Asymptotic Expansion of the Risk Difference of the Bayesian Spectral Density in the ARMA mo
2017年8月19日雅思阅读考试回忆及解析
2017年8月19日雅思阅读考试回忆及解析The August 19th, 2017 IELTS reading exam was a challenging yet stimulating experience for many test-takers. The exam covered a wide range of topics, from environmental issues to historical events, showcasing the diversity and breadth of knowledge required for success in the test. One particular passage that stood out to many was the discussion on climate change and its impact on global ecosystems. This topic not only tested the candidates' reading comprehension skills but also required them to critically analyze and evaluate the information presented.The passage on climate change delved into the various factors contributing to this pressing issue, such as greenhouse gas emissions and deforestation. It highlighted the urgent need for global cooperation and concerted efforts to mitigate the effects of climate change before irreversible damage is done to the planet. Many test-takers found this passage thought-provoking and relevant, as climate change continues to be a major concern in today's world. The questionsthat followed this passage were designed to assess the candidates' ability to identify key information, make inferences, and draw conclusions based on the text.Another challenging aspect of the reading exam was the passage on a historical event, such as the Industrial Revolution. This passage required candidates to have a good grasp of historical events and their impact on society and the environment. Test-takers had to navigate through complex language and unfamiliar terms to extract the necessary information to answer the questions effectively. This passage tested not only their reading skills but also their ability to contextualize historical events and understand their significance in a broader context.In addition to the content-related challenges, the reading exam also tested the candidates' time management skills and ability to stay focused under pressure. With a limited amount of time to complete the exam, test-takers had to prioritize their reading and question-answering strategies to maximize their chances of success. This aspect of the exam added an extra layer of difficulty, as candidateshad to balance speed and accuracy to ensure they completed all sections within the allotted time.Overall, the August 19th, 2017 IELTS reading exam was a comprehensive test of the candidates' reading comprehension skills, critical thinking abilities, and knowledge of a wide range of topics. It challenged test-takers to think analytically, draw connections between different pieces of information, and demonstrate their understanding of complex ideas. While the exam was undoubtedly demanding, it also provided an opportunity for candidates to showcase their academic prowess and ability to excel in a high-pressure testing environment.。
principles of risk analysis
principles of risk analysisPrinciples of Risk AnalysisRisk analysis is a crucial process that helps organizations and individuals identify, assess, and mitigate potential threats to their objectives, assets, and well-being. It involves a systematic approach to understanding the nature of risks, their likelihood of occurrence, and the potential consequences they may entail. Here are the fundamental principles that underlie effective risk analysis: Identification of Risks: The first step in risk analysis is to identify all possible risks that could affect the organization or individual. This involves identifying internal and external factors that could lead to negative outcomes.Quantification of Risks: Quantifying risks involves estimating the probability of their occurrence and the potential impact they could have. This helps in prioritizing risks and allocating resources for their management.Evaluation of Risks: Evaluation involves assessing the significance of each risk based on its probability and impact. This helps in identifying the most critical risks that require immediate attention.Risk Treatment: Once risks are evaluated, it is necessary to develop strategies to treat or manage them. This may include risk avoidance, risk reduction, risk transfer, or risk acceptance based on the organization's risk tolerance and objectives.Monitoring and Review: Risks are dynamic and constantly evolving. It is, therefore, essential to monitor and review the risk management plan regularly to ensure its effectiveness. This involves tracking changes in risk profiles and updating the risk management plan accordingly.Transparency and Communication: Effective risk analysis requires transparent communication among all stakeholders. This ensures that everyone is aware of the risks facing the organization and the strategies being implemented to manage them.Compliance with Standards and Regulations: Risk analysis must comply withrelevant standards, guidelines, and regulations to ensure its legitimacy and credibility. This helps in building trust with external stakeholders and maintaining a positive reputation.In conclusion, principles of risk analysis provide a framework for organizations and individuals to manage risks effectively. By adhering to these principles, they can identify, assess, and mitigate risks, protecting their assets, objectives, and well-being.。
风险沟通模型
Developing A Risk Communication Model to Encourage Community Safetyfrom Natural HazardsJ UNE 2004Peter O’NeillThe author acknowledges the contributions of Joan Young and Les Robinson in the development of this model.Com Safety Program 4 04.doc 27/04/04- 1 -“T HE PURPOSE OF (RISK) COMMUNICATION IS TO ASSIST PEOPLE TO OBTAIN THE INFORMATION THEY NEED TO MAKE INFORMED CHOICES ABOUT THE POSSIBLE RISK THEY FACE.”(Wade, C R, Molony, S T, Durbin, M E, Klein S H, and Wahl L E, (1992), P1)“H UMAN BEINGS DO NOT HAVE THE TIME OR THE ABILITY TO BE CONCERNED ABOUT EVERY PROBLEM IN THE WORLD. T HEY DEVOTE THEIR TIME AND ENERGY TO PROBLEMS THAT INVOLVE THEM AND FOR WHICH THEY CAN MAKE A DIFFERENCE.”J E Grunig quoted in Leffler (1998)Com Safety Program 4 04.doc 27/04/04- 2 -TABLE OF CONTENTS1) INTRODUCTION (4)2) TRADITIONAL APPROACHES TO COMMUNITY SAFETY (5)3) FINDING BETTER RISK COMMUNICATION APPROACHES FOR COMMUNITY SAFETY (8)4) FACTORS THAT INFLUENCE COMMUNITY SAFETY (10)A. T HE NATURE OF THE HAZARD AND ASSOCIATED RISK (10)B. R ISK P ERCEPTION (11)C. S TAGES OF R ISK C OMMUNICATION (14)D. I DENTIFYING AUDIENCES AND ASSOCIATED MESSAGES (15)i. Demographic factors (16)ii. Psychological traits (17)iii. Experience of the hazard (20)E) R ESILIENCE (24)5) AN INTEGRATED COMMUNICATION FRAMEWORK (26)6) FOSTERING BEHAVIOURAL CHANGE (32)7) SUMMARY: THE PROPOSED MODEL (38)8) DETAILS OF THE INTEGRATED PROGRAM (40)A) C OMMUNITY DEVELOPMENT (41)B) C OMMUNITY EDUCATION (41)C) S OCIAL MARKETING AND PUBLIC AWARENESS - ONE-WAY PERSUASION (42)D) E DUCATION ABOUT MANDATORY DIRECTIONS (EMERGENCY WARNINGS) (43)9) CONCLUSION (43)REFERENCES (45)Com Safety Program 4 04.doc 27/04/04- 3 -1) I NTRODUCTIONThis discussion paper will outline issues relating to developing a risk communication model in the context of a severe but infrequent hazard such as a significant flood or storm. It will also investigate the concept of risk perception and the elements that contribute to an integrated community safety campaign. The paper will review traditional approaches to community education used by emergency agencies. It will suggest a need for a more integrated risk communication model that acknowledges community perceptions about the risks they face, and while encouraging self-reliance acknowledges the limitations of this approach. It will then present a coherent conceptual framework for communicating and involving the public, focusing on adopting protective behaviour for the pre-disaster phase. Hopefully, this paper will generate vigorous debate over future directions for community safety within the SES and lead to the development of rigorous and effective safety programs for flood and storm education.Emergency managers are in the midst of historic changes. The focus of expectations has changed dramatically, from a pure emergency response to a proactive 'risk management' approach involving disaster mitigation, prevention, and risk communication (Keys 1999a, Buckle 1998, Granger 1999).These shifts involve:- a whole-of-government approach that sees community safety as a total system;- locally focused and integrated planning;- the need for greater community participation;- community-centric, rather than agency-centric approaches;- risk management and multi-disciplinary approaches;- improved use of technology;- the need for greater cost effectiveness and public accountability;- the need to form and enhance partnerships and to reduce organisations’isolation;- the need for sophisticated skills in risk management and communication(EMA 1999a, Hodges 1999).A changing publicAt the same time as expectations of emergency services are changing, so is too the nature of the public changing:- the changing nature of 'community', from communities-of-place to dispersed communities-of-interest;- the demand for greater community participation (EMA 1999a);Com Safety Program 4 04.doc 27/04/04- 4 -- increasingly low tolerance of risk and increasing expectation of emergency services;- a declining level of trust in government and authorities;- a community that is shifting its concerns from the public to the private and personal (Quantum Market Research 2002);- an increasingly complex and competitive communication environment;- an increasing urbanisation and an increase in communities of older people living along the coast (Salt 2003); and- a community that is sophisticated in reading and interpreting communications. These factors reinforce the need for innovation, rigorous planning and an evidence-based culture in the design of community safety programs.However, while there has been extensive education resources developed in Australia, there has been little research to substantiate a link to an appropriate risk communication model: one that explains the relationship between vulnerable communities and their willingness to become involved with community safety programs (Boura, 1998).2) Traditional approaches to community safetyTraditional education approaches, often called public awareness programs, are increasingly being questioned. In the flood and storm safety arena, there remains a lack of clarity about what approaches are appropriate in different situations. As Keys (1999b) noted, "It has been apparent for some time that creating community awareness of floods and storms is not easy, and that our various pamphlets and guides do not 'move' in large numbers. Most of the time, people are not particularly interested in them.”From a national multi-hazard perspective, the outlook is just as bleak."…there is currently no nationally accepted theory which provides the basis for determining 'good practice' and programs and activities have been developed from a basis of intuition, past experience or adoption and adaption of activities from other areas…." (AMEC 2002 p7).Historically, when emergency services have undertaken community education, they have informed the community about hazards and their risks, through distribution of prepared material emphasising actions residents can undertake to protect themselves and their property during emergencies. The communication process was often one-off and one-way, and assumed that the audience was an indistinguishable group of individuals who had the same needs and values.The effectiveness of this traditional approach and the extent to which individuals implemented safety messages was often measured by the number of resources distributed, or the public recalling the message. The indicators used to determine a Com Safety Program 4 04.doc 27/04/04- 5 -successful campaign focussed on the ability of the individual to demonstrate an awareness of the safety messages presented (eg. Mountford and Davidson 1999, storm safety evaluation).This traditional model is one where the emergency professional is the ‘active agent’and the community member is the passive recipient of appropriate messages (Macdonald, 1998). The deficits of this model are at last being recognised and research has questioned the effectiveness of these education strategies in changing people’s behaviour. "One of the most puzzling findings … was that many people did not implement strategies that would improve their safety, despite understanding the issues associated with safety and acknowledging that safety was their own responsibility" (Esmund et al. 2000, p5).Implicit in this traditional approach was the assumption that there was a direct correlation between awareness raising and behavioural change. "It is frequently assumed that providing the public with information on hazards and their mitigation will encourage preparation. This assumption is unfounded." (Paton et al., undated). This