1986-Thomsen-Weak elastic anisotropy
Thomsen_86弱各向异性理论
1954
Weak Elastic Anisotropy
1955
there are nine such relations, each involving one component of stress and nine components of strain. These nine equations may be written compactly as
*Amoco Production Company. P.O. Box 3385, Tulsa, OK 74102. (‘ 1986 Society of Exploration Geophysicists. All rights reserved.
Manuscriptreceived by the Editor September 9. 1985; revised manuscript received February 24, 1986.
(2) The mathematical equations for anisotropic wave propagation are algebraically daunting, even for this simple case. The purpose of this paper is to point out that in most cases of interest to geophysicists the anisotropy is weak (l&20 percent). allowing the equations to simplify considerably. In fact, the equations become so simple that certain basic conclusions
冉绍尔-汤森效应实验
冉绍尔-汤森效应实验【摘要】加速电子与充氙闸流管中的氙原子碰撞,电子被散射,把闸流管先后浸入77K 液氮和在室温下测俩观众的栅极及板极电流。
得出散射概率、散射截面与电子能量的关系,低能电子与气体原子的散射几率与电子速度的关系,验证冉绍尔-汤森效应。
用量子力学解释这一效应测量氙原子的电离电位。
【实验原理】当灯丝加热后,就有电子自阴极逸出,设阴极电流为K I ,电子在加速电压的作用下,有一部分电子在到达栅极之前,被屏极接收,形成电流1S I ;有一部分穿越屏极上的矩形孔,形成电流0I ,由于屏极上的矩形孔与板极P 之间是一个等势空间,所以电子穿越矩形孔后就以恒速运动,受到气体原子散射的电子则到达屏极,形成散射电流2S I ;而未受到散射的电子则到达板极P ,形成板流P I ,因此有10S K I I I +=21S S S I I I +=20S P I I I +=电子在等势区内的散射概率为:01I I P PS -= (1)可见,只要分别测量出P I 和0I 即可以求得散射几率。
从上面论述可知,P I 可以直接测得,至于0I 则需要用间接的方法测定。
由于阴极电流K I 分成两部分1S I 和0I ,它们不仅与K I 成比例,而且他们之间也有一定的比例关系,这一比值称为几何因子f ,即有10S I I f =(2)几何因子f 是由电极间相对张角及空间电荷效应所决定,即f 与管子的几何结构及所用的加速电压、阴极电流有关。
将式(2)带入(1)式得到111S PS I I f P -= (3)为了测量几何因子f ,我们把电子碰撞管的管端部分浸入温度为77K 的液氮中,这时,管内掉气体冻结,在这种低温状态下,气体原子的密度很小,对电子的散射可以忽略不计,几何因子f 就等于这时的板流*P I 与屏流*S I 之比,即**=SP I I f (4)如果这时阴极电流和加速电压保持与式(1)和(2)时的相同,那么上式中的f 值与式(3)中掉相等,因此有**-=PS S P S I II I P 11 (5)设L 为出射孔S 到板极P 之间的距离,则)exp(1QL P S --= (6)当f<<1时,由(5)、(6)两式得⎪⎪⎭⎫ ⎝⎛-=**P S S P I I I I L Q ln 1 测量不同的加速电压Ea 下的Ps 的值,即可由上式得到总有效散射截面Q 与a E 的关系曲线。
全面创新质量管理——一个整合性概念
全面创新质量管理——一个整合性概念杨幽红;宋明顺【摘要】考察全面质量管理起源与发展、创新的概念与分类,回顾全面质量管理对创新的影响,分析全面质量管理与创新辩证关系,提出一个整合性概念框架,全面创新质量管理概念、特点和三个重要推力.【期刊名称】《科技管理研究》【年(卷),期】2019(039)010【总页数】5页(P243-247)【关键词】全面质量管理;创新;全面创新质量管理;推力【作者】杨幽红;宋明顺【作者单位】中国计量大学标准化学院,浙江杭州 310018;中国计量大学经济与管理学院,浙江杭州 310018【正文语种】中文在过去半个多世纪,全面质量管理(Total Quality Management,简称TQM)一直被认为是企业维系生存与获得竞争优势的重要工具并在全球范围内广泛传播[1-2]。
在西方国家,很少有企业(特别是制造业)可以忽略TQM[3]。
随着经济全球化和技术的飞速发展,创新在确保企业可持续性经营方面发挥关键作用,被认为是目前市场上的获得竞争优势的源泉[4]。
而在二者之间关系研究上,不同情境下研究结论并不收敛[5-6]。
一些国外学者定量的研究结果显现了两种截然不同的观点:TQM对创新有积极的影响作用[7-11]和有阻碍作用[12-13],另一些学者研究表明:TQM是创新的先决条件,TQM虽然不足以实现创新,但仍是必要的[1]。
国内学者的研究表明,质量管理实践可以直接或间接影响企业的技术创新[2,14-16]。
许庆瑞等[15]分析了全面质量管理对创新正面负面影响,提出了将TQM分为管理控制模式和全员创新模式两个维度[16]。
总体而言,现有关于“TQM”与“创新”关系的研究,认为两者是企业内部不同类型的活动,前者对后者有正面影响、反面影响或一致性关系,而将两者作为一个整体性概念的研究不多。
本文在重新审视TQM对企业创新绩效的影响基础上,探讨TQM与创新的辩证关系,提出了全面创新质量管理的一个整合性概念,并分析了它的特点及三个重要推力。
由含有Mg的钛酸锂构成的锂离子电池用活性物质和锂离子电池[发明专利]
专利名称:由含有Mg的钛酸锂构成的锂离子电池用活性物质和锂离子电池
专利类型:发明专利
发明人:远藤大辅,稻益德雄,温田敏之,片山祯弘
申请号:CN200680009969.9
申请日:20060322
公开号:CN101151746A
公开日:
20080326
专利内容由知识产权出版社提供
摘要:本发明的课题是提供电位平坦部的放电容量优良的锂离子电池用活性物质和提供高容量、高寿命的锂离子电池,特别是提供提高电压平坦性的技术。
本发明的锂离子电池用活性物质是用组成式:Li[LiMgTi]O(0<x<1/2)表示,该组成式是用Mg来取代钛酸锂的元素的一部分而得的,锂离子电池采用该活性物质作为负极活性物质。
申请人:株式会社杰士汤浅
地址:日本京都府
国籍:JP
代理机构:北京银龙知识产权代理有限公司
代理人:钟晶
更多信息请下载全文后查看。
Empirical processes of dependent random variables
2
Preliminaries
n i=1
from R to R. The centered G -indexed empirical process is given by (P n − P )g = 1 n
n
the marginal and empirical distribution functions. Let G be a class of measurabrocesses that have been discussed include linear processes and Gaussian processes; see Dehling and Taqqu (1989) and Cs¨ org˝ o and Mielniczuk (1996) for long and short-range dependent subordinated Gaussian processes and Ho and Hsing (1996) and Wu (2003a) for long-range dependent linear processes. A collection of recent results is presented in Dehling, Mikosch and Sorensen (2002). In that collection Dedecker and Louhichi (2002) made an important generalization of Ossiander’s (1987) result. Here we investigate the empirical central limit problem for dependent random variables from another angle that avoids strong mixing conditions. In particular, we apply a martingale method and establish a weak convergence theory for stationary, causal processes. Our results are comparable with the theory for independent random variables in that the imposed moment conditions are optimal or almost optimal. We show that, if the process is short-range dependent in a certain sense, then the limiting behavior is similar to that of iid random variables in that the limiting distribution is a Gaussian process and the norming √ sequence is n. For long-range dependent linear processes, one needs to apply asymptotic √ expansions to obtain n-norming limit theorems (Section 6.2.2). The paper is structured as follows. In Section 2 we introduce some mathematical preliminaries necessary for the weak convergence theory and illustrate the essence of our approach. Two types of empirical central limit theorems are established. Empirical processes indexed by indicators of left half lines, absolutely continuous functions, and piecewise differentiable functions are discussed in Sections 3, 4 and 5 respectively. Applications to linear processes and iterated random functions are made in Section 6. Section 7 presents some integral and maximal inequalities that may be of independent interest. Some proofs are given in Sections 8 and 9.
The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time seri
904N.E.Huang and others10.Discussion98711.Conclusions991References993 A new method for analysing has been devel-oped.The key part of the methodany complicated data set can be decomposed intoof‘intrinsic mode functions’Hilbert trans-This decomposition method is adaptive,and,highly efficient.Sinceapplicable to nonlinear and non-stationary processes.With the Hilbert transform,Examplesthe classical nonlinear equation systems and dataare given to demonstrate the power new method.data are especially interesting,for serve to illustrate the roles thenonlinear and non-stationary effects in the energy–frequency–time distribution.Keywords:non-stationary time series;nonlinear differential equations;frequency–time spectrum;Hilbert spectral analysis;intrinsic time scale;empirical mode decomposition1.Introductionsensed by us;data analysis serves two purposes:determine the parameters needed to construct the necessary model,and to confirm the model we constructed to represent the phe-nomenon.Unfortunately,the data,whether from physical measurements or numerical modelling,most likely will have one or more of the following problems:(a)the total data span is too short;(b)the data are non-stationary;and(c)the data represent nonlinear processes.Although each of the above problems can be real by itself,the first two are related,for a data section shorter than the longest time scale of a sta-tionary process can appear to be non-stationary.Facing such data,we have limited options to use in the analysis.Historically,Fourier spectral analysis has provided a general method for examin-the data analysis has been applied to all kinds of data.Although the Fourier transform is valid under extremely general conditions(see,for example,Titchmarsh1948),there are some crucial restrictions of Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis905the Fourier spectral analysis:the system must be linear;and the data must be strict-ly periodic or stationary;otherwise,the resulting spectrum will make little physicalsense.to the Fourier spectral analysis methods.Therefore,behoves us review the definitions of stationarity here.According to the traditional definition,a time series,X (t ),is stationary in the wide sense,if,for all t ,E (|X (t )2|)<∞,E (X (t))=m,C (X (t 1),X (t 2))=C (X (t 1+τ),X (t 2+τ))=C (t 1−t 2),(1.1)in whichE (·)is the expected value defined as the ensemble average of the quantity,and C (·)is the covariance function.Stationarity in the wide sense is also known as weak stationarity,covariance stationarity or second-order stationarity (see,forexample,Brockwell &Davis 1991).A time series,X (t ),is strictly stationary,if the joint distribution of [X (t 1),X (t 2),...,X (t n )]and [X (t 1+τ),X (t 2+τ),...,X (t n +τ)](1.2)are the same for all t i and τ.Thus,a strictly stationaryprocess with finite second moments is alsoweakly stationary,but the inverse is not true.Both definitions arerigorous but idealized.Other less rigorous definitions have also beenused;for example,that is stationary within a limited timespan,asymptotically stationary is for any random variableis stationary when τin equations (1.1)or (1.2)approaches infinity.In practice,we can only have data for finite time spans;these defini-tions,we haveto makeapproximations.Few of the data sets,from either natural phenomena or artificial sources,can satisfy these definitions.It may be argued thatthe difficulty of invoking stationarity as well as ergodicity is not on principlebut on practicality:we just cannot have enough data to cover all possible points in thephase plane;therefore,most of the cases facing us are transient in nature.This is the reality;we are forced to face it.Fourier spectral analysis also requires linearity.can be approximated by linear systems,the tendency tobe nonlinear whenever their variations become finite Compounding these complications is the imperfection of or numerical schemes;theinteractionsof the imperfect probes even with a perfect linear systemcan make the final data nonlinear.For the above the available data are ally of finite duration,non-stationary and from systems that are frequently nonlinear,either intrinsicallyor through interactions with the imperfect probes or numerical schemes.Under these conditions,Fourier spectral analysis is of limited use.For lack of alternatives,however,Fourier spectral analysis is still used to process such data.The uncritical use of Fourier spectral analysis the insouciant adoption of the stationary and linear assumptions may give cy range.a delta function will giveProc.R.Soc.Lond.A (1998)906N.E.Huang and othersa phase-locked wide white Fourier spectrum.Here,added to the data in the time domain,Constrained bythese spurious harmonics the wide frequency spectrum cannot faithfully represent the true energy density in the frequency space.More seri-ously,the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give thefinal delta function. Thus,the Fourier components might make mathematical sense,but do not really make physical sense at all.Although no physical process can be represented exactly by a delta function,some data such as the near-field strong earthquake records areFourier spectra.Second,tions;wave-profiles.Such deformations,later,are the direct consequence of nonlinear effects.Whenever the form of the data deviates from a pure sine or cosine function,the Fourier spectrum will contain harmonics.As explained above, both non-stationarity and nonlinearity can induce spurious harmonic components that cause energy spreading.The consequence is the misleading energy–frequency distribution forIn this paper,modemode functions The decomposition is based on the direct extraction of theevent on the time the frequency The decomposition be viewed as an expansion of the data in terms of the IMFs.Then,based on and derived from the data,can serve as the basis of that expansion linear or nonlinear as dictated by the data,Most important of all,it is adaptive.As will locality and adaptivity are the necessary conditions for the basis for expanding nonlinear and non-stationary time orthogonality is not a necessary criterionselection for a nonlinearon the physical time scaleslocal energy and the instantaneous frequencyHilbert transform can give us a full energy–frequency–time distribution of the data. Such a representation is designated as the Hilbert spectrum;it would be ideal for nonlinear and non-stationary data analysis.We have obtained good results and new insights by applying the combination of the EMD and Hilbert spectral analysis methods to various data:from the numerical results of the classical nonlinear equation systems to data representing natural phe-nomena.The classical nonlinear systems serve to illustrate the roles played by the nonlinear effects in the energy–frequency–time distribution.With the low degrees of freedom,they can train our eyes for more complicated cases.Some limitations of this method will also be discussed and the conclusions presented.Before introducing the new method,we willfirst review the present available data analysis methods for non-stationary processes.Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis9072.Review of non-stationary data processing methodsWe willfirstgivea brief survey of themethodsstationary data.are limited to linear systems any method is almost strictly determined according to the special field in which the application is made.The available methods are reviewed as follows.(a )The spectrogramnothing but a limited time window-width Fourier spectral analysis.the a distribution.Since it relies on the tradition-al Fourier spectral analysis,one has to assume the data to be piecewise stationary.This assumption is not always justified in non-stationary data.Even if the data are piecewise stationary how can we guarantee that the window size adopted always coincides with the stationary time scales?What can we learn about the variations longer than the local stationary time scale?Will the collection of the locally station-ary pieces constitute some longer period phenomena?Furthermore,there are also practical difficulties in applying the method:in order to localize an event in time,the window width must be narrow,but,on the other hand,the frequency resolu-tion requires longer time series.These conflicting requirements render this method of limited usage.It is,however,extremely easy to implement with the fast Fourier transform;thus,ithas attracted a wide following.Most applications of this methodare for qualitative display of speech pattern analysis (see,for example,Oppenheim &Schafer 1989).(b )The wavelet analysisThe wavelet approach is essentially an adjustable window Fourier spectral analysiswith the following general definition:W (a,b ;X,ψ)=|a |−1/2∞−∞X (t )ψ∗ t −b ad t,(2.1)in whichψ∗(·)is the basic wavelet function that satisfies certain very general condi-tions,a is the dilation factor and b is the translationof theorigin.Although time andfrequency do not appear explicitly in the transformed result,the variable 1/a givesthe frequency scale and b ,the temporal location of an event.An intuitive physical explanation of equation (2.1)is very simple:W (a,b ;X,ψ)is the ‘energy’of X ofscale a at t =b .Because of this basic form of at +b involvedin thetransformation,it is also knownas affinewavelet analysis.For specific applications,the basic wavelet function,ψ∗(·),can be modified according to special needs,but the form has to be given before the analysis.In most common applications,however,the Morlet wavelet is defined as Gaussian enveloped sine and cosine wave groups with 5.5waves (see,for example,Chan 1995).Generally,ψ∗(·)is not orthogonalfordifferent a for continuous wavelets.Although one can make the wavelet orthogonal by selecting a discrete set of a ,thisdiscrete wavelet analysis will miss physical signals having scale different from theselected discrete set of a .Continuous or discrete,the wavelet analysis is basically a linear analysis.A very appealing feature of the wavelet analysis is that it provides aProc.R.Soc.Lond.A (1998)908N.E.Huang and othersuniform resolution for all the scales.Limited by the size of thebasic wavelet function,the downside of the uniform resolution is uniformly poor resolution.Although wavelet analysis has been available only in the last ten years or so,it hasbecome extremelypopular.Indeed,it is very useful in analysing data with gradualfrequency changes.Since it has an analytic form for the result,it has attracted extensive attention of the applied mathematicians.Most of its applications have been in edge detection and image compression.Limited applications have also been made to the time–frequency distribution in time series (see,for example,Farge 1992;Long et al .1993)andtwo-dimensionalimages (Spedding et al .1993).Versatile as the wavelet analysis is,the problem with the most commonly usedMorlet wavelet is its leakage generated by the limited length of the basic wavelet function,whichmakesthe quantitativedefinitionof the energy–frequency–time dis-tribution difficult.Sometimes,the interpretation of the wavelet can also be counter-intuitive.For example,to define a change occurring locally,one must look for theresult in the high-frequencyrange,for the higher the frequency the more localized thebasic wavelet will be.If a local event occurs only in the low-frequency range,one willstill be forced to look for its effects inthe high-frequencyrange.Such interpretationwill be difficultif it is possible at all (see,for example,Huang et al .1996).Another difficulty of the wavelet analysis is its non-adaptive nature.Once the basic waveletis selected,one will have to use it to analyse all the data.Since the most commonlyused Morlet wavelet is Fourier based,it also suffers the many shortcomings of Fouri-er spectral analysis:it can only give a physically meaningful interpretation to linear phenomena;it can resolve the interwave frequency modulation provided the frequen-cy variationis gradual,but it cannot resolve the intrawave frequency modulation because the basic wavelet has a length of 5.5waves.Inspite of all these problems,wavelet analysisisstillthe bestavailable non-stationary data analysis method so far;therefore,we will use it in this paper as a reference to establish the validity and thecalibration of the Hilbert spectrum.(c )The Wigner–Ville distributionThe Wigner–Ville distribution is sometimes alsoreferred toas the Heisenberg wavelet.By definition,it is the Fourier transform of the central covariance function.For any time series,X (t ),we can define the central variance as C c (τ,t )=X (t −12τ)X ∗(t +12τ).(2.2)Then the Wigner–Ville distribution is V (ω,t )=∞−∞C c (τ,t )e −i ωτd τ.(2.3)This transform has been treated extensively by Claasen &Mecklenbr¨a uker (1980a ,b,c )and by Cohen (1995).It has been extremely popular with the electrical engi-neering community.The difficulty with this method is the severe cross terms as indicated by the exis-tence of negativepowerfor some frequency ranges.Although this shortcoming canbe eliminated by using the Kernel method (see,for example,Cohen 1995),the resultis,then,basically that of a windowed Fourier analysis;therefore,itsuffers all thelim-itations of the Fourier analysis.An extension of this method has been made by Yen(1994),who used the Wigner–Ville distribution to define wave packets that reduce Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis909 a complicated data set to afinite number of simple components.This extension is very powerful and can be applied to a variety of problems.The applications to complicated data,however,require a great amount of judgement.(d)Evolutionary spectrumThe evolutionary spectrum wasfirst proposed by Priestley(1965).The basic idea is to extend the classic Fourier spectral analysis to a more generalized basis:from sine or cosine to a family of orthogonal functions{φ(ω,t)}indexed by time,t,and defined for all realω,the frequency.Then,any real random variable,X(t),can beexpressed asX(t)= ∞−∞φ(ω,t)d A(ω,t),(2.4)in which d A(ω,t),the Stieltjes function for the amplitude,is related to the spectrum asE(|d A(ω,t)|2)=dµ(ω,t)=S(ω,t)dω,(2.5) whereµ(ω,t)is the spectrum,and S(ω,t)is the spectral density at a specific time t,also designated as the evolutionary spectrum.If for eachfixedω,φ(ω,t)has a Fourier transformφ(ω,t)=a(ω,t)e iΩ(ω)t,(2.6) then the function of a(ω,t)is the envelope ofφ(ω,t),andΩ(ω)is the frequency.If, further,we can treatΩ(ω)as a single valued function ofω,thenφ(ω,t)=α(ω,t)e iωt.(2.7) Thus,the original data can be expanded in a family of amplitude modulated trigono-metric functions.The evolutionary spectral analysis is very popular in the earthquake communi-ty(see,for example,Liu1970,1971,1973;Lin&Cai1995).The difficulty of its application is tofind a method to define the basis,{φ(ω,t)}.In principle,for this method to work,the basis has to be defined a posteriori.So far,no systematic way has been offered;therefore,constructing an evolutionary spectrum from the given data is impossible.As a result,in the earthquake community,the applications of this method have changed the problem from data analysis to data simulation:an evo-lutionary spectrum will be assumed,then the signal will be reconstituted based on the assumed spectrum.Although there is some general resemblance to the simulated earthquake signal with the real data,it is not the data that generated the spectrum. Consequently,evolutionary spectrum analysis has never been very useful.As will be shown,the EMD can replace the evolutionary spectrum with a truly adaptive representation for the non-stationary processes.(e)The empirical orthogonal function expansion(EOF)The empirical orthogonal function expansion(EOF)is also known as the principal component analysis,or singular value decomposition method.The essence of EOF is briefly summarized as follows:for any real z(x,t),the EOF will reduce it toz(x,t)=n1a k(t)f k(x),(2.8)Proc.R.Soc.Lond.A(1998)910N.E.Huang and othersin whichf j·f k=δjk.(2.9)The orthonormal basis,{f k},is the collection of the empirical eigenfunctions defined byC·f k=λk f k,(2.10)where C is the sum of the inner products of the variable.EOF represents a radical departure from all the above methods,for the expansion basis is derived from the data;therefore,it is a posteriori,and highly efficient.The criticalflaw of EOF is that it only gives a distribution of the variance in the modes defined by{f k},but this distribution by itself does not suggest scales or frequency content of the signal.Although it is tempting to interpret each mode as indepen-dent variations,this interpretation should be viewed with great care,for the EOF decomposition is not unique.A single component out of a non-unique decomposition, even if the basis is orthogonal,does not usually contain physical meaning.Recently, Vautard&Ghil(1989)proposed the singular spectral analysis method,which is the Fourier transform of the EOF.Here again,we have to be sure that each EOF com-ponent is stationary,otherwise the Fourier spectral analysis will make little sense on the EOF components.Unfortunately,there is no guarantee that EOF compo-nents from a nonlinear and non-stationary data set will all be linear and stationary. Consequently,singular spectral analysis is not a real improvement.Because of its adaptive nature,however,the EOF method has been very popular,especially in the oceanography and meteorology communities(see,for example,Simpson1991).(f)Other miscellaneous methodsOther than the above methods,there are also some miscellaneous methods such as least square estimation of the trend,smoothing by moving averaging,and differencing to generate stationary data.Methods like these,though useful,are too specialized to be of general use.They will not be discussed any further here.Additional details can be found in many standard data processing books(see,for example,Brockwell &Davis1991).All the above methods are designed to modify the global representation of the Fourier analysis,but they all failed in one way or the other.Having reviewed the methods,we can summarize the necessary conditions for the basis to represent a nonlinear and non-stationary time series:(a)complete;(b)orthogonal;(c)local;and (d)adaptive.Thefirst condition guarantees the degree of precision of the expansion;the second condition guarantees positivity of energy and avoids leakage.They are the standard requirements for all the linear expansion methods.For nonlinear expansions,the orthogonality condition needs to be modified.The details will be discussed later.But even these basic conditions are not satisfied by some of the above mentioned meth-ods.The additional conditions are particular to the nonlinear and non-stationary data.The requirement for locality is the most crucial for non-stationarity,for in such data there is no time scale;therefore,all events have to be identified by the time of their occurences.Consequently,we require both the amplitude(or energy) and the frequency to be functions of time.The requirement for adaptivity is also crucial for both nonlinear and non-stationary data,for only by adapting to the local variations of the data can the decomposition fully account for the underlying physics Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis911of the processes and not just to fulfil the mathematical requirements for fitting the data.This is especially important for the nonlinear phenomena,for a manifestation of nonlinearity is the ‘harmonic distortion’in the Fourier analysis.The degree of distortion depends on the severity of nonlinearity;therefore,one cannot expect a predetermined basis to fit all the phenomena.An easy way to generate the necessary