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Candidates may use any calculator allowed by the regulations of theJoint Council for Qualifications. Calculators must not have the facilityfor symbolic algebra manipulation, differentiation and integration, orhave retrievable mathematical formulae stored in them.Instructions• Use black ink or ball-point pen.•I f pencil is used for diagrams/sketches/graphs it must be dark (HB or B).Coloured pencils and highlighter pens must not be used.•Fill in the boxes at the top of this page with your name,centre number and candidate number.•A nswer all questions and ensure that your answers to parts of questions areclearly labelled.•A nswer the questions in the spaces provided– there may be more space than you need.•Y ou should show sufficient working to make your methods clear. Answerswithout working may not gain full credit.•V alues from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.Information•The total mark for this paper is 75.•T he marks for each question are shown in brackets– use this as a guide as to how much time to spend on each question.Advice•Read each question carefully before you start to answer it.•Try to answer every question.•Check your answers if you have time at the end.*P48241A0128*Turn over P48241A©2016 Pearson Education Ltd.1/1/1/1/1/Leave blank 2*P48241A0228*DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 1. A mobile phone company claims that each year 5% of its customers have their mobile phone stolen. An insurance company claims this percentage is higher. A random sample of 30 of the mobile phone company’s customers is taken and 4 of them have had their mobile phone stolen during the last year.(a) Test the insurance company’s claim at the 10% level of significance. State your hypotheses clearly.(6)A new random sample of 90 customers is taken. A test is carried out using these 90 customers, to see if the percentage of customers who have had a mobile phone stolen in the last year is more than 5%(b) Using a suitable approximation and a 10% level of significance, find the critical region for this 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The lifetime of a particular battery, T hours, is modelled using the cumulative distribution functiont < 88 t 12t > 12F()()t t t k =−+ 0741196522(a) Show that k = –432(2)(b) Find the probability density function of T , for all values of t.(2)(c) Write down the mode of T .(1)(d) Find the median of T.(3) (e) Find the probability that a randomly selected battery has a lifetime of less than 9 hours.(2)A battery is selected at random. Given that its lifetime is at least 9 hours,(f) find the probability that its lifetime is no more than 11 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14 marks)Leave blank 10*P48241A01028*DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 3. A large number of students sat an examination. All of the students answered the first question. The first question was answered correctly by 40% of the students.In a random sample of 20 students who sat the examination, X denotes the number of students who answered the first question correctly.(a) Write down the distribution of the random variable X (1)(b) Find P(4 X < 9)(2)Students gain 7 points if they answer the first question correctly and they lose 3 points if they do not answer it correctly. (c) Find the probability that the total number of points scored on the first question by the 20 students is more than 0(4)(d) Calculate the variance of the total number of points scored on the first question by a random sample of 20 students.(3)____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________D O N O T W R I TE I N T H I S A R E A D O N O T W R I T E I N T H I S A R E A D O N O T W R I T E I N T H I S A R E A _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________。
PART IIANSWERS TO END-OF-CHAPTER QUESTIONSCHAPTER 14: INTERNATIONAL LOGISTICS14-1. Discuss some of the key political restrictions on cross-border trade.Political restrictions on cross-border trade can take a variety of forms. Many nations ban certain types of shipments that might jeopardize their national security. Likewise, individual nations may band together to pressure another country from being an active supplier of materials that could be used to build nuclear weapons. Some nations restrict the outflow of currency because a nation,s economy will suffer if it imports more than it exports over a long term. A relatively commonpolitical restriction on trade involves tariffs, or taxes that governments place on the importation of certain items. Another group of political restrictions can be classified as nontariff barriers, which refer to restrictions other than tariffs that are placed upon imported products. Another political restriction involves embargoes, or the prohibition of trade between particular countries.14-2. How might a particular country,s government be involved in international trade?Governments may exert strong control over ocean and air traffic because they operate as extensions of a nation,s economy and most of the revenue flows into that nation's economy. In some cases, import licenses may restrict movement to a vessel or plane owned or operated by the importing country. In addition, some nations provide subsidies to develop and/or maintain their ocean and air carriers. Governments also support their own carriers through cargo preference rules, which require a certain percentage of traffic to move on a nation,s flag vessels. Although federal governments have often owned ocean carriers and international airlines, some government-owned international carriers are moving toward the private sector.14-3. Discuss how a nation,s market size might impact international trade and, in turn,international logistics.Population is one proxy for market size, and China and India account for about one-third of the world's population. As such, these two countries might be potentially attractive markets because of their absolute and relative size. Having said this, India has a relatively low gross domestic product per capita, and because of this some customers buy singleuse packets of products called sachets. From a logistical perspective, single-use packets require different packaging, and are easier to lose and more prone to theft than products sold in larger quantities.14-4. How might economic integration impact international logistics?Potential logistical implications of economic integration include reduced documentation requirements, reduced tariffs, and the redesign of distribution networks. For example,Poland and the Czech Republic have become favorite distribution sites with the eastward expansion of the European Union.14-5. How can language considerations impact the packaging and labeling of international shipments?With respect to language, cargo handlers may not be able to read and understand the language of the exporting country, and it would not be unusual for cargo handlers in some countries to beilliterate. Hence, cautionary symbols, rather than writing, must be used. A shipper,s mark, which looks like a cattle brand, is used in areas where dockworkers cannot read but need a method to keep documents and shipments together.14-6. What is a certificate of origin, a commercial invoice, and a shipper,s export declaration?A certificate of origin specifies the country or countries in which a product is manufactured .This document can be required by governments for control purposes or by an exporter to verify thelocation of manufacture. A commercial invoice is similar in nature to a domestic bill of lading in the sense that a commercial invoice summarizes the entire transaction and contains (or should contain) key information to include a description of the goods, the terms of sale and payment, the shipment quantity, the method of shipment, and so on. A shipper,s export declaration contains relevant export transaction data such as the transportation mode(s), transaction participants, and a description of what is being exported.14-7. Discuss international terms of sale and Incoterms.International terms of sale determine where and when buyers and seller will transfer 1) the physical goods; 2) payment for goods, freight charges, and insurance for the in-transit goods; 3) legal title to the goods; 4) required documentation; and 5) responsibility for controlling or caring for the goods in transit. The International Chamber of Commerce is in charge of establishing, and periodically revising, the terms of sale for international shipments, commonly referred to as Incoterms. The most recent revision, Incoterms 2010, reflects the rapid expansion of global trade with a particular focus on improved cargo security and new trends in cross-border transportation. Incoterms 2010 are now organized by modes of transport and the terms can be used in both international and domestic transportation.14-8. Name the four methods of payment for international shipments. Which method is riskiest for the buyer? For the seller?Four distinct methods of payment exist for international shipments: cash in advance, letters of credit, bills of exchange, and the open account. Cash in advance is of minimal risk to the seller, but is the riskiest for the buyer-what if the paid-for product is never received? The open account involves tremendous potential risk for the seller and minimal risk for the buyer.14-9. Discuss four possible functions that might be performed by international freight forwarders.The text describes eight functions, such as preparing an export declaration and booking space on carriers, so discussion of any four would be appropriate.14-10. What is an NVOCC?An NVOCC (nonvessel-operating common carrier) is often confused with an international freight forwarder. Although both NVOCCs and international freight forwarders must be licensed by the Federal Maritime Commission, NVOCCs are common carriers and thus have common carrier obligations to serveand deliver, among other obligations. NVOCCs consolidate freight from different shippers and leverage this volume to negotiate favorable transportation rates from ocean carriers. From the shipper's perspective, an NVOCC is a carrier; from an ocean carrier,s perspective, an NVOCC is a shipper.14-11. What are the two primary purposes of export packing?One function is to allow goods to move easily through customs. For a country assessing duties on the weight of both the item and its container, this means selecting lightweight packing materials. The second purpose of export packing is to protect products in what almost always is a more difficult journey than they would experience if they were destined for domestic consignees.14-12. Discuss the importance of water transportation for international trade.A frequently cited statistic is that approximately 60 percent of cross-border shipments move by water transportation. Another example of the importance of water