failing of the traditional Information-Action model is the belief that merely informing the individual or community about a hazard, will lead to risk awareness and awareness to actions, and then to sustained behavioural change. Boura (1998) identifies the weakness in the belief that there is a strong and direct causal link between receiving information and appropriate actions.The literature on risk communication indicates that distribution of information on the hazard and associated risk will not by itself make a significant difference in attitude, perception or behaviour (Boura, 1998). Keys (1996, p3) noted, “…public awareness strategies have had a low profile in the emergency management field. Their potential as tools for reducing the costs which floods impose has also been little developed, despite the fact that a flood-aware community is recognised in the floodplain management literature as being important in this regard.”Health Promotion and Injury Prevention CampaignsAlthough emergency education programs are customarily under-funded, additional resources alone will not improve residents’ ability to prepare and cope with a major disaster. It is instructive to review other behavioural change campaigns that have received greater funding and have a more empirically rigorous model to support their strategies. In spite of health promotion campaigns making some advancement in lessening dangerous behaviours these measures are still the subject of constant revisions, as more effective strategies are discovered.Researchers are still investigating why so many people continue to maintain unhealthy and potentially dangerous lifestyles. Even well funded programs in related areas, such as road safety campaigns, have been criticised for a lack of success. According to the 2002 Safety Strategy Report, “It is likely that millions of dollars have been wasted each year on road safety advertising in Victoria since 1989” (Sinclair, 2003).Com Safety Program 4 04.doc 27/04/04- 6 -The 1997/98 NSW Sun Protection Campaign (the Seymour Snowman campaign) is an example of this in a related health area (NSW Cancer Council, 1998). The campaign focussed on encouraging appropriate sun protection behaviour through a social marketing campaign and information distribution. The campaign used radio, TV, billboards, posters and leaflets to give positive information about appropriate sun protection to children and their carers. The campaign generated significant increase in awareness of the main character (Seymour Snowman) used in the campaign, with 73% recalling they had seen the commercial on TV. However, over the same time there was only a 3% increase in children engaging in the desired sun protection behaviour, while adults reported a 2% fall in appropriate sun protection behaviour.The success and possible new directions for the smoking cessation campaigns were recently reviewed in the Sydney Morning Herald. The campaigns use a system-based intervention approach consisting of macro and micro projects such as pricing increases, mass-media projects, restricting cigarettes availability, restricting advertising, targeting vulnerable groups, bans in public areas and work places, commercial cessation programs, pack warnings, the Quitline and GP guidelines.The multiple players in these campaigns, such as state and federal health departments, commercial companies and various independent agencies (eg. Cancer Councils) make it difficult to evaluate individual programs. However, as these various programs employ common strategies, their success is ultimately measured by the reduction in smoking and the number of new smokers. While the smoking cessation campaigns had been successful in lowering the rate of smoking, in the last few years the cessation rate has stalled at just below 20 per cent. Professor Simon Chapman of the Cancer Council, believes that while mass-media campaigns can be effective in reducing smoking rates, there will always be smokers and that the messages won’t reach everybody. To further reduce the smoking rate within 10 years in NSW, Dr Penman, CEO of the Cancer Council believes a $15 million a year anti-smoking campaign could significantly decrease the smoking rate within 10 years (SMH, 16 Oct, 2003).Clearly current disaster education programs are not as sophisticated or as well resourced as health promotion and injury prevention campaigns. Yet these campaigns are now struggling to have a significant impact on their target audiences. What approach should emergency managers take in encouraging safety preparation for disasters?Com Safety Program 4 04.doc 27/04/04- 7 -3) F INDING BETTER RISK COMMUNICATION APPROACHES FOR COMMUNITY SAFETYIt is apparent that new approaches are needed to create desirable behavioural changes. The focus in the current methodology on individual behavioural change through conveying information needs to be broadened. The Institute of Medicine (2002) has identified three major determinates of intention to undertake behavioural change. They are:• Attitudes of a person;• community norms; and• the degree of self-efficacy of a person.Macdonald (1998) also includes the social setting in which people make decisions about their risks.In response to these challenges, alternative approaches are emerging. Fortunately, rather than re-inventing the wheel, we are able to learn and adapt approaches and models which have been proven in other jurisdictions -- notably health promotion, social marketing, community safety and adult education -- to present a community safety model that identifies and addresses the concerns of all affected groups. The report of a national flood warning workshop (Proudley and Handmer, 2003) identified many of the issues that need to be addressed in developing effective warning systems, including:• The necessity for community engagement through increased awareness and engagement;• The need to improve the communication of risk;• The importance of recognising the target audience of flood warnings; and• The need for policy improvements in the area of flash flood warnings.Other approaches that have proved useful in improving health and safety outcomes include:•Comprehensive systems-based intervention. This approach recognises community behaviour is the outcomes of interaction between legislation, organisational policy and practice, social networks, engineering solutions, and community norms. System-based intervention approaches have been widely applied in health promotion, notably in community safety and injury prevention work (Lindquist et al 2002, Cohen and Swift 2003, Jensen 1999, Esmund et al 2000). An example of this is the smoking cessation campaign. The programs are supported by federal and state governments and by community health groups. Strategies include legislation to ban smoking in workplaces, individual Quit packs, powerful advertisements to alert smokers to the danger of smoking and measures to protect non-smokers (SMH, 16 Oct, 2003).Com Safety Program 4 04.doc 27/04/04- 8 -•Greater use of "bottom-up" (participative) strategies. These focus on empowering and resourcing local groups and networks, to identify problems, define solutions and initiate action plans. Examples in the emergency management field include: Community Fire Guard (Vic CFA), Community Fire Units (NSW FB), AWARE (WA FESA) and the American Red Cross's Disaster Resistant Neighbourhood program.•Greater use of social marketing methods. Mass persuasion methods originally developed in the commercial marketing field are now widely used to foster positive behaviours. These are being applied to improve community resilience to natural hazards, e.g. FloodSafe (NSW SES). The National Flood Warning Centre (UK) ran a social marketing and health promotion campaign that is credited with raising flood awareness from 48% to 79% over the past five years (Proudley and Handmer, 2003).•Greater use of evidence-based approaches. Social research is replacing gut feeling in emergency risk communication. The last few years have seen a dramatic increase in the commissioning of quantitative and qualitative social research: for instance, FESA's Community Safety Survey 2000, the Queensland Department of Emergency Services'focus group research (AC Neislen 2003), and NSW SES research into flood knowledge and perceptions.The shift from a public awareness approach to one of community safety alters the traditional top-down, 'command and control' relationship with the community. In this new model, the community is seen as an active participant in its own safety, rather than a passive recipient of services. This requires emergency agencies to become specialists, facilitators and supporters of the community, while maintaining their traditional disaster response functions. These are challenging roles which requiring flexibility, new skills and new approaches (AMEC 2002).The behavioural models that have influenced the development of community safety programs are summarised in Speaking of Health (Institute of Medicine 2002). The first of these health promotion models is the Health Belief Model. According to this model, two main factors contribute to a person’s willingness to adopt appropriate health behaviours. First, the person must believe that there is a significant risk to them and the suggested benefits will compensate for the cost of undertaking the appropriate behaviours. The second is the Social Cognitive Theory, that emphasises the importance of individual self-efficacy, or self-confidence that they can exercise some degree over their behaviour and the outcomes they want to achieve. The Theory of Reasoned Action asserts that the extent of behaviour change can be viewed as a function of a person’s attitude towards performing the action and a person’s perception of what his/her peers’ attitude is towards performing the task. The identification of community norms as an incentive or hindrance to change is an important factor, especially in low perceived/high actual risk environments, and highlights the need to work closely with community expectations.Com Safety Program 4 04.doc 27/04/04- 9 -4) F ACTORS THAT I NFLUENCE C OMMUNITY S AFETYOne of the most contentious issues in the risk communication area is the identification of factors that contribute to a successful community safety program. Previous programs had centred on the Information/Action model; however, research carried out by the NSW SES in Kempsey and by Pfister (2001) at Grafton, have demonstrated that hazard and risk information, when distributed in isolation from the social setting, will have little significant impact on awareness or behavioural change.The principal factors that contribute to an effective community safety program include:a) The nature of the hazard and associated risk;b) The perception of the risk and people’s willingness to act;c) Identifying the stages of risk communication;d) Identifying audiences and associated messages; ande) Community resilience.a. The nature of the hazard and associated riskEmergency managers frequently express frustration with the public when they demonstrate a lack of concern when experts identify an extreme risk that threatens a community. This is particularly so when managers need to communicate the risk resulting from an infrequent but severe hazard. "Arguably, the flood threat is neither frequent enough in its impact nor severe enough in its usual consequences for experience of it to generate deliberate protective behaviour in most people" (Keys 1999b).Severe floods are an example of a risk that most non-experts would see as unlikely to have an impact on their lives. However floods are one of most costly natural disasters in Australia. In NSW from 1967-99 (in 1998 $A), floods cost $128,000,000 per annum; about 26.5% of the total cost of all disasters. For Australia as a whole, the cost was $314,000,000 per annum or 28% of all disaster costs (Bureau of Transport Economics, 2001).So while severe floods are a real problem for many communities, there is evidence that the public does not agree with this assessment. A recent survey of Queenslanders showed that floods are generally perceived as less risky than other hazards such as cyclones.Flooding:• It is low risk unless you live near a river;• You can do little to prepare until you receive a flood warning; and• A lot of clean-up is needed, but little damage (AC Neilsen 2003).Com Safety Program 4 04.doc 27/04/04- 10 -As risk managers one of the greatest dilemmas in flood and storm communication is how do you alert the public to the risk of low-probability, high-consequence disasters such as severe floods? The conventional wisdom is that people need to be convinced of the risks. It is therefore our role as risk managers, to give them sufficient details of the hazard, so that they will be prepared to protect themselves from the consequences.However, conventional wisdom runs into a wall of public indifference – an