adaptive basis is to derive the basis from the data.In this paper,we will introduce a general method which requires two steps in analysing the data as follows.The first step is to preprocess the data by the empirical mode decomposition method,with which the data are decomposed into a number of intrinsic mode function components.Thus,we will expand the data in a basis derived from the data.The second step is to apply the Hilbert transform to the decomposed IMFs and construct the energy–frequency–time distribution,designated as the Hilbert spectrum,from which the time localities of events will be preserved.In other words,weneed the instantaneous frequency and energy rather than the global frequency and energy defined by the Fourier spectral analysis.Therefore,before goingany further,we have to clarify the definition of the instantaneous frequency.3.Instantaneous frequencyis to accepting it only for special ‘monocomponent’signals 1992;Cohen 1995).Thereare two basicdifficulties with accepting the idea of an instantaneous fre-quency as follows.The first one arises from the influence of theFourier spectral analysis.In the traditional Fourier analysis,the frequency is defined for thesineor cosine function spanning the whole data length with constant ampli-tude.As an extension of this definition,the instantaneous frequencies also have torelate to either a sine or a cosine function.Thus,we need at least one full oscillationof a sineor a cosine wave to define the local frequency value.According to this logic,nothing full wave will do.Such a definition would not make sense forThe secondarises from the non-unique way in defining the instantaneousfrequency.Nevertheless,this difficulty is no longer serious since the introduction ofthe meanstomakethedata analyticalthrough the Hilbert transform.Difficulties,however,still exist as ‘paradoxes’discussed by Cohen (1995).For an arbitrary timeseries,X (t ),we can always have its Hilbert Transform,Y (t ),as Y (t )=1πP∞−∞X (t )t −t d t,(3.1)where P indicates the Cauchy principal value.This transformexists forallfunctionsof class L p(see,for example,Titchmarsh 1948).With this definition,X (t )and Y (t )form the complex conjugate pair,so we can have an analytic signal,Z (t ),as Z (t )=X (t )+i Y (t )=a (t )e i θ(t ),(3.2)in which a (t )=[X 2(t )+Y 2(t )]1/2,θ(t )=arctanY (t )X (t ).(3.3)Proc.R.Soc.Lond.A (1998)912N.E.Huang andothers Theoretically,there are infinitely many ways of defining the imaginary part,but the Hilbert transform provides a unique way of defining the imaginary part so that the result is ananalyticfunction.A brief tutorial on the Hilbert transform with theemphasis on its physical interpretation can be found in Bendat &Piersol is the bestlocal fitan amplitude and phase varying trigonometric function to X (t ).Even with the Hilbert transform,there is still controversy in defining the instantaneous frequency as ω=d θ(t )d t .(3.4)This leads Cohen (1995)to introduce the term,‘monocomponent function’.In prin-ciple,some limitations on the data are necessary,forthe instantaneous frequencygiven in equation (3.4)is a single value function of time.At any given time,thereis only one frequency value;therefore,it can only represent one component,hence ‘monocomponent’.Unfortunately,no cleardefinition of the ‘monocomponent’signalwas given to judge whether a function is or is not ‘monocomponent’.For lack ofa precise definition,‘narrow band’was adopted a on the data for the instantaneous frequency to make sense (Schwartz et al .1966).There are two definitions for bandwidth.The first one is used in the study of the probability properties of the signalsand waves,wherethe processes are assumed tobe stationary and Gaussian.Then,the bandwidth can be defined in spectral moments The expected number of zero crossings per unit time is given byN 0=1π m 2m 0 1/2,(3.5)while the expected number of extrema per unit time is given byN 1=1π m 4m 2 1/2,(3.6)in which m i is the i th moment of the spectrum.Therefore,the parameter,ν,definedas N 21−N 20=1π2m 4m 0−m 22m 2m 0=1π2ν2,(3.7)offers a standard bandwidth measure (see,for example,Rice 1944a,b ,1945a,b ;Longuet-Higgins 1957).For a narrow band signal ν=0,the expected numbers extrema and zero crossings have to equal.the spectrum,but in a different way.coordinates as z (t )=a (t )e i θ(t ),(3.8)with both a (t )and θ(t )being functions of time.If this function has a spectrum,S (ω),then the mean frequency is given byω = ω|S (ω)|2d ω,(3.9)Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis913which can be expressed in another way asω =z ∗(t )1i dd tz (t )d t=˙θ(t )−i ˙a (t )a (t )a 2(t )d t =˙θ(t )a 2(t )d t.(3.10)Based on this expression,Cohen (1995)suggested that ˙θbe treated as the instanta-neous frequency.With these notations,the bandwidth can be defined asν2=(ω− ω )2 ω 2=1 ω 2(ω− ω )2|S (ω)|2d ω=1 ω 2z ∗(t ) 1i d d t− ω 2z (t )d t =1 ω 2 ˙a 2(t )d t +(˙θ(t )− ω )2a 2(t )d t .(3.11)For a narrow band signal,this value has to be small,then both a and θhave to begradually varying functions.Unfortunately,both equations (3.7)and (3.11)defined the bandwidth in the global sense;they are both overly restrictive and lack preci-sion at the same time.Consequently,the bandwidth limitation on the Hilbert trans-form to give a meaningful instantaneous frequency has never been firmly established.For example,Melville (1983)had faithfully filtered the data within the bandwidth requirement,but he still obtained many non-physical negative frequency values.It should be mentioned here that using filtering to obtain a narrow band signal is unsat-isfactory for another reason:the filtered data have already been contaminated by the spurious harmonics caused by the nonlinearity and non-stationarity as discussed in the introduction.In order to obtain meaningful instantaneous frequency,restrictive conditions have to be imposed on the data as discussed by Gabor (1946),Bedrosian (1963)and,more recently,Boashash (1992):for any function to have a meaningful instantaneous frequency,the real part of its Fourier transform has to have only positive frequency.This restriction can be proven mathematically as shown in Titchmarsh (1948)but it is still global.For data analysis,we have to translate this requirement into physically implementable steps to develop a simple method for applications.For this purpose,we have to modify the restriction condition from a global one to a local one,and the basis has to satisfy the necessary conditions listed in the last section.Let us consider some simple examples to illustrate these restrictions physically,by examining the function,x (t )=sin t.(3.12)Its Hilbert transform is simply cos t .The phase plot of x –y is a simple circle of unit radius as in figure 1a .The phase function is a straight line as shown in figure 1b and the instantaneous frequency,shown in figure 1c ,is a constant as expected.If we move the mean offby an amount α,say,then,x (t )=α+sin t.(3.13)Proc.R.Soc.Lond.A (1998)。
高吸收性复合组合物,该组合物制备的吸收性片材,和它们的制备方法[发明专利]
专利名称:高吸收性复合组合物,该组合物制备的吸收性片材,和它们的制备方法
专利类型:发明专利
发明人:铃木磨,松本良一,森真吾
申请号:CN97181473.2
申请日:19971215
公开号:CN1244881A
公开日:
20000216
专利内容由知识产权出版社提供
摘要:主要由微纤维形式的可水合性细纤维和水可溶胀性固体组成的复合结构,细纤维是从纤维素或其衍生物获得,固体的表面的至少一部分被细纤维覆盖。
该吸收性复合材料能够以各种形式获得,例如颗粒、圆粒、片材等,尤其有无纺织物的支持片材的片材型。
本发明还提供制造复合结构的方法。
申请人:株式会社日本吸收体技术研究所
地址:日本东京都
国籍:JP
代理机构:中国国际贸易促进委员会专利商标事务所
代理人:黄泽雄
更多信息请下载全文后查看。
Vitali’s theorem and WWKL
VITALI’S THEOREM AND WWKLDOUGLAS K.BROWNMARIAGNESE GIUSTOSTEPHEN G.SIMPSONAbstract.Continuing the investigations of X.Yu and others,westudy the role of set existence axioms in classical Lebesgue mea-sure theory.We show that pairwise disjoint countable additivityfor open sets of reals is provable in RCA0.We show that sev-eral well-known measure-theoretic propositions including the VitaliCovering Theorem are equivalent to WWKL over RCA0.1.IntroductionThe purpose of Reverse Mathematics is to study the role of set ex-istence axioms,with an eye to determining which axioms are needed in order to prove specific mathematical theorems.In many cases,it is shown that a specific mathematical theorem is equivalent to the set existence axiom which is needed to prove it.Such equivalences are often proved in the weak base theory RCA0.RCA0may be viewed as a kind of formalized constructive or recursive mathematics,with full clas-sical logic but severely restricted comprehension and induction.The program of Reverse Mathematics has been developed in many publica-tions;see for instance[5,10,11,12,20].In this paper we carry out a Reverse Mathematics study of some aspects of classical Lebesgue measure theory.Historically,the subject of measure theory developed hand in hand with the nonconstructive, set-theoretic approach to mathematics.Errett Bishop has remarked that the foundations of measure theory present a special challenge to the constructive mathematician.Although our program of Reverse Mathematics is quite different from Bishop-style constructivism,we feel that Bishop’s remark implicitly raises an interesting question:Which nonconstructive set existence axioms are needed for measure theory?VITALI’S THEOREM AND WWKL 2This paper,together with earlier papers of Yu and others [21,22,23,24,25,26],constitute an answer to that question.The results of this paper build upon and clarify some early results of Yu and Simpson.The reader of this paper will find that familiarity with Yu–Simpson [26]is desirable but not essential.We begin in section 2by exploring the extent to which measure theory can be developed in RCA 0.We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA 0.This is in contrast to a result of Yu–Simpson [26]:countable additivity for open sets of reals is equivalent over RCA 0to a nonconstructive set existence axiom known as Weak Weak K¨o nig’s Lemma (WWKL).We show in sections 3and 4that several other basic propositions of measure theory are also equivalent to WWKL over RCA 0.Finally in section 5we show that the Vitali Covering Theorem is likewise equivalent to WWKL over RCA 0.2.Measure Theory in RCA 0Recall that RCA 0is the subsystem of second order arithmetic with∆01comprehension and Σ01induction.The purpose of this section is toshow that some measure-theoretic results can be proved in RCA 0.Within RCA 0,let X be a compact separable metric space.We define C (X )= A,the completion of A ,where A is the vector space of rational “polynomials”over X under the sup-norm, f =sup x ∈X |f (x )|.For the precise definitions within RCA 0,see [26]and section III.E of Brown’s thesis [4].The construction of C (X )within RCA 0is inspired by the constructive Stone–Weierstrass theorem in section 4.5of Bishop and Bridges [2].It is provable in RCA 0that there is a natural one-to-one correspondence between points of C (X )and continuous functions f :X →R which are equipped with a modulus of uniform continuity ,that is to say,a function h :N →N such that for all n ∈N and x ,y ∈Xd (x,y )<12n .Within RCA 0we define a measure (more accurately,a nonnegative Borel probability measure)on X to be a nonnegative bounded linear functional µ:C (X )→R such that µ(1)=1.(Here µ(1)denotes µ(f ),f ∈C (X ),f (x )=1for all x ∈X .)For example,if X =[0,1],the unit interval,then there is an obvious measure µL :C ([0,1])→R given by µL (f )= 10f (x )dx ,the Riemann integral of f from 0to 1.We refer to µL as Lebesgue measure on [0,1].There is also the obvious generalization to Lebesgue measure µL on X =[0,1]n ,the n -cube.VITALI’S THEOREM AND WWKL 3Definition 2.1(measure of an open set).This definition is made in RCA 0.Let X be any compact separable metric space,and let µbe any measure on X .If U is an open set in X ,we defineµ(U )=sup {µ(f )|f ∈C (X ),0≤f ≤1,f =0on X \U }.Within RCA 0this supremum need not exist as a real number.(Indeed,the existence of µ(U )for all open sets U is equivalent to ACA 0over RCA 0.)Therefore,when working within RCA 0,we interpret assertions about µ(U )in a “virtual”or comparative sense.For example,µ(U )≤µ(V )is taken to mean that for all >0and all f ∈C (X )with 0≤f ≤1and f =0on X \U ,there exists g ∈C (X )with 0≤g ≤1and g =0on X \V such that µ(f )≤µ(g )+ .See also [26].Some basic properties of Lebesgue measure are easily proved in RCA 0.For instance,it is straightforward to show that the Lebesgue measure of the union of a finite set of pairwise disjoint open intervals is equal to the sum of the lengths of the intervals.We define L 1(X,µ)to be the completion of C (X )under the L 1-norm given by f 1=µ(|f |).