Smooth dependence on parameters of solution of cohomology equations over Anosov systems and
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2020年7月英语六级真题及参考答案【完整版】ListeningCONEVERSATION 1-Tonight,we have a very special guest,Mrs.An a Sanchez is a three time Olympic champion and author of the new book To the Edge. Mrs.Sanchez,thankyou for joining us.-Thankyou for having me.-Let's start with your book. What does the title to the Edge mean? What are you referring to?-The book is about how science and technology has helped push humans to the edge of their physical abilities. l argue that in thepast 20 years we have had the best athletes the world has everseen. But is this a fair comparison? How do you know how,say,a football player from 50 years ago would compare to one today?-Well,you are right.-That comparison would be perhaps impossible to make. But thepoint is more about our knowledge today of human biochemistry,nutrition and mechanics. I believe that while our bodies have not changed in thousands of years,what has changed is the scientific knowledge. This has allowed athletes to push the limits of whatwas previously thought possible.-That's interesting. Please tell us more about these perceivedlimits.-The world has seen sports records being broken. That could onlybe broken with the aid of technology,whether this be the speedof a tennis serve or the fastest time in 100 meter dash or 200meter swimming race.-Is there any concern that technology is giving some athletes an unfair advantage over others?-That is an interesting question and one that has to be considered very carefully.Skis,for example,went from being made of wood to a metal alloy,which allows for better control and faster speed. There is no stopping technological progress. But as I said,each sit- uation should be considered carefully on a case by case basis. Questions one to four are based on the conversation you have just heard.1.What do we learn about Anna Sanchez?2.What is the woman's book mainly about?3.What has changed in the past thousands of years?4.What is the man's concern about the use of technology in sports competitions?CONE VERSATION 2-I've worked in international trade all my life. My father did so to before me .So l guess you could say it runs in the family.-What products have you worked with?-Allsorts,really. I've imported textiles,machinery,toys,solar panels,all kinds of things. Over the years,trends in demand come and go. So what needs to be very flexible to succeed in this indus-try?-l see.What goods are you trading now?-I now import furniture from China into Italy and foods from Italyinto China.l even use the same container.It's a very efficient wayof conducting trade.-The same container.You mean you owna40footcargo container?-Yeah,that's right.I have a warehouse in Genoa,Italy,and another in Shanghai.I source midcentury modern furniture from different factories in China.It's a very good value for money.I collect it allin my warehouse and then dispatch it to my other warehouse inItaly over there.I do the same,but with Italian foods instead of furniture,things like pasta,cheese,wine,chocolate.And Is end all that to my warehouse in China in the same freight container l usefor the furniture.-Sol presume you sell both lines of products wholesale in each re- spective country?-Of course,I possess a network of clients and partners in both countries.That's the main benefit of having done this for so long.I've made great business contacts overtime.-How many times do you ship?-l did 12 shipments last year,18 this year,and I hope to grow to around 25 next year.That's both ways.There and back again,Demand for authentic Italian food in China is growing rapidly.And similarly,sales of affordable,yet stylish wooden furniture are also increasing in Italy.Furniture is marginally more profitable,mostly because it enjoys lower customs duties.Questions five to eight are based on the conversation you havejust heard.5.What does the woman think is required to be successful in the international trade?6.What does the woman say is special about her way of doing change?7.What does the woman have in both Italy and China?8.What does the woman say makes furniture marginally more profitable?ListeningLECTURE 1Qualities of a relationship such as openness,compassion and mental stimulation are of concern to most of us regardless of sex,but-judging from the questionnaire response-they are more im-portant to women than to men.Asked to consider the ingredientsof close friendship,women rated these qualities above all others. Men assigned a lower priority to them in favor of similarity in in- terests,selected by 77 percent of men,and responsiveness in a crisis,chosen by 61 percent of male respondents.Mental stimula- tion,ranked third in popularity by men as well as women,was the only area of overlap.Among men,only 28 percent named open nessa san important quality;caring was picked by just 23per- cent.It is evident by their selections that when women speak ofclose friendships,they are referring to emotional factors,while men emphasize the pleasure they find in a friend's company.Thatis,when a man speaks of“a friend”he is likely to be talking about someone he does things with-a teammate,a fellow hobbyist,a drinking buddy.These activities are the fabric of the friendship;itis a“doing”relationsh ip in which similarity in interests is the key bond.This factor was a consideration of less than 11 percent of women.Women opt for a warm,emotional atmosphere where communi- cation flows freely;activity is mere stly,men,as we have seen,have serious questions about eachother's loyalty. Perhaps this is why they placed such strong emphasis on respon- siveness in a crisisSomeone l can call on for help.Women,as their testimonies indi- cate,are generally more secure with each other and consequently are more likely to treat this issue lightly.In follow-up interviewsthis was confirmed numerous times as woman after woman indi cated that“being there when needed was taken for granted.”Asfor the hazards of friendship,more than a few relationships have been shattered because of cutthroat competition and feelings of betrayal.This applies to both men and women,but unequally.In comparison,nearly twice as many men complained about these issues as women.Further,while competition and betrayal are the main thorns to female friendship,men are plagued in almost equal amounts by two additional issues,lack of frankness and a fear of appearing unmanly.Obviously,for a man,a good friend- ship is hard to findQuestion 16 to 18.Based on the recording you have just heard16.What quality do men value most concerning friendship accord- ing to a questionnaire response?17.What do women refer to when speaking of close friendships?18.What may threaten a friendship for both men and women? LECTURE 2Recording to the partial skeletons of more than 20 dinosaurs and the scattered bones of about 300 more have been discovered in Utah and Colorado.At what is now the Dinosaur National Monu- ment.Many of the best specimens maybe seen today at museumsof natural history in the largest cities of the United States and Canada.This dinosaur pit is the largest and best preserved depos-it of dinosaurs known today.Many people get the idea from the massive