indifference with its own logic. “Why should I concern myself with risks that -- while they may be severe -- are rare and usually low-intensity, and which the government and emergency agencies are practiced at managing?”The fact is, we may be asking the public to act on someone else's problem – in this case, the risk communication manager’s problem. The issue then becomes one of not only identifying the actual risk from a severe hazard, but also understanding how people will perceive the risk and be willing to adopt protective behaviours.b. Risk PerceptionIntegral to the community safety approach is the belief that people do not categorise all risks as the same. In other words, they will underestimate or overestimate the risk according to their perception or understanding of the impact of the risk on their own lives. In situations such as an infrequent but severe hazard, the decision-making process is made harder by the complex variables that influence an individual’s perception of the risk.Research suggests that when people feel threatened when confronted with health and safety messages, they become defensive and believe that it won’t affect them. Sandman (1994) found that people were often hostile to the idea that they are at risk. People judged themselves less at risk than the ‘average’ person to a variety of natural and technological hazards. This psychological bias is well known: people believe that they are impervious to events that affect the average person and is referred to as Optimism Bias (Amber, 2003). This view dominates most responses to risk, and people support it by devising a rationale for the conviction that the hazard will pass them by, or that it will only inflict minor damage to their property. Carney (1993) hypothesised that when communicating about risk there is a need to develop a contingency model that takes into consideration both the actual risk of the situation (Factor 1) and the perceived risk (Factor 2). This determines the best communication strategies to use in the situation it represents.Risk Contingency Factors1. Low actual risk/ low perceived risk (e.g., volcanic eruption in Sydney)2. Low actual risk/ high perceived risk (e.g., attack by bees)3. High actual risk/ high perceived risk (e.g., motor vehicle accident)Com Safety Program 4 04.doc 27/04/04- 11 -4. High actual risk/ low perceived risk (e.g., severe flood).In essence, people usually underestimate risks because they would rather believe they are safe, free to live their lives without the responsibility of feeling vulnerable and obliged to make difficult or unpopular decisions that would affect their lifestyle. Festinger (1964) identified this conflict in his Theory of Cognitive Dissonance. Festinger examined situations where there are often mutually incompatible alternatives that ensure conflict in the decision-making process. The greater the conflict before the decision, the greater the dissonance. To reduce this dissonance, a person may try to justify the decision by increasing the attractiveness of the chosen alternative and decreasing the attractiveness of the rejected alternative. For example, people who are confronted with the devastating news of a future severe flood may deny that this level of flooding could occur and reject the information as well as assistance to reduce the risk. This is because they may consider there is a low risk from a severe flood, coupled with low benefits from becoming flood prepared and a high cost in terms of their time and effort. Thus, they would consider their vulnerability as being low and would make a decision not to become involved in any risk-management programs. When a severe flood occurs, these people would be ill prepared and require the assistance of emergency agencies to evacuateHow then, do people determine the degree of risk that they are willing to accept when going about their lives? There are many empirical studies that attempt to establish an objective comparison between risks that communities are exposed to, people’s attitude towards risk and their willingness to act to reduce the risk are more subjective. Wade et al. (1992) identified several of the variables that a person will use to determine their reaction to a specific risk (see Table 1). Thus, a person may have a high vulnerability to a specific risk because of their belief that the risk will not affect their life, which in turn will influence their willingness to adopt safety messages.Table 1. Variables That Influence Risk Perception Model (adapted by O’Neill from Wade et al. [1992])Com Safety Program 4 04.doc 27/04/04- 12 -H IGH B ENEFITSPeople will accept a risk if they can identify corresponding benefits (eg., car travel).V L ITTLE B ENEFITSPeople are less accepting if theysee no benefits or high costs fromaccepting the risk (e.g., industrialpollution).F AMILIARIf a risk is an everyday occurrence, it may be accepted into a person’s schema (e.g., smoking).V U NKNOWNIf the risk is unknown or rare, thereis likely to be resistance in acceptingit (e.g., GM foods).TRUSTEDThe risk is more likely to be accepted if people know and trust the organisation helping to manage the risk (e.g., SES).V N OT TRUSTEDIf the organisation is not trusted, themessage about the risk may not beaccepted (e.g., a bank)1.N ATURALPeople are more likely to accept what is regarded as a natural hazard (this perception gives a sense of inevitability about the risk).V T ECHNOLOGICALPeople have a higher expectationthat technological or industrial riskswill be managed.V OLUNTARYPeople are more willing to accept a risk when they make the decision about their own exposure to it. (e.g., smoking)V I MPOSEDPeople may react negatively if theyfeel they have little choice inaccepting the risk. (e.g., pollution)M EMORABLEPeople are less willing to accept a risk concerning a hazard that will attract wide public and media attention.V F ORGETTABLEPeople may accept a riskconcerning an event not likely tocreate community or media interest.C ATASTROPHIC P OTENTIAL People are more concerned about risks from hazards that are capable of causing dread because of the significant impact on a community.V C HRONIC P OTENTIALPeople may be unconcerned aboutthe risk from hazards that seem tohave little potential to significantlyaffect a community.F OCUSED THREATAn event that occurs over a brief period concentrates media and community interest.V D ISPERSED THREATThere is less media and communityattention if an event occurs over along period (e.g., drought).U NCERTAIN TIME AND SEVERITY A vague or undefined threat can make people reject safety messages as too hard to implement.V C ERTAIN TIME AND SEVERITYPeople feel more comfortable andwilling to listen to safety messages ifa threat can be defined andprepared for.M ANAGEDPeople are more willing to accept V H APHAZARDPeople are unwilling to accept risksCom Safety Program 4 04.doc 27/04/04- 13 -。
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ity to air pollution among different population subgroups. (Am. J. Respir. Crit. Care Med. .197,155, ,68-76) Dioxin trends in fish Concentrations of PCBs, polychlori-nated dibenzo-p-dioxins, and poly-chlorinated dibenzofurans are often gauged in terms of toxic equivalence factors. S.Y. Huestis and co-workers reported temporal (1977-93) and age-related trends in concentration and toxic equivalencies of these compounds in Lake Ontario lake trout. Analysis of the stored frozen fish tissue used a single analysis pro-tocol, which allowed improved com-parison of data from different time periods. Results showed that contami-nant levels have declined substantially since 1977 but concentration levels have stabilized approaching a steady state or a very slow decline The pro-portion of the total toxic equivalency ascribed to each compound has changed little in each of the three sets examined (Environ Toxicol Chem 1997 16(2) 154-64) Herbicide extractionEfficient extraction and analysis of phenoxyacid herbicides are difficult because of their high polarity and low volatility. T. S. Reighard and S. V Olesik reported the use of methanol/ C02 mixtures at elevated tempera-tures and pressures to extract these herbicides from household dust. The experiments were done at conditions covering both subcritical and super-critical regimes. They found that the highest recoveries (between 83% and 95% for the four herbicides studied) were observed at 20 mol % methanol and at temperatures of 100 °C or 150 °C. In addition, when a 200-uLvolume of hexane was added to the1-g dust sample, a preextraction withC02 and no methanol removed muchof the extraneous matrix. These ma-trix compounds, when present, cre-ate a more complex chromatogramand require more reagent. (Anal.Chem. 1997, 69, 566-74)Overcoming NIMBYPublic participation programs canhelp citizens get past "not-in-my-backyard" (NIMBY) responses to thesiting of hazardous waste facilities.J. J. Duffield and S. E Depoe de-scribed the effects of citizen partici-pation in the storage of 2.4 millioncubic yards of low-level radioactivewaste from the Fernald, Ohio, nu-clear weapons complex. Among theparticipants were labor representa-tives, academicians, 8X63. residents,and activists. Because the task forcehad the opportunity to questiontechnical experts and dispute evi-dence a democratic formatated for two-way communicationbetween officials and citizens (RiskPol Rev 1997 3(2) 31-34)RISKProbabilistic techniquesEfforts to improve risk characteriza-tion emphasize the use of uncer-tainty analyses and probabilistictechniques. K. M. Thompson andJ. D. Graham describe how MonteCarlo analysis and other probabilis-tic techniques can be used to im-prove risk assessments. A probabilis-tic approach to risk assessmentincorporates uncertainty and vari-ability, calculating risk with variablesthat include resources expended andpolicy mandates. Despite these ad-vantages, there are significant barri-ers to its widespread use, includingrisk managers' inexperience withprobabilistic risk assessment resultsand general suspicion of themethod. The authors describe waysto promote the proper use of proba-bilistic risk assessment. (Hum. Ecol.Risk Assess. 1996,2(4), 1008-34)Uncertainty modelingMonte Carlo modeling is a powerfulmathematical tool with many advan-tages over traditional point estimatesfor assessing uncertainty and vari-ability with hazardous waste site ex-posure and risk. However, a lack ofvariability information hindersMonte Carlo modeling. As a solu-tion, W. J. Brattin and colleaguesproposed running repeated MonteCarlo simulations using differentcombinations of uncertainty param-eters. The amount of variationamong the different simulationsshows how certain or uncertain anyindividual estimate of exposure orrisk may be An example of thisproach is provided including an es-timation of the average exposure toradon daughter products in indoorair (Hum Ecol Risk Assess .1992(4) 820-40)SOILDecomposition modelClay has a stabilizing effect on or-ganic matter in soil and thus reducesthe rate of decomposition. Currentcomputer simulation models, how-ever, do not adequately describe theprotection of organic matter by clay.J. Hassink and A. P. Whitmore devel-oped and tested a model that pre-dicts the preservation of added or-ganic matter as a function of theclay fraction and the degree of or~ganic matter saturation of the soil. Itclosely followed the buildup and de-cline of organic matter in 10 soils towhich organic matter was addedBetter than conventional modelsthis model was able to predict theaccumulation and degradation oforganic matter in soils of differenttextures and the contents of initialorganic matter (Soil Sci Soc Am J1997 61 131-39)Tracing trace metal complexationChemical reactions determine the fate of trace metals released into aquatic envi-ronments. J. M. Gamier and colleagues investigated the kinetics of trace metal complexation by monitoring chemical reactions in suspended matter from a French river. Radiotracer experiments on Mn, Co, Fe, Cd, Zn, Ag, and Cs identified sea-sonal variations in the predominant species. In summer, Cd and Zn were com-plexed by specific natural organic matter ligands. Cs remained in an inorganicform, whereas Fe and Ag were either organic complexes or colloidal species. In winter, a two-step process occurred for Mn and Co. They were rapidly complexed by weak ligands, followed by slower complexation by stronger ligands. The authors conclude that low concentrations of natural ligands may control the speciation of trace elements. (Environ. Sci. Techno!..,his issue, pp. .597-1606)2 60 A • VOL. 31, NO. 6, 1997 / ENVIRONMENTAL SCIENCE & TECHNOLOGY / NEWS。
最新理论试题及答案英语
