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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
中国科学院人事教育局制
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createshapecontextdistanceextractor 参数解释-回复"createshapecontextdistanceextractor 参数解释"Shape context is a powerful feature descriptor used in computer vision and image analysis tasks, particularly in shape matching and recognition. It provides a representation that captures the important geometric and topological properties of a shape. The ShapeContextDistanceExtractor is a function or tool used in OpenCV, an open-source computer vision library, to extract the shape context distance between two shapes. In this article, we will explore the parameters and explain their roles in the ShapeContextDistanceExtractor function.1. Histogram Bins:The parameter `nAngularBins` refers to the number of bins used to represent the angular information in the shape context. It divides the angular range into equal intervals to capture the distribution of edge points around the shape's reference point. Increasing the number of bins can lead to a higher level of detail in angular information. However, a larger number of bins increase computational complexity and may lead to overfitting for simpleshapes.2. Radial Range:The parameter `nRadialBins` determines the number of concentric rings used to represent the distances between the reference point and the edge points. It divides the radial range into equal intervals, capturing variations in scale. Higher values of`nRadialBins` enable the extraction of more detailed information about the shape's contour variations but increases computational complexity.3. Radial Width:The parameter `radialRadius` determines the width of each radial bin. It defines the distance between the concentric rings used for angular histogram computation. A smaller value for`radialRadius` increases the resolution of radial distances but may not capture the scale variations accurately. Conversely, a larger value can smoothen out variations in scale but may lead to information loss.4. Angle Clusters:The parameter `innerRadius` specifies the innermost radius forthe annulus (ring), where the shape context is calculated. It helps to discard irrelevant or noisy background information present near the shape's boundary. A smaller `innerRadius` ensures that only relevant shape information within a certain distance from the reference point is considered. Too small a value may exclude some important boundary details, while too large a value may include irrelevant information.5. Iterative Refinement:The parameter `iterations` denotes the number of iterations used to refine the shape context distance calculation. The iterative refinement process helps to progressively improve the distance estimation by adjusting the shape context representation to better align the shapes. Increasing the number of iterations allows for a more accurate shape matching but increases computational complexity.6. Density Corrections:The parameter `stdDev` controls the standard deviation of the Gaussian kernel used for the density correction. Density correction helps to equalize the impact of points in more densely populated regions compared to sparser regions. A lower `stdDev` results in amore focused correction, while a higher value provides a broader correction. It is essential to balance this parameter based on the density of points in the shape.In conclusion, the ShapeContextDistanceExtractor function in OpenCV is a powerful tool for shape matching and recognition tasks. By understanding and appropriately setting the parameters such as histogram bins, radial range, radial width, angle clusters, iterative refinement, and density corrections, users can achieve accurate and efficient shape analysis. Selecting optimal parameter values becomes crucial to ensure meaningful shape comparison and recognition results. Adjusting these parameters based on the characteristics of the shapes being analyzed and the desired level of detail is essential for obtaining satisfactory outcomes.。
Biophysical Chemistry 109(2004)105–1120301-4622/04/$-see front matter ᮊ2003Elsevier B.V .All rights reserved.doi:10.1016/j.bpc.2003.10.021Cloud-point temperature and liquid–liquid phase separation ofsupersaturated lysozyme solutionJie Lu *,Keith Carpenter ,Rui-Jiang Li ,Xiu-Juan Wang ,Chi-Bun Ching a ,a a b bInstitute of Chemical and Engineering Sciences,Ayer Rajah Crescent 28,࠻02-08,Singapore 139959,Singapore aChemical and Process Engineering Center,National University of Singapore,Singapore 117576,SingaporebReceived 31July 2003;received in revised form 8October 2003;accepted 16October 2003AbstractThe detailed understanding of the structure of biological macromolecules reveals their functions,and is thus important in the design of new medicines and for engineering molecules with improved properties for industrial applications.Although techniques used for protein crystallization have been progressing greatly,protein crystallization may still be considered an art rather than a science,and successful crystallization remains largely empirical and operator-dependent.In this work,a microcalorimetric technique has been utilized to investigate liquid–liquid phase separation through measuring cloud-point temperature T for supersaturated lysozyme solution.The effects of cloud ionic strength and glycerol on the cloud-point temperature are studied in detail.Over the entire range of salt concentrations studied,the cloud-point temperature increases monotonically with the concentration of sodium chloride.When glycerol is added as additive,the solubility of lysozyme is increased,whereas the cloud-point temperature is decreased.