transportation in international trade involves the world,s busiest container ports as measured by TEUs (twenty-foot equivalent units) handled; 9 of the 10 busiest container ports are located in Asia, with 7 of the busiest ports located in China.14-13. Explain the load center concept. How might load centers affect the dynamics of international transportation?Load centers are major ports where thousands of containers arrive and depart each week. As vessel sizes increase, it becomes more costly to stop at multiple ports in a geographic area, and as a result, operators of larger container ships prefer to call at only one port in a geographic area. Load centers might impact the dynamics of international trade in the sense that some ports will be relegated to providing feeder service to the load centers.14-14. Discuss the role of ocean carrier alliances in international logistics.In the mid-1990s, ocean carrier alliances, in which carriers retain their individual identities but cooperate in the area of operations, began forming in the container trades. These alliances provide two primary benefits to participating members, namely, the sharing of vessel space and the ability to offer shippers a broader service network. The size of the alliance allows them to exercise considerable clout in their dealings with shippers, port terminal operators, and connecting land carriers.14-15. How do integrated air carriers impact the effectiveness and efficiency of international logistics?Integrated air carriers own all their vehicles and the facilities that fall in-between. These carriers often provide the fastest service between many major points. They are also employed to carry the documentation that is generated by—and is very much a part of— the international movement of materials. The integrated carriers also handle documentation services for their clients.14-16. How do open-skies agreements differ from bilateral agreements?Bilateral agreements generally involved two countries and tended to be somewhat restrictive in nature. For example, the bilateral agreements would specify the carriers that were to serve particular city pairs. By contrast, open skies agreements liberalize aviation opportunities andlimit federal government involvement. For example, the Open Aviation Agreement between the United States and 27 European Union (EU) member states allows any EU airline as well as any U.S. airline to fly between any point in the EU and any point in the United States.14-17. Discuss the potential sources of delays in certain countries with respect to motor carrier shipments that move across state borders.One source of delays is that certain countries limit a motor carrier,s operations to within a particular state,s borders; as a result, multi-state shipments must be transferred from one company,s vehicle to another company,s vehicle whenever crossing into another state. Another source of delays is that certain countries conduct inspections of trucks as they move from one state to another. This can include physical counting and inspection of all shipments, inspection of documentation, and vehicle inspection, as well as driver inspection.14-18. Define what is meant by short-sea shipping (SSS), and discuss some advantages of SSS.Short-sea shipping (SSS) refers to waterborne transportation that utilizes inland and coastal waterways to move shipments from domestic ports to their destination. Potential benefits to SSS include reduced rail and truck congestion, reduced highway damage, a reduction in truck-related noise and air pollution, and improved waterways utilization.14-19. What are some challenges associated with inventory management in cross-border trade?Because greater uncertainties, misunderstandings, and delays often arise in international movements, safety stocks must be larger. Furthermore, inventory valuation on an international scale isdifficult because of continually changing exchange rates. When a nation,s (or the world's) currency is unstable, investments in inventories rise because they are believed to be less risky than holding cash or securities.Firms involved in international trade must give careful thought to their inventory policies, in part because inventory available for sale in one nation may not necessarily serve the needs of markets in nearby nations. Product return policies are another concern with respect to international inventory management. One issue is that, unlike the United States where products can be returned for virtually reason, some countries don,t allow returns unless the product is defective in some respect.14-20. What is the Logistics Performance Index? How can it be used?The Logistics Performance Index (LPI) was created in recognition of the importance of logistics in global trade. The LPI measures a country,s performance across six logistical dimensions:•Efficiency of the clearance process (i.e., speed, simplicity, and predictability of formalities) by border control agencies, including customs;•Quality of trade- and transport-related infrastructure (e.g., ports, railroads, roads, and information technology);•Ease of arranging competitively priced shipments;•Competence and quality of logistics services (e.g., transport operators and customs brokers);•Capability to track and trace consignments;•Timeliness of shipments in reaching the destination within the scheduled or expected delivery time.The LPI is a potentially valuable international logistics tool because the data can be analyzed from several different perspectives. First, the LPI can be analyzed for all countries according to the overall LPI score as well as according to scores on each of the six dimensions. Second, the LPI can be analyzed in terms of an individual country's performance over time, relative to its geographic region, and relative to its income group.PART IIICASE SOLUTIONSCASE 14-1: Nurnberg Augsburg Maschinenwerke (N.A.M.)Question 1: Assume that you are Weiss. How many viable alternatives do you have to consider regarding the initial shipment of 25 buses?The answer to this question can vary depending on how students define “viable alternatives.” If we take a broad perspective and just focus on the primary cities, Bremerhaven does not appear to be an option because there is no scheduled liner service in the desired time frame. That leaves us with Prague to Santos through Hamburg and Prague to Santos through Rotterdam. Several of the vessel departure dates for both alternatives are not feasible. For example, the 18-day transit time from Hamburg eliminates both the October 31 and November 3 departures; likewise, the 17-day transit time from Rotterdam eliminates the November 2 departure. And although the October 27 departure from Hamburg or the October 28 departure from Rotterdam should get the buses to Santos by November 15, neither departure leaves much room for potential transit delays (e.g., a late season hurricane). As such, it appears that Weiss has but two viable alternatives: the October 24 departure from Hamburg and the October 23 departure from Rotterdam.Question 2: Which of the routing alternatives would you recommend to meet the initial 90-day deadline for the 25-bus shipment? Train or waterway? To which port(s)? What would it cost?If one assumes that rail transport is used from Prague to either Hamburg or Rotterdam, then thetotal transportation costs of the two alternatives are virtually identical. Although rail costs to Rotterdam are €300 higher than to Hamburg, the shipping costs from Rotterdam are €300 lower than from Hamburg (based on €6000 x .95). Because the total transportation costs are essentially the same, the decision likely needs to be based on service considerations. The initial shipment is extremely important. It might be suggested that Prague to Hamburg by rail and Hamburg to Santos by ocean vessel is the preferred alternative. Our rationale is that the provided transit times with Hamburg are definitive— that is, 3 days by rail and 18 days by water. With Rotterdam, by contrast, the rail transit time is either 4 or 5 days, although water transportation is 17 days.Question 3: What additional information would be helpful for answering Question 2?A variety of other information would be helpful for answering Question 2. For example, the case offers no insight about port congestion issues and how this congestion might impact the timeliness of shipment loadings. There also is no information about port performance in terms of loss and damage metrics. In addition, although the case indicates that rail transit time from Prague iseither four or five days, it might be helpful to know what percentage of shipments is completed in four days. Students are likely to come up with more suggestions.Question 4: How important, in fact, are the transport costs for the initial shipment of 25 buses?Clearly, with ocean shipping costs of either €5700 or €6000 per bus, transportation costs cannot be ignored. Having said this, the initial shipment holds the key to the remainder of the order (another 199 buses) and appears to be instrumental in securing another order for 568 buses (for a total of 767 more buses). As such, N.A.M might be somewhat flexible with respect to transportation costs for the initial shipment. Suppose, for example, that N.A.M. can earn a profit of €5000 per bus (such profit on a €120000 bus is by no means exorbitant). A profit of €5000 x 767 buses yields a total profit of €3,835,000. Because of such a large upside with respect to additional orders,N.A.M. might focus on achieving the specified metrics for the initial shipment without being overly concerned with transportation costs.Question 5: What kinds of customer service support must be provided for this initial shipment of 25 buses? Who is responsible?Although a number of different constituencies is involved in the initial shipment (e.g., railroads, dock workers, ocean carrier, etc.), the particular customers—the public transit authorities—are buying product from N.A.M. Because of this, N.A.M. should be the responsible party with respect to customer service support. There are myriad customer service support options that might be provided. Real-time shipment tracking should be an option so that the customers can know, at any time, the location of the shipment. N.A.M. might also provide regular updates of shipment progress; perhaps N.A.M. could email or fax important progress points (e.g., the shipment has left Prague; the shipment has arrived in Hamburg, etc.) to the customers. Because successful performance on theinitial shipment is crucial to securing future business, N.A.M. might have one of its managers actually accompany the shipment.Question 6: The Brazilian buyer wants the buses delivered at Santos. Weiss looks up theInternational Chamber of Commerce,s Incoterms and finds three categories of “delivered” terms:DAT (Delivered at Terminal). In this type of transaction, the seller