Quantization noise caused by analog-to-digital converter(ADC)gives rise to the reliability performance degradation of communication systems.In this paper,a quantized non-Hermitian symmetry(NHS)orthogonal frequency-div...Quantization noise caused by analog-to-digital converter(ADC)gives rise to the reliability performance degradation of communication systems.In this paper,a quantized non-Hermitian symmetry(NHS)orthogonal frequency-division multiplexing-based visible light communication(OFDM-VLC)system is presented.In order to analyze the effect of the resolution of ADC on NHS OFDM-VLC,a quantized mathematical model of NHS OFDM-VLC is established.Based on the proposed quantized model,a closed-form bit error rate(BER)expression is derived.The theoretical analysis and simulation results both confirm the effectiveness of the obtained BER formula in high-resolution ADC.In addition,channel coding is helpful in compensating for the BER performance loss due to the utilization of lower resolution ADC.展开更多
The rapid growth of digital data necessitates advanced natural language processing(NLP)models like BERT(Bidi-rectional Encoder Representations from Transformers),known for its superior performance in text classificati...The rapid growth of digital data necessitates advanced natural language processing(NLP)models like BERT(Bidi-rectional Encoder Representations from Transformers),known for its superior performance in text classification.However,BERT’s size and computational demands limit its practicality,especially in resource-constrained settings.This research compresses the BERT base model for Bengali emotion classification through knowledge distillation(KD),pruning,and quantization techniques.Despite Bengali being the sixth most spoken language globally,NLP research in this area is limited.Our approach addresses this gap by creating an efficient BERT-based model for Bengali text.We have explored 20 combinations for KD,quantization,and pruning,resulting in improved speedup,fewer parameters,and reduced memory size.Our best results demonstrate significant improvements in both speed and efficiency.For instance,in the case of mBERT,we achieved a 3.87×speedup and 4×compression ratio with a combination of Distil+Prune+Quant that reduced parameters from 178 to 46 M,while the memory size decreased from 711 to 178 MB.These results offer scalable solutions for NLP tasks in various languages and advance the field of model compression,making these models suitable for real-world applications in resource-limited environments.展开更多
This paper proposes a novel method for the automatic diagnosis of keratitis using feature vector quantization and self-attention mechanisms(ADK_FVQSAM).First,high-level features are extracted using the DenseNet121 bac...This paper proposes a novel method for the automatic diagnosis of keratitis using feature vector quantization and self-attention mechanisms(ADK_FVQSAM).First,high-level features are extracted using the DenseNet121 backbone network,followed by adaptive average pooling to scale the features to a fixed length.Subsequently,product quantization with residuals(PQR)is applied to convert continuous feature vectors into discrete features representations,preserving essential information insensitive to image quality variations.The quantized and original features are concatenated and fed into a self-attention mechanism to capture keratitis-related features.Finally,these enhanced features are classified through a fully connected layer.Experiments on clinical low-quality(LQ)images show that ADK_FVQSAM achieves accuracies of 87.7%,81.9%,and 89.3% for keratitis,other corneal abnormalities,and normal corneas,respectively.Compared to DenseNet121,Swin transformer,and InceptionResNet,ADK_FVQSAM improves average accuracy by 3.1%,11.3%,and 15.3%,respectively.These results demonstrate that ADK_FVQSAM significantly enhances the recognition performance of keratitis based on LQ slit-lamp images,offering a practical approach for clinical application.展开更多
Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of qua...Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrdinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems, such as momentum, energy and Hamiltonian, so Schrdinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrdinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.展开更多
A fast encoding algorithm based on the mean square error （MSE） distortion for vector quantization is introduced. The vector, which is effectively constructed with wavelet transform （WT） coefficients of images, can...A fast encoding algorithm based on the mean square error （MSE） distortion for vector quantization is introduced. The vector, which is effectively constructed with wavelet transform （WT） coefficients of images, can simplify the realization of the non-linear interpolated vector quantization （NLIVQ） technique and make the partial distance search （PDS） algorithm more efficient. Utilizing the relationship of vector L2-norm and its Euclidean distance, some conditions of eliminating unnecessary codewords are obtained. Further, using inequality constructed by the subvector L2-norm, more unnecessary codewords are eliminated. During the search process for code, mostly unlikely codewords can be rejected by the proposed algorithm combined with the non-linear interpolated vector quantization technique and the partial distance search technique. The experimental results show that the reduction of computation is outstanding in the encoding time and complexity against the full search method.展开更多
Information hiding schemes based on vector quantization （VQ） usually require lengthy VQ encoding and decoding processes. In this paper, we propose an efficient information hiding method based on closest paired tree ...Information hiding schemes based on vector quantization （VQ） usually require lengthy VQ encoding and decoding processes. In this paper, we propose an efficient information hiding method based on closest paired tree structure vector quantization （CPTSVQ）. The simulation result shows that the execution time of the proposed scheme is much shorter than that attained by previous approaches.展开更多