bones and the pitbull that some disaster,such as a vol ca nic explosion or a sudden flood,killed a whole herd of dinosaurs in this area.This could have happened,but it probably did not. The main reasons for thinking otherwise are the scattered bones and the thickness of the deposit. In other deposits where the an i- mals were thought to have died together,the skeletons were usu- ally complete and often all the bones were in their proper places. Rounded pieces of fossil bones have been found here.These frag ments got their smooth round shape,though,rolling along the stream bottom.In a mass killing,the bones would have been left on the stream or lake bottom together at the same level.But inthis deposit,the bones occur throughout a zone of sandstone about 12feet thick.The mixture of swamp dwellers and dryland types also seems to indicate that the deposit is a mixture from dif- ferent places.The pit area is a large dinosaur graveyard,not aplace where they died.Most of the remains probably floated downon eastward flowing river until they were left on a shallows and- bar.Some of them may have come from faraway dryland areas tothe west.Perhaps they drowned trying to cross a small stream or washed away during floods.Some of the swamp dwellers mayhave got stuck in the very sandbar that became their grave.Others may have floated for miles before being stranded.Even today,similar events take place when floods come in the spring Sheep,castling,deer are often trapped by rising waters and often drown.Their dead bodies float downstream until the flood re-cedes and leaves them stranded on a bar or shore where they liehalf buried in the sand until they decay.Early travelers on theM issour iRiver reported that shores and bars were often lined withthe decaying bodies of Buffalo that had died during spring floods. Questions 19 to 21 are based on the recording you have just heard 19.Where can many of the best dinosaur specimens be found in NorthAmerica?20.What occurs to many people when they see the massive bonesin the pit wall?21.What does the speaker suggest about the large number of di-no saur bones found in the pet?LECTURE 3I would like particularly to talkabout the need to develop a newstyle of aging in our own society.Young people in this countryhave been accused of not caring for their parents the way theywould have in the old country.And this is true.But it is also truethat old people have been influenced by an American ideal of in- dependence and autonomy.So we live alone,perhaps on theverge of starvation in time without friends.But we are indepen- dent.This standard American style has been forced on everyethnic group,although there are many groups for whom the ideal is not practical.It is a poor ideal in pursuing it does a great deal of harm.This ideal of independence also contains a tremendousamount of unselfishness.In talking to today's young mothers.Ihave asked them what kind of grandmothers they think they are going to be.l hear devoted,loving mothers say that when they are through raising their children,they have no intention of becoming grandmothers.They were astonished to hear that in most of the world,throughout most of its history,families have been three or four generation families living under the same roof We have over- emphasized the small family unit,father,mother,small children. We think it is wonderful if grandma and grandpa,if they're still alive,can live alone.We have reached the point where we thinkthe only thing we can do for our children is to stay out of their way.And the only thing we can do for our daughter in law is to see as little of her as possible.All peoples nursing homes.Eventhe best run are filled with older people who believe the only thing they can do for their children is to look cheerful when they come to visit.So in the end,older people have to devote all their energies to not being a burden.We are beginning to see what a tremendous price we've paid for emphasis on independence and autonomy.We've isolated old people and we've cutoff the chil- dren from their grandparents.One of the reasons we have as bad a generation gap today as we do is that grandparents have stepped out.Young people are being deprived of the thing they need most perspective to know why their parents behaves opec u- liar ly and why their grandparents say the things they do. Questions 22 to 25 based on recording you have just heard.22.What of young Americans being accused of?23 What does the speaker say about old people in the United States?24.What is astonishing to the young mothers interviewed by the speaker?25.What does the speaker say old people try their best to do?答案Listening1.A She is a great athlete解析:同义替换Olympic Champion=athlete2.D How technology has helped athlete to scale new heights.解析:视听一致+同义替换push humans to their edge of physical abili- ty=scale new heights.3.B Our scientific knowledge.解析:视听一致4.CIt may give an unfair advantage to some athletes.解析:视听一致5.B Flexibility解析:视听一致6.D Using the same container back and forth.解析:视听一致7.A Warehouses.解析:视听一致8.C Lower import duties.解析:视听一致+同义替换import duties=customs duties9.A It helps employees to reduce their stress.解析:视听一致(乱序全篇有stress reducing=reduce their stress)10.D Humor can help workers excel at routine tasks解析:视听一致11.B Take the boss doll apart as long as they reassemble it.解析:视听一致+同义替换put...back in place=reassemble it 12.A The recent finding of a changed gene in obese mice.解析:视听一致+同义替换the latest discovery=recent finding 13.DIt renders mice unable to sense when to stop eating.解析:视听一致+同义替换can't tell=unable to sense14.C People are born with a tendency to have a certain weight. 解析:视听一致15.B The abundant provision of rich foods解析:视听一致16.A Similarity in interests.解析:视听一致17.D Emotional factors.解析:视听一致(问女生,要通过问题判断)18.C Feelings of betrayal解析:视听一致(问男女共同点,要通过问题判断)19.DAt museums of natural history in large cities解析:视听一致(要通过问题判断)20.B Some natural disaster killed a whole herd of dinosaurs in the area.解析:视听一致21.A The floated down an eastward of flowing river.解析:视听一致22.C Failing to care for parents in the traditional way.解析:视听一致+同义替换(not caring=failing to care,in an old coun try=in the traditional way)23.D The have a sense of independence and autonomy.解析:视听一致24.B There have been extended families in most parts of theworld.解析:视听一致+同义替换(three or four-generation family=extended families)25.B Avoid being a burden to their children解析:视听一致+同义替换(not=avoid)Reading【1】选词填空26.G grabbed27.B declaration28.Ms take29.K overwhelming30.C deteriorating31.F eroding32.E disaster33.D determined34.0 urgent35.A capacity段落匹配36.C Historically,children didn't receive...37.JIna set of experiments...38.F Part of the motivation...39.A Until a few decades ago...40.G To prove that infants know more...41.E Today,avery different picture...42.M There'sno consensus...43.HInst aed of engaging babies...44.B Much of the subsequent research...45.L Despite these obvious advances...仔细阅读46.B They hold a different view on stress from the popular one.47.D They apply extreme tactics.48.A They help him combat stress from work.49.DIt does not help buildup one's tolerance.50.C Its effect varies considerably from person to person.51.B Hunting may also be a solution to the problem caused byhunting.52.CIt leads to ecological imbalance.53.A Overpopulation is not an issue for most hunted animals.54.A When it benefits animals and their ecosystem.55.C Coordinated efforts of hunters and environmentalists.Translation【1】《三国演义》(The Romance of the Three Kingdoms) 是中国一部著名的历史小说,写于十四世纪。