clears the goods forexport and bears all risks and costs associated with delivering the goods and unloading them at the terminal at the named port or place of destination. The buyer is responsible for all costs and risks from this point forward including clearing the goods for import at the named country of destination.DAP (Delivered at Place). The seller clears the goods for export and bears all risks and costs associated with delivering goods to the named place of destination not unloaded. The buyer is responsible for all costs and risks associated with unloading the goods and clearing customs to import goods into the named country of destination.DDP (Delivered Duty Paid). The seller bears all risks and costs associated with delivering the goods to the named place of destination ready for unloading and clearing for import.How should he choose? Why?Again, given the importance of the initial shipment, it would appear that the more control thatN.A.M. has over the process, the better. Although the DDP option is likely the costliest option, it also affords N.A.M. more control later into the shipment process. Moreover, a willingness by N.A.M. to take on the additional costs associated with DDP might be viewed in a positive fashion by the customers.Question 7: Would you make the same routing recommendation for the second, larger (199 buses) component of the order, after the initial 90-day deadline is met? Why or why not?Time pressures do not appear to be as critical for the larger component of the order, so this might argue for use of water transportation between Prague and Hamburg. The rationale would be that even though water transportation is slower, it saves money (€48 per bus) over rail shipments. Alternatively, given that the selling price per bus is likely to be around €120000, trading off three days of transit time in exchange for a savings of €48 might not be such a good idea.Question 8：How important, if at all, is it for N.A.M. to ship via water to show its support of the European Union,s Motorways of the Seas concept?This question may generate a variety of opinions from students. For example, some students might argue that Question 75s answer also applies to Question 8. Having said this, the case doesn,t delve too deeply into potential environmental considerations associated with water transportation, so a pure cost-benefit analysis (such as Question 7) might be insufficient. Furthermore, because the European Union (EU) continues to be a contentious issue for many Europeans, the answer to Question 8 might depend upon one,s view of the EU. Thus, someone who is supportive of the EU might lean toward supporting the Motorways of the Seas concept, while someone not supportive of the EU might lean against supporting the Motorways of the Seas concept.。
a rXiv:c ond-ma t/9411013v12Nov1994Concentration and energy fluctuations in a critical polymer mixture M.M¨u ller and N.B.Wilding Institut f¨u r Physik,Johannes Gutenberg Universit¨a t,Staudinger Weg 7,D-55099Mainz,Germany Abstract A semi-grand-canonical Monte Carlo algorithm is employed in conjunction with the bond fluctuation model to investigate the critical properties of an asymmetric binary (AB)polymer mixture.By applying the equal peak-weight criterion to the concentration distribution,the coexistence curve separating the A-rich and B-rich phases is identified as a function of temperature and chemical potential.To locate the critical point of the model,the cumulant intersection method is used.The accuracy of this approach for determining the critical parameters of fluids is assessed.Attention is then focused on the joint distribution function of the critical concentration and energy,which is analysed using a mixed-field finite-size-scaling theory that takes due account of the lack of symme-try between the coexisting phases.The essential Ising character of the binary polymer critical point is confirmed by mapping the critical scaling operator distributions onto in-dependently known forms appropriate to the 3D Ising universality class.In the process,estimates are obtained for the field mixing parameters of the model which are compared both with those yielded by a previous method,and with the predictions of a mean field calculation.PACS numbers 64.70Ja,05.70.Jk1IntroductionThe critical point of binary liquid and binary polymer mixtures,has been a subject of abidinginterest to experimentalists and theorists alike for many years now.It is now well established that the critical point properties of binary liquid mixtures fall into the Ising universality class(the default for systems with short ranged interactions and a scalar order parameter)[1]. Recent experimental studies also suggest that the same is true for polymer mixtures[2]–[11].However,since Ising-like critical behaviour is only apparent when the correlation length farexceeds the polymer radius of gyrationξ≫ R g ,the Ising regime in polymer mixtures is confined(for all but the shortest chain lengths)to a very narrow temperature range nearthe critical point.Outside this range a crossover to mean-field type behaviour is seen.Theextent of the Ising region is predicted to narrow with increasing molecular weight in a manner governed by the Ginsburg criterion[12],disappearing entirely in the limit of infinite molecular weight.Although experimental studies of mixtures with differing molecular weights appear to confirm qualitatively this behaviour[5],there are severe problems in understanding the scaling of the so-called“Ginsburg number”(which marks the centre of the crossover region and is empirically extracted from the experimental data[13])with molecular weight Computer simulation potentially offers an additional source of physical insight into polymer critical behaviour,complementing that available from theory and experiment.Unfortunately, simulations of binary polymer mixtures are considerable more exacting in computational terms than those of simple liquid or magnetic systems.The difficulties stem from the problems of deal-ing with the extended physical structure of polymers.In conventional canonical simulations, this gives rise to extremely slow polymer diffusion rates,manifest in protracted correlation times[14,15].Moreover,the canonical ensemble does not easily permit a satisfactory treat-ment of concentrationfluctuations,which are an essential feature of the near-critical region in polymer mixture.In this latter regard,semi-grand-canonical ensemble(SGCE)Monte Carlo schemes are potentially more attractive than their canonical counterparts.In SGCE schemes one attempts to exchange a polymer of species A for one of species B or vice-versa,thereby permitting the concentration of the two species tofluctuate.Owing,however,to excluded volume restrictions,the acceptance rate for such exchanges is in general prohibitively small, except in the restricted case of symmetric polymer mixtures,where the molecular weights of the two coexisting species are identical(N A=N B).All previous simulation work has therefore focussed on these symmetric systems,mapping the phase diagram as a function of chain length and confirming the Ising character of the critical point[16,17,18].Tentative evidence for a crossover from Ising to meanfield behaviour away from the critical point was also obtained [19].Hitherto however,no simulation studies of asymmetric polymer mixtures(N A=N B) have been reported.Recently one of us has developed a new type of SGCE Monte Carlo method that amelio-rates somewhat the computational difficulties of dealing with asymmetric polymer mixtures [20].The method,which is described briefly in section3.1,permits the study of mixtures of polymer species of molecular weight N A and N B=kN A,with k=2,3,4···.In this paper we shall employ the new method to investigate the critical behaviour of such an asymmet-ric polymer mixture.In particular we shall focus on those aspects of the critical behaviour of asymmetric mixtures that differ from those of symmetric mixtures.These difference are rooted in the so called‘field mixing’phenomenon,which manifests the basic lack of energetic(Ising) symmetry between the coexisting phases of all realisticfluid systems.Although it is expected to have no bearing on the universal properties offluids,field mixing does engender certain non-universal effects in near-criticalfluids.The most celebrated of these is a weak energy-like critical singularity in the coexistence diameter[21,22],the existence of which constitutes afailure for the‘law of rectilinear diameter’.As we shall demonstrate however,field mixing has a far more legible signature in the interplay of the near-critical energy and concentration fluctuations,which are directly accessible to computer simulation.In computer simulation of critical phenomena,finite-size-scaling(FSS)techniques are of great utility in allowing one to extract asymptotic data from simulations offinite size[23].One particularly useful tool in this context is the order parameter distribution function[24,25,26]. Simulation studies of magnetic systems such as the Ising[25]andφ4models[27],demonstrate that the critical point form of the order parameter distribution function constitutes a useful hallmark of a university class.Recently however,FSS techniques have been extended tofluids by incorporatingfield mixing effects[28,29].The resulting mixed-field FSS theory has been successfully deployed in Monte Carlo studies of critical phenomena in the2D Lennard-Jones fluid[29]and the2D asymmetric lattice gas model[30].The present work extends this programme offield mixing studies to3D complexfluids with an investigation of an asymmetric polymer mixture.The principal features of our study are as follows.We begin by studying the order parameter(concentration)distribution as a function of temperature and chemical potential.The measured distribution is used in conjunction with the equal peak weight criterion to obtain the coexistence curve of the model.Owing to the presence offield mixing contributions to the concentration distribution,the equal weight criterion is found to break down near thefluid critical point.Its use to locate the coexistence curve and critical concentration therefore results in errors,the magnitude of which we gauge using scaling arguments.Thefield mixing component of the critical concentration distribution is then isolated and used to obtain estimates for thefield mixing parameters of the model. These estimates are compared with the results of a meanfield calculation.We then turn our attention to thefinite-size-scaling behaviour of the critical scaling op-erator distributions.This approach generalises that of previousfield mixing studies which concentrated largely on thefield mixing contribution to the order parameter distribution func-tion.We show that for certain choices of the non-universal critical parameters—the critical temperature,chemical potential and the twofield mixing