Potential electric and gravitational fields do not change steady quantized states of electrons in chemical bonds, microscopic clusters of charges and macroscopic superconducting rings. There are no theoretical grounds...Potential electric and gravitational fields do not change steady quantized states of electrons in chemical bonds, microscopic clusters of charges and macroscopic superconducting rings. There are no theoretical grounds to create Squid-type instruments to measure electric and gravitational fields with quantum accuracy basing on the Bohr-Sommerfeld quantization of charged particles. Squid-verified spatial flatness for superfluid electrons corresponds to the material space paradigm for reality.展开更多
Many classical encoding algorithms of vector quantization （VQ） of image compression that can obtain global optimal solution have computational complexity O（N）. A pure quantum VQ encoding algorithm with probability...Many classical encoding algorithms of vector quantization （VQ） of image compression that can obtain global optimal solution have computational complexity O（N）. A pure quantum VQ encoding algorithm with probability of success near 100% has been proposed, that performs operations 45√N times approximately. In this paper, a hybrid quantum VQ encoding algorithm between the classical method and the quantum algorithm is presented. The number of its operations is less than √N for most images, and it is more efficient than the pure quantum algorithm.展开更多
This work studies the stabilization of a class of control systems that use communication networks as signal transmission medium. The lateral motion of independently actuated four-wheel vehicle is modeled as an uncerta...This work studies the stabilization of a class of control systems that use communication networks as signal transmission medium. The lateral motion of independently actuated four-wheel vehicle is modeled as an uncertain-linear system. Time delay and quantization density are modeled as Markov chains.The networked control systems(NCSs) with plants being lateral motion are first transformed to switched linear systems with uncertain parameters. Sufficient and necessary conditions for the stochastic stability of closed-loop networked control systems are then established. By solving the matrix inequalities, this work presents an output-feedback controller that depends on the modes of time delay and quantization density. The controller performance is illustrated via a vehicular lateral motion system.展开更多
Conventional quantization index modulation (QIM) watermarking uses the fixed quantization step size for the host signal.This scheme is not robust against geometric distortions and may lead to poor fidelity in some are...Conventional quantization index modulation (QIM) watermarking uses the fixed quantization step size for the host signal.This scheme is not robust against geometric distortions and may lead to poor fidelity in some areas of content.Thus,we proposed a quantization-based image watermarking in the dual tree complex wavelet domain.We took advantages of the dual tree complex wavelets (perfect reconstruction,approximate shift invariance,and directional selectivity).For the case of watermark detecting,the probability of false alarm and probability of false negative were exploited and verified by simulation.Experimental results demonstrate that the proposed method is robust against JPEG compression,additive white Gaussian noise (AWGN),and some kinds of geometric attacks such as scaling,rotation,etc.展开更多
The quantum object is in general considered as displaying both wave and particle nature. By particle is understood an item localized in a very small volume of the space, and which cannot be simultaneously in two disjo...The quantum object is in general considered as displaying both wave and particle nature. By particle is understood an item localized in a very small volume of the space, and which cannot be simultaneously in two disjoint regions of the space. By wave, to the contrary, is understood a distributed item, occupying in some cases two or more disjoint regions of the space. The quantum formalism did not explain until today the so-called “collapse” of the wave-function, i.e. the shrinking of the wave-function to one small region of the space, when a macroscopic object is encountered. This seems to happen in “which-way” experiments. A very appealing explanation for this behavior is the idea of a particle, localized in some limited part of the wave-function. The present article challenges the concept of particle. It proves in the base of a variant of the Tan, Walls and Collett experiment, that this concept leads to a situation in which the particle has to be simultaneously in two places distant from one another—situation that contradicts the very definition of a particle. Another argument is based on a modified version of the Afshar experiment, showing that the concept of particle is problematic. The concept of particle makes additional difficulties when the wave-function passes through fields. An unexpected possibility to solve these difficulties seems to arise from the cavity quantum electrodynamics studies done recently by S. Savasta and his collaborators. It involves virtual particles. One of these studies is briefly described here. Though, experimental results are needed, so that it is too soon to conclude whether it speaks in favor, or against the concept of particle.展开更多