ᮊ2003Elsevier B.V .All rights reserved.Keywords:Biocrystallization;Microcalorimetry;Cloud-point temperature;Liquid–liquid phase separation1.IntroductionKnowledge of detailed protein structure is essen-tial for protein engineering and the design of pharmaceuticals.Production of high-quality pro-tein crystals is required for molecular structure determination by X-ray crystallography.Although considerable effort has been made in recent years,obtaining such crystals is still difficult in general,and predicting the solution conditions where pro-*Corresponding author.Tel.:q 65-6874-4218;fax:q 65-6873-4805.E-mail address:lujie@.sg (J.Lu ).teins successfully crystallize remains a significant obstacle in the advancement of structural molecu-lar biology w 1x .The parameters affecting protein crystallization are typically reagent concentration,pH,tempera-ture,additive,etc.A phase diagram can provide the method for quantifying the influence of solu-tion parameters on the production of crystals w 2,3x .To characterize protein crystallization,it is neces-sary to first obtain detailed information on protein solution phase behavior and phase diagram.Recently physics shows that there is a direct relationship between colloidal interaction energy106J.Lu et al./Biophysical Chemistry109(2004)105–112and phase diagram.Gast and Lekkerkerker w4,5x have indicated that the range of attraction between colloid particles has a significant effect on the qualitative features of phase diagram.A similar relationship should hold for biomacromolecules, i.e.the corresponding interaction potentials govern the macromolecular distribution in solution,the shape of the phase diagram and the crystallization process w6x.Many macromolecular crystallizations appear to be driven by the strength of the attractive interactions,and occur in,or close to,attractive regimes w7,8x.Recent intensive investigation has revealed that protein or colloidal solution possesses a peculiar phase diagram,i.e.liquid–liquid phase separation and sol–gel transition exists in general in addition to crystallization w9,10x.The potential responsible for the liquid–liquid phase separation is a rather short range,possibly van der Waals,attractive potential w11,12x.The measurement of cloud-point temperature T can provide useful informationcloudon the net attractive interaction between protein molecules,namely,the higher the cloud-point tem-perature,the greater the net attractive interaction. Herein Taratuta et al.w13x studied the effects of salts and pH on the cloud-point temperature of lysozyme.Broide et al.w14x subsequently meas-ured the cloud-point temperature and crystalliza-tion temperature for lysozyme as a function of salt type and concentration.From these works the cloud-point temperature was found to be typically 15–458C below the crystallization temperature. Furthermore,Muschol and Rosenberger w15x deter-mined the metastable coexistence curves for lyso-zyme through cloud-point measurements,and suggested a systematic approach to promote pro-tein crystallization.In general,an effective way to determine the strength of protein interactions is to study temperature-induced phase transitions that occur in concentrated protein solutions.Liquid–liquid phase separation can be divided into two stages w11x:(1)the local separation stage at which the separation proceeds in small regions and local equilibrium is achieved rapidly;and(2) the coarsening stage at which condensation of these small domains proceeds slowly to reduce the loss of interface free energy w16x.The coexisting liquid phases both remain supersaturated but differ widely in protein concentration.The effect of a metastable liquid–liquid phase separation on crystallization remains ambiguous w17x.Molecular dynamics simulations and analyt-ical theory predict that the phase separation will affect the kinetics and the mechanisms of protein crystal nucleation w18x.tenWolde and Frenkel w19x have demonstrated that the free energy barrier for crystal nucleation is remarkably reduced at the critical point of liquid–liquid phase separation, thus in general,after liquid–liquid phase separa-tion,crystallization occurs much more rapidly than in the initial solution,which is typically too rapid for the growth of single crystal with low defect densities w15x.The determination of the location of liquid–liquid phase separation curve is thus crucial for efficiently identifying the optimum solution conditions for growing protein crystals. Microcalorimetry has the potential to be a useful tool for determining:(1)the metastable-labile zone boundary;(2)the temperature-dependence of pro-tein solubility in a given solvent;and(3)the crystal-growth rates as a function of supersatura-tion w20x.Microcalorimeters can detect a power signal as low as a few microwatts whereas standard calorimeters detect signals in the milliwatt range. Because of this greater sensitivity,samples with small heat effects can be analyzed.In addition, microcalorimetry has the advantage of being fast, non-destructive to the protein and requiring a relatively small amount of material.The present work is concerned with the analysis of the transient heat signal from microcalorimeter to yield liquid–liquid phase separation information for lysozyme solutions at pH4.8.To further examine the role of salt and additive on interprotein interactions, cloud-point temperature T has been determinedcloudexperimentally as a function of the concentrations of salt,protein and glycerol.2.Materials and methods2.1.MaterialsSix times crystallized lysozyme was purchased from Seikagaku Kogyo,and used without further107J.Lu et al./Biophysical Chemistry 109(2004)105–112purification.All other chemicals used were of reagent grade,from Sigma Chemical