parameters—these operator distri-butions can be mapped into close correspondence with independently known universal forms representative of the Ising universality class.This data collapse serves two purposes.Firstly,it acts as a powerful means for accurately determining the critical point andfield mixing param-eters of modelfluid systems.Secondly and more generally,it serves to clarify the sense of the universality linking the critical polymer mixture with the critical Ising magnet.We compare the ease and accuracy with which the critical parameters can be determined from the data collapse of the operator distributions,with that possible from studies of the order parameter distribution alone.It is argued that for criticalfluids the study of the scaling operator distri-butions represent the natural extension of the order parameter distribution analysis developed for models of the Ising symmetry.2BackgroundIn this section we review and extend the mixed-fieldfinite-size-scaling theory,placing it within the context of the present study.The system we consider comprises a mixture of two polymer species which we denote A and B,having lengths N A and N B monomers respectively.The configurational energyΦ(which we express in units of k B T)resides in the intra-and inter molecular pairwise interactions between monomers of the polymer chains:N i<j=1v(|r i−r j|)(2.1)Φ({r})=where N=n A N A+n b N B with n A and n B the number of A and B type polymers respectively. N is therefore the total number of monomers(of either species),which in the present study is maintained strictly constant.The inter-monomer potential v is assigned a square-well formv(r)=−ǫr≤r m(2.2)v(r)=0r>r mwhereǫis the well depth and r m denotes the maximum range of the potential.In accordance with previous studies of symmetric polymer mixtures[16,17],we assignǫ≡ǫAA=ǫBB=−ǫAB>0.The independent model parameters at our disposal are the chemical potential difference per monomer between the two species∆µ=µA−µB,and the well depthǫ(both in units of k B T). These quantities serve to control the observables of interest,namely the energy density u and the monomer concentrationsφA andφB.Since the overall monomer densityφN=φA+φB is fixed,however,it is sufficient to consider only one concentration variableφ,which we take as the concentration of A-type monomers:φ≡φA=L−d n A N A(2.3) The dimensionless energy density is defined as:u=L−dǫ−1Φ({r})(2.4) with d=3in the simulations to be chronicled below.The critical point of the model is located by critical values of the reduced chemical potential difference∆µc and reduced well-depthǫc.Deviations ofǫand∆µfrom their critical values control the sizes of the two relevant scalingfield that characterise the critical behaviour.In the absence of the special symmetry prevailing in the Ising model,onefinds that the relevant scalingfields comprise(asymptotically)linear combinations of the well-depth and chemical potential difference[21]:τ=ǫc−ǫ+s(∆µ−∆µc)h=∆µ−∆µc+r(ǫc−ǫ)(2.5) whereτis the thermal scalingfield and h is the ordering scalingfield.The parameters s and r are system-specific quantities controlling the degree offield mixing.In particular r is identifiable as the limiting critical gradient of the coexistence curve in the space of∆µandǫ. The role of s is somewhat less tangible;it controls the degree to which the chemical potential features in the thermal scalingfield,manifest in the widely observed critical singularity of the coexistence curve diameter offluids[1,22,31].Conjugate to the two relevant scalingfields are scaling operators E and M,which comprise linear combinations of the concentration and energy density[28,29]:M=11−sr[u−rφ](2.6) The operator M(which is conjugate to the orderingfield h)is termed the ordering operator, while E(conjugate to the thermalfield)is termed the energy-like operator.In the special caseof models of the Ising symmetry,(for which s=r=0),M is simply the magnetisation while E is the energy density.Near criticality,and in the limit of large system size L,the probability distributions p L(M) and p L(E)of the operators M and E are expected to be describable byfinite-size-scaling relations having the form[25,29]:p L(M)≃a M−1L d−λM˜p M(a M−1L d−λMδM,a M LλM h,a E LλEτ)(2.7a)p L(E)≃a E−1L d−λE˜p E(a E−1L d−λEδE,a M LλM h,a E LλEτ)(2.7b) whereδM≡M−M c andδE≡E−E c.The functions˜p M and˜p E are predicted to be universal,modulo the choice of boundary conditions and the system-specific scale-factors a M and a E of the two relevantfields,whose scaling indices areλM=d−β/νandλE=1/νrespectively.Precisely at criticality(h=τ=0)equations2.7a and 2.7b implyp L(M)≃a M−1Lβ/ν˜p⋆M(a M−1Lβ/νδM)(2.8a)p L(E)≃a E−1L(1−α)/ν˜p⋆E(a E−1L(1−α)/νδE)(2.8b) where˜p⋆M(x)≡˜p M(x,y=0,z=0)˜p⋆E(x)≡˜p E(x,y=0,z=0)(2.9) are functions describing the universal and statistically scale-invariantfluctuation spectra of the scaling operators,characteristic of the critical point.The claim that the binary polymer critical point belongs to the Ising universality class is expressed in its fullest form by the requirement that the critical distribution of thefluid scaling operators p L(M)and p L(E)match quantitatively their respective counterparts—the magneti-sation and energy distributions—in the canonical ensemble of the critical Ising magnet.As we shall demonstrate,these mappings also permit a straightforward and accurate determination of the values of thefield mixing parameters s and r of the model.An alternative route to obtaining estimates of thefield mixing parameters is via thefield mixing correction to the order parameter(i.e.concentration)distribution p L(φ).At criticality, this distribution takes the form[29,30]:p L(φ)≃a M−1Lβ/ν ˜p⋆M(x)−sa E a M−1L−(1−α−β)/ν∂∂x (˜p⋆M(x)˜ω⋆(x)),is a function characterising the mixing of the critical energy-like operator into the order parameter distribution.Thisfield mixing term is down on the first term by a factor L−(1−α−β)/νand therefore represents a correction to the large L lim-iting behaviour.Given further the symmetries of˜ω(x)and˜p⋆M(x),both of which are even (symmetric)in the scaling variable x[29],thefield mixing correction is the leading antisym-metric contribution to the concentration distribution.Accordingly,it can be isolated frommeasurements of the critical concentration distribution simply by antisymmetrising around φc= φ c.The values of s and r are then obtainable by matching the measured critical func-tion−s∂simply by cutting a B-type polymer into k equal segments.Conversely,a B-type polymer is manufactured by connecting together the ends of k A-type polymers.This latter operation is,of course,subject to condition that the connected ends satisfy the bond restrictions of the BFM.Consequently it represents the limiting factor for the efficiency of the method,since for large values of k and N A,the probability that k polymer ends simultaneously satisfy the bond restrictions becomes prohibitively small.The acceptance rate for SGCE moves is also further reduced by factors necessary to ensure that detailed balance is satisfied.In view of this we have chosen k=3,N A=10for the simulations described below,resulting in an acceptance rate for SGCE moves of approximately14%.In addition to the compositionalfluctuations associated with SGCE moves,it is also nec-essary to relax the polymer configurations at constant composition.This is facilitated by monomer moves which can be either of the local displacement form,or of the reptation(‘slith-ering snake’)variety[2].These moves were employed in conjunction with SGCE moves,in the following ratios:local displacement:reptation:semi-grandcanonical=4:12:1the choice of which was found empirically to relax the configurational and compositional modes of the system on approximately equal time scales.In the course of the simulations,a total offive system sizes were studied having linear extent L=32,40,50,64and80.An overall monomerfilling fraction of8φN=0.5was chosen, representative of a dense polymer melt[15].Here the factor of8constitutes the monomeric volume,each monomer occupying8lattice sites.The cutoffrange of the inter-monomeric√square well potential was set at r m=whereφ∗is a parameter defining the boundary between the two peaks.Well below criticality,the value of∆µcx obtained from the equal weight criterion is in-sensitive to the choice ofφ∗,provided it is taken to lie approximately midway between the peaks and well away from the tails.As criticality is approached however,the tails of the two peaks progressively overlap making it impossible to unambiguously define a peak in the manner expressed by equation3.1.For models of the Ising symmetry,for which the peaks are symmetric about the coexistence concentrationφcx,the correct value of∆µcx can nevertheless be obtained by choosingφ∗= φ in equation3.1.In near-criticalfluids,however,the imposed equal weight rule forces a shift in the chemical potential away from its coexistence value in or-der to compensate for the presence of thefield mixing component.Only in the limit as L→∞(where thefield mixing component dies away),will the critical order parameter distribution be symmetric allowing one to chooseφ∗= φ and still obtain the correct coexistence chemical potential.Thus forfinite-size systems,use of the equal weight criterion is expected to lead to errors in the determination of∆µcx near the critical point.Although this error is much smaller than the uncertainty in the location of the critical point along the coexistence curve (see below),it can lead to significant errors in estimates of the critical concentrationφc.To quantify the error inφc it is necessary to match the magnitude of thefield mixing component of the concentration distribution w(δp L),to the magnitude of the peak weight asymmetry w′(δµ)associated with small departuresδµ=∆µ−∆µcx from coexistence:w(δp L)=w′(δµ)(3.2) Now from equation2.10w(δp L)≈ φNφ∗dφδp L(φ)∼L−(1−α−β)/ν(3.3) whilew′(δµ)≈ φNφ∗dφ∂p L(φ)3 m2 2(3.7)where m2and m4are the second and fourth moments respectively of the order parameter m=φ− φ .To the extent thatfield mixing corrections can be neglected,the critical order parameter distribution function is expected to assume a universal scale invariant form. Accordingly,when plotted as a function ofǫ,the coexistence values of G L for different system sizes are expected to intersect at the critical well depthǫc[26].This method is particularly attractive for locating the critical point influid systems because the even moments of the order parameter distribution are insensitive to the antisymmetric(odd)field mixing contribution. Figure2displays the results of performing this cumulant analysis.A well-defined intersection point occurs for a value G L=0.47,in accord with previously published values for the3D Ising universality class[36].The corresponding estimates for the critical well depth and critical chemical potential areǫc=0.02756(15)∆µc=0.003603(15)It is important in this context,that a