Localized Solutions of the Non-Linear Klein-Gordon Equation in Many Dimensions

英语作文震惊句子1. Opening with a Bang: Begin your essay with a sentence that grabs attention. For example, "Amidst the chaos of the city,a single act of kindness can be the beacon of hope that illuminates the path to humanity."2. Thought-Provoking Questions: Pose a question that encourages readers to ponder. "Have we become so engrossed in our digital lives that we've forgotten the value of a genuine conversation?"3. Powerful Statistics: Use statistics to back up your points. "An alarming 90% of plastic in the ocean comes from only ten rivers, highlighting the urgent need for localized solutions."4. Emotionally Resonant Statements: Tap into emotions tocreate a connection. "The laughter of children, the warmth of a family meal, these are the simple joys that make life's struggles worthwhile."5. Contrasting Ideas: Present contrasting ideas to create a dynamic narrative. "While some see the night as a canvas of darkness, others view it as a tapestry of stars, each twinkle a story waiting to be told."6. Metaphors and Similes: Use figurative language to paint a vivid picture. "Her voice was like a melody that danced onthe wind, carrying the whispers of a thousand tales."7. Strong Conclusions: End with a sentence that leaves a lasting impression. "As the sun sets on the horizon, it does not bid adieu but promises a new dawn, a fresh start for us all."8. Call to Action: Encourage readers to take action. "It is not enough to be a spectator in the theater of life; we must step onto the stage and play our part."9. Complex Ideas Simplified: Break down complex ideas into understandable sentences. "The concept of time is not alinear path but a web of possibilities, each choice a thread that weaves a different future."10. Historical References: Bring in history to provide context. "Just as the Gutenberg press revolutionized the spread of knowledge, the internet today is reshaping the way we communicate and learn."11. Quotes from Authorities: Use quotes to add weight to your argument. "As Einstein once said, 'The only source of knowledge is experience,' and this rings true in our questfor understanding."12. Reflective Statements: Encourage self-reflection. "In the quiet moments, when the world slows down, it is the echo of our actions that resounds the loudest."13. Provocative Assertions: Make bold statements to challengethe status quo. "The greatest threat to our future is not the unknown but the unwillingness to question the known."14. Descriptive Language: Engage the senses to create a vivid experience. "The aroma of fresh coffee wafts through the air, a comforting embrace on a cold winter morning."15. Rhetorical Devices: Employ rhetorical questions, alliteration, or anaphora for stylistic flair. "Do we dream to escape reality, or do we dream to find it?"By incorporating these types of sentences into your English composition, you can create a piece that is not only informative but also engaging, thought-provoking, and memorable.。

微环谐振腔中非线性开关效应的理论研究丰昀;李齐良;王哲【摘要】该文利用耦合器的耦合矩阵,研究了非线性微环谐振腔的开关特性.研究表明,光场在微环谐振腔中的自相位调制,会引起谐振频率的偏移.利用谐振腔的谐振以及非线性效应,可以实现在不同的频率处,由输入功率来诱导直通/下路型微环谐振腔输出的开关特性.同时频率失谐量较小时,实现非线性开关效应所需的最小阈值功率也较小.【期刊名称】《杭州电子科技大学学报》【年(卷),期】2014(034)003【总页数】4页(P9-12)【关键词】微环谐振腔;开关特性;透射率【作者】丰昀;李齐良;王哲【作者单位】杭州电子科技大学通信工程学院,浙江杭州310018;杭州电子科技大学通信工程学院,浙江杭州310018;杭州电子科技大学通信工程学院,浙江杭州310018【正文语种】中文【中图分类】TN2530 引言全光网络中，全部采用光波技术完成信息的传输和交换，不需要经过光电转换，避免了“电子瓶颈”现象，同时还能降低转换的噪声和误码率，因此一些体积小，结构简单，性能稳定的光子器件，像微环谐振腔等，得到了广泛研究。

微环谐振腔以其独特的优点长期以来一直在光无源，有源器件的设计和制作中有着广泛的应用。

微环谐振腔最基本的单环结构有全通型微环谐振腔(All Pass Filter，APF)和直通/下路型微环谐振腔［1－2］。

由于单环微谐振腔器件的工作性能不够完善，对于该器件的优化层出不穷。

近年来研究的许多复杂结构的谐振腔系统，例如耦合谐振腔光波导，边耦合集成空间谐振腔系列，环增强型的马赫－曾德尔干涉仪［1，3－5］，都是以最基本的单环结构为原型排列组合而成的。

不同结构的微环谐振腔系统可以表现出不同的特性。

目前国内外对于微环谐振腔的研究，主要集中在不同结构微环谐振腔中产生的快、慢光现象，全光开关等。

并且已经研制了多种以光纤耦合微谐振腔为载体的光学器件，比如光调制器件等［6］。

本文分析了直通/下路型微环谐振腔系统在输入一路信号情况下，光信号在微环中传输的自相位调制对系统谐振频率的影响;并主要研究了由输入功率诱导系统的非线性开关特性。

Chapter 3 Vector Spaces1.Suppose that(6204),(3157)αβ=-=-,find vector v , let βα43=-v2.Determine whether the vector β can be a linear combination of the vectors123,,ααα，if it can be ，write down the expressions 。

（1）123(83125),(1305),(2073),(4126)βααα=--=-=-=--3. Prove that any vector 1234()b b b b β=can be a linear combination of the vectors 1234(1000),(1100)(1110),(1111)αααα====。

4.Determine whether the following vectors are linearly independent 。

（1）123(1203),(2510),(3412).ααα=-=-=。

（2）1234(3425),(2503),(5012),(3335)αααα=-=--=-=-。

（3）232323123(1),(1),(1)aa ab b bc c c ααα===。

5. If 123,,ααα are linearly independent ，prove that whether 12233123,4,5αααααα+++ are linearly independent 。

6. Suppose that 1234,,,αααα are linearly independent ，then determine whether 12233441,,,αααααααα++++ are linearly independent, show the reason.7. Prove that in n R ，if 12,,,n ααα are linearly independent ，then any vector n R ∈β can be a linear combination of12,,,n ααα .8. If 1234,,,αααα are linearly dependent ，and any three of then are linearly independent, prove that there must have nonzero numbers 1234,,,k k k k such that112233440k k k k αααα+++=。

