Asymptotic behaviors of solutions for dissipative quantum Zakharov equations
创新思维的重要性及其发展途径英语作文
创新思维的重要性及其发展途径英语作文The Importance of Innovative Thinking and Its Development PathwaysInnovation is the cornerstone of progress in our ever-evolving world. It is the driving force behind advancements in technology, science, and various other fields, shaping the way we live, work, and interact with one another. The ability to think innovatively is a crucial skill that not only sets individuals apart but also propels organizations and societies forward. In this essay, we will explore the importance of innovative thinking and the pathways that can foster its development.Firstly, innovative thinking is essential for addressing the complex challenges we face in the modern era. From climate change to global health crises, the problems we encounter often require novel solutions that go beyond traditional approaches. Innovative thinkers are able to step outside the confines of conventional wisdom, challenge existing paradigms, and devise innovative strategies to tackle these pressing issues. By thinking outside the box, they can uncover new opportunities and create transformative change.Moreover, innovative thinking is a key driver of economic growthand competitiveness. In today's rapidly changing business landscape, organizations that embrace innovation are more likely to thrive and maintain a competitive edge. Innovative companies are able to develop new products, services, and business models that cater to evolving customer needs and market demands. This not only allows them to stay ahead of the curve but also opens up new revenue streams and growth opportunities.Furthermore, innovative thinking fosters personal and professional development. Individuals who cultivate their ability to think innovatively are better equipped to adapt to change, solve problems creatively, and seize new opportunities. This mindset can lead to enhanced problem-solving skills, increased creativity, and a greater capacity for risk-taking and entrepreneurship. As a result, innovative thinkers are often more valuable assets in the workplace and are better positioned to advance their careers and contribute to the success of their organizations.To nurture the development of innovative thinking, there are several key pathways that individuals and organizations can explore. One crucial aspect is the cultivation of a supportive and collaborative environment. Organizations that encourage open communication, risk-taking, and the exchange of diverse ideas are more likely to foster innovative thinking among their employees. By creating a culture that celebrates experimentation, acknowledges failures aslearning opportunities, and values interdisciplinary collaboration, organizations can unleash the innovative potential of their workforce.Another important pathway is the promotion of lifelong learning and continuous skill development. Innovative thinkers are often those who are curious, adaptable, and constantly seeking new knowledge and experiences. By engaging in ongoing education, whether through formal training, self-directed learning, or exposure to diverse perspectives, individuals can expand their knowledge, challenge their assumptions, and develop the cognitive flexibility required for innovative thinking.Additionally, the integration of design thinking principles can be a powerful tool for cultivating innovative thinking. Design thinking emphasizes a human-centered approach to problem-solving, encouraging individuals to deeply understand the needs and perspectives of the end-users or stakeholders involved. By applying this iterative process of empathizing, defining, ideating, prototyping, and testing, innovators can develop solutions that are more responsive to real-world challenges and needs.Furthermore, the fostering of interdisciplinary collaboration can be a catalyst for innovative thinking. By bringing together individuals from diverse backgrounds, disciplines, and areas of expertise, organizations can leverage the power of cross-pollination. Thisintegration of different perspectives, skills, and knowledge can lead to the emergence of novel ideas and innovative solutions that would not have been possible through a siloed approach.In conclusion, the importance of innovative thinking cannot be overstated. It is a crucial skill that drives progress, fuels economic growth, and empowers individuals to navigate the complexities of the modern world. By cultivating a supportive environment, promoting lifelong learning, embracing design thinking principles, and fostering interdisciplinary collaboration, individuals and organizations can unlock the transformative potential of innovative thinking. As we continue to face unprecedented challenges and opportunities, the ability to think innovatively will be a key determinant of success and the shaping of a better future.。
好奇心是科学突破的秘诀英语作文
好奇心是科学突破的秘诀英语作文Curiosity is the key to scientific breakthroughs. Throughout history, countless discoveries and innovations have been made possible by individuals who were driven by a relentless curiosity to explore the unknown. From the invention of the light bulb to the discovery of penicillin, curiosity has played a fundamental role in pushing the boundaries of human knowledge and understanding.One of the greatest examples of the power of curiosity in driving scientific breakthroughs is the story of Sir Isaac Newton. As a young student, Newton was constantly asking questions and seeking answers to the mysteries of the world around him. His curiosity led him to develop the laws of motion and gravitation, revolutionizing our understanding of the universe and laying the foundation for modern physics.In a similar vein, the pioneering research of Marie Curie in the field of radioactivity was fueled by her insatiable curiosity. Despite facing numerous obstacles and challenges as a woman in a male-dominated field, Curie's relentless pursuit of knowledge ultimately led to groundbreaking discoveries that transformed our understanding of the nature of matter and energy.More recently, the curiosity of scientists like Jane Goodall has led to new insights into the behavior of chimpanzees and other primates. Through decades of observation and research, Goodall has challenged conventional thinking and deepened our understanding of the complex social structures and behaviors of these animals.In today's rapidly evolving world, curiosity continues to drive scientific progress in fields ranging from space exploration to biotechnology. The Mars Curiosity rover, for example, is a testament to humanity's insatiable desire toexplore and discover new horizons. By analyzing the Martian surface and collecting valuable data, the rover is helping scientists unlock the mysteries of the Red Planet and pave the way for future missions to Mars.In conclusion, curiosity remains a powerful force that propels scientific breakthroughs and drives innovation. By embracing our natural inclination to question, explore, and discover, we can unlock the secrets of the universe and push the boundaries of human knowledge further than ever before. As the famous physicist Richard Feynman once said, "I would rather have questions that can't be answered than answers that can't be questioned." It is this spirit of curiosity that will continue to inspire and guide us on our quest for understanding and discovery.。
具有Hassell-Varley型反应函数的捕食者－食饵系统的随机建模
具有Hassell-Varley型反应函数的捕食者－食饵系统的随机建模费清;许超群;原三领【摘要】基于具有 Hassell-Varley 型功能反应函数的确定性捕食系统建立了两类新的随机捕食者－食饵模型：连续时间马尔科夫链模型和伊藤型随机微分方程模型。
分析了不同形式的出生率和死亡率对模型动力学的影响，并通过数值模拟讨论了模型的渐近性态。