(For the precise definitions,see [5]and[26].)In RCA 0we see that L 1(X,µ)is a separable Banach space,but to assert within RCA 0that points of the Banach space L 1(X,µ)represent measurable functions f :X →R is problematic.We shall comment further on this question in section 4below.Lemma 2.2.The following is provable in RCA 0.If U n ,n ∈N ,is a sequence of open sets,then µ∞ n =0U n ≥lim k →∞µ k n =0U n .Proof.Trivial.Lemma 2.3.The following is provable in RCA 0.If U 0,U 1,...,U k is a finite,pairwise disjoint sequence of open sets,then µ k n =0U n ≥k n =0µ(U n ).Proof.Trivial.An open set is said to be connected if it is not the union of two disjoint nonempty open sets.Let us say that a compact separable metric space X is nice if for all sufficiently small δ>0and all x ∈X ,the open ballB (x,δ)={y ∈X |d (x,y )<δ}VITALI’S THEOREM AND WWKL4 is connected.Such aδis called a modulus of niceness for X.For example,the unit interval[0,1]and the n-cube[0,1]n are nice, but the Cantor space2N is not nice.Theorem2.4(disjoint countable additivity).The following is prov-able in RCA0.Assume that X is nice.If U n,n∈N,is a pairwise disjoint sequence of open sets in X,thenµ∞n=0U n=∞n=0µ(U n).Proof.Put U= ∞n=0U n.Note that U is an open set.By Lemmas2.2and2.3,we have in RCA0thatµ(U)≥ ∞n=0µ(U n).It remainsto prove in RCA0thatµ(U)≤ ∞n=0µ(U n).Let f∈C(X)be suchthat0≤f≤1and f=0on X\U.It suffices to prove thatµ(f)≤∞n=0µ(U n).Claim1:There is a sequence of continuous functions f n:X→R, n∈N,defined by f n(x)=f(x)for all x∈U n,f n(x)=0for all x∈X\U n.To prove this in RCA0,recall from[6]or[20]that a code for a continuous function g from X to Y is a collection G of quadruples (a,r,b,s)with certain properties,the idea being that d(a,x)<r im-plies d(b,g(x))≤s.Also,a code for an open set U is a collection of pairs(a,r)with certain properties,the idea being that d(a,x)<r im-plies x∈U.In this case we write(a,r)<U to mean that d(a,b)+r<s for some(b,s)belonging to the code of U.Now let F be a code for f:X→R.Define a sequence of codes F n,n∈N,by putting(a,r,b,s) into F n if and only if1.(a,r,b,s)belongs to F and(a,r)<U n,or2.(a,r,b,s)belongs to F and b−s≤0≤b+s,or3.b−s≤0≤b+s and(a,r)<U m for some m=n.It is straightforward to verify that F n is a code for f n as required by claim1.Claim2:The sequence f n,n∈N,is a sequence of elements of C(X). To prove this in RCA0,we must show that the sequence of f n’s has a sequence of moduli of uniform continuity.Let h:N→N be a modulus of uniform continuity for f,and let k be so large that1/2k is a modulus of niceness for X.We shall show that h :N→N defined by h (m)=max(h(m),k)is a modulus of uniform continuity for all of the f n’s.Let x,y∈X and m∈N be such that d(x,y)<1/2h (m). To show that|f n(x)−f n(y)|<1/2m,we consider three cases.Case1:VITALI’S THEOREM AND WWKL5 x,y∈U n.In this case we have|f n(x)−f n(y)|=|f(x)−f(y)|<1VITALI’S THEOREM AND WWKL 6From (1)we see that for each >0there exists k such that µ(f )− ≤ kn =0µ(f n ).Thus we haveµ(f )− ≤kn =0µ(f n )≤k n =0µ(U n )≤∞ n =0µ(U n ).Since this holds for all >0,it follows that µ(f )≤ ∞n =0µ(U n ).Thus µ(U )≤ ∞n =0µ(U n )and the proof of Theorem 2.4is complete.Corollary 2.5.The following is provable in RCA 0.If (a n ,b n ),n ∈N is a sequence of pairwise disjoint open intervals,then µL ∞ n =0(a n ,b n ) =∞ n =0|a n −b n |.Proof.This is a special case of Theorem 2.4.Remark 2.6.Theorem 2.4fails if we drop the assumption that X is nice.Indeed,let µC be the familiar “fair coin”measure on the Cantor space X =2N ,given by µC ({x |x (n )=i })=1/2for all n ∈N and i ∈{0,1}.It can be shown that disjoint finite additivity for µC is equivalent to WWKL over RCA 0.(WWKL is defined and discussed in the next section.)In particular,disjoint finite additivity for µC is not provable in RCA 0.3.Measure Theory in WWKL 0Yu and Simpson [26]introduced a subsystem of second order arith-metic known as WWKL 0,consisting of RCA 0plus the following axiom:if T is a subtree of 2<N with no infinite path,thenlim n →∞|{σ∈T |length(σ)=n }|VITALI’S THEOREM AND WWKL 7see also Sieg [18].In this sense,every mathematical theorem provable in WKL 0or WWKL 0is finitistically reducible in the sense of Hilbert’s Program;see [19,6,20].Remark 3.2.The study of ω-models of WWKL 0is closely related to the theory of 1-random sequences,as initiated by Martin-L¨o f [16]and continued by Kuˇc era [7,13,14,15].At the time of writing of [26],Yu and Simpson were unaware of this work of Martin-L¨o f and Kuˇc era.The purpose of this section and the next is to review and extend the results of [26]and [21]concerning measure theory in WWKL 0.A measure µ:C (X )→R on a compact separable metric space X is said to be countably additive if µ∞ n =0U n =lim k →∞µ k n =0U n for any sequence of open sets U n ,n ∈N ,in X .The following theorem is implicit in [26]and [21].Theorem 3.3.The following assertions are pairwise equivalent over RCA 0.1.WWKL.2.(countable additivity)For any compact separable metric space Xand any measure µon X ,µis countably additive.3.For any covering of the closed unit interval [0,1]by a sequence of open intervals (a n ,b n ),n ∈N ,we have ∞n =0|a n −b n |≥1.Proof.That WWKL implies statement 2is proved in Theorem 1of [26].The implication 2→3is trivial.It remains to prove that statement 3implies WWKL.Reasoning in RCA 0,let T be a subtree of 2<N with no infinite path.PutT ={σ i |σ∈T,σ i /∈T,i <2}.For σ∈2<N put lh(σ)=length of σanda σ=lh(σ)−1n =0σ(n )2lh(σ).Note that |a σ−b σ|=1/2lh(σ).Note also that σ,τ∈2<N are incompa-rable if and only if (a σ,b σ)∩(a τ,b τ)=∅.In particular,the intervals (a τ,b τ),τ∈ T,are pairwise disjoint and cover [0,1)except for some of the points a σ,σ∈2<N .Fix >0and put c σ=a σ− /4lh(σ),d σ=a σ+ /4lh(σ).Then the open intervals (a τ,b τ),τ∈ T,(c σ,d σ),VITALI’S THEOREM AND WWKL 8σ∈2<N and (1− ,1+ )form a covering of [0,1].Applying statement 3,we see that the sum of the lengths of these intervals is ≥1,i.e. τ∈ T12lh(τ)=1.From this,equation (2)follows easily.Thus we have proved that state-ment 3implies WWKL.This completes the proof of the theorem.It is possible to take a somewhat different approach to measure the-ory in RCA 0.Note that the definition of µ(U )that we have given (Definition 2.1)is extensional in RCA 0.This means that if U and V contain the same points then µ(U )=µ(V ),provably in RCA 0.An alternative approach is the intensional one,embodied in Definition 3.4below.Recall that an open set U is given in RCA 0as a sequence of basic open sets.In the case of the real line,basic open sets are just intervals with rational endpoints.Definition 3.4(intensional Lebesgue measure).We make this defini-tion in RCA 0.Let U = (a n ,b n ) n ∈N be an open set in the real line.The intensional Lebesgue measure of U is defined by µI (U )=lim k →∞µL k n =0(a n ,b n ) .Theorem 3.5.It is provable in RCA 0that intensional Lebesgue mea-sure µI is countably additive on open sets.In other words,if U n ,n ∈N ,is a sequence of open sets,then µI∞ n =0U n =lim k →∞µI k n =0U n .Proof.This is immediate from the definitions,since ∞n =0U n is defined as the union of the sequences of basic open intervals in U n ,n ∈N .Returning now to WWKL 0,we can prove that intensional Lebesgue measure concides with extensional Lebesgue measure.In fact,we have the following easy result.Theorem 3.6.The following assertions are pairwise equivalent over RCA 0.VITALI’S THEOREM AND WWKL91.WWKL.2.µI(U)=µL(U)for all open sets U⊆[0,1].3.µI is extensional on open sets.In other words,for all open setsU,V⊆[0,1],if∀x(x∈U↔x∈V)thenµI(U)=µI(V).4.For all open sets U⊇[0,1],we haveµI(U)≥1.Proof.This is immediate from Theorems3.3and3.5.4.More Measure Theory in WWKL0In this section we show that a good theory of measurable functions and measurable sets can be developed within WWKL0.Wefirst consider pointwise values of measurable functions.Our ap-proach is due to Yu[21,24].Let X be a compact separable metric space and letµ:C(X)→R be a positive Borel probability measure on X.Recall that L1(X,µ)is defined within RCA0as the completion of C(X)under the L1-norm.In what sense or to what extent can we prove that a point of the Banach space L1(X,µ)gives rise to a function f:X→R?In order to answer this question,recall that f∈L1(X,µ)is given by a sequence f n∈C(X),n∈N,which converges to f in the L1-norm; more preciselyf n−f n+1 1≤12nfor all n,and|f m(x)−f m (x)|≤12k.VITALI’S THEOREM AND WWKL10 Then for x∈C fnand m ≥m≥n+2k+2we have|f m(x)−f m (x)|≤m −1i=m|f i(x)−f i+1(x)|≤∞i=n+2k+2|f i(x)−f i+1(x)|≤12k.We need a lemma:Lemma4.2.The following is provable in RCA0.For f∈C(X)and >0,we haveµ({x|f(x)> })≤ f 1/ .Proof.Put U={x|f(x)> }.Note that U is an open set.If g∈C(X),0≤g≤1,g=0on X\U,then we have g≤|f|, hence µ(g)=µ( g)≤µ(|f|)= f 1,henceµ(g)≤ f 1/ .Thus µ(U)≤ f 1/ and the lemma is proved.Using this lemma we haveµ(X\C fnk )=µx∞i=n+2k+2|f i(x)−f i+1(x)|>12i=1VITALI’S THEOREM AND WWKL 11hence by countable additivityµ(X \C f n )≤∞ k =0µ(X \C f nk )≤∞k =012n .This completes the proof of Proposition 4.1.Remark 4.3(Yu [21]).In accordance with Proposition 4.1,forf = f n n ∈N ∈L 1(X,µ)and x ∈ ∞n =0C f n ,we define f (x )=lim n →∞f n (x ).Thus we see thatf (x )is defined on an F σset of measure 1.Moreover,if f =g in L 1(X,µ),i.e.if f −g 1=0,then f (x )=g (x )for all x in an F σset of measure 1.These facts are provable in WWKL 0.We now turn to a discussion of measurable sets within WWKL 0.We sketch two approaches to this topic.Our first approach is to identify measurable sets with their characteristic functions in L 1(X,µ),accord-ing to the following definition.Definition 4.4.This definition is made within WWKL 0.We say that f ∈L 1(X,µ)is a measurable characteristic function if there exists a sequence of closed setsC 0⊆C 1⊆···⊆C n ⊆...,n ∈N ,such that µ(X \C n )≤1/2n for all n ,and f (x )∈{0,1}for all x ∈ ∞n =0C n .Here f (x )is as defined in Remark 4.3.Our second approach is more direct,but in its present form it applies only to certain specific situations.For concreteness we consider only Lebesgue measure µL on the unit interval [0,1].Our discussion can easily be extended to Lebesgue measure on the n -cube [0,1]n ,the “fair coin”measure on the Cantor space 2N ,etc .Definition 4.5.The following definition is made within RCA 0.Let S be the Boolean algebra of finite unions of intervals in [0,1]with rational endpoints.For E 1,E 2∈S we define the distanced (E 1,E 2)=µL ((E 1\E 2)∪(E 2\E 1)),the Lebesgue measure of the symmetric difference of E 1and E 2.Thus d is a pseudometric on S ,and we define S to be the compact separable metric space which is the completion of S under d .A point E ∈ S is called a Lebesgue measurable set in [0,1].VITALI’S THEOREM AND WWKL 12We shall show that these two approaches to measurable sets (Defi-nitions 4.4and 4.5)are equivalent in WWKL 0.Begin by defining an isometry χ:S →L 1([0,1],µL )as follows.For 0≤a <b ≤1defineχ([a,b ])= f n n ∈N ∈L 1([0,1],µL )where f n (0)=f n (a )=f n (b )=f n (1)=0and f n a +b −a 2n +1=1and f n ∈C ([0,1])is piecewise linear otherwise.Thus χ([a,b ])is a measurable characteristic function corresponding to the interval [a,b ].For 0≤a 1<b 1<···<a k <b k ≤1defineχ([a 1,b 1]∪···∪[a k ,b k ])=χ([a 1,b 1])+···+χ([a k ,b k ]).It is straightforward to prove in RCA 0that χextends to an isometryχ: S→L 1([0,1],µL ).Proposition 4.6.The following is provable in WWKL 0.If E ∈ Sis a Lebesgue measurable set,then χ(E )is a measurable characteristic function in L 1([0,1],µL ).Conversely,given a measurable characteristic function f ∈L 1([0,1],µL ),we can find E ∈ Ssuch that χ(E )=f in L 1([0,1],µL ).Proof.It is straightforward to prove in RCA 0that for all E ∈ S , χ(E )is a measurable characteristic function.For the converse,let f be a measurable characteristic function.By Definition 4.4we have that f (x )∈{0,1}for all x ∈ ∞n =0C n .ByProposition 4.1we have |f (x )−f 3n +3(x )|<1/2n for all x ∈C f n .Put U n ={x ||f 3n +3(x )−1|<1/2n }and V n ={x ||f 3n +3(x )|<1/2n }.Then for n ≥1,U n and V n are disjoint open sets.Moreover C n ∩C f n ⊆U n ∪V n ,hence µL (U n ∪V n )≥1−1/2n −1.By countable additivity(Theorem 3.3)we can effectively find E n ,F n ∈S such that E n ⊆U n and F n ⊆V n and µL (E n ∪F n )≥1−1/2n −2.Put E = E n +5 n ∈N .It is straightforward to show that E belongs to S and that χ(E )=f in L 1([0,1],µL ).This completes the proof.Remark 4.7.We have presented two notions of Lebesgue measurable set and shown that they are equivalent in WWKL 0.Our first notion (Definition 4.4)has the advantage of generality in that it applies to any measure on a compact separable metric space.Our second no-tion (Definition 4.5)is advantageous in other ways,namely it is more straightforward and works well in RCA 0.It would be desirable to find a single definition which combines all of these advantages.VITALI’S THEOREM AND WWKL 135.Vitali’s TheoremLet S be a collection of sets.A point x is said to be Vitali covered by S if for all >0there exists S ∈S such that x ∈S and the diameter of S is less than .The Vitali Covering Theorem in its simplest form says the following:if I is a sequence of intervals which Vitali covers an interval E in the real line,then I contains a countable,pairwise disjoint set of intervals I n ,n ∈N ,such that ∞n =0I n covers E except for a set of Lebesgue measure 0.The purpose of this section is to show that various forms of the Vitali Covering Theorem are provable in WWKL 0and in fact equivalent to WWKL over RCA 0.Throughout this section,we use µto denote Lebesgue measure.Lemma 5.1(Baby Vitali Lemma).The following is provable in RCA 0.Let I 0,...,I n be a finite sequence of intervals.Then we can find a pair-wise disjoint subsequence I k 0,...,I k m such thatµ(I k 0∪···∪I k m )≥1VITALI’S THEOREM AND WWKL 14I =[2a −b,2b −a ].)Thusµ(I 0∪···∪I n )≤µ(I k 0∪···∪I k m )≤µ(I k 0)+···+µ(I k m )=3µ(I k 0)+···+3µ(I k m )=3µ(I k 0∪···∪I k m )and the lemma is proved.Lemma 5.2.The following is provable in WWKL 0.Let E be an in-terval,and let I n ,n ∈N ,be a sequence of intervals.If E ⊆ ∞n =0I n ,then µ(E )≤lim k →∞µ k n =0I n .Proof.If the intervals I n are open,then the desired conclusion follows immediately from countable additivity (Theorem 3.3).Otherwise,fix >0and let I n be an open interval with the same midpoint as I n andµ(I n )=µ(I n )+µ(E \A ).