LEOPOLD-FRANZENS UNIVERSITYChair of Engineering Mechanicso.Univ.-Prof.Dr.-Ing.habil.G.I.Schu¨e ller,Ph.D.G.I.Schueller@uibk.ac.at Technikerstrasse13,A-6020Innsbruck,Austria,EU Tel.:+435125076841Fax.:+435125072905 mechanik@uibk.ac.at,http://mechanik.uibk.ac.atIfM-Publication2-407G.I.Schu¨e ller.Developments in stochastic structural mechanics.Archive of Applied Mechanics,published online,2006.Archive of Applied Mechanics manuscript No.(will be inserted by the editor)Developments in Stochastic Structural MechanicsG.I.Schu¨e llerInstitute of Engineering Mechanics,Leopold-Franzens University,Innsbruck,Aus-tria,EUReceived:date/Revised version:dateAbstract Uncertainties are a central element in structural analysis and design.But even today they are frequently dealt with in an intuitive or qualitative way only.However,as already suggested80years ago,these uncertainties may be quantified by statistical and stochastic procedures. In this contribution it is attempted to shed light on some of the recent advances in the now establishedfield of stochastic structural mechanics and also solicit ideas on possible future developments.1IntroductionThe realistic modeling of structures and the expected loading conditions as well as the mechanisms of their possible deterioration with time are un-doubtedly one of the major goals of structural and engineering mechanics2G.I.Schu¨e ller respectively.It has been recognized that this should also include the quan-titative consideration of the statistical uncertainties of the models and the parameters involved[56].There is also a general agreement that probabilis-tic methods should be strongly rooted in the basic theories of structural en-gineering and engineering mechanics and hence represent the natural next step in the development of thesefields.It is well known that modern methods leading to a quantification of un-certainties of stochastic systems require computational procedures.The de-velopment of these procedures goes in line with the computational methods in current traditional(deterministic)analysis for the solution of problems required by the engineering practice,where certainly computational pro-cedures dominate.Hence,their further development within computational stochastic structural analysis is a most important requirement for dissemi-nation of stochastic concepts into engineering practice.Most naturally,pro-cedures to deal with stochastic systems are computationally considerably more involved than their deterministic counterparts,because the parameter set assumes a(finite or infinite)number of values in contrast to a single point in the parameter space.Hence,in order to be competitive and tractable in practical applications,the computational efficiency of procedures utilized is a crucial issue.Its significance should not be underestimated.Improvements on efficiency can be attributed to two main factors,i.e.by improved hard-ware in terms of ever faster computers and improved software,which means to improve the efficiency of computational algorithms,which also includesDevelopments in Stochastic Structural Mechanics3 utilizing parallel processing and computer farming respectively.For a con-tinuous increase of their efficiency by software developments,computational procedure of stochastic analysis should follow a similar way as it was gone in the seventieth and eighties developing the deterministic FE approach. One important aspect in this fast development was the focus on numerical methods adjusted to the strength and weakness of numerical computational algorithms.In other words,traditional ways of structural analysis devel-oped before the computer age have been dropped,redesigned and adjusted respectively to meet the new requirements posed by the computational fa-cilities.Two main streams of computational procedures in Stochastic Structural Analysis can be observed.Thefirst of this main class is the generation of sample functions by Monte Carlo simulation(MCS).These procedures might be categorized further according to their purpose:–Realizations of prescribed statistical information:samples must be com-patible with prescribed stochastic information such as spectral density, correlation,distribution,etc.,applications are:(1)Unconditional simula-tion of stochastic processes,fields and waves.(2)Conditional simulation compatible with observations and a priori statistical information.–Assessment of the stochastic response for a mathematical model with prescribed statistics(random loading/system parameters)of the param-eters,applications are:(1)Representative sample for the estimation of the overall distribution.4G.I.Schu¨e ller Indiscriminate(blind)generation of samples.Numerical integration of SDE’s.(2)Representative sample for the reliability assessment by gen-erating adverse rare events with positive probability,i.e.by:(a)variance reduction techniques controlling the realizations of RV’s,(b)controlling the evolution in time of sampling functions.The other main class provides numerical solutions to analytical proce-dures.Grouping again according to the respective purpose the following classification can be made:Numerical solutions of Kolmogorov equations(Galerkin’s method,Finite El-ement method,Path Integral method),Moment Closure Schemes,Compu-tation of the Evolution of Moments,Maximum Entropy Procedures,Asymp-totic Stability of Diffusion Processes.In the following,some of the outlined topics will be addressed stressing new developments.These topics are described within the next six subject areas,each focusing on a different issue,i.e.representation of stochastic processes andfields,structural response,stochastic FE methods and parallel processing,structural reliability and optimization,and stochastic dynamics. In this context it should be mentioned that aside from the MIT-Conference series the USNCCM,ECCM and WCCM’s do have a larger part of sessions addressing computational stochastic issues.Developments in Stochastic Structural Mechanics5 2Representation of Stochastic ProcessesMany quantities involving randomfluctuations in time and space might be adequately described by stochastic processes,fields and waves.Typical ex-amples of engineering interest are earthquake ground motion,sea waves, wind turbulence,road roughness,imperfection of shells,fluctuating prop-erties in random media,etc.For this setup,probabilistic characteristics of the process are known from various measurements and investigations in the past.In structural engineering,the available probabilistic characteristics of random quantities affecting the loading or the mechanical system can be often not utilized directly to account for the randomness of the structural response due to its complexity.For example,in the common case of strong earthquake motion,the structural response will be in general non-linear and it might be too difficult to compute the probabilistic characteristics of the response by other means than Monte Carlo simulation.For the purpose of Monte Carlo simulation sample functions of the involved stochastic pro-cess must be generated.These sample functions should represent accurately the characteristics of the underlying stochastic process orfields and might be stationary and non-stationary,homogeneous or non-homogeneous,one-dimensional or multi-dimensional,uni-variate or multi-variate,Gaussian or non-Gaussian,depending very much on the requirements of accuracy of re-alistic representation of the physical behavior and on the available statistical data.6G.I.Schu¨e ller The main requirement on the sample function is its accurate represen-tation of the available stochastic information of the process.The associ-ated mathematical model can be selected in any convenient manner as long it reproduces the required stochastic properties.Therefore,quite different representations have been developed and might be utilized for this purpose. The most common representations are e.g.:ARMA and AR models,Filtered White Noise(SDE),Shot Noise and Filtered Poisson White Noise,Covari-ance Decomposition,Karhunen-Lo`e ve and Polynomial Chaos Expansion, Spectral Representation,Wavelets Representation.Among the various methods listed above,the spectral representation methods appear to be most widely used(see e.g.[71,86]).According to this procedure,samples with specified power spectral density information are generated.For the stationary or homogeneous case the Fast Fourier Transform(FFT)techniques is utilized for a dramatic improvements of its computational efficiency(see e.g.[104,105]).Advances in thisfield provide efficient procedures for the generation of2D and3D homogeneous Gaus-sian stochasticfields using the FFT technique(see e.g.[87]).The spectral representation method generates ergodic sample functions of which each ful-fills exactly the requirements of a target power spectrum.These procedures can be extended to the non-stationary case,to the generation of stochastic waves and to incorporate non-Gaussian stochasticfields by a memoryless nonlinear transformation together with an iterative procedure to meet the target spectral density.Developments in Stochastic Structural Mechanics7 The above spectral representation procedures for an unconditional simula-tion of stochastic processes andfields can also be extended for Conditional simulations techniques for Gaussianfields(see e.g.