Hirota equation and Bethe ansatz

物理化学概念及术语A B C D E F G H I J K L M N O P Q R S T U V W X Y Z概念及术语 (16)BET公式BET formula (16)DLVO理论 DLVO theory (16)HLB法hydrophile—lipophile balance method (16)pVT性质 pVT property (16)ζ电势 zeta potential (16)阿伏加德罗常数 Avogadro’number (16)阿伏加德罗定律 Avogadro law (16)阿累尼乌斯电离理论Arrhenius ionization theory (16)阿累尼乌斯方程Arrhenius equation (17)阿累尼乌斯活化能 Arrhenius activation energy (17)阿马格定律 Amagat law (17)艾林方程 Erying equation (17)爱因斯坦光化当量定律 Einstein’s law of photochemical equivalence (17)爱因斯坦－斯托克斯方程 Einstein-Stokes equation (17)安托万常数 Antoine constant (17)安托万方程 Antoine equation (17)盎萨格电导理论Onsager’s theory of conductance (17)半电池half cell (17)半衰期half time period (18)饱和液体 saturated liquids (18)饱和蒸气 saturated vapor (18)饱和吸附量 saturated extent of adsorption (18)饱和蒸气压 saturated vapor pressure (18)爆炸界限 explosion limits (18)比表面功 specific surface work (18)比表面吉布斯函数 specific surface Gibbs function (18)比浓粘度 reduced viscosity (18)标准电动势 standard electromotive force (18)标准电极电势 standard electrode potential (18)标准摩尔反应焓 standard molar reaction enthalpy (18)标准摩尔反应吉布斯函数 standard Gibbs function of molar reaction (18)标准摩尔反应熵 standard molar reaction entropy (19)标准摩尔焓函数 standard molar enthalpy function (19)标准摩尔吉布斯自由能函数 standard molar Gibbs free energy function (19)标准摩尔燃烧焓 standard molar combustion enthalpy (19)标准摩尔熵 standard molar entropy (19)标准摩尔生成焓 standard molar formation enthalpy (19)标准摩尔生成吉布斯函数 standard molar formation Gibbs function (19)标准平衡常数 standard equilibrium constant (19)标准氢电极 standard hydrogen electrode (19)标准态 standard state (19)标准熵 standard entropy (20)标准压力 standard pressure (20)标准状况 standard condition (20)表观活化能apparent activation energy (20)表观摩尔质量 apparent molecular weight (20)表面活性剂surfactants (21)表面吸附量 surface excess (21)表面张力 surface tension (21)表面质量作用定律 surface mass action law (21)波义尔定律 Boyle law (21)波义尔温度 Boyle temperature (21)波义尔点 Boyle point (21)玻尔兹曼常数 Boltzmann constant (22)玻尔兹曼分布 Boltzmann distribution (22)玻尔兹曼公式 Boltzmann formula (22)玻尔兹曼熵定理 Boltzmann entropy theorem (22)泊Poise (22)不可逆过程 irreversible process (22)不可逆过程热力学thermodynamics of irreversible processes (22)不可逆相变化 irreversible phase change (22)布朗运动 brownian movement (22)查理定律 Charle's law (22)产率 yield (23)敞开系统 open system (23)超电势 over potential (23)沉降 sedimentation (23)沉降电势 sedimentation potential (23)沉降平衡 sedimentation equilibrium (23)触变 thixotropy (23)粗分散系统 thick disperse system (23)催化剂 catalyst (23)单分子层吸附理论 mono molecule layer adsorption (23)单分子反应 unimolecular reaction (23)单链反应 straight chain reactions (24)弹式量热计 bomb calorimeter (24)道尔顿定律 Dalton law (24)道尔顿分压定律 Dalton partial pressure law (24)德拜和法尔肯哈根效应Debye and Falkenhagen effect (24)德拜立方公式 Debye cubic formula (24)德拜－休克尔极限公式 Debye-Huckel’s limiting equation (24)等焓过程 isenthalpic process (24)等焓线isenthalpic line (24)等几率定理 theorem of equal probability (24)等温等容位Helmholtz free energy (25)等温等压位Gibbs free energy (25)等温方程 equation at constant temperature (25)低共熔点 eutectic point (25)低共熔混合物 eutectic mixture (25)低会溶点 lower consolute point (25)低熔冰盐合晶 cryohydric (26)第二类永动机 perpetual machine of the second kind (26)第三定律熵 Third—Law entropy (26)第一类永动机 perpetual machine of the first kind (26)缔合化学吸附 association chemical adsorption (26)电池常数 cell constant (26)电池电动势 electromotive force of cells (26)电池反应 cell reaction (27)电导 conductance (27)电导率 conductivity (27)电动势的温度系数 temperature coefficient of electromotive force (27)电化学极化 electrochemical polarization (27)电极电势 electrode potential (27)电极反应 reactions on the electrode (27)电极种类 type of electrodes (27)电解池 electrolytic cell (28)电量计 coulometer (28)电流效率current efficiency (28)电迁移 electro migration (28)电迁移率 electromobility (28)电渗 electroosmosis (28)电渗析 electrodialysis (28)电泳 electrophoresis (28)丁达尔效应 Dyndall effect (28)定容摩尔热容 molar heat capacity under constant volume (28)定容温度计 Constant voIume thermometer (28)定压摩尔热容 molar heat capacity under constant pressure (29)定压温度计 constant pressure thermometer (29)定域子系统 localized particle system (29)动力学方程kinetic equations (29)动力学控制 kinetics control (29)独立子系统 independent particle system (29)对比摩尔体积 reduced mole volume (29)对比体积 reduced volume (29)对比温度 reduced temperature (29)对比压力 reduced pressure (29)对称数 symmetry number (29)对行反应reversible reactions (29)对应状态原理 principle of corresponding state (29)多方过程polytropic process (30)多分子层吸附理论 adsorption theory of multi—molecular layers (30)二级反应second order reaction (30)二级相变second order phase change (30)法拉第常数 faraday constant (31)法拉第定律 Faraday’s law (31)反电动势back E。

