospective ice-detection system (IDS) able to supply an array of the heating elements of an SDS with the element-individual spatiotemporal data and procedures for identification of the material parameters of atmospheric-ice (AI) layer accreted on the outer surfaces of the BSs. Both the SDS and IDS flexibly allow for complex, curvilinear and space-time-varying shapes of BSs. The proposed IDS presumes monitoring of the QE components of the normal stresses in BSs. The IDS is supposed to include an array of pressure-sensing resistors, also known as force-sensing resistors (FSRs), and communication hardware, as well as the parameter-identification software package (PISP), which provides the identification on the basis of the aforementioned PDE and the data measured by the FSRs. The IDS does not have hardware components located outside the outer surfaces of, or implanted in, BSs. The FSR array and communication hardware are reliable, and both cost- and energy-efficient. The present work extends methods of structural-health/operational-load monitoring (SH/OL-M) with measurements of the operational-load-caused stress in closed solid shells and, if the prospective PISP is used, endows the methods with identification of material parameters of the shells. The identification algorithms that can underlie the PISP are computationally efficient and suitable for implementation in the real-time mode. The identification model and algorithms can deal with not only the single-layer systems such as the BS layer without the AI layer or two-layer systems but also multi-layer systems. The outcomes can be applied to not only BSs of wind turbines but also non-QE closed single- or multi-layer deformable solid shells of various engineering systems (e.g., the shells of driver or passenger compartments of ships, cars, busses, airplanes, and other vehicles). The proposed monitoring of the normal-stress QE component in the mentioned shells extends the methods of SH/OL-M. The topic for the nearest research is a better adjustment of the settings for the FSR-based measurement of the mentioned components and a calibration of the parameter-identification model and algorithms, as well as the resulting improvement of the PISP.
