New light fields based on integration theory within the Weyl ordering product of operators

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.

Approach to obtaining weights of uncertain ordered weighted geometric averaging operator

The ordered weighted geometric averaging（OWGA） operator is extended to accommodate uncertain conditions where all input arguments take the forms of interval numbers. First, a possibility degree formula for the comparison between interval numbers is introduced. It is proved that the introduced formula is equivalent to the existing formulae, and also some desired properties of the possibility degree is presented. Secondly, the uncertain OWGA operator is investigated in which the associated weighting parameters cannot be specified, but value ranges can be obtained and the associated aggregated values of an uncertain OWGA operator are known. A linear objective-programming model is established; by solving this model, the associated weights vector of an uncertain OWGA operator can be determined, and also the estimated aggregated values of the alternatives can be obtained. Then the alternatives can be ranked by the comparison of the estimated aggregated values using the possibility degree formula. Finally, a numerical example is given to show the feasibility and effectiveness of the developed method.

Wigner function of optical cumulant operator and its dissipation in thermo-entangled state representation

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.

New Bosonic Operator Ordering Identities Gained by the Entangled State Representation and Two－Variable Hermite Polynomials

Based on the technique of integration within an ordered product of operators, we derive new bosonic operators, ordering identities by using entangled state representation and the properties of two-variable Hermite polynomials , and vice versa. In doing so, some concise normally (antinormally) ordering operator identities, such as : are obtained.

A New Kind of Two-Fold Integration Transformation in Phase Space and Its Uses in Weyl Ordering of Operators

We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation to the Weyl ordering of operators, especially those Q-P ordered and P-Q ordered operators.

Generalized ordered weighted averaging operators based methods for MADM in intuitionistic fuzzy set setting

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.

BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRODINGER TYPE OPERATORS

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.

A generalized Weyl-Wigner quantization scheme unifying P-Q and Q-P ordering and Weyl ordering of operators

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.

New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

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.

Hardy Spaces H_L^p(R^n) Associated with Higher-Order Schrdinger Type Operators

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.

Re-explanation of Weyl Quantization Scheme via Weyl Ordering Approach

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl orderedproduct of operators,i.e.,the Weyl correspondence rule can be reconstructed by classical functions' Fourier transfor-mation followed by an inverse Fourier transformation within Weyl ordering of operators.As an application of thisreconstruction,we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.

Operator Methods in High Order Partial Differential Equation

For linear partial differential equation 〔 2t 2-a 2P( x)〕 m u=f(x,t), where m1,X∈R n,t∈R 1, the author gives the analytic solution of the initial value problem using the operators sh(tP( x) 1/2 )P( x) 1/2 . By representing the operators with integrals, explicit solutions are obtained with an integral form of a given function.

Linear goal programming approach to obtaining the weights of intuitionistic fuzzy ordered weighted averaging operator

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.

Gauge Transformation and Constraint Induced Operator Ordering for Charged Rigid Planar Rotator in Uniform Magnetic Field

When the motion of a particle is constrained, excess terms exist using hermitian form of Cartesian momentum pi （i=1, 2,3） in usual kinetic energy （1/2/μ）∑ pi^2 , and the correct kinetic energy turns out to be （1/2μ） ∑1/ fipi f ipi, where fi are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this paper the explicit form of the dummy functions fi is given for a charged rigid planar rotator in the uniform magnetic field with different gauge chosen. Under different gauges, we have different sets of dummy factors. It means that these factors do not have direct observable effect.

Weyl Ordering Operator Formula Derived by IWOP Technique and Its Application for Fresnel Operator

Based on the technique of integration within an ordered product of operators, the Weyl ordering operator formula is derived and the Fresnel operators＇ Weyl ordering is also obtained, which together with the Weyl transformation can immediately lead to Eresnel transformation kernel in classical optics.

Weyl Ordering Expansion of Power Product of Coordinate and Momentum Operators

By virtue of the technique of integration within Weyl ordered of operators we derive the formula of Weyl ordering expansion of power product of coordinate and momentum operators （√2Q）^m（√2iP） ^τ=：： Hm,r （√2Q, √2iP）：：, the introduction of two-variable Hermite polynomial Hm,r brings much convenience to the study of Weyl correspondence.

New approach to Q-P (P-Q) ordering of quantum mechanical operators and its applications

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.

Recapitulating Quantum Phase Space Representation by Virtue of Normally Ordered Gaussian Form of Wigner Operator

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.

Quantum entangled fractional Fourier transform based on the IWOP technique

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.

An anomaly detection method for spacecraft solar arrays based on the ILS-SVM model

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.