最新理论试题及答案英语一、选择题(每题1分,共10分)1. The word "phenomenon" is most closely related to which of the following concepts?A. EventB. FactC. TheoryD. Hypothesis答案:C2. In the context of scientific research, what does the term "hypothesis" refer to?A. A proven factB. A testable statementC. A final conclusionD. An unverifiable assumption答案:B3. Which of the following is NOT a characteristic of scientific theories?A. They are based on empirical evidence.B. They are subject to change.C. They are always universally applicable.D. They are supported by a body of evidence.答案:C4. The scientific method typically involves which of the following steps?A. Observation, hypothesis, experimentation, conclusionB. Hypothesis, observation, conclusion, experimentationC. Experimentation, hypothesis, observation, conclusionD. Conclusion, hypothesis, observation, experimentation答案:A5. What is the role of experimentation in the scientific process?A. To confirm a hypothesisB. To disprove a hypothesisC. To provide evidence for or against a hypothesisD. To replace the need for a hypothesis答案:C6. The term "paradigm shift" in the philosophy of science refers to:A. A minor change in scientific theoryB. A significant change in the dominant scientific viewC. The process of scientific discoveryD. The end of scientific inquiry答案:B7. Which of the following is an example of inductive reasoning?A. Observing a pattern and making a general ruleB. Drawing a specific conclusion from a general ruleC. Making a prediction based on a hypothesisD. Testing a hypothesis through experimentation答案:A8. Deductive reasoning is characterized by:A. Starting with a specific observation and drawing a general conclusionB. Starting with a general rule and applying it to a specific caseC. Making assumptions without evidenceD. Relying on intuition rather than logic答案:B9. In scientific research, what is the purpose of a control group?A. To provide a baseline for comparisonB. To test an alternative hypothesisC. To increase the number of participantsD. To confirm the results of previous studies答案:A10. The principle of falsifiability, introduced by Karl Popper, suggests that:A. Scientific theories must be proven trueB. Scientific theories must be able to withstand attempts at being disprovenC. Scientific theories are never wrongD. Scientific theories are always based on personal beliefs答案:B二、填空题(每题1分,共5分)1. The scientific method is a systematic approach to__________ knowledge through observation, experimentation, and __________.答案:gaining; logical reasoning2. A scientific law is a statement that describes a__________ pattern observed in nature, while a scientific theory explains the __________ behind these patterns.答案:recurring; underlying principles3. The process of peer review in scientific publishing is important because it helps to ensure the __________ and__________ of research findings.答案:validity; reliability4. In the context of scientific inquiry, an __________ is a tentative explanation for an aspect of the natural world that is based on a limited range of __________.答案:hypothesis; observations5. The term "empirical" refers to knowledge that is based on __________ and observation, rather than on theory or__________.答案:experimentation; speculation三、简答题(每题5分,共10分)1. Explain the difference between a scientific theory and a scientific law.答案:A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experimentation. It is a broad framework that can encompass multiple laws and observations. A scientific law, on the other hand, is a concise verbal or mathematical statement that describes a general pattern observed in nature. Laws summarize specific phenomena, while theories explain the broader principles behind those phenomena.2. What is the significance of the falsifiability criterionin the philosophy of science?答案:The falsifiability criterion, proposed byphilosopher of science Karl Popper, is significant because it provides a way to distinguish between scientific and non-scientific theories. For a theory to be considered scientific, it must be testable and potentially refutable by empirical evidence. This criterion ensures that scientific theories are open。
asymptotic analysis缩写
asymptotic analysis缩写Asymptotic analysis is a mathematical method used to analyze the behavior of an algorithm as the input size approaches infinity. It is widely used in computer science and engineering to compare different algorithms and design efficient algorithms. In this article, we will discuss the basics of asymptotic analysis and its important concepts.Firstly, we need to understand the importance of asymptotic analysis. In computer science and engineering, we often deal with huge datasets and complex algorithms. Therefore, it is important to know how the algorithm will behave as the input size becomes large. Asymptotic analysis helps us to estimate the computational time and space complexity of an algorithm for large inputs. This estimation can help us to choose the best algorithm for a given problem.Asymptotic analysis is based on the concept of limits. A limit is the value a function approaches as the input value approaches a certain point. We use big O, big Omega, and big Theta notations to express the growth rate of a function. These notations give us a rough idea about the behavior of the function.Big O notation: The big O notation gives the upper bound of the running time of an algorithm. We say that algorithm A has a time complexity of O(f(n)) if the running time of the algorithm does not exceed a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is less than or equal to 2n^2+3n+4, we can say that the time complexity of the algorithm A is O(n^2).Big Omega notation: The big Omega notation gives the lower bound of the running time of an algorithm. We say that algorithm A has a time complexity of Omega(f(n)) if the running time of the algorithm is not less than a constant multiple of f(n) for large n. For example, if the running time of the algorithm A is greater than or equal to n^2/2, we can say that the time complexity of the algorithm A is Omega(n^2).Big Theta notation: The big Theta notation gives the tight bounds of the running time of an algorithm. We say that algorithm A has a time complexity of Theta(f(n)) if the running time of the algorithm is between a constant multiple of f(n) and another constant multiple of f(n) for large n. For example, if the running time of the algorithm A is between 5n^2+3n+4 and 7n^2+5n+6, we can say that the time complexity of the algorithm A is Theta(n^2).Asymptotic analysis also covers the space complexity of an algorithm. We use the same notations to express the growth rate of the space usage of an algorithm. For example, if the space used by the algorithm A is less than or equal to 3n+4, we can say that the space complexity of the algorithm A is O(n).In conclusion, asymptotic analysis is an important concept in computer science and engineering. It helps us to estimate the computational time and space complexity of an algorithm for large inputs. By using the big O, big Omega, and big Theta notations, we can compare different algorithms and choose the best algorithm for a given problem.。
Asymptotic expansions for ratios of products of gamma functions
a r X i v :m a t h /0206200v 2 [m a t h .C A ] 27 M a y 2003Asymptotic expansions for ratios of productsof gamma functionsWolfgang B¨u hringPhysikalisches Institut,Universit¨a t Heidelberg,Philosophenweg 1269120Heidelberg,Germanybuehring@physi.uni-heidelberg.deAbstractAn asymptotic expansion for a ratio of products of gamma func-tions is derived.2000Mathematics Subject Classification:Primary 33B15;Secondary 33C20Keywords and phrases:Gamma function,generalized hypergeometric functions1IntroductionAn asymptotic expansion for a ratio of products of gamma functions has recently been found [2],which,withs 1=b 1−a 1−a 2,(1)may be writtenΓ(a 1+n )Γ(a 2+n )(1)m (1+s 1−n )m +O (n −M −1)(2)1as n→∞.Here use is made of the Pochhammer symbol(x)n=x(x+1)···(x+n−1)=Γ(x+n)/Γ(x).The special case when b1=1of this formula(2)had been stated earlierby Dingle[3],and there wereproofs by Paris[8]and Olver[6][7].The proof of(2)is based on the formula for the analytic continuation near unit argument of the Gaussian hypergeometric function.For the more general hypergeometric functionsp+1F p a1,a2,...,a p+1b1,...,b p z=∞n=0(a1)n(a2)n···(a p+1)nΓ(b1)···Γ(b p)p+1F p a1,a2,...,a p+1b1,...,b p z(4)=∞m=0g m(0)(1−z)m+(1−z)s p∞ m=0g m(s p)(1−z)m,wheres p=b1+···+b p−a1−a2−···−a p+1(5) and the coefficients g m are known.While the g m(0)are not needed for the present purpose,the g m(s p)are[1]g m(s p)=(−1)m(a1+s p)m(a2+s p)mΓ(−s p−m)(a1+s p)k(a2+s p)kA(p)k,2where the coefficients A(p)k will be shown below.The left-hand side L of(4)isL=∞n=0Γ(a1+n)Γ(a2+n)···Γ(a p+1+n)Γ(1+n)z n.(9)Interchanging the order of summation(and making use of the reflection for-mula of the gamma function)we may getR=∞n=0Γ(−s p+n)Γ(−s p−m)(1+s p−n)m z n.(10)Comparison of the coefficients of the two power series for R and L,which asymptotically,as n→∞,should agree,then leads toΓ(a1+n)Γ(a2+n)···Γ(a p+1+n)Γ(−s p−m)(1+s p−n)m.Inserting g m from(6)and keeping thefirst M+1terms of the asymptotic series,we getTheorem1Γ(a1+n)Γ(a2+n)···Γ(a p+1+n)(1)m(1+s p−n)mmk=0(−m)kThe simple formula(2)above corresponding to p=1can be recovered from this theorem if we define A(1)0=1,A(1)k=0for k>0,so that then the sum over k is equal to1and disappears.The coefficients for larger p can be found in[1],but a few of them are here displayed again for convenience:A(2)k=(b2−a3)k(b1−a3)k(k−k2)!k2!,(14)A(4)k=kk2=0(b4+b3+b2−a5−a4−a3+k2)k−k2(b1−a3)k−k2(k2−k3)!×(b4−a5)k3(b3−a5)k3k!(16)×3F2 b3−a4,b2−a4,−kb3+b2−a4−a3,1+a3−b1−k 1orA(3)k=(b1+b3−a3−a4)k(b2+b3−a3−a4)kΓ(b1+n)Γ(b2+n)Γ(−s2+n)=1(18)+Mm=1(a1+s2)m(a2+s2)m3Additional commentsThe derivation of the theorem is based on the continuation formula(4)which holds,as it stands,only if s p is not equal to an integer.Nevertheless,the the-orem is valid without such a restriction.This can be verified if the derivation is repeated starting from any of the continuation formulas for the exceptional cases[1].Instead of or in addition to the binomial theorem,the expansion(1−z)m ln(1−z)=∞n=1c n z n,(19)is then needed for integer m≥0,wherec n=−1Γ(n)(20)for n>m,while the coefficients are not needed here for n≤m.The theorem has been proved here for any sufficiently large positive in-teger n only.On the basis of the discussion in[2],it can be suspected that the theorem may be theoretically valid(although less useful)in the larger domain of the complex n-half-plane Re(s p+a1+a2−1+n)≥0.Expansions for ratios of even more general products of gamma functions are treated in a recent monograph by Paris and Kaminski[9]. References[1]W.B¨u hring,Generalized hypergeometric functions at unit argument,Proc.Amer Math.Soc.114(1992),145–153.[2]W.B¨u hring,An asymptotic expansion for a ratio of products ofgamma functions,Internat.J.Math.&Math.Sci.24(2000),504–510.[3]R.B.Dingle,Asymptotic Expansions:Their Derivation and Inter-pretation(Academic Press,London,1973).[4]P.Flajolet and A.Odlyzko,Singularity analysis of generating func-tions,SIAM J.Discrete Math.3(1990),216–240.[5]F.W.J.Olver,Asymptotics and Special Functions(Academic Press,New York,1974).5[6]F.W.J.Olver,Asymptotic expansions of the coefficients in asymp-totic series solutions of linear differential equations,Methods and Applications of Analysis1(1994),1–13.[7]F.W.J.Olver,On an asymptotic expansion of a ratio of gammafunctions,Proc.Royal Irish Acad.A95(1995),5–9.[8]R.B.Paris,Smoothing of the Stokes phenomenon using Mellin-Barnes integrals,put.Appl.Math.41(1992),117–133. [9]R.B.Paris and D.Kaminski,Asymptotics and Mellin-Barnes inte-grals,Cambridge University Press,Cambridge,2001.[10]R.Sch¨a fke and D.Schmidt,The connection problem for general lin-ear ordinary differential equations at two regular singular points with applications in the theory of special functions,SIAM J.Math.Anal.11(1980),848–862.6。
斯特林公式Stirling Formular
题目:关于阶乘的近似公式1.相关历史与进程历史上对阶乘的估计在数学上有着重要的作用,首先是它在概率论与数理统计中,最早可以追溯到1733年一位法国的数学家de Moivre 的工作,同时也是第一次遇到对整数阶乘的估计问题。
在他研究Gauss 分布和中心极限定理时发现了如下公式:!constant nn n e ⎛⎫≈ ⎪⎝⎭然后,瑞典数学家Stirling在试图给出二项分布的一般的近似值时,发现了未知的常数:constant =Stirling 公式:!nn n n e σ⎫≈=⎪⎭紧接着他就得到如下的结果,并发表在了Miscellaneis Analyticis Supplementum 中:221111ln[(1)!]~ln()ln(2)222(21)k k k B n n n n k k nπ-≥⎛⎫---++ ⎪-⎝⎭∑ (1)公式(1)也被称为Stirling 级数,其中的2k B 称为Bernoulli 数,定义如下:0011,0kj j k B B j =+⎛⎫== ⎪⎝⎭∑其中1k ≥。
将(1)式的前m 项记为2211exp 2(21)nm k m k k B n e k k x τ-=⎛⎫⎫= ⎪⎪-⎭⎝⎭∑同时Euler 提出了一个函数,它可以作为整数的阶乘在正实数中的拟合。
这函数便是Γ-函数:10()t z z e t dt +∞-Γ=⎰,也可以定义为极限的形式:!()lim(1)()zn n n z z z z n →∞Γ=++而且显然有(1)!n n Γ+=,而且目前对阶乘的估计也或多或少的用Γ-函数来描述,甚至利用Γ-函数的性质来发现新的更好的渐进函数。
之后,关于!n 的渐进公式的探索逐渐缓慢下来。
直到最近才有了新的突破。
2.第一种有关!n 的渐进形式——含有幂级数的渐进公式依靠幂级数来求数值解的思想一直是较好的方法。
其中在Stirling 所处的时期便已经有了一个幂级数展开,而且拥有着各种相似的形式,如在Abramowitz 和Stegun [1]的书中记载着:3571111!exp 1236012601680nn n e n n n n ⎫⎛⎫=-+-+⎪ ⎪⎭⎝⎭但是在1763年Bayes [5]在给Canton 的信中说:Stirling 给出的这个幂级数展开并不是一个收敛级数。
老年急性缺血性脑卒中患者神经功能缺损加重的危险因素
老年急性缺血性脑卒中患者神经功能缺损加重的危险因素任行龙,聂贝贝,马丽霞,孙冬丽郑州大学第一附属医院神经ICU ,河南郑州450000【摘要】目的探讨老年急性缺血性脑卒中(AIS)患者发病1周内神经功能缺损加重的危险因素。
方法选取2020年6月至2022年8月郑州大学第一附属医院收治的135例老年AIS 患者作为研究对象,依据发病1周内神经功能缺损程度分为加重组38例和非加重组97例。