(3)VITALI’S THEOREM AND WWKL 15To prove the claim,use Lemma 5.2and the Vitali property to find a finite set of intervals J 1,...,J l ∈I such that J 1,...,J l ⊆E \A andµ(E \(A ∪J 1∪···∪J l ))<13µ(J 1∪···∪J l ).We then have µ(E \(A ∪I 1∪···∪I k ))<212µ(E \A )≤212µ(E \A )=34nµ(E ).Then by countable additivity we have µ E \∞ n =1A n =0and the lemma is proved.Remark 5.4.It is straightforward to generalize the previous lemma to the case of a Vitali covering of the n -cube [0,1]n by closed balls or n -dimensional cubes.In the case of closed balls,the constant 3in the Baby Vitali Lemma 5.1is replaced by 3n .Theorem 5.5.The Vitali theorem for the interval [0,1](as stated in Lemma 5.3)is equivalent to WWKL over RCA 0.Proof.Lemma 5.3shows that,in RCA 0,WWKL implies the Vitali theorem for intervals.It remains to prove within RCA 0that the Vitali theorem for [0,1]implies WWKL.Instead of proving WWKL,we shall prove the equivalent statement 3.3.3.Reasoning in RCA 0,suppose thatVITALI’S THEOREM AND WWKL 16(a n ,b n ),n ∈N ,is a sequence of open intervals which covers [0,1].Let I be the countable set of intervals (a nki ,b nki )= a n +i k(b n −a n ) where i,k,n ∈N and 0≤i <k .Then I is a Vitali covering of [0,1].By the Vitali theorem for intervals,I contains a sequence of pairwise disjoint intervals I m ,m ∈N ,such that µ ∞ m =0I m ≥1.By disjoint countable additivity (Corollary 2.5),we have∞m =0µ(I m )≥1.From this it follows easily that∞n =0|a n −b n |≥1.Thus we have 3.3.3and our theorem is proved.We now turn to Vitali’s theorem for measurable sets.Recall our discussion of measurable sets in section 4.A sequence of intervals I is said to almost Vitali cover a Lebesgue measurable set E ⊆[0,1]if for all >0we have µL (E \O )=0,where O = {I |I ∈I ,diam(I )< }.Theorem 5.6.The following is provable in WWKL 0.Let E ⊆[0,1]be a Lebesgue measurable set with µ(E )>0.Let I be a sequence of intervals which almost Vitali covers E .Then I contains a pairwise disjoint sequence of intervals I n ,n ∈N ,such that µ E \∞ n =0I n =0.Proof.The proof of this theorem is similar to that of Lemma 5.3.The only modification needed is in the proof of the claim.Recall from Definition 4.5that E =lim n →∞E n where each E n is a finite union of intervals in [0,1].Fix m so large thatµ((E \E m )∪(E m \E ))<1VITALI’S THEOREM AND WWKL 17andµ(E m \(A ∪J 1∪···∪J l ))<136µ(E \A )<236µ(E \A )≤236µ(E \A )<236µ(E \A )=3,The Baire category theorem in weak subsystems of second order arith-metic ,Journal of Symbolic Logic 58(1993),557–578.7.O.Demuth and A.Kuˇc era,Remarks on constructive mathematical analysis ,[3],1979,pp.81–129.8.H.-D.Ebbinghaus,G.H.M¨u ller,and G.E.Sacks (eds.),Recursion Theory Week ,Lecture Notes in Mathematics,no.1141,Springer-Verlag,1985,IX +418pages.VITALI’S THEOREM AND WWKL189.Harvey Friedman,unpublished communication to Leo Harrington,1977.10.Harvey Friedman,Stephen G.Simpson,and Rick L.Smith,Countable algebraand set existence axioms,Annals of Pure and Applied Logic25(1983),141–181.11.,Randomness and generalizations offixed point free functions,[1],1990, pp.245–254.15.,Subsystems of Second Order Arithmetic,Perspectives in Mathematical Logic,Springer-Verlag,1998,XIV+445pages.21.Xiaokang Yu,Measure Theory in Weak Subsystems of Second Order Arithmetic,Ph.D.thesis,Pennsylvania State University,1987,vii+73pages.22.,Riesz representation theorem,Borel measures,and subsystems of sec-ond order arithmetic,Annals of Pure and Applied Logic59(1993),65–78. 24.,A study of singular points and supports of measures in reverse mathe-matics,Annals of Pure and Applied Logic79(1996),211–219.26.Xiaokang Yu and Stephen G.Simpson,Measure theory and weak K¨o nig’slemma,Archive for Mathematical Logic30(1990),171–180.E-mail address:dkb5@,giusto@dm.unito.it,simpson@ The Pennsylvania State University。
应用岩石物理不等式(RPI)验证各向异性岩石物理的预测和测量
应用岩石物理不等式(RPI)验证各向异性岩石物理的预测和测量ZakirHossain*,MichelleEllisRock Solid Images Inc, 2600 South Gessner Road, Houston, TX 77063, USA摘要这项研究的目的是描述各向异性岩石物理的不均匀。
各向异性岩石物理提供岩石地震各向异性和各向异性属性之间的联系。
然而,各向异性岩石物理预测和测量的局限性是不能很好地理解。
在这项研究中,我们提供了岩石物理不等式为指导检查各向异性岩石物理预测和实验室测量的有效性。
起初,我们在TI 介质模型数据中应用Rudzki 不等式;我们在各向同性介质证明了这些不等式,并加以扩展。
除此之外,我们验证了这些不等式在各向同性体中的应用,最后我们用这些不等式检查岩石物理预测和测量。
对于球形孔隙结构,各向同性自洽(SC)岩石物理属性与各向异性自洽岩石物理属性近似,不等式满足岩石物理预测孔隙度达到60%。
随着孔隙结构的复杂性增加,其中各向同性不等式与各向异性不等式近似,岩石物理不等式描述的各向异性SC 岩石物理预测的一部分对于横向各向同性介质而言是无效的。
我们发现这些无效的预测都具有较高的各向异性常数相关性。
实验室测量中,具有较低的各向异性常数(小于0.6)的各向异性速度满足这个不等式的。
然而,实测结果为粘土矿物(如伊利石和高岭石)不符合这些不等式,它具有较高的各向异性常数(0.6以上)。
我们得出结论,这些不满足各向异性岩石物理预测和测量的,应被视为各向异性介质(斜方晶系,单斜晶),它们比横向各向同性介质的各向异性常数要高。
介绍岩石物理在地球物理勘查中的作用已在过去的20年显著增加。
在这段时间很多岩石物理学理论得到发展,许多实验室进行实验来更好地了解岩石性质及其与地震和油藏特性的关系。
然而,在实际应用前检查这些理论关系的有效性是非常重要的。
以同样的方式检查测量实验室数据的有效性进一步的分析是有必要的。
第4章2:地层各向异性及其评估方法
6.12 9.86 10.79 11.39 8.17 8.99 10.29 10.78 9.21 8.22 8.57 9.08 10.2 10.21
4.55 8.57 9.89 10.78 7.82 8.48 8.61 9.52 8.22 9.55 6.06 8.14 8.18 7.93
4.34 10.65 10.06 10.9 7.4 8.47 9.14 9.86 9.03 7.31 7.71 8.95 8.64 8.77
钻头与地层相互作用模型
f e rN e r p d e d I r 1 p u e u I r 2 p s e s
式中,r 表示沿 N 位力
er
方向的“正规化”钻井效率;
fe
称为在单
作 用 下 的 有 效 钻 力 ; Ir1 和 Ir2 表 示 正 交 各 向 异 性 地 层 的 岩 石 各 向 异 性 指 数 ; p d , p u 及 p s可 表 达 如 下 :
9 10
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0 .6 0 .9 1 .2 0 .6 1 .2 1 .8
6
20
30
1400
1600
井1 8 0 0 深
2000
2200
2400
3.岩石各向异性实验分析
3.1 rock drillability anisotropy:
• The transversely isotropic formation is a typical anisotropic formation. Its rock drillability anisotropy can be expressed by an anisotropy index( Ir ) defined as:
基于正交各向异性叠前地震反演的裂缝预测
摘要裂缝不仅是油气储集空间的一种重要类型,还是油气的主要运移通道。
随着非常规油气地震勘探开发的精细水平不断提高以及广泛应用的宽方位采集处理技术的迅速发展,储层裂缝描述在页岩气勘探中发挥着越来越重要的作用。
页岩储层发育高角度裂缝(垂直裂缝),利用叠前方位地震属性以及裂缝敏感弹性参数的地震反演方法,可以提高裂缝预测精细程度,为勘探开发提供可靠的依据。
本文从地震各向异性基本原理出发,将含高角度裂缝的页岩储层近似为OA介质,并将其作两个正交对称面内的分解。
对前人推导的正交各向异性介质的纵波反射系数近似方程进行整理,根据反射系数对方位角、入射角以及各个参数的敏感程度,将其展示为适于高角度裂缝的近似方程,计算获得裂缝发育的方位角、两个正交对称面内各向异性梯度,分析指出能够利用各向异性梯度来指示裂缝发育部位、程度。
以叠前地震反演理论为基础,依据正交各向异性纵波反射系数近似方程,利用岩石物理建模估算的纵横波阻抗以及裂缝各向异性参数,实现了参数去相关的线性反演方法,得到各向异性参数可以反映裂缝发育部位、程度,通过模型测试和实际数据处理,证实了提出的方法能取得较好的反演结果。
结合贝叶斯理论,研究了基于柯西约束的方位弹性阻抗反演方法,并利用正则化多元线性拟合求取裂缝各向异性参数。
模型测试结果和实际工区处理,都说明正交各向异性介质中基于柯西约束的方位弹性阻抗反演方法具有较高的准确性和稳定性,能较好反映页岩储层中的裂缝发育情况。
关键词:正交各向异性,敏感性分析,方位叠前属性分析,裂缝预测,方位弹性阻抗反演The Crack Prediction in Pre-stack Seismic Inversion Based onOrthorhombic AnisotropyWang Xiuwei (Geological Resources and Geological Engineering)Directed by Prof. Zhang GuangzhiAbstractThe fracture is not only the reservoir space in the underground reservoir, but also the main transport channel. With the improvement of meticulous standards of the seismic exploration and development of unconventional oil and gas, and the rapid improvement of the wide range acquisition and processing technology, the description of reservoirs fracture is playing an increasingly important role in shale gas exploration. For the high angle fracture (vertical cracks) of shale reservoir, it is necessary to do pre-stack azimuth seismic attributes analysis technology and study the azimuth pre-stack seismic inversion method, which can improve the fracture prediction accuracy and provide a reliable basis for the exploration.First, this paper introduces the basic theory of seismic anisotropy, and the shale reservoirs with high-angle fractures are approximately equivalent to orthotropic media. The vertical fracture is decomposed into two orthogonal symmetric planes. Sensitivity analysis is made on the formula of the approximate longitudinal wave reflection coefficient of the orthotropic media deduced by the predecessors. According to the sensitivity of the azimuth, the incidence and the parameters of the formula, the formula is further deduced into the expression of the approximate P-wave reflection coefficient for the high-angle fractures. And it is given the computing method of the azimuth of fracture, before the method of extraction of anisotropic gradient in two different symmetric surfaces of the orthotropic media, which explains the anisotropy gradient can be used to determine the degree of fracture development.Second, this paper introduces the basic theory of pre-stack seismic inversion. According to the orthogonal anisotropic P-wave reflection coefficient approximate formula, the linear inversion method of parameter de-correlation is realized by using rock physical model to estimate the vertical and horizontal wave impedances and the anisotropic parameters which can reflect fracture development. A single well test and a practical work area test have been carried it out successively.Then according to the orientation data of this area with bad quality, this paper proposed azimuth elastic impedance inversion with Cauchy-constraint, based on the Bayesian theory, and the anisotropic parameters related to the development degree of cracks are obtained by multivariate linear fitting. Both the well data test and the actual work area test are conducted successively to show that the inversion method of azimuth elastic impedance with Cauchy-constraint for orthotropic media has higher Stability and reliability, which can more accurately reflect the shale reservoir fractures.Key words: Orthorhombic anisotropy, Sensitivity analysis, Azimuth pre-stack attribute analysis, Fracture prediction, Azimuthal elastic impedance inversion目录第一章引言 (1)1.1研究目的与意义 (1)1.2国内外研究现状 (2)1.3论文主要研究内容 (4)第二章地震各向异性基本理论 (6)2.1各向异性的形成机理 (6)2.2各向异性介质弹性波的波动方程 (7)2.2.1几何方程 (7)2.2.2广义胡克定律 (7)2.2.3纳维尔方程 (8)2.2.4各向异性介质波动方程 (8)2.3各向异性介质的分类 (9)2.3.1各向异性介质的基本类型 (9)2.3.2地层中的各向异性介质 (11)2.4T HOMSEN各向异性参数 (14)2.4.1 VTI模型的Thomsen各向异性参数 (15)2.4.2 HTI模型的各向异性参数 (16)2.4.3 OA模型的各向异性参数 (17)2.5小结 (18)第三章正交各向异性反射特征及属性分析 (19)3.1正交各向异性介质的反射系数 (19)3.2正交各向异性反射系数敏感性分析 (24)3.2.1OA介质反射系数方程的方位特征分析 (24)3.2.2OA介质反射系数方程的参数敏感性分析 (27)3.3基于正交各向异性介质的叠前属性分析 (28)3.4小结 (36)第四章参数去相关的线性反演方法 (37)4.1正交各向异性介质的叠前线性反演 (37)4.1.1基于阻尼最小二乘的线性反演方法 (37)4.1.2参数去相关与协方差矩阵的建立 (38)4.2数据测试 (40)4.2.1模型测试 (40)4.2.2实例分析 (42)4.3小结 (44)第五章基于柯西约束的方位弹性阻抗反演方法 (45)5.1正交各向异性介质中的弹性阻抗及其标准化 (45)5.2贝叶斯基本理论 (47)5.2.1贝叶斯公式 (47)5.2.2高斯先验分布 (48)5.2.3稀疏解 (48)5.2.4柯西先验分布的解 (49)5.3弹性阻抗反演及其参数提取 (51)5.3.1弹性阻抗反演流程 (51)5.3.2岩性及其裂缝敏感性参数提取 (53)5.4数据测试 (54)5.4.1模型测试 (54)5.4.2实例分析 (63)5.5小结 (69)结论与认识 (70)参考文献 (71)攻读硕士学位期间取得的学术成果 (77)致谢 (78)中国石油大学(华东)硕士学位论文第一章引言1.1 研究目的与意义裂缝预测的主要方法有,基于叠后数据的断裂检测技术以及基于叠前数据拟合各向异参数识别裂缝发育强度。
Frequency characteristic equalization circuit