distinction be drawn between the errors on the location of the critical point,and the error with which the coexistence curve can be determined.The uncertainty in the position of the critical point along the coexistence curve,as determined from the cumulant intersection method,is in general considerably greater than the uncertainty in the location of the coexistence curve itself.This is because the order parameter distribution function is much more sensitive to small deviations offcoexistence(due tofiniteǫ−ǫcx orfinite ∆µ−∆µcx)than it is for deviations along the coexistence curve,(ǫand∆µtuned together to maintain equal weights).In the present case,wefind that the errors on∆µc andǫc are approximately10times those of the coexistence valuesǫcx and∆µcx near the critical point.The concentration distribution function at the assigned value ofǫc and the corresponding value of∆µcx,(determined according to the equal weight rule withφ∗=<φ>),is shown infigure3for the L=40and L=64system sizes.Also shown in thefigure is the critical magnetisation distribution function of the3D Ising model obtained in a separate study[37]. Clearly the L=40and L=64data differ from one another and from the limiting Ising form.These discrepancies manifest both the pure antisymmetricfield mixing component of the true(finite-size)critical concentration distribution,and small departures from coexistence associated with the inability of the equal weight rule to correctly identify the coexistence chemical potential.To extract the infinite-volume value ofφc from thefinite-size data,it is therefore necessary to extrapolate to the thermodynamic limit.To this end,and in accordance with equation3.6,we have plottedφcx(L),representing thefirst moment of the concentration distribution determined according to equal weight criterion at the assigned value ofǫc,against L(1−α)/ν.This extrapolation(figure4)yields the infinite-volume estimate:φc=0.03813(19)corresponding to a reduced A-monomer densityφc/φN=0.610(3).Thefinite-size shift in the value ofφcx(L)is of order2%.We turn next to the determination of thefield mixing parameters r and s.The value of r represents the limiting critical gradient of the coexistence curve which,to a good approxima-tion,can be simply read offfromfigure1with the result r=−0.97(3).Alternatively(and as detailed in[29])r may be obtained as the gradient of the line tangent to the measured critical energy function(equation2.11)atφ=φc.Carrying out this procedure yields r=−1.04(6).The procedure for extracting the value of thefield mixing parameter s from the concentra-tion distribution is rather more involved,and has been described in detail elsewhere[29,30]. The basic strategy is to choose s such as to satisfy∂δp L(φ)=−swhereδp L(φ)is the measured antisymmetricfield mixing component of the critical concen-tration distribution[30],obtained by antisymmetrising the concentration distribution about φc(L)and subtracting additional corrections associated with small departures from coexistence resulting from the failure of the equal weight rule.Carrying out this procedure for the L=40 and L=64critical concentration distributions yields thefield mixing components shown in figure5.The associated estimate for s is0.06(1).Also shown infigure5(solid line)is the predicted universal form of the3D order parameterfield mixing correction−∂u−rφ1−sr p L(E)=p Lsection2).This discrepancy implies that the system size is still too small to reveal the asymp-totic behaviour Nevertheless the data do afford a test of the approach to the limiting regime, via the FSS behaviour of the variance of the energy distribution.Recalling equation2.14, we anticipate that this variance exhibits the same FSS behaviour as the Ising susceptibility, namely:L d( u2 −u2c)∼Lγ/ν(3.10) By contrast,the variance of the scaling operator E is expected to display the FSS behaviour of the Ising specific heat:L d( E2 −E2c)∼Lα/ν.(3.11) Figure9shows the measured system size dependence of these two quantities at criticality.Also shown is the scaled variance of the ordering operator L d( M2 −M2c)∼Lγ/ν.Straight lines of the form Lγ/νand Lα/ν,(indicative of the FSS behaviour of the Ising susceptibility and specific heat respectively)have also been superimposed on the data.Clearly for large L,the scaling behaviour of the variance of the energy distribution does indeed appear to approach that of the ordering operator distribution.4Meanfield calculationsIn this section we derive approximate formulae for the values of thefield mixing parameters s and r on the basis of a meanfield calculation.Within the well-known Flory-Huggins theory of polymer mixtures,the mean-field equation of state takes the form:∆µ=1N Bln(1−ρ)−2zǫ(2ρ−1)+C(4.1)In this equation,z≈2.7is the effective monomer coordination number,whose value we have obtained from the measured pair correlation function.ρ=φ/φN is the density of A-type monomers and the constant C is the entropy density difference of the pure phases,which is independent of temperature and composition.In what follows we reexpressρby the concen-trationφ.The critical point is defined by the condition:∂∂φ2∆µc=0(4.2) where∆µc=∆µ(φc,ǫc).This relation can be used to determine the critical concentration and critical well-depth,for which onefindsφc1+1/√ǫc=z4N A N BN A+√where the expansion coefficients take the formr ′=−2z (2φc φN b =(1+√3√k )4φ+−φ−(4.6)where φ−and φ+denote the concentration of A monomers in the A-poor phase and A-rich phases respectively.Thus to leading order in ǫ,the phase boundary is given by :∆µcx (ǫ)=∆µc +r ′δǫc +···(4.7)Consequently we can identify the expansion coefficient r ′with the field mixing parameter r (c.f.equation 2.5)that controls the limiting critical gradient of the coexistence curve in the space of ∆µand ǫ.Substituting for ∆µc and ǫc in equation 4.7and setting k =3,we find r =−1.45,in order-of-magnitude agreement with the FSS analysis of the simulation data.In order to calculate the value of the field mixing parameter s ,it is necessary to obtain the concentration and energy densities of the coexisting phases near the critical point.The concentration of A-type monomers in each phase is given byδφ±=φ±−φc =± b −2acδǫ2−φc =−2acδǫ2 z s +z (2ρ−1)2 =−φN φN −1)2+rδφ−2z2−u (φc )=−2za5zb δǫ+···(4.11)Now since (1−rs ) M = δφ −s δu vanishes along the coexistence line,equations 4.9and4.11yield the following estimate for the field mixing parameter s :s = δφ 5zb 1+r cφN 20z √20z √[30].The sign of the product rs differs however from that found at the liquid-vapour critical point.In the present context this product is given byrs10(√NB)2N A N B(4.13)However an analogous treatment of the van der Waalsfluid predicts a positive sign rs,in agreement with that found at the liquid vapour critical point[29,30].5Concluding remarksIn summary we have employed a semi-grand-canonical Monte Carlo algorithm to explore the critical point behaviour of a binary polymer mixture.The near-critical concentration and scal-ing operator distributions have been analysed within the framework of a mixed-fieldfinite-size scaling theory.The scaling operator distributions were found to match independently known universal forms,thereby confirming the essential Ising character of the binary polymer critical point.Interestingly,this universal behaviour sets in on remarkably short length scales,being already evident in systems of linear extent L=32,containing only an average of approximately 100polymers.Regarding the specific computational issues raised by our study,wefind that the concen-tration distribution can be employed in conjunction with the cumulant intersection method and the equal weight rule to obtain a rather accurate estimate for the critical temperature and chemical potential.The accuracy of this estimate is not adversely affected by the anti-symmetric(odd)field mixing contribution to the order parameter distribution,since only even moments of the distribution feature in the cumulant ratio.Unfortunately,the method can lead to significant errors in estimates of the critical concentrationφc,which are sensitive to the magnitude of thefield mixing contribution.The infinite-volume value ofφc must therefore be estimated by extrapolating thefinite-size data to the thermodynamic limit(where thefield mixing component vanishes).Estimates of thefield mixing parameters s and r can also be extracted from thefield mixing component of the order parameter distribution,although in practice wefind that they can be determined more accurately and straightforwardly from the data collapse of the scaling operators onto their universalfixed point forms.In addition to clarifying the universal aspects of the binary polymer critical point,the results of this study also serve more generally to underline the crucial role offield mixing in the behaviour of criticalfluids.This is exhibited most strikingly in the form of the critical energy distribution,which in contrast to models of the Ising symmetry,is doubly peaked with variance controlled by the Ising susceptibility exponent.Clearly therefore close attention must be paid tofield mixing effects if one wishes to perform a comprehensive simulation study of criticalfluids.In this regard,the scaling operator distributions are likely to prove themselves of considerable utility in future simulation studies.These operator distributions represent the natural extension tofluids of the order parameter distribution analysis deployed so successfully in critical phenomena studies of(Ising)magnetic systems.Provided therefore that one works within an ensemble that affords adequate sampling of the near-criticalfluctuations,use of the operator distribution functions should also permit detailed studies offluid critical behaviour. AcknowledgementsThe authors thank K.Binder for helpful discussions.NBW acknowledges thefinancial sup-port of a Max Planck fellowship from the Max Planck Institut f¨u r Polymerforschung,Mainz.。