化工进展Chemical Industry and Engineering Progress2023 年第 42 卷第 9 期具有pH 响应性的自愈合蓝光水凝胶王少凡1,周颖1,郝康安2,黄安荣3,张如菊1,吴翀4,左晓玲1(1 贵州民族大学材料科学与工程学院,贵州 贵阳550025;2 贵州民族大学机械与电子工程学院,贵州 贵阳 550025;3国家复合改性聚合物材料工程技术研究中心,贵州 贵阳 550014;4 贵州中医药大学药学院,贵州 贵阳550025)摘要:通过分子设计,以改性壳聚糖与改性海藻酸钠通过席夫碱键交联,制备了一种低成本且具有多功能集成特性的新型荧光自愈合水凝胶(CSA 水凝胶)。
研究分析结果表明,该水凝胶表现出快速的凝胶化能力,最短仅在3min 内即可成胶。
并且水凝胶还表现出优异的自愈合能力,在室温下最快2h 即可实现自主型自愈合,且愈合效率高。
同时,在365nm 紫外光的照射下,CSA 水凝胶能够稳定释放出强烈的蓝色荧光,同时表现出荧光激发波长依赖特性。
此外,调节水凝胶的pH 能够很好地实现水凝胶的溶胶-凝胶相转变,进而实现凝胶的动态重组。
这种同时具备pH 响应性、自愈合特性以及荧光性能的水凝胶材料,为开发可用于生物影像和信息防伪领域的新一代智能材料提供了新的思路和重要指导作用。
关键词:水凝胶;自愈合;荧光;pH 响应;壳聚糖;海藻酸钠中图分类号:O636.9 文献标志码:A 文章编号:1000-6613(2023)09-4837-10Self-healing and blue-light hydrogel with pH responsivenessWANG Shaofan 1,ZHOU Ying 1,HAO Kang ’an 2,HUANG Anrong 3,ZHANG Ruju 1,WU Chong 4,ZUO Xiaoling 1(1 College of Materials Science and Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China; 2 College of Mechanical and Electrical Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China; 3 National Engineering Research Center for Compounding and Modification of Polymeric Materials, Guiyang 550014, Guizhou, China;4College of Pharmacy, Guizhou University of Traditional Chinese Medicine, Guiyang 550025, Guizhou, China)Abstract: Based on the molecular design, a novel fluorescent self-healing hydrogel (CSA Hydrogel) with low cost and multi-functional integration properties was synthesized by the modified chitosan and sodium alginate via Schiff base crosslinking. The result was found that the hydrogel had a rapid gelatinizationability and the shortest time was 3 minutes. The hydrogel also showed the excellent self-healing ability, concretely, the self-healing phenomenon could be realized with the shortest time of 2 hours at room temperature and the self-healing efficiency was high. It was worth noting that CSA hydrogels can stably release strong blue fluorescence under 365nm UV light, and show the wavelength dependence of fluorescence excitation. In addition, the sol-gel phase transition of hydrogels could be realized by adjusting the pH of hydrogels, and then the dynamic recombination of hydrogels could be achieved. The研究开发DOI :10.16085/j.issn.1000-6613.2022-1944收稿日期:2022-10-19;修改稿日期:2022-12-19。
MARCH 2023SolSmart – Standard Pathway Program Guide Welcome to SolSmart!in your community! In the next ten years, the amount of solar energyin the U.S. is expected grow dramatically- by 2033 there is likely tobe five times more solar installed than there is today!1Byimplementing solar-friendly policies, not only can you helpaccelerate this transition to clean energy, but you can also ensureyour community is poised to take advantage of the many benefits.Becoming SolSmart-designated means you are helping yourresidents save money, protecting natural resources, bolstering localresilience and increasing job opportunities in the clean energysector. Through SolSmart, your community will get access to freetechnical assistance and learn how to implement strategies thatmake solar more affordable and accessible to all residents. YourSolSmart designation will send a signal that your community is“open for solar business,”encouraging growth of local solarcompanies and other sustainability-minded businesses.This guide is a comprehensive resource to help you implement solar best practices in your community andgain national recognition by earning Solsmart designation! This guide will help you to navigate the “Standard Pathway,” which is applicable to most local governments that have authority over permitting, planning, zoning and/or inspection processes. Local governments that do not control these processes should refer to the SolSmart Modified Pathway Program Guide. Regional Organizations, including Regional Planning Commissions and Councils of Governments, should refer to the Regional Organization Program Guide.The SolSmart program will connect you with solar best practices from across the country and provide clear guidance on how to implement these actions. Along the way you will receive points for the actions you take and achieve recognition as a Bronze, Silver, Gold or Platinum SolSmart-designated community! Throughout this process, our technical assistance providers are available to provide support at no cost. Please complete this form to connect with a technical assistance provider and get started on the path to Solsmart designation.ContentsI.SolSmart Overview, pg. 2An introduction to the SolSmart Program and the designation process.II. Criteria Overview, pg. 5A list of SolSmart criteria organized by category.III. SolSmart Technical Assistance and Designation Process, pg. 10A summary of the designation process and how to use the information in the guide to achievedesignation.IV. Criteria Detail and Verification Guidance, pg. 11A detailed description of each SolSmart criteria with guidance and examples to assist you inimplementing solar best practices and achieving points toward designation.1resources/solar-market-insight-report-2022-year-reviewI. SolSmart OverviewAcross the United States, communities are increasingly using solar energy to power their homes and businesses and enjoying the benefits of clean, reliable, and affordable electricity. Rapidly declining prices for solar and related technologies have brought vast amounts of solar energy into the mainstream within a few short years. Homeowners, businesses, schools and local governments are using solar energy to drastically reduce their utility costs, while also reducing the environmental impact of their energy use. As natural disasters become more frequent and intense, distributed solar and energy storage is also bolstering energy resilience. Local and regional governments play an important role in establishing policies, procedures and programs that impact solar deployment in communities. When local governments create barriers to solar in their local plans, permitting and other policies, either intentionally or unintentionally, they can hinder solar development. Alternately, when local governments provide a supportive environment for solar energy and take steps to streamline permitting, inspection and zoning processes, they expedite the installation of solar PV systems and help make it more affordable for residents and businesses.Action at the local level is also fundamental to ensuring that solar programs are equitable and inclusive and ultimately deliver shared benefits to all Americans. SolSmart is committed to the goals of the federal Justice40 initiative to provide equitable opportunities for underserved communities which face barriers including fossil dependence, energy burden, environmental and climate hazards, and socio-economic vulnerabilities. SolSmart criteria reflect the importance of developing equitable and inclusive solar policies and programs. The SolSmart program has two key components. First, the program provides no-cost technical assistance to help local governments follow national best practices to expand solar energy use in their jurisdictions. Second, it recognizes