%Based on a deterministic predator-prey system with Hassell-Varley type response function,two new stochastic predator-prey models,a continuous time Markov chain model and an Ito type stochastic predator-prey model were established.The influences of different forms of birth and death rate on the dynamics of the model were ing numerical simulations,the asymptotic behaviors of the models were discussed.【期刊名称】《上海理工大学学报》【年(卷),期】2016(038)002【总页数】6页(P103-108)【关键词】捕食者-食饵模型;Hassell-Varley型反应函数;连续时间马尔科夫链;随机建模【作者】费清;许超群;原三领【作者单位】上海理工大学理学院，上海 200093;上海理工大学理学院，上海200093;上海理工大学理学院，上海 200093【正文语种】中文【中图分类】O175在种群动力学中,捕食者-食饵模型一直是研究者重点关注、研究最多的模型.一般的捕食者-食饵模型具有如下结构[1]:式中:x与y分别为食饵和捕食者的数量;r为食饵种群的内禀增长率;K为环境最大容纳量;ε为食饵转化为捕食者的转化率;μ为捕食者的死亡率;g(x,y)为捕食者对食饵的功能性反应函数.在模型(1)中,捕食者对食饵的功能反应函数g(x,y)起着重要的作用.陈兰荪等[1]总结了g(x,y)常见的几种形式,例如,Holling I-III型[2]、Beddington-DeAngelis型[3]、Crowley-Martin型[4]、比率依赖型[5]等.根据具体的生物背景,研究者们选取不同的功能反应函数建立了不同的模型[6-7].例如,张拥军等[6]建立了捕食者具有传染病且功能性反应函数为仅依赖被捕食者的捕食模型,通过构造Liapunov函数得到平衡点的局部稳定性和全局渐近稳定性的充分条件;孙凯玲等[7]研究了捕食者具有性别偏食且功能反应函数为Holling III型的捕食模型,通过构造Liapunov函数证明了系统周期解的存在唯一性,讨论了周期解全局稳定性的充分条件.最近,Hsu等[8]研究了如下具有Hassell-Varley(HV)型功能反应函数的捕食模型:式中:e为单位时间内捕食者与所有食饵相遇的概率;c为每次相遇捕食成功的概率;h 为捕食者与食饵搏斗的时间;α为HV常量,α∈[0,1].在文献[8]中,给出了平衡点局部稳定的条件,讨论了系统的一致持久性,证明了当α=1/2时,系统存在唯一极限环.Wang[9]考虑了一类具有Hassell-Varley型功能反应函数的时滞捕食者-食饵模型的周期解,利用正重合度定理得到了正周期解存在的充分条件.上述研究均忽略了环境波动对捕食系统的影响,然而,种群所处的环境会受到各种随机因素(如气候、温度等)的影响[10-11],而这些因素可能会对模型的动力学产生很大的影响.因此,本文以模型(2)为基础,建立了两类具有Hassell-Varley型功能反应函数的随机捕食者-食饵模型:一类是连续时间马尔科夫链模型,另一类是伊藤型随机微分方程模型.其中,随机微分方程模型的建立是以前者为基础,并通过数值模拟讨论模型的渐近性态.基于常微分模型(2)建立其相应的连续时间马尔科夫链模型.设X(t)和Y(t)为离散随机变量,分别表示t时刻食饵与捕食者数量,其中,X(t),Y(t)∈{0,1,2,…,N},t∈+.在从t 时刻变化到t+Δt时刻时,随机变量X(t)和Y(t)的变化为ΔX和ΔY,其中,ΔX=X(t+Δt)-X(t),ΔY=Y(t+Δt)-Y(t),且Δt是充分小的时间段,使得,即在充分短的时间内,至少存在出生或者死亡,且出生和死亡仅发生一次.给定初值X(0)=x0>0,Y(0)=y0>0,则二元随机过程{(X(t),Y(t))}的联合概率函数为现分别用随机变量X(t)和Y(t)的生灭过程来建立连续时间马尔科夫链模型.设ai和bi(i=1,2)分别为食饵和捕食者的出生率和死亡率,其中则其无穷小转移概率为[12]概率px,y满足柯尔莫哥向前方程当y=0时,有当x=0,又有且两个变量过程的分布阶数可以由柯尔莫哥向前方程推导出,于是,矩母函数是有如下形式:其中,θ1,θ2∈.其中,(ε1,ε2)∈(0,x)×(0,y),δ∈(ε1,x),η∈(ε2,y),可以得出矩母函数是如下偏微分方程的一个解:上式两端同时对θ1求偏导,可以得到令上式中θ1=θ2=0,可以得出x期望满足的微分方程为又由于同理,可以得到y期望满足的微分方程为类似的方法得出n阶矩所满足的微分方程为可以直观地看出x的均值被最大环境容纳量所约束,由x∈{0,1,2,…},x的期望是非负的,满足不等式进而有若0<x0≤K,则得到随机变量X满足{0≤E(X)≤K},即X期望具有类似确定模型不变子集的性质.现以确定性模型(2)为基础,利用上述马尔科夫链模型等[12]建立相应的伊藤型随机微分方程.设x(t)和y(t)为两个连续随机变量,分别表示t时刻食饵与捕食者数量,令Z(t)=(x(t),y(t))T,其中,,t∈(0,),表示随机变量在Δt时间段变化的随机向量,根据式(4)可求得在充分小时间Δt内的期望和协方差Var(ΔZ).充分小时间Δt内Δx与Δy的方差分别为Δx与Δy的协方差为因此,ΔZ的协方差为又因为所以,Var(ΔZ)≈M2Δt.由上述计算可知,当Δt充分小时,ΔZ近似于一个均值和协方差分别为μΔt和M2Δt 的正态分布.令ψ=(φ1,φ2)T～N(0,I),则).因此,Z(t+Δt)可以近似表达为由文献[12]可知,方程(5)是伊藤型随机微分方程的一个欧拉近似,即式(5)收敛到如下伊藤型随机微分方程:dZ=μdt+MdB式中:B=(B1,B2)T,B1和B2分别表示相互独立的标准布朗运动;μ为漂移系数;M为扩散矩阵.注意到μ与M的表达式,方程(6)即为由文献[13]可知,方程(6)的形式可以不唯一,且与之等价的随机微分方程具有相同的联合概率密度函数.与方程(6)等价的随机微分方程具有如下形式:dZ=μdt+CdB*其中,扩散矩阵C满足CCT=M2.例如,其中,分别表示相互独立的标准布朗运动.上述假设出生率和死亡率分别为模型(2)的正项和负项部分,得到了伊藤型随机微分方程(7).当出生率和死亡率取其他形式时,可以得到与方程(7)不同形式的伊藤型随机微分方程.例如,选取其中,aij,bij>0,i,j=1,2,且满足以下关系对于重新定义的ai和bi,通过类似方法可以得到如下的伊藤型随机微分方程:由ai1+bi1>αi,ai2+bi2>βi可知,方程(9)中的扩散系数比方程(7)中的扩散系数大. 基于具有Hassell-Varley型功能反应函数的确定性捕食系统建立了两类新的随机捕食者-食饵模型:一类是连续时间马尔科夫链模型;另一类是伊藤型随机微分方程模型.分析了不同形式的出生率和死亡率对随机模型的影响.现通过数值模拟讨论模型的渐近性态.选取初值(x0,y0)=(10,20),参数α=0.5,r=1.5,K=16.67,c=0.14,ε=0.6,e=0.5,h=13,μ=0.035.其中,ε>hμ,计算得,此时确定性模型(2)的正平衡点E*稳定.图1和图2(见下页)中的蓝色虚线和红色实线分别表示确定性模型和随机模型的解曲线.从图1和图2察到随机模型(7)和随机模型(9)的解都围绕确定模型(2)的正平衡点E*振荡.比较图1和图2可以发现,随机模型(9)解的振荡幅度明显大于随机模型(7)解的振荡幅度,这与理论结果相符.图3(见下页)是马尔科夫链模型(4)的二维随机游走,容易看出,马尔科夫链模型(4)的随机游走是围绕确定性模型(2)的正平衡点E*进行的.【相关文献】[1] 陈兰荪,宋新宇,陆征一.数学生态学模型与研究方法[M].成都:四川科学技术出版社,2003.[2] Walley G S.The Odonata of Canada and Alaska[J].The CanadianEntomologist,1959,91(5):291-292.[3] Beddington J R.Mutual interference between parasites or predators and its effect on searching efficiency[J].Journal of Animal Ecology,1975,44(1):331-340.[4] Crowley P H,Martin E K.Functional responses and interference within and between year classes of a dragonfly population[J].Journal of the North American Benthological Society,1989,8(3):211-221.[5] Arditi R,Ginzburg L R.Coupling in predator-prey dynamics:ratiodependence[J].Journal of Theoretical Biology,1989,139(3):311-326.[6] 张拥军,王美娟,徐金瑞.捕食者具有传染病的捕食系统模型的分析[J].上海理工大学学报,2009,31(5):409-413.[7] 孙凯玲,王美娟,朱春娟.具有性别偏食和Holling III类功能反应的食饵捕食者模型[J].上海理工大学学报,2009,31(1):6-10.[8] Hsu S B,Hwang T W,Kuang Y.Global dynamics of a predator-prey model with Hassell-Varley type functional response[J].Discrete and Continuous Dynamical Systems-Series B,2008,10(4):857-871.[9] Wang K.Periodic solutions to a delayed predator-prey model with Hassell-Varley type functional response[J].Nonlinear Analysis:Real World Applications,2011,12(1):137-145. [10] Ji C Y,Jiang D Q,Shi N Z.Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation[J].Journal of Mathematical Analysis and Applications,2009,359(2):482-498.[11] Mandal P S,Banerjee M.Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model[J].Physica A,2012,391(4):1216-1233.[12] Mandal P S,Allen L J S,Banerjee M.Stochastic modeling of phytoplankton allelopathy[J].Applied Mathematical Modelling,2014,38(5/6):1583-1596.[13] Allen E J,Allen L J S,Arciniega A,et al.Construction of equivalent stochastic differential equation models[J].Stochastic Analysis and Applications,2008,26(2):274-297.。
cfd中的asymptotic theory -回复
cfd中的asymptotic theory -回复[cfd中的asymptotic theory]Asymptotic theory plays a vital role in Computational Fluid Dynamics (CFD) as it provides a mathematical framework to understand the behavior of fluid flow in various situations. In this article, we will explore the fundamentals of asymptotic theory in CFD and its applications.What is asymptotic theory?Asymptotic theory is a branch of mathematics that deals with the behavior of functions as a variable approaches a particular value. It provides a systematic way to approximate complex functions and simplify their analysis. In CFD, asymptotic theory is used to solve fluid flow problems by approximating complex equations and simplifying their solutions.The foundation of asymptotic theory in CFD lies in the concept of a small parameter. A small parameter is a dimensionless quantity that represents the scale of a physical phenomenon relative to other quantities in the problem. It allows us to analyze thebehavior of the fluid flow as the small parameter tends to zero or infinity.The key idea behind asymptotic theory is that we can approximate the solution to a problem by expanding it as a series in powers of the small parameter. The resulting series is then truncated at an appropriate order to obtain an approximate solution. This approach is particularly useful when the problem involves a wide range of scales, such as in turbulent flows or flows with thin boundary layers.Applications of asymptotic theory in CFD1. Boundary layer theory: Boundary layers are thin layers of fluid that form near solid boundaries in a fluid flow. They play a crucial role in many engineering applications, such as aerodynamics and heat transfer. Asymptotic theory allows us to derive simplified equations for the boundary layer and study its behavior.By assuming the small parameter to be the ratio of the boundary layer thickness to the characteristic length scale of the problem, we can derive the famous Prandtl's boundary layer equations.These equations provide a simplified description of the flow near the boundary and have been widely used to analyze various flow problems.2. Perturbation methods: Perturbation methods involve solving a problem by treating the small parameter as a small perturbation to an already known solution. This approach is particularly useful when the small parameter is not related to a physical scale but appears in the governing equations due to simplifications or assumptions.For example, in the study of unsteady flows, the small parameter can be the ratio of the unsteadiness time scale to a characteristic time scale of the problem. By assuming a known steady solution and perturbing it in terms of the small parameter, we can develop a systematic procedure to obtain the unsteady solution.3. Homogenization theory: Homogenization theory is concerned with problems where the governing equations have oscillatory coefficients or rapidly varying parameters. These problems arise in various fields, such as porous media flow and composite materials.Asymptotic theory provides a powerful tool to derive effective (homogenized) equations that capture the behavior of the system on a macroscopic scale. By assuming the small parameter as the ratio of the periodicity length to the characteristic length scale of the problem, we can develop an asymptotic expansion to obtain the macroscopic equations.ConclusionAsymptotic theory is an essential tool in CFD for studying complex fluid flow problems. It allows us to approximate the solution to a problem by expanding it as a series in powers of a small parameter and truncating the series at an appropriate order. This approach provides simplified equations that capture the essential behavior of the flow and enable efficient computational analysis. Through examples such as boundary layer theory, perturbation methods, and homogenization theory, we have seen how asymptotic theory finds applications in various areas of CFD. By using asymptotic theory, researchers can gain valuable insights into the behavior of fluid flow and develop efficient numericalmethods for practical engineering applications.。
Scattering of solitons on resonance
We found that the scattering of solitary waves on resonance is a general effect for nonlinear equations described the wave propagation. In this work we investigate this effect for the simplest model. It allows to show the essence of this effect without unnecessary details.
