The development of quantum optics theory based on the method of integration within an ordered product of operators(IWOP)has greatly stimulated the study of quantum states in the light field,especially non-Gaussian sta...The development of quantum optics theory based on the method of integration within an ordered product of operators(IWOP)has greatly stimulated the study of quantum states in the light field,especially non-Gaussian states with various non-classical properties.In this paper,the two-mode squeezing operator is derived with integral theory within the Weyl ordering product of operators using a combinatorial field in which one mode is a chaotic field and the other mode is a vacuum field.The density operator of the new light field,its entanglement property and photon number distribution are analyzed.We also note that tracing a three-mode pure state can yield this new light field.These methods represent a theoretical approach to investigating new density operators of light fields.展开更多
To conveniently calculate the Wigner function of the optical cumulant operator and its dissipation evolution in a thermal environment, in this paper, the thermo-entangled state representation is introduced to derive t...To conveniently calculate the Wigner function of the optical cumulant operator and its dissipation evolution in a thermal environment, in this paper, the thermo-entangled state representation is introduced to derive the general evolution formula of the Wigner function, and its relation to Weyl correspondence is also discussed. The method of integration within the ordered product of operators is essential to our discussion.展开更多
Multiattribute decision making(MADM) problems, in which the weights and ratings of alternatives are expressed with intuitionistic fuzzy(IF) sets, are investigated.Firstly, the relative degrees of membership and th...Multiattribute decision making(MADM) problems, in which the weights and ratings of alternatives are expressed with intuitionistic fuzzy(IF) sets, are investigated.Firstly, the relative degrees of membership and the relative degrees of non-membership are formulated as IF sets, the weights and values of alternatives on both qualitative and quantitative attributes may be expressed as IF sets in a unified way.Then a MADM method based on generalized ordered weighted averaging operators is proposed.The proposed method is illustrated with a numerical example.展开更多
In this article,we derive the Lp-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrodinger type operator(—Δ)^2+V%2 in R%n(n≥5)with V being a nonnegat...In this article,we derive the Lp-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrodinger type operator(—Δ)^2+V%2 in R%n(n≥5)with V being a nonnegative potential satisfying the reverse Holder inequality.Furt her more,we prove the boundedness of the variation operators on associated Morrey spaces.In the proof of the main results,we always make use of the variation inequalities associated with the hea t semigroup genera ted by the biharmonic operator(-Δ)2.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...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
Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, J. Phys.A 25 (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator △( q',p) q-number transf...Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, J. Phys.A 25 (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator △( q',p) q-number transform) in phase space quantum mechanics,∫∫∞-∞dp'dq'/π △(q',p')e-2i( p-p')( q-q')=δ( p-P)δ( q-Q),∫∫∞-∞dqdpδ(p-P)δ(q-Q)e2i(p-p')(q-q')=△(q',p'),whereQ,P are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among Q-P ordering, P-Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.展开更多
Let L = L0 + V be the higher order Schrodiger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coeffi- cients and V is a real measurable function as multiplication...Let L = L0 + V be the higher order Schrodiger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coeffi- cients and V is a real measurable function as multiplication operator (e.g., including (-△)m+v (m∈N) as special examples). In this paper, assume that V satisfies a strongly subcritical form condition associated with L0, the authors attempt to establish a the- HL (R) (0 〈 p ≤ 1) associated with the higher order Schrodinger ory of Hardy space P n type operator L. Specifically, we first define the molecular Hardy space Hp (JRn) by the so-called (p, q,ε, M) molecule associated to L and then establish its characterizations by the area integral defined by the heat semigroup e-tL.展开更多
The multiple attribute decision making problems are studied, in which the information about attribute weights is partly known and the attribute values take the form of intuitionistic fuzzy numbers. The operational law...The multiple attribute decision making problems are studied, in which the information about attribute weights is partly known and the attribute values take the form of intuitionistic fuzzy numbers. The operational laws of intuitionistic fuzzy numbers are introduced, and the score function and accuracy function are presented to compare the intuitionistic fuzzy numbers. The intuitionistic fuzzy ordered weighted averaging (IFOWA) operator which is an extension of the well-known ordered weighted averaging (OWA) operator is investigated to aggregate the intuitionistic fuzzy information. In order to determine the weights of intuitionistic fuzzy ordered weighted averaging operator, a linear goal programming procedure is proposed for learning the weights from data. Finally, an example is illustrated to verify the effectiveness and practicability of the developed method.展开更多
In this paper, we introduce a new way to obtain the Q-P (P-Q) ordering of quantum mechanical operators, i.e., from the classical correspondence of Q-P (P-Q) ordered operators by replacing q and p with coordinate a...In this paper, we introduce a new way to obtain the Q-P (P-Q) ordering of quantum mechanical operators, i.e., from the classical correspondence of Q-P (P-Q) ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single (two-) mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator’s Q-P (P-Q) ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q-P ordering of operators.展开更多
Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessar...Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessary and sufficient conditions are obtained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
Solar arrays are important and indispensable parts of spacecraft and provide energy support for spacecraft to operate in orbit and complete on-orbit missions.When a spacecraft is in orbit,because the solar array is ex...Solar arrays are important and indispensable parts of spacecraft and provide energy support for spacecraft to operate in orbit and complete on-orbit missions.When a spacecraft is in orbit,because the solar array is exposed to the harsh space environment,with increasing working time,the performance of its internal electronic components gradually degrade until abnormal damage occurs.This damage makes solar array power generation unable to fully meet the energy demand of a spacecraft.Therefore,timely and accurate detection of solar array anomalies is of great significance for the on-orbit operation and maintenance management of spacecraft.In this paper,we propose an anomaly detection method for spacecraft solar arrays based on the integrated least squares support vector machine(ILS-SVM)model:it selects correlated telemetry data from spacecraft solar arrays to form a training set and extracts n groups of training subsets from this set,then gets n corresponding least squares support vector machine(LS-SVM)submodels by training on these training subsets,respectively;after that,the ILS-SVM model is obtained by integrating these submodels through a weighting operation to increase the prediction accuracy and so on;finally,based on the obtained ILS-SVM model,a parameterfree and unsupervised anomaly determination method is proposed to detect the health status of solar arrays.We use the telemetry data set from a satellite in orbit to carry out experimental verification and find that the proposed method can diagnose solar array anomalies in time and can capture the signs before a solar array anomaly occurs,which reflects the applicability of the method.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 notion of the interval-valued intuitionistic fuzzy set (IVIFS) is a generalization of that of the Atanassov's intuitionistic fuzzy set. The fundamental characteristic of IVIFS is that the values of its membersh...The notion of the interval-valued intuitionistic fuzzy set (IVIFS) is a generalization of that of the Atanassov's intuitionistic fuzzy set. The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. There are various averaging operators defined for IVlFSs. These operators are not monotone with respect to the total order of IVIFS, which is undesirable. This paper shows how such averaging operators can be represented by using additive generators of the product triangular norm, which simplifies and extends the existing constructions. Moreover, two new aggregation operators based on the t.ukasiewicz triangular norm are proposed, which are monotone with respect to the total order of IVIFS. Finally, an application of the interval-valued intuitionistic fuzzy weighted averaging operator is given to multiple criteria decision making.展开更多
In this paper, we investigate the value distribution of the difference counterpart △f(z)- af(z)^n of f′(z)- af(z)^n and obtain an almost direct difference analogue of result of Hayman.
By virtue of the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators. Several important examples are considered as the applications of this...By virtue of the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators. Several important examples are considered as the applications of this equation, which shows that our approach is effective and convenient for deducing these entangled state representations.展开更多
In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By u...In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.展开更多
By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials wh...By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.展开更多
We newly construct two mutually-conjugate tripartite entangled state representations, based on which we propose the formulation of three-mode entangled fractional Fourier transformation (EFFT) and derive the transfo...We newly construct two mutually-conjugate tripartite entangled state representations, based on which we propose the formulation of three-mode entangled fractional Fourier transformation (EFFT) and derive the transformation kernel. The EFFT's additivity property is proved and the eigenmode of EFFT is derived. As an application, we calculate the EFFT of the three-mode squeezed vacuum state.展开更多
Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =(z1, z′2,…, z′k)′∈ C × C^n2×…...Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =(z1, z′2,…, z′k)′∈ C × C^n2×…× Cnk: |z1|^2+ ||z2||2^p2+ … + ||zk ||k^pk〈 1} given〈1} give by F(z)=(f(z1)+f'(z1)∑j=2 kPj(zj,(f'(z1))1/p2 z2',…,(f'(z1))1/pkzk')', where f is a normaljized biholomorphic function k(z) =(f(z1) + f′(z1)=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj : C^nj→ C is a homogeneous polynomial of degree pj and zj =(zj1, …, zjnj)′∈ C^nj, nj ≥ 1, pj ≥ 1,||zj||j =(∑l=1 nj|zjl|^pj)1/pj. In this paper, some conditions for Pjare found under which the loperator p |zjl|pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.11775208)the Foundation for Young Talents in 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The development of quantum optics theory based on the method of integration within an ordered product of operators(IWOP)has greatly stimulated the study of quantum states in the light field,especially non-Gaussian states with various non-classical properties.In this paper,the two-mode squeezing operator is derived with integral theory within the Weyl ordering product of operators using a combinatorial field in which one mode is a chaotic field and the other mode is a vacuum field.The density operator of the new light field,its entanglement property and photon number distribution are analyzed.We also note that tracing a three-mode pure state can yield this new light field.These methods represent a theoretical approach to investigating new density operators of light fields.