and celebrates these communities with SolSmart designations of Bronze, Silver, Gold or Platinum. SolSmart is led by the International City/County Management Association (ICMA) and the Interstate Renewable Energy Council (IREC) and funded by the U.S. Department of Energy Solar Energy Technologies Office (SETO).Designation LevelsThe SolSmart program has developed a set of designation criteria based on established best practices that encourage the growth of solar energy at the local level. The criteria for the Standard Pathway are organized into five categories – Permitting and Inspection, Planning and Zoning, Government Operations, Community Engagement and Market Development. Within each category, SolSmart provides clear guidance and templates to help communities put these practices into action. Some of the criteria are prerequisites, while others are elective. Each criterion has a corresponding point value. Upon meeting the prerequisites and reaching a sufficient number of points in each category, a participant qualifies for SolSmart designation. There are four levels of SolSmart designation for local governments. Below are the requirements for each level. Communities that earn 60% of the available points in a category are additionally eligible for a special recognition award.Criteria CategoriesBelow is a summary of each category and the types of actions that are recognized as best practices in each. Permitting and Inspection | 28 Criteria | 275 PointsMost local governments have direct oversight of the permitting and inspection policies and procedures within their jurisdiction. Communities that implement permitting best practices provide solar developers and installers with a transparent, efficient, and cost-effective approval process. Well-trained staff and simplified permit applications can reduce staff time needed to review permits which allows them to focus on other priorities. Clear inspection procedures ensure compliance with applicable state and local codes while protecting public health and safety. Many of the criteria in the permitting and inspection category can be verified by providing information in a detailed permitting checklist. Verification of trainings for permitting and inspection staff and documented improvements to inspection processes are also part of ensuring a transparent and efficient permitting and inspection process.Planning and Zoning | 26 Criteria | 215 PointsLocal government planning and zoning regulations can help facilitate the rapid expansion of solar energy and associated technologies, including energy storage and electric vehicles, within a community. Communities can utilize planning and zoning regulations to increase opportunities for rooftop and ground-mounted solar energy while also advancing other community goals including affordable housing, economic development, clean transportation and the protection of natural and cultural resources. Plans should set forth a vision for the community’s clean energy future, while zoning codes should provide clear and transparent regulations on the development and use of solar energy within the jurisdiction. Many of the criteria in the planning and zoning category can be verified by providing a link to a community’s codes, or dinances, and community plans. Government Operations | 14 Criteria | 185 PointsLocal governments can lead the way by installing solar energy on public facilities and land. Communities can engage with their local utility to discuss goals for solar energy, net metering, interconnection, and community solar. These actions are high impact and can directly lead to an increase in solar energy deployment. Many of the criteria in the government operations category can be verified by providing documents demonstrating installed solar capacity such as news articles about solar installations, dashboards/metrics showing solar production, and contracts that demonstrate solar project construction.Community Engagement | 13 Criteria | 90 PointsLocal governments can be an important and trusted source of information for residents, businesses, and solar installers. Providing clear, high-quality information, public education and inclusive engagement opportunities can help residents and businesses interested in solar energy make informed decisions. Local governments can support more equitable outcomes by partnering with community organizations and developing goals and strategies that meet the needs of disadvantaged communities. Many of the criteria in the community engagement category can be verified by providing information about a community’s solar energy goals, strategies and partnerships on a local government’s solar webpage.Market Development | 10 Criteria | 155 PointsLocal governments can collaborate and partner with organizations to promote solar development within their jurisdiction. Supporting a community solar program, promoting a solarize group-buy campaign, or partnering with a local financial institution can make solar energy more affordable and accessible for homes and businesses while improving business opportunities for solar installers. Many of the criteria in the market development category can be verified by providing news articles about the local government's role in supporting solar development or by providing official documents that established policies or programs.I. Criteria OverviewThe SolSmart Standard Pathway contains 92 criteria, each of which is a specific action that local governments can implement to encourage solar energy development in their community. Each criterion has a corresponding point value ranging from 5 to 20. A detailed description with relevant templates, examples and resources to help you achieve each criterion is available in Section IV.II. SolSmart Technical Assistance and Designation ProcessAny local government, regardless of previous solar experience, is eligible for SolSmart designation. To request a call with a member of the SolSmart program, please complete the contact form on .Once the local government decides to pursue SolSmart designation, Array they need to complete a Solar Statement and submit it to theSolSmart team. The Solar Statement demonstrates the community’scommitment to work with the SolSmart program and achievedesignation. The local government will be connected with one of ourtechnical assistance providers who will work with community toreview communi ty’s goals and processes. This review helpsdetermine how close the community is to designation and anyadditional technical assistance to achieve designation. The localgovernment with work with their technical assistance provider todevelop a plan, identify which criteria they will meet to achieve theirdesired designation level, and implement best practices in thecommunity. Once they have completed the required actions, the localgovernment can submit for designation.To earn national recognition from the SolSmart Program, acommunity must provide documentation of the actions it hasimplemented. This may include a combination of signed memos, weblinks, program materials, policy documents, etc. as appropriate.Section IV of this Program Guide provides a detailed description ofeach SolSmart criterion with resources to support implementationand guidance on