Co.2.2.Preparation of solutionsSodium acetate buffer (0.1M )at pH 4.8was prepared with ultrafiltered,deionized water.Sodi-um azide,at a concentration of 0.05%(w y v ),was added to the buffer solution as an antimicrobial agent.Protein stock solution was prepared by dissolving protein powder into buffer.To remove undissolved particles,the solution was centrifuged in a Sigma centrifuge at 12000rev.y min for 5–10min,then filtered through 0.22-m m filters (Mil-lex-VV )into a clean sample vial and stored at 48C for further experiments.The concentration of protein solution was determined by measuring the absorbance at 280nm of UV spectroscopy (Shi-madzu UV-2550),with an extinction coefficient of 2.64ml y (mg cm )w 21x .Precipitant stock solution was prepared by dissolving the required amount of sodium chloride together with additive glycerol into buffer.The pH of solutions was measured by a digital pH meter (Mettler Toledo 320)and adjusted by the addition of small volumes of NaOH or HAc solution.2.3.Measurement of solubilitySolubility of lysozyme at various temperatures and precipitant y additive concentrations was meas-ured at pH 4.8in 0.1M acetate buffer.Solid–liquid equilibrium was approached through both crystallization and dissolution.Dissolving lasted 3days,while the period of crystallization was over 2weeks.The supernatant in equilibrium with a macroscopically observable solid was then filtered through 0.1-m m filters (Millex-VV ).The concen-tration of diluted supernatant was determined spec-troscopically and verified by refractive meter(Kruss)until refractive index remained unchanged ¨at equilibrium state.Solubility of each sample was measured in duplicate.2.4.Differential scanning microcalorimetry Calorimetric experiments were performed with a micro-differential scanning calorimeter with anultra sensitivity,micro-DSC III,from Setaram SA,France.The micro-DSC recorded heat flow in microwatts vs.temperature,thus can detect the heat associated with phase transition during a temperature scan.The sample made up of equal volumes of protein solution and precipitant solu-tion was filtered through 0.1-m m filters to remove dust particles further.To remove the dissolved air,the sample was placed under vacuum for 3min while stirring at 500rev.y min by a magnetic stirrer.The degassed sample was placed into the sample cell of 1.0ml,and a same concentration NaCl solution was placed into the reference cell.The solutions in the micro-DSC were then cooled at the rate of 0.28C y min.After every run,the cells were cleaned by sonicating for 10–15min in several solutions in the following order:deionized water,methanol,ethanol,acetone,1M KOH and finally copious amounts of deionized water.This protocol ensured that lysozyme was completely removed from the cells.The cells were then placed in a drying oven for several hours.The rubber gaskets were cleaned in a similar manner except acetone and 1M KOH were omitted and they were allowed to dry at low temperature.3.Results and discussionA typical micro-DSC scanning experiment is shown in Fig.1.The onset of the clouding phe-nomenon is very dramatic and easily detected.The sharp increase in the heat flow is indicative of a liquid–liquid phase separation process producing a latent heat.This is much consistent with many recent investigations of the liquid–liquid phase separation of lysozyme from solution w 22,23x .In fact,such a liquid–liquid phase separation is a phase transition with an associated latent heat of demixing.In this work,the cloud-point tempera-tures at a variety of lysozyme,NaCl and glycerol concentrations are determined by the micro-DSC at the scan rate of 128C y h.3.1.Effect of protein concentrationIn semilogarithmic Fig.2we plot the solid–liquid and liquid–liquid phase boundaries for lyso-108J.Lu et al./Biophysical Chemistry 109(2004)105–112Fig.1.Heat flow of a typical micro-DSC scan of lysozyme solution,50mg y ml,0.1M acetate buffer,pH 4.8,3%NaCl.The scan rate 128C y h is chosen referenced to the experimental results of Darcy and Wiencek w 23x .Note the large deflection in the curve at approximately 4.38C indicating a latent heat resulting from demixing (i.e.liquid–liquid phase separation )process.Fig.2.Cloud-point temperature and solubility determination for lysozyme in 0.1M acetate buffer,pH 4.8:solubility (5%NaCl )(s );T (5%NaCl,this work )(d );T (5%cloud cloud NaCl,the work of Darcy and Wiencek w 23x )(*);solubility (3%NaCl )(h );T (3%NaCl )(j ).cloud Fig.3.Cloud-point temperature determination for lysozyme as a function of the concentration of sodium chloride,50mg y ml,0.1M acetate buffer,pH 4.8.zyme in 0.1M acetate buffer,pH 4.8,for a range of protein concentrations.It is worth noting that,at 5%NaCl,our experimental data of T from cloud micro-DSC are quite consistent with those from laser light scattering and DSC by Darcy and Wiencek w 23x ,with difference averaging at approx-imately 0.88C.This figure demonstrates that liquid–liquid phase boundary is far below solid–liquid phase boundary,which implies that the liquid–liquid phase separation normally takes place in a highly metastable solution.In addition,cloud-point temperature T increases with the cloud concentration of protein.3.2.Effect of salt concentrationFig.3shows how cloud-point temperature changes as the concentration of NaCl is varied from 2.5to 7%(w y v ).The buffer is 0.1M acetate (pH 4.8);the protein concentration is fixed at 50mg y ml.Over the entire range of salt concentrations studied,the cloud-point temperature strongly depends on the ionic strength and increases monotonically with the concentration of NaCl.Crystallization is driven by the difference in chemical potential of the solute in solution and in the crystal.The driving