In this paper, a labelled transition semantics for higher-order processcalculi is studied. The labelled transition semantics is relatively clean and simple, andcorresponding bisimulation equivalence can be easily form...In this paper, a labelled transition semantics for higher-order processcalculi is studied. The labelled transition semantics is relatively clean and simple, andcorresponding bisimulation equivalence can be easily formulated based on it. And the congruenceproperties of the bisimulation equivalence can be proved easily. To show the correspondence betweenthe proposed semantics and the well-established ones, the bisimulation is characterized as a versionof barbed equivalence and a version of context bisimulation.展开更多
Modal logic characterization in a higher-order setting is usually not a trivial task because higher-order process-passing is quite different from first-order name-passing. We study the logical characterization of high...Modal logic characterization in a higher-order setting is usually not a trivial task because higher-order process-passing is quite different from first-order name-passing. We study the logical characterization of higherorder processes constrained by linearity. Linearity respects resource-sensitiveness and does not allow processes to duplicate themselves arbitrarily. We provide a modal logic that characterizes linear higher-order processes,particularly the bisimulation called local bisimulation over them. More importantly, the logic has modalities for higher-order actions downscaled to resembling first-order ones in Hennessy-Milner logic, based on a formulation exploiting the linearity of processes.展开更多
Invertibility is one of the desirable properties of moving average processes. This study derives consequences of the invertibility condition on the parameters of a moving average process of order three. The study also...Invertibility is one of the desirable properties of moving average processes. This study derives consequences of the invertibility condition on the parameters of a moving average process of order three. The study also establishes the intervals for the first three autocorrelation coefficients of the moving average process of order three for the purpose of distinguishing between the process and any other process (linear or nonlinear) with similar autocorrelation structure. For an invertible moving average process of order three, the intervals obtained are , -0.5ρ2ρ1<0.5.展开更多
Statistics of order 2 (variance, auto and cross-correlation functions, auto and cross-power spectra) and 3 (skewness, auto and cross-bicorrelation functions, auto and cross-bispectra) are used to analyze the wave-part...Statistics of order 2 (variance, auto and cross-correlation functions, auto and cross-power spectra) and 3 (skewness, auto and cross-bicorrelation functions, auto and cross-bispectra) are used to analyze the wave-particle interaction in space plasmas. The signals considered here are medium scale electron density irregularities and ELF/ULF electrostatic turbulence. Nonlinearities are mainly observed in the ELF range. They are independently pointed out in time series associated with fluctuations in electronic density and in time series associated with the measurement of one electric field component. Peaks in cross-bicorrelation function and in mutual information clearly show that, in well delimited frequency bands, the wave-particle interactions are nonlinear above a certain level of fluctuations. The way the energy is transferred within the frequencies of density fluctuations is indicated by a bi-spectra analysis.展开更多
This paper addresses the problem of adaptively estimating the consistent parameters for non Gaussian nonminimum MA processes with symmetric PDF using the fourth order cumulant of the underlying processes. The process...This paper addresses the problem of adaptively estimating the consistent parameters for non Gaussian nonminimum MA processes with symmetric PDF using the fourth order cumulant of the underlying processes. The processes may be corrupted by additive noise.展开更多
The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for th...The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for the estimation error and its variance, as well as their limiting large sample values are derived. Finally, a special case relevant to cubic phase coupling is considered, and these results are stated for this case, the variance is shown to comprise an ergodic and a nonergodic part.展开更多
Four phenoxysilicon networks for nonlinear optical (NLO) applications were designed and prepared by an extended sol-gel process without additional H2O and catalyst. All poled polymer network films possess high second-...Four phenoxysilicon networks for nonlinear optical (NLO) applications were designed and prepared by an extended sol-gel process without additional H2O and catalyst. All poled polymer network films possess high second-order nonlinear optical coefficients (d(33)) Of 10(-7)similar to 10(-8) esu. The investigation of NLO temporal stability at room temperature and elevated temperature (120 degreesC) indicated that these films exhibit high d(33) stability because the orientation of the chromophores are locked in the phenoxysilicon organic/inorganic networks.展开更多
