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Multi-Time Scale Operation and Simulation Strategy of the Park Based on Model Predictive Control
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作者 Jun Zhao Chaoying Yang +1 位作者 Ran Li Jinge Song 《Energy Engineering》 EI 2024年第3期747-767,共21页
Due to the impact of source-load prediction power errors and uncertainties,the actual operation of the park will have a wide range of fluctuations compared with the expected state,resulting in its inability to achieve... Due to the impact of source-load prediction power errors and uncertainties,the actual operation of the park will have a wide range of fluctuations compared with the expected state,resulting in its inability to achieve the expected economy.This paper constructs an operating simulation model of the park power grid operation considering demand response and proposes a multi-time scale operating simulation method that combines day-ahead optimization and model predictive control(MPC).In the day-ahead stage,an operating simulation plan that comprehensively considers the user’s side comfort and operating costs is proposed with a long-term time scale of 15 min.In order to cope with power fluctuations of photovoltaic,wind turbine and conventional load,MPC is used to track and roll correct the day-ahead operating simulation plan in the intra-day stage to meet the actual operating operation status of the park.Finally,the validity and economy of the operating simulation strategy are verified through the analysis of arithmetic examples. 展开更多
关键词 Demand response model predictive control multiple time scales operating simulation
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Establishment of an astronomical time scale for the Shizigou Formation in the Qaidam Basin,Inner Asia and orbital forced evolution of lakes during The Pliocene
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作者 DunZhu JiaoBa MingZhen Zhang +6 位作者 GuoLong Liu JianGuo Hui ShaoHua Lin AiJing Li Jing Zhang Jing Peng YiQiao Fu 《Research in Cold and Arid Regions》 CSCD 2023年第5期239-252,共14页
The Qaidam Basin,as the largest inland basin within the Tibetan Plateau,has accumulated more than 10,000 m of Cenozoic continental sediments.It serves as a crucial research area for documenting Cenozoic climate change... The Qaidam Basin,as the largest inland basin within the Tibetan Plateau,has accumulated more than 10,000 m of Cenozoic continental sediments.It serves as a crucial research area for documenting Cenozoic climate changes and plateau uplift processes in the Asian interior.Additionally,the basin holds vast reserves of oil and gas resources,making high-resolution drilling data invaluable for studying paleoclimate.In this study,the longsequence lacustrine deposits of JS1 drill core across the Shizigou Formation in the Yiliping Depression at the western center of the basin were studied,aiming to establish an astronomical timescale for the Shizigou Formation and investigate the characteristics of paleoclimatic changes during the late Miocene to the Pliocene for the Asian interior.The analysis was carried out using high-resolution natural gamma ray(GR)data sequences,employing techniques such as spectral analysis,filtering,and wavelet analysis in cyclostratigraphy.The results indicated the presence of a stable Milankovitch orbital signal was perfectly recorded in the Shizigou Formation,primarily influenced by eccentricity cycles,with weaker obliquity and precession cycles.Using the stable and continuous 405 ka eccentricity cycle in astronomical tuning,a"floating"astronomical timescale with a duration of 6.1 Ma for the Yiliping depression's Shizigou Formation has been established.With reference to previously established stratigraphic age anchor points,an absolute astronomical timescale(2.5–8.6 Ma)has been ultimately provided for the Shizigou Formation.Simultaneously,a clear 100 ka short eccentricity cycle record has been identified during the Pliocene(5.3–2.5 Ma),which corresponds in time with the aridification within the basin during this Pliocene period.In addition,a comparison of the Pliocene natural gamma ray curve of the Qaidam Basin with global ice volume variations indicated that the basin's aridification was influenced by global cooling,with eccentricity-modulated precession cycles controlling solar radiation and subsequently affecting the evolution of lakes in the arid region of Inner Asia. 展开更多
关键词 Qaidam basin Shizigou Formation Astronomical time scale Astronomical tuning~100 ka orbital eccentricity
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Two-phase micro-and macro-time scales in particle-laden turbulent channel flows
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作者 Bing Wang Michael Manhart 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2012年第3期595-604,共10页