收集并比较两组患者的一般资料、血清学指标[总胆固醇(TC)、甘油三酯(TG)、高密度脂蛋白胆固醇(HDL-C)、低密度脂蛋白胆固醇(LDL-C)、超敏C 反应蛋白(hs-CRP)、基质金属蛋白酶-9(MMP -9)、脂蛋白相关磷脂酶A2(Lp-PLA2)]。
采用单因素和二元Logistic 回归分析老年AIS 患者发病1周内神经功能缺损加重的危险因素。
结果单因素分析结果显示,加重组患者的年龄、缺血性脑卒中史占比、入院时美国国立卫生研究院卒中评分量表(NIHSS)评分、血清TG 、LDL-C 、hs-CRP 、MMP -9、Lp-PLA2水平明显高于非加重组,差异均有统计学意义(P <0.05);经二元Logistic 回归分析结果显示,高龄、既往缺血性脑卒中史、较高入院时NIHSS 评分及血清TG 、LDL-C 、hs-CRP 、MMP -9、Lp-PLA2水平升高均为老年AIS 患者发病1周内神经功能缺损加重的独立危险因素(P <0.05)。
结论临床应针对上述老年AIS 患者发病1周内神经功能缺损加重的危险因素进行早期干预,以降低老年AIS 患者发病1周内神经功能缺损加重的风险,改善老年AIS 患者的预后。
【关键词】老年;急性缺血性脑卒中;神经功能缺损;危险因素【中图分类号】R743.3【文献标识码】A【文章编号】1003—6350(2023)24—3544—04Risk factors analysis of aggravation of neurological deficit in elderly patients with acute ischemic stroke.REN Xing-long,NIE Bei-bei,MA Li-xia,SUN Dong-li.Neurological ICU,the First Affiliated Hospital of Zhengzhou University,Zhengzhou 450000,Henan,CHINA【Abstract 】ObjectiveTo analyze the risk factors of aggravation of neurological deficits within one week afteronset in elderly patients with acute ischemic stroke (AIS).MethodsA total of 135elderly patients with AIS admittedto the First Affiliated Hospital of Zhengzhou University from June 2020to August 2022were selected as the study sub-jects.They were divided into an exacerbation group (n =38)and a non-exacerbation group (n =97)based on the degree of neurological deficit within one week after onset.The general information and serological indicators [total cholesterol (TC),triglyceride (TG),high-density lipoprotein cholesterol (HDL-C),low-density lipoprotein cholesterol (LDL-C),high-sensitivity C-reactive protein (hs-CRP),matrix metalloproteinase -9(MMP -9),lipoprotein-associated phospholi-pase A2(Lp-PLA2)]of the two groups were collected and compared.Single-factor and binary logistic regression were used to analyze the risk factors for aggravation of neurological deficits in elderly AIS patients within one week after on-set.ResultsThe results of univariate analysis showed that age,proportion of patients with ischemic stroke history,andNIHSS score,serum TG,LDL-C,hs-CRP,MMP -9,Lp-PLA2levels at admission in the exacerbation group were signifi-cantly higher than those in the non-exacerbation group (P <0.05).The results of binary logistic regression analysis showed that advanced age,previous history of ischemic stroke,higher NIHSS score at admission,and elevated serum TG,LDL-C,hs-CRP,MMP -9,and Lp-PLA2levels were all independent risk factors for aggravation of neurological def-icit within one week after onset in elderly patients with AIS (P <0.05).ConclusionEarly clinical intervention shouldbe carried out according to the above risk factors for aggravation of neurological deficits within one week after onset in elderly patients with AIS,so as to reduce the risk of aggravation and improve the prognosis of the patients.【Key words 】Elderly;Acute ischemic stroke;Neurological deficit;Risk factors ·论著·doi:10.3969/j.issn.1003-6350.2023.24.010基金项目:河南省科学技术厅重点研发与推广专项(科技攻关)(编号:202102310490)。
具有奇性方程的非线性奇摄动问题的正解英文
应 用 数 学MAT HEMA TICA AP PL ICA TA2010,23(1):13217Posit ive Solut ion of N onlinear Singular lyPer t ur bed Pr oblems wit h Singular Equa t ionMO Jia 2qi (莫嘉琪)1,2(1.D ep a rtment of M at hem ati cs ,A n hui N or mal U ni versit y ,W u h u 241000,Chi na ;2.D i vis ion o f Com p ut at ion al Sci ence ,E 2I nst it utes of S ha n gh ai U ni vers it ies at SJ TU ,Sh an g h ai 200240,Chi n a)A bstract :A cla ss of singularly pe rturbed initial value problem with singular equation is consider ed.Under suita ble co nditions ,by using the theor y of diffe rential inequalities the exi stence and a symptot 2ic behavior of solution fo r initial value problem are studied ,a nd obtaine d unifor mly valid asymptotic expa nsion of solution wit h an initial la yer.K ey w or ds :Nonlinear ;Singular equation ;Singular pert urbationCL C N umber :O175.14 AMS(2000)Subject Cla ssif ica t ion :34E15Document code :A A r t icle ID :100129847(2010)01200132051.IntroductionRecent ly ,many approxi mat e met hods for t he nonli near si ngularly pert urbed p roble m have been developed and refined [1],and ma ny schola rs have done a great deal of work [228].U 2si ng t he different ial inequali ties and ot her met hods Mo et al.considered al so a cla ss of si ngu 2larly pert ur bed nonli near boundary value probl ems for ordi nary differential equat ions [9210],t he reaction diff usion equations [11213],t he bo undary val ue proble ms of t he elliptic equa 2t ions [14],t he shock l ayer solution of nonli nea r equat ions for t he singularly pert urbed p rob 2lem s [15],t he singularly pert urbed ecological model [16],t he act ivator i nhi hitor syst em [17]and t he probl ems of at mo spheric p hysic s [18220].In t hi s paper ,using a speci al singula rl y pert urbed met hod ,we st udy a cl ass of t he initi al value probl em wit h singular equation.Now we consi der t he following nonlinear problem :εd y d x=f (x ,y ),0<x ≤T (<∞),(1)y (0)=0,(2)3Rece ived date :Sep 1,2008Found a tion item :Supporte d by the Na tional Na tural Scie nce Foundation of China (40676016),the Na 2tional Key Projec t for Basics Resea rch (2003CB415101203a nd 2004CB418304),the K ey Projec t of the y f S (KZ X 32SW 2)y 2I f S M 2(3)B y MO 2q ,,,Z j ,f ,j Chinese Academ o cience sC 221and in part b E nsitutes o hanghai unicipal Edu cation Commission E0004iogra ph :Jia i male Han he iang pro essor ma or in applied mathe matics.where εi s a posi tive small pa ramet er ,f i s a conti nuous f unction and has singula rit ies at y =0and x =0.The probl em (1)(2)i s a nonli near si ngular ini tial val ue problem.We need t he following hypot heses :(H 1)The f unct io n f ∶(0,T]×(0,∞)→R is a sufficientl y smoot h f unct ion wit h respect t o t heir ar gument s i n correspondi ng regions.(H 2)There i s a positive constant δ,suc h t hat5f 5y(x ,y)≤-δ. (H 3)The reduced equat ion of problem (1)(2):f (q n (x),y)=0,x ∈(0,T ]has a posit ive solution Y n0(t )∈C ∞[0,T],w here q n (x )=max {x ,T/n},n i s a la rge enough po sit ive i nte ge r.2.Outer Solut ionWe now const ruct t he formal asymptotic solution of p roble m (1)(2).Fi rst ly ,consi der t he equationεd y n d x=f (q n (x ),y n ),x ∈(0,T].(3)Let t he outer sol ut ion Y n (x ,ε)of probl em (3)(2)isY n (x ,ε)=∑∞i =0Y n i (x )εi .(4) Substi t uti ng (4)into (3),developing f i n εand equating coefficient s of t he sa me power s of εbot h si des of (3)and not ing f (q n (x ),Y n 0)=0,we obt ai nY ni (x )=1f y (q n (x ),Y n 0)d Y n(i-1)d x+F i ,i =1,2,…,(5)where F i ,i =1,2,…are determined functions ,which const ructions are omitt ed.From Y n 0(x)and (5),we obt ain t he outer solution (4)for the problem (3)(2).But it may not satisfy t he initial c on 2di tion (2),so t hat we need to construct t he initial layer c orrective t erm near x =0.3.Initial Layer CorrectionWe lead i nto a st ret ched variable [1]ξ=x/ε.Lety n =Y n +V n (6)sati sfies equation (3)and ini tial co ndit ion y n (0)=ρn ,(7)where ρn =T/n.Substi t uti ng (6)into equat io n (3)a nd initi al condition (7),we haved V n d ξ=[f (q n (ξε),Y n +V n )-f (q n (ξε),Y n )],(8)V n |ξ=0=ρn -Y n (0).(9)LetV n =∑∞=v n i (ξ)εi .(10) S ()(6)()()(),f εq ff f f εf ()(),41MAT HEMA TICA AP PL ICA TA 2010i 0ubstituting 4and 10into 89developing in and e uating coe icient s o the sa m e power s o b oth sides o 89we obtaind v n 0d ξ=f T n ,Y n0+v n 0-f T n ,Y n 0,v n 0(0)=ρn -Y n 0(0),(11)d v n i dξ=f y T n ,Y n 0+v n0v n i +F n i ,v n i (0)=-Y n i (0),i =1,2,…,(12)where F n i ,i =1,2,…are det ermined f unct io ns ,w hi ch const ruct ions are omit ted too.Fro m (11)(12),we ca n obt ai n v n i ,i =0,1,2,…,and posse ss behaviorv n i =O(exp (-k n i ξ))=O exp -k n i xε,i =0,1,2,…,0<εν1,(13)where k n i ≥k n(i+1)>0,i =0,1,2,…are po sit ive constant s.Substi t uti ng v ni i nto (10),we t hen have t he i niti al layer cor rective ter m V n near x =0.Thus we ca n co nst r uct asymptoti c expa nsiony n ~∑∞i =0(Y n i +v n i )εi ,0<εν1.(14)Letlim n →∞y n =y ,li m n →∞Y ni =Y i ,li m n →∞v n i =v i ,i =0,1,2,….A nd from (14),we have t he followi ng formal asymptot ic expansion y of t he problem (1)(2):y (x )~∑mi =0(Y i +v i )εi +O (εm +1),0<εν1.(15)Fro m hypot hesi s (H 3),it i s easy to see t hat y(x)∈[0,T]i s a posit ive f unction as εsmall e 2nough.4.U n ifor m V alidityIn order t o prove (15)i s a uniformly valid a symptotic expansion ,we need al so t he fol 2lowing hypot hesis :(H 4)f (x ,y)/g (y)i s a conti nuous on (0,T ]×[0,∞),where g >0is conti nuous on (0,∞)a nd sati sfie s ∫T0sup 0≤y ≤N |f (q n (x),y)|g (y)d x <∞,N =sup x ∈[0,T ]β(x). We have t he following t heorem :Theorem U nder t he hypot hese s (H 1)2(H 4),t he re exi st s a sol ution y of si ngularl y pe r 2t urbed p roble m (1)(2)wit h si ngular equation and t he sol ut io n may be e xpanded i nto t he u 2niformly vali d a symptotic e xpa nsion (15)on x ∈[0,T ].Pr oof We const ruct t he auxi li ary functions αn and βn :αn =y n -rx εm +1,βn =y n +rxεm+1,x ∈[0,T ],(16)where r is a posit ive consta nt large enough to be chosen below ,m is a fixed arbi t rar y con 2stant.Obviously ,for εsmall enough ,we haveα(x)=lim n →∞αn (x)>0,x ∈(0,T ],α(0)=0,α>0,x ∈(0,T ],α≤αn ≤βn ,x ∈[0,T ].(17)Now we prove t hatεd αn xf (q (x ),α)≤,x ∈(,T],()εβx f (q (x),β)≥,x ∈(,T ]()51No.1MO Jia 2qi :Po sitive Solution of Nonlinear Singularly Pe rtur bed Problems d -n n 0018d n d -n n 00.19Fro m t he hypot heses ,i t follows t hat for εsmall enough ,t here e xi st s a posit ive consta nt M ,such t hatεd αn d x -f (q n (x ),αn )=εd d x ∑m i =0(Y n i +v n i )εi -f q n (x ),∑m i =0(Y ni +v n i )εi +f q n (x ),∑mi =0(Y n i +v n i )εi -f (q n (x ),αn )-r εm+1≤-f (p n (x),Y n 0)+∑mi =0d Y n (i-1)d x -f y (q n (x ),Y n 0)Y ni +F i εi +d v n 0d ξ-f T n ,Y n 0+v n0+f T n ,Y n 0+∑m i =1d v n i d ξ-f yT n ,Y n 0+v n 0v n i -F ni εi +M εm +1-r εm+1-r δεm +1=(M -r (1+δ))εm +1.Selecti ngr ≥M 1+δ,we have prove d t he i nequalit y (18).Simila rl y ,we can prove t he inequalit y (19)too.Fro m(17)2(19),by usi ng t he t heory of differenti al i nequalitie s [7],t here i s a solution y of p roble m (1)(2),such t hatαn ≤y ≤βn ,x ∈[0,T].Thus f ro m (14)(16),we get t he final result (15).The proof of t he Theore m is complet ed.R eferences :[1] Bar bu Morosa nu L.Singularly Pe rturbed Bounda ry 2Value Pro ble ms[M ].Ba sel :Birkha userm Ve rla g AG,2007.[2] Bar tier J P.G lobal behavior of solutions of a reaction 2diff usion equation with gradient bsorption in un 2bounded d o mains[J ].Asymptotic Anal.,2006,46(324):3252347.[3] Libre J ,Da Silva P R ,Teixeira M A.Regula riza tion of disco ntinuous vector fields on R 3via singula r pe r 2t ur bation[J ].J.Dyn.Diff er.Equa tions ,2007,19(2):3092331.[4] Duehring D ,Huang We nzha ng.Periodic traveling wave s fo r diff usion equations with time dela yed andn o n 2local responding reaction[J ].J.Dyn.Differ.Eqns.,2007,19(2):4572477.[5] Guar guaglini F R ,Natalini R.Fa st reaction limit and la rge time be havior of s olutions to a nonlinear modelof sulp hation p henomena [J ].Commun.Partial Diff er.Equations ,2007,32(2):1632189.[6] Alva rez 2Dio s J A ,Chipot M ,G arcia J C ,Muuiz M C.On a singular pe rturba tion p roblem for a class ofvaria tio nal inequalitie s[J ].Z eitsc hrif t f ur Anal.Und Ihre Anwendungen ,2008,27(1):79294.[7] Agar wal R P ,O ’Rega n D ,La kshmikantha m V ,Leela S.Existence of po sitive solutio ns f or singular initiala nd bounda ry val ue p ro ble ms via the classical upper a nd lower solution.approach[J ].Non.Anal.,2002,50(1):2152222.[8] Aarwal R P ,Regan D.Singular problems a n uppe r a nd lower solution app roach[J ].J.Mat h.Anal.Appl.,,5()325[] M q y y []M ,3,()2361MAT HEMA TICA AP PL ICA TA 20102000211:2020.9o Jia i.A singularl per turbe d nonlinea r b o unda r value problem J .J.ath.Anal.Appl.1991781:28929.[10] Mo Jiaqi ,Wa ng Hui ,Lin Wantao.The solvability for a class of singularly per turbed qua si 2linear diffe r 2ential system[J ].Acta Ma th.Sci.,2008,28B (3):4952500.[11] Mo Jia qi.Singula r pe rturbation for a class of nonlinea r reactio n diff usio n system s[J ].Sci.in China ,SerA ,1989,32(11):130621315.[12] Mo Jiaqi ,Lin Wantao.A n o nlinea r singula r pertur bed problem for reaction diff usion e quations withb o unda ry pert ur bation[J ].Acta Mat h.Appl.Sinica ,2005,21(1):1012104.[13] Mo Jia qi ,Zhang Weijia ng ,He Ming.Asymptopic met hod of traveling wave sol utions for a class of no n 2linea r reaction diff usion equation[J ].Acta Ma th.Sci.,2007,27(4):7772780.[14] Mo Jia qi.The singularly pert ur bed generalized Dirichlet problems fo r semilinea r elliptic equa tio n ofhigher order[J ].Advances in Math.,2006,35(1):75281.[15] Mo Jiaqi ,Zhu Jiang ,Wang Hui.Asymptotic be havior of t he shock solution for a class of nonlinear eqaut 2ions[J ].Prog.Na t.Sci.,2003,13(10):7682770.[16] Mo Jiaqi ,Wa ng Hui.Nonlinea r singularly pe rtur bed approximate solution for gener alized Lotke 2Volte rraecolo gical model[J ].Acta Ecologica Sinica ,2007,27(10):436624370.[17] Mo Jiaqi ,Lin Wantao.Asymptotic sol ution of activator inhibitor system s for n onlinear reactio n diff usionequations[J ].J.Sys.Sci.&Co mple xit y ,2008,20(1):1192128.[18] Mo Jia qi ,Lin Wangtao ,Wa ng Hui.Varia tional iteration solving met hod of a sea 2air oscillator model fort he ENSO[J ].Progre ss in Natural Sci.,2007,17(2):2302232.[19] Mo J Q ,Wang H ,Lin W T ,Lin Y H.Va ritional iteration metho d for solv i ng the mecha nism of the equa 2torial eastern Pacific El Nino 2Southe rn oscillation[J ].Chin.Phys.,2006,15(4):6712675.[20] Mo Jia qi ,Lin Wangtao ,Wang Hui.Variational itera tion method for solving pert ur bed mecha nism ofwestern boundary unde rcurre nts in the Pacif ic [J ].Chin.Phys.,2007,16(4):9512964.具有奇性方程的非线性奇摄动问题的正解莫嘉琪1,2(1.安徽师范大学数学系,安徽芜湖241000;2.上海高校计算科学E 2研究院SJ TU 研究所,上海200240)摘要:本文讨论了一类具有奇性方程的奇摄动初值问题.在适当条件下,利用微分不等式理论,研究了初值问题解的存在性及其渐近性态,并且得到了具有初始层的一致有效解的渐近展开式.关键词:非线性;奇性方程;奇摄动71No.1MO Jia 2qi :Po sitive Solution of Nonlinear Singularly Pe rtur bed Problems。