1 Statement of the problem and result
Let us consider the perturbed NLSE
i∂tΨ + ∂x2Ψ + |Ψ|2Ψ = ε2f eiS/ε2,
2017年考研英语一作文
2017年考研英语一作文范文：Critical thinking is an essential skill that everyone should possess in today's complex and rapidly changing world. It is the ability to analyze and evaluate information and ideas in a logical and systematic way, and to make well-informed decisions. In this essay, I will discuss the importance of critical thinking and how it can be developed and applied in various aspects of our lives.First and foremost, critical thinking is crucial for academic success. In an era where information is abundant and easily accessible, students need to be able to discern the reliability and validity of the information they encounter. With critical thinking skills, they can evaluate the credibility of sources, analyze the evidence presented, and draw well-reasoned conclusions. This not only helps them excel in their studies but also prepares them for the challenges of the real world.Furthermore, critical thinking is essential for effective problem-solving. In today's complex and interconnected world, we are constantly faced with a myriad of problems and challenges. Whether it's in the workplace, in our personal lives, or in society as a whole, theability to think critically is invaluable. By carefully analyzing the root causes of problems, considering alternative solutions, and weighing the potential consequences, individuals can make informed decisions and find effective solutions.Moreover, critical thinking is vital for fostering open-mindedness and tolerance. In an increasingly diverse and globalized world, it is important to be able to consider different perspectives and to engage in constructive dialogue with others. Critical thinking enables individuals to question their own assumptions, challenge their biases, and appreciate the complexity of issues. This not only promotes intellectual growth but also fosters mutual understanding and respect.In addition, critical thinking is essential for making informed decisions as citizens and consumers. In a democratic society, it is crucial for people to be able to critically evaluate political rhetoric, media messages, and advertising claims. By applying critical thinking skills, individuals can separate fact from fiction, identify hidden agendas, and make well-informed choices. This is essential for maintaining a healthy democracy and for promoting ethical and responsible consumption.In conclusion, critical thinking is a vital skill that is essential for success in today's world. It is crucialfor academic achievement, effective problem-solving, open-mindedness, and informed decision-making. Therefore, it is important for individuals to actively develop and apply their critical thinking skills in all aspects of their lives. By doing so, they can navigate the complexities of the modern world with confidence and competence.高质量仿写：The Significance of Critical Thinking in the 21stCentury。
Topics in conformally compact Einstein metrics
a r X i v :m a t h /0503243v 2 [m a t h .D G ] 2 N o v 2005TOPICS IN CONFORMALLY COMPACT EINSTEIN METRICSMICHAEL T.ANDERSON1.Introduction.Conformal compactifications of Einstein metrics were introduced by Penrose [38],as a means to study the behavior of gravitational fields at infinity,i.e.the asymptotic behavior of solutions to the vacuum Einstein equations at null infinity.This has remained a very active area of research,cf.[27],[19]for recent surveys.In the context of Riemannian metrics,the modern study of conformally compact Einstein metrics began with the work of Fefferman-Graham [26],in connection with their study of conformal invariants of Riemannian metrics.Recent mathematical work in this area has been significantly influenced by the AdS/CFT (or gravity-gauge)correspondence in string theory,introduced by Maldacena [36].We will only comment briefly here on aspects of the AdS/CFT correspondence,and refer to [2],[42],[7]for general surveys.In this paper,we discuss recent mathematical progress in this area,focusing mainly on global aspects of conformally compact Einstein metrics and the global existence question for the Dirichlet problem.One reason for this is that it now appears that the beginnings of a general existence theory for such metrics may be emerging,at least in dimension 4.Of course to date there is no general theory for the existence of complete Einstein metrics on manifolds,with two notable exceptions;the existence theory for K¨a hler-Einstein metrics due to Calabi,Yau,Aubin and others,and the existence theory in dimension 3,due to Perelman,Hamilton and Thurston.In contrast to the situation for compact 4-manifolds,an existence theory for conformally compact Einstein metrics may not be that far beyond the current horizon.We discuss numerous open problems on this topic;some new results are also presented,cf.in particular Theorem 3.4and the discussion and results in Sections 4and 5.In brief,the contents of the paper are as follows.The groundwork is laid in §2,where we discuss the moduli space of conformally compact Einstein metrics and the boundary map tothe space of conformal infinities.The general situation is also illustrated by the discussion of a simple but important class of examples,the static AdS black hole metrics.Section 3deals with the general asympototic behavior of the metrics near conformal infinity,and the control of the asymptotic behavior by the metric at infinity.It will be seen that at least in even dimensions,this issue is now quite well understood.Then in Section 4we turn to the analysis of the behavior of the metrics on compact regions,away from infinity,mostly in dimension 4where the possible degenerations can be described in terms of orbifold and cusp degenerations.In Section 5,we conclude with a discussion of the possibility of actually finding examples where orbifold or cusp degenerations occur.I would like to thank David Calderbank,Tom Farrell,Lowell Jones,Claude LeBrun,Rafe Mazzeo and Michael Singer for discussions related to various issues in the paper.Thanks also to Vestislav Apostolov and collegues for organizing an interesting workshop at the CRM,Montreal in July,04.22.Conformally compact Einstein metrics.Let M be the interior of a compact(n+1)-dimensional manifold¯M with boundary∂M.A complete Riemannian metric g on M is C m,αconformally compact if there is a defining functionρon¯M such that the conformally equivalent metric(2.1) g=ρ2gextends to a C m,αmetric on the compactification¯M.Hereρis a smooth,non-negative function on¯M withρ−1(0)=∂M and dρ=0on∂M.The induced metricγ= g|∂M is the boundary metric associated to the compactification g.Since there are many possible defining functions,there are many conformal compactifications of a given metric g,and so only the conformal class[γ]ofγon∂M,called conformal infinity,is uniquely determined by(M,g).Clearly any manifold M carries many conformally compact metrics but we are mainly concerned here with Einstein metrics g,normalized so that(2.2)Ric g=−ng.A simple computation for conformal changes of metric shows that if g is at least C2confor-mally compact,then the sectional curvature K g of g satisfies(2.3)|K g+1|=O(ρ2).Thus,the local geometry of(M,g)approaches that of hyperbolic space,and conformally compact Einstein metrics are frequently called asymptotically hyperbolic(AH),or also Poincar´e-Einstein.All these notions will be used here interchangeably.The natural“thresh-old level”for smoothness is C2,since even if g is C m,αconformally compact,m>2,(2.3) cannot be improved to|K g+1|=o(ρ2)in general.Mathematically,an obviously basic issue in this area is the Dirichlet problem for confor-mally compact Einstein metrics:given the topological data(M,∂M),and a conformal class [γ]on∂M,does there exist a conformally compact Einstein metric g on M,with conformal infinity[γ]?In one form or another,this question is the basic leitmotiv throughout this paper.As will be seen later,uniqueness of solutions with a given conformal infinity fails in general.To set the stage,wefirst examine the structure of the moduli space of Poincar´e-Einstein metrics on a given(n+1)-manifold M.Let E m,αbe the space of Poincar´e-Einstein metrics on M which admit a C2conformal compactification¯g as in(2.1),with C m,αboundary metricγon∂M.Here0<α<1,m≥2,and we allow m=∞or m=ω,the latter corresponding to real-analytic.The topology on E m,αis given by a weighted H¨o lder norm, cf.