Conventional PI control encounters some problems when dealing with large lag process in the presence of parameter uncertainties.For the typical first-order process,an observerbased linear active disturbance rejection ...Conventional PI control encounters some problems when dealing with large lag process in the presence of parameter uncertainties.For the typical first-order process,an observerbased linear active disturbance rejection control(LADRC) scheme is presented to cope with the difficulties,and a reduced-order observer scheme is proposed further.Some quantitative dynamic results with regard to non-overshoot characteristics are obtained.Finally,the performance boundaries of LADRC and PI control are explicitly compared with each other,which shows that the former is more superior in most cases.展开更多
This paper addresses a unified approach of the PID controller design for low as well as high order unstable processes with time delay.The design method is based on the direct synthesis(DS)approach to achieve the enhan...This paper addresses a unified approach of the PID controller design for low as well as high order unstable processes with time delay.The design method is based on the direct synthesis(DS)approach to achieve the enhanced load disturbance rejection.To improve the servo response,a two-degree of freedom control scheme has been considered.A suitable guideline has been provided to select the desired reference model in the DS scheme.The direct synthesis controller has been approximated to the PID controller using the frequency response matching method.A consistently better performance has been obtained in comparison with the recently reported methods.展开更多
针对传统方法子指标的不确定性问题,提出了一种基于三角直觉模糊网络分析法(triangular intuitionistic fuzzy analytic network process,TIFANP)、优劣解距离法(technique for order preference by similarity to an ideal solution,TO...针对传统方法子指标的不确定性问题,提出了一种基于三角直觉模糊网络分析法(triangular intuitionistic fuzzy analytic network process,TIFANP)、优劣解距离法(technique for order preference by similarity to an ideal solution,TOPSIS)和三角直觉模糊综合评价(triangular intuitionistic fuzzy comprehensive evaluation,TIFCE)的大跨桥梁安全状态评估方法。首先,建立大跨桥梁安全状态分层指标体系,采用TIFANP法确定考虑指标相互影响后的权重;其次,引入TOPSIS法分配指标各截面权重,进而获得子指标取值;然后,将桥梁安全状态划分为5级,通过TIFCE法构建指标在不同等级的隶属度和非隶属度函数,据此建立对应判断矩阵,并进行桥梁安全状态指数综合计算;最后,以某大跨悬索桥监测数据为依托,验证了所提方法的有效性。结果表明:所提方法能够更合理地处理评价指标间的相互影响以及专家评分时的不确定,可为运营期大跨桥梁安全状态的准确评估提供新思路。展开更多
It is important to identify and remove the wastes not only from manufacturing process, but also from nonmanufacturing process. In the last several decades, significant research achievements and practice benefits have ...It is important to identify and remove the wastes not only from manufacturing process, but also from nonmanufacturing process. In the last several decades, significant research achievements and practice benefits have been achieved about removing wastes from manufacturing process. Since the1990 s, some researchers and lean practitioners have paid more attention to removing waste from non-manufacturing process.Based on the authors' research work and industrial practice, the paper introduces a kind of lean approach for removing waste from non-manufacturing process. In its case study, the order handling process in a value chain is described with respect to a factory and its downstream distribution centers(DCs). The paper proposes a lean approach solution for creating the improved order handling process, and analyze how great improvements in performance can be achieved. As a result, the significant achievement has created a win-win scenario for both the nonmanufacturing process in a factory and non-manufacturing facilities(like DCs) across the value chain. It demonstrates that improvements have been made by removing waste from the non-manufacturing process that takes place within a factory as well as with external participants through the whole value chain. Likewise, the proposed lean approach has helped the case companies to achieve greater levels of efficiency and more benefits. Finally, some conclusions are drawn.展开更多
This paper addresses the problem of adaptive,consistent parameter estimation for a MA model from the 3rd order cumulant of the system output. The proposed adaptive algorithm is derived by using the new linear equation...This paper addresses the problem of adaptive,consistent parameter estimation for a MA model from the 3rd order cumulant of the system output. The proposed adaptive algorithm is derived by using the new linear equation system (J. K. Tugnait, 1990), which is proved to have unique solution,and hence guarantees the consistence of the MA parameters. Simulation results are provided to show the performance of the new algorithm.展开更多
Based on Duan's topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitl...Based on Duan's topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitly the fine topological structure of defects.The branch processes of defects in the vector order parameter system have also been investigated with this method.展开更多
To effectively implement order fulfillment, we present an integrated framework model focusing on the whole process of order fulfillment. Firstly, five aims of the OFS (order fulfillment system) are built. Then after...To effectively implement order fulfillment, we present an integrated framework model focusing on the whole process of order fulfillment. Firstly, five aims of the OFS (order fulfillment system) are built. Then after discussing three major processes of order fulfillment, we summarize functional and quality attributes of the OFS. Subsequently, we investigate SOA (Service Oriented Architecture) and present a SOA meta-model to be an integrated framework and to fulfill quality requirements. Moreover,based on the SOA meta-model, we construct a conceptual framework model that aims to conveniently integrate other functions fiom different systems into the order fulfillment system. This model offers enterprises a new approach to implementing order fulfillment.展开更多