documentation and verification that will be requiredby SolSmart.Once the local government is ready for designation review, thesubmission is reviewed by the Designation Program Administratorwithin two weeks and the local government is notified of their designation by email.Local governments are encouraged to celebrate and publicize their designations and to post information about SolSmart on their own websites. Many SolSmart designees have held events, shared photos and videos, and taken other actions to publicize their achievements. The designation email contains a Designation Toolkit with template press release, sample social media, and SolSmart Designation logos. SolSmart will also recognize local governments on the SolSmart website, on social media, and in the SolSmart newsletter.III. Criteria Detail and Verification GuidanceThe SolSmart criteria are based on specific best practices that local governments and community stakeholders can implement to encourage solar energy development in their community. This section provides a detailed description of each criterion, recommended verification for designation review, community examples, templates, and/or resources.The following provides an overview of the information that is provided for each SolSmart criterion:Solar StatementPermitting and InspectionPlanning and ZoningGovernment OperationsCommunity EngagementMarket DevelopmentInnovative ActionAcknowledgmentThis material is based upon work supported by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office Award Numbers DE-EE0009950 & DE-EE009951.Full Legal DisclaimerThis report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.。
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。
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中国科学院“百人计划”入选者
终期考核评估表
聘用单位___ 中科院上海光机所_ _
姓名__________张龙__________
从事专业_________材料学__________
入选年度______ ___2006
是否获得择优支持是资助年度 2007
入选者现任职务院重点实验室主任
到岗工作时间_ 2006 __年___1 _月
年在岗工作时间_________12______个月
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a rXiv:089.1235v1[math.D S]8Se p28SMOOTH DEPENDENCE ON PARAMETERS OF SOLUTION OF COHOMOLOGY EQUATIONS OVER ANOSOV SYSTEMS AND APPLICATIONS TO COHOMOLOGY EQUATIONS ON DIFFEOMORPHISM GROUPS R.DE LA LLAVE AND A.WINDSOR Abstract.We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms.We show that the solutions depend on param-eters as smoothly as the data.As a consequence we prove optimal regularity results for the solutions of equations taking value in diffeomorphism groups.These results are motivated by applications to rigidity theory,dynamical systems,and geometry.In particular,in the context of diffeomorphism groups we show:Let f be a transitive Anosov diffeomorphism of a compact manifold M .Suppose that η∈C k +α(M,Diffr (N ))for a compact manifold N ,k,r ∈N ,r ≥1,and 0<α≤Lip.We show that if there exists a ϕ∈C k +α(M,Diff1(N ))solving ϕf (x )=ηx ◦ϕx then in fact ϕ∈C k +α(M,Diffr (N )).1.Introduction Let f be a diffeomorphism of a compact manifold M .A cohomology equation over f is an equation of the form ϕf (x )=ηx ·ϕx (1.1)where ηis given and we are to determine ϕ.The interpretation of ·varies depending on the context but is typically either a group operation or the composition of linear maps.Equations of the form (1.1)have been studied for many different maps (including rotations,and horocycle flows).In this paper we will always consider f to be an Anosovdiffeomorphism,usually a transitive Anosov diffeomorphism.It is relevant to point out that if f n (p )=p then,by applying (1.1)repeatedly,we obtainϕf n (p )=ηf n −1(p )···ηp ·ϕp .2R.DE LA LLAVE AND A.WINDSORHence,if there is a solutionϕthen,sinceϕf n(p)=ϕp,we must haveηf n−1p···ηp=Id.(1.2)When f is an Anososv diffeomorphism a great deal of effort,starting with the pioneer-ing work[Liv71],has been devoted to showing that(1.2),supplemented with regularity and“localization”assumptions,is sufficient for the existence and regularity ofϕ.The main goal of this paper is to study the parameter dependence of these solutions given that the data depends smoothly on parameters.In this paper,we will not be concerned with the existence of solutions of(1.1),rather, we will assume that a solution exists and establish smoothness with respect to parameters. Of course,there is an extensive literature establishing the existence of solution,see for example the references in[dW07].There are several contexts in which to study cohomological equations(1.1).The most classical one is whenϕ,ηtake values in a Lie group G.In dynamical systems and geometry,(1.1)also appears in some different contexts.Given a bundle E over M,we takeϕto be an object defined on thefiber E x andηx is an action transporting objects in E x to objects in E f(x).For example,in[dW07],one canfind an application whereϕx are conformal structures on E x(a subspace of T x M)andηx is the natural transportation of the conformal structure by the differential.We note that it is not necessary to assume that the bundle E isfinite dimensional.It suffices to assume that the linear operators lie on a Banach algebra[BN98].Of course,when the bundle E is trivial,the linear operators on E x can be identified with a matrix group,so that this geometric framework reduces to the Lie group framework with G a matrix group(or,more generally,a Banach algebra of operators).One very interesting example,which indeed serves as the main motivation for this paper is whenϕx andηx are supposed to be C r diffeomorphisms of a compact manifold N and the operation in the right hand side of(1.1)is just composition.Though Diffr(N) is certainly a group under composition,it is not a Lie group since the group operation is not differentiable as a map from Diffr(N)to Diffr(N).The problem was considered in [NT96]where it was shown that results on(1.1)have implications for rigidity.In[NT96] one canfind results on existence of solutions when N=T d and,in[dW07]for a general N.Both in[NT96],[dW07],from the fact thatηx∈Diffr(N)one can only reach the conclusion thatϕx∈Diffr−R(N)(in[dW07]the regularity loss R=3,but in[NT96]R depends on the dimension of N).SMOOTH DEPENDENCE3The main goal of this paper is to overcome this loss of regularity and show that if ηx∈Diffr(N)for r≥1,andϕ∈Diff1(N),thenϕ∈Diffr(N).See Theorem6for a more precise formulation.As it turns out,the main technical tool in the proof of Theorem6is to establish results on the smooth dependence on parameters for the solutions of(1.1)which may be of independent interest.We formulate them as Theorem3,Theorem5.When G is a commutative group smooth dependence on parameters is rather elemen-tary,see Proposition2,and our main technique is to reduce to the commutative case.1.1.Sketch of the proofs.In this section we informally present the main ideas behind the proof omitting several of the details which we will treat later.1.1.1.Some remarks on the relation between bootstrap of regularity and dependence on parameters.Going over the proofs in[NT96]and,more explicitly,in[dW07]it becomes apparent that,the loss of regularity of the solutions is closely related to the fact that the group operation is not differentiable.Hence,wefind it convenient to turn the tables.Rather than consideringϕ:M×N→N as a function from M to a space of mappings on N,we considerϕas a