425 BibliographyH.A KAIKE(1974).Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes.Annals Institute Statistical Mathematics,vol.26,pp.363-387. B.D.O.A NDERSON and J.B.M OORE(1979).Optimal rmation and System Sciences Series, Prentice Hall,Englewood Cliffs,NJ.T.W.A NDERSON(1971).The Statistical Analysis of Time Series.Series in Probability and Mathematical Statistics,Wiley,New York.R.A NDRE-O BRECHT(1988).A new statistical approach for the automatic segmentation of continuous speech signals.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-36,no1,pp.29-40.R.A NDRE-O BRECHT(1990).Reconnaissance automatique de parole`a partir de segments acoustiques et de mod`e les de Markov cach´e s.Proc.Journ´e es Etude de la Parole,Montr´e al,May1990(in French).R.A NDRE-O BRECHT and H.Y.S U(1988).Three acoustic labellings for phoneme based continuous speech recognition.Proc.Speech’88,Edinburgh,UK,pp.943-950.U.A PPEL and A.VON B RANDT(1983).Adaptive sequential segmentation of piecewise stationary time rmation Sciences,vol.29,no1,pp.27-56.L.A.A ROIAN and H.L EVENE(1950).The effectiveness of quality control procedures.Jal American Statis-tical Association,vol.45,pp.520-529.K.J.A STR¨OM and B.W ITTENMARK(1984).Computer Controlled Systems:Theory and rma-tion and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.M.B AGSHAW and R.A.J OHNSON(1975a).The effect of serial correlation on the performance of CUSUM tests-Part II.Technometrics,vol.17,no1,pp.73-80.M.B AGSHAW and R.A.J OHNSON(1975b).The influence of reference values and estimated variance on the ARL of CUSUM tests.Jal Royal Statistical Society,vol.37(B),no3,pp.413-420.M.B AGSHAW and R.A.J OHNSON(1977).Sequential procedures for detecting parameter changes in a time-series model.Jal American Statistical Association,vol.72,no359,pp.593-597.R.K.B ANSAL and P.P APANTONI-K AZAKOS(1986).An algorithm for detecting a change in a stochastic process.IEEE rmation Theory,vol.IT-32,no2,pp.227-235.G.A.B ARNARD(1959).Control charts and stochastic processes.Jal Royal Statistical Society,vol.B.21, pp.239-271.A.E.B ASHARINOV andB.S.F LEISHMAN(1962).Methods of the statistical sequential analysis and their radiotechnical applications.Sovetskoe Radio,Moscow(in Russian).M.B ASSEVILLE(1978).D´e viations par rapport au maximum:formules d’arrˆe t et martingales associ´e es. Compte-rendus du S´e minaire de Probabilit´e s,Universit´e de Rennes I.M.B ASSEVILLE(1981).Edge detection using sequential methods for change in level-Part II:Sequential detection of change in mean.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-29,no1,pp.32-50.426B IBLIOGRAPHY M.B ASSEVILLE(1982).A survey of statistical failure detection techniques.In Contribution`a la D´e tectionS´e quentielle de Ruptures de Mod`e les Statistiques,Th`e se d’Etat,Universit´e de Rennes I,France(in English). M.B ASSEVILLE(1986).The two-models approach for the on-line detection of changes in AR processes. In Detection of Abrupt Changes in Signals and Dynamical Systems(M.Basseville,A.Benveniste,eds.). Lecture Notes in Control and Information Sciences,LNCIS77,Springer,New York,pp.169-215.M.B ASSEVILLE(1988).Detecting changes in signals and systems-A survey.Automatica,vol.24,pp.309-326.M.B ASSEVILLE(1989).Distance measures for signal processing and pattern recognition.Signal Process-ing,vol.18,pp.349-369.M.B ASSEVILLE and A.B ENVENISTE(1983a).Design and comparative study of some sequential jump detection algorithms for digital signals.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-31, no3,pp.521-535.M.B ASSEVILLE and A.B ENVENISTE(1983b).Sequential detection of abrupt changes in spectral charac-teristics of digital signals.IEEE rmation Theory,vol.IT-29,no5,pp.709-724.M.B ASSEVILLE and A.B ENVENISTE,eds.(1986).Detection of Abrupt Changes in Signals and Dynamical Systems.Lecture Notes in Control and Information Sciences,LNCIS77,Springer,New York.M.B ASSEVILLE and I.N IKIFOROV(1991).A unified framework for statistical change detection.Proc.30th IEEE Conference on Decision and Control,Brighton,UK.M.B ASSEVILLE,B.E SPIAU and J.G ASNIER(1981).Edge detection using sequential methods for change in level-Part I:A sequential edge detection algorithm.IEEE Trans.Acoustics,Speech,Signal Processing, vol.ASSP-29,no1,pp.24-31.M.B ASSEVILLE, A.B ENVENISTE and G.M OUSTAKIDES(1986).Detection and diagnosis of abrupt changes in modal characteristics of nonstationary digital signals.IEEE rmation Theory,vol.IT-32,no3,pp.412-417.M.B ASSEVILLE,A.B ENVENISTE,G.M OUSTAKIDES and A.R OUG´E E(1987a).Detection and diagnosis of changes in the eigenstructure of nonstationary multivariable systems.Automatica,vol.23,no3,pp.479-489. M.B ASSEVILLE,A.B ENVENISTE,G.M OUSTAKIDES and A.R OUG´E E(1987b).Optimal sensor location for detecting changes in dynamical behavior.IEEE Trans.Automatic Control,vol.AC-32,no12,pp.1067-1075.M.B ASSEVILLE,A.B ENVENISTE,B.G ACH-D EVAUCHELLE,M.G OURSAT,D.B ONNECASE,P.D OREY, M.P REVOSTO and M.O LAGNON(1993).Damage monitoring in vibration mechanics:issues in diagnos-tics and predictive maintenance.Mechanical Systems and Signal Processing,vol.7,no5,pp.401-423.R.V.B EARD(1971).Failure Accommodation in Linear Systems through Self-reorganization.Ph.D.Thesis, Dept.Aeronautics and Astronautics,MIT,Cambridge,MA.A.B ENVENISTE and J.J.F UCHS(1985).Single sample modal identification of a nonstationary stochastic process.IEEE Trans.Automatic Control,vol.AC-30,no1,pp.66-74.A.B ENVENISTE,M.B ASSEVILLE and G.M OUSTAKIDES(1987).The asymptotic local approach to change detection and model validation.IEEE Trans.Automatic Control,vol.AC-32,no7,pp.583-592.A.B ENVENISTE,M.M ETIVIER and P.P RIOURET(1990).Adaptive Algorithms and Stochastic Approxima-tions.Series on Applications of Mathematics,(A.V.Balakrishnan,I.Karatzas,M.Yor,eds.).Springer,New York.A.B ENVENISTE,M.B ASSEVILLE,L.E L G HAOUI,R.N IKOUKHAH and A.S.W ILLSKY(1992).An optimum robust approach to statistical failure detection and identification.IFAC World Conference,Sydney, July1993.B IBLIOGRAPHY427 R.H.B ERK(1973).Some asymptotic aspects of sequential analysis.Annals Statistics,vol.1,no6,pp.1126-1138.R.H.B ERK(1975).Locally most powerful sequential test.Annals Statistics,vol.3,no2,pp.373-381.P.B ILLINGSLEY(1968).Convergence of Probability Measures.Wiley,New York.A.F.B ISSELL(1969).Cusum techniques for quality control.Applied Statistics,vol.18,pp.1-30.M.E.B IVAIKOV(1991).Control of the sample size for recursive estimation of parameters subject to abrupt changes.Automation and Remote Control,no9,pp.96-103.R.E.B LAHUT(1987).Principles and Practice of Information Theory.Addison-Wesley,Reading,MA.I.F.B LAKE and W.C.L INDSEY(1973).Level-crossing problems for random processes.IEEE r-mation Theory,vol.IT-19,no3,pp.295-315.G.B ODENSTEIN and H.M.P RAETORIUS(1977).Feature extraction from the encephalogram by adaptive segmentation.Proc.IEEE,vol.65,pp.642-652.T.B OHLIN(1977).Analysis of EEG signals with changing spectra using a short word Kalman estimator. Mathematical Biosciences,vol.35,pp.221-259.W.B¨OHM and P.H ACKL(1990).Improved bounds for the average run length of control charts based on finite weighted sums.Annals Statistics,vol.18,no4,pp.1895-1899.T.B OJDECKI and J.H OSZA(1984).On a generalized disorder problem.Stochastic Processes and their Applications,vol.18,pp.349-359.L.I.B ORODKIN and V.V.M OTTL’(1976).Algorithm forfinding the jump times of random process equation parameters.Automation and Remote Control,vol.37,no6,Part1,pp.23-32.A.A.B OROVKOV(1984).Theory of Mathematical Statistics-Estimation and Hypotheses Testing,Naouka, Moscow(in Russian).Translated in French under the title Statistique Math´e matique-Estimation et Tests d’Hypoth`e ses,Mir,Paris,1987.G.E.P.B OX and G.M.J ENKINS(1970).Time Series Analysis,Forecasting and Control.Series in Time Series Analysis,Holden-Day,San Francisco.A.VON B RANDT(1983).Detecting and estimating parameters jumps using ladder algorithms and likelihood ratio test.Proc.ICASSP,Boston,MA,pp.1017-1020.A.VON B RANDT(1984).Modellierung von Signalen mit Sprunghaft Ver¨a nderlichem Leistungsspektrum durch Adaptive Segmentierung.Doctor-Engineer Dissertation,M¨u nchen,RFA(in German).S.B RAUN,ed.(1986).Mechanical Signature Analysis-Theory and Applications.Academic Press,London. L.B REIMAN(1968).Probability.Series in Statistics,Addison-Wesley,Reading,MA.G.S.B RITOV and L.A.M IRONOVSKI(1972).Diagnostics of linear systems of automatic regulation.Tekh. Kibernetics,vol.1,pp.76-83.B.E.B RODSKIY and B.S.D ARKHOVSKIY(1992).Nonparametric Methods in Change-point Problems. Kluwer Academic,Boston.L.D.B ROEMELING(1982).Jal Econometrics,vol.19,Special issue on structural change in Econometrics. L.D.B ROEMELING and H.T SURUMI(1987).Econometrics and Structural Change.Dekker,New York. D.B ROOK and D.A.E VANS(1972).An approach to the probability distribution of Cusum run length. Biometrika,vol.59,pp.539-550.J.B RUNET,D.J AUME,M.L ABARR`E RE,A.R AULT and M.V ERG´E(1990).D´e tection et Diagnostic de Pannes.Trait´e des Nouvelles Technologies,S´e rie Diagnostic et Maintenance,Herm`e s,Paris(in French).428B IBLIOGRAPHY S.P.B RUZZONE and M.K AVEH(1984).Information tradeoffs in using the sample autocorrelation function in ARMA parameter estimation.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-32,no4, pp.701-715.A.K.C AGLAYAN(1980).Necessary and sufficient conditions for detectability of jumps in linear systems. IEEE Trans.Automatic Control,vol.AC-25,no4,pp.833-834.A.K.C AGLAYAN and R.E.L ANCRAFT(1983).Reinitialization issues in fault tolerant systems.Proc.Amer-ican Control Conf.,pp.952-955.A.K.C AGLAYAN,S.M.A LLEN and K.W EHMULLER(1988).Evaluation of a second generation reconfigu-ration strategy for aircraftflight control systems subjected to actuator failure/surface damage.Proc.National Aerospace and Electronic Conference,Dayton,OH.P.E.C AINES(1988).Linear Stochastic Systems.Series in Probability and Mathematical Statistics,Wiley, New York.M.J.C HEN and J.P.N ORTON(1987).Estimation techniques for tracking rapid parameter changes.Intern. Jal Control,vol.45,no4,pp.1387-1398.W.K.C HIU(1974).The economic design of cusum charts for controlling normal mean.Applied Statistics, vol.23,no3,pp.420-433.E.Y.C HOW(1980).A Failure Detection System Design Methodology.Ph.D.Thesis,M.I.T.,L.I.D.S.,Cam-bridge,MA.E.Y.C HOW and A.S.W ILLSKY(1984).Analytical redundancy and the design of robust failure detection systems.IEEE Trans.Automatic Control,vol.AC-29,no3,pp.689-691.Y.S.C HOW,H.R OBBINS and D.S IEGMUND(1971).Great Expectations:The Theory of Optimal Stop-ping.Houghton-Mifflin,Boston.R.N.C LARK,D.C.F OSTH and V.M.W ALTON(1975).Detection of instrument malfunctions in control systems.IEEE Trans.Aerospace Electronic Systems,vol.AES-11,pp.465-473.A.C OHEN(1987).Biomedical Signal Processing-vol.1:Time and Frequency Domain Analysis;vol.2: Compression and Automatic Recognition.CRC Press,Boca Raton,FL.J.C ORGE and F.P UECH(1986).Analyse du rythme cardiaque foetal par des m´e thodes de d´e tection de ruptures.Proc.7th INRIA Int.Conf.Analysis and optimization of Systems.Antibes,FR(in French).D.R.C OX and D.V.H INKLEY(1986).Theoretical Statistics.Chapman and Hall,New York.D.R.C OX and H.D.M ILLER(1965).The Theory of Stochastic Processes.Wiley,New York.S.V.C ROWDER(1987).A simple method for studying run-length distributions of exponentially weighted moving average charts.Technometrics,vol.29,no4,pp.401-407.H.C S¨ORG¨O and L.H ORV´ATH(1988).Nonparametric methods for change point problems.In Handbook of Statistics(P.R.Krishnaiah,C.R.Rao,eds.),vol.7,Elsevier,New York,pp.403-425.R.B.D AVIES(1973).Asymptotic inference in stationary gaussian time series.Advances Applied Probability, vol.5,no3,pp.469-497.J.C.D ECKERT,M.N.D ESAI,J.J.D EYST and A.S.W ILLSKY(1977).F-8DFBW sensor failure identification using analytical redundancy.IEEE Trans.Automatic Control,vol.AC-22,no5,pp.795-803.M.H.D E G ROOT(1970).Optimal Statistical Decisions.Series in Probability and Statistics,McGraw-Hill, New York.J.D ESHAYES and D.P ICARD(1979).Tests de ruptures dans un mod`e pte-Rendus de l’Acad´e mie des Sciences,vol.288,Ser.A,pp.563-566(in French).B IBLIOGRAPHY429 J.D ESHAYES and D.P ICARD(1983).Ruptures de Mod`e les en Statistique.Th`e ses d’Etat,Universit´e deParis-Sud,Orsay,France(in French).J.D ESHAYES and D.P ICARD(1986).Off-line statistical analysis of change-point models using non para-metric and likelihood methods.In Detection of Abrupt Changes in Signals and Dynamical Systems(M. Basseville,A.Benveniste,eds.).Lecture Notes in Control and Information Sciences,LNCIS77,Springer, New York,pp.103-168.B.D EVAUCHELLE-G ACH(1991).Diagnostic M´e canique des Fatigues sur les Structures Soumises`a des Vibrations en Ambiance de Travail.Th`e se de l’Universit´e Paris IX Dauphine(in French).B.D EVAUCHELLE-G ACH,M.B ASSEVILLE and A.B ENVENISTE(1991).Diagnosing mechanical changes in vibrating systems.Proc.SAFEPROCESS’91,Baden-Baden,FRG,pp.85-89.R.D I F RANCESCO(1990).Real-time speech segmentation using pitch and convexity jump models:applica-tion to variable rate speech coding.IEEE Trans.Acoustics,Speech,Signal Processing,vol.ASSP-38,no5, pp.741-748.X.D ING and P.M.F RANK(1990).Fault detection via factorization approach.Systems and Control Letters, vol.14,pp.431-436.J.L.D OOB(1953).Stochastic Processes.Wiley,New York.V.D RAGALIN(1988).Asymptotic solutions in detecting a change in distribution under an unknown param-eter.Statistical Problems of Control,Issue83,Vilnius,pp.45-52.B.D UBUISSON(1990).Diagnostic et Reconnaissance des Formes.Trait´e des Nouvelles Technologies,S´e rie Diagnostic et Maintenance,Herm`e s,Paris(in French).A.J.D UNCAN(1986).Quality Control and Industrial Statistics,5th edition.Richard D.Irwin,Inc.,Home-wood,IL.J.D URBIN(1971).Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test.Jal Applied Probability,vol.8,pp.431-453.J.D URBIN(1985).Thefirst passage density of the crossing of a continuous Gaussian process to a general boundary.Jal Applied Probability,vol.22,no1,pp.99-122.A.E MAMI-N AEINI,M.M.A KHTER and S.M.R OCK(1988).Effect of model uncertainty on failure detec-tion:the threshold selector.IEEE Trans.Automatic Control,vol.AC-33,no12,pp.1106-1115.J.D.E SARY,F.P ROSCHAN and D.W.W ALKUP(1967).Association of random variables with applications. Annals Mathematical Statistics,vol.38,pp.1466-1474.W.D.E WAN and K.W.K EMP(1960).Sampling inspection of continuous processes with no autocorrelation between successive results.Biometrika,vol.47,pp.263-280.G.F AVIER and A.S MOLDERS(1984).Adaptive smoother-predictors for tracking maneuvering targets.Proc. 23rd Conf.Decision and Control,Las Vegas,NV,pp.831-836.W.F ELLER(1966).An Introduction to Probability Theory and Its Applications,vol.2.Series in Probability and Mathematical Statistics,Wiley,New York.R.A.F ISHER(1925).Theory of statistical estimation.Proc.Cambridge Philosophical Society,vol.22, pp.700-725.M.F ISHMAN(1988).Optimization of the algorithm for the detection of a disorder,based on the statistic of exponential smoothing.In Statistical Problems of Control,Issue83,Vilnius,pp.146-151.R.F LETCHER(1980).Practical Methods of Optimization,2volumes.Wiley,New York.P.M.F RANK(1990).Fault diagnosis in dynamic systems using analytical and knowledge based redundancy -A survey and new results.Automatica,vol.26,pp.459-474.430B IBLIOGRAPHY P.M.F RANK(1991).Enhancement of robustness in observer-based fault detection.Proc.SAFEPRO-CESS’91,Baden-Baden,FRG,pp.275-287.P.M.F RANK and J.W¨UNNENBERG(1989).Robust fault diagnosis using unknown input observer schemes. In Fault Diagnosis in Dynamic Systems-Theory and Application(R.Patton,P.Frank,R.Clark,eds.). International Series in Systems and Control Engineering,Prentice Hall International,London,UK,pp.47-98.K.F UKUNAGA(1990).Introduction to Statistical Pattern Recognition,2d ed.Academic Press,New York. S.I.G ASS(1958).Linear Programming:Methods and Applications.McGraw Hill,New York.W.G E and C.Z.F ANG(1989).Extended robust observation approach for failure isolation.Int.Jal Control, vol.49,no5,pp.1537-1553.W.G ERSCH(1986).Two applications of parametric time series modeling methods.In Mechanical Signature Analysis-Theory and Applications(S.Braun,ed.),chap.10.Academic Press,London.J.J.G ERTLER(1988).Survey of model-based failure detection and isolation in complex plants.IEEE Control Systems Magazine,vol.8,no6,pp.3-11.J.J.G ERTLER(1991).Analytical redundancy methods in fault detection and isolation.Proc.SAFEPRO-CESS’91,Baden-Baden,FRG,pp.9-22.B.K.G HOSH(1970).Sequential Tests of Statistical Hypotheses.Addison-Wesley,Cambridge,MA.I.N.G IBRA(1975).Recent developments in control charts techniques.Jal Quality Technology,vol.7, pp.183-192.J.P.G ILMORE and R.A.M C K ERN(1972).A redundant strapdown inertial reference unit(SIRU).Jal Space-craft,vol.9,pp.39-47.M.A.G IRSHICK and H.R UBIN(1952).A Bayes approach to a quality control model.Annals Mathematical Statistics,vol.23,pp.114-125.A.L.G OEL and S.M.W U(1971).Determination of the ARL and a contour nomogram for CUSUM charts to control normal mean.Technometrics,vol.13,no2,pp.221-230.P.L.G OLDSMITH and H.W HITFIELD(1961).Average run lengths in cumulative chart quality control schemes.Technometrics,vol.3,pp.11-20.G.C.G OODWIN and K.S.S IN(1984).Adaptive Filtering,Prediction and rmation and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.R.M.G RAY and L.D.D AVISSON(1986).Random Processes:a Mathematical Approach for Engineers. Information and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.C.G UEGUEN and L.L.S CHARF(1980).Exact maximum likelihood identification for ARMA models:a signal processing perspective.Proc.1st EUSIPCO,Lausanne.D.E.G USTAFSON, A.S.W ILLSKY,J.Y.W ANG,M.C.L ANCASTER and J.H.T RIEBWASSER(1978). ECG/VCG rhythm diagnosis using statistical signal analysis.Part I:Identification of persistent rhythms. Part II:Identification of transient rhythms.IEEE Trans.Biomedical Engineering,vol.BME-25,pp.344-353 and353-361.F.G USTAFSSON(1991).Optimal segmentation of linear regression parameters.Proc.IFAC/IFORS Symp. Identification and System Parameter Estimation,Budapest,pp.225-229.T.H¨AGGLUND(1983).New Estimation Techniques for Adaptive Control.Ph.D.Thesis,Lund Institute of Technology,Lund,Sweden.T.H¨AGGLUND(1984).Adaptive control of systems subject to large parameter changes.Proc.IFAC9th World Congress,Budapest.B IBLIOGRAPHY431 P.H ALL and C.C.H EYDE(1980).Martingale Limit Theory and its Application.Probability and Mathemat-ical Statistics,a Series of Monographs and Textbooks,Academic Press,New York.W.J.H ALL,R.A.W IJSMAN and J.K.G HOSH(1965).The relationship between sufficiency and invariance with applications in sequential analysis.Ann.Math.Statist.,vol.36,pp.576-614.E.J.H ANNAN and M.D EISTLER(1988).The Statistical Theory of Linear Systems.Series in Probability and Mathematical Statistics,Wiley,New York.J.D.H EALY(1987).A note on multivariate CuSum procedures.Technometrics,vol.29,pp.402-412.D.M.H IMMELBLAU(1970).Process Analysis by Statistical Methods.Wiley,New York.D.M.H IMMELBLAU(1978).Fault Detection and Diagnosis in Chemical and Petrochemical Processes. Chemical Engineering Monographs,vol.8,Elsevier,Amsterdam.W.G.S.H INES(1976a).A simple monitor of a system with sudden parameter changes.IEEE r-mation Theory,vol.IT-22,no2,pp.210-216.W.G.S.H INES(1976b).Improving a simple monitor of a system with sudden parameter changes.IEEE rmation Theory,vol.IT-22,no4,pp.496-499.D.V.H INKLEY(1969).Inference about the intersection in two-phase regression.Biometrika,vol.56,no3, pp.495-504.D.V.H INKLEY(1970).Inference about the change point in a sequence of random variables.Biometrika, vol.57,no1,pp.1-17.D.V.H INKLEY(1971).Inference about the change point from cumulative sum-tests.Biometrika,vol.58, no3,pp.509-523.D.V.H INKLEY(1971).Inference in two-phase regression.Jal American Statistical Association,vol.66, no336,pp.736-743.J.R.H UDDLE(1983).Inertial navigation system error-model considerations in Kalmanfiltering applica-tions.In Control and Dynamic Systems(C.T.Leondes,ed.),Academic Press,New York,pp.293-339.J.S.H UNTER(1986).The exponentially weighted moving average.Jal Quality Technology,vol.18,pp.203-210.I.A.I BRAGIMOV and R.Z.K HASMINSKII(1981).Statistical Estimation-Asymptotic Theory.Applications of Mathematics Series,vol.16.Springer,New York.R.I SERMANN(1984).Process fault detection based on modeling and estimation methods-A survey.Auto-matica,vol.20,pp.387-404.N.I SHII,A.I WATA and N.S UZUMURA(1979).Segmentation of nonstationary time series.Int.Jal Systems Sciences,vol.10,pp.883-894.J.E.J ACKSON and R.A.B RADLEY(1961).Sequential and tests.Annals Mathematical Statistics, vol.32,pp.1063-1077.B.J AMES,K.L.J AMES and D.S IEGMUND(1988).Conditional boundary crossing probabilities with appli-cations to change-point problems.Annals Probability,vol.16,pp.825-839.M.K.J EERAGE(1990).Reliability analysis of fault-tolerant IMU architectures with redundant inertial sen-sors.IEEE Trans.Aerospace and Electronic Systems,vol.AES-5,no.7,pp.23-27.N.L.J OHNSON(1961).A simple theoretical approach to cumulative sum control charts.Jal American Sta-tistical Association,vol.56,pp.835-840.N.L.J OHNSON and F.C.L EONE(1962).Cumulative sum control charts:mathematical principles applied to their construction and use.Parts I,II,III.Industrial Quality Control,vol.18,pp.15-21;vol.19,pp.29-36; vol.20,pp.22-28.432B IBLIOGRAPHY R.A.J OHNSON and M.B AGSHAW(1974).The effect of serial correlation on the performance of CUSUM tests-Part I.Technometrics,vol.16,no.1,pp.103-112.H.L.J ONES(1973).Failure Detection in Linear Systems.Ph.D.Thesis,Dept.Aeronautics and Astronautics, MIT,Cambridge,MA.R.H.J ONES,D.H.C ROWELL and L.E.K APUNIAI(1970).Change detection model for serially correlated multivariate data.Biometrics,vol.26,no2,pp.269-280.M.J URGUTIS(1984).Comparison of the statistical properties of the estimates of the change times in an autoregressive process.In Statistical Problems of Control,Issue65,Vilnius,pp.234-243(in Russian).T.K AILATH(1980).Linear rmation and System Sciences Series,Prentice Hall,Englewood Cliffs,NJ.L.V.K ANTOROVICH and V.I.K RILOV(1958).Approximate Methods of Higher Analysis.Interscience,New York.S.K ARLIN and H.M.T AYLOR(1975).A First Course in Stochastic Processes,2d ed.Academic Press,New York.S.K ARLIN and H.M.T AYLOR(1981).A Second Course in Stochastic Processes.Academic Press,New York.D.K AZAKOS and P.P APANTONI-K AZAKOS(1980).Spectral distance measures between gaussian pro-cesses.IEEE Trans.Automatic Control,vol.AC-25,no5,pp.950-959.K.W.K EMP(1958).Formula for calculating the operating characteristic and average sample number of some sequential tests.Jal Royal Statistical Society,vol.B-20,no2,pp.379-386.K.W.K EMP(1961).The average run length of the cumulative sum chart when a V-mask is used.Jal Royal Statistical Society,vol.B-23,pp.149-153.K.W.K EMP(1967a).Formal expressions which can be used for the determination of operating character-istics and average sample number of a simple sequential test.Jal Royal Statistical Society,vol.B-29,no2, pp.248-262.K.W.K EMP(1967b).A simple procedure for determining upper and lower limits for the average sample run length of a cumulative sum scheme.Jal Royal Statistical Society,vol.B-29,no2,pp.263-265.D.P.K ENNEDY(1976).Some martingales related to cumulative sum tests and single server queues.Stochas-tic Processes and Appl.,vol.4,pp.261-269.T.H.K ERR(1980).Statistical analysis of two-ellipsoid overlap test for real time failure detection.IEEE Trans.Automatic Control,vol.AC-25,no4,pp.762-772.T.H.K ERR(1982).False alarm and correct detection probabilities over a time interval for restricted classes of failure detection algorithms.IEEE rmation Theory,vol.IT-24,pp.619-631.T.H.K ERR(1987).Decentralizedfiltering and redundancy management for multisensor navigation.IEEE Trans.Aerospace and Electronic systems,vol.AES-23,pp.83-119.Minor corrections on p.412and p.599 (May and July issues,respectively).R.A.K HAN(1978).Wald’s approximations to the average run length in cusum procedures.Jal Statistical Planning and Inference,vol.2,no1,pp.63-77.R.A.K HAN(1979).Somefirst passage problems related to cusum procedures.Stochastic Processes and Applications,vol.9,no2,pp.207-215.R.A.K HAN(1981).A note on Page’s two-sided cumulative sum procedures.Biometrika,vol.68,no3, pp.717-719.B IBLIOGRAPHY433 V.K IREICHIKOV,V.M ANGUSHEV and I.N IKIFOROV(1990).Investigation and application of CUSUM algorithms to monitoring of sensors.In Statistical Problems of Control,Issue89,Vilnius,pp.124-130(in Russian).G.K ITAGAWA and W.G ERSCH(1985).A smoothness prior time-varying AR coefficient modeling of non-stationary covariance time series.IEEE Trans.Automatic Control,vol.AC-30,no1,pp.48-56.N.K LIGIENE(1980).Probabilities of deviations of the change point estimate in statistical models.In Sta-tistical Problems of Control,Issue83,Vilnius,pp.80-86(in Russian).N.K LIGIENE and L.T ELKSNYS(1983).Methods of detecting instants of change of random process prop-erties.Automation and Remote Control,vol.44,no10,Part II,pp.1241-1283.J.K ORN,S.W.G ULLY and A.S.W ILLSKY(1982).Application of the generalized likelihood ratio algorithm to maneuver detection and estimation.Proc.American Control Conf.,Arlington,V A,pp.792-798.P.R.K RISHNAIAH and B.Q.M IAO(1988).Review about estimation of change points.In Handbook of Statistics(P.R.Krishnaiah,C.R.Rao,eds.),vol.7,Elsevier,New York,pp.375-402.P.K UDVA,N.V ISWANADHAM and A.R AMAKRISHNAN(1980).Observers for linear systems with unknown inputs.IEEE Trans.Automatic Control,vol.AC-25,no1,pp.113-115.S.K ULLBACK(1959).Information Theory and Statistics.Wiley,New York(also Dover,New York,1968). K.K UMAMARU,S.S AGARA and T.S¨ODERSTR¨OM(1989).Some statistical methods for fault diagnosis for dynamical systems.In Fault Diagnosis in Dynamic Systems-Theory and Application(R.Patton,P.Frank,R. Clark,eds.).International Series in Systems and Control Engineering,Prentice Hall International,London, UK,pp.439-476.A.K USHNIR,I.N IKIFOROV and I.S AVIN(1983).Statistical adaptive algorithms for automatic detection of seismic signals-Part I:One-dimensional case.In Earthquake Prediction and the Study of the Earth Structure,Naouka,Moscow(Computational Seismology,vol.15),pp.154-159(in Russian).L.L ADELLI(1990).Diffusion approximation for a pseudo-likelihood test process with application to de-tection of change in stochastic system.Stochastics and Stochastics Reports,vol.32,pp.1-25.T.L.L A¨I(1974).Control charts based on weighted sums.Annals Statistics,vol.2,no1,pp.134-147.T.L.L A¨I(1981).Asymptotic optimality of invariant sequential probability ratio tests.Annals Statistics, vol.9,no2,pp.318-333.D.G.L AINIOTIS(1971).Joint detection,estimation,and system identifirmation and Control, vol.19,pp.75-92.M.R.L EADBETTER,G.L INDGREN and H.R OOTZEN(1983).Extremes and Related Properties of Random Sequences and Processes.Series in Statistics,Springer,New York.L.L E C AM(1960).Locally asymptotically normal families of distributions.Univ.California Publications in Statistics,vol.3,pp.37-98.L.L E C AM(1986).Asymptotic Methods in Statistical Decision Theory.Series in Statistics,Springer,New York.E.L.L EHMANN(1986).Testing Statistical Hypotheses,2d ed.Wiley,New York.J.P.L EHOCZKY(1977).Formulas for stopped diffusion processes with stopping times based on the maxi-mum.Annals Probability,vol.5,no4,pp.601-607.H.R.L ERCHE(1980).Boundary Crossing of Brownian Motion.Lecture Notes in Statistics,vol.40,Springer, New York.L.L JUNG(1987).System Identification-Theory for the rmation and System Sciences Series, Prentice Hall,Englewood Cliffs,NJ.。