基金Project supported by the Foundation for Young Talents in College of Anhui Province, China (Grant Nos. gxyq2021210 and gxyq2019077)the Natural Science Foundation of the Anhui Higher Education Institutions, China (Grant Nos. 2022AH051580 and 2022AH051586)。
文摘To conveniently calculate the Wigner function of the optical cumulant operator and its dissipation evolution in a thermal environment, in this paper, the thermo-entangled state representation is introduced to derive the general evolution formula of the Wigner function, and its relation to Weyl correspondence is also discussed. The method of integration within the ordered product of operators is essential to our discussion.
基金supported by the National Natural Science Foundation of China (70871117 70571086)
文摘Multiattribute decision making(MADM) problems, in which the weights and ratings of alternatives are expressed with intuitionistic fuzzy(IF) sets, are investigated.Firstly, the relative degrees of membership and the relative degrees of non-membership are formulated as IF sets, the weights and values of alternatives on both qualitative and quantitative attributes may be expressed as IF sets in a unified way.Then a MADM method based on generalized ordered weighted averaging operators is proposed.The proposed method is illustrated with a numerical example.
基金supported by the National Natural Science Foundation of China(11701453)Fundamental Research Funds for the Central Universities(31020180QD05)+2 种基金The second author was supported by the National Natural Science Foundation of China(11971431,11401525)the Natural Science Foundation of Zhejiang Province(LY18A010006)and the first Class Discipline of Zhejiang-A(Zhejiang Gongshang University-Statistics).
文摘In this article,we derive the Lp-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrodinger type operator(—Δ)^2+V%2 in R%n(n≥5)with V being a nonnegative potential satisfying the reverse Holder inequality.Furt her more,we prove the boundedness of the variation operators on associated Morrey spaces.In the proof of the main results,we always make use of the variation inequalities associated with the hea t semigroup genera ted by the biharmonic operator(-Δ)2.
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.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10775097 and 10874174)Specialized Research Fund for the Doctoral Program of Higher Education of China
文摘Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, J. Phys.A 25 (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator △( q',p) q-number transform) in phase space quantum mechanics,∫∫∞-∞dp'dq'/π △(q',p')e-2i( p-p')( q-q')=δ( p-P)δ( q-Q),∫∫∞-∞dqdpδ(p-P)δ(q-Q)e2i(p-p')(q-q')=△(q',p'),whereQ,P are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among Q-P ordering, P-Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.
基金supported by NSFC(No.11301203)NSFC(No.11371057,11471033)+5 种基金NSFC(No.11371158)the Fundamental Research Funds for the Central Universities(CCNU-14A05037)the Fundamental Research Funds for the Central Universities(No.2014KJJCA10)SRFDP(No.20130003110003)the program for Changjiang ScholarsInnovative Research Team in University(No.IRT13066)
文摘Let L = L0 + V be the higher order Schrodiger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coeffi- cients and V is a real measurable function as multiplication operator (e.g., including (-△)m+v (m∈N) as special examples). In this paper, assume that V satisfies a strongly subcritical form condition associated with L0, the authors attempt to establish a the- HL (R) (0 〈 p ≤ 1) associated with the higher order Schrodinger ory of Hardy space P n type operator L. Specifically, we first define the molecular Hardy space Hp (JRn) by the so-called (p, q,ε, M) molecule associated to L and then establish its characterizations by the area integral defined by the heat semigroup e-tL.
基金supported by the National Natural Science Foundation of China (70771025)the Fundamental Research Funds for the Central Universities of Hohai University (2009B04514)Humanities and Social Sciences Foundations of Ministry of Education of China(10YJA630067)
文摘The multiple attribute decision making problems are studied, in which the information about attribute weights is partly known and the attribute values take the form of intuitionistic fuzzy numbers. The operational laws of intuitionistic fuzzy numbers are introduced, and the score function and accuracy function are presented to compare the intuitionistic fuzzy numbers. The intuitionistic fuzzy ordered weighted averaging (IFOWA) operator which is an extension of the well-known ordered weighted averaging (OWA) operator is investigated to aggregate the intuitionistic fuzzy information. In order to determine the weights of intuitionistic fuzzy ordered weighted averaging operator, a linear goal programming procedure is proposed for learning the weights from data. Finally, an example is illustrated to verify the effectiveness and practicability of the developed method.
基金Project supported by the National Natural Science Foundation of China(Grant No.11264018)the Natural Science Foundation of Jiangxi Province of China(Grant No.20132BAB212006)the Fund from the Key Laboratory of Optoelectronics and Telecommunication of Jiangxi Province,China
文摘In this paper, we introduce a new way to obtain the Q-P (P-Q) ordering of quantum mechanical operators, i.e., from the classical correspondence of Q-P (P-Q) ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single (two-) mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator’s Q-P (P-Q) ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q-P ordering of operators.