Home Search Collections Journals About Contact us My IOPscienceCasimir Effect for Dielectric PlatesThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2006 Commun. Theor. Phys. 45 499(/0253-6102/45/3/024)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 222.20.93.118The article was downloaded on 16/02/2012 at 04:11Please note that terms and conditions apply.Commun.Theor.Phys.(Beijing,China)45(2006)pp.499–504c International Academic Publishers Vol.45,No.3,March15,2006Casimir Effect for Dielectric Plates∗SHAO Cheng-Gang,TONG Ai-Hong,and LUO Jun†Department of Physics,Huazhong University of Science and Technology,Wuhan430074,China(Received May30,2005;Revised July12,2005)Abstract We generalize Kupisewska method to the three-dimensional system and another derivation of the Casimir effect between two dielectric plates is presented based on the explicit quantization of the electromagneticfield in the presence of dielectrics,where the physical meaning of“evanescent mode”is discussed.The Lifshitz’s formula is rederived using all the vacuum mode functions,which include the contribution of the‘evanescent modes’.Only in the case of the perfect metallic plates will the evanescent modes become unimportant.PACS numbers:12.20.DsKey words:Casimir effect,electromagneticfield quantization,evanescent mode1IntroductionDuring the last few years the Casimir effect has at-tracted a lot of experimental and theoretical attention as a nontrivial macroscopic evidence for the existence of zero-point oscillations of electromagneticfield.[1−4]This is in part due to the resurgence of the interest in the fun-damental problems connected with the physical vacuum. Except for the familiar mode summation method of de-riving Casimir forces as employed by Casimir himself,[5] various concepts and calculational techniques have been developed,but only a few of them have turned out to be capable of dealing with realistic dielectric bodies.Lifshitzfirst introduced a macroscopic theory for the evaluation of the Casimir force between two dielectric plates by calculating the Maxwell stress tensor,where the electromagneticfield and the material bodies are treated macroscopically,and explicitfield quantization is avoided.[6]The dielectric media are characterized by randomlyfluctuating sources of current densities hav-ing aδ-function correlation.Lifshitz’s theory is very complicated and developed by many authors,such as Schwinger,[7,8]Matloob,[9−11]and Tomas.[12]In these methods,the dissipation-fluctuation relation or thefield correlation function plays an important role.Another derivation of the Lifshitz results based on the surface mode is also discussed by some authors,starting directly from the zero-point energy of electromagneticfield.[1,13] Kupisewska has presented an instructive treatment of the Casimir effect between dielectric plates,[14,15]where the electromagneticfield is quantized in the presence of dielectric material and the notion of physical photons is introduced.The Casimir force can be calculated from the pressure of the electromagneticfield in the vacuum state on the slab with the help of the Maxwell stress tensor.In this approach,the difficulty is in quantizing the electro-magneticfield in the presence of dielectric material,where the physical meaning of“evanescent mode”is not clear. The intricate structure of thefield operators has restricted approach to one-dimensional systems so far.[11,12,16−18]In the present paper,we formally generalize Kupi-sewska method to the three-dimensional system.A rig-orous derivation of Casimir effect using three-dimensional mode functions,including the contribution of the evanes-cent modes,is given and the results are the same as the Lifshitz’s formula.We also discuss the physical meaning of evanescent mode andfind that only in the case of the perfect metallic plates,the evanescent modes will become unimportant.2Quantization of Electromagnetic Field and Physical Meaning of“Evanescent Mode”The most popular interpretation of the Casimir force is based upon the electromagnetic vacuum pressure.[19,20] The magnetic intensity B and the electric intensity E can be expressed by the vector potential A( r,t)in the rela-tions B=∇× A, E=−∂ A/∂t under the Coulomb gauge ∇·ε A=0,whereε( r)is the dielectric function.The equation of motion for the vector potential is∇×∇× A+( r,ω)−ε( r,ω)ω2c2A+( r,ω)=0(1) with A( r,t)=∞(dω/2π) A+( r,ω)e−iωt+c.c.Let us consider the general process of the electromag-neticfield quantization.Roughly speaking,there have been two routes to study the behavior of the quantized electromagneticfield in the presence of dielectrics.One is to use the Green-tensor formation by employing the dissipation-fluctuation theory,[21,22]which has successfully been applied to the study of the Casimir effect.[12]The other is to perform explicitfield quantization.Here we pay attention on the second one.By introducing the anni-hilation and the creation operators b kλand b†kλ,the vector potential operator can be expanded as a linear combina-tion of the quantum vacuum mode functions,A( r,t)=k,λc2ωkε01/2b kλe−iωt f kλ( r)+b†ke iωt f∗kλ( r).(2)∗The project partially supported by the State Key Basic Research Program of China under Grant No.2003CB716300and National Natural Science Foundation of China under Grant No.10121503†E-mail:junluo@500SHAO Cheng-Gang,TONG Ai-Hong,and LUO Jun Vol.45The operators b kλand b †kλact in the Fock space.The physical photons are associated with the mode functions f kλ( r ).The photon frequency is ωk and polarization la-bels by λ=s (s -polarized or TE)and by λ=p (p -polarized or TM).The mode functions obey the equation∇×∇× f kλ( r )−ω2k ε( r )c2 f kλ( r )=0,(3)together with gauge condition and orthonormality condi-tions:∇·[ε( r ) f kλ( r )]=0,(4)d 3 r [ε( r ) f kλ( r ) f −k λ ( r )]=δkk δλλ .(5)In free space, fkλ=(1/√V ) e λe i k · r ,V is the normaliza-tion volume, e λis the unit polarization vector,which is perpendicular to the wave vector k ,apparently.Let us apply the above considerations to the configu-ration with two parallel infinite dielectric plates each of thickness d ,a distance of a apart (Fig.1).The two di-electric plates situated in (y ,z )planes has the frequency-dependent dielectric permittivity ε.The complex refrac-tive index is assumed to be equal to n with n 2=ε.Then the vector potential may be written asA ( r ,t )= λc 1(2π)d 3 k 2ωk ε0 1/2×e λ b kλe −i ωtf kλ( r )+b †kλe i ωt f kλ∗( r ).(6)The parameter λ=s ,and p labels the two possible po-larizations.If εis constant in the region of the dielectric,then the mode function satisfies the equation∇2 f kλ( r )+ω2k ε( r )c 2f kλ( r )=0(7)with the appropriate boundary condition and the or-thonormality conditions.Fig.1Configuration of the dielectric plates with per-mittivity ε( r ).Fig.2The TE modes polarization λ=s .Electric vec-tor Epoints along z axis.Exploiting the translational and rotational symmetry in the y –z plane,we may select the incident wave vec-tor in the x –y plane for simplicity,and the angle between the incident direction and the x axis is θ.Therefore,wemay write the mode function in the form of f kλ( r )=f kλ(x )e i k y ywith k y =k sin θ.f kλ(x )is given in Table 1.θis confirmed by the refraction law sin θ=n sin θ .Table 1Mode functions f kλ(x ).k cos θ=k x >0k cos θ=k x <0e i k cos θx+R kλe −i k cos θxT −kλe i k cos θxx ∈−∞,−a2−dA kλe i kn cos θx+B kλe −i kn cos θxE −kλe i kn cos θx +F −kλe −i kn cos θxx ∈−a2−d,−a 2 C kλe i k cos θx +D kλe −i k cos θx C −kλe i k cos θx +D −kλe −i k cos θxx ∈ −a 2,a2E kλe i kn cos θx+F kλe −i kn cos θxA −kλe i kn cos θx+B −kλe −i kn cos θxx ∈ a 2,a 2+dT kλe i k cos θxe i k cos θx +R −kλe −i k cos θxx ∈ a 2+d,∞When the electric field intensity is perpendicular to the incident plane as shown in Fig.2,polarization direction is in z axis,labeling by λ=s .The boundary conditions of E z and H y at the points x =−a/2−d ,−a/2,a/2,and a/2+d correspond to a system of eight linear equations,e −i k (cos θ)(a/2+d )+R ks e i k (cos θ)(a/2+d )=A ks e −i kn (cos θ)(a/2+d )+B ks e i kn (cos θ)(a/2+d ),cos θ(e −i k (cos θ)(a/2+d )−R ks e i k (cos θ)(a/2+d ))=n cos θ (A ks e −i kn (cos θ)(a/2+d )−B ks e i kn (cos θ)(a/2+d )),A ks e −i kn (cos θ)a/2+B ks e i kn (cos θ)a/2=C ks e −i k (cos θ)a/2+D ks e i k (cos θ)a/2,n cos θ (A ks e −i kn (cos θ)a/2−B ks e i kn (cos θ)a/2)=cos θ(C ks e −i k (cos θ)a/2−D ks e i k (cos θ)a/2),No.3Casimir Effect for Dielectric Plates501C ks e i k(cosθ)a/2+D ks e−i k(cosθ)a/2=E ks e i kn(cosθ )a/2+F ks e−i kn(cosθ )a/2,cosθ(C ks e i k(cosθ)a/2−D ks e−i k(cosθ)a/2)=n cosθ (E ks e i kn(cosθ )a/2−F ks e−i kn(cosθ )a/2),E ks e i kn(cosθ )(a/2+d)+F ks e−i kn(cosθ )(a/2+d)=T ks e i k(cosθ)(a/2+d),n cosθ (E ks e i kn(cosθ )(a/2+d)−F ks e−i kn(cosθ )(a/2+d))=cosθT ks e i k(cosθ)(a/2+d).(8) The coefficients R ks,T ks,A ks,B ks,C ks,D ks,E ks,and F ks areR ks=r s+r s e2ik(cosθ)a(e2i kn(cosθ )d−r2ds)/(1−r2dse2i kn(cosθ )d)1−r2s e2i k(cosθ)ae−i k(cosθ)(a+2d),A ks=t s(1+r ds r s e2i k(cosθ)a)t ds(1−r2s e2i k(cosθ)a)e−i k(cosθ)(a/2+d)e i kn(cosθ )a/2,B ks=t s(r ds+r s e2i k(cosθ)a)t ds(1−r2s e2i k(cosθ)a)e−i k(cosθ)(a/2+d)e−i kn(cosθ )a/2,C ks=t s e−i k(cosθ)d1−r s e,D ks=t s r s e i k(cosθ)(a−d)1−r s e,E ks=t2st ds(1−r2s e2i k(cosθ)a)e i k(cosθ)(a/2+d)e−i kn(cosθ )(a/2+d),F ks=t2s r dst ds(1−r s e)e i k(cosθ)(a/2+d)e i kn(cosθ )(a/2+d),T ks=t2s e−2i k(cosθ)d1−r2s e2i k(cosθ)a,(9)with r s and t s the reflection and refraction coefficients for one dielectric plater s=r ds(e2i kn(cosθ )d−1)1−r2dse2i kn(cosθ )d,t s=4n cosθ cosθ(n cosθ +cosθ)2e i kn(cosθ )d1−r2dse2i kn(cosθ )d,r ds=n cosθ −cosθn cosθ +cosθ,t ds=2n cosθn cosθ +cosθ.(10)Fig.3The TM modes polarizationλ=p.Magnetic vector H points along z axis.When the electricfield intensity is parallel to the incident plane as shown in Fig.3,the magneticfield intensity points along z axis and the polarization labels byλ=p.The boundary conditions of E y and H z at the points x=−a/2−d,−a/2,a/2,and a/2+d also correspond to a system of eight linear equations.Here we only write down two equations corresponding to the plane of x=−a/2−d,e−i k(cosθ)(a/2+d)+R kp e i k(cosθ)(a/2+d)=n(A kp e−i kn(cosθ )(a/2+d)+B kp e i kn(cosθ )(a/2+d)),cosθ(e−i k(cosθ)(a/2+d)−R kp e i k(cosθ)(a/2+d))=cosθ (A kp e−i kn(cosθ )(a/2+d)−B kp e i kn(cosθ )(a/2+d)).(11) The coefficients R kp,T kp,A kp,B kp,C kp,D kp,E kp,and F kp have the same form as Eq.(9),by replacing the subscript labelλ=s byλ=p withr p=r dp(e2ikn(cosθ )d−1)1−r2dpe,t p=4n cosθ cosθ(cosθ+n cosθ)e i kn(cosθ )d1−r2dpe,r dp=cosθ −n cosθcosθ +n cosθ,t dp=2n cosθcosθ +n cosθ.(12)502SHAO Cheng-Gang,TONG Ai-Hong,and LUO Jun Vol.45The results derived above have only considered the normal modes of k 2x =k 2p 2>0with the parameter p =cos θ.Moreover,additional kinds of modes of k 2x =k 2p 2<0also exist,which may be called ‘evanescent modes’.For thesemodes the inequality k 2<k 2y +k 2z =k 2||holds and p can take the imaginary number.The term ‘evanescent’means that the reflection and transmission waves are exponentially damping for x <−a/2−d and x >a/2+d .However,according to Table 1,the incident wave e −κx is divergent for x →−∞if k x =i κ,κ>0.This divergence can be avoided by using the formalism of surface modes,[1]where the mode function behaves like e −κ|x |and is exponentially damping for |x |→∞,κmay take discrete values.These surface modes in physics describe wave propagating parallel to the surface of the plates.However,any exponentially damping (or exponentially increasing)mode functions cannot be normality in the infinite volume V →∞,which is not consistent with the orthonormality condition (5).In physical sense,it is well known that the k x space and the x space are physically equivalent in quantum theory.If momentum k x takes the imaginary value k x =i κ,κ>0,position x should also be replaced by χ=i x .The evanescent mode function outside the plates can be written asf k (x )=f κ(χ)= e i κχ+R e −i κχ,χ<−a a −dT e i κχ,χ>a a+dwith k x =i κ,x =−i χ.