It is currently believed that light quantum or the quantization of light energy is beyond classical physics, and the picture of wave-particle duality, which was criticized by Einstein but has attracted a number of exp...It is currently believed that light quantum or the quantization of light energy is beyond classical physics, and the picture of wave-particle duality, which was criticized by Einstein but has attracted a number of experimental researches, is necessary for the description of light. It is shown in this paper, however, that the quantization of light energy in vacuum, which is the same as that in quantum electrodynamics, can be derived directly from the classical electromagnetic theory through the consideration of statistics based on classical physics. Therefore, the quantization of energy is an intrinsic property of light as a classical electromagnetic wave and has no need of being related to particles.展开更多
We study quantum classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the ...We study quantum classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than h is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 27r-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.展开更多
Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to ...Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to the common goals of any quantization, such as Schroedinger’s representation and Schroedinger’s equation. Careful attention is paid toward seeking favored classical variables, which are those that should be promoted to the principal quantum operators. This effort leads toward classical variables that have a constant positive, zero, or negative curvature, which typically characterize such favored variables. This focus leans heavily toward affine variables with a constant negative curvature, which leads to a surprisingly accommodating analysis of non-renormalizable scalar models as well as Einstein’s general relativity.展开更多
This paper is concerned with the problem of finitetime control for a class of discrete-time networked systems.The measurement output and control input signals are quantized before being transmitted in communication ne...This paper is concerned with the problem of finitetime control for a class of discrete-time networked systems.The measurement output and control input signals are quantized before being transmitted in communication network.The quantization density of the network is assumed to be variable depending on the throughputs of network for the sake of congestion avoidance.The variation of the quantization density modes satisfies persistent dwell-time(PDT)switching which is more general than dwell-time switching in networked channels.By using a quantization-error-dependent Lyapunov function approach,sufficient conditions are given to ensure that the quantized systems are finite-time stable and finite-time bounded with a prescribed H∞performance,upon which a set of controllers depending on the mode of quantization density are designed.In order to show the effectiveness of the designed H_∞controller,we apply the developed theoretical results to a numerical example.展开更多
By extending the usual Weyl transformation to the s-parameterized Weyl transformation with s being a real parameter,we obtain the s-parameterized quantization scheme which includes P–Q quantization, Q–P quantization...By extending the usual Weyl transformation to the s-parameterized Weyl transformation with s being a real parameter,we obtain the s-parameterized quantization scheme which includes P–Q quantization, Q–P quantization, and Weyl ordering as its three special cases. Some operator identities can be derived directly by virtue of the s-parameterized quantization scheme.展开更多
基金supported by the National Natural Science Foundation of China(No.62201508)the Zhejiang Provincial Natural Science Foundation of China(Nos.LZ21F010001 and LQ23F010004)the State Key Laboratory of Millimeter Waves of Southeast University,China(No.K202212).
文摘Quantization noise caused by analog-to-digital converter(ADC)gives rise to the reliability performance degradation of communication systems.In this paper,a quantized non-Hermitian symmetry(NHS)orthogonal frequency-division multiplexing-based visible light communication(OFDM-VLC)system is presented.In order to analyze the effect of the resolution of ADC on NHS OFDM-VLC,a quantized mathematical model of NHS OFDM-VLC is established.Based on the proposed quantized model,a closed-form bit error rate(BER)expression is derived.The theoretical analysis and simulation results both confirm the effectiveness of the obtained BER formula in high-resolution ADC.In addition,channel coding is helpful in compensating for the BER performance loss due to the utilization of lower resolution ADC.
文摘The rapid growth of digital data necessitates advanced natural language processing(NLP)models like BERT(Bidi-rectional Encoder Representations from Transformers),known for its superior performance in text classification.However,BERT’s size and computational demands limit its practicality,especially in resource-constrained settings.This research compresses the BERT base model for Bengali emotion classification through knowledge distillation(KD),pruning,and quantization techniques.Despite Bengali being the sixth most spoken language globally,NLP research in this area is limited.Our approach addresses this gap by creating an efficient BERT-based model for Bengali text.We have explored 20 combinations for KD,quantization,and pruning,resulting in improved speedup,fewer parameters,and reduced memory size.Our best results demonstrate significant improvements in both speed and efficiency.For instance,in the case of mBERT,we achieved a 3.87×speedup and 4×compression ratio with a combination of Distil+Prune+Quant that reduced parameters from 178 to 46 M,while the memory size decreased from 711 to 178 MB.These results offer scalable solutions for NLP tasks in various languages and advance the field of model compression,making these models suitable for real-world applications in resource-limited environments.