The micro- and macro-time scales in two-phase turbulent channel flows are investigated using the direct nu- merical simulation and the Lagrangian particle trajectory methods for the fluid- and the particle-phases, res... The micro- and macro-time scales in two-phase turbulent channel flows are investigated using the direct nu- merical simulation and the Lagrangian particle trajectory methods for the fluid- and the particle-phases, respectively. Lagrangian and Eulerian time scales of both phases are cal- culated using velocity correlation functions. Due to flow anisotropy, micro-time scales are not the same with the theo- retical estimations in large Reynolds number (isotropic) tur- bulence. Lagrangian macro-time scales of particle-phase and of fluid-phase seen by particles are both dependent on particle Stokes number. The fluid-phase Lagrangian inte- gral time scales increase with distance from the wall, longer than those time scales seen by particles. The Eulerian inte- gral macro-time scales increase in near-wall regions but de- crease in out-layer regions. The moving Eulerian time scales are also investigated and compared with Lagrangian integral time scales, and in good agreement with previous measure- ments and numerical predictions. For the fluid particles the micro Eulerian time scales are longer than the Lagrangian ones in the near wall regions, while away from the walls the micro Lagrangian time scales are longer. The Lagrangian integral time scales are longer than the Eulerian ones. The results are useful for further understanding two-phase flow physics and especially for constructing accurate prediction models of inertial particle dispersion. 展开更多
关键词 Micro-time scale Lagrangian integral timescale ~ Moving Eulerian time scale Particle-laden turbulentflow ~ Particle Stokes number Direct numerical simulation(DNS) Lagrangian trajectory method
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Lagrangian time scales and its relationship to Eulerian equivalents in turbulent channel flow
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作者 罗剑平 卢志明 刘宇陆 《Journal of Shanghai University(English Edition)》 2010年第1期71-75,共5页
Lagrangian and Eulerian time scales were obtained from the direct numerical simulation of turbulent channel flow at two Reynolds numbers based on the friction velocity and channel half-height, Rer= 80, 100. The Lagran... Lagrangian and Eulerian time scales were obtained from the direct numerical simulation of turbulent channel flow at two Reynolds numbers based on the friction velocity and channel half-height, Rer= 80, 100. The Lagrangian integral time scales and time microscales were compared to their Eulerian equivalents. It is found that the ratio of Lagrangian to TL Eulerian integral time scales is given by TE/TiE= 1 + 0.1y+ for y+ ≤ 10, and that the ratios between the Lagrangian to theEulerian time microscales are almost the same irrespective of the components. Those increase with y+ are approximated by ≈ 2.75 - 1.75 exp (-v+/a) . These results also show that these expressions are independent of the Reynolds number. 展开更多
关键词 turbulent channel flow Lagrangian time scale Eulerian time scale direct numerical simulation (DNS)
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To Improve the Accuracy of Laser Pulse Range Finding by Time Scale-Up
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作者 殷聪 韩绍坤 +2 位作者 刘巽亮 张化朋 赵跃进 《Journal of Beijing Institute of Technology》 EI CAS 2002年第3期259-262,共4页
A method of improving the accuracy of laser pulse range finding from ±10 m to ±1 m inexpensively by means of time scale up is described. Time scale up can stretch the entire flight time by a factor of 1 00... A method of improving the accuracy of laser pulse range finding from ±10 m to ±1 m inexpensively by means of time scale up is described. Time scale up can stretch the entire flight time by a factor of 1 000 and then the stretched result is counted to calculate the distance. The use of this technique decreases the resolution of counting from nanosecond to microsecond, therefore a separate counting oscillator followed by an interpolation operation is unnecessary. This technique can improve the accuracy of laser pulse range finding inexpensively and effectively. 展开更多
关键词 time scale up range finding CAPACITOR charging and discharging
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Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales 被引量:11
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作者 宋静 张毅 《Chinese Physics B》 SCIE EI CAS CSCD 2017年第8期201-209,共9页