(2.7)below;briefly,the topology is somewhat stronger than the C2topology on metricson¯M under a conformal compactification g as in(2.1).Let E m,α=E m,α/Diffm+1,α1(¯M),where Diffm+1,α1(¯M)is the group of C m+1,αdiffeomorphisms of¯M inducing the identityon∂M,acting on E in the usual way by pullback.Next,let Met m,α(∂M)be the space of C m,αmetrics on∂M and C m,α=C m,α(∂M)the corresponding space of pointwise conformal classes.The natural boundary map,(2.4)Π:E m,α→C m,α,Π[g]=[γ],takes a conformally compact Einstein metric g on M to its conformal infinity on∂M. Thus,global existence for the Dirichlet problem is equivalent to the surjectivity ofΠ,while uniqueness is equivalent to the injectivity ofΠ.The following result describes the general structure of E and the mapΠ,building on previous work of Graham-Lee[29]and Biquard[15].3 Theorem2.1.(Manifold structure[5],[6])Let M be a compact,oriented(n+1)-manifold with boundary∂M with n≥3.If E m,αis non-empty,then E m,αis a smooth infinite dimensional manifold.Further,the boundary mapΠ:E m,α→C m,αis a C∞smooth Fredholm map of index0.When m<∞,E m,αhas the structure of a Banach manifold,while E∞has the structure of a Fr´e chet manifold.For n=3,one expects that Theorem2.1also holds for m≥2,but this is an open problem.Theorem2.1shows that the moduli space E has a very satisfactory global structure. In particular if M carries some Poincar´e-Einstein metric,then it also carries a large set of them,mapping underΠto at least a variety offinite codimension in C.Recall that a metric g∈E is a regular point ofΠif D gΠis surjective.SinceΠis Fredholm of index 0,D gΠis injective at regular points;hence,by the inverse function theorem,Πis a local diffeomorphism in a neighborhood of each regular point.Remark2.2.Note that Theorem2.1does not hold,as stated,when n=1,i.e.in dimension 2.In this case,the space E as defined above is infinite dimensional,but it becomesfinite dimensional when one divides out by the larger group of diffeomorphisms isotopic to the identity on¯M.This space of conformally compact(geometricallyfinite)hyperbolic metrics on a surfaceΣis a smooth,finite dimensional manifold,but the conformal infinity is unique. The boundary∂Σis a collection of circles and there is only one conformal structure on S1 up to diffeomorphism.In particular,Πis not of index0.When n=2,Einstein metrics are again hyperbolic,and the space of such metrics, modulo diffeomorphisms isotopic to the identity,is parametrized by the Teichm¨u ller space of conformal classes on Riemann surfaces forming∂M.Thus,Theorem2.1does hold for n=2.However,we point out that the mapΦin(2.7)below used in constructing E is not Fredholm when n=1,2.Thus,the proof of Theorem2.1does not extend to the case n=2.It is worthwhile to examine the local structure of the boundary mapΠnear singular points in more detail.To do this,we need to discuss some background material,related to the proof of Theorem2.1.Given a boundary metricγ,one may form the standard “hyperbolic cone”metric onγby setting,in a neighborhood of∂M,gγ=ρ−2(dρ2+γ),and extending gγto M in afixed but arbitrary way.Given afixed background metric g0∈E m,αwith boundary metricγ0,forγnearγ0,let g(γ)=g0+η(gγ-gγ0),whereηis a cutofffunction supported near∂M.Thus g(γ)is close to g0and consider metrics g near g0 of the form(2.5)g=g(γ)+h,where h is a symmetric bilinear form on M which decays as O(ρ2).Essentially following[15],the Bianchi-gauged Einstein operator at g0is defined by (2.6)Φ(g)=Ric g+ng+δ∗gβg(γ)(g).We viewΦas a map(2.7)Φ:Met m,α(∂M)×S m,α2(M)→S m−2,α2(M),Φ(γ,h)=Φ(g(γ)+h),4where S m,α2(M )is the space of symmetric bilinear forms h on M ,of the form h =ρ2¯h ,with ¯hbounded in C m,α(M ).It turns out that if g 0∈E m,αthen the variety Φ−1(0)forms a local slice for the action of diffeomorphisms on E m,αnear g 0.The derivative of Φat g 0with respect to the second factor is the linearized Einstein operator(2.8)L (h )=D ∗Dh −2R (h ),h ∈S m,α2(M ).By [29],this map is Fredholm,and so has finite dimensional kernel and cokernel.Let K be the kernel of L on S m,α2(M );K is also the kernel of L on L 2(M,g ).To prove Theorem 2.1,it suffices to show that Φis a submersion at any g 0∈E m,α,and for this one needs to show that the pairing (2.9) MD Φ g (˙γ0),κ dV g 0is non-degenerate,in the sense that for any κ∈K g 0,there exists a variation ˙γ0of γ0=Π(g 0)such that (2.9)is non-zero.This is actually not so easy in general,and we refer to [6]for details.The boundary map Πis locally,near g 0,just the projection map on the first factor ofΦ−1(0)in (2.7).Thus,locally,a slice for E m,αthrough g 0is written as a (possibly multi-valued and singular)graph over Met m,α(∂M ).The kernel K of D Πat g is the subspace at which the graph is vertical,and corresponds to the kernel K of the operator L in (2.8).To understand the singularities of Πin more detail,note that since Πis Fredholm,it is locally proper,i.e.for any g ∈E m,α,there exists an open set U with g ∈U such that Π|U is a proper map onto its image V ⊂C .This means that Πhas a local degree,deg g Π∈Z ,cf.[41],[13];in fact if U is chosen sufficiently small,then deg g Π=−1,0or +1.If deg g Π=0,then Πis locally surjective onto a neighborhood of γ=Π(g );this may or not be the case if deg g Π=0.Observe however that (of course)deg g Πis not continuous in g .The local degree can be calculated by examining the behavior of Πon generic,finite dimensional slices.Thus,let B be any p -dimensional local affine subspace (or submanifold)of Met m,α(∂M )with γ=Π(g )∈B and consider the restriction of Φto B ×S m,α2(M ),and correspondingly,the graph E m,αB =Φ−1(0)∩Π−1(B )of E m,αover B .For a generic choice of B ,E m,αB is a p -dimensional manifold,and thus one can examine the behavior of Π|E m,αB in the context of the study of singularities of smooth mappings between equidimensional manifolds.By construction,cf.[13]for instance,one has for generic B ,deg g Π=deg g Π|E m,αB.Consider for example the situation where B is 1-dimensional.Then E m,αB is a local curve in Met m,α(∂M )×S m,α2(M )graphed over the interval B =(−ε,ε),with 0correspondingto γ.One sees that if deg g Π|E m,αB=±1,then Πis locally surjective near γ,while if deg g Π|E m,αB =0,then locally Π|E m,αB is a fold map,equivalent to x →x 2on (−ε,ε).In this case,at least in a small neighborhood U of g ,Πis not surjective onto a neighborhood of γ;there is a local “wall”in C ,(the image of the fold locus),which Π(U )does not cross.Some natural questions related to this discussion are the following:is the set of critical points of Πa non-degenerate critical submanifold (in sense of Bott)?Is it possible that Πmaps a connected manifold or variety of dimension ≥1onto a point γ∈C ?At this point,it is useful to illustrate the discussion on the basis of some concrete exam-ples.5 Example2.3.(Static AdS black hole metrics).Let N n−1be any closed(n−1)-dimensional manifold,which carries an Einstein metric g N satisfying(2.10)Ric gN=k(n−2)g N,where k=+1,0or−1.We assume n≥3.Consider the metric g m on R2×N defined by (2.11)g m=V−1dr2+V dθ2+r2g N,where(2.12)V(r)=k+r2−2mn )1/2,and for every m=m0,there are two values m±of m giving the same value ofβ.Thus two metrics have the same conformal infinity;in particular,the boundary mapΠin(2.4)is not 1-1along this curve.This behavior is thefirst example of non-uniqueness for the Dirichlet problem,and was discovered in[32]in the context of the AdS Schwarzschild metrics,where N=S2(1).The mapΠis a fold map,(of the form x→x2),in a neighborhood of the curve g m nearm=m0.The local degree at g m0is0andΠis not locally surjective.In fact,Theorem2.4below implies thatΠis globally not surjective,in that the conformal class of S1(L)×(N,g N),for L>β0,is not in ImΠ,cf.