force can be simplified as w 24xf sy Dm s kT ln C y C (1)Ž.eq109J.Lu et al./Biophysical Chemistry 109(2004)105–112Fig.4.The driving force required by liquid–liquid phase sep-aration as a function of the concentration of sodium chloride,50mg y ml lysozyme solution,0.1M acetate buffer,pH 4.8.In the same way,we plot the driving force,f ,required by liquid–liquid phase separation as a function of the concentration of sodium chloride in Fig.4.At the moderate concentration of sodium chloride,the driving force required by liquid–liquid phase separation is higher than that at low or high salt concentration.As shown in Fig.3,with NaCl concentration increasing,the cloud-point temperature increases,which is in accord with the results of Broide et al.w 14x and Grigsby et al.w 25x .It is known that protein interaction is the sum of different potentials like electrostatic,van der Waals,hydrophobic,hydration,etc.The liquid–liquid phase separation is driven by a net attraction between protein molecules,and the stronger the attraction,the higher the cloud-point temperature.Ionic strength is found to have an effect on the intermolecular forces:attractions increase with ionic strength,solubility decreases with ionic strength,resulting in the cloud-point temperature increases with ionic strength.It is worth noting that,the effect of ionic strength on cloud-point temperature depends strongly on the specific nature of the ions w 13x .Kosmotropic ions bind adjacent water molecules more strongly than water binds itself.When akosmotropic ion is introduced into water,the entro-py of the system decreases due to increased water structuring around the ion.In contrast,chaotropes bind adjacent water molecules less strongly than water binds itself.When a chaotrope is introduced into water,the entropy of the system increases because the water structuring around the ion is less than that in salt-free water.This classification is related to the size and charge of the ion.At high salt concentration ()0.3M ),the specific nature of the ions is much more important w 25x .The charges on a protein are due to discrete positively and negatively charged surface groups.In lysozyme,the average distance between thesecharges is approximately 10Aw 26x .As to the salt ˚NaCl used as precipitant,Na is weakly kosmo-q tropic and Cl is weakly chaotropic w 27x .At low y NaCl concentrations,as the concentration of NaCl increases,the repulsive electrostatic charge–charge interactions between protein molecules decrease because of screening,resulting in the increase of cloud-point temperature.While at high NaCl con-centrations,protein molecules experience an attrac-tion,in which differences can be attributed to repulsive hydration forces w 14,25x .That is,as the ionic strength increases,repulsive electrostatic or hydration forces decrease,protein molecules appear more and more attractive,leading to higher cloud-point temperature.At various salt concentra-tions,the predominant potentials reflecting the driving force for liquid–liquid phase separation are different.Fig.4shows that the driving force,f ,is parabolic with ionic strength,while Grigsby et al.w 25x have reported that f y kT is linear with ionic strength for monovalent salts.The possible reasons for that difference include,their model is based on a fixed protein concentration of 87mg y ml,which is higher than that used in our study,yet f y kT is probably dependent on protein concentration,besides the solutions at high protein and salt concentrations are far from ideal solutions.3.3.Effect of glycerolFig.5compares cloud-point temperature data for 50mg y ml lysozyme solutions in absence of glycerol and in presence of 5%glycerol,respec-110J.Lu et al./Biophysical Chemistry109(2004)105–112parison of cloud-point temperatures for lysozyme at different glycerol concentrations as a function of the con-centration of sodium chloride,50mg y ml,0.1M acetate buffer, pH4.8:0%glycerol(s);5%glycerol(j).Fig.6.Cloud-point temperatures for lysozyme at different glycerol concentrations,50mg y ml lysozyme,5%NaCl,0.1M acetate buffer,pH4.8.Fig.7.Cloud-point temperature and solubility determination for lysozyme at different concentrations of glycerol in0.1M acetate buffer,5%NaCl,pH4.8:solubility(0%glycerol)(s); T(0%glycerol)(d);solubility(5%glycerol)(h);cloudT(5%glycerol)(j).cloudtively.Fig.6shows the cloud-point temperature as a function of the concentration of glycerol.The cloud-point temperature is decreased as the addi-tion of glycerol.In semilogarithmic Fig.7we plot the solid–liquid and liquid–liquid phase boundaries at dif-ferent glycerol concentrations for lysozyme in0.1 M acetate buffer,5%NaCl,pH4.8,for a range of protein concentration.This figure demonstrates that liquid–liquid and solid–liquid phase bounda-ries in the presence of glycerol are below those in absence of glycerol,and the region for growing crystals is narrowed when glycerol is added. Glycerol has the property of stabilizing protein structure.As a result,if crystallization occurs over a long period of time,glycerol is a useful candidate to be part of the crystallization solvent and is often included for this purpose w28x.In addition,glycerol is found to have an effect on the intermolecular forces:repulsions increase with glycerol concentra-tion w29x.Our experiment results of solubility and cloud-point temperature can also confirm the finding.The increased repulsions induced by glycerol can be explained by a number of possible mecha-nisms,all of which require small changes in the protein or the solvent in its immediate vicinity.The addition of glycerol decreases the volume of protein core w30x,increases hydration and the size of hydration layer at the particle surface w31,32x. In this work,we