On-chip interconnect buses consume tens of percents of dynamic power in a nanometer scale integrated circuit and they will consume more power with the rapid scaling down of technology size and continuously rising cloc...On-chip interconnect buses consume tens of percents of dynamic power in a nanometer scale integrated circuit and they will consume more power with the rapid scaling down of technology size and continuously rising clock frequency, therefore it is meaningful to lower the interconnecting bus power in design. In this paper, a simple yet accurate interconnect parasitic capacitance model is presented first and then, based on this model, a novel interconnecting bus optimization method is proposed. Wire spacing is a process for spacing wires for minimum dynamic power, while wire ordering is a process that searches for wire orders that maximally enhance it. The method, i.e., combining wire spacing with wire ordering, focuses on bus dynamic power optimization with a consideration of bus performance requirements. The optimization method is verified based on various nanometer technology parameters, showing that with 50% slack of routing space, 25.71% and 32.65% of power can be saved on average by the proposed optimization method for a global bus and an intermediate bus, respectively, under a 65-nm technology node, compared with 21.78% and 27.68% of power saved on average by uniform spacing technology. The proposed method is especially suitable for computer-aided design of nanometer scale on-chip buses.展开更多
Kinetics of homogeneous degradation of Eosin Y (EY), also known as Acid red 87 (CI 45380), are studied, mostly using Fenton’s process, at 30℃ by monitoring its absorbance at 517 nm (λ<sub>max</sub> of E...Kinetics of homogeneous degradation of Eosin Y (EY), also known as Acid red 87 (CI 45380), are studied, mostly using Fenton’s process, at 30℃ by monitoring its absorbance at 517 nm (λ<sub>max</sub> of EY). This process is one of the advanced oxidation processes (AOPs). Mixture of H<sub>2</sub>O<sub>2</sub> and Fe(II) ion in acetate buffer medium (pH 2.74 - 4.56) generates hydroxyl free radicals (?OH) which attack the dye molecules, resulting in degradation of the dye molecules. Results show that the initial rate of EY degradation decreases with the increasing of solution pH because of removal of kinetically important Fe (iron) species through formation of ferric hydroxide. On the other hand, the rate increases with increasing the concentrations of H<sub>2</sub>O<sub>2</sub>, Fe(II) and EY at low solution pH. The initial rate increases with increasing of concentration of H2O2 and, subsequently remains unaffected with further increase of its concentration at a constant Fe(II) concentration because of the enhanced scavenging environment created by H<sub>2</sub>O<sub>2</sub> at its higher concentration. The initial rate also increases with increasing of concentration of Fe(II) at a constant H<sub>2</sub>O<sub>2</sub> concentration and remains unaffected with its further increase. EY concentration also enhances the initial rate at low pH. However, the initial rate is significantly enhanced by UV light. This is because of formation of additional hydroxyl radicals through excitation of the dye molecules by UV light. During the period of experiment, EY in aqueous solution alone hardly suffered any degradation. Degradation mechanism of EY by the Fenton and photo-Fenton’s processes is also discussed. Statistical analysis was used to validate the experimental results. Low values of the standard deviation for both the initial rate and % degradation indicated the consistency of the experimental data.展开更多
1. Introduction In quantum optics, optical frequency conversion is a typical nonlinear process and is worth studying, for example, a second harmonic frequency generation will generate a squeezed state.[1'2l In this ...1. Introduction In quantum optics, optical frequency conversion is a typical nonlinear process and is worth studying, for example, a second harmonic frequency generation will generate a squeezed state.[1'2l In this work, we tackle the evolution of an initial coherent state in a Raman dispersion process which is also a nonlinear process. The process involves the inelastic scattering of a pho- ton when it is incident on a molecule. The photon loses some of its energy to the molecule or gains some from it, and so leaves the molecule with a lower or a higher frequency. The lower frequency components of the scattered radiation are called the Stokes lines and the higher frequency components are called the anti- Stokes lines. The Hamiltonian governing its dynamics is[3]展开更多
文摘In this paper, a labelled transition semantics for higher-order processcalculi is studied. The labelled transition semantics is relatively clean and simple, andcorresponding bisimulation equivalence can be easily formulated based on it. And the congruenceproperties of the bisimulation equivalence can be proved easily. To show the correspondence betweenthe proposed semantics and the well-established ones, the bisimulation is characterized as a versionof barbed equivalence and a version of context bisimulation.