[5].Observe that this result requires global smoothness of the Einstein metrics;if one allows cone singularities along the horizon N={r=r+},i.e. ifβis allowed to be arbitrary,then one can go past the“wall”through S1(β0)×(N,g N).This clearly illustrates the global nature of the global existence or surjectivity problem.II.Suppose k=0.In this caseβ=4πr+/(nr2+)is a monotone function of r+or m,so that it assumes all values in R+as m∈(0,∞).On the curve g m,Πis1-1.However,the actual situation is somewhat more subtle than this.Suppose for instancethat N=T n−1,so that M=R2×T n−1is a solid torus.Topologically,the disc D2=R2 can be attached onto any simple closed curve in the boundary∂M=T n instead of just the“trivial”S1factor in the product T n=S1×T n−1.The resulting manifolds are all diffeomorphic.This can also be done metrically,preserving the Einstein condition,cf.[4], and leads to the existence of infinitely many distinct Einstein metrics on R2×T n−1with the same conformal infinity(T n,[g0]),where g0is anyflat metric.Each of these metrics lies in a distinct component of the moduli space E,so that E hasinfinitely many components.This situation is closely related to the mapping class groupSL(n,Z)of T n,i.e.the group of diffeomorphisms of T n modulo those homotopic to the6identity map,(so called “large diffeomorphisms”).Any element of SL (n,Z )extends to a diffeomorphism of the solid torus R 2×T n −1,and while SL (n,Z )acts trivially on the moduli space of flat metrics on T n ,the action on E is highly non-trivial,giving rise to the distinct components of E .Similar constructions can obviously be carried out for manifolds N of the form N =T k ×N ′,k ≥1,but it would be interesting to investigate the most general version of this phenomenon.III.Suppose k =−1.Again βis a monotone function of m ,and so takes on all values in R +;the boundary map Πis 1-1on the curve g m .Further aspects of this case are discussed later in §5.These simple examples already show a number of subtle features of the global behavior of the boundary map Π.With regard to the global surjectivity question,the basic property that one needs to make progress is to understand whether Πis a proper map;if Πis not proper,it is important to understand exactly what possible degenerations of Poincar´e -Einstein metrics can or do occur with controlled conformal infinity.Recall that Πis proper if and only if Π−1(K )is compact in E ,whenever K is compact in C .If Πis proper,then one has a well-defined Z 2-valued degree,cf.[41].In fact,since the spaces E and C can be given a well-defined orientation,one has a Z -valued degree,given by (2.14)deg Π= g i ∈Π−1[γ](−1)ind g i ,where [γ]is a regular value of Πand ind g i is the L 2index of D g i Π,i.e.the number of negative eigenvalues of the operator L in (2.8)at g i acting on L 2,cf.[5].Of course if deg Π=0,then Πis surjective;(if deg Π=0,then Πmay or may not be surjective).Note that deg Πis defined on each component E 0of E and may differ on different components.Let M =M 4be a 4-manifold,satisfying(2.15)H 2(∂M,R )→H 2(M,R )→0.It is proved in [5]that Πis then proper,when restricted to the space E 0of Einstein metrics whose conformal infinity is of non-negative scalar curvature.More precisely,(2.16)Π0:E 0→C 0is proper,where C 0is the space of conformal classes having a non-flat representative of non-negative scalar curvature and E 0=Π−1(C 0);in particular there are only finitely many components to E 0,compare with Example 2.3,Case II above.In situations where Πis proper,the degree can be calculated in a number of concrete situations by the following:Theorem 2.4.(Isometry Extension,[5])Let (M n +1,g)be a C 2conformally compact Ein-stein metric with C ∞boundary metric γ,n ≥3.Then any connected group G of conformal isometries of (∂M,γ)extends to a group G of isometries of (M,g ).This result has a number of immediate consequences.For instance,it implies that the Poincar´e (or hyperbolic)metric is the unique C 2conformally compact Einstein metric on an (n +1)-manifold with conformal infinity given by the round metric on S n ;see also [12],[39]for previous special cases of this result.In particular,one has on (B 4,S 3),deg Π0=1,so that Πis surjective onto C 0.On the other hand,on (M 4,S 3),M 4=B 4,deg Π0=0,7 sinceΠcannot be surjective in this case.Another application of Theorem2.4is the follow-ing:Corollary2.5.Let M be any compact(n+1)-manifold with boundary∂M,n≥3,and let ˆM=M∪∂M M be the closed manifold obtained by doubling M across its boundary.Suppose ∂M admits an effective S1action,butˆM admits no effective S1action.ThenΠ=Π(M) is not surjective;in factImΠ∩Met S1(∂M)=∅,where Met S1(∂M)is the space of S1invariant metrics on∂M.The space Met S1(∂M)is of infinite dimension and codimension in Met(∂M).8but is otherwise undetermined byγand the Einstein equations;it depends on the particular structure of the AH Einstein metric(M,g)near infinity.If n is even,one has(3.4)gρ∼g(0)+ρ2g(2)+....+ρn−2g(n−2)+ρn g(n)+ρn logρH+ρn+1g(n+1)+... Again(via the Einstein equations)the terms g(2k)up to order n−2are explicitly computable from the boundary metricγ,as is the coefficient H of thefirst logρterm.The term H is transverse-traceless.The term g(n)satisfies(3.5)trγg(n)=τ,δγg(n)=δ,where againτandδare explicitly determined by the boundary metricγand its derivatives; however,as before g(n)is otherwise undetermined byγ.There are(logρ)k terms that appear in the expansion at order>n.Note also that these expansions(3.2)and(3.4)depend on the choice of boundary metric. Transformation properties of the coefficients g(i),i≤n,under conformal changes have been explicitly studied in the physics literature,cf.[24].As discovered by Fefferman-Graham [26],the term H is conformally invariant,or more precisely covariant:if γ=φ2γ,then H=φ2−n H.Remark 3.1.Analogous to the Fefferman-Graham expansion above,there is a formal expansion of a vacuum solution to the Einstein equations near null infinity,although this has been carried out in detail only in dimension3+1,cf.[16].This expansion is closely related to the properties of the Penrose conformal compactification.More recently,as discussed in[20],logarithmic terms appear in the expansion in general,and these play an important role in understanding the global structure of the space-time. Mathematically,it is of some importance to keep in mind that the expansions(3.2),(3.4) are only formal,obtained by conformally compactifiying the Einstein equations and taking iterated Lie derivatives of¯g atρ=0;1(3.6)g(k)=9 To begin to make some of the discussion above more rigorous,we next discuss the bound-ary regularity issue;many aspects of this have been resolved over the past few years.Sup-posefirst n=3,so dim M=4.If g∈E m,α,m≥2,then by definition g has a C2conformal compactification to a C m,αboundary metricγ.In[4],it is proved that there isa C m,αconformal compactification g∈C m,α(¯M)of g,cf.also[6].This result also holds ifm=∞or m=ω.It is proved using the fact that4-dimensional Einstein metrics satisfythe Bach equations,cf.[14],which are conformally invariant.In suitable gauges,the Bachequation can be recast as a non-degenerate elliptic system of equations for a conformalcompactification g,and the result follows from elliptic boundary regularity.In dimension4,the Bach tensor is the Fefferman-Graham obstruction tensor H above.In any even dimension,the system of equations(3.8)H=0is conformally invariant,and is satisfied by metrics conformal to Einstein metrics.Thus,one might expect that the method using the Bach equation in[4],[6]when n=3canbe extended to all n odd.This is in fact the case,and has been worked out in detail byHelliwell[31].Thus,essentially the same regularity results hold for n odd.When n is even,so that dim M is odd,this type of boundary regularity cannot hold ofcourse,due to the presence of the logarithmic terms in the FG expansion.A result of Lee[35]shows that if g∈E m,αand m<n,then g is C m,αconformally compact.This is optimal, but does not reach the important threshold level m=n,where logarithmic terms and theimportant g(n)termfirst appear.Recently,Chru´s ciel et al.