confirm that glycerol shifts the solid–liquid and liquid–liquid phase boundaries. The effect of glycerol on the phase diagram strong-111 J.Lu et al./Biophysical Chemistry109(2004)105–112ly depends on its concentration and this canprovide opportunities for further tuning of nuclea-tion rates.4.ConclusionsGrowing evidence suggests protein crystalliza-tion can be understood in terms of an order ydisorder phase transition between weakly attractiveparticles.Control of these attractions is thus keyto growing crystals.The study of phase transitionsin concentrated protein solutions provides one witha simple means of assessing the effect of solutionconditions on the strength of protein interactions.The cloud-point temperature and solubility datapresented in this paper demonstrate that salt andglycerol have remarkable effects on phase transi-tions.The solid–liquid and liquid–liquid bounda-ries can be shifted to higher or lower temperaturesby varying ionic strength or adding additives.Ourinvestigation provides further information upon therole of glycerol used in protein crystallization.Glycerol can increase the solubility,and decreasethe cloud-point temperature,which is of benefit totuning nucleation and crystal growth.In continuingstudies,we will explore the effects of other kindsof additives like nonionic polymers on phasetransitions and nucleation rates.Much more theo-retical work will be done to fully interpret ourexperimental results.AcknowledgmentsThis work is supported by the grant from theNational Natural Science Foundation of China(No.20106010).The authors also thank Professor J.M.Wiencek(The University of Iowa)for kinddiscussion with us about the thermal phenomenaof liquid–liquid phase separation.Referencesw1x A.McPherson,Current approaches to macromolecular crystallization,Eur.J.Biochem.189(1990)1–23.w2x A.M.Kulkarni, C.F.Zukoski,Nanoparticle crystal nucleation:influence of solution conditions,Langmuir18(2002)3090–3099.w3x E.E.G.Saridakis,P.D.S.Stewart,L.F.Lloyd,et al., Phase diagram and dilution experiments in the crystal-lization of 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收稿日期：2002-03-15.作者简介：汤懿芬（1970-），女，硕士研究生；武汉，华中科技大学控制科学与工程系（430074）.基金项目：国家自然科学基金资助项目（60074033）.关于S en 个人主权不可能性定理的注记汤懿芬罗云峰（华中科技大学控制科学与工程系）摘要：研究了社会选择中的个人主权问题.在定义个人主权的形式化描述的基础上，探讨了个人主权与社会选择规则合理指标体系的相容性.通过分析个人主权与无关方案独立性条件、P areto 准则和正响应条件的关系，得到基于A rrow 不可能性定理、寡头独裁不可能性定理和否决者存在性定理的个人主权不可能性定理的变式.关键词：社会选择；个人主权；不可能性定理中图分类号：F 062.6文献标识码：A文章编号：1671-4512（2002）09-0001-02S en 个人主权不可能性定理是社会选择理论的三大不可能性结论之一，它分析了个人主权与Pareto 准则的相容性［1!3］.本文将在定义个人主权的形式化描述的基础上，探讨个人主权与社会选择规则合理指标体系的相容性，给出个人主权不可能性定理的变式.!符号与定义用N =｛1，2，…，7｝表示所有个体的集合，X =｛a ，b ，c ，…｝表示备选方案集，U 表示所有偏好断面的集合.所谓社会选择函数C 是定义在偏好断面集U 和可行的被选集集合V 上的函数关系［3，4］C ：U >V !V ，其中可行的被选集集合V 由非空的X 的子集组成.在本文的讨论中，用C （·）表示在给定偏好断面下的社会选择函数C ，并假设C （·）定义在无约束域上［3］.定义!理性选择函数C （·）［3］.在给定的偏好断面下，若存在备选方案集X 上的二元关系R ，"S #V ，有C （S ）=｛I #S "$#S ，IR $｝.（1）在式（1）中，若R 连通、自反和传递，则称C（·）为完全理性的选择函数；若R 连通、自反和拟传递，则称C （·）为拟传递理性的选择函数；若R 连通、自反和非循环传递，则称C （·）为非循环传递理性的选择函数."个人主权的形式化描述所谓个人主权（ri g ht ）是指个人自由选择的权利［2］.例如，个人在选举中的选举权就是个人主权，这种个人主权包含个人自由决定是否参加选举及自由决定选举中投谁的票等.在某些情形下，对备选方案集｛a ，b ｝的选择结果仅对个体a 的私人利益产生影响，以致个体a 对备选方案集｛a ，b ｝拥有选择权，即当对备选方案集｛a ，b ｝进行选择时，个体a 有权根据其偏好确定最后的选择结果［2］.如方案对（a ，b ）仅仅涉及到个体a 的私人利益.上述个人主权可以用以下形式描述（记为R 1）：设a #N ，a ，b #X ，若a 对方案a 和b 拥有R 1形式的个人主权，则：!.若aP a b ，有C （｛a ，b ｝）=｛a ｝；".若bP a a ，有C （｛a ，b ｝）=｛b ｝.#S en 个人主权不可能性定理的变式引理!［3，4］同时满足无关方案独立性条件、Pareto 准则的完全理性选择函数C （·）中存在独裁者.引理"［3，4］同时满足无关方案独立性条件、Pareto 准则的拟传递理性选择函数C（·）中存在第30卷第9期华中科技大学学报（自然科学版）V o l .30N o.92002年9月J .~uazhon g U n iv.o f S ci .&T ech.（N ature S cience e d ition ）S e p .2002寡头独裁.引理!［3，4］同时满足无关方案独立性条件、Pareto准则和正响应条件的非循环理性选择函数C（·）中存在否决者.设个体k（k G N）拥有Rl形式的个人主权，即】a，b G X，当aPkb时，C（｛a，b｝）=｛a｝；当bP k a时，C（｛a，b｝）=｛b｝.下面讨论个体k的个人主权与选择函数C（·）的合理性质的相容性.A.假设C（·）为完全理性的选择函数且满足无关方案独立性条件和Pareto准则.由引理l可知［3］：在社会选择函数C（·）中，存在独裁者，即】G N，对于任意的偏好断面和V I，$G X，使：a.若IP i$，则｛I｝=C（｛I，$｝）；b.若$P i I，则｛$｝=C（｛I，$｝）.假设k不是独裁者，即k羊，不失一般性，设bP a且aP k b.由于为独裁者，因此C（｛a，b｝）=｛b｝.（2）由于k对方案a和b拥有Rl形式的个人主权，因此，C（｛a，b｝）=｛a｝.（3）式（2）与式（3）矛盾，因此，假设不成立，k= .这说明：当k对备选方案集X中某一方案对（a，b）（a，b G X）拥有Rl形式的个人主权时，k 将对X中的任一方案对（I，$）（I，$G X）拥有R l形式的个人主权，即k为独裁者.B.假设C（·）为拟传递理性的选择函数且满足无关方案独立性条件和Pareto准则.由引理2可知［3］：在社会选择函数C（·）中，存在寡头独裁，即】G二N，对于任意的偏好断面和V I，$G X，使：a.若V i G G，IP i$，则｛I｝=C（｛I，$｝）；b.若】i G G，IP i$，则｛I｝=C（｛I，$｝）.假设k奄G，不失一般性，设V i G G，bPia且aPkb.由于k对方案a和b拥有Rl形式的个人主权，因此，C（｛a，b｝）=｛a｝，（4）由寡头独裁的定义可得C（｛a，b｝）=｛b｝.（5）式（4）与式（5）矛盾，因此，假设不成立，k G G.假设】G G且羊k，不失一般性，设aPkb 且bP a.由于k对方案a和b拥有Rl形式的个人主权，因此，C（｛a，b｝）=｛a｝，（6）由寡头独裁的定义可得b G C（｛a，b｝）.（7）式（6）与式（7）矛盾，假设不成立.所以，G=｛k｝.（8）式（8）说明：k不仅是寡头独裁者，而且还是独裁者.这意味着当社会（集体）中某一成员拥有Rl 形式的个人主权时，寡头独裁不可能性定理（即引理2）将转化为A rroW不可能性定理（引理l）.C.假设C（·）为非循环传递理性的选择函数且满足无关方案独立性条件、Pareto准则和正响应条件.由引理3可知［3］，在社会选择函数C（·）中，存在否决者k.这说明：当k对备选方案集X中某一方案对（a，b）（a，b G X）拥有Rl形式的个人主权时，k不仅对该方案对拥有Rl形式的个人主权，而且对备选方案集中的任一方案对（I，$）（V I，$G X）拥有否决权，即若IPk$，则I G C（｛I，$｝）.由以上的分析可得如下结论：定理"设k（k G N）拥有Rl形式的个人主权，若完全理性的选择函数C（·）满足无关方案独立性条件和Pareto准则，则k为独裁者.定理#设k（k G N）拥有Rl形式的个人主权，若拟传递理性的选择函数C（·）满足无关方案独立性条件和Pareto准则，则k为寡头独裁者.定理!设k（k G N）拥有Rl形式的个人主权，若非循环传递理性的选择函数C（·）满足无关方案独立性条件、Pareto准则和正响应条件，则k为否决者.定理l，定理2和定理3实际上就是引理l，引理2和引理3的个人主权不可能性定理变式.参考文献［l］S en A K.the i m p oss i b ilit y o f a P aretian li beral.Journal o f Po litical S tud ies，l970，30：307!3l7［2］C raven J.S ocial cho ice.C a m bri d g e：C a m bri d g e U n iver-s it y P ress，l994.［3］S en A K.S ocial cho ice t heor y：a re-exa m i nation.E-conom etrica，l977，45：53!89［4］S uzu m ura K.rational cho ice，co llective decis ion，and social W e lf are.C a m bri d g e：C a m bri d g e U n ivers it y P ress，l983.（下转第5页）2华中科技大学学报（自然科学版）第30卷Cho ice W e lf are ，1994，11：283!285［2］C raven l.S ocial cho ice.C a m bri d g e ：C a m bri d g e U n iver-s it y P ress ，1994.［3］罗云峰，肖人彬，岳超源.关于极大偏好断面约束的分析.