On the asymptotic expansion of the solutions of the separated nonlinear Schroedinger equati
a r X i v :n l i n /0012025v 3 [n l i n .S I ] 10 M a y 2001On the Asymptotic Expansion of the Solutions of the Separated Nonlinear Schr¨o dinger EquationA.A.Kapaev,St Petersburg Department of Steklov Mathematical Institute,Fontanka 27,St Petersburg 191011,Russia,V.E.Korepin,C.N.Yang Institute for Theoretical Physics,State University of New York at Stony Brook,Stony Brook,NY 11794-3840,USAAbstractNonlinear Schr¨o dinger equation with the Schwarzian initial data is important in nonlinear optics,Bose condensation and in the theory of strongly correlated electrons.The asymptotic solutions in the region x/t =O (1),t →∞,can be represented as a double series in t −1and ln t .Our current purpose is the description of the asymptotics of the coefficients of the series.MSC 35A20,35C20,35G20Keywords:integrable PDE,long time asymptotics,asymptotic expansion1IntroductionA coupled nonlinear dispersive partial differential equation in (1+1)dimension for the functions g +and g −,−i∂t g +=12∂2x g −+4g 2−g +,(1)called the separated Nonlinear Schr¨o dinger equation (sNLS),contains the con-ventional NLS equation in both the focusing and defocusing forms as g +=¯g −or g +=−¯g −,respectively.For certain physical applications,e.g.in nonlin-ear optics,Bose condensation,theory of strongly correlated electrons,see [1]–[9],the detailed information on the long time asymptotics of solutions with initial conditions rapidly decaying as x →±∞is quite useful for qualitative explanation of the experimental phenomena.Our interest to the long time asymptotics for the sNLS equation is inspired by its application to the Hubbard model for one-dimensional gas of strongly correlated electrons.The model explains a remarkable effect of charge and spin separation,discovered experimentally by C.Kim,Z.-X.M.Shen,N.Motoyama,H.Eisaki,hida,T.Tohyama and S.Maekawa [19].Theoretical justification1of the charge and spin separation include the study of temperature dependent correlation functions in the Hubbard model.In the papers[1]–[3],it was proven that time and temperature dependent correlations in Hubbard model can be described by the sNLS equation(1).For the systems completely integrable in the sense of the Lax representa-tion[10,11],the necessary asymptotic information can be extracted from the Riemann-Hilbert problem analysis[12].Often,the fact of integrability implies the existence of a long time expansion of the generic solution in a formal series, the successive terms of which satisfy some recurrence relation,and the leading order coefficients can be expressed in terms of the spectral data for the associ-ated linear system.For equation(1),the Lax pair was discovered in[13],while the formulation of the Riemann-Hilbert problem can be found in[8].As t→∞for x/t bounded,system(1)admits the formal solution given byg+=e i x22+iν)ln4t u0+∞ n=12n k=0(ln4t)k2t −(1t nv nk ,(2)where the quantitiesν,u0,v0,u nk and v nk are some functions ofλ0=−x/2t.For the NLS equation(g+=±¯g−),the asymptotic expansion was suggested by M.Ablowitz and H.Segur[6].For the defocusing NLS(g+=−¯g−),the existence of the asymptotic series(2)is proven by P.Deift and X.Zhou[9] using the Riemann-Hilbert problem analysis,and there is no principal obstacle to extend their approach for the case of the separated NLS equation.Thus we refer to(2)as the Ablowitz-Segur-Deift-Zhou expansion.Expressions for the leading coefficients for the asymptotic expansion of the conventional NLS equation in terms of the spectral data were found by S.Manakov,V.Zakharov, H.Segur and M.Ablowitz,see[14]–[16].The general sNLS case was studied by A.Its,A.Izergin,V.Korepin and G.Varzugin[17],who have expressed the leading order coefficients u0,v0andν=−u0v0in(2)in terms of the spectral data.The generic solution of the focusing NLS equation contains solitons and radiation.The interaction of the single soliton with the radiation was described by Segur[18].It can be shown that,for the generic Schwarzian initial data and generic bounded ratio x/t,|c−xthese coefficients as well as for u n,2n−1,v n,2n−1,wefind simple exact formulaeu n,2n=u0i n(ν′)2n8n n!,(3)and(20)below.We describe coefficients at other powers of ln t using the gener-ating functions which can be reduced to a system of polynomials satisfying the recursion relations,see(24),(23).As a by-product,we modify the Ablowitz-Segur-Deift-Zhou expansion(2),g+=exp i x22+iν)ln4t+i(ν′)2ln24t2] k=0(ln4t)k2t −(18t∞n=02n−[n+1t n˜v n,k.(4)2Recurrence relations and generating functions Substituting(2)into(1),and equating coefficients of t−1,wefindν=−u0v0.(5) In the order t−n,n≥2,equating coefficients of ln j4t,0≤j≤2n,we obtain the recursion−i(j+1)u n,j+1+inu n,j=νu n,j−iν′′8u n−1,j−2−−iν′8u′′n−1,j+nl,k,m=0l+k+m=nα=0, (2)β=0, (2)γ=0, (2)α+β+γ=ju l,αu k,βv m,γ,(6) i(j+1)v n,j+1−inv n,j=νv n,j+iν′′8v n−1,j−2++iν′8v′′n−1,j+nl,k,m=0l+k+m=nα=0, (2)β=0, (2)γ=0, (2)α+β+γ=ju l,αv k,βv m,γ,(7)where the prime means differentiation with respect toλ0=−x/(2t).Master generating functions F(z,ζ),G(z,ζ)for the coefficients u n,k,v n,k are defined by the formal seriesF(z,ζ)= n,k u n,k z nζk,G(z,ζ)= n,k v n,k z nζk,(8)3where the coefficients u n,k,v n,k vanish for n<0,k<0and k>2n.It is straightforward to check that the master generating functions satisfy the nonstationary separated Nonlinear Schr¨o dinger equation in(1+2)dimensions,−iFζ+izF z= ν−iν′′8zζ2 F−iν′8zF′′+F2G,iGζ−izG z= ν+iν′′8zζ2 G+iν′8zG′′+F G2.(9) We also consider the sectional generating functions f j(z),g j(z),j≥0,f j(z)=∞n=0u n,2n−j z n,g j(z)=∞n=0v n,2n−j z n.(10)Note,f j(z)≡g j(z)≡0for j<0because u n,k=v n,k=0for k>2n.The master generating functions F,G and the sectional generating functions f j,g j are related by the equationsF(zζ−2,ζ)=∞j=0ζ−j f j(z),G(zζ−2,ζ)=∞j=0ζ−j g j(z).(11)Using(11)in(9)and equating coefficients ofζ−j,we obtain the differential system for the sectional generating functions f j(z),g j(z),−2iz∂z f j−1+i(j−1)f j−1+iz∂z f j==νf j−z iν′′8f j−ziν′8f′′j−2+jk,l,m=0k+l+m=jf k f lg m,2iz∂z g j−1−i(j−1)g j−1−iz∂z g j=(12)=νg j+z iν′′8g j+ziν′8g′′j−2+jk,l,m=0k+l+m=jf kg l g m.Thus,the generating functions f0(z),g0(z)for u n,2n,v n,2n solve the systemiz∂z f0=νf0−z (ν′)28g0+f0g20.(13)The system implies that the product f0(z)g0(z)≡const.Since f0(0)=u0and g0(0)=v0,we obtain the identityf0g0(z)=−ν.(14) Using(14)in(13),we easilyfindf0(z)=u0e i(ν′)28n n!z n,4g0(z)=v0e−i(ν′)28n n!z n,(15)which yield the explicit expressions(3)for the coefficients u n,2n,v n,2n.Generating functions f1(z),g1(z)for u n,2n−1,v n,2n−1,satisfy the differential system−2iz∂z f0+iz∂z f1=νf1−z iν′′8f1−ziν′8g0−z(ν′)24g′0+f1g20+2f0g0g1.(16)We will show that the differential system(16)for f1(z)and g1(z)is solvable in terms of elementary functions.First,let us introduce the auxiliary functionsp1(z)=f1(z)g0(z).These functions satisfy the non-homogeneous system of linear ODEs∂z p1=iν4−ν′′4f′0z(p1+q1)−i(ν′)28−ν′g0,(17)so that∂z(q1+p1)=−(ν2)′′8z,p1(z)= −iνν′′8−ν′u′032z2,g1(z)=q1(z)g0(z),g0(z)=v0e−i(ν′)24−ν′′4v0 z+i(ν′)2ν′′4−ν′′4u0 ,v1,1=v0 iνν′′8−ν′v′0u n,2n −1=−2u 0i n −1(ν′)2(n −1)n −1ν′′u 0,n ≥2,v n,2n −1=−2v 0(−i )n −1(ν′)2(n −1)n −1ν′′v 0,n ≥2.Generating functions f j (z ),g j (z )for u n,2n −j ,v n,2n −j ,j ≥2,satisfy the differential system (12).Similarly to the case j =1above,let us introduce the auxiliary functions p j and q j ,p j =f jg 0.(21)In the terms of these functions,the system (12)reads,∂z p j =iνz(p j +q j )+b j ,(22)wherea j =2∂z p j −1+i (ν′)28−j −14(p j −1f 0)′8f 0+iν4−ν′′zq j −1−−ν′g 0+i(q j −2g 0)′′zj −1 k,l,m =0k +l +m =jp k q l q m .(23)With the initial condition p j (0)=q j (0)=0,the system is easily integrated and uniquely determines the functions p j (z ),q j (z ),p j (z )= z 0a j (ζ)dζ+iνzdζζζdξ(a j (ξ)+b j (ξ)).(24)These equations with expressions (23)together establish the recursion relationfor the functions p j (z ),q j (z ).In terms of p j (z )and q j (z ),expansion (2)readsg +=ei x22+iν)ln 4t +i(ν′)2ln 24tt2t−(18tv 0∞ j =0q j ln 24tln j 4t.(25)6Let a j (z )and b j (z )be polynomials of degree M with the zero z =0of multiplicity m ,a j (z )=M k =ma jk z k,b j (z )=Mk =mb jk z k .Then the functions p j (z )and q j (z )(24)arepolynomials of degree M +1witha zero at z =0of multiplicity m +1,p j (z )=M +1k =m +11k(a j,k −1+b j,k −1)z k ,q j (z )=M +1k =m +11k(a j,k −1+b j,k −1) z k.(26)On the other hand,a j (z )and b j (z )are described in (23)as the actions of the differential operators applied to the functions p j ′,q j ′with j ′<j .Because p 0(z )=q 0(z )≡1and p 1(z ),q 1(z )are polynomials of the second degree and a single zero at z =0,cf.(19),it easy to check that a 2(z )and b 2(z )are non-homogeneous polynomials of the third degree such thata 2,3=−(ν′)4(ν′′)2210(2+iν),(27)a 2,0=−iνν′′8−ν′u ′08u 0,b 2,0=iνν′′8−ν′v ′08v 0.Thus p 2(z )and q 2(z )are polynomials of the fourth degree with a single zero at z =0.Some of their coefficients arep 2,4=q 2,4=−(ν′)4(ν′′)24−(1+2iν)ν′′8u 0−ν(u ′0)24−(1−2iν)ν′′8v 0−ν(v ′0)22.Proof .The assertion holds true for j =0,1,2.Let it be correct for ∀j <j ′.Then a j ′(z )and b j ′(z )are defined as the sum of polynomials.The maximal de-grees of such polynomials are deg (p j ′−1f 0)′/f 0 =2j ′−1,deg (q j ′−1g 0)′/g 0 =72j′−1,anddeg 1z j′−1 α,β,γ=0α+β+γ=j′pαqβqγ =2j′−1. Thus deg a j′(z)=deg b j′(z)≤2j′−1,and deg p j′(z)=deg q j′(z)≤2j′.Multiplicity of the zero at z=0of a j′(z)and b j′(z)is no less than the min-imal multiplicity of the summed polynomials in(23),but the minor coefficients of the polynomials2∂z p j′−1and−(j−1)p j′−1/z,as well as of2∂z q j′−1and −(j−1)q j′−1/z may cancel each other.Let j′=2k be even.Thenm j′=min m j′−1;m j′−2+1;minα,β,γ=0,...,j′−1α+β+γ=j′mα+mβ+mγ =j′2 . Let j′=2k−1be odd.Then2m j′−1−(j′−1)=0,andm j′=min m j′−1+1;m j′−2+1;minα,β,γ=0,...,j′−1α+β+γ=j′mα+mβ+mγ =j′+12]p j,k z k,q j(z)=2jk=[j+12]z nn−[j+18k k!,g j(z)=v0∞n=[j+12]k=max{0;n−2j}q j,n−k(−i)k(ν′)2k2]k=max{0;n−2j}p j,n−ki k(ν′)2k2]k=max{0;n−2j}q j,n−k(−i)k(ν′)2kIn particular,the leading asymptotic term of these coefficients as n→∞and j fixed is given byu n,2n−j=u0p j,2j i n−2j(ν′)2(n−2j)n) ,v n,2n−j=v0q j,2j (−i)n−2j(ν′)2(n−2j)n) .(32)Thus we have reduced the problem of the evaluation of the asymptotics of the coefficients u n,2n−j v n,2n−j for large n to the computation of the leading coefficients of the polynomials p j(z),q j(z).In fact,using(24)or(26)and(23), it can be shown that the coefficients p j,2j,q j,2j satisfy the recurrence relationsp j,2j=−i (ν′)2ν′′2jj−1k,l,m=0k+l+m=jp k,2k p l,2l q m,2m++ν(ν′)2ν′′4j2j−1k,l,m=0k+l+m=jp k,2k(p l,2l−q l,2l)q m,2m,q j,2j=i (ν′)2ν′′2jj−1k,l,m=0k+l+m=jp k,2k q l,2l q m,2m−(33)−ν(ν′)2ν′′4j2j−1k,l,m=0k+l+m=jp k,2k(p l,2l−q l,2l)q m,2m.Similarly,the coefficients u n,0,v n,0for the non-logarithmic terms appears from(31)for j=2n,and are given simply byu n,0=u0p2n,n,v n,0=v0q2n,n.(34) Thus the problem of evaluation of the asymptotics of the coefficients u n,0,v n,0 for n large is equivalent to computation of the asymptotics of the minor coeffi-cients in the polynomials p j(z),q j(z).However,the last problem does not allow a straightforward solution because,according to(8),the sectional generating functions for the coefficients u n,0,v n,0are given byF(z,0)=∞n=0u n,0z n,G(z,0)=∞n=0v n,0z n,and solve the separated Nonlinear Schr¨o dinger equation−iFζ+izF z=νF+18zG′′+F G2.(35)93DiscussionOur consideration based on the use of generating functions of different types reveals the asymptotic behavior of the coefficients u n,2n−j,v n,2n−j as n→∞and jfixed for the long time asymptotic expansion(2)of the generic solution of the sNLS equation(1).The leading order dependence of these coefficients on n is described by the ratio a n2+d).The investigation of theRiemann-Hilbert problem for the sNLS equation yielding this estimate will be published elsewhere.Acknowledgments.We are grateful to the support of NSF Grant PHY-9988566.We also express our gratitude to P.Deift,A.Its and X.Zhou for discussions.A.K.was partially supported by the Russian Foundation for Basic Research under grant99-01-00687.He is also grateful to the staffof C.N.Yang Institute for Theoretical Physics of the State University of New York at Stony Brook for hospitality during his visit when this work was done. References[1]F.G¨o hmann,V.E.Korepin,Phys.Lett.A260(1999)516.[2]F.G¨o hmann,A.R.Its,V.E.Korepin,Phys.Lett.A249(1998)117.[3]F.G¨o hmann,A.G.Izergin,V.E.Korepin,A.G.Pronko,Int.J.Modern Phys.B12no.23(1998)2409.[4]V.E.Zakharov,S.V.Manakov,S.P.Novikov,L.P.Pitaevskiy,Soli-ton theory.Inverse scattering transform method,Moscow,Nauka,1980.[5]F.Calogero,A.Degasperis,Spectral transforms and solitons:toolsto solve and investigate nonlinear evolution equations,Amsterdam-New York-Oxford,1980.[6]M.J.Ablowitz,H.Segur,Solitons and the inverse scattering trans-form,SIAM,Philadelphia,1981.10[7]R.K.Dodd,J.C.Eilbeck,J.D.Gibbon,H.C.Morris,Solitons andnonlinear wave equations,Academic Press,London-Orlando-San Diego-New York-Toronto-Montreal-Sydney-Tokyo,1982.[8]L.D.Faddeev,L.A.Takhtajan,Hamiltonian Approach to the Soli-ton Theory,Nauka,Moscow,1986.[9]P.Deift,X.Zhou,Comm.Math.Phys.165(1995)175.[10]C.S.Gardner,J.M.Greene,M.D.Kruskal,R.M.Miura,Phys.Rev.Lett.19(1967)1095.[11]x,Comm.Pure Appl.Math.21(1968)467.[12]V.E.Zakharov,A.B.Shabat,Funkts.Analiz Prilozh.13(1979)13.[13]V.E.Zakharov,A.B.Shabat,JETP61(1971)118.[14]S.V.Manakov,JETP65(1973)505.[15]V.E.Zakharov,S.V.Manakov,JETP71(1973)203.[16]H.Segur,M.J.Ablowitz,J.Math.Phys.17(1976)710.[17]A.R.Its,A.G.Izergin,V.E.Korepin,G.G.Varzugin,Physica D54(1992)351.[18]H.Segur,J.Math.Phys.17(1976)714.[19]C.Kim,Z.-X.M.Shen,N.Motoyama,H.Eisaki,hida,T.To-hyama and S.Maekawa Phys Rev Lett.82(1999)802[20]A.R.Its,SR Izvestiya26(1986)497.11。