基金Supported in part by the National Natural Science Foundation of China (1 0 0 71 0 2 1 ) the Foundationfor University Key Teacher by MEC and Shanghai Priority Academic Discipline Foundation
文摘Based on the method of symplectic geometry and variational calculation,the method for some PDEs to be ordered and analytically represented by Hamiltonian canonical system is discussed.Meanwhile some related necessary and sufficient conditions are obtained
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.
基金supported by the National Natural Science Foundation of China(7190121061973310).
文摘Solar arrays are important and indispensable parts of spacecraft and provide energy support for spacecraft to operate in orbit and complete on-orbit missions.When a spacecraft is in orbit,because the solar array is exposed to the harsh space environment,with increasing working time,the performance of its internal electronic components gradually degrade until abnormal damage occurs.This damage makes solar array power generation unable to fully meet the energy demand of a spacecraft.Therefore,timely and accurate detection of solar array anomalies is of great significance for the on-orbit operation and maintenance management of spacecraft.In this paper,we propose an anomaly detection method for spacecraft solar arrays based on the integrated least squares support vector machine(ILS-SVM)model:it selects correlated telemetry data from spacecraft solar arrays to form a training set and extracts n groups of training subsets from this set,then gets n corresponding least squares support vector machine(LS-SVM)submodels by training on these training subsets,respectively;after that,the ILS-SVM model is obtained by integrating these submodels through a weighting operation to increase the prediction accuracy and so on;finally,based on the obtained ILS-SVM model,a parameterfree and unsupervised anomaly determination method is proposed to detect the health status of solar arrays.We use the telemetry data set from a satellite in orbit to carry out experimental verification and find that the proposed method can diagnose solar array anomalies in time and can capture the signs before a solar array anomaly occurs,which reflects the applicability of the method.
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 (71171048)the Scientific Research and Innovation Project for College Graduates of Jiangsu Province (CXZZ11 0185)+1 种基金the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1135)the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (RCS2011K002)
文摘The notion of the interval-valued intuitionistic fuzzy set (IVIFS) is a generalization of that of the Atanassov's intuitionistic fuzzy set. The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. There are various averaging operators defined for IVlFSs. These operators are not monotone with respect to the total order of IVIFS, which is undesirable. This paper shows how such averaging operators can be represented by using additive generators of the product triangular norm, which simplifies and extends the existing constructions. Moreover, two new aggregation operators based on the t.ukasiewicz triangular norm are proposed, which are monotone with respect to the total order of IVIFS. Finally, an application of the interval-valued intuitionistic fuzzy weighted averaging operator is given to multiple criteria decision making.
基金supported by the National Natural Science Foundation of China(11171119)
文摘In this paper, we investigate the value distribution of the difference counterpart △f(z)- af(z)^n of f′(z)- af(z)^n and obtain an almost direct difference analogue of result of Hayman.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10874174 and 90203002)
文摘By virtue of the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators. Several important examples are considered as the applications of this equation, which shows that our approach is effective and convenient for deducing these entangled state representations.
基金supported by Natural Science Foundation of China(No.11171220) Support Projects of University of Shanghai for Science and Technology(No.14XPM01)
文摘In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.
基金supported by the National Natural Science Foundation of China(Grant No.11175113)the Fundamental Research Funds for the Central Universities of China(Grant No.WK2060140013)
文摘By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.
基金Project supported by the Specialized Research Fund for Doctoral Program of High Education of Chinathe National Natural Science Foundation of China (Grant Nos. 10874174 and 10947017/A05)
文摘We newly construct two mutually-conjugate tripartite entangled state representations, based on which we propose the formulation of three-mode entangled fractional Fourier transformation (EFFT) and derive the transformation kernel. The EFFT's additivity property is proved and the eigenmode of EFFT is derived. As an application, we calculate the EFFT of the three-mode squeezed vacuum state.
基金Supported by the National Natural Science Foundation of China(11001074,11061015,11101124)
文摘Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =(z1, z′2,…, z′k)′∈ C × C^n2×…× Cnk: |z1|^2+ ||z2||2^p2+ … + ||zk ||k^pk〈 1} given〈1} give by F(z)=(f(z1)+f'(z1)∑j=2 kPj(zj,(f'(z1))1/p2 z2',…,(f'(z1))1/pkzk')', where f is a normaljized biholomorphic function k(z) =(f(z1) + f′(z1)=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj : C^nj→ C is a homogeneous polynomial of degree pj and zj =(zj1, …, zjnj)′∈ C^nj, nj ≥ 1, pj ≥ 1,||zj||j =(∑l=1 nj|zjl|^pj)1/pj. In this paper, some conditions for Pjare found under which the loperator p |zjl|pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.