专利名称:Frequency characteristic equalization circuit 发明人:本多 隆,福田 督也,山下 紀之,三宅 仁毅申请号:JP特願昭61-129423申请日:19860604公开号:JP特公平7-60987B2公开日:19950628专利内容由知识产权出版社提供摘要:PURPOSE:To vary the gain at an optional frequency without changing the phase characteristic and the delay characteristic by constituting the titled circuit with a 1st, a2nd and a 3rd amplifiers whose output signal is fed to an inverting input respectively, a capacitor and a control circuit. CONSTITUTION:The frequency characteristic adjusting circuit suitable for the sharpness control of a VTR is provided with the 1st and the 2nd amplifiers 1, 2 whose noninverting input receives an input signal and whose output signal is fed to the inverting input 5, the 3rd amplifier 3 whose noninverting input 5 receives a noninverting output of the 2nd amplifier 2 and whose output signal is fed to the inverting input 5, a 1st capacitor 6 inserted between the input and output, a capacitor 7 provided between the noninverting output of the 2nd amplifier 2 and ground, and a control circuit 4 controlling the mutual conductance of the lst amplifier 1. Thus, the gain of an optional frequency is varied without changing the phase characteristic and the delay characteristic.申请人:ソニー株式会社地址:東京都品川区北品川6丁目7番35号国籍:JP代理人:杉浦 正知更多信息请下载全文后查看。
反演基础inversion theory课件 Cambridge1
f(m) g(o) o = o(m) L(m) = g(o)/µ(o) o=o(m) h(m) = k f(m) L(m)
Jorge Landa (Chevron-Texaco)
300
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0 Cumulative Oil Production (STB)
h(m) = k f(m) L(m)
Region of interest for 1994 Northridge earthquake simulation
SCEC geological model provides 3D soil properties in Greater LA Basin
Adaptive grid resolves up to 1Hz freq. w/100 million grid pts; uniform grid would require 2000x more points
Assume that we have a black box that generates samples m1, m2... of the a priori probability density f(m) . In the Metropolis method, some samples of the prior are accepted, others are rejected, so that the accepted points are samples of the posterior probability density h(m) .
Snapshot of simulated ground motion (simulation requires 3hr on 6Tflops PSC machine, running at >80% parallel eff)
各向异性
各向异性在地球物理中的应用 二、Cascadia
Continental crust anisotropy measurements from tectonic tremor in Cascadia
Region:
Method:
13 broadband 3C station Filter 1-8 Hz 1 min window Near-vertical raypath Zero-lag covariance matrix
各向异性介质中波速是传播方向的函数:
群速度/相速度: 群速度是地震波能量的传播速度; 相速度是波前面的传播速度;
• 弹性体的各向异性:
• 各向同性介质的弹性常数矩阵:
• ������33 = ������ + 2������; ������44 = ������,其中������和������为拉梅系数;
弱各向异性、各项异性的测量、 各向异性的地学应用
Weak elastic anisotropy
1.TI介质及地震波在各向异性介质中的传播; 2.用弹性系数组合得到的各向异性参数来描述各向异性; 3. 弱各向异性的条件会极大简化地震波的传播公式;
最常见的各向异性介质为横向各向同性介质(TI),可分为:
Continental crust Upper ocean crust Lower ocean crust
Estimation of anisotropy parameter:
Result and discussion Continental crust:
1. A: microcrack 2. BC: confining pressure
reduction 3. D-I: open crack
TheoryOfElasticStability.SecondEdition.ByStephen…
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TTI各向异性逆时偏移技术及应用
TTI各向异性逆时偏移技术及应用王咸彬【摘要】地下介质广泛存在各向异性,传统各向同性地震偏移成像技术往往会导致成像精度不高甚至深度偏差问题,宽方位采集技术和高精度逆时偏移(RTM)成像技术的应用更是突显了各向异性的影响.从弱各向异性弹性波波动方程出发,首先采用拟声波近似得到VTI各向异性伪声波控制方程,然后引入交叉导数项进行坐标旋转得到TTI各向异性伪声波控制方程,再由高阶有限差分方法得到TTI-RTM偏移算子,最后采用波场校正消除横波分量影响,提高各向异性偏移算子的精度.模型试算和实际资料处理结果表明,该技术在处理各向异性介质地震资料时具有更高的精度,是高精度地震成像理想的技术手段.%Anisotropy is widespread in the subsurface formation medium.The conventional isotropic seismic migration imaging technique often leads to low imaging accuracy and even depth error.The application of wide azimuth seismic acquisition technique and high precision RTM imaging technology have highlighted the influence of anisotropy.In this paper we begin with the weakly anisotropic elastic wave equation,First,the VTI anisotropic pseudo acoustic wave control equation is obtained by quasi acoustic approximation.Then,the cross derivative term is introduced to coordinate rotation to get the TTI anisotropic pseudo acoustic wave control equation.Next,the TTI-RTM migration operator is obtained by high order finite difference.Finally,the influence of the S-wave component is eliminated by wave field correction to improve the accuracy of the anisotropic migration operator.The model test and field data processing results show that,the technique is more beneficial to theanisotropic medium with higher precision and it's an ideal technique for high precision seismic imaging.【期刊名称】《石油物探》【年(卷),期】2017(056)004【总页数】9页(P534-542)【关键词】各向异性;拟声波近似;控制方程;TTI逆时偏移【作者】王咸彬【作者单位】中国石油化工股份有限公司石油物探技术研究院,江苏南京211103【正文语种】中文【中图分类】P631地下介质具有广泛的各向异性特性,如我国东部陆相砂泥岩薄互层具有长波长各向异性特征,西部海相岩溶—裂缝型碳酸盐岩储层具有裂隙诱导各向异性特征。
New Perspectives on the Structure of Graphitic Carbons
Critical Reviews in Solid State and Materials Sciences,30:235–253,2005 Copyright c Taylor and Francis Inc.ISSN:1040-8436printDOI:10.1080/10408430500406265New Perspectives on the Structure of Graphitic CarbonsPeter J.F.Harris∗Centre for Advanced Microscopy,University of Reading,Whiteknights,Reading,RG66AF,UKGraphitic forms of carbon are important in a wide variety of applications,ranging from pollutioncontrol to composite materials,yet the structure of these carbons at the molecular level ispoorly understood.The discovery of fullerenes and fullerene-related structures such as carbonnanotubes has given a new perspective on the structure of solid carbon.This review aims toshow how the new knowledge gained as a result of research on fullerene-related carbons canbe applied to well-known forms of carbon such as microporous carbon,glassy carbon,carbonfibers,and carbon black.Keywords fullerenes,carbon nanotubes,carbon nanoparticles,non-graphitizing carbons,microporous carbon,glassy carbon,carbon black,carbonfibers.Table of Contents INTRODUCTION (235)FULLERENES,CARBON NANOTUBES,AND CARBON NANOPARTICLES (236)MICROPOROUS(NON-GRAPHITIZING)CARBONS (239)Background (239)Early Models (241)Evidence for Fullerene-Like Structures in Microporous Carbons (242)New Models for the Structure of Microporous Carbons (242)Carbonization and the Structural Evolution of Microporous Carbon (243)GLASSY CARBON (244)CARBON FIBERS (245)CARBON BLACK (248)Background (248)Structure of Carbon Black Particles (249)Effect of High-Temperature Heat Treatment on Carbon Black Structure (250)CONCLUSIONS (250)ACKNOWLEDGMENTS (251)REFERENCES (251)INTRODUCTIONUntil quite recently we knew for certain of just two allotropes of carbon:diamond and graphite.The vast range of carbon ma-∗E-mail:p.j.f.harris@ terials,both natural and synthetic,which have more disordered structures have traditionally been considered as variants of one or other of these two allotropes.Because the great majority of these materials contain sp2carbon rather than sp3carbon,their struc-tures have been thought of as being made up from tiny fragments235236P.J.F.HARRISFI G.1.(a)Model of PAN-derived carbon fibres from the work of Crawford and Johnson,1(b)model of a non-graphitizing carbon by Ban and colleagues.2of crystalline graphite.Examples of models for the structures of carbons in which the basic elements are graphitic are reproduced in Figure 1.The structure shown in Figure 1(a)is a model for the structure of carbon fibers suggested by Crawford and Johnson in 1971,1whereas 1(b)shows a model for non-graphitizing car-bon given by Ban and colleagues in 1975.2Both structures are constructed from bent or curved sheets of graphite,containing exclusively hexagonal rings.Although these models probably provide a good first approximation of the structures of these car-bons,in many cases they fail to explain fully the properties of the materials.Consider the example of non-graphitizing carbons.As the name suggests,these cannot be transformed into crystalline graphite even at temperatures of 3000◦C and above.I nstead,high temperature heat treatments transform them into structures with a high degree of porosity but no long-range crystalline order.I n the model proposed by Ban et al.(Figure 1(b)),the structure is made up of ribbon-like sheets enclosing randomly shaped voids.It is most unlikely that such a structure could retain its poros-ity when subjected to high temperature heat treatment—surface energy would force the voids to collapse.The shortcomings of this and other “conventional”models are discussed more fully later in the article.The discovery of the fullerenes 3−5and subsequently of re-lated structures such as carbon nanotubes,6−8nanohorns,9,10and nanoparticles,11has given us a new paradigm for solid car-bon structures.We now know that carbons containing pentago-nal rings,as well as other non-six-membered rings,among the hexagonal sp 2carbon network,can be highly stable.This new perspective has prompted a number of groups to take a fresh look at well-known forms of carbon,to see whether any evidence can be found for the presence of fullerene-like structures.12−14The aim of this article is to review this new work on the structure of graphitic carbons,to assess whether models that incorporate fullerene-like elements could provide a better basis for under-standing these materials than the conventional models,and to point out areas where further work is needed.The carbon ma-terials considered include non-graphitizing carbon,glassy car-bon,carbon fibers,and carbon black.The article begins with an outline of the main structural features of fullerenes,carbon nanotubes,and carbon nanoparticles,together with a brief dis-cussion of their stability.FULLERENES,CARBON NANOTUBES,AND CARBON NANOPARTICLESThe structure of C 60,the archetypal fullerene,is shown in Figure 2(a).The structure consists of twelve pentagonal rings and twenty hexagons in an icosahedral arrangement.I t will be noted that all the pentagons are isolated from each other.This is important,because adjacent pentagonal rings form an unstable bonding arrangement.All other closed-cage isomers of C 60,and all smaller fullerenes,are less stable than buck-minsterfullerene because they have adjacent pentagons.For higher fullerenes,the number of structures with isolated pen-tagonal rings increases rapidly with size.For example,C 100has 450isolated-pentagon isomers.16Most of these higher fullerenes have low symmetry;only a very small number of them have the icosahedral symmetry of C 60.An example of a giant fullerene that can have icosahedral symmetry is C 540,as shown in Figure 2(b).There have been many studies of the stability of fullerenes as a function of size (e.g.,Refs.17,18).These show that,in general,stability increases with size.Experimentally,there is evidence that C 60is unstable with respect to large,multiwalled fullerenes.This was demonstrated by Mochida and colleagues,who heated C 60and C 70in a sublimation-limiting furnace.19They showed that the cage structure broke down at 900◦C–1000◦C,although at 2400◦C fullerene-like “hollow spheres”with diameters in the range 10–20nm were formed.We now consider fullerene-related carbon nanotubes,which were discovered by Iijima in 1991.6These consist of cylinders of graphite,closed at each end with caps that contain precisely six pentagonal rings.We can illustrate their structure by considering the two “archetypal”carbon nanotubes that can be formed by cutting a C 60molecule in half and placing a graphene cylinder between the two halves.Dividing C 60parallel to one of the three-fold axes results in the zig-zag nanotube shown in Figure 3(a),whereas bisecting C 60along one of the fivefold axes produces the armchair nanotube shown in Figure 3(b).The terms “zig-zag”and “armchair”refer to the arrangement of hexagons around the circumference.There is a third class of structure in which the hexagons are arranged helically around the tube axis.Ex-perimentally,the tubes are generally much less perfect than the idealized versions shown in Figure 3,and may be eitherNEW PERSPECTIVES ON GRAPHITIC CARBONS STRUCTURE237FI G.2.The structure of (a)C 60,(b)icosahedral C 540.15multilayered or single-layered.Figure 4shows a high resolu-tion TEM image of multilayered nanotubes.The multilayered tubes range in length from a few tens of nm to several microns,and in outer diameter from about 2.5nm to 30nm.The end-caps of the tubes are sometimes symmetrical in shape,but more often asymmetric.Conical structures of the kind shown in Fig-ure 5(a)are commonly observed.This type of structure is be-lieved to result from the presence of a single pentagon at the position indicated by the arrow,with five further pentagons at the apex of the cone.Also quite common are complex cap struc-tures displaying a “bill-like”morphology such as thatshownFI G.3.Drawings of the two nanotubes that can be capped by one half of a C 60molecule.(a)Zig-zag (9,0)structure,(b)armchair (5,5)structure.20in Figure 5(b).21This structure results from the presence of a single pentagon at point “X”and a heptagon at point “Y .”The heptagon results in a saddle-point,or region of negative curvature.The nanotubes first reported by Iijima were prepared by va-porizing graphite in a carbon arc under an atmosphere of helium.Nanotubes produced in this way are invariably accompanied by other material,notably carbon nanoparticles.These can be thought of as giant,multilayered fullerenes,and range in size from ∼5nm to ∼15nm.A high-resolution image of a nanopar-ticle attached to a nanotube is shown in Figure 6(a).22In this238P.J.F.HARRISFI G.4.TEM image of multiwalled nanotubes.case,the particle consists of three concentric fullerene shells.A more typical nanoparticle,with many more layers,is shown in Figure 6(b).These larger particles are probably relatively im-perfect instructure.FI G.5.I mages of typical multiwalled nanotube caps.