This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilt... This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilton prin- ciple based on the action with non-standard Lagrangians on time scales is established, with which the corresponding Euler-Lagrange equation is given. Secondly, according to the invariance of the Hamilton action under the infinitesimal transformation, the Noether theorem for the dynamical system with non-standard Lagrangians on time scales is established. The proof of the theorem consists of two steps. First, it is proved under the infinitesimal transformations of a special one-parameter group without transforming time. Second, utilizing the technique of time-re-parameterization, the Noether theorem in a general form is obtained. The Noether-type conserved quantities with non-standard Lagrangians in both clas- sical and discrete cases are given. Finally, an example in Friedmann-Robertson-Walker spacetime and an example about second order Duffing equation are given to illustrate the application of the results. 展开更多
关键词 time scale non-standard Lagrangian Noether symmetry conserved quantity
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Multiple Time Scale Analysis of River Runoff Using Wavelet Transform for Dagujia River Basin, Yantai, China 被引量:10
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作者 LIU Delin LILT Xianzhao +2 位作者 LI Bicheng ZHAO Shiwei LI Xiguo 《Chinese Geographical Science》 SCIE CSCD 2009年第2期158-167,共10页
Based on monthly river runoff and meteorological data, a method of Morlet wavelet transform was used to analyze the multiple time scale characteristics of river runoff in the Dagujia River Basin, Yantai City, Shandong... Based on monthly river runoff and meteorological data, a method of Morlet wavelet transform was used to analyze the multiple time scale characteristics of river runoff in the Dagujia River Basin, Yantai City, Shandong Province. The results showed that the total annual river runoffin the Dagujia River Basin decreased significantly from 1966 to 2004, and the rate of decrease was 48× 10^6ma/10yr, which was higher than the mean value of most rivers in China. Multiple time scale characteristics existed, which accounted for different aspects of the changes in annual river runoff, and the major periods of the runofftime series were identified as about 28 years, 14 years and 4 years with decreasing levels of fluctuation. The river runoff evolution process was controlled by changes in precipitation to a certain extent, but it was also greatly influenced by human activities. Also, for different time periods and scales, the impacts of climate changes and human activities on annual river runoff evolution occurred at the same time. Changes in the annual river runoffwere mainly associated with climate change before the 1980s and with human activities after 1981. 展开更多
关键词 multiple time scale river runoff climate change Morlet wavelet transform Dagujia River
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OSCILLATION FOR NONAUTONOMOUS NEUTRAL DYNAMIC DELAY EQUATIONS ON TIME SCALES 被引量:6
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作者 刘爱莲 吴洪武 +1 位作者 朱思铭 Ronald M.Mathsen 《Acta Mathematica Scientia》 SCIE CSCD 2006年第1期99-106,共8页
The article is concerned with oscillation of nonautonomous neutral dynamic delay equations on time scales. Sufficient conditions are established for the existence of bounded positive solutions and for oscillation of a... The article is concerned with oscillation of nonautonomous neutral dynamic delay equations on time scales. Sufficient conditions are established for the existence of bounded positive solutions and for oscillation of all solutions of this equation. Some results extend known results for difference equations when the time scale is the set Z^+ of positive integers and for differential equations when the time scale is the set IR of real numbers. 展开更多
关键词 time scale Δ-derivative neutral delay equation OSCILLATION
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Oscillation of Second-order Nonlinear Dynamic Equation on Time Scales 被引量:7
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作者 YANG Jia-shan 《Chinese Quarterly Journal of Mathematics》 CSCD 2013年第2期172-179,共8页
: The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati tech... : The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many knownresults for second order dynamic equations. Some examples are given to illustrate the main results of this article. 展开更多
关键词 time scales dynamic equations nonlinear neutral term damping term variable delay Riccati transformation OSCILLATION
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Generalized Chaplygin equations for nonholonomic systems on time scales 被引量:7
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作者 金世欣 张毅 《Chinese Physics B》 SCIE EI CAS CSCD 2018年第2期267-272,共6页
The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamil- ton principle for nonholonomic systems on time scales is established, and the corresponding generalized ... The generalized Chaplygin equations for nonholonomic systems on time scales are proposed and studied. The Hamil- ton principle for nonholonomic systems on time scales is established, and the corresponding generalized Chaplygin equa- tions are deduced. The reduced Chaplygin equations are also presented. Two special cases of the generalized Chaplygin equations on time scales, where the time scales are equal to the set of real numbers and the integer set, are discussed. Finally, several examples are given to illustrate the application of the results. 展开更多