mapping from M to N with parameters from N.In this application,N plays two roles:as the ambient space for the diffeomorphisms and as the parameter.For the sake of clarity,we will develop our main technical results denoting the target space as a N and the set of parameters as U.This is,of course,more general,and in many cases,the parameters appear independently of the target space.More precisely,if wefix u∈U and suppose thatϕis differentiable with respect to u then the chain rule tells us thatD uϕf(x)(u)=D uηx◦ϕx(u)·D uϕx(u)(1.3)where D u denotes the derivative with respect to the parameter u∈U.Note that for typographical convenience we will often useηu x in place ofη(x,u),andϕu x in place of ϕ(x,u).We see that for each u∈U(1.3)is an equation of the form(1.1)for the cocycle D uϕx(u)with generator˜ηu x=D uηx·ϕu x(1.4)N.In this case E is the bundle of linear maps from T u N to Tϕx(u)Ifϕu x depends C1on u andηx depends C2on u,then˜ηx is C1in ing the regularity theory for commutative cohomology equations we obtain that D uϕu x is C1with respect to4R.DE LA LLAVE AND A.WINDSORu.This means thatϕu x is C2with respect to u.This argument to improve the regularity can be repeated so long as we can differentiateηx.1.1.2.The dependence on parameters.The idea of the proof of the smooth dependence on parameters for(1.1)is more involved that the bootstrap of regularity since we need to justify the existence of thefirst derivative.For simplicity of notation,we interpret(1.1)in the linear bundle maps framework. This will be the crucial technical result for the bootstrap of regularity in diffeomorphism groups.The case of Lie group valued cocycles will be dealt with in Section2.3.We want to show that ifϕu,ηu solve(1.1)andηu depends smoothly on parameters, thenϕu depends smoothly on parameters.However solutions of(1.1)are not unique.Taking advantage of this it is very easy to construct solutions which depend badly on parameters.It is therefore necessary to impose some additional condition such as thatϕu p is smooth in u∈U for somefixed p∈M.For convenience we will take p∈M to be a periodic point.In order to prove differentiability wefirstfind a candidate for the derivative using Livˇs ic methods and then argue that this candidate is the true derivative.The key observation is that if we formally differentiate(1.1)with respect to the parameter u we obtainD uϕu f(x)=D uηu x·ϕu x+ηu x·D uϕu xUsing(1.1)we getD uϕu f(x)=D uηu x·ϕu x+ϕu f(x)·(ϕu x)−1·D uϕu x(1.5) Multiplying on the right both sides of(1.5)by(ϕu)−1,we obtainf(x)(ϕu f(x))−1·D uϕu f(x)=(ϕu f(x))−1·D uηu x·ϕu x+(ϕu x)−1·D uϕu x(1.6) where(ϕu x)−1refers to the inverse of the linear mapϕu x.This equation(1.6)is a coho-mology equation over a commutative group for the new unknownξx=(ϕu x)−1D uϕu x.We will show that the periodic obstruction(1.2)corresponding to(1.6)is the derivative with respect to parameters of the periodic obstruction corresponding to(1.1). Hence,applying the results on Livˇs ic equations,we obtain a solution of(1.6).It remains to show that this candidate is the true derivative.As mentioned before,the solutions of(1.6)are not unique,but,under the hypothesis thatϕp(u)=:γ(u)is smooth in u for some p,which we made at the outset,it is naturalSMOOTH DEPENDENCE5 to impose that the solution to(1.6)satisfy the normalizationD uϕp=D uγ.Once we have a candidate for a derivative,we will use a comparison argument,Propo-sition4to show that indeed it is a derivative.Of course,once we have established the existence of thefirst derivative,we have also shown that it satisfies(1.6).Hence,to establish the existence of higher derivatives it suffices to use the(much simpler)theory of dependence of parameters in commutative cohomology equations.The above argument uses very heavily that we are dealing with the framework of linear operators on bundles.The adaptation of the above argument to general Lie groups requires some adaptations of the geometry,see Section2.3.2.Smooth dependence on parameters of cohomology equationIn this section,we make precise the previous arguments and establish smooth depen-dence on parameters for the solutions of(1.1).In Section2.1we make precise what we mean by smooth dependence on parameters.In Section2.2we present the results when (1.1)is interpreted as an equation between linear bundle maps.This section will be crucial for the case of diffeomorphism groups.In Section2.3we present the results for cocycles taking values in a Lie group.2.1.Notation on regularity.We will understand that a function is C r when it admits r continuous derivatives.When the spaces we consider are not compact,we will assume that all the derivatives are bounded.For0<α≤Lip we will define real valued Cαfunctions on a metric space N in the usual way|f(x)−f(y)|Hα(N)=supx=y6R.DE LA LLAVE AND A.WINDSORWe will write h∈C k+α,r if(1)for all0≤i≤r and all0≤j≤k the derivative D j x D i uϕexists and is continuouson M×U.(2)for all0≤i≤r and all u∈U the derivative D i uϕ(·,u)∈Cα(M)and the H¨o lderconstant does not depend on u.We also defineˆC k+α,r(U,M)=C r U,C k+α(M) .That,is the set of functions h: M×U→N such that u→h(·,u)is C r when we giveϕ(·,u)the C k+αtopology.The reason to introduce these spaces is that C k+α,r is natural when performing geo-metric constructions involving derivatives.The spaceˆC k+α,r is natural when we consider dependence on parameters of commutative Livˇs ic equations.Of course,the two spaces are closely related.The relation is formulated in the following proposition. Proposition1.For any0<α′<α≤Lip,k,r∈N,we have:ˆC k+α,r⊂C k+α,r⊂ˆC k+α′,r(2.1) The simple exampleϕ(u,x)=(x−u)k+αsatisfiesϕ∈C k+α,0,ϕ/∈ˆC k+α,0.Integrating with respect to u one can obtain similar examples for all r∈N.Proof.Thefirst inclusion of(2.1)is obvious.Letϕ∈C k+α,r.Letα′=α·θfor0<θ< 1.We wish to show thatϕ∈C r(U,C k+α′(M)).This is equivalent to∂i uϕ∈C0(U,C k+α′(M)for all multi-indices i with0≤|i|≤r.Fix u∈U and consider a compact K with u∈K⊂U.Restricted to M×K the functionϕand all its partial derivatives are uniformly continuous.So we have∂j x∂i uϕ(·,u)−∂j x∂i uϕ(·,u′) 0is controlled by d(u,u′),for all multi-indices j with0≤|j|≤k.It remains to show that for j=k we have∂j x∂i uϕ(·,u)−∂j x∂i uϕ(·,u′) α′is controlled.For any multi-index j with|j|=k we have from the Kolmogorov-Hadamard inequalities[dlLO99]∂j x∂i uϕ(·,u)−∂j x∂i uϕ(·,u′) α′≤C ∂j x∂i uϕ(·,u)−∂j x∂i uϕ(·,u′) θα ∂j x∂i uϕ(·,u)−∂j x∂i uϕ(·,u′) 1−θSMOOTH DEPENDENCE7By definition ofϕ∈C k+α,r we have that ∂j x∂i uϕ(·,u) αis bounded independent of u∈U and hence∂j x∂i uϕ(·,u)−∂j x∂i uϕ(·,u′) 0is controlled since∂j x∂i uϕis uniformly continuous.Consequentlyϕ∈C r(U,C k+α′(M)). Note that we do not claim uniform continuity in u.Next,we discuss the regularity properties of the commutative cohomology equation. The results for k=0appear in[Liv71,Liv72]and the results for k>0appear in [dlLMM86].The following result will be one of our basic tools later.Proposition2.Let M be a compact manifold and f a transitive Anosov diffeomorphism on M.Let p∈M be a periodic point for f.Consider the commutative cohomology equationϕu◦f(x)−ϕu(x)=ηu(x)(2.2)ϕ(p,·)∈C r(U)Assume that for all values of u,the periodic orbit obstruction withηu vanishes.Then,(1)ifη∈ˆC k+α,r,thenϕ∈ˆC k+α,r,and(2)ifη∈C k+α,r,thenϕ∈C k+α,r.Proof.Thefirst statement is obvious since we are considering the solution of a linear equation and[dlLMM86]showed that the solution operator is continuous from C k+αto C k+α/R.The second statement follows because,by the previous argument,we have thatϕ∈ˆC k+α′,r.In particular for|i|≤r,∂iϕ(·,u)∈C k+α′but this function satisfiesu∂i uϕu◦f−∂i uϕu=∂i uηuSince we assumed thatη∈C k+α,r we have that∂i uη(·,u)∈C k+α(M).and,by the results of[dlLMM86],∂i uϕ(·,u)∈C k+α(M).2.2.Cohomology Equations for Bundle Maps.As we have mentioned before,coho-mology equations for bundle maps appear naturally when we consider the linearization of a dynamical system or cohomology equations for cocycles taking values in diffeomorphism groups.Theorem3.Let M and N be compact smooth manifolds,U⊆R n open,and E a bundle over N.Let0<k+α≤1and r∈N with r≥1.8R.DE LA LLAVE AND A.WINDSORLet f:M→M be a transitive C r Anosov diffeomorphism,σ:U→N withσ∈C r(N) andτ:M×U→N withτ∈C k+α,r(resp.