[43,44])employing the conditional probability density method.The aim of this procedure is the generation of Gaussian random variates U n under the condition that(n−1) realizations u i of U i,i=1,2,...,(n−1)are specified and the a priori known covariances are satisfied.An alternative procedure is based on the so called Kriging method used in geostatistical application and applied also to con-ditional simulation problems in earthquake engineering(see e.g.[98]).The Kriging method has been improved significantly(see e.g.[36])that has made this method theoretically clearer and computationally more efficient.The differences and similarities of the conditional probability density methods and(modified)Kriging methods are discussed in[37]showing the equiva-lence of both procedures if the process is Gaussian with zero mean.A quite general spectral representation utilized for Gaussian random pro-cesses andfields is the Karhunen-Lo`e ve expansion of the covariance function (see e.g.[54,33]).This representation is applicable for stationary(homoge-neous)as well as for non-stationary(inhomogeneous)stochastic processes (fields).The expansion of a stochastic process(field)u(x,θ)takes the formu(x,θ)=¯u(x)+∞i=1ξ(θ) λiφi(x)(1)where the symbolθindicates the random nature of the corresponding quan-tity and where¯u(x)denotes the mean,φi(x)are the eigenfunctions andλi the eigenvalues of the covariance function.The set{ξi(θ)}forms a set of8G.I.Schu¨e ller orthogonal(uncorrelated)zero mean random variables with unit variance.The Karhunen-Lo`e ve expansion is mean square convergent irrespective of its probabilistic nature provided it possesses afinite variance.For the im-portant special case of a Gaussian process orfield the random variables{ξi(θ)}are independent standard normal random variables.In many prac-tical applications where the random quantities vary smoothly with respectto time or space,only few terms are necessary to capture the major part of the randomfluctuation of the process.Its major advantage is the reduction from a large number of correlated random variables to few most important uncorrelated ones.Hence this representation is especially suitable for band limited colored excitation and stochastic FE representation of random me-dia where random variables are usually strongly correlated.It might also be utilized to represent the correlated stochastic response of MDOF-systems by few most important variables and hence achieving a space reduction.A generalization of the above Karhunen-Lo`e ve expansion has been proposed for application where the covariance function is not known a priori(see[16, 33,32]).The stochastic process(field)u(x,θ)takes the formu(x,θ)=a0(x)Γ0+∞i1=1a i1(x)Γ1(ξi1(θ))+∞i1=1i1i2=1a i1i2(x)Γ2(ξi1(θ),ξi2(θ))+ (2)which is denoted as the Polynomial Chaos Expansion.Introducing a one-to-one mapping to a set with ordered indices{Ψi(θ)}and truncating eqn.2Developments in Stochastic Structural Mechanics9 after the p th term,the above representations reads,u(x,θ)=pj=ou j(x)Ψj(θ)(3)where the symbolΓn(ξi1,...,ξin)denotes the Polynomial Chaos of order nin the independent standard normal random variables.These polynomialsare orthogonal so that the expectation(or inner product)<ΨiΨj>=δij beingδij the Kronecker symbol.For the special case of a Gaussian random process the above representation coincides with the Karhunen-Lo`e ve expan-sion.The Polynomial Chaos expansion is adjustable in two ways:Increasingthe number of random variables{ξi}results in a refinement of the random fluctuations,while an increase of the maximum order of the polynomialcaptures non-linear(non-Gaussian)behavior of the process.However,the relation between accuracy and numerical efforts,still remains to be shown. The spectral representation by Fourier analysis is not well suited to describe local feature in the time or space domain.This disadvantage is overcome in wavelets analysis which provides an alternative of breaking a signal down into its constituent parts.For more details on this approach,it is referred to[24,60].In some cases of applications the physics or data might be inconsistent with the Gaussian distribution.For such cases,non-Gaussian models have been developed employing various concepts to meet the desired target dis-tribution as well as the target correlation structure(spectral density).Cer-tainly the most straight forward procedures is the above mentioned memo-ryless non-linear transformation of Gaussian processes utilizing the spectralrepresentation.An alternative approach utilizes linear and non-linearfil-ters to represent normal and non-Gaussian processes andfields excited by Gaussian white noise.Linearfilters excited by polynomial forms of Poisson white noise have been developed in[59]and[34].These procedures allow the evaluation of moments of arbitrary order without having to resort to closure techniques. Non-linearfilters are utilized to generate a stationary non-Gaussian stochas-tic process in agreement with a givenfirst-order probability density function and the spectral density[48,15].In the Kontorovich-Lyandres procedure as used in[48],the drift and diffusion coefficients are selected such that the solutionfits the target probability density,and the parameters in the solu-tion form are then adjusted to approximate the target spectral density.The approach by Cai and Lin[15]simplifies this procedure by matching the spec-tral density by adjusting only the drift coefficients,which is the followed by adjusting the diffusion coefficient to approximate the distribution of the pro-cess.The latter approach is especially suitable and computationally highly efficient for a long term simulation of stationary stochastic processes since the computational expense increases only linearly with the number n of dis-crete sample points while the spectral approach has a growth rate of n ln n when applying the efficient FFT technique.For generating samples of the non-linearfilter represented by a stochastic differential equations(SDE), well developed numerical procedures are available(see e.g.[47]).3Response of Stochastic SystemsThe assessment of the stochastic response is the main theme in stochastic mechanics.Contrary to the representation of of stochastic processes and fields designed tofit available statistical data and information,the output of the mathematical model is not prescribed and needs to be determined in some stochastic sense.Hence the mathematical model can not be selected freely but is specified a priori.The model involves for stochastic systems ei-ther random system parameters or/and random loading.Please note,due to space limitations,the question of model validation cannot be treated here. For the characterization of available numerical procedures some classifi-cations with regard to the structural model,loading and the description of the stochastic response is most instrumental.Concerning the structural model,a distinction between the properties,i.e.whether it is determinis-tic or stochastic,linear or non-linear,as well as the number of degrees of freedom(DOF)involved,is essential.As a criterion for the feasibility of a particular numerical procedure,the number of DOF’s of the structural system is one of the most crucial parameters.Therefore,a distinction be-tween dynamical-system-models and general FE-discretizations is suggested where dynamical systems are associated with a low state space dimension of the structural model.FE-discretization has no essential restriction re-garding its number of DOF’s.The stochastic loading can be grouped into static and dynamic loading.Stochastic dynamic loading might be charac-terized further by its distribution and correlation and its independence ordependence on the response,resulting in categorization such as Gaussian and non-Gaussian,stationary and non-stationary,white noise or colored, additive and multiplicative(parametric)excitation properties.Apart from the mathematical model,the required terms in which the stochastic re-sponse should be evaluated play an essential role ranging from assessing thefirst two moments of the response to reliability assessments and stabil-ity analysis.The large number of possibilities for evaluating the stochas-tic response as outlined above does not allow for a discussion of the en-tire subject.Therefore only some selected advances and new directions will be addressed.As already mentioned above,one could distinguish between two main categories of computational procedures treating the response of stochastic systems.Thefirst is based on Monte Carlo simulation and the second provides numerical solutions of analytical procedures for obtaining quantitative results.Regarding the numerical solutions of analytical proce-dures,a clear distinction between dynamical-system-models and FE-models should be made.Current research efforts in stochastic dynamics focus to a large extent on dynamical-system-models while there are few new numerical approaches concerning the evaluation of the stochastic dynamic response of e.g.FE-models.Numerical solutions of the Kolmogorov equations are typical examples of belonging to dynamical-system-models where available approaches are computationally feasible only for state space dimensions one to three and in exceptional cases for dimension four.Galerkin’s,Finite El-ement(FE)and Path Integral methods respectively are generally used tosolve numerically the forward(Fokker-Planck)and backward Kolmogorov equations.For example,in[8,92]the FE approach is employed for stationary and transient solutions respectively of the mentioned forward and backward equations for second order systems.First passage probabilities have been ob-tained employing a Petrov-Galerkin FE method to solve the backward and the related Pontryagin-Vitt equations.An instructive comparison between the computational efforts using Monte Carlo simulation and the FE-method is given e.g.in an earlier IASSAR report[85].The Path Integral method follows the evolution of the(transition)prob-ability function over short time intervals,exploiting the fact that short time transition probabilities for normal white noise excitations are locally Gaus-sian distributed.All existing path integration procedures utilize certain in-terpolation schemes where the probability density function(PDF)is rep-resented by values at discrete grid points.In a wider sense,cell mapping methods(see e.g.