关键词 nonholonomic system generalized Chaplygin equations time scales
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Astronomical time scale of the Turonian constrained by multiple paleoclimate proxies 被引量:6
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作者 Chao Ma Mingsong Li 《Geoscience Frontiers》 SCIE CAS CSCD 2020年第4期1345-1352,共8页
One of the clocks that record the Earth history is(quasi-) periodic astronomical cycles.These cycles influence the climate that can be ultimately stored in sedimentary rocks.By cracking these(quasi-) periodic sediment... One of the clocks that record the Earth history is(quasi-) periodic astronomical cycles.These cycles influence the climate that can be ultimately stored in sedimentary rocks.By cracking these(quasi-) periodic sedimentation signals,high resolution astronomical time scale(ATS) can be obtained.Paleoclimate proxies are widely used to extract astronomical cycles.However different proxies may respond differently to astronomical signals and nonastronomical noises including tectonics,diagenesis,and measurement error among others.Astronomical time scale constructed based on a single proxy where its signal-to-noise ratio is low may have uncertainty that is difficult to evaluate but can be revealed by utilizing other proxies.Here,we test eight astronomical age models using two astrochro no logical methods from four paleoclimate proxies(i.e.,color reflection L~* and b~*,natural gamma radiation,and bulk density) from the Turonian to the Coniacian of the Cretaceous Period at the Demerara Rise in the equatorial Atlantic.The two astrochronological methods are time calibration using long eccentricity bandpass filtering(E1 bandpass) and tracking the long eccentricity from evolutive harmonic analysis(tracking EHA).The statistical mean and standard deviation of four age models from the four proxies are calculated to construct one integrated age model with age uncertainty in each method.Results demonstrate that extracting astronomical signals from multiple paleoclimate proxies is a valid method to estimate age model uncertainties.Anchored at the Cenomanian/Turonian boundary with an age of 93.9 ± 0.15 Ma from biostratigraphy,the ages for CC11/CC12(calcareous nannofossil zones),Turonian/Coniacian(CC12/CC13),CC13/CC14,and Coniacian/Santonian boundaries are 91.25±0.20 Ma,89.87±0.20 Ma,86.36±0.33 Ma,and 86.03±0.32 Ma in E1 bandpass method,compared with 91.17±0.36 Ma,89.74±0.38 Ma,86.13±1.31 Ma,and 85.80±1.33 Ma respectively in tracking EHA method.These results are consistent with previous studies within error and provide a reliable estimation of uncertainties of the ages. 展开更多
关键词 Astrochronology Astronomical time scale Uncertainty CRETACEOUS TURONIAN Demerara rise
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Oscillation of Third-order Delay Dynamic Equations on Time Scales 被引量:6
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作者 YANG Jia-shan 《Chinese Quarterly Journal of Mathematics》 CSCD 2014年第3期447-456,共10页
This paper is concerned with the oscillatory behavior of a class of third-order noonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequal... This paper is concerned with the oscillatory behavior of a class of third-order noonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequality technique, we establish some new oscilla- tion criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of third-order nonlinear variable delay functional differential equations and functional difference equations with a nonlinear neutral term. Some examples are given to illustrate the importance of our results. 展开更多
关键词 OSCILLATION time scales delay dynamic equations l^iccati transformation non-linear neutral term
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Methods of reduction for Lagrange systems on time scales with nabla derivatives 被引量:4
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作者 Shi-Xin Jin Yi Zhang 《Chinese Physics B》 SCIE EI CAS CSCD 2017年第1期243-249,共7页
The Routh and Whittaker methods of reduction for Lagrange system on time scales with nabla derivatives are studied.The equations of motion for Lagrange system on time scales are established, and their cyclic integrals... The Routh and Whittaker methods of reduction for Lagrange system on time scales with nabla derivatives are studied.The equations of motion for Lagrange system on time scales are established, and their cyclic integrals and generalized energy integrals are given. The Routh functions and Whittaker functions of Lagrange system are constructed, and the order of differential equations of motion for the system are reduced by using the cyclic integrals or the generalized energy integrals with nabla derivatives. The results show that the reduced Routh equations and Whittaker equations hold the form of Lagrnage equations with nabla derivatives. Finally, two examples are given to illustrate the application of the results. 展开更多