黎曼猜想英语The Riemann Hypothesis, named after the 19th-century mathematician Bernhard Riemann, is one of the most profound and consequential conjectures in mathematics. It is concerned with the distribution of the zeros of the Riemann zeta function, a complex function denoted as $$\zeta(s)$$, where $$s$$ is a complex number. The hypothesis posits that all non-trivial zeros of this analytical function have their real parts equal to $$\frac{1}{2}$$.To understand the significance of this conjecture, one must delve into the realm of number theory and the distribution of prime numbers. Prime numbers are the building blocks of arithmetic, as every natural number greater than 1 is either a prime or can be factored into primes. The distribution of these primes, however, has puzzled mathematicians for centuries. The Riemann zeta function encodes information about the distribution of primes through its zeros, and thus, the Riemann Hypothesis is directly linked to understanding this distribution.The zeta function is defined for all complex numbers except for $$s = 1$$, where it has a simple pole. For values of $$s$$ with a real part greater than 1, it converges to a sum over the positive integers, as shown in the following equation:$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$。

a r X i v:solv-int/99713v18J ul1999Lattice geometry of the Hirota equation Adam Doliwa Instytut Fizyki Teoretycznej,Uniwersytet Warszawski ul.Ho˙z a 69,00-681Warszawa,Poland e-mail:doliwa@.pl Abstract Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadri-lateral lattices.Different forms of the equation are given together with their geometric interpretation:(i)the discrete coupled Volterra sys-tem for the coefficients of the Laplace equation,(ii)the gauge invariant form of the Hirota equation for projective invariants of the Laplace sequence,(iii)the discrete Toda system for the rotation coefficients and (iv)the original form of the Hirota equation for the τ-function of the quadrilateral lattice.Keywords:Integrable discrete geometry;Hirota equation 1IntroductionThe integrable discrete geometry deals with lattice submanifolds described by integrable equations.Among the integrable discrete (difference)equations an important role is played by the Hirota equation [12]which is the discrete analog of the two-dimensional Toda system [17].Both the Toda system and the Hirota equation are important in the theory of integrable equations and in their applications.It turns out that the two-dimensional Toda system was studied in classical differential geometry and describes the so called Laplace sequences of two-dimensional conjugate nets [3].The lattice geometric interpretation of the discrete Toda system was found in [6]and is based on the observation that the discrete analog of1the conjugate net on a surface,which is given by the two-dimensional lattice made of planar quadrilaterals[18],allows for definition of the correspond-ing Laplace sequence of discrete conjugate nets.Since then,the multidi-mensional quadrilateral lattice[10](multidimensional lattice made of planar quadrilaterals–the integrable discrete analog of a multidimensional conju-gate net)became one of the central notions of the integrable discrete geom-etry.In particular,many classical results of the theory of conjugate nets and of their reductions have been generalized recently to the discrete level [1,16,2,7,11,14,5,9].The goal of the present article is to reinterpret and expand results ob-tained in[6]using new notions provided by the general theory of quadrilateral lattices.Different formulations of the Hirota equations are reviewed and,for each of them,the geometric interpretation of the corresponding functions is given.In Sections2and3we reformulate,using more convenient notation,re-sults found in[6].In Section4we present the discrete analog of the standard version of the Toda system as an equation governing Laplace transformations of the rotation coefficients.Then in Section5,basing on the geometric inter-pretation of theτ–function of the quadrilateral lattice given in[9],we show that theτ–functions of the Laplace sequence of quadrilateral lattices solve the original Hirota’s form of the discrete Toda system—this last resultfills out the missing point of paper[6].2Laplace transformations of quadrilateral lat-ticesDefinition2.1.Two dimensional quadrilateral lattice is a mapping of the two dimensional integer lattice into M dimensional projective space such that elementary quadrilaterals of the lattice are planar:x:Z2→P M,T1T2x∈ x,T1x,T2x .In the above definition T i,i=1,2,denotes the shift operator along the i-th direction of the lattice.In the non-homogenous coordinates of the projective space a two dimen-sional quadrilateral lattice is represented by the mapping x:Z2→R M2L (x )L (x )T i ij x i T x T j xj -1x T i T j x ij T ij L (x )j T Figure 1:Laplace transformation of quadrilateral latticessatisfying the Laplace equation [6]∆1∆2x =(T 1A 12)∆2x +(T 2A 21)∆2x ,(2.1)which is equivalent to the planarity condition;here ∆i =T i −1,i =1,2,is the partial difference operator,and the functions A 12,A 21:Z 2→R ,define the position of the point T 1T 2x with respect to the points x ,T 1x and T 2x .Planarity of elementary quadrilaterals implies that the ”tangent”lines containing opposite sides of an elementary quadrilateral intersect.The Laplace transformation L ij ,i =j of the lattice x is defined [6,11]as intersection of the line x,T i x with the line T −1j x,T i T −1j x (see Figure 1).Using elementary calculations one can show the following results [6,11].Proposition 2.1.In the non-homogenous representation the Laplace trans-formation L ij (x )of the quadrilateral lattice x is given byL ij (x )=x −1T j A ji (T i A ij +1)−1,(2.3)L ij (A ji )=T −1j T i L ij (A ij )Proposition2.3.Under the assumption that the transformed lattices are non-degenerate,i.e.their quadrilaterals do not degenerate to segments or points,we haveL ij◦L ji=L ji◦L ij=id.(2.5) In this way given two dimensional quadrilateral lattice x one can define a sequence of quadrilateral latticesx(l)=L l12(x),l∈N,L−112=L21.In analogy to the Laplace sequence of conjugate nets,the above sequence can be called the Laplace sequence of quadrilateral lattices.Equations(2.3)-(2.4) can be then rewritten in the form∆2A(l)21(T1A(l)12+1)(A(l+1)12+1),(2.6)∆1A(l)12(T2A(l)21+1)(A(l−1)21+1),(2.7)which is the discrete analog of the coupled Volterra system.3Projective invariants of the Laplace sequence The planarity of elementary quadrilaterals of the quadrilateral lattice and the construction of the Laplace sequence are essentially of the projective nature[6].It would be therefore interesting to know the pure projective-geometric version of the equation describing the Laplace sequence of quadri-lateral lattices.The basic numeric invariant of projective transformations is the so called cross-ratio of four collinear points,which is given in the affine representation ascr(a,b;c,d)= c−a d−b ;notice the simple identitycr(a,b;c,d)=cr(b,a;d,c).(3.1)4Define the function K ij as the cross-ratio of x ,L ij (x ),T i x and T j L ij (x ).Elementary calculations show thatT i x −L ij (x )=1+A jiT j A ji ∆i x ,T j L ij (x )−L ij (x )=1T j A ji ∆i x ,and,therefore,K ij =cr(x ,L ij (x );T i x ,T j L ij (x ))=A ji (T i A ij +1)−T j A ji(T i K ij )(T j K ij )(T i K ij +1)(T j K ij +1)K (l )+1 T 1 K (l −1)+1(T 1K (l ))(T 2K (l )),(3.4)known as the gauge invariant form of the Hirota equation.4Rotation coefficients of the quadrilaterallatticeAs it was shown in[10]it is convenient to reformulate the Laplace equation (2.1)as a first order system.We introduce the suitably scaled tangent vectors X i ,i =1,2,∆i x =(T i H i )X i ,(4.1)5x X i X j T j X i(T j Q ij )X j T jx T i xT i X jT i T j xFigure 2:Definition of the rotation coefficientsin such a way that the j -th variation of X i is proportional to X j only (see Figure 2)∆j X i =(T j Q ij )X j ,i =j,(4.2)the coefficients Q ij in equation (4.2)are called the rotation coefficients.The scaling factors H i in equation (4.1),called the Lam´e coefficients,satisfy the linear equations∆i H j =(T i H i )Q ij ,i =j ,adjoint to (4.2);moreoverA ij =∆j H iQ ij,L ij (H j )=T −1j Q ij ∆jH jthe tangent vectors of the new lattice are given byL ij(X i)=−∆i X i+∆i Q ijQ ij Xi.From above formulas follow transformation rules for the rotation coeffi-cients.Proposition4.2.The rotation coefficients transform according toL ij(Q ij)=T−1j T i Q ij−Q ij T i T j Q ijQ ij.(4.4)Equations(4.3)and(4.4)can be rewritten in terms of the function Q= Q12as∆2∆1Q(l)Q(l−1) −T2Q(l+1)T j -1X j ~T i -1i X ~X j~T j -1X i~T i -1Q ~ij ()X i ~T i -1T i -1T j -1xx xx Figure 3:Definition of the backward dataadjoint to system (5.1)Since the forward and backward rotation coefficients Q ij and ˜Qij describe the same lattice x but from different points of view,then one cannot expect that they are independent.Indeed,defining the functions ρi :Z 2→R ,i =1,2,as the proportionality factors between X i and T i ˜Xi (both vectors are proportional to ∆i x ):X i =−ρi (T i ˜X i ),T i H i =−1ρi =1−(T i Q ji )(T j Q ij ),i =j .(5.2)The right hand side of equation (5.2)is symmetric with respect to the interchange of i and j ,which implies the existence of a potential τ:Z 2→R ,8such thatρi=T iτ(T iτ)(T jτ)=1−(T i Q ji)(T j Q ij),i=j.(5.3)The potentialτconnecting the forward and backward data is theτ-function of the quadrilateral lattice[9].Let usfind the Laplace transformation of theτ-function.Formulas(4.3) and(4.4)imply that1−(T i L ij(Q ji))(T j L ij(Q ij))=Q ij T i T j Q ijT iτT jτ,(5.4)which,due to equation(5.3),allows for identificationL ij(τ)=τQ ij.(5.5) It should be mentioned here that the above formula was strongly suggested by the identification of the Schlesinger transformation of the theory of the mul-ticomponent Kadomtsev–Petviashvili hierarchy[4,13,15]with the Laplace transformation of conjugate nets[8].Corollary5.2.The geometric meaning ofτij as the Laplace transformation L ij(τ)of theτ-function applies for any dimension of the quadrilateral lattice.Finally,equation(5.3)rewritten in terms of theτ-function and its Laplace transformations take the following formτ(l)T1T2τ(l)=(T1τ(l))(T2τ(l))−(T1τ(l−1))(T2τ(l+1)),(5.6) which is the original Hirota’s bilinear form of the discrete Toda system.6ConclusionThe geometric interpretation of the Hirota equation(integrable discrete ana-log of the Toda system)is given by the Laplace sequence of quadrilateral9lattices,therefore various representations of the lattice give different versions of the equation.In the paper we presented four different forms of the Hi-rota equation:(i)the discrete coupled Volterra system(2.6)-(2.7)for the coefficients of the Laplace equations,(ii)the gauge invariant form of the Hi-rota equation(3.4)for projective invariants of the Laplace sequence,(iii)the discrete Toda system for the rotation coefficients(4.5),and(iv)the origi-nal form of the Hirota equation(5.6)for theτ-function of the quadrilateral lattice.AcknowledgementsThe author would like to thank the organizers of the SIDE III meeting for invitation and support.References[1]A.Bobenko and U.Pinkall,Discrete isothermic surfaces,J.reine angew.Math.475(1996),187–208.[2]J.Cie´s li´n ski,A.Doliwa,and P.M.Santini,The integrable discrete ana-logues of orthogonal coordinate systems are multidimensional circular lattices,Phys.Lett.A235(1997),480–488.[3]G.Darboux,Le¸c ons sur la th´e orie g´e n´e rale des surfaces I–IV,Gauthier–Villars,Paris,1887–1896.[4]E.Date,M.Kashiwara,M.Jimbo,and T.Miwa,Transformation groupsfor soliton equations,Nonlinear integrable systems—classical the-ory and quantum theory,Proc.of RIMS Symposium(M.Jimbo and T.Miwa,eds.),World Scientific,Singapore,1983,pp.39–119.[5]A.Doliwa,Quadratic reductions of quadrilateral lattices,J.Geom.Phys.30(1999)169–186.[6][7]A.Doliwa,S.V.Manakov,and P.M.Santini,¯∂-reductions of the multi-dimensional quadrilateral lattice:the multidimensional circular lattice, Comm.Math.Phys.196(1998),1–18.[8]A.Doliwa,M.Ma˜n as,L.Mart´ınez Alonso,E.Medina,and P.M.Santini,Multicomponent KP hierarchy and classical transformations of conjugate nets,J.Phys.A32(1999)1197–1216.[9]A.Doliwa and P.M.Santini,The symmetric and Egorov reductions ofthe quadrilateral lattice,solv-int/9907012.[10]。