[38,39])can be regarded as special setups of the path integral procedure.As documented in[9],cumulant neglect closure described in section7.3 has been automated putational procedures for the automated generation and solutions of the closed set of moment equations have been developed.The method can be employed for an arbitrary number of states and closed at arbitrary levels.The approach,however,is limited by available computational resources,since the computational cost grows exponentially with respect to the number of states and the selected closurelevel.The above discussed developments of numerical procedures deal with low dimensional dynamical systems which are employed for investigating strong non-linear behavior subjected to(Gaussian)white noise excitation. Although dynamical system formulations are quite general and extendible to treat non-Gaussian and colored(filtered)excitation of larger systems,the computational expense is growing exponentially rendering most numerical approaches unfeasible for larger systems.This so called”curse of dimen-sionality”is not overcome yet and it is questionable whether it ever will be, despite the fast developing computational possibilities.For this reason,the alternative approach based on Monte Carlo simu-lation(MCS)gains importance.Several aspects favor procedures based on MCS in engineering applications:(1)Considerably smaller growth rate of the computational effort with dimensionality than analytical procedures.(2) Generally applicable,well suited for parallel processing(see section5.1)and computationally straight forward.(3)Non-linear complex behavior does not complicate the basic procedure.(4)Manageable for complex systems.Contrary to numerical solutions of analytical procedures,the employed structural model and the type of stochastic loading does for MCS not play a deceive role.For this reason,MCS procedures might be structured ac-cording to their purpose i.e.where sample functions are generated either for the estimation of the overall distribution or for generating rare adverse events for an efficient reliability assessment.In the former case,the prob-ability space is covered uniformly by an indiscriminate(blind)generationof sample functions representing the random quantities.Basically,at set of random variables will be generated by a pseudo random number generator followed by a deterministic structural analysis.Based on generated random numbers realizations of random processes,fields and waves addressed in section2,are constructed and utilized without any further modification in the following structural analysis.The situation may not be considered to be straight forward,however,in case of a discriminate MCS for the reliability estimation of structures,where rare events contributing considerably to the failure probability should be gener-ated.Since the effectiveness of direct indiscriminate MCS is not satisfactory for producing a statistically relevant number of low probability realizations in the failure domain,the generation of samples is restricted or guided in some way.The most important class are the variance reduction techniques which operate on the probability of realizations of random variables.The most widely used representative of this class in structural reliability assess-ment is Importance Sampling where a suitable sampling distribution con-trols the generation of realizations in the probability space.The challenge in Importance Sampling is the construction of a suitable sampling distribu-tion which depends in general on the specific structural system and on the failure domain(see e.g.[84]).Hence,the generation of sample functions is no longer independent from the structural system and failure criterion as for indiscriminate direct MCS.Due to these dependencies,computational procedures for an automated establishment of sampling distributions areurgently needed.Adaptive numerical strategies utilizing Importance Direc-tional sampling(e.g.[11])are steps in this direction.The effectiveness of the Importance sampling approach depends crucially on the complexity of the system response as well as an the number of random variables(see also section5.2).Static problems(linear and nonlinear)with few random vari-ables might be treated effectively by this approach.Linear systems where the randomness is represented by a large number of RVs can also be treated efficiently employingfirst order reliability methods(see e.g.[27]).This ap-proach,however,is questionable for the case of non-linear stochastic dynam-ics involving a large set of random variables,where the computational effort required for establishing a suitable sampling distribution might exceed the effort needed for indiscriminate direct MCS.Instead of controlling the realization of random variables,alternatively the evolution of the generated sampling can be controlled[68].This ap-proach is limited to stochastic processes andfields with Markovian prop-erties and utilizes an evolutionary programming technique for the genera-tion of more”important”realization in the low probability domain.This approach is especially suitable for white noise excitation and non-linear systems where Importance sampling is rather difficult to apply.Although the approach cannot deal with spectral representations of the stochastic processes,it is capable to make use of linearly and non-linearlyfiltered ex-citation.Again,this is just contrary to Importance sampling which can be applied to spectral representations but not to white noisefiltered excitation.4Stochastic Finite ElementsAs its name suggests,Stochastic Finite Elements are structural models rep-resented by Finite Elements the properties of which involve randomness.In static analysis,the stiffness matrix might be random due to unpredictable variation of some material properties,random coupling strength between structural components,uncertain boundary conditions,etc.For buckling analysis,shape imperfections of the structures have an additional impor-tant effect on the buckling load[76].Considering structural dynamics,in addition to the stiffness matrix,the damping properties and sometimes also the mass matrix might not be predictable with certainty.Discussing numerical Stochastic Finite Elements procedures,two cat-egories should be distinguished clearly.Thefirst is the representation of Stochastic Finite Elements and their global assemblage as random structural matrices.The second category addresses the evaluation of the stochastic re-sponse of the FE-model due to its randomness.Focusingfirst on the Stochastic FE representation,several representa-tions such as the midpoint method[35],the interpolation method[53],the local average method[97],as well as the Weighted-Integral-Method[94,25, 26]have been developed to describe spatial randomfluctuations within the element.As a tendency,the midpoint methods leads to an overestimation of the variance of the response,the local average method to an underestima-tion and the Weighted-Integral-Method leads to the most accurate results. Moreover,the so called mesh-size problem can be resolved utilizing thisrepresentation.After assembling all Finite Elements,the random structural stiffness matrix K,taken as representative example,assumes the form,K(α)=¯K+ni=1K Iiαi+ni=1nj=1K IIijαiαj+ (4)where¯K is the mean of the matrix,K I i and K II ij denote the determinis-ticfirst and second rate of change with respect to the zero mean random variablesαi andαj and n is the total number of random variables.For normally distributed sets of random variables{α},the correlated set can be represented advantageously by the Karhunen-Lo`e ve expansion[33]and for non-Gaussian distributed random variables by its Polynomial chaos ex-pansion[32],K(θ)=¯K+Mi=0ˆKiΨi(θ)(5)where M denotes the total number of chaos polynomials,ˆK i the associated deterministicfluctuation of the matrix andΨi(θ)a polynomial of standard normal random variablesξj(θ)whereθindicates the random nature of the associated variable.In a second step,the random response of the stochastic structural system is determined.The most widely used procedure for evaluating the stochastic response is the well established perturbation approach(see e.g.[53]).It is well adapted to the FE-formulation and capable to evaluatefirst and second moment properties of the response in an efficient manner.The approach, however,is justified only for small deviations from the center value.Since this assumption is satisfied in most practical applications,the obtainedfirst two moment properties are evaluated satisfactorily.However,the tails of the。