(a)cap with asymmetric cone structure,(b)cap with bill-like structure.21Single-walled nanotubes were first prepared in 1993using a variant of the arc-evaporation technique.23,24These are quite different from multilayered nanotubes in that they generally have very small diameters (typically ∼1nm),and tend to be curledNEW PERSPECTIVES ON GRAPHITIC CARBONS STRUCTURE239FI G.6.I mages of carbon nanoparticles.(a)small nanoparticle with three concentric layers on nanotube surface,22(b)larger multilayered nanoparticle.and looped rather than straight.They will not be considered further here because they have no parallel among well-known forms of carbon discussed in this article.The stability of multilayered carbon nanotubes and nanopar-ticles has not been studied in detail experimentally.However,we know that they are formed at the center of graphite electrodes during arcing,where temperatures probably approach 3000◦C.I t is reasonable to assume,therefore,that nanotubes and nanopar-ticles could withstand being re-heated to such temperatures (in an inert atmosphere)without significant change.MICROPOROUS (NON-GRAPHITIZING)CARBONS BackgroundIt was demonstrated many years ago by Franklin 25,26that carbons produced by the solid-phase pyrolysis of organic ma-terials fall into two distinct classes.The so-called graphitizing carbons tend to be soft and non-porous,with relatively high den-sities,and can be readily transformed into crystalline graphite by heating at temperatures in the range 2200◦C–3000◦C.I n con-trast,“non-graphitizing”carbons are hard,low-densitymateri-FI G.7.(a)High resolution TEM image of carbon prepared by pyrolysis of sucrose in nitrogen at 1000◦C,(b)carbon prepared bypyrolysis of anthracene at 1000◦C.I nsets show selected area diffraction patterns.30als that cannot be transformed into crystalline graphite even at temperatures of 3000◦C and above.The low density of non-graphitizing carbons is a consequence of a microporous struc-ture,which gives these materials an exceptionally high internal surface area.This high surface area can be enhanced further by activation,that is,mild oxidation with a gas or chemical pro-cessing,and the resulting “activated carbons”are of enormous commercial importance,primarily as adsorbents.27−29The distinction between graphitizing and non-graphitizing carbons can be illustrated most clearly using transmission elec-tron microscopy (TEM).Figure 7(a)shows a TEM image of a typical non-graphitizing carbon prepared by the pyrolysis of sucrose in an inert atmosphere at 1000◦C.30The inset shows a diffraction pattern recorded from an area approximately 0.25µm in diameter.The image shows the structure to be disordered and isotropic,consisting of tightly curled single carbon layers,with no obvious graphitization.The diffraction pattern shows symmetrical rings,confirming the isotropic structure.The ap-pearance of graphitizing carbons,on the other hand,approxi-mates much more closely to that of graphite.This can be seen in the TEM micrograph of a carbon prepared from anthracene,240P.J.F.HARRI Swhich is shown in Figure 7(b).Here,the structure contains small,approximately flat carbon layers,packed tightly together with a high degree of alignment.The fragments can be considered as rather imperfect graphene sheets.The diffraction pattern for the graphitizing carbon consists of arcs rather than symmetrical rings,confirming that the layers are preferentially aligned along a particular direction.The bright,narrow arcs in this pattern correspond to the interlayer {0002}spacings,whereas the other reflections appear as broader,less intense arcs.Transmission electron micrographs showing the effect of high-temperature heat treatments on the structure of non-graphitizing and graphitizing carbons are shown in Figure 8(note that the magnification here is much lower than for Figure 7).I n the case of the non-graphitizing carbon,heating at 2300◦C in an inert atmosphere produces the disordered,porous material shown in Figure 8(a).This structure is made up of curved and faceted graphitic layer planes,typically 1–2nm thick and 5–15nm in length,enclosing randomly shaped pores.A few somewhat larger graphite crystallites are present,but there is no macroscopic graphitization.n contrast,heat treatment of the anthracene-derived carbon produces large crystals of highly or-dered graphite,as shown in Figure 8(b).Other physical measurements also demonstrate sharp dif-ferences between graphitizing and non-graphitizing carbons.Table 1shows the effect of preparation temperature on the sur-face areas and densities of a typical graphitizing carbon prepared from polyvinyl chloride,and a non-graphitizing carbon prepared from cellulose.31It can be seen that the graphitizing carbon pre-pared at 700◦C has a very low surface area,which changes lit-tle for carbons prepared at higher temperatures,up to 3000◦C.The density of the carbons increases steadily as thepreparationFI G.8.Micrographs of (a)sucrose carbon and (b)anthracene carbon following heat treatment at 2300◦C.TABLE 1Effect of temperature on surface areas and densities of carbonsprepared from polyvinyl chloride and cellulose 31(a)Surface areas Specific surface area (m 2/g)for carbons prepared at:Starting material 700◦C 1500◦C 2000◦C 2700◦C 3000◦C PVC 0.580.210.210.710.56Cellulose 4081.601.172.232.25(b)Densities Helium density (g/cm 3)for carbons prepared at:Starting material 700◦C 1500◦C 2000◦C 2700◦C 3000◦C PVC 1.85 2.09 2.14 2.21 2.26Cellulose1.901.471.431.561.70temperature is increased,reaching a value of 2.26g/cm 3,which is the density of pure graphite,at 3000◦C.The effect of prepara-tion temperature on the non-graphitizing carbon is very different.A high surface area is observed for the carbon prepared at 700◦C (408m 2/g),which falls rapidly as the preparation temperature is increased.Despite this reduction in surface area,however,the density of the non-graphitizing carbon actually falls when the temperature of preparation is increased.This indicates that a high proportion of “closed porosity”is present in the heat-treated carbon.NEW PERSPECTIVES ON GRAPHITIC CARBONS STRUCTURE241FI G.9.Franklin’s representations of(a)non-graphitizing and(b)graphitizing carbons.25Early ModelsThefirst attempt to develop structural models of graphitizingand non-graphitizing carbons was made by Franklin in her1951paper.25In these models,the basic units are small graphitic crys-tallites containing a few layer planes,which are joined togetherby crosslinks.The precise nature of the crosslinks is not speci-fied.An illustration of Franklin’s models is shown in Figure9.Using these models,she put forward an explanation of whynon-graphitizing carbons cannot be converted by heat treatmentinto graphite,and this will now be summarized.During car-bonization the incipient stacking of the graphene sheets in thenon-graphitizing carbon is largely prevented.At this stage thepresence of crosslinks,internal hydrogen,and the viscosity ofthe material is crucial.The resulting structure of the carbon(at ∼1000◦C)consists of randomly ordered crystallites,held to-gether by residual crosslinks and van der Waals forces,as inFigure9(a).During high-temperature treatment,even thoughthese crosslinks may be broken,the activation energy for themotion of entire crystallites,required for achieving the struc-ture of graphite,is too high and graphite is not formed.Onthe other hand,the structural units in a graphitizing carbon areapproximately parallel to each other,as in Figure9(b),and thetransformation of such a structure into crystalline graphite wouldbe expected to be relatively facile.Although Franklin’s ideason graphitizing and non-graphitizing carbons may be broadlycorrect,they are in some regards incomplete.For example,thenature of the crosslinks between the graphitic fragments is notspecified,so the reasons for the sharply differing properties ofgraphitizing and non-graphitizing carbons is not explained.The advent of high-resolution transmission electron mi-croscopy in the early1970s enabled the structure of non-graphitizing carbons to be imaged directly.n a typical study,Ban,Crawford,and Marsh2examined carbons prepared frompolyvinylidene chloride(PVDC)following heat treatments attemperatures in the range530◦C–2700◦C.I mages of these car-bons apparently showed the presence of curved and twistedgraphite sheets,typically two or three layer planes thick,enclos-ing voids.These images led Ban et al.to suggest that heat-treatednon-graphitizing carbons have a ribbon-like structure,as shownin Figure1(b).This structure corresponds to a PVDC carbonheat treated at1950◦C.This ribbon-like model is rather similar to an earlier model of glassy carbon proposed by Jenkins andKawamura.32However,models of this kind,which are intendedto represent the structure of non-graphitizing carbons follow-ing high-temperature heat treatment,have serious weaknesses,as noted earlier.Such models consist of curved and twistedgraphene sheets enclosing irregularly shaped pores.However,graphene sheets are known to be highlyflexible,and wouldtherefore be expected to become ever more closely folded to-gether at high temperatures,in order to reduce surface energy.Indeed,tightly folded graphene sheets are quite frequently seenin carbons that have been exposed to extreme conditions.33Thus,structures like the one shown in Figure1(b)would be unlikelyto be stable at very high temperatures.It has also been pointed out by Oberlin34,35that the modelsput forward by Jenkins,Ban,and their colleagues were basedon a questionable interpretation of the electron micrographs.In most micrographs of partially graphitized carbons,only the {0002}fringes are resolved,and these are only visible when they are approximately parallel to the electron beam.Therefore,such images tend to have a ribbon-like appearance.However,because only a part of the structure is being imaged,this appear-ance can be misleading,and the true three-dimensional structuremay be more cagelike than ribbon-like.This is a very importantpoint,and must always be borne in mind when analyzing imagesof graphitic carbons.Oberlin herself believes that all graphiticcarbons are built up from basic structural units,which comprisesmall groups of planar aromatic structures,35but does not appearto have given a detailed explanation for the non-graphitizabilityof certain carbons.The models of non-graphitizing carbons described so farhave assumed that the carbon atoms are exclusively sp2and arebonded in hexagonal rings.Some authors have suggested thatsp3-bonded atoms may be present in these carbons(e.g.,Refs.36,37),basing their arguments on an analysis of X-ray diffrac-tion patterns.The presence of diamond-like domains would beconsistent with the hardness of non-graphitizing carbons,andmight also explain their extreme resistance to graphitization.Aserious problem with these models is that sp3carbon is unsta-ble at high temperatures.Diamond is converted to graphite at1700◦C,whereas tetrahedrally bonded carbon atoms in amor-phousfilms are unstable above about700◦C.Therefore,the242P.J.F.HARRI Spresence of sp 3atoms in a carbon cannot explain the resistance of the carbon to graphitization at high temperatures.I t should also be noted that more recent diffraction studies of non-graphitizing carbons have suggested that sp 3-bonded atoms are not present,as discussed further in what follows.Evidence for Fullerene-Like Structures in Microporous CarbonsThe evidence that microporous carbons might have fullerene-related structures has come mainly from high-resolution TEM studies.The present author and colleagues initiated a series of studies of typical non-graphitizing microporous carbons using this technique in the mid 1990s.30,38,39The first such study in-volved examining carbons prepared from PVDC and sucrose,after heat treatments at temperatures in the range 2100◦C–2600◦C.38The carbons subjected to very high temperatures had rather disordered structures similar to that shown in Figure 8(a).Careful examination of the heated carbons showed that they often contained closed nanoparticles;examples can be seen in Figure 10.The particles were usually faceted,and often hexagonal or pentagonal in shape.Sometimes,faceted layer planes enclosed two or more of the nanoparticles,as shown in Figure 10(b).Here,the arrows indicate two saddle-points,similar to that shown in Figure 5(b).The closed nature of the nanoparticles,their hexagonal or pentagonal shapes,and other features such as the saddle-points strongly suggest that the parti-cles have fullerene-like structures.I ndeed,in many cases the par-ticles resemble those produced by arc-evaporation in a fullerene generator (see Figure 6)although in the latter case the particles usually contain many more layers.The observation of fullerene-related nanoparticles in the heat treated carbons suggested that the original,freshly prepared car-bons may also have had fullerene-related structures (see next section).However,obtaining direct evidence for this is diffi-cult.High resolution electron micrographs of freshly prepared carbons,such as that shown in Figure 7(a),are usuallyratherFI G.10.