基金supported by the National Natural Science Foundation of China(Nos.62276210,82201148 and 62376215)the Key Research and Development Project of Shaanxi Province(No.2025CY-YBXM-044)+3 种基金the Natural Science Foundation of Zhejiang Province(No.LQ22H120002)the Medical Health Science and Technology Project of Zhejiang Province(Nos.2022RC069 and 2023KY1140)the Natural Science Foundation of Ningbo(No.2023J390)the Ningbo Top Medical and Health Research Program(No.2023030716).
文摘This paper proposes a novel method for the automatic diagnosis of keratitis using feature vector quantization and self-attention mechanisms(ADK_FVQSAM).First,high-level features are extracted using the DenseNet121 backbone network,followed by adaptive average pooling to scale the features to a fixed length.Subsequently,product quantization with residuals(PQR)is applied to convert continuous feature vectors into discrete features representations,preserving essential information insensitive to image quality variations.The quantized and original features are concatenated and fed into a self-attention mechanism to capture keratitis-related features.Finally,these enhanced features are classified through a fully connected layer.Experiments on clinical low-quality(LQ)images show that ADK_FVQSAM achieves accuracies of 87.7%,81.9%,and 89.3% for keratitis,other corneal abnormalities,and normal corneas,respectively.Compared to DenseNet121,Swin transformer,and InceptionResNet,ADK_FVQSAM improves average accuracy by 3.1%,11.3%,and 15.3%,respectively.These results demonstrate that ADK_FVQSAM significantly enhances the recognition performance of keratitis based on LQ slit-lamp images,offering a practical approach for clinical application.
文摘Formal state space models of quantum control systems are deduced and a scheme to establish formal state space models via quantization could been obtained for quantum control systems is proposed. State evolution of quantum control systems must accord with Schrdinger equations, so it is foremost to obtain Hamiltonian operators of systems. There are corresponding relations between operators of quantum systems and corresponding physical quantities of classical systems, such as momentum, energy and Hamiltonian, so Schrdinger equation models of corresponding quantum control systems via quantization could been obtained from classical control systems, and then establish formal state space models through the suitable transformation from Schrdinger equations for these quantum control systems. This method provides a new kind of path for modeling in quantum control.
基金the National Natural Science Foundation of China (60602057)the NaturalScience Foundation of Chongqing Science and Technology Commission (2006BB2373).
文摘A fast encoding algorithm based on the mean square error （MSE） distortion for vector quantization is introduced. The vector, which is effectively constructed with wavelet transform （WT） coefficients of images, can simplify the realization of the non-linear interpolated vector quantization （NLIVQ） technique and make the partial distance search （PDS） algorithm more efficient. Utilizing the relationship of vector L2-norm and its Euclidean distance, some conditions of eliminating unnecessary codewords are obtained. Further, using inequality constructed by the subvector L2-norm, more unnecessary codewords are eliminated. During the search process for code, mostly unlikely codewords can be rejected by the proposed algorithm combined with the non-linear interpolated vector quantization technique and the partial distance search technique. The experimental results show that the reduction of computation is outstanding in the encoding time and complexity against the full search method.
基金supported by the National Natural Science Foundation of China under Grant No.60133012 and No.661272374
文摘Information hiding schemes based on vector quantization （VQ） usually require lengthy VQ encoding and decoding processes. In this paper, we propose an efficient information hiding method based on closest paired tree structure vector quantization （CPTSVQ）. The simulation result shows that the execution time of the proposed scheme is much shorter than that attained by previous approaches.
文摘Potential electric and gravitational fields do not change steady quantized states of electrons in chemical bonds, microscopic clusters of charges and macroscopic superconducting rings. There are no theoretical grounds to create Squid-type instruments to measure electric and gravitational fields with quantum accuracy basing on the Bohr-Sommerfeld quantization of charged particles. Squid-verified spatial flatness for superfluid electrons corresponds to the material space paradigm for reality.
文摘Many classical encoding algorithms of vector quantization （VQ） of image compression that can obtain global optimal solution have computational complexity O（N）. A pure quantum VQ encoding algorithm with probability of success near 100% has been proposed, that performs operations 45√N times approximately. In this paper, a hybrid quantum VQ encoding algorithm between the classical method and the quantum algorithm is presented. The number of its operations is less than √N for most images, and it is more efficient than the pure quantum algorithm.
基金supported by Humanities and Social Sciences Youth Foundation of the Ministry of Education(17YJC630093)
文摘This work studies the stabilization of a class of control systems that use communication networks as signal transmission medium. The lateral motion of independently actuated four-wheel vehicle is modeled as an uncertain-linear system. Time delay and quantization density are modeled as Markov chains.The networked control systems(NCSs) with plants being lateral motion are first transformed to switched linear systems with uncertain parameters. Sufficient and necessary conditions for the stochastic stability of closed-loop networked control systems are then established. By solving the matrix inequalities, this work presents an output-feedback controller that depends on the modes of time delay and quantization density. The controller performance is illustrated via a vehicular lateral motion system.