(13)In the above ‘evanescent modes’is not evanescent in the space χ.The integral of the orthonormality conditions (5)can be taken in the space χ.In mathematical sense,the evanescent modes and the normal modes can be expressed in the same form as in Table 1,which form a complete set of functions in the space satisfying the orthonormality conditions.The formal definition (13)is a simple generalization of the normal mode function,which did not affect the distri-bution, λ1(2π)3 d k y d k z d k x f kλ( r ) f kλ∗( r )=λe λ e λδ3( r − r ).(14)Using the commutation relations[b kλ,b †k λ ]=(2π)3δλλ δ(k x −k x )δ(k y −k y )δ(k z −k z ),(15)we get equal-time commutation relations between the vector potential operator and its conjugate momentum,[ A ( r ,t ),−ε0 E ( r ,t )]=iλe λ e λδ3( r − r ).(16)3Casimr Force for Dielectric PlatesWith the explicit form of the mode functions given,we may derive the interaction between two dielectric plates asthe effect of vacuum pressure,which may be expressed by Maxwell tensor,[14,19]T ij =−12 2ε(r )ε0E i E j +21µ0B i B j −( E 2+ B 2)δij .(17)The vacuum field correlations E i E j and B i B j are built by taking the appropriate derivatives of the vector potential operator.The vacuum pressure on a body can be calculated by integrating of T ij over the surrounding surface of the volume,f i =ΣT ij n j d s ,(18)where n j is normal to the integral surface.In our case of two parallel plates the pressure will be the difference between T xx component of the stress tensor outside the plates and that between the plates,calculated on the surface:f =T xx x =−a2−d −T xx x =−a 2.(19)Here,T xx =[ε0(E 2y +E 2z −E 2x )+(1/µ0)(B 2y +B 2z −B 2x )]/2.For the single mode function with the angel θbetween the wave vector and the dielectric surface,we have T xx =[ε0E 2+(1/µ0)B 2(1−2sin 2θ)]for λ=s and T xx =[ε0E 2(1−2sin 2θ)+(1/µ0)B 2]/2for λ=p .The electromagnetic field operators expand as a linear combination of mode functions.As an example let us calculate the T xx in region I,which is equal to that in region V .E ( r ,t )= λ1(2π)3∞−∞d 2 k ⊥ i k ⊥→0→∞d k x 2ωk ε01/2 e λb kλi ωk e −i ωk t e i k ⊥· r (e i k x x +R kλe −i k x x )+c.c.+λ1(2π)3∞ −∞d 2k ⊥−∞→0→−i k ⊥d k x2ωk ε01/2 e λb kλi ωk e −i ωk t e i k ⊥· r T −kλe i k x x+c.c. ,No.3Casimir Effect for Dielectric Plates503B( r,t)=1(2π)3∞−∞d2 k⊥ik⊥→0→∞d k x2ωkε01/2e p b ks i k e−iωk t e i k⊥· r(e i k x x−R ks e−i k x x)+c.c.+1(2π)3∞−∞d2 k⊥i k⊥→0→∞d k x2ωkε01/2e s b kp i k e−iωk t e i k⊥· r(−e i k x x+R kp e−i k x x)+c.c.+1(2π)3∞−∞d2 k⊥−∞→0→−k⊥d k x2ωkε01/2e p b ks i k e−iωk t e i k⊥· r T∗ks e−i k x x+c.c.+1(2π)3∞−∞d2 k⊥−∞→0→−k⊥d k x2ωkε01/2e s b kp i k e−iωk t e i k⊥· r T∗kp e−i k x x+c.c.,(20)where vectors e s and e p point,respectively,to the directions as E shown in Figs.2and3.The integration of the k xspace can be changed to the integration of the frequencyi k⊥→0→∞k x d k x=∞(ω/c2)dωwithω2/c2=k2x+k2⊥,whichcoincides with Fourier time transform in Eq.(1).The physical vacuum is given by b k|0 =0and 0|b k b†k|0 =1.Atfinite temperature,the physical vacuum|0 shouldbe replaced by the thermodynamical vacuum state|Ω with Ω|b k b†k |Ω =coth( ωk/2kBT),which represents a thermalPlanck distribution including the zero-point term.From Eq.(20)and taking into account R kλR∗kλ+T kλT∗kλ=1,themean expectation value of operator T xx in region I will beΩ|T xx|ΩI =12λc(2π)d k yd k zi k⊥→0→∞d k xk2xk2x+k2y+k2zcothωk2kBT+c.c.(21)Similarly for the region between the plates,Ω|T xx|ΩIII =12λc(2π)3d k yd k zi k⊥→0→∞d k xk2xk2x+k2y+k2z(C kλC−kλ+D kλD−kλ)×cothωk2kBT+c.c.(22)Finally we get the expression for the interaction between two parallel plates of unit area,f=12Reλc(2π)3d k yd k zi k⊥→0→∞d k xk2xk2x+k2y+k2z(1−C kλC−kλ−D kλD−kλ)cothωk2kBT.(23)Let usfirstly discuss the Casimir effect between two perfect metallic plates,which means the reflection coefficients |r dλ|→1and|rλ|→1.Setting2k x a=µ,the term in the brackets of the above formula is nonzero only forµn=2πn, n=0,1,2,...Forµnear resonances we may writelim rλ→1(C kλC−kλ+D kλD−kλ)=limrλ→11−r4λ1+r4λ−2r2λcosµ≈2πδ(µ−µn).(24)In this limit the Casimir force isF c= cπ3∞d k x∞d k y∞d k zk2xk2x+k2y+k2z[1−2πδ(µ−µn)]cothωk2kBT=kBT4πa3∞l=0∞2aξl/cy2d ye y−1.(25)Here the prime adds a multiple1/2near the term l=0andξl=2πkBT l/ is the Matsubara frequencies.Since k x=nπ/a takes discrete value only,the evanescent modes will not contribute to the Casimir effect in the case of the perfect metallic plates.However the evanescent modes makes an important role in the case of non-perfectly-reflecting ing the relation r kλr−kλ+t kλt−kλ=1,equation(23)leads tof=Reλ c2π2∞d k1→0→i∞d p·k3p211−r−2λ(k)e−2i kpacothωk2kBT.(26)The integration of p over(0,1)represents the contribution of the oscillatory modes while the integration over the imaginary axis represents the contribution of the evanescent modes,which was recognized previously.[23,24] In order to compare with the Lifshitz result,we change the integral path in the k and p complex plane and use the residue theorem as done by Lifshitz.[6,25,26]Noting the relations r ds=(K−ε)/(K+ε)and r dp=(K−pε)(K+pε)504SHAO Cheng-Gang,TONG Ai-Hong,and LUO Jun Vol.45with K=2it is easy to obtain the Casimir force per unit area between two plates,f=kBTπc3∞l=0ξ3l∞1p2d p1−Q1(iξl)Q1(iξl)+1−Q2(iξl)Q2(iξl),(27)where a more detailed representation for the function Q1,2is[1,13]Q1(iξ)=1− (K−εp)(K+εp)−(K−εp)(K+εp)e−2ξKd/c(K+εp)(K+εp)−(K−εp)(K−εp)e−2ξKd/ce−2ξpa/c,Q2(iξ)=1− (K−p)(K+p)−(K−p)(K+p)e−2ξKd/c(K+p)(K+p)−(K−p)(K−p)e−2ξKd/ce−2ξpa/c.(28)For T→0in the sense that kBT c/a the sum over l turns to an integralfT=0=2π2c3∞1p2d p∞ξ3dξ1−Q1(iξl)Q1(iξl)+1−Q2(iξl)Q2(iξl).(29)The above expression coincides with Lifshitz result[1,6]for the force per unit area between semispaces with a dielectric permittivityε.To obtain this limiting case from Eq.(27)one should put d→∞,f Lifshitz=kBTπc3∞l=0ξ3l∞1p2d pK(iξl)+pK(iξl)−p2e2ξl pa/c−1−1+K(iξl)+pεK(iξl)−pε2e2ξl pa/c−1−1.(30)It can also be obtained directly from Eq.(26)by replacing rλwith its mean value r dλdue to its rapid vibration since d a.4Summary and DiscussionWe have generalized the Kupisewska’s method and presented another derivation of the Lifshitz results starting directly from the quantization of the electromagneticfield in the presence of dielectrics,where the physical meaning of ‘evanescent mode’is discussed.The Casimir force between two dielectric plates can be regarded as the pressure of the electromagnetic vacuum,which characterized by the quantum vacuum mode functions.The calculation of the Casimir force needs to use all the vacuum mode functions,which include the contribution of the evanescent modes.Only in the case of the perfect metallic plates,will the evanescent modes become unimportant,since the metallic plates discrete the modes k x=nπ/a.References[1] A.Bordag,U.Mohideen,and V.M.Mostepanenko,Phys.Rep.353(2001)1.[2]moreaux,Am.J.Phys.67(1999)850.[3]U.Mohideen and A.Roy,Phys.Rev.Lett.81(1998)4549.[4]G.Bressi,et al.,Phys.Rev.Lett.88(2002)041804.[5]H.B.G.Casimir,Proc.K.Ned.Akad.Wei.51(1948)793.[6] E.M.Lifshitz,Sov.Phys.JETP2(1956)73.[7]J.Schwinger,Lett.Math.Phys.1(1975)43.[8]J.Schwinger,L.L.DeRaad,Jr.,and ton,Ann.Phys.(Leipzig)115(1978)1.[9]R.Matloob,A.Keshavarz,and 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New Programs From OldG.RAMALINGAM and THOMAS REPSUniversity of Wisconsin−MadisonThere is often a need in the program-development process to generate a new version of a program that relates to existing versions of the program in some specific way.Program-development tools that assist in the generation of the new version,or possibly even automatically generate the new version from the exist-ing versions,will obviously be of great use in this process.The problems of merging software extensions, program integration,separating consecutive edits,and propagating changes through multiple versions are all instances of situations where such tools would be welcome.These problems give rise to various questions concerning the relationship between these problems and between various strategies that may be used to tackle these problems.The goal of this paper is to address these questions.Previous work has shown how programs may be treated as elements of a double Brouwerian algebra, and how program modifications may be treated as functions from the set of programs to itself.In this paper we study the algebraic properties of program modifications,and use them to establish various results con-cerning the problem of program integration and the problem of separating consecutive edits.We also show how a particular category can be constructed from programs and program modifications,and address the question of whether integration is a pushout in this category.CR Categories and Subject Descriptors:D.2.2[Software Engineering]:Tools and Techniques−program-mer workbench;D.2.6[Software Engineering]:Programming Environments;D.2.7[Software Engineer-ing]:Distribution and Maintenance−enhancement,restructuring,version control; D.2.9[Software Engineering]:Management−programming teams,software configuration management; F.4.m [Mathematical Logic and Formal Languages]:Miscellaneous;G.2.m[Discrete Mathematics]:Miscel-laneousGeneral Terms:TheoryAdditional Key Words and Phrases:Brouwerian algebra,dependence graph,program slice,program modification,program integration,separating consecutive editsThis work was supported in part by an IBM Graduate Fellowship,by a David and Lucile Packard Fellowship for Science and En-gineering,by the National Science Foundation under grants DCR-8552602and CCR-9100424,by the Defense Advanced Research Projects Agency,monitored by the Office of Naval Research under contract N00014-88-K-0590,as well as by a grant from the Digital Equipment Corporation.An abridged version of this work appeared in the Proceedings of the Second International Conference on Algebraic Methodology and Software Technology(AMAST),(Iowa City,Iowa,May22-25,1991)[Rama92].Authors’address:Computer Sciences Department,University of Wisconsin−Madison,1210W.Dayton St.,Madison,WI53706.E-mail:{ramali,reps}@Copyright©1991by G.Ramalingam and Thomas W.Reps.All rights reserved.At afirst approximation,a program derivation is likely to be a directed graph in which nodes are programs and arcs are fundamental program derivation steps.W.L.Scherlis and D.S.Scott[Sche83]1.IntroductionIn the program-development process,there is often the need to generate a new version of a program that relates to the existing versions of the program in some specific way.Examples of program-development steps during which such a need arises include merging software extensions,program integration,separat-ing consecutive edits,and propagating changes through multiple versions.Program-development tools that provide assistance during such steps—by aiding in the creation of the new version,or possibly even gen-erating the new version from the existing versions automatically—would obviously be of great value.The problem of merging software extensions,first addressed by Berzins[Berz86],is as follows:given two programs a and b,generate a program that“has all the capabilities of each of the original programs”, that is,a program that“combines the features of both a and b”.Berzins formally defines the problem to be that of generating a program whose semantics is the least upper bound(in the approximation ordering of Scott[Stoy77])of the semantics of the two given programs.Such a program is called a least common extension of the given programs.The least common extension of two programs is not computable in gen-eral,and Berzins outlines(conservatively approximate)ways of generating a common extension of two programs written in a simple functional language.The problem of program integration,first formalized by Horwitz et al.