cfd中的asymptotic theory -回复
cfd中的asymptotic theory -回复[cfd中的asymptotic theory]Asymptotic theory plays a vital role in Computational Fluid Dynamics (CFD) as it provides a mathematical framework to understand the behavior of fluid flow in various situations. In this article, we will explore the fundamentals of asymptotic theory in CFD and its applications.What is asymptotic theory?Asymptotic theory is a branch of mathematics that deals with the behavior of functions as a variable approaches a particular value. It provides a systematic way to approximate complex functions and simplify their analysis. In CFD, asymptotic theory is used to solve fluid flow problems by approximating complex equations and simplifying their solutions.The foundation of asymptotic theory in CFD lies in the concept of a small parameter. A small parameter is a dimensionless quantity that represents the scale of a physical phenomenon relative to other quantities in the problem. It allows us to analyze thebehavior of the fluid flow as the small parameter tends to zero or infinity.The key idea behind asymptotic theory is that we can approximate the solution to a problem by expanding it as a series in powers of the small parameter. The resulting series is then truncated at an appropriate order to obtain an approximate solution. This approach is particularly useful when the problem involves a wide range of scales, such as in turbulent flows or flows with thin boundary layers.Applications of asymptotic theory in CFD1. Boundary layer theory: Boundary layers are thin layers of fluid that form near solid boundaries in a fluid flow. They play a crucial role in many engineering applications, such as aerodynamics and heat transfer. Asymptotic theory allows us to derive simplified equations for the boundary layer and study its behavior.By assuming the small parameter to be the ratio of the boundary layer thickness to the characteristic length scale of the problem, we can derive the famous Prandtl's boundary layer equations.These equations provide a simplified description of the flow near the boundary and have been widely used to analyze various flow problems.2. Perturbation methods: Perturbation methods involve solving a problem by treating the small parameter as a small perturbation to an already known solution. This approach is particularly useful when the small parameter is not related to a physical scale but appears in the governing equations due to simplifications or assumptions.For example, in the study of unsteady flows, the small parameter can be the ratio of the unsteadiness time scale to a characteristic time scale of the problem. By assuming a known steady solution and perturbing it in terms of the small parameter, we can develop a systematic procedure to obtain the unsteady solution.3. Homogenization theory: Homogenization theory is concerned with problems where the governing equations have oscillatory coefficients or rapidly varying parameters. These problems arise in various fields, such as porous media flow and composite materials.Asymptotic theory provides a powerful tool to derive effective (homogenized) equations that capture the behavior of the system on a macroscopic scale. By assuming the small parameter as the ratio of the periodicity length to the characteristic length scale of the problem, we can develop an asymptotic expansion to obtain the macroscopic equations.ConclusionAsymptotic theory is an essential tool in CFD for studying complex fluid flow problems. It allows us to approximate the solution to a problem by expanding it as a series in powers of a small parameter and truncating the series at an appropriate order. This approach provides simplified equations that capture the essential behavior of the flow and enable efficient computational analysis. Through examples such as boundary layer theory, perturbation methods, and homogenization theory, we have seen how asymptotic theory finds applications in various areas of CFD. By using asymptotic theory, researchers can gain valuable insights into the behavior of fluid flow and develop efficient numericalmethods for practical engineering applications.。
无症状心律失常的管理
• 在无症状的室早中潜在的心脏病是一个预后不良 的标志。
室性早搏
(premature ventricular capture,PVC)
ACE、β-受体阻滞剂、醛固酮受体拮抗剂、ICD
最佳医疗治疗 (血管紧张素转换酶抑制剂、β-受体阻断剂和皮质激素受体拮抗剂) ICD如果LVEF <30%见相关指南
β受体阻断剂 LVEF<30%和排除急性期心肌炎时考虑ICD
二尖瓣脱垂心肌病 HCM
可能增加
不确定 心脏MRI识别心肌瘢痕
β-受体阻滞剂的好处尚不清楚 可以在特定的情况下考虑ICD
者,或对房颤症状已经适应了的患者。 • 5、对于无症状房颤且有快速室率的患者,应处方速率控制药物,以降低心动
过速性心肌病风险。 • 6、在详细知情同意后,可根据患者的偏好,无症状房颤患者也可考虑消融。
建议:
建议:
室性早搏
(premature ventricular capture,PVC)
• 孤立和稀疏的室早在大多数个体中是正常的。然 而,在某些个体中,可能存在较多的室早。
室性早搏诱导的心肌病(PVC-induced cardiomyopathy)
• 频发室性早搏(通常定义为每>10-15%)可损害 左室功能(室性早搏引起的心肌病),可通过药 物治疗或导管消融术及标准HF治疗逆转
• 与左室功能障碍相关的因素包括:宽 QRS波、 心外膜室早、室早逆传房、插入性室早。
初三英语完形填空深度理解单选题60题
初三英语完形填空深度理解单选题60题1. In the classic novel, the character was in a dilemma. He had to ______ between staying with his family and pursuing his dream.A. chooseB. selectC. electD. pick答案:A。
解析:choose是普通用词,侧重根据个人意愿和判断从众多的对象中进行选择,这里表达在家庭和梦想之间做出选择,比较通用。
select较正式,强调经过认真考虑后的挑选,通常是从多个类似事物中进行挑选,在这里语境没有那么正式。
elect主要用于选举,是选举某人担任某职位,不符合语境。
pick通常用于口语,有挑选、采、摘等意思,不如choose正式且常用于这种两难抉择的语境。
2. The scientific article mentioned that the new species ______ a unique feature that distinguishes it from others.A. hasB. ownsC. possessesD. holds答案:C。
解析:possess表示拥有,常指拥有抽象的东西,如品质、特征等,这里说新物种拥有独特的特征,用possess最合适。
has 是最普通的表示“有”,比较口语化。
own强调合法地拥有某物,多指所属关系,如拥有财产等,这里不是指所属关系。
hold表示握住、持有(具体东西)或者举行(会议等),不符合这里表示拥有特征的语境。
3. In the adventure story, the hero ______ his courage when facing the dangerous situation.A. showedB. displayedC. exhibitedD. demonstrated答案:A。
risk a sociological theory -回复
risk a sociological theory -回复Risk: A Sociological TheoryIn modern society, the concept of risk has gained significant attention and is widely discussed in various fields such as sociology, psychology, economics, and public policy. From social interactions and everyday activities to global threats like climate change and pandemics, the notion of risk has become a fundamental aspect of our lives. In this article, we will delve into the sociological theory of risk, exploring its origins, key concepts, and implications for understanding social behavior and dynamics.Origins of the Sociological Theory of RiskThe sociological theory of risk emerged in the late 20th century as a response to the growing recognition of the omnipresence and complexity of risks in modern societies. Building upon the work of scholars like Ulrich Beck and Anthony Giddens, this theory aimed to understand how risks are socially constructed and how they shape individual behavior and societal dynamics.Key Concepts of the Sociological Theory of RiskCentral to the sociological theory of risk is the idea that risks are not objective, natural phenomena but are socially constructed. Risk is defined as the possibility of harm, loss, or negative consequences associated with a particular action, behavior, or situation. It is not an inherent property of an activity or event, but rather a product of social interactions, perceptions, and judgments.The theory emphasizes three key concepts: risk society, reflexive modernization, and manufactured uncertainty.1. Risk Society: According to Ulrich Beck, modern societies have transformed into risk societies, where risks are pervasive and play a central role in shaping socioeconomic structures and individual lives. In risk societies, traditional sources of authority, such as religion and tradition, have diminished, and decision-making is increasingly influenced by the evaluation and management of risks.2. Reflexive Modernization: Anthony Giddens introduced the concept of reflexive modernization, which refers to the ongoing process of individuals and societies reflecting on and adapting to new risks and uncertainties. As individuals become more aware ofrisks, they engage in reflexive behavior, constantly questioning and reevaluating their actions and choices.3. Manufactured Uncertainty: This concept highlights the role of institutions and powerful actors in manufacturing uncertainty and obscuring risks. Through processes such as risk assessment, risk communication, and risk management, certain groups control and shape the perception of risks, often favoring their own interests. This manufactured uncertainty can lead to distorted understandings of risks and unequal distribution of their consequences.Implications for Understanding Social Behavior and DynamicsThe sociological theory of risk has important implications for understanding social behavior and dynamics. It highlights the ongoing negotiation and management of risks within society, revealing the intricate interplay between individuals, institutions, and power relations.1. Risk Perception and Decision-making: The theory emphasizes that risk perception is socially constructed and influenced byfactors such as media representation, cultural values, and personal experiences. Different social groups may have divergent perceptions of risks, which shape their decision-making processes. Sociologists studying risk examine how these perceptions and decisions are socially mediated and reflect broader social dynamics.2. Social Inequality and Risk Burdens: The theory draws attention to the unequal distribution of risks within society. Marginalized and disadvantaged groups often bear a disproportionate burden of risks and their negative consequences. For example, low-income neighborhoods may face higher risks of environmental hazards due to discriminatory policies and structural inequalities. Sociologists analyze how social structures and power differentials contribute to the unequal distribution of risks.3. Social Movements and Risk Protest: The theory highlights the role of social movements in challenging and contesting risks and their management. Activist groups often mobilize around specific risks to raise awareness, demand accountability, and advocate for change. Sociologists explore the dynamics of risk protest, examining how social movements shape public discourse and influence policy decisions.Overall, the sociological theory of risk provides a valuable framework for understanding the social dimensions of risks in contemporary society. By acknowledging the socially constructed nature of risks and examining their implications for individual behavior and social dynamics, sociologists contribute to broader discussions on risk governance, social justice, and sustainable development.。
院内获得耐碳青霉烯类抗菌药物肺炎克雷伯菌感染的危险因素分析
院内获得耐碳青霉烯类抗菌药物肺炎克雷伯菌感染的危险因素分析杨健(阜宁县人民医院检验科,江苏阜宁224400)[摘要]目的:了解院内获得耐碳青霉烯类抗菌药物肺炎克雷伯菌(CRKP)感染的危险因素。
方法:选取2017年3月~2019年3月收治的CRKP院内感染患者86例,随机分为两组,各43例,观察组以亚胺培南或美罗培南MIC逸4滋g/L为标准,对照组以亚胺培南或美罗培南MICW1滋g/L为标准。
比较两组CRKP感染的单因素分析与分布率、CRKP院内感染的独立危险因素。
结果:观察组住院时间逸14d、抗菌药物暴露时间逸14d、暴露抗菌药物种类逸2种、暴露于碳青霉烯类抗菌药物、暴露于糖肽类抗菌药物、合并其他细菌感染逸2种等其分布率均高于对照组,差异有统计学意义(P<0.05);观察组CRKP院内感染的独立因素为暴露于碳青霉烯类抗菌药物,差异有统计学意义(P<0.05)。
结论:以亚胺培南或美罗培南MICW1滋g/L为标准对院内获得耐碳青霉烯类抗菌药物肺炎克雷伯菌感染的危险因素进行分析可知,CRKP院内感染的独立因素为暴露于碳青霉烯类抗菌药物。