辅助函数法求解非线性偏微分方程精确解杨健;赖晓霞【摘要】在数学和物理学领域,将含有非线性项的偏微分方程称为非线性偏微分方程.非线性偏微分方程用于描述物理学中许多不同的物理模型,范围涉及从引力到流体动力学的众多领域,还在数学中用于验证庞加莱猜想和卡拉比猜想.在求解非线性偏微分方程的过程中,几乎没有通用的求解方法能够应用于所有的方程.通常,可依据模型方程的数学物理背景来先验地假设非线性偏微分方程解的形式,并根据解的特点给出辅助方程.非线性偏微分方程可通过行波变换转化为常微分方程,再借助辅助方程来求解常微分方程.为此,借助行波变换及辅助方程的求解思路对BBM方程和Burgers方程进行了研究,并获得了其双曲正切函数及三角函数形式的精确解.研究结果表明,所采用的方法可广泛应用于若干在数学物理中有典型应用背景的非线性偏微分方程的精确解求解中.%In mathematics and physics,a nonlinear partial differential equation is a partial differential equation with nonlinear terms,which can describe many different physical models ranging from gravitation to fluid dynamics,and have been adopted in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. There are almost no general solutions that can be applied for all equa-tions. Nonlinear partial differential equation usually originates from mathematical and physical fields,such that the ansatz of the solutions has been given and an auxiliary function has been provided according to its mathematical and physical features. They can be transmitted to an ordinary differential equations via a traveling wave transformation. Through introduction of the auxiliary function into the ordinary dif-ferential equation a set of nonlinear algebra equations is acquired,which can supply solutions original partial differential equation in sol-ving process. Therefore,BBM equation and Burgers equation can be solved with the auxiliary function. The exact solutions include tan-gent function and trigonometric functions. The research shows that the proposed auxiliary function method can be applied to solve some other nonlinear partial differential equations with mathematical and physical background.【期刊名称】《计算机技术与发展》【年(卷),期】2017(027)011【总页数】5页(P196-200)【关键词】非线性偏微分方程;辅助函数法;BBM方程;Burgers方程;精确解【作者】杨健;赖晓霞【作者单位】陕西师范大学计算机科学学院,陕西西安 710119;陕西师范大学计算机科学学院,陕西西安 710119【正文语种】中文【中图分类】TP39非线性方程广泛应用于物理学和应用数学的许多分支，尤其在流体力学、固态物理学、等离子物理和非线性光学等。

More informationNONLINEAR TIME SERIES ANALYSISThis book represents a modern approach to time series analysis which is based onthe theory of dynamical systems.It starts from a sound outline of the underlyingtheory to arrive at very practical issues,which are illustrated using a large number ofempirical data sets taken from variousfields.This book will hence be highly usefulfor scientists and engineers from all disciplines who study time variable signals,including the earth,life and social sciences.The paradigm of deterministic chaos has influenced thinking in manyfields ofscience.Chaotic systems show rich and surprising mathematical structures.In theapplied sciences,deterministic chaos provides a striking explanation for irregulartemporal behaviour and anomalies in systems which do not seem to be inherentlystochastic.The most direct link between chaos theory and the real world is the anal-ysis of time series from real systems in terms of nonlinear dynamics.Experimentaltechnique and data analysis have seen such dramatic progress that,by now,mostfundamental properties of nonlinear dynamical systems have been observed in thelaboratory.Great efforts are being made to exploit ideas from chaos theory where-ver the data display more structure than can be captured by traditional methods.Problems of this kind are typical in biology and physiology but also in geophysics,economics and many other sciences.This revised edition has been significantly rewritten an expanded,includingseveral new chapters.In view of applications,the most relevant novelties will be thetreatment of non-stationary data sets and of nonlinear stochastic processes insidethe framework of a state space reconstruction by the method of delays.Hence,non-linear time series analysis has left the rather narrow niche of strictly deterministicsystems.Moreover,the analysis of multivariate data sets has gained more atten-tion.For a direct application of the methods of this book to the reader’s own datasets,this book closely refers to the publicly available software package TISEAN.The availability of this software will facilitate the solution of the exercises,so thatreaders now can easily gain their own experience with the analysis of data sets.Holger Kantz,born in November1960,received his diploma in physics fromthe University of Wuppertal in January1986with a thesis on transient chaos.InJanuary1989he obtained his Ph.D.in theoretical physics from the same place,having worked under the supervision of Peter Grassberger on Hamiltonian many-particle dynamics.During his postdoctoral time,he spent one year on a Marie Curiefellowship of the European Union at the physics department of the University ofMore informationFlorence in Italy.In January1995he took up an appointment at the newly foundedMax Planck Institute for the Physics of Complex Systems in Dresden,where heestablished the research group‘Nonlinear Dynamics and Time Series Analysis’.In1996he received his venia legendi and in2002he became adjunct professorin theoretical physics at Wuppertal University.In addition to time series analysis,he works on low-and high-dimensional nonlinear dynamics and its applications.More recently,he has been trying to bridge the gap between dynamics and statis-tical physics.He has(co-)authored more than75peer-reviewed articles in scien-tific journals and holds two international patents.For up-to-date information seehttp://www.mpipks-dresden.mpg.de/mpi-doc/kantzgruppe.html.Thomas Schreiber,born1963,did his diploma work with Peter Grassberger atWuppertal University on phase transitions and information transport in spatio-temporal chaos.He joined the chaos group of Predrag Cvitanovi´c at the Niels BohrInstitute in Copenhagen to study periodic orbit theory of diffusion and anomaloustransport.There he also developed a strong interest in real-world applications ofchaos theory,leading to his Ph.D.thesis on nonlinear time series analysis(Univer-sity of Wuppertal,1994).As a research assistant at Wuppertal University and duringseveral extended appointments at the Max Planck Institute for the Physics of Com-plex Systems in Dresden he published numerous research articles on time seriesmethods and applications ranging from physiology to the stock market.His habil-itation thesis(University of Wuppertal)appeared as a review in Physics Reportsin1999.Thomas Schreiber has extensive experience teaching nonlinear dynamicsto students and experts from variousfields and at all levels.Recently,he has leftacademia to undertake industrial research.NONLINEAR TIME SERIES ANALYSIS HOLGER KANTZ AND THOMAS SCHREIBERMax Planck Institute for the Physics of Complex Systems,DresdenMore informationMore informationpublished by the press syndicate of the university of cambridgeThe Pitt Building,Trumpington Street,Cambridge,United Kingdomcambridge university pressThe Edinburgh Building,Cambridge CB22RU,UK40West20th Street,New York,NY10011–4211,USA477Williamstown Road,Port Melbourne,VIC3207,AustraliaRuiz de Alarc´o n13,28014Madrid,SpainDock House,The Waterfront,Cape Town8001,South AfricaC Holger Kantz and Thomas Schreiber,2000,2003This book is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2000Second edition published2003Printed in the United Kingdom at the University Press,CambridgeTypeface Times11/14pt.System L A T E X2ε[tb]A catalogue record for this book is available from the British LibraryLibrary of Congress Cataloguing in Publication dataKantz,Holger,1960–Nonlinear time series analysis/Holger Kantz and Thomas Schreiber.–[2nd ed.].p.cm.Includes bibliographical references and index.ISBN0521821509–ISBN0521529026(paperback)1.Time-series analysis.2.Nonlinear theories.I.Schreiber,Thomas,1963–II.TitleQA280.K3552003519.5 5–dc212003044031ISBN0521821509hardbackISBN0521529026paperbackThe publisher has used its best endeavours to ensure that the URLs for external websites referred to in this bookare correct and active at the time of going to press.However,the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.More informationContentsPreface to thefirst edition page xiPreface to the second edition xiiiAcknowledgements xvI Basic topics11Introduction:why nonlinear methods?32Linear tools and general considerations132.1Stationarity and sampling132.2Testing for stationarity152.3Linear correlations and the power spectrum182.3.1Stationarity and the low-frequency component in thepower spectrum232.4Linearfilters242.5Linear predictions273Phase space methods303.1Determinism:uniqueness in phase space303.2Delay reconstruction353.3Finding a good embedding363.3.1False neighbours373.3.2The time lag393.4Visual inspection of data393.5Poincar´e surface of section413.6Recurrence plots434Determinism and predictability484.1Sources of predictability484.2Simple nonlinear prediction algorithm504.3Verification of successful prediction534.4Cross-prediction errors:probing stationarity564.5Simple nonlinear noise reduction58vMore informationvi Contents5Instability:Lyapunov exponents655.1Sensitive dependence on initial conditions655.2Exponential divergence665.3Measuring the maximal exponent from data696Self-similarity:dimensions756.1Attractor geometry and fractals756.2Correlation dimension776.3Correlation sum from a time series786.4Interpretation and pitfalls826.5Temporal correlations,non-stationarity,and space timeseparation plots876.6Practical considerations916.7A useful application:determination of the noise level using thecorrelation integral926.8Multi-scale or self-similar signals956.8.1Scaling laws966.8.2Detrendedfluctuation analysis1007Using nonlinear methods when determinism is weak1057.1Testing for nonlinearity with surrogate data1077.1.1The null hypothesis1097.1.2How to make surrogate data sets1107.1.3Which statistics to use1137.1.4What can go wrong1157.1.5What we have learned1177.2Nonlinear statistics for system discrimination1187.3Extracting qualitative information from a time series1218Selected nonlinear phenomena1268.1Robustness and limit cycles1268.2Coexistence of attractors1288.3Transients1288.4Intermittency1298.5Structural stability1338.6Bifurcations1358.7Quasi-periodicity139II Advanced topics1419Advanced embedding methods1439.1Embedding theorems1439.1.1Whitney’s embedding theorem1449.1.2Takens’s delay embedding theorem1469.2The time lag148More informationContents vii9.3Filtered delay embeddings1529.3.1Derivative coordinates1529.3.2Principal component analysis1549.4Fluctuating time intervals1589.5Multichannel measurements1599.5.1Equivalent variables at different positions1609.5.2Variables with different physical meanings1619.5.3Distributed systems1619.6Embedding of interspike intervals1629.7High dimensional chaos and the limitations of the time delayembedding1659.8Embedding for systems with time delayed feedback17110Chaotic data and noise17410.1Measurement noise and dynamical noise17410.2Effects of noise17510.3Nonlinear noise reduction17810.3.1Noise reduction by gradient descent17910.3.2Local projective noise reduction18010.3.3Implementation of locally projective noise reduction18310.3.4How much noise is taken out?18610.3.5Consistency tests19110.4An application:foetal ECG extraction19311More about invariant quantities19711.1Ergodicity and strange attractors19711.2Lyapunov exponents II19911.2.1The spectrum of Lyapunov exponents and invariantmanifolds20011.2.2Flows versus maps20211.2.3Tangent space method20311.2.4Spurious exponents20511.2.5Almost two dimensionalflows21111.3Dimensions II21211.3.1Generalised dimensions,multi-fractals21311.3.2Information dimension from a time series21511.4Entropies21711.4.1Chaos and theflow of information21711.4.2Entropies of a static distribution21811.4.3The Kolmogorov–Sinai entropy22011.4.4The -entropy per unit time22211.4.5Entropies from time series data226More informationviii Contents11.5How things are related22911.5.1Pesin’s identity22911.5.2Kaplan–Yorke conjecture23112Modelling and forecasting23412.1Linear stochastic models andfilters23612.1.1Linearfilters23712.1.2Nonlinearfilters23912.2Deterministic dynamics24012.3Local methods in phase space24112.3.1Almost model free methods24112.3.2Local linearfits24212.4Global nonlinear models24412.4.1Polynomials24412.4.2Radial basis functions24512.4.3Neural networks24612.4.4What to do in practice24812.5Improved cost functions24912.5.1Overfitting and model costs24912.5.2The errors-in-variables problem25112.5.3Modelling versus prediction25312.6Model verification25312.7Nonlinear stochastic processes from data25612.7.1Fokker–Planck equations from data25712.7.2Markov chains in embedding space25912.7.3No embedding theorem for Markov chains26012.7.4Predictions for Markov chain data26112.7.5Modelling Markov chain data26212.7.6Choosing embedding parameters for Markov chains26312.7.7Application:prediction of surface wind velocities26412.8Predicting prediction errors26712.8.1Predictability map26712.8.2Individual error prediction26812.9Multi-step predictions versus iterated one-step predictions27113Non-stationary signals27513.1Detecting non-stationarity27613.1.1Making non-stationary data stationary27913.2Over-embedding28013.2.1Deterministic systems with parameter drift28013.2.2Markov chain with parameter drift28113.2.3Data analysis in over-embedding spaces283More informationContents ix13.2.4Application:noise reduction for human voice28613.3Parameter spaces from data28814Coupling and synchronisation of nonlinear systems29214.1Measures for interdependence29214.2Transfer entropy29714.3Synchronisation29915Chaos control30415.1Unstable periodic orbits and their invariant manifolds30615.1.1Locating periodic orbits30615.1.2Stable/unstable manifolds from data31215.2OGY-control and derivates31315.3Variants of OGY-control31615.4Delayed feedback31715.5Tracking31815.6Related aspects319A Using the TISEAN programs321A.1Information relevant to most of the routines322A.1.1Efficient neighbour searching322A.1.2Re-occurring command options325A.2Second-order statistics and linear models326A.3Phase space tools327A.4Prediction and modelling329A.4.1Locally constant predictor329A.4.2Locally linear prediction329A.4.3Global nonlinear models330A.5Lyapunov exponents331A.6Dimensions and entropies331A.6.1The correlation sum331A.6.2Information dimension,fixed mass algorithm332A.6.3Entropies333A.7Surrogate data and test statistics334A.8Noise reduction335A.9Finding unstable periodic orbits336A.10Multivariate data336B Description of the experimental data sets338B.1Lorenz-like chaos in an NH3laser338B.2Chaos in a periodically modulated NMR laser340B.3Vibrating string342B.4Taylor–Couetteflow342B.5Multichannel physiological data343More informationx ContentsB.6Heart rate during atrialfibrillation343B.7Human electrocardiogram(ECG)344B.8Phonation data345B.9Postural control data345B.10Autonomous CO2laser with feedback345B.11Nonlinear electric resonance circuit346B.12Frequency doubling solid state laser348B.13Surface wind velocities349References350Index365More informationPreface to thefirst editionThe paradigm of deterministic chaos has influenced thinking in manyfields of sci-ence.As mathematical objects,chaotic systems show rich and surprising structures.Most appealing for researchers in the applied sciences is the fact that determinis-tic chaos provides a striking explanation for irregular behaviour and anomalies insystems which do not seem to be inherently stochastic.The most direct link between chaos theory and the real world is the analysis oftime series from real systems in terms of nonlinear dynamics.On the one hand,experimental technique and data analysis have seen such dramatic progress that,by now,most fundamental properties of nonlinear dynamical systems have beenobserved in the laboratory.On the other hand,great efforts are being made to exploitideas from chaos theory in cases where the system is not necessarily deterministicbut the data displays more structure than can be captured by traditional methods.Problems of this kind are typical in biology and physiology but also in geophysics,economics,and many other sciences.In all thesefields,even simple models,be they microscopic or phenomenological,can create extremely complicated dynamics.How can one verify that one’s model isa good counterpart to the equally complicated signal that one receives from nature?Very often,good models are lacking and one has to study the system just from theobservations made in a single time series,which is the case for most non-laboratorysystems in particular.The theory of nonlinear dynamical systems provides new toolsand quantities for the characterisation of irregular time series data.The scope ofthese methods ranges from invariants such as Lyapunov exponents and dimensionswhich yield an accurate description of the structure of a system(provided thedata are of high quality)to statistical techniques which allow for classification anddiagnosis even in situations where determinism is almost lacking.This book provides the experimental researcher in nonlinear dynamics with meth-ods for processing,enhancing,and analysing the measured signals.The theorist willbe offered discussions about the practical applicability of mathematical results.ThexiMore informationxii Preface to thefirst editiontime series analyst in economics,meteorology,and otherfields willfind inspira-tion for the development of new prediction algorithms.Some of the techniquespresented here have also been considered as possible diagnostic tools in clinical re-search.We will adopt a critical but constructive point of view,pointing out ways ofobtaining more meaningful results with limited data.We hope that everybody whohas a time series problem which cannot be solved by traditional,linear methodswillfind inspiring material in this book.Dresden and WuppertalNovember1996More informationPreface to the second editionIn afield as dynamic as nonlinear science,new ideas,methods and experimentsemerge constantly and the focus of interest shifts accordingly.There is a continuousstream of new results,and existing knowledge is seen from a different angle aftervery few years.Five years after thefirst edition of“Nonlinear Time Series Analysis”we feel that thefield has matured in a way that deserves being reflected in a secondedition.The modification that is most immediately visible is that the program listingshave been be replaced by a thorough discussion of the publicly available softwareTISEAN.Already a few months after thefirst edition appeared,it became clearthat most users would need something more convenient to use than the bare libraryroutines printed in the book.Thus,together with Rainer Hegger we prepared stand-alone routines based on the book but with input/output functionality and advancedfeatures.Thefirst public release was made available in1998and subsequent releasesare in widespread use now.Today,TISEAN is a mature piece of software thatcovers much more than the programs we gave in thefirst edition.Now,readerscan immediately apply most methods studied in the book on their own data usingTISEAN programs.By replacing the somewhat terse program listings by minuteinstructions of the proper use of the TISEAN routines,the link between book andsoftware is strengthened,supposedly to the benefit of the readers and users.Hencewe recommend a download and installation of the package,such that the exercisescan be readily done by help of these ready-to-use routines.The current edition has be extended in view of enlarging the class of data sets to betreated.The core idea of phase space reconstruction was inspired by the analysis ofdeterministic chaotic data.In contrast to many expectations,purely deterministicand low-dimensional data are rare,and most data fromfield measurements areevidently of different nature.Hence,it was an effort of our scientific work over thepast years,and it was a guiding concept for the revision of this book,to explore thepossibilities to treat other than purely deterministic data sets.xiiiMore informationxiv Preface to the second editionThere is a whole new chapter on non-stationary time series.While detectingnon-stationarity is still briefly discussed early on in the book,methods to deal withmanifestly non-stationary sequences are described in some detail in the secondpart.As an illustration,a data source of lasting interest,human speech,is used.Also,a new chapter deals with concepts of synchrony between systems,linear andnonlinear correlations,information transfer,and phase synchronisation.Recent attempts on modelling nonlinear stochastic processes are discussed inChapter12.The theoretical framework forfitting Fokker–Planck equations to datawill be reviewed and evaluated.While Chapter9presents some progress that hasbeen made in modelling input–output systems with stochastic but observed inputand on the embedding of time delayed feedback systems,the chapter on mod-elling considers a data driven phase space approach towards Markov chains.Windspeed measurements are used as data which are best considered to be of nonlinearstochastic nature despite the fact that a physically adequate mathematical model isthe deterministic Navier–Stokes equation.In the chapter on invariant quantities,new material on entropy has been included,mainly on the -and continuous entropies.Estimation problems for stochastic ver-sus deterministic data and data with multiple length and time scales are discussed.Since more and more experiments now yield good multivariate data,alternativesto time delay embedding using multiple probe measurements are considered at var-ious places in the text.This new development is also reflected in the functionalityof the TISEAN programs.A new multivariate data set from a nonlinear semicon-ductor electronic circuit is introduced and used in several places.In particular,adifferential equation has been successfully established for this system by analysingthe data set.Among other smaller rearrangements,the material from the former chapter“Other selected topics”,has been relocated to places in the text where a connectioncan be made more naturally.High dimensional and spatio-temporal data is now dis-cussed in the context of embedding.We discuss multi-scale and self-similar signalsnow in a more appropriate way right after fractal sets,and include recent techniquesto analyse power law correlations,for example detrendedfluctuation analysis.Of course,many new publications have appeared since1997which are potentiallyrelevant to the scope of this book.At least two new monographs are concerned withthe same topic and a number of review articles.The bibliography has been updatedbut remains a selection not unaffected by personal preferences.We hope that the extended book will prove its usefulness in many applicationsof the methods and further stimulate thefield of time series analysis.DresdenDecember2002More informationAcknowledgementsIf there is any feature of this book that we are proud of,it is the fact that almost allthe methods are illustrated with real,experimental data.However,this is anythingbut our own achievement–we exploited other people’s work.Thus we are deeplyindebted to the experimental groups who supplied data sets and granted permissionto use them in this book.The production of every one of these data sets requiredskills,experience,and equipment that we ourselves do not have,not forgetting thehours and hours of work spent in the laboratory.We appreciate the generosity ofthe following experimental groups:NMR laser.Our contact persons at the Institute for Physics at Z¨u rich University were Leci Flepp and Joe Simonet;the head of the experimental group is E.Brun.(See AppendixB.2.)Vibrating string.Data were provided by Tim Molteno and Nick Tufillaro,Otago University, Dunedin,New Zealand.(See Appendix B.3.)Taylor–Couetteflow.The experiment was carried out at the Institute for Applied Physics at Kiel University by Thorsten Buzug and Gerd Pfister.(See Appendix B.4.) Atrialfibrillation.This data set is taken from the MIT-BIH Arrhythmia Database,collected by G.B.Moody and R.Mark at Beth Israel Hospital in Boston.(See Appendix B.6.) Human ECG.The ECG recordings we used were taken by Petr Saparin at Saratov State University.(See Appendix B.7.)Foetal ECG.We used noninvasively recorded(human)foetal ECGs taken by John F.Hofmeister as the Department of Obstetrics and Gynecology,University of Colorado,Denver CO.(See Appendix B.7.)Phonation data.This data set was made available by Hanspeter Herzel at the Technical University in Berlin.(See Appendix B.8.)Human posture data.The time series was provided by Steven Boker and Bennett Bertenthal at the Department of Psychology,University of Virginia,Charlottesville V A.(SeeAppendix B.9.)xvMore informationxvi AcknowledgementsAutonomous CO2laser with feedback.The data were taken by Riccardo Meucci and Marco Ciofini at the INO in Firenze,Italy.(See Appendix B.10.)Nonlinear electric resonance circuit.The experiment was designed and operated by M.Diestelhorst at the University of Halle,Germany.(See Appendix B.11.)Nd:YAG laser.The data we use were recorded in the University of Oldenburg,where we wish to thank Achim Kittel,Falk Lange,Tobias Letz,and J¨u rgen Parisi.(See AppendixB.12.)We used the following data sets published for the Santa Fe Institute Time SeriesContest,which was organised by Neil Gershenfeld and Andreas Weigend in1991:NH3laser.We used data set A and its continuation,which was published after the contest was closed.The data was supplied by U.H¨u bner,N.B.Abraham,and C.O.Weiss.(SeeAppendix B.1.)Human breath rate.The data we used is part of data set B of the contest.It was submitted by Ari Goldberger and coworkers.(See Appendix B.5.)During the composition of the text we asked various people to read all or part of themanuscript.The responses ranged from general encouragement to detailed technicalcomments.In particular we thank Peter Grassberger,James Theiler,Daniel Kaplan,Ulrich Parlitz,and Martin Wiesenfeld for their helpful remarks.Members of ourresearch groups who either contributed by joint work to our experience and knowl-edge or who volunteered to check the correctness of the text are Rainer Hegger,Andreas Schmitz,Marcus Richter,Mario Ragwitz,Frank Schm¨u ser,RathinaswamyBhavanan Govindan,and Sharon Sessions.We have also considerably profited fromcomments and remarks of the readers of thefirst edition of the book.Their effortin writing to us is gratefully appreciated.Last but not least we acknowledge the encouragement and support by SimonCapelin from Cambridge University Press and the excellent help in questions ofstyle and English grammar by Sheila Shepherd.。