基金the National Natural Science Foundation of China(Nos.61202023,61261130589 and61173048)the PACE Project(No.12IS02001)the Specialized Research Fund for the Doctoral Program of Higher Edueation of China(No.20120073120031)
文摘Modal logic characterization in a higher-order setting is usually not a trivial task because higher-order process-passing is quite different from first-order name-passing. We study the logical characterization of higherorder processes constrained by linearity. Linearity respects resource-sensitiveness and does not allow processes to duplicate themselves arbitrarily. We provide a modal logic that characterizes linear higher-order processes,particularly the bisimulation called local bisimulation over them. More importantly, the logic has modalities for higher-order actions downscaled to resembling first-order ones in Hennessy-Milner logic, based on a formulation exploiting the linearity of processes.
文摘Invertibility is one of the desirable properties of moving average processes. This study derives consequences of the invertibility condition on the parameters of a moving average process of order three. The study also establishes the intervals for the first three autocorrelation coefficients of the moving average process of order three for the purpose of distinguishing between the process and any other process (linear or nonlinear) with similar autocorrelation structure. For an invertible moving average process of order three, the intervals obtained are , -0.5ρ2ρ1<0.5.
文摘Statistics of order 2 (variance, auto and cross-correlation functions, auto and cross-power spectra) and 3 (skewness, auto and cross-bicorrelation functions, auto and cross-bispectra) are used to analyze the wave-particle interaction in space plasmas. The signals considered here are medium scale electron density irregularities and ELF/ULF electrostatic turbulence. Nonlinearities are mainly observed in the ELF range. They are independently pointed out in time series associated with fluctuations in electronic density and in time series associated with the measurement of one electric field component. Peaks in cross-bicorrelation function and in mutual information clearly show that, in well delimited frequency bands, the wave-particle interactions are nonlinear above a certain level of fluctuations. The way the energy is transferred within the frequencies of density fluctuations is indicated by a bi-spectra analysis.
文摘This paper addresses the problem of adaptively estimating the consistent parameters for non Gaussian nonminimum MA processes with symmetric PDF using the fourth order cumulant of the underlying processes. The processes may be corrupted by additive noise.
文摘The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for the estimation error and its variance, as well as their limiting large sample values are derived. Finally, a special case relevant to cubic phase coupling is considered, and these results are stated for this case, the variance is shown to comprise an ergodic and a nonergodic part.
文摘Four phenoxysilicon networks for nonlinear optical (NLO) applications were designed and prepared by an extended sol-gel process without additional H2O and catalyst. All poled polymer network films possess high second-order nonlinear optical coefficients (d(33)) Of 10(-7)similar to 10(-8) esu. The investigation of NLO temporal stability at room temperature and elevated temperature (120 degreesC) indicated that these films exhibit high d(33) stability because the orientation of the chromophores are locked in the phenoxysilicon organic/inorganic networks.