华中理工大学学报，1998，26（8）：53!56［4］A rrow K l ，R a y nard H.S ocial cho ice and m ulti-criteriadecis ion-m aki n g .C a m bri d g e ：T he M i T P ress ，1987.S i n g led-p eaked and si n g led-trou g hed p referenceW an g Jin gy uLuO yun f en gAbstract ：T he conce p t o f si n g led-p eaked and si n g led-trou g hed constrai nt condition is descri bed.T he char-acteristics o f si n g le-p eaked and si n g le-trou g hed constrai nt condition are anal y zed.B y di vi di n g t he p ref erence se C uence i nto t w o sets ，t he characteristics o f si n g le-p eaked and si n g le-trou g hed p ref erences i n each are st ud-ied and t he f or m s o f si n g le-p eaked and si n g le-trou g hed p ref erence se C uence i n each set are g i ven.in so do-i n g ，t he f or m s o f all t he si n g le-p eaked and si n g le-trou g hed p ref erences are obtai ned ，f acilitati n g t he so lution o f t he p roble m o f t he m ax i m u m nu m ber o f ad m issi ble p ro files w it h si n g le-p eaked and si n g le-trou g hed p ref-erence.K e y words ：social cho ice ；si n g le-p eaked and si n g le-trou g hed ；ad m issi ble p ref erence p ro file W an g Ji n gy u M aster ；D e p t .o f C ontro l S ci .&En g .，H uazhon g U ni v .o f S ci .&T ech .，W uhan430074，Chi na .（上接第2页）A note of Sen li bertarian i m p ossi bilit y t heore mT an g y i f enLuO yun f en gAbstract ：T he i ndi vi dual li beral ri g ht i n social cho ice is st udied.O n t he basis o f t he f or m o f t he i ndi vi dual li beral ri g ht ，t he com p ati bilit y o f t he i ndi vi dual li beral ri g ht and t he ot her p ro p erties o f t he social cho ice rule are i nvesti g ated.T hrou g h an anal y sis o f t he relationshi p bet w een t he li beral ri g ht and t he conditions f or t he i nde p endence o f irrelevant alternati ves ，and t hat bet w een t he Pareto criterion and t he conditions f or p ositi ve res p onse ，t he variant o f t he li bertarian i m p ossi bilit y t heore m based on t he A rrow i m p ossi bilit y T heore m ，t he O li g archic i m p ossi bilit y T heore m and v etoer i m p ossi bilit y T heore mis g i ven.K e y words ：social cho ice ；i ndi vi dual li beral ri g ht ；i m p ossi bilit y t heore m Tan g Y ifen Post g raduate ；D e p t .o f C ontro l S ci .&En g .，H uazhon g U ni v .o f S ci .&T ech .，W uhan430074，Chi na .5第9期王镜宇等：关于单峰单凹混合约束的特性分析。
2012-2013 Academic Year, Fall SemesterFinal Exam of Physical Chemistry (2) — Paper APart I Choose the best answers (3 points for each problem, totally 30 points)[1] Which of the following electrolytes obeys the Debye-Hückel limitinglaw in a widest concentration range?(A) MgCl2; (B) MgSO4; (C) NaCl; (D)AlCl3[2] For As2S3sol, if the precipitation value of SnCl2is x mol/L, theprecipitation of MgSO4 is(A) Equal to x mol/L (B) Less than x mol/L;(C) Larger than x mol/L; (D) Uncertain;[3] For an overall reaction A + B →P, its rate law is experimentallydetermined to be r = k[A][B], what conclusion can you reach?(A) This is a simple reaction; (B) This is a bimolecular reaction;(C) The unit of rate constant k is s-1; (D) uncertain.[4] During formation of HBr, Br⋅and H⋅usually exist as activeintermediates. For the following reaction, which is the most possible chainterminating step?(A) Br⋅+ H⋅→ HBr; (B) Br⋅+Br⋅→Br2;(C) H⋅+H⋅→ H2; (D) H⋅ + HBr → H2 + Br⋅.[5] Which of the following factors does not affect surface tension of aliquid?(A) Temperature; (B) External pressure over pressure;(C) Contacting phase; (D) Surface area of the liquid.[6] For a firs-order reaction, the unit of its rate constant is:(A) s-1; (B)dm3 mol-1 s-1; (C) mol dm-3 s-1; (D) dm-6 mol-2 s-1;[7] Activation energy can be of different nature. The activation energy defined by simple collision theory is(A) Potential energy; (B) vibrational energy;(C) Transitional energy; (D) photo energy.[8] The electrode potential of which electrode is the highest?(A) Ag(s)⎪Ag2O(s)⎪OH-(m); (B) Ag(s)⎪Ag+ (m);(C) Ag(s)⎪AgCl(s)⎪Cl-(m); (D) Ag(s)⎪[Ag(NH3)2]+ (m)[9] The quantum efficiency of photochemical formation of HCl is 106 times higher than that of HBr. What cause this great difference?(A) The primary photochemical process of HCl formation takes place much easier;(B) Unreactive decay in HBr formation is rather serious;(C) For HCl formation, the secondary photochemical process is chain reaction, but for HBr formation, it is not;(D) The life of Br⋅ is much longer.[10] Which of the following parameters of a colloidal particle can not be determined by ultramicroscopy?(A) The shape; (B) The molar mass;(C) The size; (D) The electrokinetic potential.Part II Fill in the blanks (totally 22 points)[1] (3 points) At 298 K, the solubility of AgCl in water is 1.27⨯10-5 mol⋅kg-1. Its solubility in 0.010 mol⋅kg-1 KNO3 solution is ______ mol⋅kg-1.[2] (3 points) The cell notation of a reversible cell that can be used to measure the decomposition equilibrium constant of Ag 2O is __________.[3] (2 points) List at least two ways to reduce the polarization of an electrode: _________________; _____________________.[4] (3 points) Sulfur, a gel which is always negatively charged due to adsorption of HS -, can be prepared by purging SO 2 into H 2S solution. Write the diagram of its colloidal structure _____________________________.[5] (3 points) During formation of HCl, the newly formed HCl usually possess higher energy. If a light quantum passes through the reaction system, a laser radiation with wavelength of 760 nm can be emitted. The difference between the activated HCl and the HCl molecule at ground state is ____________kJ ⋅mol -1.[6] (3 points) If 0.100 m NaCl solution is added into the CuSO 4 solution, the electrode potential of Cu(s)⎢CuSO 4(0.010 m) will _______ (increase or decrease).[7] (2 points) Heating can destroy the stability of sol. Give at least two reasons to explain this fact. _____________; _________________.