τ∈ˆC k+α,r).Suppose that we have linear mapsϕu x:Eσ(u)→Eτ(x,u),ηu x:Eτ(x,u)→Eτ(f(x),u).such that for all u∈U we haveϕu f(x)=ηu x·ϕu x.(2.3) Ifη∈C k+α,r(resp.η∈ˆC k+α,r)and there exists a periodic point p∈M such that ϕ(p,·)∈C r(U)thenϕ∈C k+α,r(resp.ϕ∈ˆC k+α,r).The most difficult case of Theorem3is the case when r=1.The higher differentia-bility cases follow from this one rather straightforwardly.The case r=1of Theorem3 will be the inductive step for the bootstrap of regularity for cohomology equations on diffeomorphism groups.The proof we present of the r=1case uses that f is transitive but the subsequent bootstrap argument does not require that f is transitive.As indicated before,we will obtain a candidate for a derivative and,then,we argue it is indeed a true derivative.Smooth dependence is a local question.For that reason we consider a local trivial-ization of the bundle E about the pointsσ(u)andτ(x,u).The base space portion of the mapϕu x is as smooth as the minimum smoothness ofσandτ.We will concentrate therefore on the smoothness of the linear operator in thefibers.Proof.We note that if we could take derivatives in(2.3)then D uϕu x,the derivative ofϕu x with respect to u,would satisfyD uϕu f(x)=D uηu x·ϕu x+ηu x·D uϕu x(2.4) Observing thatηu x=ϕu·(ϕu x)−1we getf(x)(ϕu f(x))−1·D uϕu f(x)=(ϕu f(x))−1·D uηu x·ϕu x+(ϕu x)−1·D uϕu x.(2.5) Writingξu x:=(ϕu x)−1·D uϕu x(2.6) we see thatξu x would satisfyξu f(x)−ξu x=(ϕu f(x))−1·D uηu x·ϕu x(2.7)SMOOTH DEPENDENCE9Note that(2.7)is a commutative cohomology equation with the group operation being addition on a vector space.In this case,as we will show,the methods of[Liv72,dlLMM86]to obtain existence and regularity forξ.Using(2.6)we will obtain a candidate for D uϕu x.Since we are assuming that there are solutions of(2.3)for all u in an open set U,we have that for any p,f n(p)=pηu f n−1(p)·····ηu p=Id.(2.8)Differentiating(2.8)with respect to u we obtain0=D uηu f n−1(p)·ηu f n−2(p)···ηu p+ηu f n−1(p)·D uηu f n−2(p)·ηu f n−3(p)···ηu p+···+ηu f n−1(p)·ηu f n−2(p)···ηu f(p)·D uηu p.Using the(2.3)to express the products ofη,we obtain0=D uηu f n−1(p)·ϕu f n−1(p)·(ϕu p)−1+ϕu p·(ϕu f n−1(p))−1·D uηu f n−2(p)·ϕu f n−2(p)·(ϕu p)−1(2.9)+···+ϕu p·(ϕu f(p))−1·D uηu p.Multiplying(2.9)on the left by(ϕu p)−1and multiplying byϕu p on the right we obtain 0=(ϕu p)−1·D uηu f n−1(p)·ϕu f n−1(p)+(ϕu f n−1(p))−1·D uηu f n−2(p)·ϕu f n−2(p)(2.10)+···+(ϕu f(p))−1·D uηu p·ϕu pwhich shows that the periodic orbit obstruction for the existence of a solution to the cohomology equation(2.7)vanishes.Hence we have a solutionξu x to(2.7).Using(2.6)we see that we have a candidate for the derivative,D uϕu x=ϕu x·ξu x.It remains to show that this candidate,which we obtained by a formal argument is actually the derivative.We start by proving that all the solutions of(2.3)are differentiable in the stable manifold of a periodic point.For the sake of notation,and without any loss of generality, we will consider afixed point p.Applying(2.3)repeatedly we haveϕu f n+1(x)=ηu f n(x)···ηu x·ϕu x.(2.11)For x∈W s p we have d(f n(x),p)≤C xλn for some0<λ<1,n≥0.Sinceηu x is H¨o lder in x,andηu p=Id is continuous,in any smooth local trivialization around p we obtainηu f n(x)−Id ≤Cλαn.10R.DE LA LLAVE AND A.WINDSORTherefore we can pass to the limit in(2.11)and obtain for x∈W s pηu f n(x)···ηu x·ϕu x(2.12)ϕu p=limn→∞where the convergence in(2.12)is uniform in bounded sets of W s p(in the topology of W s p).We will show that(2.12)can be differentiated with respect to u.By the product rule D u ηu f n(x)···ηu x =D uηu f n(x)·ηu f n−1(x)···ηu x+ηu f n(x)·D uηu f n−1(x)·ηu f n−2(x)···ηu x+···+ηu f n(x)···ηu f(x)D uηu x=ϕu f n+1(x)·(ϕu f n+1(x))−1·D uηu f n(x)·ϕu f n(x)·(ϕu x)−1+ϕu f n+1(x)·(ϕu f n(x))−1·)−1·D uηu x·ϕu x·(ϕu x)−1D uηu f n−1(x)·ϕu f n−1(x)·(ϕu x)−1+···+ϕu f n+1(x)·(ϕuf(x)=ϕu f n+1(x)· n−1 i=0(ϕu f i+1(x))−1·D uηu f i(x)·ϕu f i(x) ·(ϕu x)−1(2.13) where we have used again(2.3).Since p is afixed point using(2.9)we obtain that(ϕu p)−1·D uηu p·ϕu p=0.Because of the assumed regularity onϕx,D uηu x and the exponentially fast convergence of f n(x)to p,we obtain that the general term in(2.13)converges to zero exponentially. By the Weierstrass M-test we obtain that D u ηu f n(x)···ηu x converges uniformly on compact subsets of W s p.Therefore,the limit is the derivative of lim n→∞ηf n(x)···ηx and from that,it follows immediately thatϕu x is differentiable with respect to u for x∈W s p. Of course,the argument so far allows only to conclude that D uϕu x is bounded on bounded sets of W s p.This does not show that it is bounded on M since W s p is unbounded. Nevertheless,we realize that the derivative solves(2.4)on W s p.Of course,so does the candidate for the derivative that we produced earlier.From the fact that these two functions satisfy the same functional equation in W s p,and that they agree at zero,we will show that they agree.This will allow us to conclude that the candidate is indeed the derivative in W s ing that it is a continuous function on the whole manifold,there is a standard argument that shows it is the derivative everywhere.Proposition4.Let p∈M be afixed point of f.Letξ1,ξ2be continuous functions on W s p.Assume thatξ1(p)=ξ2(p)and that both of them satisfy(2.7).Thenξ1(x)=ξ2(x) for all x∈W s p.Proof.Note that A(x)≡ξ1(x)−ξ2(x)satisfies A(x)=A(f(x)).For every x∈W s p we have lim n→∞f n(x)=p.Since A is continuous,and A(p)=0we see thatA(x)=A f n(x) =lim n→∞A f n(x) =A lim n→∞f n(x) =A(p)=0.Thusξ1(x)=ξ2(x)for all x∈W s x. We have now shown that the function is differentiable in a leaf of the foliation which is dense.Furthermore,D uϕu x extends continuously–indeed H¨o lder continuously–to the whole manifold.In this case,an elementary real analysis argument,which we will present now,shows thatϕu x is everywhere differentiable.Since we just want to show that the continuous function is indeed a derivative in the stable leaf,we can just take a trivialization of the bundles in a small neighborhood.We will use the same notation for the objects in the trivialization and the corresponding one in the bundles.This allows us to subtract objects in differentfibers.Ifψ(s)is any smooth path in U contained in the trivializing neighborhood withψ(0)= u0,ψ1=u1we have,for x∈W s pϕu1x−ϕu2x= 10D uϕψ(s)xψ′(s)ds(2.14) Both sides of the equation are H¨o lder in x.Therefore,we can pass to the limit in(2.14) and conclude that the set of points x for which(2.14)holds is closed.But we had already shown that(2.14)held for x∈W s p,which is a dense set since f is transitive.Thus(2.14) holds for every x∈M.This shows immediately that our candidate is indeed the true derivative and concludes the proof of the case r=1of Theorem3.It is now relatively simple to obtain higher derivatives.The auxiliary functionξsatisfies the commutative cohomology equation(2.7).By assumption D uη∈C k+α,r−1(resp.D uη∈ˆC k+α,r−1)hence,ifϕ∈C k+α,m(resp.