(a)Micrograph showing closed structure in PVDC-derived carbon heated at 2600◦C,(b)another micrograph of same sample,with arrows showing regions of negative curvature.38featureless,and do not reveal the detailed structure.Occasion-ally,however,very small closed particles can be found in the carbons.30The presence of such particles provides circumstan-tial evidence that the surrounding carbon may have a fullerene-related structure.Direct imaging of pentagonal rings by HRTEM has not yet been achieved,but recent developments in TEM imaging techniques suggest that this may soon be possible,as discussed in the Conclusions.As well as high-resolution TEM,diffraction methods have been widely applied to microporous and activated carbons (e.g.,Refs.40–44).However,the interpretation of diffraction data from these highly disordered materials is not straightforward.As already mentioned,some early X-ray diffraction studies were interpreted as providing evidence for the presence of sp 3-bonded atoms.More recent neutron diffraction studies have suggested that non-graphitizing carbons consist entirely of sp 2atoms.40It is less clear whether diffraction methods can establish whether the atoms are bonded in pentagonal or hexagonal rings.Both Petkov et al .42and Zetterstrom and colleagues 43have interpreted neutron diffraction data from nanoporous carbons in terms of a structure containing non-hexagonal rings,but other interpreta-tions may also be possible.Raman spectroscopy is another valuable technique for the study of carbons.45Burian and Dore have used this method to analyze carbons prepared from sucrose,heat treated at tem-peratures from 1000◦C–2300◦C.46The Raman spectra showed clear evidence for the presence of fullerene-and nanotube-like elements in the carbons.There was also some evidence for fullerene-like structures in graphitizing carbons prepared from anthracene,but these formed at higher temperatures and in much lower proportions than in the non-graphitizing carbons.New Models for the Structure of Microporous Carbons Prompted by the observations described in the previous section,the present author and colleagues proposed a model for the structure of non-graphitizing carbons that consists ofNEW PERSPECTIVES ON GRAPHITIC CARBONS STRUCTURE243FI G.11.Schematic illustration of a model for the structure of non-graphitizing carbons based on fullerene-like elements.discrete fragments of curved carbon sheets,in which pentagons and heptagons are dispersed randomly throughout networks of hexagons,as illustrated in Figure11.38,39The size of the micropores in this model would be of the order of0.5–1.0nm, which is similar to the pore sizes observed in typical microp-orous carbons.The structure has some similarities to the“ran-dom schwarzite”network put forward by Townsend and col-leagues in1992,47although this was not proposed as a model for non-graphitizing carbons.I f the model we have proposed for non-graphitizing carbons is correct it suggests that these carbons are very similar in structure to fullerene soot,the low-density, disordered material that forms on walls of the arc-evaporation vessel and from which C60and other fullerenes may be ex-tracted.Fullerene soot is known to be microporous,with a sur-face area,after activation with carbon dioxide,of approximately 700m2g−1,48and detailed analysis of high resolution TEM mi-crographs of fullerene soot has shown that these are consis-tent with a structure in which pentagons and heptagons are dis-tributed randomly throughout a network of hexagons.49,50It is significant that high-temperature heat treatments can transform fullerene soot into nanoparticles very similar to those observed in heated microporous carbon.51,52Carbonization and the Structural Evolutionof Microporous CarbonThe process whereby organic materials are transformed into carbon by heat treatment is not well understood at the atomic level.53,54In particular,the very basic question of why some organic materials produce graphitizing carbons and others yield non-graphitizing carbons has not been satisfactorily answered. It is known,however,that both the chemistry and physical prop-erties of the precursors are important in determining the class of carbon formed.Thus,non-graphitizing carbons are formed, in general,from substances containing less hydrogen and more oxygen than graphitizing carbons.As far as physical properties are concerned,materials that yield graphitizing carbons usu-ally form a liquid on heating to temperatures around400◦C–500◦C,whereas those that yield non-graphitizing carbons gen-erally form solid chars without melting.The liquid phase pro-duced on heating graphitizing carbons is believed to provide the mobility necessary to form oriented regions.However,this may not be a complete explanation,because some precursors form non-graphitizing carbons despite passing through a liquid phase.The idea that non-graphitizing carbons contain pentagons and other non-six-membered rings,whereas graphitizing car-bons consist entirely of hexagonal rings may help in understand-ing more fully the mechanism of carbonization.Recently Kumar et al.have used Monte Carlo(MC)simulations to model the evo-lution of a polymer structure into a microporous carbon structure containing non-hexagonal rings.55They chose polyfurfuryl al-cohol,a well-known precursor for non-graphitizing carbon,as the starting material.The polymer was represented as a cubic lattice decorated with the repeat units,as shown in Figure12(a). All the non-carbon atoms(i.e.,hydrogen and oxygen)were then removed from the polymer and this network was used in the。
On the asymptotic expansion of the solutions of the separated nonlinear Schroedinger equati
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。
马古斯方程模型参数表
马古斯方程模型参数表摘要:一、引言二、马古斯方程的背景与基本概念三、马古斯方程模型的参数表1.模型参数的分类2.参数的具体数值及意义3.参数之间的关系四、马古斯方程模型的应用领域五、结论正文:一、引言马古斯方程模型是描述物质传输过程的一个重要模型,广泛应用于物理、化学、生物等多个领域。
为了更好地理解和应用该模型,我们需要了解其参数表以及参数之间的关系。
本文将详细介绍马古斯方程模型的参数表及其应用。
二、马古斯方程的背景与基本概念马古斯方程(Marcus equation)是由美国科学家鲁道夫·马古斯(Rudolph A.Marcus)于1956年提出的,用于描述化学反应中电子转移过程的速率。
马古斯方程模型包括一个微观描述和一个宏观描述,其中微观描述涉及电子转移过程,宏观描述涉及物质传输过程。
三、马古斯方程模型的参数表1.模型参数的分类马古斯方程模型参数主要分为以下几类:(1)电子转移参数:包括电子亲和势(EA)、电离势(EI)、电子转移数(n)等。
(2)传输参数:包括扩散系数(D)、迁移率(μ)、扩散系数与电场的关系(Einstein relation)等。
(3)反应物和生成物的浓度:包括反应物A和B的浓度([A]、[B])、生成物C的浓度([C])等。
2.参数的具体数值及意义(1)电子转移参数:电子亲和势(EA)表示一个物质吸引电子的能力,单位为电子伏特(eV);电离势(EI)表示一个物质失去电子的能力,单位为电子伏特(eV);电子转移数(n)表示一个物质转移的电子数量。
(2)传输参数:扩散系数(D)表示物质在电场中扩散的速度,单位为米每秒(m/s);迁移率(μ)表示物质在电场中迁移的速度,单位为米每秒(m/s);扩散系数与电场的关系(Einstein relation)表示扩散系数与电场强度成正比。
(3)反应物和生成物的浓度:反应物A和B的浓度([A]、[B])表示反应物A和B的浓度,单位通常为摩尔每升(mol/L);生成物C的浓度([C])表示生成物C的浓度,单位通常为摩尔每升(mol/L)。
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I I
3
3
CijktEkt,
i,j = 1,2,3,
(1)
k= I t= I
where the 3 x 3 x 3 x 3 elastic modulus tensor C ij ki completely characterizes the elasticity of the medium. Because of the symmetry of stress (crij = <5 j ;) , only six of these equations are independent. Because of the symmetry of strain (E kt = Ei k), only six of the terms on the right side of each set of equations (1) are independent. Hence, without loss of generality, the elasticity may be represented more compactly with a change of indices, following the Voigt recipe:
REVIEW OF ELASTIC ANISOTROPY
INTRODUCTION
In most applications of elasticity theory to problems in petroleum geophysics, the elastic medium is assumed to be isotropic. On the other hand, most crustal rocks are found experimentally to be anisotropic. Further, it is known that if a layered sequence of different media (isotropic or not) is probed with an elastic wave of wavelength much longer than the typical layer thickness (i.e., the normal seismic exploration context), the wave propagates as though it were in a homogeneous, but anisotropic, medium (Backus, 1962). Hence, there is a fundamental inconsistency between practice on the one hand and reality on the other. Two major reasons for the continued existence of this inconsistency come readily to mind: (1) The most commonly occurring type of anisotropy (transverse isotropy) masquerades as isotropy in nearvertical reflection profiling, with the angular dependence disguised in the uncertainty of the depth to each reflector (cf., Krey and Helbig, 1956). (2) The mathematical equations for anisotropic wave propagation are algebraically daunting, even for this simple case. The purpose of this paper is to point out that in most cases of interest to geophysicists the anisotropy is weak (10-20 percent), allowing the equations to simplify considerably. In fact, the equations become so simple that certain basic conclusions
(1) The most common measure of anisotropy (contrasting vertical and horizontal velocities) is not very relevant to problems of near-vertical P-wave propagation. (2) The most critical measure of anisotropy (denoted 8) does not involve the horizontal velocity at all in its definition and is often undetermined by experimental programs intended to measure anisotropy of rock samples. (3) A common approximation used to simplify the anisotropic wave-velocity equations (elliptical anisotropy) is usually inappropriate and misleading for Pand SV-waves. (4) Use of Poisson's ratio, as determined from vertical P and S velocities, to estimate horizontal stress usually leads to significant error. These conclusions apply irrespective of the physical cause of the anisotropy. Specifically, anisotropy in sedimentary rock sequences may be caused by preferred orientation of anisotropic mineral grains (such as in a massive shale formation), preferred orientation of the shapes of isotropic minerals (such as flat-lying platelets), preferred orientation of cracks (such as parallel cracks, or vertical cracks with no preferred azimuth), or thin bedding of isotropic or anisotropic layers. The conclusions stated here may be applied to rocks with any or all of these physical attributes, with the sole restriction that the resulting anisotropy is "weak" (this condition is given precise meaning below). To establish these conclusions, some elementary facts about anisotropy are reviewed in the next section. This is followed by a presentation of the simplified angular dependence of wave velocities appropriate for weak anisotropy. In the following section, the anisotropic parameters thus identified are used to analyze several common problems in petroleum geophysics. Finally, further discussion and conclusions are presented.
1955
there are nine such relations, each involving one component of stress and nine components of strain. These nine equations may be written compactly as
1954
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Weak Elastic Anisotropy
GEOPHYSICS, VOL. 51, NO. 10 (OCTOBER 1986); P. 1954-1966, 5 FIGS., 1 TABLE.
Weak elastic anisotropy
Leon Thomsen*
are immediately obvious:
ABSTRACT
Most bulk elastic media are weakly anisotropic. The equations governing weak anisotropy are much simpler than those governing strong anisotropy, and they are much easier to grasp intuitively. These equations indicate that a certain anisotropic parameter (denoted 8) controls most anisotropic phenomena of importance in exploration geophysics, some of which are nonnegligible even when the anisotropy is weak. The critical parameter 8 is an awkward combination of elastic parameters, a combination which is totally independent of horizontal velocity and which may be either positive or negative in natural contexts.