关键词 reduction of dynamical system cyclic integral energy integral time scales
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Mei’s symmetry theorem for time scales nonshifted mechanical systems 被引量:7
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作者 Yi Zhang 《Theoretical & Applied Mechanics Letters》 CSCD 2021年第5期246-251,共6页
We focus on Mei symmetry for time scales nonshifted mechanical systems within Lagrangian framework and its resulting new conserved quantities.Firstly,the dynamic equations of time scales nonshifted holonomic systems a... We focus on Mei symmetry for time scales nonshifted mechanical systems within Lagrangian framework and its resulting new conserved quantities.Firstly,the dynamic equations of time scales nonshifted holonomic systems and time scales nonshifted nonholonomic systems are derived from the generalized Hamilton’s principle.Secondly,the definitions of Mei symmetry on time scales are given and its criterions are deduced.Finally,Mei’s symmetry theorems for time scales nonshifted holonomic conservative systems,time scales nonshifted holonomic nonconservative systems and time scales nonshifted nonholonomic systems are established and proved,and new conserved quantities of above systems are obtained.Results are illustrated with two examples. 展开更多
关键词 Mei’s symmetry theorem Nonholonomic system Nonconservative system time scales Nonshifted calculus of variation
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Southern Ocean SST Variability and Its Relationship with ENSO on Inter-Decadal Time Scales 被引量:4
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作者 YAN Li DU Yan ZHANG Lan 《Journal of Ocean University of China》 SCIE CAS 2013年第2期287-294,共8页
Empirical orthogonal function (EOF) analysis reveals a co-variability of Sea surface temperatures (SSTs) in the Southern Hemisphere (0°-60°S). In the South Indian and Atlantic Oceans, there is a subtro... Empirical orthogonal function (EOF) analysis reveals a co-variability of Sea surface temperatures (SSTs) in the Southern Hemisphere (0°-60°S). In the South Indian and Atlantic Oceans, there is a subtropical dipole pattern slanted in the southwest-northeast direction. In the South Pacific Ocean, a meridional tripole structure emerges, whose middle pole co-varies with the dipoles in the South Indian and Atlantic Oceans and is used in this study to track subtropical Pacific variability. The South Indian and Atlantic Ocean dipoles and the subtropical Pacific variability are phase-locked in austral summer. On the inter-decadal time scales, the dipoles in the South Indian and Atlantic Oceans weaken in amplitude after 1979/1980. No such weakening is found in the subtropical South Pacific Ocean. Interestingly, despite the reduced amplitude, the correlation of the Indian Ocean and Atlantic dipoles with E1 Nino and Southern Oscillation (ENSO) are enhanced after 1979/1980. The same increase in correlation is found for subtropical South Pacific variability after 1979/1980. These inter-decadal modulations imply that the Southern Hemisphere participates in part of the climate shift in the late 1970s. The correlation between Southern Hemisphere SST and ENSO reduces after 2000. 展开更多
关键词 Southern Ocean SST ENSO subtropical dipole inter-decadal time scales
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Bursting oscillations in a hydro-turbine governing system with two time scales 被引量:3
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作者 韩青爽 陈帝伊 张浩 《Chinese Physics B》 SCIE EI CAS CSCD 2017年第12期609-616,共8页
A fast-slow coupled model of the hydro-turbine governing system (HTGS) is established by introducing frequency disturbance in this paper. Based on the proposed model, the performances of two time scales for bursting... A fast-slow coupled model of the hydro-turbine governing system (HTGS) is established by introducing frequency disturbance in this paper. Based on the proposed model, the performances of two time scales for bursting oscillations in the HTGS are investigated and the effect of periodic excitation of frequency disturbance is analyzed by using the bifurcation diagrams, time waveforms and phase portraits. We find that stability and operational characteristics of the HTGS change with the value of system parameter kd. Furthermore, the comparative analyses for the effect of the bursting oscillations on the system with different amplitudes of the periodic excitation a are carried out. Meanwhile, we obtain that the relative deviation of the mechanical torque mt rises with the increase of a. These methods and results of the study, combined with the performance of two time scales and the fast-slow coupled engineering model, provide some theoretical bases for investigating interesting physical phenomena of the engineering system. 展开更多