ϕ∈ˆC k+α,m)for m≤r−1then the right hand side of(2.7)is in C k+α,m(resp.ˆC k+α,m).Applying Proposition2we then obtain D uϕ∈C k+α,m(resp.D uϕ∈ˆC k+α,m)and thusϕ∈C k+α,m+1(resp.ϕ∈C k+α,m+1).The induction stops at m=r−1when we have exhausted the regularity of D uη.At this pointϕ∈C k+α,r(resp.ϕ∈ˆC k+α,r)as required.2.3.Cohomology Equations for Lie Group Valued Cocycles.In this section we consider the dependence on parameters of the solution to(1.1)whenη,andϕare func-tions taking values in a Lie group G.We denote the Lie algebra of the Lie group G by g and we denote the identity in G by e.The proof follows along the same lines as the proof of Theorem3.We derive a func-tional equation for a candidate for thefirst derivative,show that there is a solution for this equation,and that the candidate is a true derivative.Finally we use a bootstrapping argument to get full regularity.The main difference with the previous section is that we have to deal with the fact that the group operation is not just the product of linear operations,so that the derivatives of the functional equation involve the derivatives of the group operation.We introduce the notationL g h=g·hR g h=h·g(2.15) where·denotes the group operation.It follows directly from the definitions of L,R thatL g◦L h=L g·hR g◦R h=R h·g(2.16) It is immediate to show that if g u,h u are smooth familiesD u(g u·h u)=DL g u(h u)D u h u+DR h u(g u)D u g u(2.17) Theorem 5.Let M be a compact manifold,U⊂R d open,G be a Lie group,f a transitive Anosov diffeomorphism of M,and p∈M a periodic point for f.Suppose that η:M×U→G withη∈C k+α,r(resp.η∈ˆC k+α,r)for0<α≤Lip and k,r∈N with r≥1.Ifϕ:M×U→G solvesϕu f(x)=ηu x·ϕu xandϕp∈C r(U,G)thenϕ∈C k+α,r(resp.ϕ∈ˆC k+α,r).The definition of the spaces C k+α,r andˆC k+α,r appears in Section2.1.Proof.Taking derivatives of(1.1)and applying the product rule we obtain that if there is a derivative of D uϕu x,it should satisfy:D uϕu f(x)=DLηux (ϕu x)D uϕu x+DRϕux(ηu x)D uηu x(2.18)We introduce a functionξ:M×U→g byD uϕu x=DLϕux(e)ξu x(2.19) Introducing the notation(2.19)is geometrically natural because we want to transport all the infinitesimal derivatives to the identity,so thatξu x takes values in the Lie algebra g. In terms ofξ,the equation(2.18)becomesDLϕuf(x)(e)ξu f(x)=DLηux(ϕu x)DLϕux(e)ξu x+DRϕux(ηu x)D uηu x.(2.20)Thefirst factor of thefirst term in(2.20)can be simplifiedDLηux (ϕu x)·DLϕux(e)=D u Lηu x◦Lϕu x (e)=D u Lηu x·ϕu x (e)=DLϕu f(x)(e).(2.21)Substituting(2.21)into(2.20)we obtainDLϕuf(x)(e)ξu f(x)=DLϕuf(x)(e)ξu x+DRϕux(ηu x)D uηu x.(2.22)Multiplying(2.22)in the left by DLϕu f(x)(e) −1we obtain:ξu f(x)=ξu x+ DLϕu f(x)(e) −1DRϕu x(ηu x)D uηu x(2.23) Equation(2.23)can be simplified further;from(2.16),we haveId=L(ϕuf(x))−1◦Lϕu f(x)where Id is the identity map on G.Thus,by the chain rule,we haveId=DL(ϕuf(x))−1(ϕu f(x))DLϕu f(x)(e)where Id is the identity map on g.HenceDLϕu f(x)(e) −1=DL(ϕu f(x))−1(ϕu f(x)).(2.24) Therefore,(2.23)can be rewritten asξu f(x)=ξu x+DL(ϕuf(x))−1(ϕu f(x))DRϕu x(ηu x)D uηu x.(2.25) We are assumingϕu p is a C r function of u.Thus,taking derivatives,and using(2.19) we obtainD uϕu p=DLϕup(e)ξu p(2.26) so we see thatξp:U→g is C r.In summary,if the functionϕu x is differentiable then the g valued functionξintroduced in(2.19)would satisfy(2.25).Equation(2.25)is a cohomology equation for functions taking values in g.It is therefore a commutative cohomology equation.Thus a necessary and sufficient condition for the existence of a smooth solution is the vanishing of the periodic orbit obstruction.In the next paragraphs we will show that indeed this periodic orbit obstruction is met, so that indeed one canfind aξsolving(2.25).Since(1.1)holds for all u,if f n(p)=p, we have for all ue=ηu f n−1(p)···ηu p=(ϕu p)−1·ηu f n−1(p)···ηu p·ϕu pDifferentiating with respect to u,we obtain:0=DRηuf n−1(p)···ηu p·ϕu p (ϕu p)−1 D u(ϕu p)−1+DL(ϕup)−1(ηu f n−1(p)···ηu p·ϕu p)DRηu f n−2(p)···ηu p·ϕu p(ηu f n−1(p))D uηu f n−1(p)+···+DL(ϕup )−1·ηuf n−1(p)···ηuf(p)(ηu pϕu p)DRϕup(ηu p)D uηu p+DL(ϕup )−1·ηuf n−1(p)···ηu p(ϕu p)D uϕu p.(2.27)Using(1.1)to reduce the products,we transform(2.27)into0=DRϕup (ϕu p)−1 D u(ϕu p)−1+DL(ϕup)−1(ϕu p)DRϕu f n−1(p)(ηu f n−1(p))D uηu f n−1(p)+DL(ϕuf n−1(p))−1(ϕu f n−1(p))DRϕu f n−2(p)(ηu f n−2(p))D uηu f n−2(p)+···+DL(ϕuf(p))−1(ϕu f(p))DRϕu p(ηu p)D uηu p+DL(ϕup )−1(ϕu p)D uϕu p.(2.28)Finally we observe thatDRϕup (ϕu p)−1 D u(ϕu p)−1+DL(ϕu p)−1(ϕu p)D uϕu p=D u (ϕu p)−1·ϕu p =0and hence(2.28)becomes0=DL(ϕup)−1(ϕu p)DRϕu f n−1(p)(ηu f n−1(p))D uηu f n−1(p)+DL(ϕuf n−1(p))−1(ϕu f n−1(p))DRϕu f n−2(p)(ηu f n−2(p))D uηu f n−2(p)+···+DL(ϕuf2(p))−1(ϕu f2(p))DRϕu f(p)(ηu f(p))D uηu f(p)+DL(ϕuf(p))−1(ϕu f(p))DRϕu p(ηu p)D uηu pwhich shows the vanishing of the periodic orbit obstruction for(2.25).The rest of the argument does not need any modification from the argument in the previous case and we just refer to the previous section.We argue thatϕis differentiable in W s p for some p∈M and that the candidate for the drivative is indeed the true derivative.3.Cohomology equations on diffeomorphism groupsIn this section,we consider(1.1)whenηandϕtake values in the diffeomorphism group of a compact manifold.As before f is a transitive Anosov diffeomorphism.In order to emphasize that the parameter space and target space are the same smooth manifold N and to match the notation in[dW07]we use y in place of u.Theorem6.Letη∈C k+α M,Diffr(N) and p is a periodic point of f.Ifϕ∈C k+α M,Diff1(N) solvesϕf(x)=ηx◦ϕx(3.1) andϕp∈Diffr(N)thenϕ∈C k+α M,Diffr(N) .Notice that questions of differentiability is entirely local and hence the global structure of the diffemorphism group,which is quite complicated,does not enter into the argument. This should be compared with the existence argument in[dW07]for k=0which devotes considerable effort to defining the metric on Diffr(N).The local differential structure on Diffr(N)can be found in[Ban97].We remark that to show thatϕ:M→Diffr(N)is C k+αit suffices to show that all partial derivatives in N are C k+αuniformly. Proof.Taking derivatives of(1.1)with respect to the variable y in the manifold N we obtainD yϕf(x)(y)=D yηx◦ϕx(y)·D yϕx(y).This is possible since for all x we haveϕx∈Diff1(N).Now we writeψy x=D yϕx(y)and ˆηy x=D yηx◦ϕx(y).The equation is thenψyf(x)=ˆηy x·ψy xWe haveψy x:T y N→Tϕx(y)Nˆηy x:Tϕx(y)N→Tϕf(x)(y)N.Thus we have the set up of Theorem3withσ(y)=y andτ(x,y)=ϕy x.Clearlyσ∈C∞(N).If we assume thatϕ∈C k+α(M,Diffm(N))thenτ∈C k+α,m.By hypothesisη∈C k+α(M,Diffr(N))thus for m≤r−1ˆη∈C k+α,m.Applying Theorem3we obtain thatD yϕ∈C k+α,m.Thusϕ∈C k+α,m+1.We proceed by induction until m=r−1at which point we haveϕ∈C k+α,r.。