基金supported by the National Natural Science Foundation of China(60774088)the National High Technology Research and Development Program of China(863 Program)(2009AA04Z132)the Specialized Research Fund for the Doctoral Program of Higher Education of China(20090031110029)
文摘Conventional PI control encounters some problems when dealing with large lag process in the presence of parameter uncertainties.For the typical first-order process,an observerbased linear active disturbance rejection control(LADRC) scheme is presented to cope with the difficulties,and a reduced-order observer scheme is proposed further.Some quantitative dynamic results with regard to non-overshoot characteristics are obtained.Finally,the performance boundaries of LADRC and PI control are explicitly compared with each other,which shows that the former is more superior in most cases.
文摘This paper addresses a unified approach of the PID controller design for low as well as high order unstable processes with time delay.The design method is based on the direct synthesis(DS)approach to achieve the enhanced load disturbance rejection.To improve the servo response,a two-degree of freedom control scheme has been considered.A suitable guideline has been provided to select the desired reference model in the DS scheme.The direct synthesis controller has been approximated to the PID controller using the frequency response matching method.A consistently better performance has been obtained in comparison with the recently reported methods.
文摘针对传统方法子指标的不确定性问题,提出了一种基于三角直觉模糊网络分析法(triangular intuitionistic fuzzy analytic network process,TIFANP)、优劣解距离法(technique for order preference by similarity to an ideal solution,TOPSIS)和三角直觉模糊综合评价(triangular intuitionistic fuzzy comprehensive evaluation,TIFCE)的大跨桥梁安全状态评估方法。首先,建立大跨桥梁安全状态分层指标体系,采用TIFANP法确定考虑指标相互影响后的权重;其次,引入TOPSIS法分配指标各截面权重,进而获得子指标取值;然后,将桥梁安全状态划分为5级,通过TIFCE法构建指标在不同等级的隶属度和非隶属度函数,据此建立对应判断矩阵,并进行桥梁安全状态指数综合计算;最后,以某大跨悬索桥监测数据为依托,验证了所提方法的有效性。结果表明:所提方法能够更合理地处理评价指标间的相互影响以及专家评分时的不确定,可为运营期大跨桥梁安全状态的准确评估提供新思路。
基金supported in part by the National Natural Science Foundation of China (61773381,61773382,61533019,91520301)Finnish TEKES’s Project "SoMa2020:Social Manufacturing" (211560)+1 种基金Chinese Guangdong’s S&T Project (2015B010103001,2016B090910001,2017B090912001)Dongguan’s Innovation Talents Project (Gang Xiong)
文摘It is important to identify and remove the wastes not only from manufacturing process, but also from nonmanufacturing process. In the last several decades, significant research achievements and practice benefits have been achieved about removing wastes from manufacturing process. Since the1990 s, some researchers and lean practitioners have paid more attention to removing waste from non-manufacturing process.Based on the authors' research work and industrial practice, the paper introduces a kind of lean approach for removing waste from non-manufacturing process. In its case study, the order handling process in a value chain is described with respect to a factory and its downstream distribution centers(DCs). The paper proposes a lean approach solution for creating the improved order handling process, and analyze how great improvements in performance can be achieved. As a result, the significant achievement has created a win-win scenario for both the nonmanufacturing process in a factory and non-manufacturing facilities(like DCs) across the value chain. It demonstrates that improvements have been made by removing waste from the non-manufacturing process that takes place within a factory as well as with external participants through the whole value chain. Likewise, the proposed lean approach has helped the case companies to achieve greater levels of efficiency and more benefits. Finally, some conclusions are drawn.
文摘This paper addresses the problem of adaptive,consistent parameter estimation for a MA model from the 3rd order cumulant of the system output. The proposed adaptive algorithm is derived by using the new linear equation system (J. K. Tugnait, 1990), which is proved to have unique solution,and hence guarantees the consistence of the MA parameters. Simulation results are provided to show the performance of the new algorithm.
基金Project supported by the National Natural Science Foundation of China (Grant No 10275030)Cuiying Project of Lanzhou University of China (Grant No 225000-582404)
文摘Based on Duan's topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitly the fine topological structure of defects.The branch processes of defects in the vector order parameter system have also been investigated with this method.