关键词 bursting oscillations physical phenomena hydro-turbine governing system two time scales
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Mei Symmetry for Constrained Mechanical System on Time Scales 被引量:3
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作者 SONG Chuanjing 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI CSCD 2021年第2期353-360,共8页
Mei symmetry on time scales is investigated for Lagrangian system,Hamiltonian system,and Birkhoffian system.The main results are divided into three sections.In each section,the definition and the criterion of Mei symm... Mei symmetry on time scales is investigated for Lagrangian system,Hamiltonian system,and Birkhoffian system.The main results are divided into three sections.In each section,the definition and the criterion of Mei symmetry are first presented.Then the conserved quantity deduced from Mei symmetry is obtained,and perturbation to Mei symmetry and adiabatic invariant are studied.Finally,an example is given to illustrate the methods and results in each section.The conserve quantity achieved here is a special case of adiabatic invariant.And the results obtained in this paper are more general because of the definition and property of time scale. 展开更多
关键词 Mei symmetry time scale Lagrangian system Hamiltonian system Birkhoffian system
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Small-time scale network traffic prediction based on a local support vector machine regression model 被引量:10
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作者 孟庆芳 陈月辉 彭玉华 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第6期2194-2199,共6页
In this paper we apply the nonlinear time series analysis method to small-time scale traffic measurement data. The prediction-based method is used to determine the embedding dimension of the traffic data. Based on the... In this paper we apply the nonlinear time series analysis method to small-time scale traffic measurement data. The prediction-based method is used to determine the embedding dimension of the traffic data. Based on the reconstructed phase space, the local support vector machine prediction method is used to predict the traffic measurement data, and the BIC-based neighbouring point selection method is used to choose the number of the nearest neighbouring points for the local support vector machine regression model. The experimental results show that the local support vector machine prediction method whose neighbouring points are optimized can effectively predict the small-time scale traffic measurement data and can reproduce the statistical features of real traffic measurements. 展开更多
关键词 network traffic small-time scale nonlinear time series analysis support vector machine regression model
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On the McShane Integral on Time Scales 被引量:3
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作者 刘德华 赵大方 《Chinese Quarterly Journal of Mathematics》 CSCD 2012年第4期556-561,共6页
In this paper,we introduce and study the concept of McShane delta integral on time scales.We discuss the relationship between McShane delta integral and Henstock delta integral,and prove a dominated convergence theore... In this paper,we introduce and study the concept of McShane delta integral on time scales.We discuss the relationship between McShane delta integral and Henstock delta integral,and prove a dominated convergence theorems for the McShane delta integral. 展开更多
关键词 time scales McShane integral delta integral
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Existence of Positive Solutions for Higher Order Boundary Value Problem on Time Scales 被引量:2
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作者 Xie Da-peng Liu Yang +1 位作者 Sun Ming-zhe Li Yong 《Communications in Mathematical Research》 CSCD 2013年第1期1-13,共13页
In this paper, we investigate the existence of positive solutions of a class higher order boundary value problems on time scales. The class of boundary value problems educes a four-point (or three-point or two-point... In this paper, we investigate the existence of positive solutions of a class higher order boundary value problems on time scales. The class of boundary value problems educes a four-point (or three-point or two-point) boundary value problems, for which some similar results are established. Our approach relies on the Krasnosel'skii fixed point theorem. The result of this paper is new and extends previously known results. 展开更多
关键词 higher order boundary value problem positive solution SEMIPOSITONE on time scale fixed point
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