[21]have proved that wheng∈E∞,i.e.g has a C∞boundary metricγ,then g has a C∞polyhomogeneous conformal compactification,so that the expansion(3.4)exists as an asymptotic series.Moreover,ifγ∈C m,α(∂M),then the expansion exists up to order k,where k can be made large bychoosing m sufficiently large;(in general m must be much larger than k).Finally,it hasrecently been proved by Kichenassamy[34]that when g∈Eωand g(n)is real-analytic,the formal series(3.4)exists,i.e.it is summable,and it converges to gρ.These results have the following immediate consequence.Suppose n is odd.Given anyreal-analytic symmetric bilinear forms g(0)and g(n)on∂M,satisfying(3.3),there exists a unique Cωconformally compact Einstein metric g defined in a thickening∂M×[0,ε)of ∂M.If instead n is even,given any analytic symmetric bilinear forms g(0)and g(n)on∂M, satisfying(3.5),there exists a unique C∞polyhomogeneous conformally compact Einstein metric g defined in a thickening∂M×[0,ε)of∂M.In both cases,the expansions(3.2)or (3.4)converge to the metric gρ.These results follow from the work in[4],[6],[31]when n is odd,and[34]when n is even.Since analytic data g(0)and g(n)may be specified arbitrarily and independently of each other,subject only to the constraint(3.3)or(3.5),to give“local”AH Einstein metrics,defined in a neighborhood of∂M,this shows that the correspondence (3.7)must depend highly on global properties of Poincar´e-Einstein metrics.On the other hand,it is well-known that the use of analytic data to solve elliptic-typeproblems is misleading.While the Dirichlet or Neumann problem is formally well-posed,the Cauchy problem is not.Standard examples involving Laplace operator and harmonicfunctions show that even if Cauchy data on a boundary converge smoothly to limit Cauchydata on the boundary,the corresponding solutions do not converge to a limit in any neigh-borhood of the boundary.To pass from analytic to smooth boundary data,one needs apriori estimates or equiva-lently a stability result.In this respect,one has the following:Theorem3.2.(Local Stability,[4],[6])Let g be a C2conformally compact Einstein metric,defined in a regionΩ=[0,ρ0]×∂M containing∂M,whereρis a geodesic compactification.10Suppose there exists a compactification g with C m,αboundary metricγ,such that (3.9)|| g||C1,α(Ω)≤K.If n=3,then there is a(possibly different)compactification,also called g,such that,in Ω′=[0,ρ0compact,i.e.M is the interior of a compact manifold with boundary,and g is complete and globally defined on M.Theorem3.4.(Control near Boundary)Let(M n+1,g)be a globally conformally compact Poincar´e-Einstein metric,with n odd,so that dim M is even.Suppose that g is C2con-formally compact,with C m,αboundary metricγ,with m>n and m≥6if n=3.Then there exists a neighborhoodΩ=[0,ρ0]×∂M of∂M,depending only on the boundary data (∂M,γ)such that(3.11)|| g||C m,α(Ω)≤C,in some compactification g.The bound(3.11)implies that the boundary mapΠis proper near conformal infinity,in the sense that if one has afixed boundary metricγ,or compact set of boundary metrics γ∈Γ,then the set of Poincar´e-Einstein metrics with boundary metricγ,(orγ∈Γ), is compact,as far as their behavior inΩis concerned;any sequence has a convergent subsequence on afixed domainΩ,whereΩonly depends on the boundary data.Proof:This result is proved for n=3in[5],and the proof for arbitrary n odd is very similar.Thus,we refer to[5]for much of the proof,and only discuss those situations where the proof needs to be modified in higher dimensions.There are several steps in the proof.First,let¯g be the geodesic compactification of g determined byγ,and letτbe the distance to the cutlocus of the normal exponential map from(∂M,γ)into(M,¯g).Here of course g is any Poincar´e-Einstein metric on M with boundary metricγ,(orγ∈Γ).Thefirst(and most important)step is to prove that there is a constantτ0>0,depending only on n andγ(orΓ)such that(3.12)τ(x)≥τ0.The estimate(3.12)already implies for instance that the topology of M cannot become non-trivial too close to the boundary∂M.The proof of(3.12)in[5,Prop.4.5]holds with only minor and essentially obvious changes in all even dimensions,given the local stability result,Theorem3.2.As noted in[5,Remark2.4],one should use the renormalized action in place of the renormalized volume or L2norm of the Weyl curvature.Also,the classification of R n-invariant solutions as AdS toral black holes is given in[9],(again the proof of this holds in all dimensions).Next,letζ(x)=ζn,α(x)be the C n,αharmonic radius of(M,¯g)at x,for afixedα<1. The next claim,(cf.[5,Prop.4.4])is that there is a constantζ0,depending only on n and γ,such that(3.13)ζ(x)≥ζ0τ(x).(The proof in[5,Prop.4.4]uses the L p curvature radius,but the proof works equally well for the much stronger C n,αharmonic radius).The proof of(3.13)is by contradiction.If(3.13)does not hold,then there exist x i∈(M i,g i)such thatξ(x i)<<τ(x i).Choose x i to realize the minimum of the ratioξ/τ.One then takes a blow-up limit of the rescalings g′i=ζ(x i)−2¯g i based at x i.Sinceζ′(x i)=1,ζ′(y i)≥1I.dist g′i (x i,∂M i)≤D,for some D<∞.In this case,the limit N has a boundary(∂N,γ′).Since this limit is the blow-up of(∂M i,γi),it is clear that(∂N,γ′)isflat(R n,δ), whereδis theflat metric.(Here we use of course the fact thatΓis compact).Moreover,∂N is totally geodesic in N.As in[5],(N,g′)is Ricci-flat,Ric g′=0.The proof that N is actuallyflat in[5]used the fact that N contains a line;when n=3,i.e.in dimension4, this implies N isflat.This of course does not hold in higher dimensions.Instead,one can argue as follows.Since N is Ricci-flat and hasflat and totally geodesic boundary R n,the reflection double of N across R n is a weak C1solution of the Einstein equations Ric g′=0 on R n+1.Elliptic regularity implies that g′is then real-analytic across R n.It then follows easily from the Cauchy-Kovalevsky theorem that(N,g′)isflat.This gives the required contradiction in this case.II.dist g′i (x i,∂M i)→∞as i→∞.For this case,we give a different and simpler proofthan that in[5,Prop.4.4,Case II].Let d i(x)=dist g′i(x i,∂M i).It follows easily from Case I above thatζi(y i)≥(1−δ)d i(y i),for y i within bounded distance to(∂M,g′i),withδ→1as i→∞.This just corresponds to the statement that the geometry becomesflat near∂M i with respect to g′i,which has been proved in Case I.Now by hypothesis,at x i,ζi(x i)/d i(x i)→0,(since the ratio is scale-invariant andζi(x i)=1in the scale g′i).Therefore,by continuity,there are points y isuch thatζ(y i)=12d(z i),for all z i such that d(z i)≤d(y i).One now works in the scaleˆg i=ζi(y i)−2g i whereˆζi(y i)=1and hence distˆg i(y i,∂M)=2.The proof is now completed just as in Case I.Thus,one may pass to a limit(N,ˆg,y).On the one hand,the limit(N,ˆg)is notflat,since,by Theorem3.2the convergence to the limit is in C n,αandζis continuous in this topology,so thatˆζ(y)=1.As before,(N,ˆg)hasflat and totally geodesic boundary,and the same proof as in Case I implies that(N,ˆg)isflat,giving again a contradiction.Taken together,(3.12)and(3.13)imply thatζ(x)≥τ1>0,for all x in a neighborhood of∂M offixed size in(M,¯g).The bound(3.11)is then a consequence of the local stability result,Theorem3.2.。
中国工业与应用数学学会第十一届年会
13:30—15:20
材料科学
海韵宾馆六会议室
15:30—17:20 17:20—17:30 17:30 18:30
大会报告 闭幕式 候车（重庆夜景旅游） 晚饭
海韵宾馆六会议室 海韵宾馆六会议室 海韵宾馆六会议室外 两江游游船 请凭代表证上车
7 月 11 日 会议参观——赤水
时间 6:45 7:00 7:30 事项 叫醒 早饭 候车 海琴酒店中餐厅(酒店负一楼) 海琴酒店一楼大厅 请凭旅游牌上车 地点 备注
海琴二会议室
13:30—15:30
海韵宾馆多功能厅
小 讨 论 会 15:30—15:45
海琴三会议室 海琴五会议室
海琴二会议室
15:45—18:30
海韵宾馆多功能厅
小 讨 论 会 18:30—17:30
海琴三会议室 海琴五会议室 海琴酒店中餐厅(酒店负一楼) 请凭餐劵就餐
20:00—22:00
海琴三会议室
新加坡国立大学
—15:20 为小讨论会（Minisymposium） 注：13:30 13:30—
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中国工业与应用数学学会第十一届年会暨中国工业与应用数学学会成立 20 周年庆典
圆桌会议
Round Tables
—22:00 July 8, 2010 Evening 20:00 20:00—
主题一：应用数学教育和数模竞赛 主持人:谢金星 清华大学 地点：海琴三会议室 地点：海琴二会议室