机器学习题库一、 极大似然1、 ML estimation of exponential model (10)A Gaussian distribution is often used to model data on the real line, but is sometimesinappropriate when the data are often close to zero but constrained to be nonnegative. In such cases one can fit an exponential distribution, whose probability density function is given by()1xb p x e b-=Given N observations x i drawn from such a distribution:(a) Write down the likelihood as a function of the scale parameter b.(b) Write down the derivative of the log likelihood.(c) Give a simple expression for the ML estimate for b.2、换成Poisson 分布：()|,0,1,2,...!x e p x y x θθθ-==()()()()()1111log |log log !log log !N Ni i i i N N i i i i l p x x x x N x θθθθθθ======--⎡⎤=--⎢⎥⎣⎦∑∑∑∑3、二、 贝叶斯假设在考试的多项选择中，考生知道正确答案的概率为p ，猜测答案的概率为1-p ，并且假设考生知道正确答案答对题的概率为1，猜中正确答案的概率为1，其中m 为多选项的数目。

一类具局部化源的快扩散方程解的熄灭孟繁慧;高文杰【摘要】研究一类具局部化源的快扩散方程解的熄灭性质.通过分析非线性扩散项和非线性源项对解熄灭的影响并借助一些特殊构造的上下解,得到了该问题解的临界熄灭指标.结果表明:当源项指标适当大时,该问题的解对于适当小的初值在有限时刻熄灭;当源项指标适当小时,其最大解不熄灭.【期刊名称】《吉林大学学报（理学版）》【年(卷),期】2013(051)006【总页数】5页(P1041-1045)【关键词】快扩散方程;局部化源;临界熄灭指标【作者】孟繁慧;高文杰【作者单位】长春金融高等专科学校,长春130028;吉林大学数学学院,长春130012【正文语种】中文【中图分类】O175.80 引言考虑如下快扩散方程：解的熄灭性质.其中：0＜m＜1；a，q＞0；Ω是ℝN（N≥1）中的有界光滑区域；x0是Ω中的一个固定点；非负函数u0∈L∞（Ω）.该模型是一个耦合局部化源的快扩散方程，具有较强的物理和生物学背景，可以用来描述具有内部源的流体在多孔介质中的扩散或种群密度的变化规律［1－3］.解的有限时刻熄灭与解的有限时刻爆破相对应，也是发展型方程的一个重要性质.自Kalashnikov［4］通过比较 Cauchy问题：的解及其显式解提出解的有限时刻熄灭概念以来，关于该问题的研究已引起人们广泛关注.目前，发展方程解的熄灭理论已日趋完善［5－12］.特别地，如下形式的快扩散方程：得到了广泛研究，其中0＜m＜1.当f（u）＝auq，a，q＞0时，借助积分估计和Sobolev嵌入定理，李玉祥等［13］证明了：如果q＞m，则对适当小的初值u0，问题（2）的唯一解在有限时刻熄灭；如果q＜m，问题（2）的最大解在Ω内是恒正的；对于临界情形q＝m，问题（2）的解是否熄灭取决于－Δ在Ω上第一特征值的大小.田娅等［14］和尹景学等［15］分别对p－Laplace方程证明了类似的结论.当时，韩玉柱等［16］证明了该问题的临界熄灭指标为q＝m.与具局部源相对应的问题为：当q＝m时，问题（2）的解是否熄灭取决于aμ的大小.这里μ＝∫Ωφ（x）dx，φ（x）是如下线性椭圆方程的唯一正解：基于上述结果，本文研究问题（1）解的临界熄灭指标.1 预备知识当m＞1时，问题（1）是退化的；当0＜m＜1时，问题（1）是奇异的，所以它没有古典解.为了对问题（1）定义恰当的弱解，对任意的0＜T＜∞，记引入如下检验函数类：定义1 如果函数u∈L∞（QT）满足下列条件：1）u（x，0）≤（≥）u0（x），x∈Ω；2）u（x，t）≤（≥）0，（x，t）∈ΓT；3）对任意的t∈（0，T）和任意的ξ∈F，则称函数u是问题（1）在QT上的一个弱下解（弱上解）.如果u既是弱上解、又是弱下解，则称u是问题（1）在QT上的一个弱解.线性椭圆方程解具有相关性质：本文令φ（x）为式（3）的唯一正解，并记问题（1）弱解的局部存在性证明过程是标准的［17］.考虑如下正则化问题：这里选取T＞0适当小，使得对任意的k∈ℕ，问题（4）在QT上存在唯一正解uk（x，t），且‖uk‖L∞（QT）关于k是一致有界的.事实上，对任意的k∈ℕ，下述常微分方程Cauchy问题的解都是问题（4）的一个上解：选取T＞0为问题（5）的最大存在区间即可.此外，由一致抛物型方程的比较原理［18］可知，若k＜l，则1／l≤ul≤uk.由于uk关于k单调递减，故可定义易验证U（x，t）是问题（1）的一个弱解.如果u是问题（1）的任意解，则有下面定义有界函数Φk和Fk，使其满足进一步可得用类似文献［17］的方法选取一个恰当的检验函数ξ并借助Gronwall不等式可得u≤uk，从而u（x，t）≤U（x，t）.如果v是问题（1）的一个下解，则同理可得v≤U.于是，U（x，t）是问题（1）的最大解，且满足下解比较原理.类似上述过程可得：命题1 假设v和u分别是问题（1）在QT上的非负上、下解，且存在δ＞0，使得u≥δ，则u≤v于QT.2 主要结果定理1 如果q＞m，则当初值u0适当小时，问题（1）的解在有限时刻熄灭；如果q＝m且aμ＜1，则问题（1）的解对任意有界非负初值都在有限时刻熄灭.证明：首先证明q＝m且aμ＜1的情形.通过构造一个在有限时刻熄灭的上解完成证明.设φ1（x）是问题的唯一正解，这里区域Ω1满足Ω⊂Ω1.如果记则由线性椭圆方程的强极值原理可知μ1＞μ，δ＞0.其中μ＝∫Ωφ（x）dx，φ（x）是问题（3）的唯一正解.注意到方程（6）的解对区域Ω1有连续依赖性及假设aμ＜1，从而可以选取适当的区域Ω1⊃Ω，使得aμ1＜1.设g（t）是下述常微分方程初值问题的唯一解：这里常数A＞0满足对任意的由0＜m＜1可知，g（t）是单调不增的，且在时刻熄灭.定义易见v（x，t）也在T0 时刻熄灭.直接验证可知，v（x，t）是问题（1）的一个上解.事实上，v（x，t）（在弱的意义下）满足：此外，对任意给定的0＜T＜T0，存在正常数C1，C2，使得于是在×［0，T］上对v（x，t）和u（x，t）应用命题1可得u（x，t）≤v（x，t）.由T＜T0的任意性和v（x，T0）＝0可知u（x，T0）＝0.定义（x，t）＝u（x，t＋T0），则（x，t）是问题（1）的满足初值条件（x，0）＝0的解.同理对任意的A＞0，（x，t）≤v（x，t）.由v（x，t）的熄灭时间T0与A的关系可知，对任意的t＞0，有（x，t）＝0，即对任意的t≥T0，u（x，t）＝0.表明u（x，t）在T0 时刻熄灭.下面考虑q＞m的情形.令易验证，对于充分小的正常数k，如下关系式成立：固定满足上述条件的常数k可知，当初值u0（x）适当小使得u0（x）≤v（x）成立时，v（x）即为问题（1）的一个上解.此外，它还是有正下界的.由命题1可得从而u（x，t）是下述问题的弱下解：其中注意到a1 和具有正相关性，故当u0 充分小使得a1μ＜1时，w（x，t）在有限时刻熄灭，从而u（x，t）也在有限时刻熄灭.证毕.定理2 如果q＜m或q＝m且aμ＞1，则对任意非负初值u0（x），问题（1）的最大解U（x，t）是不熄灭的.证明：由于问题（1）的最大解U（x，t）满足下解比较原理，所以为证明定理的结论，只需构造一个不熄灭的下解即可.设v（x，t）＝g（t）φ1／m（x），其中φ（x）是问题（3）的唯一正解.如果g（t）满足如下常微分方程：则g（t）是单调不减且上方有界的.此外，容易验证v（x，t）是问题（1）当q＜m时的一个不熄灭下解.事实上，当x∈Ω时，v（x，0）＝0≤u0（x）；当x∈∂Ω时，v（x，t）＝0，且v（x，t）（在弱的意义下）满足：如果q＝m且aμ＞1，选取g（t）使其满足：同理可验证v（x，t）是问题（1）的一个不熄灭下解.于是U（x，t）不可能在任何有限时刻熄灭.证毕.注1 当q＝m且aμ＝1时，易验证对任意的k＞0，kφ1／m（x）是问题（1）的一个稳态解.从而对任意光滑的正初值u0（x），问题（1）的最大解U（x，t）是不熄灭的.参考文献【相关文献】［1］Vazquez J L.The Porous Medium Equation［M］.Oxford：Clarendon Press，2007. ［2］CHEN You－peng，XIE Chun－hong.Blow－up for a Porous Medium Equation with a Localized Source［J］.Appl Math Comput，2004，159（1）：79－93.［3］CHEN You－peng，LIU Qi－lin，GAO Hong－jun.Boundedness of Global Solutions of a Porous Medium Equation with a Localized Source［J］.Non Anal：Theory，Methods ＆Applications，2006，64（10）：2168－2182.