[8] (3 points) Electrochemical reduction is commonly adopted for organic synthesis in aqueous solution. Why Pb or Pb alloy is usually used as the cathode material ______________.Part III Answer the following questions (totally 28 points)[1] (12 points) For some reactions, their temperature-dependence of reaction rate can be expressed as:⎪⎭⎫ ⎝⎛=RT E AT k c m -exp (Eq. 1) where E c is a threshold energy independent of temperature. The empirical equation proposed by vant ’ Hoff in the middle 19 century isC TB dT k d +=2ln (Eq. 2) (1) Please relate the empirical constant in Eq. 2, i.e., B andC , to the physical parameters in Eq. 1;(2) Why can Arrhenius neglect C in his empirical equation?(3) For deduction the relation in (1), we usually assume that A is independent of T . Is this assumption true? Which theory can explain this dependence? At elevated temperature, does A increase or decrease?[2] (16 points) At 298 K, reactionAg (s) + 21Hg 2Cl 2(s) −→ AgCl(s) + Hg(l) ∆r H m = 7950 J ⋅mol -1. The standard entropies of these substances are listed(1) Write the notation of the cell in which this reaction takes place;(2) Write the electrode reactions;(3) Calculate the electromotive force of this reversible cell at 298 K;(4) Determine the temperature coefficient of electromotive force.Part IV calculation (totally 20 points)[1] (10 points) A glass capillary of 250 mm long with different diameters (R 1, R 2) as shown in the right figure is plugged vertically in water. This glass capillary can be completely wettedby water. The surface tension anddensity of water at 298 K is 0.07275N/m and 1.00 g/ml respectively. If watercan rise 200 mm high in the capillary,(1) Determine the largest diameter ofthe thick part (R 1);(2) Determine the diameter of the thinpart (R 2).(3) If we turn the capillary upsidedown, how high will water to rise?[2] (10 points) The bromination of acetone is acid catalyzed:-+++−→−++Br H Br COCH CH Br COCH CH 23H 233The rate of disappearance of bromine was measured at different(1) Determine the rate law for the reaction.(2) Determine the rate constant.(3) The following mechanism has been proposed for this reaction:A BB CC DShow whether or not this mechanism fit the rate law obtained in (1).(4) If E a, 1, E a, -1, E a, 2, E a, 3 stands for the activation energy of corresponding reaction, express the apparent activation energy (E a ) of the overall reaction by activation energy of the separate steps involved.2012-2013 Academic Year, Fall SemesterFinal Exam of Physical Chemistry (2) — Paper AANSWER SHEETPart I Choose the best answers (3 points for each problem, totally 30Part II Fill in the blanks (totally 22 points)[1] ________________ mol⋅kg-1.[2] __________________________________________________.[3] ___________________________; __________________________.[4] _________________________________________________.[5] ________________________kJ⋅mol-1.[6] _________________________________.[7] __________________________; ___________________________.[8] ______________________________________________________.。
For an exploration of the molecular dimensions of IDPs, coupled with thorough control of their monomeric state, the combination of dynamic light scattering (DLS) and static light scattering (SLS) is a method of choice.IDPs ：intrinsically disordered proteins静态光散射：如果将溶液中的聚合物分子看作一个个各向同性的粒子,以一定频率的入射光照射这些粒子,它们不吸收入射光能量,而仅作为二次波源向各个方向发射与入射光频率相同的球面散射光,这种没有频率位移(即无能量变化)的散射称为弹性光散射( elastic light scattering) ,也常称为经典光散射( classical light scattering)、静态光散射( static light scattering) 、time-averaged light scattering在静态光散射情况下,通常用瑞利比来表征一个体系的散射能力和角度依赖性。
当qRg<<1A perceptible angular dependence of the scattering intensity （可见的光强的角度依赖）can only be expected for particles with R G> 10 nm. Thus, light scattering is not an appropriate method for monitoring changes of R G during unfolding and refolding of small monomeric proteins. However a substantial angular dependence is observed for large protein aggregates.The optical constant K=4π2n02(dn/dc)2/λ4N A depends only on experimental parameters and the scattering properties of the molecules in the particular solvent, which is reflected by dn/dc.The exact knowledge of dn/dc, the dependence of n on protein concentration in the present case, is very important for absolute measurements of the molecular mass.在一定温度下，测得不同角度时标准物、溶剂和溶液的散射光强I0（θ）、Is（θ）、I（θ），可得到相应得R（θ），实测或从高分子手册上查知n和dn/dC的数值，于是K成为已知的常数。
Package‘interpretR’August20,2023Type PackageTitle Binary Classifier and Regression Model Interpretation FunctionsVersion0.2.5Date2023-08-19Depends randomForestImports AUC,stats,graphicsAuthor Michel Ballings and Dirk Van den PoelMaintainer Michel Ballings<*************************>Description Compute permutation-based performance measures and create partialdependence plots for(cross-validated)'randomForest'and'ada'models.License GPL(>=2)RoxygenNote7.2.3NeedsCompilation noRepository CRANDate/Publication2023-08-1922:22:31UTCR topics documented:interpretR-package (2)interpretRNews (2)parDepPlot (3)variableImportance (5)Index712interpretRNews interpretR-package Partial Dependence Plots and Permutation-Based Performance Mea-suresDescriptionCompute permutation-based performance meaures(for binary classification)and create partial de-pendence plots(cross-validated classification and regression models).Currently only binary clas-sification and regression models estimated with the package randomForest are supported.Binary classification models estimated with ada are also supported.Author(s)Authors:Michel Ballings,and Dirk Van den Poel,Maintainer:<*************************>See AlsoparDepPlot,variableImportanceinterpretRNews Display the NEWSfileDescriptioninterpretRNews shows the NEWSfile of the interpretR