大 学 化 学Univ. Chem. 2022, 37 (7), 2204035 (1 of 7)收稿：2022-04-12；录用：2022-05-09；网络发表：2022-05-20 *通讯作者，Email:***************.cn•师生笔谈• doi: 10.3866/PKU.DXHX202204035 关于Arrhenius 公式的几点讨论张晨曦，苏涵，张树永*山东大学化学与化工学院，济南 250100摘要：Arrhenius 公式是最重要的化学动力学经验公式，提出百余年来，科学家对其的讨论从未停止。

本文溯源了该公式产生的历史过程，进一步说明其与van’t Hoff 方程的联系。

针对普遍接受的Tolman 活化能的物理意义进行了讨论，给出了修正建议。

对活化能随温度变化的情况进行了讨论，对Arrhenius 公式的适用温度范围进行了量化说明，建议对于分子结构较复杂的反应，即便在通常温度范围内，亦应采用三常数公式。

对物理化学教材中与Arrhenius 公式具有相似形式的公式进行了概括，说明了其相似性的本质。

相关讨论有助于师生更正确地理解和使用Arrhenius 公式。

关键词：Arrhenius 公式；Tolman 活化能；温度适用范围；热力学关系 中图分类号：G64；O6Discussions on the Arrhenius EquationChenxi Zhang, Han Su, Shuyong Zhang *School of Chemistry and Chemical Engineering, Shandong University, Jinan 250100, ChinaAbstract: The Arrhenius equation is one of the most important equations in chemical kinetics and has been discussed by scientists for more than a century since its establishment. The historical process of the equation and further explanations of its relationship with the van’t Hoff equation are traced. The physical meaning of the Tolman activation energy is discussed with some suggested modifications. The variation of the activation energy with temperature is discussed, and the temperature range for applying the Arrhenius equation is quantitatively analyzed. It is suggested that the three-constant equation be used even in the normal temperature range for reactants with a complex molecular structure. The similarity between some thermodynamic equations and the Arrhenius equation in physical chemistry textbooks are explained. These discussions will help with correctly understanding and applying the Arrhenius equation.Key Words: Arrhenius equation; Tolman activation energy; Temperature validity range;Thermodynamic relationArrhenius 公式于1889年由实验结果得出，是最重要的化学动力学经验公式，是支撑化学动力学理论发展的两大实验基础之一，其所建立的活化能概念更是影响深远。

电子商务论文参考文献精选3篇篇一：电子商务毕业论文参考文献电子商务毕业论文参考文献电子商务毕业论文参考文献(一)世纪的典型特征之一是信息经济时代的到来，信息化的浪潮正在深刻影响着全社会的各个方面。