会议参观——黑山谷
时间 7:00 7:30 8:00 事项 叫醒 早饭 候车 海琴酒店中餐厅(酒店负一楼) 海琴酒店一楼大厅 请凭旅游牌上车 地点 备注
3
中国工业与应用数学学会第十一届年会暨中国工业与应用数学学会成立 20 周年庆典
创新思维重要性和提升方法英语作文
创新思维重要性和提升方法英语作文The Importance of Innovative Thinking and Ways to Enhance ItIn today's rapidly changing world, the ability to think innovatively has become more important than ever. With technology advancing at an unprecedented rate and competition growing fiercer in the global marketplace, individuals and organizations need to constantly adapt and innovate in order to stay ahead. In this essay, I will discuss the importance of innovative thinking and provide some tips on how to enhance this crucial skill.First and foremost, innovative thinking is crucial for problem-solving. In a world where new challenges and complexities arise every day, the ability to think outside the box and come up with creative solutions is invaluable. Whether it's developing a new product, improving an existing process, or finding a way to stay competitive in a crowded market, innovative thinking is what sets successful individuals and organizations apart from the rest.Moreover, innovative thinking is essential for driving progress and fostering growth. By challenging the status quoand pushing boundaries, innovators have the power to change the world for the better. From groundbreaking scientific discoveries to revolutionary technological advancements, innovation has been the driving force behind some of the greatest achievements in human history.So, how can one enhance their innovative thinking? Here are some tips:1. Embrace curiosity: Curiosity is the driving force behind innovative thinking. By staying curious and asking questions, you can uncover new ideas and opportunities that others may have overlooked.2. Think divergently: Instead of following conventional thinking patterns, try to think in new and different ways. Consider multiple perspectives and explore unconventional solutions to problems.3. Cultivate a growth mindset: A growth mindset is essential for fostering innovation. By believing in your ability to learn and grow, you can overcome challenges and embrace new opportunities for creativity.4. Collaborate with others: Innovation thrives in a collaborative environment. By working with others who bringdiverse perspectives and skills to the table, you can spark new ideas and push the boundaries of what is possible.In conclusion, innovative thinking is a crucial skill that is essential for success in today's fast-paced world. By embracing curiosity, thinking divergently, cultivating a growth mindset, and collaborating with others, you can enhance your ability to innovate and drive positive change in the world. So, let's all strive to think innovatively and push the boundaries of what is possible.。
asymptotic analysis缩写
asymptotic analysis缩写Asymptotic analysis is a mathematical method used to analyze the behavior of an algorithm as the input size approaches infinity. It is widely used in computer science and engineering to compare different algorithms and design efficient algorithms. In this article, we will discuss the basics of asymptotic analysis and its important concepts.Firstly, we need to understand the importance of asymptotic analysis. In computer science and engineering, we often deal with huge datasets and complex algorithms. Therefore, it is important to know how the algorithm will behave as the input size becomes large. Asymptotic analysis helps us to estimate the computational time and space complexity of an algorithm for large inputs. This estimation can help us to choose the best algorithm for a given problem.Asymptotic analysis is based on the concept of limits. A limit is the value a function approaches as the input value approaches a certain point. We use big O, big Omega, and big Theta notations to express the growth rate of a function. These notations give us a rough idea about the behavior of the function.Big O notation: The big O notation gives the upper bound of the running time of an algorithm. We say that algorithm A has a time complexity of O(f(n)) if the running time of the algorithm does not exceed a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is less than or equal to 2n^2+3n+4, we can say that the time complexity of the algorithm A is O(n^2).Big Omega notation: The big Omega notation gives the lower bound of the running time of an algorithm. We say that algorithm A has a time complexity of Omega(f(n)) if the running time of the algorithm is not less than a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is greater than or equal to n^2/2, we can say that the time complexity of the algorithm A is Omega(n^2).Big Theta notation: The big Theta notation gives the tight bounds of the running time of an algorithm. We say that algorithm A has a time complexity of Theta(f(n)) if the running time of the algorithm is between a constant multiple of f(n) and another constant multiple of f(n) for large n. For example, if the running time of the algorithm A is between 5n^2+3n+4 and 7n^2+5n+6, we can say that the time complexity of the algorithm A is Theta(n^2).Asymptotic analysis also covers the space complexity of an algorithm. We use the same notations to express the growth rate of the space usage of an algorithm. For example, if the space used by the algorithm A is less than or equal to 3n+4, we can say that the space complexity of the algorithm A is O(n).In conclusion, asymptotic analysis is an important concept in computer science and engineering. It helps us to estimate the computational time and space complexity of an algorithm for large inputs. By using the big O, big Omega, and big Theta notations, we can compare different algorithms and choose the best algorithm for a given problem.。
Chapter 1 Derivation of Reaction-Diffusion Equations
where J is the flux of P , d(x) is called diffusion coefficient at x, and ∇x is the gradient operator ∇x f (x) = (∂f /∂x1 , ∂f /∂x2 , · · · , ∂f /∂xn ). On the other hand, the number of particles at any point may change because of other reasons like birth, death, hunting, or chemical reactions. We assume that the rate of change of the density function due to these reasons is f (t, x, P ), which we usually call the reaction rate. Now we derive a differential equation using the balanced law. We choose any region O, then the total population in O is O P (t, x)dx, and the rate of change of the total population is d dt P (t, x)dx.
Chapter 1
Derivation of Reaction-Diffusion Equations
1.1 Fick’s Law
Diffusion mechanism models the movement of many individuals in an environment or media. The individuals can be very small such as basic particles in physics, bacteria, molecules, or cells, or very large objects such as animals, plants, or certain kind of events like epidemics, or rumors. The particles reside in a region, which we call Ω, and we assume that Ω is open subset of Rn (the n-th dimensional space with Cartesian coordinate system) with n ≥ 1. In particular, we are interested in the cases of n = 1, 2 and 3, but most material here are true regardless of the dimensions of the space (sometimes n = 1 and n ≥ 2 may be different as we will see.) The main mathematical variable we consider here is the density function of the particles: P (t, x), where t is the time, and x ∈ Ω is the location. The dimension of the population density usually is number of particles or organisms per unit area (if n = 2) or unit volume (if n = 3). For example, the human population density is often expressed in number of people per square kilometer. A list of world population and population density can be found at /wiki/List_of_countries_by_population_density. However in such data table, we can only find the population density for countries, and a country is not a “point” on our map. In reality, population density is always associated with a scale, like country, city, county, town, and street. But as in many other mathematical models, we will assume that the function P (t, x) has nicer properties, like continuity and differentiability, which is in fact reasonable, when a population with a large number of organisms is considered. Technically, we define the population density function P (t, x) as follows: let x be a point in the habitat Ω, and let {On }∞ n=1 be a sequence of spatial regions (which have the same dimension as Ω) surrounding x; here On is chosen in a way that the spatial measurement |On | of On (length, area, volume, or 1