［4］Kalashnikov A S.The Nature of the Propagation of Perturbations in Problems of Non－linear Heat Conduction with Absorption［J］.USSR Comp Math Math Phys，1974，14（4）：891－905.［5］Diaz G，Diaz I.Finite Extinction Time for a Class of Non－linear Parabolic Equations ［J］.Comm Part Differ Equations，1979，4（11）：1213－1231.［6］Lair A V，Oxley M E.Extinction in Finite Time for a Nonlinear Absorption－Diffusion Equation［J］.J Math Anal Appl，1994，182（3）：857－866.［7］Galaktionov V A，Vazquez J L.Asymptotic Behavior of Nonlinear Parabolic Equations with Critical Exponents.A Dynamical System Approach［J］.J Funct Anal，1991，100（2）：435－462.［8］Galaktionov V A，Vazquez J L.Extinction for a Quasilinear Heat Equation with AbsorptionⅠ.Technique of Intersection Comparison［J］.Comm Part Differ Equations，1994，19（7／8）：1075－1106.［9］Galaktionov V A，Vazquez J L.Extinction for a Quasilinear Heat Equation with AbsorptionⅡ.A Dynamical System Approach［J］.Comm Part Differ Equations，1994，19（7／8）：1107－1137.［10］Ferreira R，Vazquez J L.Extinction Behavior for Fast Diffusion Equations with Absorption ［J］.Non Anal：Theory，Methods ＆ Applications，2001，43（8）：943－985.［11］Friedman A，Herrero M A.Extinction Properties of Semilinear Heat Equations with Strong Absorption ［J］.J Math Anal Appl，1987，124（2）：530－546.［12］Herrero M A，Velazquezj J J L.Approaching an Extinction Point in One－Dimensional Semilinear Heat Equations with Strong Absorptions［J］.J Math Anal Appl，1992，170（2）：353－381.［13］LI Yu－xiang，WU Ji－chun.Extinction for Fast Diffusion Equations with Nonlinear Sources［J］.Electron J Differ Equations，2005，2005（23）：1－7.［14］TIAN Ya，MU Chun－lai.Extinction and Non－extinction for a p－Laplacian Equation with Nonlinear Source ［J］.Non Anal：Theory，Methods ＆ Applications，2008，69（8）：2422－2431.［15］YIN Jing－xue，JIN Chun－hua.Critical Extinction and Blow－up Exponents for Fast Diffusive p－Laplacian with Sources［J］.Math Method Appl Sci，2007，30（10）：1147－1167.［16］HAN Yu－zhu，GAO Wen－jie.Extinction for a Fast Diffusion Equation with a Nonlinear Nonlocal Source ［J］.Arch Math，2011，97（4）：353－363.［17］Anderson J R.Local Existence and Uniqueness of Solutions of Degenerate Parabolic Equations ［J］.Commun 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Some Properties of Solutions of Periodic Second OrderLinear Differential Equations1. Introduction and main resultsIn this paper, we shall assume that the reader is familiar with the fundamental results and the stardard notations of the Nevanlinna's value distribution theory of meromorphic functions [12, 14, 16]. In addition, we will use the notation )(f σ，)(f μand )(f λto denote respectively the order of growth, the lower order of growth and the exponent of convergence of the zeros of a meromorphic function f ，)(f e σ（[see 8]），the e-type order of f(z), is defined to berf r T f r e ),(log lim)(+∞→=σSimilarly, )(f e λ，the e-type exponent of convergence of the zeros of meromorphic function f , is defined to berf r N f r e )/1,(loglim)(++∞→=λWe say that )(z f has regular order of growth if a meromorphic function )(z f satisfiesrf r T f r log ),(log lim)(+∞→=σWe consider the second order linear differential equation0=+''Af fWhere )()(z e B z A α=is a periodic entire function with period απω/2i =. The complex oscillation theory of (1.1) was first investigated by Bank and Laine [6]. Studies concerning (1.1) have een carried on and various oscillation theorems have been obtained [2{11, 13, 17{19].When )(z A is rational in ze α，Bank and Laine [6] proved the following theoremTheorem A Let )()(z e B z A α=be a periodic entire function with period απω/2i = and rational in zeα.If )(ζB has poles of odd order at both ∞=ζ and 0=ζ, then for everysolution )0)((≠z f of (1.1), +∞=)(f λBank [5] generalized this result: The above conclusion still holds if we just suppose that both ∞=ζ and 0=ζare poles of )(ζB , and at least one is of odd order. In addition, the stronger conclusion)()/1,(l o g r o f r N ≠+ (1.2) holds. When )(z A is transcendental