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a r X i v :h e p -t h /9612048v 1 4 D e c 1996A note on the three dimensional sine–Gordon equationAhmad ShariatiInstitute for Advanced Studies in Basic Sciences,P.O.Box 159Zanjan 45195,Iran.Institute for Studies in Theoretical Physics and Mathematics,P.O.Box 5531Tehran 19395,Iran.AbstractUsing a simple ansatz for the solutions of the three dimensional generalization of the sine–Gordon and Toda model introduced by Konopelchenko and Rogers,a class of solutions is found by elementary methods.It is also shown that these equations are not evolution equations in the sense that sotution to the initial value problem is not unique.Integrable models in more than two independent variables are of interest [1].One class of such models is the generalized Loewner systems introduced by Konopelchenko and Rogers [2,3].These are third order partial differential equations which are first order in z which is to be the temporal variable.Theseequations are studied using sophisticated methods such as Lie–B¨a cklund transformation [4],¯∂dressing method [3,5],and Painlev´e analysis [6].In the following we consider the three dimensional generalizations of the sine–Gordon and Toda model asintroduced in [3].The observation is that a large class of solutions to these partial differential equations and an important property of these equations can be obtained by a very simple and elementary method.We begin with the so called three dimensional sine–Gordon equation [3]which reads asθzx sin θ y+θx θzy −θy θzxsin f+(u 2x −u 2y )f uzhappening consider the wave equation in light cone coordinates.The equationθuz=0has the known solutionθ(u,z)=f(u)+g(z),where f and g are two arbitrary functions.Now the‘initial’value problemθ(u,0)=f(u)has infinite solutions:Any function g(z)such that g(0)=0gives a solution θ(u,z)=f(u)+g(z).In other words,specifying the value ofθon a lightlike line does not lead to a unique solution;one has to specify the value ofθ,considered as a fuction of(t,x)=(u−z,u+z), and itsfirst temporal derivative on a spacelike line to get a unique solution.Therefore,the argument preceding this paragraph shows that equation1is not a good evolution equation in the sense that z is not a temporal variable.The non–uniqueness of the solution to the initial value problem of equation1can also be seen from type A solutions as follows.The sine–Gordon equation f uz=sin f has the following solutionf(u,z;α)=4tan−1eα−1u+αz(4)whereαis a non–zero real parameter.Take a function u(x,y)which is a solution to the wave equation and construct the following two solutions to equation1θ(x,y,z)=f(αu(x,y),z;α)=4tan−1e u(x,y)+αz(5)θ(x,y,z)=f(u(x,y),z;1)=4tan−1e u(x,y)+z(6) These two solutions agree at z=0but disagree at z=0.Now we turn to the three dimensional generalization of the Toda model which is given by a2×2Cartan matrix Kαβ.The same ansatz leads to solutions for this model.The differential equatin ise− βKαβψβψαzx−σ2 e− βKαβψβψαzy y=0α=1,2.(7)xHereαis an index while subscripts x,y,and z denote partial differentiation as usual.Now we use the following ansatz:ψα(x,y,z)=fα(u(x,y),z).(8) Equation7,then,becomes(u xx−σ2u yy)e− βKαβfβ+(u2x−σ2u2y) e− βKαβfβfαzu u=0.(9) From this it is evident that equation7has the following two solutions:A:ψα(x,y,z)=fα(u(x,y),z)where u(x,y)=g(x+σ−1y)+h(x−σ−1y)for two arbitrary functions g and h provided that fα(u,z)are solutions to the two dimensional Toda equation fαzu=ce− βKαβfβ. Here c is a constant.B:ψα(x,y,z)=fα(x±σ−1y,z)for arbitrary functions fαof two variables.We conclude this letter by the following remarks.1.On settingσ=0in7we get a third order partial differential equation in two variables x and z, which after afirst integration leads toψαzx=c(z)e βKαβψβ,where c is an arbitrary function of z. By redefining z,c can be considered as a constant.This is the familiar two dimensional Toda model.2Therefore,it must be true that theσ→0limit of solutions A and B lead to solutions of the two dimensional model.In doing so one must be careful because the wave equation u xx−σ2u yy=0in that limit leads to u=ax+c,for two constants a and c.In fact,c may be a function of y but this dependence is not relevant.Therefore,type A solutions actually become the solutions of the two dimensional model. For type B solutions,we note that the limit of u2x−σ2u2y=0asσ→0leads to u=constant.Now,a glance at8shows that type B solutions,in this limit,read asψα(x,y,z)=fα(z)for arbitrary fαwhich is trivially a solution of7.This shows that type B solutions are peculiar to the third order equation7.2.Settingσ2→−σ2in7leads to a three dimensional model which has solutions of type A for any harmonic function u(x,y).Solutions of type B disappear.The same is true for the three dimensional sine–Gordon equation:On changing the sign of the second term in equation1,one gets the following partial differential equation which has solutions of type A if u(x,y)is a harmonic function and f is a solution of the two dimensional sine–Gordon equation. θzxsinθ y+θxθzy−θyθzx。