基金supported by a grant from the National High Technology Research and Development Program of China (863 Program) (No.2008AA04A107)supported by a grant from the Major Programs of Guangdong-Hongkong in the Key Domain (No.2009498B21)
文摘Conventional quantization index modulation (QIM) watermarking uses the fixed quantization step size for the host signal.This scheme is not robust against geometric distortions and may lead to poor fidelity in some areas of content.Thus,we proposed a quantization-based image watermarking in the dual tree complex wavelet domain.We took advantages of the dual tree complex wavelets (perfect reconstruction,approximate shift invariance,and directional selectivity).For the case of watermark detecting,the probability of false alarm and probability of false negative were exploited and verified by simulation.Experimental results demonstrate that the proposed method is robust against JPEG compression,additive white Gaussian noise (AWGN),and some kinds of geometric attacks such as scaling,rotation,etc.
文摘The quantum object is in general considered as displaying both wave and particle nature. By particle is understood an item localized in a very small volume of the space, and which cannot be simultaneously in two disjoint regions of the space. By wave, to the contrary, is understood a distributed item, occupying in some cases two or more disjoint regions of the space. The quantum formalism did not explain until today the so-called “collapse” of the wave-function, i.e. the shrinking of the wave-function to one small region of the space, when a macroscopic object is encountered. This seems to happen in “which-way” experiments. A very appealing explanation for this behavior is the idea of a particle, localized in some limited part of the wave-function. The present article challenges the concept of particle. It proves in the base of a variant of the Tan, Walls and Collett experiment, that this concept leads to a situation in which the particle has to be simultaneously in two places distant from one another—situation that contradicts the very definition of a particle. Another argument is based on a modified version of the Afshar experiment, showing that the concept of particle is problematic. The concept of particle makes additional difficulties when the wave-function passes through fields. An unexpected possibility to solve these difficulties seems to arise from the cavity quantum electrodynamics studies done recently by S. Savasta and his collaborators. It involves virtual particles. One of these studies is briefly described here. Though, experimental results are needed, so that it is too soon to conclude whether it speaks in favor, or against the concept of particle.
文摘It is currently believed that light quantum or the quantization of light energy is beyond classical physics, and the picture of wave-particle duality, which was criticized by Einstein but has attracted a number of experimental researches, is necessary for the description of light. It is shown in this paper, however, that the quantization of light energy in vacuum, which is the same as that in quantum electrodynamics, can be derived directly from the classical electromagnetic theory through the consideration of statistics based on classical physics. Therefore, the quantization of energy is an intrinsic property of light as a classical electromagnetic wave and has no need of being related to particles.
基金supported by the National Natural Science Foundation of China (Grant No. 11075099)
文摘We study quantum classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than h is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 27r-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.
文摘Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to the common goals of any quantization, such as Schroedinger’s representation and Schroedinger’s equation. Careful attention is paid toward seeking favored classical variables, which are those that should be promoted to the principal quantum operators. This effort leads toward classical variables that have a constant positive, zero, or negative curvature, which typically characterize such favored variables. This focus leans heavily toward affine variables with a constant negative curvature, which leads to a surprisingly accommodating analysis of non-renormalizable scalar models as well as Einstein’s general relativity.
文摘This paper is concerned with the problem of finitetime control for a class of discrete-time networked systems.The measurement output and control input signals are quantized before being transmitted in communication network.The quantization density of the network is assumed to be variable depending on the throughputs of network for the sake of congestion avoidance.The variation of the quantization density modes satisfies persistent dwell-time(PDT)switching which is more general than dwell-time switching in networked channels.By using a quantization-error-dependent Lyapunov function approach,sufficient conditions are given to ensure that the quantized systems are finite-time stable and finite-time bounded with a prescribed H∞performance,upon which a set of controllers depending on the mode of quantization density are designed.In order to show the effectiveness of the designed H_∞controller,we apply the developed theoretical results to a numerical example.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11147009,11347026,and 11244005)the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2013AM012 and ZR2012AM004)the Natural Science Foundation of Liaocheng University,China
文摘By extending the usual Weyl transformation to the s-parameterized Weyl transformation with s being a real parameter,we obtain the s-parameterized quantization scheme which includes P–Q quantization, Q–P quantization, and Weyl ordering as its three special cases. Some operator identities can be derived directly by virtue of the s-parameterized quantization scheme.