in ze α, Gao [10] proved the following theoremTheorem B Let ∑=+=p j jj b g B 1)/1()(ζζζ,where )(t g is a transcendental entire functionwith 1)(<g σ, p is an odd positive integer and 0≠p b ，Let )()(ze B z A =.Then anynon-trivia solution f of (1.1) must have +∞=)(f λ. In fact, the stronger conclusion (1.2) holds.An example was given in [10] showing that Theorem B does not hold when )(g σis any positive integer. If the order 1)(>g σ , but is not a positive integer, what can we say? Chiang and Gao [8] obtained the following theoremsTheorem 1 Let )()(ze B z A α=,where )()/1()(21ζζζg g B +=,1g and 2g are entire functions with 2g transcendental and )(2g μnot equal to a positive integer or infinity, and 1g arbitrary. If Some properties of solutions of periodic second order linear differential equations )(z f and )2(i z f π+are two linearly independent solutions of (1.1), then+∞=)(f e λOr2)()(121≤+--g f e μλWe remark that the conclusion of Theorem 1 remains valid if we assume )(1g μ isnotequaltoapositiveintegerorinfinity,and2g arbitraryand stillassume )()/1()(21ζζζg g B +=,In the case when 1g is transcendental with its lower order not equal to an integer or infinity and 2g is arbitrary, we need only to consider )/1()()/1()(*21ηηηηg g B B +==in +∞<<η0,ζη/1<.Corollary 1 Let )()(z e B z A α=,where )()/1()(21ζζζg g B +=,1g and 2g are entire functions with 2g transcendental and )(2g μno more than 1/2, and 1g arbitrary.(a) If f is a non-trivial solution of (1.1) with +∞<)(f e λ,then )(z f and )2(i z f π+are linearly dependent.(b)If 1f and 2f are any two linearly independent solutions of (1.1), then +∞=)(21f f e λ.Theorem 2 Let )(ζg be a transcendental entire function and its lower order be no more than 1/2. Let )()(z e B z A =,where ∑=+=p j jj b g B 1)/1()(ζζζand p is an odd positive integer,then +∞=)(f λ for each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.We remark that the above conclusion remains valid if∑=--+=pj jjbg B 1)()(ζζζWe note that Theorem 2 generalizes Theorem D when )(g σis a positive integer or infinity but2/1)(≤g μ. Combining Theorem D with Theorem 2, we haveCorollary 2 Let )(ζg be a transcendental entire function. Let )()(z e B z A = where ∑=+=p j jj b g B 1)/1()(ζζζand p is an odd positive integer. Suppose that either (i) or (ii)below holds:(i) )(g σ is not a positive integer or infinity; (ii) 2/1)(≤g μ;then +∞=)(f λfor each non-trivial solution f to (1.1). In fact, the stronger conclusion (1.2) holds.2. Lemmas for the proofs of TheoremsLemma 1 ([7]) Suppose that 2≥k and that 20,.....-k A A are entire functions of period i π2,and that f is a non-trivial solution of0)()()(2)(=+∑-=k i j j z yz A k ySuppose further that f satisfies )()/1,(logr o f r N =+; that 0A is non-constant and rationalin ze ，and that if 3≥k ，then 21,.....-k A A are constants. Then there exists an integer qwith k q ≤≤1 such that )(z f and )2(i q z f π+are linearly dependent. The same conclusionholds if 0A is transcendental in ze ，andf satisfies )()/1,(logr o f r N =+，and if 3≥k ，thenas ∞→r through a set1L of infinite measure, wehave )),((),(j j A r T o A r T =for 2,.....1-=k j .Lemma 2 ([10]) Let )()(z e B z A α=be a periodic entire function with period 12-=απωi and betranscendental in z e α, )(ζB is transcendental and analytic on +∞<<ζ0.If )(ζB has a pole of odd order at ∞=ζ or 0=ζ(including those which can be changed into this case by varying the period of )(z A and Eq . (1.1) has a solution 0)(≠z f which satisfies )()/1,(logr o f r N =+,then )(z f and )(ω+z f are linearly independent. 