a r X i v :n l i n /0012025v 3 [n l i n .S I ] 10 M a y 2001On the Asymptotic Expansion of the Solutions of the Separated Nonlinear Schr¨o dinger EquationA.A.Kapaev,St Petersburg Department of Steklov Mathematical Institute,Fontanka 27,St Petersburg 191011,Russia,V.E.Korepin,C.N.Yang Institute for Theoretical Physics,State University of New York at Stony Brook,Stony Brook,NY 11794-3840,USAAbstractNonlinear Schr¨o dinger equation with the Schwarzian initial data is important in nonlinear optics,Bose condensation and in the theory of strongly correlated electrons.The asymptotic solutions in the region x/t =O (1),t →∞,can be represented as a double series in t −1and ln t .Our current purpose is the description of the asymptotics of the coefficients of the series.MSC 35A20,35C20,35G20Keywords:integrable PDE,long time asymptotics,asymptotic expansion1IntroductionA coupled nonlinear dispersive partial differential equation in (1+1)dimension for the functions g +and g −,−i∂t g +=12∂2x g −+4g 2−g +,(1)called the separated Nonlinear Schr¨o dinger equation (sNLS),contains the con-ventional NLS equation in both the focusing and defocusing forms as g +=¯g −or g +=−¯g −,respectively.For certain physical applications,e.g.in nonlin-ear optics,Bose condensation,theory of strongly correlated electrons,see [1]–[9],the detailed information on the long time asymptotics of solutions with initial conditions rapidly decaying as x →±∞is quite useful for qualitative explanation of the experimental phenomena.Our interest to the long time asymptotics for the sNLS equation is inspired by its application to the Hubbard model for one-dimensional gas of strongly correlated electrons.The model explains a remarkable effect of charge and spin separation,discovered experimentally by C.Kim,Z.-X.M.Shen,N.Motoyama,H.Eisaki,hida,T.Tohyama and S.Maekawa [19].Theoretical justification1of the charge and spin separation include the study of temperature dependent correlation functions in the Hubbard model.In the papers[1]–[3],it was proven that time and temperature dependent correlations in Hubbard model can be described by the sNLS equation(1).For the systems completely integrable in the sense of the Lax representa-tion[10,11],the necessary asymptotic information can be extracted from the Riemann-Hilbert problem analysis[12].Often,the fact of integrability implies the existence of a long time expansion of the generic solution in a formal series, the successive terms of which satisfy some recurrence relation,and the leading order coefficients can be expressed in terms of the spectral data for the associ-ated linear system.For equation(1),the Lax pair was discovered in[13],while the formulation of the Riemann-Hilbert problem can be found in[8].As t→∞for x/t bounded,system(1)admits the formal solution given byg+=e i x22+iν)ln4t u0+∞ n=12n k=0(ln4t)k2t −(1t nv nk ,(2)where the quantitiesν,u0,v0,u nk and v nk are some functions ofλ0=−x/2t.For the NLS equation(g+=±¯g−),the asymptotic expansion was suggested by M.Ablowitz and H.Segur[6].For the defocusing NLS(g+=−¯g−),the existence of the asymptotic series(2)is proven by P.Deift and X.Zhou[9] using the Riemann-Hilbert problem analysis,and there is no principal obstacle to extend their approach for the case of the separated NLS equation.Thus we refer to(2)as the Ablowitz-Segur-Deift-Zhou expansion.Expressions for the leading coefficients for the asymptotic expansion of the conventional NLS equation in terms of the spectral data were found by S.Manakov,V.Zakharov, H.Segur and M.Ablowitz,see[14]–[16].The general sNLS case was studied by A.Its,A.Izergin,V.Korepin and G.Varzugin[17],who have expressed the leading order coefficients u0,v0andν=−u0v0in(2)in terms of the spectral data.The generic solution of the focusing NLS equation contains solitons and radiation.The interaction of the single soliton with the radiation was described by Segur[18].It can be shown that,for the generic Schwarzian initial data and generic bounded ratio x/t,|c−xthese coefficients as well as for u n,2n−1,v n,2n−1,wefind simple exact formulaeu n,2n=u0i n(ν′)2n8n n!,(3)and(20)below.We describe coefficients at other powers of ln t using the gener-ating functions which can be reduced to a system of polynomials satisfying the recursion relations,see(24),(23).As a by-product,we modify the Ablowitz-Segur-Deift-Zhou expansion(2),g+=exp i x22+iν)ln4t+i(ν′)2ln24t2] k=0(ln4t)k2t −(18t∞n=02n−[n+1t n˜v n,k.(4)2Recurrence relations and generating functions Substituting(2)into(1),and equating coefficients of t−1,wefindν=−u0v0.(5) In the order t−n,n≥2,equating coefficients of ln j4t,0≤j≤2n,we obtain the recursion−i(j+1)u n,j+1+inu n,j=νu n,j−iν′′8u n−1,j−2−−iν′8u′′n−1,j+nl,k,m=0l+k+m=nα=0, (2)β=0, (2)γ=0, (2)α+β+γ=ju l,αu k,βv m,γ,(6) i(j+1)v n,j+1−inv n,j=νv n,j+iν′′8v n−1,j−2++iν′8v′′n−1,j+nl,k,m=0l+k+m=nα=0, (2)β=0, (2)γ=0, (2)α+β+γ=ju l,αv k,βv m,γ,(7)where the prime means differentiation with respect toλ0=−x/(2t).Master generating functions F(z,ζ),G(z,ζ)for the coefficients u n,k,v n,k are defined by the formal seriesF(z,ζ)= n,k u n,k z nζk,G(z,ζ)= n,k v n,k z nζk,(8)3where the coefficients u n,k,v n,k vanish for n<0,k<0and k>2n.It is straightforward to check that the master generating functions satisfy the nonstationary separated Nonlinear Schr¨o dinger equation in(1+2)dimensions,−iFζ+izF z= ν−iν′′8zζ2 F−iν′8zF′′+F2G,iGζ−izG z= ν+iν′′8zζ2 G+iν′8zG′′+F G2.(9) We also consider the sectional generating functions f j(z),g j(z),j≥0,f j(z)=∞n=0u n,2n−j z n,g j(z)=∞n=0v n,2n−j z n.(10)Note,f j(z)≡g j(z)≡0for j<0because u n,k=v n,k=0for k>2n.The master generating functions F,G and the sectional generating functions f j,g j are related by the equationsF(zζ−2,ζ)=∞j=0ζ−j f j(z),G(zζ−2,ζ)=∞j=0ζ−j g j(z).(11)Using(11)in(9)and equating coefficients ofζ−j,we obtain the differential system for the sectional generating functions f j(z),g j(z),−2iz∂z f j−1+i(j−1)f j−1+iz∂z f j==νf j−z iν′′8f j−ziν′8f′′j−2+jk,l,m=0k+l+m=jf k f lg m,2iz∂z g j−1−i(j−1)g j−1−iz∂z g j=(12)=νg j+z iν′′8g j+ziν′8g′′j−2+jk,l,m=0k+l+m=jf kg l g m.Thus,the generating functions f0(z),g0(z)for u n,2n,v n,2n solve the systemiz∂z f0=νf0−z (ν′)28g0+f0g20.(13)The system implies that the product f0(z)g0(z)≡const.Since f0(0)=u0and g0(0)=v0,we obtain the identityf0g0(z)=−ν.(14) Using(14)in(13),we easilyfindf0(z)=u0e i(ν′)28n n!z n,4g0(z)=v0e−i(ν′)28n n!z n,(15)which yield the explicit expressions(3)for the coefficients u n,2n,v n,2n.Generating functions f1(z),g1(z)for u n,2n−1,v n,2n−1,satisfy the differential system−2iz∂z f0+iz∂z f1=νf1−z iν′′8f1−ziν′8g0−z(ν′)24g′0+f1g20+2f0g0g1.(16)We will show that the differential system(16)for f1(z)and g1(z)is solvable in terms of elementary functions.First,let us introduce the auxiliary functionsp1(z)=f1(z)g0(z).These functions satisfy the non-homogeneous system of linear ODEs∂z p1=iν4−ν′′4f′0z(p1+q1)−i(ν′)28−ν′g0,(17)so that∂z(q1+p1)=−(ν2)′′8z,p1(z)= −iνν′′8−ν′u′032z2,g1(z)=q1(z)g0(z),g0(z)=v0e−i(ν′)24−ν′′4v0 z+i(ν′)2ν′′4−ν′′4u0 ,v1,1=v0 iνν′′8−ν′v′0u n,2n −1=−2u 0i n −1(ν′)2(n −1)n −1ν′′u 0,n ≥2,v n,2n −1=−2v 0(−i )n −1(ν′)2(n −1)n −1ν′′v 0,n ≥2.Generating functions f j (z ),g j (z )for u n,2n −j ,v n,2n −j ,j ≥2,satisfy the differential system (12).Similarly to the case j =1above,let us introduce the auxiliary functions p j and q j ,p j =f jg 0.(21)In the terms of these functions,the system (12)reads,∂z p j =iνz(p j +q j )+b j ,(22)wherea j =2∂z p j −1+i (ν′)28−j −14(p j −1f 0)′8f 0+iν4−ν′′zq j −1−−ν′g 0+i(q j −2g 0)′′zj −1 k,l,m =0k +l +m =jp k q l q m .(23)With the initial condition p j (0)=q j (0)=0,the system is easily integrated and uniquely determines the functions p j (z ),q j (z ),p j (z )= z 0a j (ζ)dζ+iνzdζζζdξ(a j (ξ)+b j (ξ)).(24)These equations with expressions (23)together establish the recursion relationfor the functions p j (z ),q j (z ).In terms of p j (z )and q j (z ),expansion (2)readsg +=ei x22+iν)ln 4t +i(ν′)2ln 24tt2t−(18tv 0∞ j =0q j ln 24tln j 4t.(25)6Let a j (z )and b j (z )be polynomials of degree M with the zero z =0of multiplicity m ,a j (z )=M k =ma jk z k,b j (z )=Mk =mb jk z k .Then the functions p j (z )and q j (z )(24)arepolynomials of degree M +1witha zero at z =0of multiplicity m +1,p j (z )=M +1k =m +11k(a j,k −1+b j,k −1)z k ,q j (z )=M +1k =m +11k(a j,k −1+b j,k −1) z k.(26)On the other hand,a j (z )and b j (z )are described in (23)as the actions of the differential operators applied to the functions p j ′,q j ′with j ′<j .Because p 0(z )=q 0(z )≡1and p 1(z ),q 1(z )are polynomials of the second degree and a single zero at z =0,cf.(19),it easy to check that a 2(z )and b 2(z )are non-homogeneous polynomials of the third degree such thata 2,3=−(ν′)4(ν′′)2210(2+iν),(27)a 2,0=−iνν′′8−ν′u ′08u 0,b 2,0=iνν′′8−ν′v ′08v 0.Thus p 2(z )and q 2(z )are polynomials of the fourth degree with a single zero at z =0.Some of their coefficients arep 2,4=q 2,4=−(ν′)4(ν′′)24−(1+2iν)ν′′8u 0−ν(u ′0)24−(1−2iν)ν′′8v 0−ν(v ′0)22.Proof .The assertion holds true for j =0,1,2.Let it be correct for ∀j <j ′.Then a j ′(z )and b j ′(z )are defined as the sum of polynomials.The maximal de-grees of such polynomials are deg (p j ′−1f 0)′/f 0 =2j ′−1,deg (q j ′−1g 0)′/g 0 =72j′−1,anddeg 1z j′−1 α,β,γ=0α+β+γ=j′pαqβqγ =2j′−1. Thus deg a j′(z)=deg b j′(z)≤2j′−1,and deg p j′(z)=deg q j′(z)≤2j′.Multiplicity of the zero at z=0of a j′(z)and b j′(z)is no less than the min-imal multiplicity of the summed polynomials in(23),but the minor coefficients of the polynomials2∂z p j′−1and−(j−1)p j′−1/z,as well as of2∂z q j′−1and −(j−1)q j′−1/z may cancel each other.Let j′=2k be even.Thenm j′=min m j′−1;m j′−2+1;minα,β,γ=0,...,j′−1α+β+γ=j′mα+mβ+mγ =j′2 . Let j′=2k−1be odd.Then2m j′−1−(j′−1)=0,andm j′=min m j′−1+1;m j′−2+1;minα,β,γ=0,...,j′−1α+β+γ=j′mα+mβ+mγ =j′+12]p j,k z k,q j(z)=2jk=[j+12]z nn−[j+18k k!,g j(z)=v0∞n=[j+12]k=max{0;n−2j}q j,n−k(−i)k(ν′)2k2]k=max{0;n−2j}p j,n−ki k(ν′)2k2]k=max{0;n−2j}q j,n−k(−i)k(ν′)2kIn particular,the leading asymptotic term of these coefficients as n→∞and j fixed is given byu n,2n−j=u0p j,2j i n−2j(ν′)2(n−2j)n) ,v n,2n−j=v0q j,2j (−i)n−2j(ν′)2(n−2j)n) .(32)Thus we have reduced the problem of the evaluation of the asymptotics of the coefficients u n,2n−j v n,2n−j for large n to the computation of the leading coefficients of the polynomials p j(z),q j(z).In fact,using(24)or(26)and(23), it can be shown that the coefficients p j,2j,q j,2j satisfy the recurrence relationsp j,2j=−i (ν′)2ν′′2jj−1k,l,m=0k+l+m=jp k,2k p l,2l q m,2m++ν(ν′)2ν′′4j2j−1k,l,m=0k+l+m=jp k,2k(p l,2l−q l,2l)q m,2m,q j,2j=i (ν′)2ν′′2jj−1k,l,m=0k+l+m=jp k,2k q l,2l q m,2m−(33)−ν(ν′)2ν′′4j2j−1k,l,m=0k+l+m=jp k,2k(p l,2l−q l,2l)q m,2m.Similarly,the coefficients u n,0,v n,0for the non-logarithmic terms appears from(31)for j=2n,and are given simply byu n,0=u0p2n,n,v n,0=v0q2n,n.(34) Thus the problem of evaluation of the asymptotics of the coefficients u n,0,v n,0 for n large is equivalent to computation of the asymptotics of the minor coeffi-cients in the polynomials p j(z),q j(z).However,the last problem does not allow a straightforward solution because,according to(8),the sectional generating functions for the coefficients u n,0,v n,0are given byF(z,0)=∞n=0u n,0z n,G(z,0)=∞n=0v n,0z n,and solve the separated Nonlinear Schr¨o dinger equation−iFζ+izF z=νF+18zG′′+F G2.(35)93DiscussionOur consideration based on the use of generating functions of different types reveals the asymptotic behavior of the coefficients u n,2n−j,v n,2n−j as n→∞and jfixed for the long time asymptotic expansion(2)of the generic solution of the sNLS equation(1).The leading order dependence of these coefficients on n is described by the ratio a n2+d).The investigation of theRiemann-Hilbert problem for the sNLS equation yielding this estimate will be published elsewhere.Acknowledgments.We are grateful to the support of NSF Grant PHY-9988566.We also express our gratitude to P.Deift,A.Its and X.Zhou for discussions.A.K.was partially supported by the Russian Foundation for Basic Research under grant99-01-00687.He is also grateful to the staffof C.N.Yang Institute for Theoretical Physics of the State University of New York at Stony Brook for hospitality during his visit when this work was done. References[1]F.G¨o hmann,V.E.Korepin,Phys.Lett.A260(1999)516.[2]F.G¨o hmann,A.R.Its,V.E.Korepin,Phys.Lett.A249(1998)117.[3]F.G¨o hmann,A.G.Izergin,V.E.Korepin,A.G.Pronko,Int.J.Modern Phys.B12no.23(1998)2409.[4]V.E.Zakharov,S.V.Manakov,S.P.Novikov,L.P.Pitaevskiy,Soli-ton theory.Inverse scattering transform method,Moscow,Nauka,1980.[5]F.Calogero,A.Degasperis,Spectral transforms and solitons:toolsto solve and investigate nonlinear evolution equations,Amsterdam-New York-Oxford,1980.[6]M.J.Ablowitz,H.Segur,Solitons and the inverse scattering trans-form,SIAM,Philadelphia,1981.10[7]R.K.Dodd,J.C.Eilbeck,J.D.Gibbon,H.C.Morris,Solitons andnonlinear wave equations,Academic Press,London-Orlando-San Diego-New York-Toronto-Montreal-Sydney-Tokyo,1982.[8]L.D.Faddeev,L.A.Takhtajan,Hamiltonian Approach to the Soli-ton Theory,Nauka,Moscow,1986.[9]P.Deift,X.Zhou,Comm.Math.Phys.165(1995)175.[10]C.S.Gardner,J.M.Greene,M.D.Kruskal,R.M.Miura,Phys.Rev.Lett.19(1967)1095.[11]x,Comm.Pure Appl.Math.21(1968)467.[12]V.E.Zakharov,A.B.Shabat,Funkts.Analiz Prilozh.13(1979)13.[13]V.E.Zakharov,A.B.Shabat,JETP61(1971)118.[14]S.V.Manakov,JETP65(1973)505.[15]V.E.Zakharov,S.V.Manakov,JETP71(1973)203.[16]H.Segur,M.J.Ablowitz,J.Math.Phys.17(1976)710.[17]A.R.Its,A.G.Izergin,V.E.Korepin,G.G.Varzugin,Physica D54(1992)351.[18]H.Segur,J.Math.Phys.17(1976)714.[19]C.Kim,Z.-X.M.Shen,N.Motoyama,H.Eisaki,hida,T.To-hyama and S.Maekawa Phys Rev Lett.82(1999)802[20]A.R.Its,SR Izvestiya26(1986)497.11。

latex洛必达法则-回复洛必达法则（L'Hôpital's Rule）是微积分中的一种重要的极限求解法则，可以有效地解决一些无法直接求解的极限问题。

它的名称来自于法国数学家洛必达（Guillaume François Antoine, Marquis de l'Hôpital），他在1696年首次提出了这个法则。

洛必达法则的表达形式是利用导数来求解极限，是微积分中的一个重要工具。

洛必达法则适用于极限形式为“\frac{0}{0}”或“\frac{\infty}{\infty}”的情况。

具体来说，设函数f(x)和g(x)在某个点a附近可导，并且在该点处满足\lim_{x \to a}{f(x)}=0和\lim_{x \to a}{g(x)}=0，或者\lim_{x \to a}{f(x)}=\infty和\lim_{x \to a}{g(x)}=\infty，则有以下两种情况：情况一：如果\lim_{x \to a}{\frac{f'(x)}{g'(x)}}存在或为\infty，那么\[\lim_{x \to a}{\frac{f(x)}{g(x)}}=\lim_{x \to a}{\frac{f'(x)}{g'(x)}}\]情况二：如果\lim_{x \to a}{\frac{f'(x)}{g'(x)}}不存在或为\infty，但\lim_{x \to a}{\frac{f(x)}{g(x)}}存在或为\infty，那么\[\lim_{x \to a}{\frac{f(x)}{g(x)}}=\lim_{x \to a}{\frac{f'(x)}{g'(x)}}\]下面将通过一些例子来逐步说明洛必达法则的应用。

例子一：求极限\lim_{x \to 0}{\frac{\sin(x)}{x}}首先我们注意到，当x接近于0时，\sin(x)和x都接近于0，属于\frac{0}{0}的形式。

Maxwell Equations and Magnetic Monopoles Sebastiano Tosto【期刊名称】《Journal of Applied Mathematics and Physics》【年(卷),期】2024(12)3【摘要】The manuscript introduces an “ab initio” quantum model to deduce the Maxwell equations. After general considerations and laying out the model’s theoretical framework, these equations can be derived alongside a broad variety of other results. Specifically, a corollary of the present model proposes a possible mechanism underlying the formation of magnetic monopoles and allows estimating their formation energy in order of magnitude.【总页数】27页(P737-763)【作者】Sebastiano Tosto【作者单位】ENEA, Rome, Italy【正文语种】中文【中图分类】O44【相关文献】grange-Maxwell equation and magnetic saturation parametric resonance of generator set2.Improved Modelling of Soil Loss in El Badalah Basin: Comparing the Performance of the Universal Soil Loss Equation,Revised Universal Soil Loss Equation and Modified Universal Soil Loss Equation Models by Using the Magnetic and Gravimetric Prospection Outcomes3.A Self-Consistent Model on Cylindrical Monopole Plasma Antenna Excited by Surface Wave Based on the Maxwell-Boltzmann Equation4.Magnetic Monopoles and the Quantum Theory of Magnetism in Matter5.New Insight into Magnetic Monopoles in Astrophysical Application因版权原因，仅展示原文概要，查看原文内容请购买。