In this paper by virtue of the technique of integration within an ordered product （IWOP） of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and anti...In this paper by virtue of the technique of integration within an ordered product （IWOP） of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator （fQ＋gP）n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.展开更多
For Hermite polynomials of radial coordinate operator in three-dimensional coordinate space we derive its normal ordering expansion, which are new operator identities. This is done by virtue of the technique of integr...For Hermite polynomials of radial coordinate operator in three-dimensional coordinate space we derive its normal ordering expansion, which are new operator identities. This is done by virtue of the technique of integration within an ordered product of operators. Application of the new formulas is briefly discussed.展开更多
Generally, open string boundary conditions play a nontrivial role in string theory. For example, in the presence of an antisymmetric tensor background field, they will lead the spacetime coordinates noncommutative. In...Generally, open string boundary conditions play a nontrivial role in string theory. For example, in the presence of an antisymmetric tensor background field, they will lead the spacetime coordinates noncommutative. In this paper, we mainly discuss how to build up a generalized Dirichlet normal ordered product of open bosonic string embedding operators that satisfies both the equations of motion and the generalized Dirichlet boundary conditions at the quantum level in the presence of an antisymmetric background field, as the generalized Neumann case has already been discussed in the literature. Further, we also give a brief check of the consistency of the theory under the newly introduced normal ordering.展开更多
By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials. We also derive the new reciprocal relation of Laguerre...By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials. We also derive the new reciprocal relation of Laguerre polynomials ∑n=0 （-1）n （n^l）Ln （x） = x^l/n, n-O and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated. Some new expansion identities about the operator Laguerre polynomial are also derived. This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.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
We show that the technique of integration within an ordered product of operators can be extended to Hilbert transform. In so doing we derive normally ordered expansion of Coulomb potential-type operators directly by u...We show that the technique of integration within an ordered product of operators can be extended to Hilbert transform. In so doing we derive normally ordered expansion of Coulomb potential-type operators directly by using the mathematical Hilbert transform formula.展开更多
Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical oper...Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector ｜q〉λ,v of linear combination of the coordinate and momentum operator, （λQ ＋ vP）, where λ,v are real numbers, and ｜q〉λv is complete, then the projector ｜q〉λ,vλ,v〈q｜ must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.展开更多
Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical...Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.展开更多
Using the photon creation operator＇s eigenstate theory we derive the normally ordered expansion of inverse of the squeezed creation operator. It turns out that using this operator a kind of excitation on the squeezed...Using the photon creation operator＇s eigenstate theory we derive the normally ordered expansion of inverse of the squeezed creation operator. It turns out that using this operator a kind of excitation on the squeezed vacuum states can be formed.展开更多
From the normally ordered form of the density operator of a squeezed coherent state（SCS）,we directly derive the compact expression of the SCS＇s photon-number distribution（PND）.Besides the known oscillation charac...From the normally ordered form of the density operator of a squeezed coherent state（SCS）,we directly derive the compact expression of the SCS＇s photon-number distribution（PND）.Besides the known oscillation characteristics,we find that the PND is a periodic function with a period of π and extremely sensitive to phase.If the squeezing is strong enough,and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values,different oscillation behaviours in PND for even and odd photon numbers appear,respectively.展开更多
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
The Grain v1 stream cipher is one of the seven finalists in the final e STREAM portfolio. Though many attacks have been published,no recovery attack better than exhaustive key search on full Grain v1 in the single key...The Grain v1 stream cipher is one of the seven finalists in the final e STREAM portfolio. Though many attacks have been published,no recovery attack better than exhaustive key search on full Grain v1 in the single key setting has been found yet. In this paper,new state recovery attacks on Grain v1 utilizing the weak normality order of the employed keystream output function in the cipher are proposed. These attacks have remarkable advantages in the offline time,online time and memory complexities,which are all better than exhaustive key search. The success probability of each new attack is 0.632. The proposed attack primarily depends on the order of weak normality of the employed keystream output function. This shows that the weak normality order should be carefully considered when designing the keystream output functions of Grain-like stream ciphers.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〉d3x = V （X1 ,X2,X3） = e-λ2/4 ：V （X1 ,X2,X3）：,where...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〉d3x = 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 sensitivities of the normal modes arrival time to solitary internal waves （IWs） are analyzed by using the SW06 environments. Simulation results show that the arrival time of mode 1 is relatively stable. But, the...The sensitivities of the normal modes arrival time to solitary internal waves （IWs） are analyzed by using the SW06 environments. Simulation results show that the arrival time of mode 1 is relatively stable. But, there are some higher-order normal modes which arrive earlier than mode 1, and fluctuate with the appearance of solitary IWs. Explanation of the phenomenon is given based on ray theory. It is shown that, when thermocline falls down to some depths, those higher-order modes with a group of definite grazing angles mainly propagate above the thermocline and arrive earlier.展开更多
For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders ...For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders of this arithmetic function.展开更多
The new generalized coherent-entangled state representation |α, p μ,ν is successfully derived via constructing the integration of unity in normally ordered Gaussian operator forms and then decomposing it as project...The new generalized coherent-entangled state representation |α, p μ,ν is successfully derived via constructing the integration of unity in normally ordered Gaussian operator forms and then decomposing it as projection operators. This is a convenient approach for obtaining new representations. We then prove that |α, p μ,ν has the completeness relation and only partly orthogonal, then discuss how to use a beamsplitter to produce such a state. As its potential application, formulas can be obtained by using the completeness of |α, p μ,ν , which is very important in mathematical physics, especially in quantum optics.展开更多
基金Project supported by the Natural Science Foundation of Shandong Province of China (Grant No Y2008A23)the Natural Science Foundation of Liaocheng University (Grant No X071049)
文摘In this paper by virtue of the technique of integration within an ordered product （IWOP） of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator （fQ＋gP）n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.