package.UsageinterpretRNews()Author(s)Authors:Michel Ballings and Dirk Van den Poel,Maintainer:<*************************>See AlsoparDepPlotExamplesinterpretRNews()parDepPlot Model interpretation functions:Partial Dependence PlotsDescriptionparDepPlot creates partial dependence plots for binary(cross-validated)classification models andregression models.Currently only binary classification models estimated with the packages randomForest and ada are supported.In addition randomForest regression models are supported.UsageparDepPlot(,object,data,rm.outliers=TRUE,fact=1.5,n.pt=50,robust=FALSE,ci=FALSE,u.quant=0.75,l.quant=0.25,xlab=substr(,1,50),ylab=NULL,main=if(any(class(object)%in%c("randomForest","ada")))paste("Partial Dependence on",substr(,1,20))elsepaste("Cross-Validated Partial Dependence on",substr(,1,10)),logit=TRUE,ylim=NULL,...)Arguments the name of the predictor as a character string for which a partial dependenceplot has to be created.object can be a model or a list of cross-validated models.Currently only binary classi-fication models built using the packages randomForest and ada are supported.data a data frame containing the predictors for the model or a list of data frames forcross-validation with length equal to the number of models.rm.outliers boolean,remove the outliers in .Outliers are values that are smaller thanmax(Q1-fact*IQR,min)or greater than min(Q3+fact*IQR,max).Overridden ifxlim is used.fact factor to use in rm.outliers.The default is1.5.n.pt if is a continuous predictor,the number of points that will be used to plotthe curve.robust if TRUE then the median is used to plot the central tendency(recommended when logit=FALSE).If FALSE the mean is used.ci boolean.Should a confidence interval based on quantiles be plotted?This only works if robust=TRUE.u.quant Upper quantile for ci.This only works if ci=TRUE and robust=TRUE.l.quant Lower quantile for ci.This only works if ci=TRUE and robust=TRUE.xlab label for the x-axis.Is determined automatically if NULL.ylab label for the y-axis.main main title for the plot.logit boolean.Should the y-axis be on a logit scale or not?If FALSE,it is recom-mended to set robust=TRUE.Only applicable for classifcation.ylim The y limits of the plot...other graphical parameters for plot.DetailsFor classification,the response variable in the model is always assumed to take on the values{0,1}.Resulting partial dependence plots always refer to class1.Whenever strange results are obtained the user has three options.First set rm.outliers=TRUE.Second,if that doesn’t help,set robust=TRUE.Finally,if that doesn’t help,the user can also try setting ci=TRUE.Areas with larger confidence intervals typically indicate problem areas.These options help the user tease out the root of strange results and converge to better parameter values.Author(s)Authors:Michel Ballings,and Dirk Van den Poel,Maintainer:<*************************> ReferencesThe code in this function uses part of the code from the partialPlot function in randomForest.It is expanded and generalized to support cross-validation and other packages.See AlsovariableImportanceExampleslibrary(randomForest)#Prepare datadata(iris)iris<-iris[1:100,]iris$Species<-as.factor(ifelse(factor(iris$Species)=="setosa",0,1))#Cross-validated models#Estimate10models and create10test setsdata<-list()rf<-list()for(i in1:10){ind<-sample(nrow(iris),50)rf[[i]]<-randomForest(Species~.,iris[ind,])data[[i]]<-iris[-ind,]}parDepPlot(="Petal.Width",object=rf,data=data)#Single model#Estimate a single modelind<-sample(nrow(iris),50)rf<-randomForest(Species~.,iris[ind,])parDepPlot(="Petal.Width",object=rf,data=iris[-ind,])variableImportance Permutation-based Variable Importance MeasuresDescriptionvariableImportance produces permutation-based variable importance measures(currently onlyfor binary classification models from the package randomForest and only for the performancemeasure AUROC)UsagevariableImportance(object=NULL,xdata=NULL,ydata=NULL,CV=3,measure="AUROC",sort=TRUE)Argumentsobject A model.Currently only binary classification models from the package randomForest.xdata A data frame containing the predictors for the model.ydata A factor containing the response variable.CV Cross-validation.How many times should the data be permuted and the decreasein performance be calculated?Afterwards the mean is taken.CV should behigher for very small samples to ensure stability.measure Currently only Area Under the Receiver Operating Characteristic Curve(AU-ROC)is supported.sort Logical.Should the results be sorted from high to low?DetailsCurrently only binary classification models from randomForest are supported.Also,currently only AUROC is supported.Definition of MeanDecreaseAUROC:for the entire ensemble the AUROC is recorded on the provided xdata.The same is subsequently done after permuting each variable (iteratively,for each variable separately).Then the latter is subtracted from the former.This is called the Decrease in AUROC.If we do this for multiple CV,it becomes the Mean Decrease in AUROC.ValueA data frame containing the variable names and the mean decrease in AUROCAuthor(s)Authors:Michel Ballings,and Dirk Van den Poel,Maintainer:<*************************> See AlsoparDepPlotExamples#Prepare datadata(iris)iris<-iris[1:100,]iris$Species<-as.factor(ifelse(factor(iris$Species)=="setosa",0,1))#Estimate modellibrary(randomForest)ind<-sample(nrow(iris),50)rf<-randomForest(Species~.,iris[ind,])#Obtain variable importancesvariableImportance(object=rf,xdata=iris[-ind,names(iris)!="Species"],ydata=iris[-ind,]$Species)IndexinterpretR(interpretR-package),2 interpretR-package,2interpretRNews,2parDepPlot,2,3,6variableImportance,2,4,57。
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.。