科学精神有助于有效解决问题初中英语作文
科学精神有助于有效解决问题初中英语作文The scientific spirit is a fundamental approach to understanding the world around us. It is a way of thinking that emphasizes the importance of empirical evidence, critical analysis, and the continuous pursuit of knowledge. This spirit has been instrumental in driving human progress, enabling us to tackle complex challenges and find innovative solutions to pressing problems. In this essay, we will explore how the scientific spirit can help us effectively solve problems.Firstly, the scientific spirit encourages a systematic and methodical approach to problem-solving. When faced with a challenge, the scientific mindset prompts us to gather relevant information, formulate hypotheses, and design experiments or observations to test these hypotheses. This process allows us to identify the root causes of a problem and develop targeted solutions, rather than relying on intuition or guesswork. By following a structured, evidence-based approach, we can ensure that our solutions are grounded in reality and have a higher likelihood of success.Moreover, the scientific spirit fosters a culture of critical thinking and skepticism. Scientists are trained to question assumptions, scrutinize existing knowledge, and challenge established beliefs. This mindset is crucial in problem-solving, as it helps us avoid biases, think outside the box, and consider alternative perspectives. By embracing a critical approach, we can uncover hidden assumptions, identify flaws in our reasoning, and explore new avenues for solving problems.Another key aspect of the scientific spirit is its emphasis on objectivity and impartiality. Scientists strive to set aside personal biases and preconceptions, focusing instead on the objective analysis of data and evidence. This objectivity is essential in problem-solving, as it allows us to make decisions based on facts rather than emotions or personal agendas. By approaching problems with an unbiased perspective, we can make more informed and rational choices, leading to more effective solutions.Furthermore, the scientific spirit encourages collaboration and the sharing of knowledge. Scientists often work in teams, sharing their findings, exchanging ideas, and building upon each other's work. This collaborative approach can be highly beneficial in solving complex problems, as it allows us to leverage the diverse expertise and perspectives of multiple individuals. By fostering a culture of knowledge-sharing, we can tap into a broader pool of resources and develop more comprehensive solutions.Finally, the scientific spirit is characterized by a willingness to learn from mistakes and a commitment to continuous improvement. Scientists understand that progress is often achieved through trial and error, and they are not afraid to acknowledge and learn from their failures. This mindset is crucial in problem-solving, as it allows us to refine our approaches, adapt to changing circumstances, and continuously improve our solutions over time. By embracing a growth mindset, we can turn setbacks into opportunities for learning and innovation.In conclusion, the scientific spirit is a powerful tool for effectively solving problems. Its emphasis on empirical evidence, critical thinking, objectivity, collaboration, and continuous learning provides a robust framework for tackling complex challenges. By adopting the scientific spirit, we can approach problems with rigor, creativity, and a commitment to finding the most effective solutions. As we continue to face global issues and emerging challenges, the scientific spirit will remain a vital asset in our quest to create a better world.。
脉冲微分方程解的存在性和稳定性
中国地质大学（武汉）硕士学位论文脉冲微分方程解的存在性和稳定性姓名：***申请学位级别：硕士专业：应用数学指导教师：***20080501“’Ｏ）＋兄甜（ｆ）＝／（ｆ，“（，一ｒ）），ｆ≠０，ｆ∈，＝【ｏ，丁】，Ａｕ（ｔ』）＝Ｉｊ（“（ｆ伪，／＝１，…Ｐ，甜（ｏ）＝“（丁）＝Ｕｏ，“（，）＝ｏ，ｆ∈卜ｆ，ｏ）．甜７（，）＋名“（，）＝厂（，，甜（ｒ））＋Ｐ（，），ｔｅＪ＝［Ｏ，丁】，ｕ（ｏ）＝甜（丁）＝‰．（７）甜’（，）＋力甜（ｆ）＝厂（ｒ，“（，））＋Ｐ（，），，∈／＝【ｏ，ｒ】，Ａｕ（ｔｊ）＝‘＠ｑ）），／＝１，２，…Ｐ，ｕ（ｏ）＝ｕ（ｒ）－Ｕｏ．对于后面的五类问题，我们均可以得到其相应的反周期边值问题解的性质。
通过讨论，我们可以清晰地看到：所研究的脉冲周期边值问题解的存在性及稳定性的结论与脉冲条件密不可分。
关键词：周期边值问题，脉冲，时滞，存在性，稳定性ＴｈｅＥｘｉｓｔｅｎｃｅａｎｄＳｔａｂｉｌｉｔｙｏｆＩｍｐｕｌｓｉＶｅＤｉｆｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎｓＭａｓｔｅｒＣａｎｄｉｄａｔｅ：ＬｉＹ砒ｉｎｇＳｕｐｅｒｖｉｓｏｒ：ＬｉｕＡｍｐｉｎｇＩｍｐｕｌｓｉｖｅｄｉ舵ｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｃ锄ｂｅｓｕｃｃｅｓｓｆｕｌｌｙ吣ｅｄｆｏｒｍａｔｈｅｍａｔｉｃａｌｓｉｍｕｌａｔｉｏｎｉｎｔｌｌｅｏｒｅｔｉｃａｌｐｈｙｓｉｃｓ，ｃｈｅｍｉｓｔⅨｂｉｏｔｅｃｈｎｏｌｏ烈ｍｅｄｉｃｉｎｅ，ｐｏｐｕＩａｔｉｏｎｄｙｎ锄ｉｃｓ，ｏｐｔｉｍａｌｃｏｎｔｍｌ，锄ｄｉｎｏｔｈｅｒｐｎｏｃｅｓｓｅｓ鲫ｄｐｈｅｎｏｍｅｍｉｎｓｃｉｅｎｃｅ觚ｄｔｅｃｈｎｏｌｏｇｙ．ＴｈｅＳｔａｂｉｌｉｔｙｔｈｅｏ搿ｏｆｉｍｐｕｌｓｉＶｅｄｉｆｆ－ｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｈ嬲ｂｅｅｎｄｅｖｅｌｏｐｅｄｂｙａｌａ呼ｎ啪ｂｅｒｏｆｍａｔｌｌｅｍａｔｉｃｉａｎｓ，ａ１１ｄｔｈｅｉｒＳｔｕｄｉｅｓｈａｖｅａｔｔｒ暑Ｉｃｔｅｄｍｕｃｈａｔｔｅｎｔｉｏｎ．Ｔｈｅｖｈａｖｅｂｅｅｎｓｕｃｃｅｓｓｆｕｌｉｎｄｉ行ｅｒｅｎｔａｐｐｒｏａｃｈｅｓｂ舔ｅｄ０ｎＬｙ印ｕｎｏＶｄｉｒｅｃｔｍｅｔｈｏｄ锄ｄｃｏｍｐ耐ｓｏｎｔｅｃｌｌｌｌｉｑｕｅ．Ｔｈｉｓｍｅｔｈｏｄｈ∞ｇｅｎｅｒａｌｉｔ），ｆ．ｒｏｍｔｈｅｔ１１ｅｏｒｅｔｉｃａｌｓｔａｎｄｐｏｉｍ，ｂｕｔｉｔｉｓｎｏｔｃｏｎｖｅｎｉｅｎｔｆｏｒｐｒａｃｔｉｃａｌｍｅｓｏｍｅｔｉｍｅｓ．Ｔｈｅｍａｉｎｄｉ师ｃｕｌｔｙｌｉｅｓｉｎｃｏｎｓｔｍｃｔｉｎｇｔｈｅＬｙａｐｕｎｏＶｆｕｎｃｔｉｏｎａｌ．Ｉｎｔｈｉｓｐ印ｅｒ，ｗｅｅｍｐｌｏｙ廿ｌｅｉｔｅｒａｔｉＶｅｍｅｔｈｏｄｗｈｉｃｈｉｓＶｅｒｙｃｏｎｃｒｅｔｅｔ０ｏｂｔａｉｎｔｈｅｅ心ｓｔｅｎｃｅ锄ｄｓｔａｂｉｌｉ锣ｏｆｐｅｒｉｏｄｉｃｂｏｕｎｄａｒｙＶａｌｕｅｐｒｏｂｌｅｍｓｗ曲ｉｍｐｕＩｓｅｓ．Ｉｎｔｈｉｓ硎ｃｌｅ，ｗｅｍａｉｎｌｙｏｂｔａｉｌｌｔｈｅｅ虹ｓｔｅｎｃｅｓ锄ｄｓ劬ｉｌｉｔｉｅｓｏｆｓｏｌｕｔｉｏｎＳｏｆｐｅｒｉｏｄｉｃｂｏｕｎｄａ巧Ｖａｌｕｅｐｒｏｂｌｅｍｓ舔ｆｏｌｌｏｗｓ：（１）材’（，）＋兄“（，）＝厂（，，甜（，）），，∈，＝【ｏ，丁】，（２）（３）“（ｏ）＝“（丁）＝‰．甜’（，）＋五材（，）＝厂（，，“（，）），，∈，＝【ｏ，丁】，“（ｏ）＝一“（ｒ）＝‰．材’（ｆ）＋勉（ｆ）＝厂（ｆ，材（ｆ））＋Ｐ（ｆ），ｆ∈‘，＝【ｏ，丁】，甜（ｏ）＝“（ｒ）＝‰．研究生学位论文原创性声明我以诚信声明：本人呈交的硕士学位论文是在刘安平教授指导下开展研究工作所取得的研究成果。
存在填隙流体时颗粒间的相互作用及其在DEM中的应用
中国农业大学硕士学位论文存在填隙流体时颗粒间的相互作用及其在DEM中的应用姓名：***申请学位级别：硕士专业：固体力学指导教师：黄文彬;徐泳2001.3.1！堕奎些查堂堡主兰些笙苎———————————————————！！—墨摘要本文主要研究了存在填隙流体时颗粒间的相互作用。
颗粒在存在填隙流体时，主要有摆动状态和浸渍状态两种状态。
文中主要针对浸渍状态下颗粒间的切向作用进行了深入细致的研究。
填隙液体的存在使得颗粒容易结块，这广泛应用于食品、化工、医药卫生、农业等方面，越来越引起人们的重视。
结块有它有利的一面，如造粒，但是又有不利的一面，比如土壤结块等，因此很有必要研究这方面的发展。
ｆ文中首先根据Ｒｃｙｎ０１ｄｓ润滑理论，导出了存在填隙幂率流体时，两刚性圆球有相对切向这动时流体压力的近似方程。
求在解压力分布时，运用试算法成功地避免了在积分区域奇异的问题，求得了流体的压力分布的数值解，进而求得了圆球所受阻力与阻力矩的近似积分表达式，还与根据Ｇ０１ｄｍａｎ等人导出的牛顿流体下的渐近解做了比较，分析表明数值解优于渐近解。
为了计算方便，这里还拟合出了圆球所受阻力和阻力矩的近似公式，与数值解相比，由近似公式算出的值非常精确，两条曲线几乎完全重合，两者偏差能够保证在２％以内。
尽管本文为简化推导采用了一个并不合理的假定，但是，至少可以证明当幂率指数为ｌ情况下，其渐近解的结果可以退化为６ｂｌｄｒｍ等人在牛顿流体下的渐近解。
除此之外，文中还对刚性圆球在存在有填隙幂率流体时，分别水平与垂直移动时所受的法向力和切向力作了比较，发现它们二者之间存在着一定的关系，这里不但给出了法向力与切向力之比的积分表达式，同时也拟合出了近似公式，与数值解相比，偏差也能够保证在２％以内。
由于颗粒间法向作用时的结果比较精确，因此可以根据求得的法向力来求解切向力。
另外，当两球其中之一的半径趋于无穷大时，可以当作半无限体问题来处理，问题可以退化到球相对于平面壁运动的特殊情况。
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科学文献
II. Norms and the Linear Problem
The approach taken is to apply the renormalization group ideas of Bricmont et al. (1993) to the equation (1.1). It is convenient to start with the linear equation (1.3) since the theory for this simple situation already deviates from that of Bricmont et al. (1993), because the main ideas appear in bold relief, avoiding the technicalities associated with the nonlinear term, and because use will be made of the linear theory in studying the nonlinear initial-value problem.
with initial data
(x 2 R; t > 1); (x 2 R ):
(1.1) (1.2)
u( ; 1) = f0 ( );
Here M is a Fourier-multiplier operator, which in Fourier transformed variables has 1 d 1, and p 2. The initial-value problem the form Mu(k) = ?jkj2 u, where 2 < b (1.1){(1.2) is always locally well posed, but for larger values of p and large initial data, it appears that it may not be globally well posed (see Bona et al. 1993). If attention is restricted to small initial data, then (1.1){(1.2) is globally well posed and it is not di cult to see in this case that solutions decay to the zero function as t becomes unboundedly large. It is our purpose here to determine the detailed structure of this evanescence. This program has already been carried out in the elegant paper of Dix (1992). We will show how some of Dix's results follow readily from a suitable adaptation
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