基金The project supported by the President Foundation of the Chinese Academy of Sciences
文摘For Hermite polynomials of radial coordinate operator in three-dimensional coordinate space we derive its normal ordering expansion, which are new operator identities. This is done by virtue of the technique of integration within an ordered product of operators. Application of the new formulas is briefly discussed.
基金supported by National Natural Science Foundation and the Doctor Education Fund of the Ministry of Education
文摘Generally, open string boundary conditions play a nontrivial role in string theory. For example, in the presence of an antisymmetric tensor background field, they will lead the spacetime coordinates noncommutative. In this paper, we mainly discuss how to build up a generalized Dirichlet normal ordered product of open bosonic string embedding operators that satisfies both the equations of motion and the generalized Dirichlet boundary conditions at the quantum level in the presence of an antisymmetric background field, as the generalized Neumann case has already been discussed in the literature. Further, we also give a brief check of the consistency of the theory under the newly introduced normal ordering.
基金supported by the National Natural Science Foundation of China (Grant No. 10874174)the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20070358009)
文摘By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials. We also derive the new reciprocal relation of Laguerre polynomials ∑n=0 （-1）n （n^l）Ln （x） = x^l/n, n-O and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated. Some new expansion identities about the operator Laguerre polynomial are also derived. This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.
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The project supported by the President Foundation of the Chinese Academy of Sciences and National Natural Science Foundation of China under Grant No. 10475056.
文摘We show that the technique of integration within an ordered product of operators can be extended to Hilbert transform. In so doing we derive normally ordered expansion of Coulomb potential-type operators directly by using the mathematical Hilbert transform formula.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10874174 and 10775097
文摘Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector ｜q〉λ,v of linear combination of the coordinate and momentum operator, （λQ ＋ vP）, where λ,v are real numbers, and ｜q〉λv is complete, then the projector ｜q〉λ,vλ,v〈q｜ must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.
文摘Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.
文摘Using the photon creation operator＇s eigenstate theory we derive the normally ordered expansion of inverse of the squeezed creation operator. It turns out that using this operator a kind of excitation on the squeezed vacuum states can be formed.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11175113)the Natural Science Foundation of Shandong Province,China (Grant No. ZR2010AQ024)the Scientific Research Foundation of Heze University of Shandong Province,China (Grant No. XYJJKJ-1)
文摘From the normally ordered form of the density operator of a squeezed coherent state（SCS）,we directly derive the compact expression of the SCS＇s photon-number distribution（PND）.Besides the known oscillation characteristics,we find that the PND is a periodic function with a period of π and extremely sensitive to phase.If the squeezing is strong enough,and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values,different oscillation behaviours in PND for even and odd photon numbers appear,respectively.
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supported in part by the National Natural Science Foundation of China (Grant No.61202491,61272041,61272488,61402523,61602514)the Science and Technology on Communication Security Laboratory Foundation of China under Grant No.9140C110303140C11051
文摘The Grain v1 stream cipher is one of the seven finalists in the final e STREAM portfolio. Though many attacks have been published,no recovery attack better than exhaustive key search on full Grain v1 in the single key setting has been found yet. In this paper,new state recovery attacks on Grain v1 utilizing the weak normality order of the employed keystream output function in the cipher are proposed. These attacks have remarkable advantages in the offline time,online time and memory complexities,which are all better than exhaustive key search. The success probability of each new attack is 0.632. The proposed attack primarily depends on the order of weak normality of the employed keystream output function. This shows that the weak normality order should be carefully considered when designing the keystream output functions of Grain-like stream ciphers.
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〉d3x = 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.
基金supported by the National Natural Science Foundation of China(11174312,11125420)the Office of Naval Research,USA
文摘The sensitivities of the normal modes arrival time to solitary internal waves （IWs） are analyzed by using the SW06 environments. Simulation results show that the arrival time of mode 1 is relatively stable. But, there are some higher-order normal modes which arrive earlier than mode 1, and fluctuate with the appearance of solitary IWs. Explanation of the phenomenon is given based on ray theory. It is shown that, when thermocline falls down to some depths, those higher-order modes with a group of definite grazing angles mainly propagate above the thermocline and arrive earlier.
文摘For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^(u)<d≤e^(u+1)}|.The Erdos-Hooley Deltafunction is then defined by Δ(n):=Max_(u∈R)Δ(n,u).We improve the current upper bounds for the average and normal orders of this arithmetic function.
基金supported by the National Natural Science Foundation of China (Grant Nos.10574060 and 11174114)the Natural Science Foundation of Shandong Province, China (Grant No.ZR2010AQ027)+1 种基金the Research Foundation of Changzhou Institute of Technology (Grant No.YN1007)the Shandong Provincal Higher Educational Science and Technology Program, China (Grant Nos.J09LA07, J10LA15)
文摘The new generalized coherent-entangled state representation |α, p μ,ν is successfully derived via constructing the integration of unity in normally ordered Gaussian operator forms and then decomposing it as projection operators. This is a convenient approach for obtaining new representations. We then prove that |α, p μ,ν has the completeness relation and only partly orthogonal, then discuss how to use a beamsplitter to produce such a state. As its potential application, formulas can be obtained by using the completeness of |α, p μ,ν , which is very important in mathematical physics, especially in quantum optics.