文摘To effectively implement order fulfillment, we present an integrated framework model focusing on the whole process of order fulfillment. Firstly, five aims of the OFS (order fulfillment system) are built. Then after discussing three major processes of order fulfillment, we summarize functional and quality attributes of the OFS. Subsequently, we investigate SOA (Service Oriented Architecture) and present a SOA meta-model to be an integrated framework and to fulfill quality requirements. Moreover,based on the SOA meta-model, we construct a conceptual framework model that aims to conveniently integrate other functions fiom different systems into the order fulfillment system. This model offers enterprises a new approach to implementing order fulfillment.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 60725415, 60971066, and 61006028)the National High-Tech Program of China (Grant Nos. 2009AA01Z258 and 2009AA01Z260)the National Key Lab Foundation,China (Grant No. ZHD200904)
文摘On-chip interconnect buses consume tens of percents of dynamic power in a nanometer scale integrated circuit and they will consume more power with the rapid scaling down of technology size and continuously rising clock frequency, therefore it is meaningful to lower the interconnecting bus power in design. In this paper, a simple yet accurate interconnect parasitic capacitance model is presented first and then, based on this model, a novel interconnecting bus optimization method is proposed. Wire spacing is a process for spacing wires for minimum dynamic power, while wire ordering is a process that searches for wire orders that maximally enhance it. The method, i.e., combining wire spacing with wire ordering, focuses on bus dynamic power optimization with a consideration of bus performance requirements. The optimization method is verified based on various nanometer technology parameters, showing that with 50% slack of routing space, 25.71% and 32.65% of power can be saved on average by the proposed optimization method for a global bus and an intermediate bus, respectively, under a 65-nm technology node, compared with 21.78% and 27.68% of power saved on average by uniform spacing technology. The proposed method is especially suitable for computer-aided design of nanometer scale on-chip buses.
文摘Kinetics of homogeneous degradation of Eosin Y (EY), also known as Acid red 87 (CI 45380), are studied, mostly using Fenton’s process, at 30℃ by monitoring its absorbance at 517 nm (λ<sub>max</sub> of EY). This process is one of the advanced oxidation processes (AOPs). Mixture of H<sub>2</sub>O<sub>2</sub> and Fe(II) ion in acetate buffer medium (pH 2.74 - 4.56) generates hydroxyl free radicals (?OH) which attack the dye molecules, resulting in degradation of the dye molecules. Results show that the initial rate of EY degradation decreases with the increasing of solution pH because of removal of kinetically important Fe (iron) species through formation of ferric hydroxide. On the other hand, the rate increases with increasing the concentrations of H<sub>2</sub>O<sub>2</sub>, Fe(II) and EY at low solution pH. The initial rate increases with increasing of concentration of H2O2 and, subsequently remains unaffected with further increase of its concentration at a constant Fe(II) concentration because of the enhanced scavenging environment created by H<sub>2</sub>O<sub>2</sub> at its higher concentration. The initial rate also increases with increasing of concentration of Fe(II) at a constant H<sub>2</sub>O<sub>2</sub> concentration and remains unaffected with its further increase. EY concentration also enhances the initial rate at low pH. However, the initial rate is significantly enhanced by UV light. This is because of formation of additional hydroxyl radicals through excitation of the dye molecules by UV light. During the period of experiment, EY in aqueous solution alone hardly suffered any degradation. Degradation mechanism of EY by the Fenton and photo-Fenton’s processes is also discussed. Statistical analysis was used to validate the experimental results. Low values of the standard deviation for both the initial rate and % degradation indicated the consistency of the experimental data.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.10775097 and 10475056)
文摘1. Introduction In quantum optics, optical frequency conversion is a typical nonlinear process and is worth studying, for example, a second harmonic frequency generation will generate a squeezed state.[1'2l In this work, we tackle the evolution of an initial coherent state in a Raman dispersion process which is also a nonlinear process. The process involves the inelastic scattering of a pho- ton when it is incident on a molecule. The photon loses some of its energy to the molecule or gains some from it, and so leaves the molecule with a lower or a higher frequency. The lower frequency components of the scattered radiation are called the Stokes lines and the higher frequency components are called the anti- Stokes lines. The Hamiltonian governing its dynamics is[3]