[关键词]肺炎克雷伯菌;碳青霉烯类抗菌药物;危险因素Risk factors of carbapenem resistant Klebsiella pneumoniae infection in hospital YANG Jian(Depart-ment of Clinical Laboratory,Suining People's Hospital,Suining224400,China)Abstract:Objective To explore the risk factors of nosocomial infection with carbapenem-resistant Klebsiella pneumoniae (CRKP).Method From March2017to March2019,86patients with CRKP nosocomial infection were randomly divided into two groups,43cases in each group.The observation group was treated with imipenem or meropenem MIC(逸4滋g/L)as the standard.The control group was treated with imipenem or meropenem MIC(臆1滋g/L)as the standard.The single factor analysis and distribution rate of CRKP infection and independent risk factors of CRKP infection in two groups were compared.Results The distribution rate of CRKP in the observation group was higher than that in the control group,and the difference was statistically significant(P<0.05).The independent factor of nosocomial infection was exposure to carbapenems,and the difference was statistically significant(P<0.05).Conclusion The risk factors of nosocomial Klebsiella pneumoniae infection with carbapenem-resistant antibiotics were analyzed by MIC臆1滋g/L of imipenem or meropenem.The independent factor of CRKP nosocomial infection was exposure to carbapenem antibiotics.Key Words:Klebsiella pneumoniae;Carbapenem antibiotics;Risk factors肺炎克雷伯菌(KP)存在于机体上呼吸道、肠道等部位,当机体抵抗力下降时,易使该细菌由呼吸道进入肺内,能够导致机体多部位出现感染[1]。
肥胖的危害英语作文
Obesity is a growing concern in modern society,affecting not only physical health but also mental wellbeing.Here are some of the key harms associated with obesity:1.Increased Risk of Chronic Diseases:Obesity is linked to a higher risk of developing chronic conditions such as type2diabetes,hypertension,and heart disease.The extra weight puts additional strain on the bodys systems,leading to a higher likelihood of these health issues.2.Respiratory Problems:Excess body weight can cause or exacerbate respiratory issues like sleep apnea,a condition where breathing repeatedly stops and starts during sleep,and chronic obstructive pulmonary disease COPD,which affects the lungs ability to function properly.3.Joint and Musculoskeletal Issues:The extra weight puts significant pressure on joints, leading to conditions such as osteoarthritis,which can be debilitating and limit mobility.4.Cancer Risk:Studies have shown that obesity is associated with an increased risk of several types of cancer,including breast,colon,and uterine cancer.5.Mental Health Impact:The psychological effects of obesity can be profound,with individuals often facing stigma and discrimination.This can lead to depression,anxiety, and low selfesteem.6.Reduced Life Expectancy:The cumulative effect of the health risks associated with obesity can lead to a reduced life expectancy.7.Quality of Life:Obesity can limit ones ability to participate in physical activities and can affect overall quality of life,making it difficult to enjoy hobbies and social activities.8.Sleep Disturbances:As mentioned with sleep apnea,obesity can lead to other sleep disturbances that affect the quality of rest and recovery the body needs.9.Fertility Issues:Obesity can affect both mens and womens fertility,making it more difficult to conceive.10.Increased Health Care Costs:The medical costs associated with treating obesityrelated conditions can be substantial,placing a financial burden on individuals and health care systems.Addressing obesity requires a multifaceted approach,including lifestyle changes,medicalinterventions when necessary,and societal efforts to promote healthier habits.It is essential to recognize the seriousness of obesity and take proactive steps to prevent and manage it for the sake of overall health and wellbeing.。
英国手术预防使用抗菌药物指南
CONTENTS
Contents
1 Introduction...................................................................................................................... 1 1.1 1.2 1.3 1.4 2 2.1 2.2 2.3 The need for a guideline.................................................................................................... 1 Remit of the guideline........................................................................................................ 1 Definitions......................................................................................................................... 3 Statement of intent............................................................................................................. 3 Key ............................................ 4 Benefits and risks of antibiotic prophylaxis......................................................................... 4 Administration of prophylactic antibiotics.......................................................................... 4 Implementing the guideline. ............................................................................................... 5
Tradeoff between source and channel coding
decreases; we therefore seek the smallest r, sometimes known as the critical rate 4], for which the maximizing equals one. From (35) we see that the maximizing equals one when r = 1 ? log ? = : As k increases, the rk satisfying E (rk ) = (p=k)rk increases. Hence, for some critical value k0, we have rk 1 ? log ? = when k > k0. From (13),
sp sp r
p p
p
29
D Proof of Lemma 4
In Appendix C it is argued that (1 ? r) ? ( + 1) log = + (1 ? ) = ] is a concave function of , and therefore any stationary point must be a maximum. De ne
1 (1+ ) 1 (1+ )
1 (k) = p k
2pC log e log + (1 ? ) log (1 ? ) ? H ( )
2 2 2
1 2 = ? log ? (1 ? ) log(1 ? ) is the binary entropy. Then, it follows from (33) that any r = r(k) for which (k) is a stationary point must satisfy
因果推断英语
因果推断英语Causal Inference in EnglishCausal inference is a fundamental concept within the realm of statistics and scientific research that revolves around establishing cause-and-effect relationships between variables. This process aims to answer the "why" behind observed phenomena, going beyond mere correlations to uncover the underlying mechanisms that drive change.At its core, causal inference involves several key principles:1. Counterfactual Thinking: This principle posits that to understand causality, we must consider what would have happened if a particular event or treatment had not occurred. Essentially, it's comparing the actual outcome with what would have been if things were different.2. Randomized Controlled Trials (RCTs): Often hailed as the gold standard in causal inference, RCTs involve randomly assigning subjects to either a treatment group or a control group. By introducing randomness, researchers can isolate the effect of the treatment from other factors, providing strong evidence for causation.3. Covariate Adjustment: In real-world scenarios where randomization isn't feasible, researchers use statisticalmethods to adjust for confounding variables—factors that could influence both the treatment and the outcome. Techniques like regression analysis help to control for these covariates, improving the accuracy of causal estimates.4. Causal Graphs: These are visual representations of causal relationships among variables, aiding in identifying pathways of causation and potential sources of bias. By diagramming assumed causal links, researchers can better design studies and interpret data.5. Instrumental Variables: In complex situations where direct manipulation isn't possible or ethical, instrumental variables can be used. This approach involves identifying a variable (the instrument) that affects the treatment but is unrelated to the outcome except through the treatment. It serves as a proxy to infer causality.6. Causal DAGs (Directed Acyclic Graphs): These diagrams represent causal assumptions in a formal way, allowing researchers to systematically evaluate different paths of causation and test for various types of causal effects.7. Potential Outcomes Framework: Introduced by Neyman and developed further by Rubin, this framework considers the potential outcomes under different treatment states for eachunit (individual or group). It forms the basis for many modern causal inference techniques, including matching and propensity score analysis.Overall, causal inference is a rigorous and multifaceted endeavor that requires careful consideration of study design, statistical analysis, and a deep understanding of the domain being studied. Its ultimate goal is to provide reliable evidence for decision-making and to advance our understanding of how the world operates.。
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If one has no knowledge on the unknown parameter θ, he or she tends to adopt a noninformative prior. It is often recommended to use the Jeffreys prior as a noninformative prior due to several reasons. However the Jeffreys prior is often improper, i.e., π (θ)dθ = ∞.
(Dated: February 2, 2008)
Abstract
The autoregressive moving average (ARMA) model is one of the most important models in time series analysis. We consider the Bayesian estimation of an unknown spectral density in the ARMA model. In the i.i.d. cases, Komaki showed that Bayesian predictive densities based on a superharmonic prior asymptotically dominate those based on the Jeffreys prior [7]. It is shown by using the asymptotic expansion of the risk difference. We obtain the corresponding result in the ARMA model.
S (ω |θ0) ˆ (ω ) S
பைடு நூலகம்
.
The above setting is proposed by Komaki [6].
B.
Bayesian framework
First, let us consider minimizing the average risk assuming that a proper prior density π (θ) is known in advance. Aitchison’s result [1] applies to this setting. The spectral density estimator minimizing the average risk, ˆ(ω ))] E Θ E X [D (S (ω |θ)||S := dθπ (θ) ˆ(ω )), dx1 . . . dxn pn (x1 , . . . , xn |θ)D (S (ω |θ)||S
A.
General setting
Let us consider a parametric model of stationary Gaussian process with mean zero. It is known that a stationary Gaussian process corresponds to its spectral density one-to-one (for proof, see, e.g., [4]). Thus, we focus on the estimation of the true spectral density S (ω |θ0) in a parametric family of spectral densities M := {S (ω |θ) : θ ∈ Θ ⊆ Rk }. 2
II.
PRELIMINARY A. Notation and assumption
In such a situation, the above result does not hold any more and other noninformative priors could be recommended. One of those is a superharmonic prior. Komaki showed that Bayesian predictive distributions based on a superharmonic prior asymptotically dominate those based on the Jeffreys prior [7]. He compared two prior distributions by using the asymptotic expansion of the risk difference. In the present paper, we extend this result to the ARMA process. We formulate the prediction problem of spectral densities in the ARMA model as described below and obtain the asymptotic expansion of the risk difference. Our conclusion is the same as in the i.i.d. cases. Since we used the properties of the ARMA model only when evaluating the expectation of the log likelihood, it can be expected that almost all our arguments hold true in general stationary Gaussian processes.
is given by the Bayesian spectral density (with respect to π (θ)), which is defined by Sπ (ω ) := S (ω |θ)π (θ|x)dθ. (1)
We call Sπ (ω ) in (1) a Bayesian spectral density even when an improper prior distribution is considered.
C.
Choice of a noninformative prior
If one has no information on the unknown parameter θ, it is natural to adopt a noninformative prior in the Bayesian framework. There is much room to argue the choice of a noninformative prior. While the Jeffreys prior is a well-known candidate from several reasons, it can be expected that it is better to adopt a superharmonic prior in some cases. The reason is that stationary Gaussian processes are getting close to the i.i.d. cases as the sample size becomes large and a superharmonic prior can be better than the Jeffreys prior in the i.i.d. cases.
ˆ(ω ) is evaluated by the Kullback-Leibler The performance of a spectral density estimator S divergence. ˆ(ω )) := D (S (ω |θ0 )||S
π −π
dω 4π
S (ω |θ0 ) − 1 − log ˆ (ω ) S
APS/123-QED
Asymptotic Expansion of the Risk Difference of the Bayesian Spectral Density in the ARMA model
arXiv:math/0510558v1 [math.ST] 26 Oct 2005
Fuyuhiko Tanaka∗ and Fumiyasu Komaki Department of Mathematical Informatics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan
3
D.
Construction
In the following section, we briefly review basic results necessary to the asymptotic expansion. The asymptotic expansion of the posterior distribution is presented. In section 3, we obtain the asymptotic expansion of the Bayesian spectral density. For the ARMA model, it can be also written in the differential-geometrical quantities as in the i.i.d. cases. In section 4, we evaluate the expectation of the KL-divergence from the true spectral density S (ω |θ0) to the Bayesian spectral density Sf up to the second order for an arbitrary prior (possibly improper) f (θ). Finally, we obtain the principal term of the risk difference between Sf and SπJ , where πJ denotes the Jeffreys prior. As a direct consequence of this result, a superharmonic prior is recommended as a noninformative one if there exists a positive superharmonic function on the corresponding model manifold.