3. Proofs of main resultsThe proof of main results are based on [8] and [15].Proof of Theorem 1 Let us assume +∞<)(f e λ.Since )(z f and )2(i z f π+are linearly independent, Lemma 1 implies that )(z f and )4(i z f π+must be linearly dependent. Let )2()()(i z f z f z E π+=,Then )(z E satisfies the differential equation222)()()(2))()(()(4z E cz E z E z E z E z A -''-'=, (2.1)Where 0≠c is the Wronskian of 1f and 2f (see [12, p. 5] or [1, p. 354]), and )()2(1z E c i z E =+πor some non-zero constant 1c .Clearly, E E /'and E E /''are both periodic functions with period i π2,while )(z A is periodic by definition. Hence (2.1) shows that 2)(z E is also periodic with period i π2.Thus we can find an analytic function )(ζΦin +∞<<ζ0,so that )()(2z e z E Φ=Substituting this expression into (2.1) yieldsΦΦ''+ΦΦ'-ΦΦ'+Φ=-2222)(43)(4ζζζζcB (2.2)Since both )(ζB and )(ζΦare analytic in }{+∞<<=ζζ1:*C ,the V aliron theory [21, p. 15] gives their representations as)()()(ζζζζb R B n =，)()()(11ζφζζζR n =Φ， （2.3）where n ，1n are some integers, )(ζR and )(1ζR are functions that are analytic and non-vanishing on }{*∞⋃C ，)(ζb and )(ζφ are entire functions. Following the same arguments as used in [8], we have),(),()/1,(),(φρρφρφρS b T N T ++=， （2.4） where )),((),(φρφρT o S =.Furthermore, the following properties hold [8])}(),(max{)()()(222E E E E f eL eR e e e λλλλλ===,)()()(12φλλλ=Φ=E eR ,Where )(2E eR λ(resp, )(2E eL λ) is defined to berE r N R r )/1,(loglim2++∞→(resp, rE r N R r )/1,(loglim2++∞→),Some properties of solutions of periodic second order linear differential equationswhere )/1,(2E r N R (resp. )/1,(2E r N L denotes a counting function that only counts the zeros of 2)(z E in the right-half plane (resp. in the left-half plane), )(1Φλis the exponent of convergence of the zeros of Φ in *C , which is defined to beρρλρlog )/1,(loglim)(1Φ=Φ++∞→NRecall the condition +∞<)(f e λ,we obtain +∞<)(φλ.Now substituting (2.3) into (2.2) yields+'+'+-'+'++=-21112111112)(43)()()()()(4φφζζφφζζζφζζζζζR R n R R n R cb R n n)222)1((1111111112112φφφφζφφζφφζζζ''+''+'''+''+'+'+-R R R R R n R R n n n （2.5）Proof of Corollary 1 We can easily deduce Corollary 1 (a) from Theorem 1 .Proof of Corollary 1 (b). Suppose 1f and 2f are linearlyindependentand +∞<)(21f f e λ,then +∞<)(1f e λ,and +∞<)(2f e λ.We deduce from the conclusion of Corollary 1 (a) that )(z f j and )2(i z f j π+are linearly dependent, j = 1; 2.Let)()()(21z f z f z E =.Then we can find a non-zero constant2c suchthat )()2(2z E c i z E =+π.Repeating the same arguments as used in Theorem 1 by using the fact that 2)(z E is also periodic, we obtain2)()(121≤+--g E e μλ,a contradiction since 2/1)(2≤g μ.Hence +∞=)(21f f e λ.Proof of Theorem 2 Suppose there exists a non-trivial solution f of (1.1) that satisfies)()/1,(logr o f r N =+. We deduce 0)(=f e λ, so )(z f and )2(i z f π+ are linearlydependent by Corollary 1 (a). However, Lemma 2 implies that )(z f and )2(i z f π+are linearly independent. This is a contradiction. Hence )()/1,(log r o f r N ≠+holds for each non-trivial solution f of (1.1). This completes the proof of Theorem 2.Acknowledgments The authors would like to thank the referees for helpful suggestions to improve this paper. References[1] ARSCOTT F M. Periodic Di®erential Equations [M]. The Macmillan Co., New Y ork, 1964. [2] BAESCH A. On the explicit determination of certain solutions of periodic differentialequations of higher order [J]. Results Math., 1996, 29(1-2): 42{55.[3] BAESCH A, STEINMETZ N. Exceptional solutions of nth order periodic linear differentialequations [J].Complex V ariables Theory Appl., 1997, 34(1-2): 7{17.[4] BANK S B. On the explicit determination of certain solutions of periodic differential equations[J]. Complex V ariables Theory Appl., 1993, 23(1-2): 101{121.[5] BANK S B. Three results in the value-distribution theory of solutions of linear differentialequations [J].Kodai Math. J., 1986, 9(2): 225{240.[6] BANK S B, LAINE I. Representations of solutions of periodic second order linear differentialequations [J]. J. Reine Angew. Math., 1983, 344: 1{21.[7] BANK S B, LANGLEY J K. Oscillation theorems for higher order linear differential equationswith entire periodic coe±cients [J]. Comment. Math. Univ. St. Paul., 1992, 41(1): 65{85.[8] CHIANG Y M, GAO Shi'an. On a problem in complex oscillation theory of periodic secondorder lineardifferential equations and some related perturbation results [J]. Ann. Acad. Sci. Fenn. Math., 2002, 27(2):273{290.一些周期性的二阶线性微分方程解的方法1． 简介和主要成果在本文中，我们假设读者熟悉的函数的数值分布理论[12，14，16]的基本成果和数学符号。