[Horw89],is a generalization of the above problem involving three versions of a program—an original base version and two independent revisions of the base version’s source code.This can happen during the concurrent development of a pro-gram,for instance,when two programmers simultaneously modify different copies of the same program, each enhancing the program in some way orfixing some bug in the program.The program-integration problem is to determine if the two changes made have an undesirable semantic interaction,and if they do not,to generate a program that incorporates the enhancements or bug-fixes made by either of the program-mers.Horwitz et al.formalized this problem for a simple programming language,and developed an integration algorithm—referred to hereafter as the HPR algorithm—for integrating programs in that language.The HPR algorithm is semantics-based:the integration algorithm is based on the assumption that any change in the behavior,rather than the text,of a program variant is significant and must be preserved in the merged program.An integration system based on this algorithm will determine whether the variants incor-porate interfering changes,and,if they do not,create an integrated program that includes all changes as well as all features of the base program that are preserved in all variants.Another problem,which is similar to that of program integration in that it involves three versions of a program,is that of separating consecutive edits.Consider the case of two consecutive edits to a program base;let a be a program obtained by modifying base,and let c be a program obtained by modifying a.By “separating consecutive edits”we mean creating a program that includes the second modification but not thefirst.Yet another problem,the problem of propagating changes through multiple versions,is the fol-lowing:a tree or dag of related versions of a program exists,and the goal is to make the same enhancement or bug-fix to all of them.This situation arises when multiple implementations of a program exist to support slightly different requirements.For instance,different versions of a program might support different operating systems.It was suggested by Horwitz et al.that the problems of separating consecutive edits and progagating changes were additional applications for a program-integration tool[Horw89].For example, they suggested that by repeatedly invoking the program-integration tool it would be possible to propagateprogram changes through multiple versions of a program.This paper studies the program-manipulation operations described above at an abstract level.We make use of an algebra that abstracts away from the details of particular(concrete)operations for the prob-lems described above.This approach permits us to study the various program-manipulation problems by investigating their algebraic properties.(A simple example of an algebraic property—which a specific program-integration operation might or might not possess—is that of associativity.In this context associa-tivity means:“If three variants of a given base are to be integrated by a pair of two-variant integrations,the same result is produced no matter which two variants are integratedfirst.”)Our algebraic approach helps us to answer questions such as the following ones:What relations(if any)exist between the various program-manipulation problems described above? Is there any single tool that can be utilized to address these and other similar problems uniformly?As mentioned above,the program-integration tool was proposed as a general tool to tackle these various problems;however,we will see later that more than one strategy exists for tackling these problems using a program-integration tool.Are these strategies equivalent?If not,which is the “correct”strategy(if any)?What is the relationship between these alternative strategies?This paper is a continuation of two earlier studies of the algebraic properties of the program-integration operation.One earlier work([Reps90])was motivated by the need to establish the algebraic properties of the HPR integration algorithm.There,the HPR integration algorithm was formalized as an operation in a Brouwerian algebra[McKi46],and the algebraic laws that hold in Brouwerian algebras were utilized in establishing properties of the HPR integration algorithm.Recently,the desire to formalize the notion of a program modification—the change made to one program to obtain another—motivated the introduction of a new algebraic structure in which integration can be formalized,called fm-algebra [Rama91].In fm-algebra,the notion of integration derives from the concepts of a program modification and an operation for combining modifications.Thus,while the earlier work reported in[Reps90]con-cerned a homogeneous algebra of programs,the later approach concerned a heterogeneous algebra of pro-grams and program modifications.The aim of this paper is to investigate further the algebraic structure of the domain of program modifications,in an attempt to explore the relationships between the program-integration problem and vari-ous related problems,and the relationships between different solutions to these problems.The paper stu-dies a particular fm-algebra that is defined in terms of a double Brouwerian algebra.Consequently,the integration operation of the fm-algebra models the HPR integration algorithm,and the results we derive apply to the HPR integration algorithm.We wish to point out that the problems of merging software extensions,program integration,separat-ing consecutive edits,and propagating changes through multiple versions must be addressed at both the syntactic and semantic levels.Let P denote the domain of programs,which are syntactic objects.Let S denote a suitable semantic domain,and let M:P→S be a semantic function assigning each program its meaning.Assume we want to formally specify an n-ary operation Merge:P×...×P→P that is meant to generate a new version of a program from the n input programs according to some criterion.At the semantic level,we need to specify what the desired semantics of the program to be generated is in terms of the semantics of the programs that are input to the Merge operation.One way this can be done is by defining a corresponding n-ary semantic operation SemanticMerge:S×...×S→S,and by requiring that the Merge operation satisfy the equation M(Merge(p1,...,p n))=SemanticMerge(M(p1),...,M(p n)).An advantage of the abstract approach taken in this paper(and our earlier ones)is that the results esta-blished can contribute to the understanding of the various program-manipulation operations at both the syn-tactic and semantic levels.Because our results are established using only algebraic identities and inequali-ties,our results hold for all structures whose elements obey a few simple axioms.For example,Berzins has considered one approach to defining the program-integration operation in a semantic domain[Berz91]. He extended the conventional semantic domain to a complete Boolean algebra,and defined a semantic integration operation as a ternary operator in this Boolean algebra.However,a Boolean algebra is also a Brouwerian algebra,and the Brouwerian integration operator defined in[Reps90]reduces to the operator defined in[Berz91]when the Brouwerian algebra under consideration is a Boolean algebra.Consequently, the results from[Reps90],[Rama91],and those that we derive in this paper are all relevant to studying the integration operation at the semantic level.The remainder of this paper is organized into seven sections.In Section2we review the frameworks of Brouwerian algebra and fm-algebra upon which this work is based.In Section3,we construct an fm-algebra from a given double Brouwerian algebra.We also construct a category from this fm-algebra,a category in which the objects denote programs and the arrows denote program modifications.Many pro-perties of the fm-algebra may be succinctly represented by commutative diagrams of this category.In Sec-tion4,we define a partial ordering among program modifications that captures the notion of subsumption among modifications and study the algebraic structure of this partially ordered set.In Section5,we relate the fm-algebraic operators to this partial ordering,define several(related)auxiliary operators that have a simple intuitive meaning,and study the properties of these operators.In Section6,we utilize the results of the earlier sections in establishing various properties of the integration operator.In Section7,we look at the problem of separating consecutive edits,and various of its solutions.In Section8,we address the ques-tion of whether program integration is a pushout in the category defined in Section3.2.Previous work2.1.The Brouwerian algebraic frameworkWe now briefly review the Brouwerian algebraic framework for program-integration developed in [Reps90].The idea behind this approach may be informally described as follows.A program consists of one or more program-components.Hence,a program may be viewed as a set of program-components. However,not every set or collection of program-components is a program.An integration algorithm like the HPR algorithm can decompose a program into its constituent components,compare two programs a and base and determine what components were added to base and what components of base were preserved in developing program a from base.Let a.−base denote the set of program-components in a but not in base.(This is not quite correct,as will be explained shortly.)Let a base denote the set of program components that are in both a and base.The integration algorithm integrates variants a and b of base by putting together the program components in a.−base,b.−base and a base b,obtaining (a.−base) (a base b) (b.−base).We may describe the Brouwerian algebraic framework more formally as follows.The HPR integra-tion algorithm makes use of a program representation called a program dependence graph [Kuck81,Ferr87].Let s be a vertex of a program dependence graph G.The slice of G with respect to s, denoted by G/s,is a graph induced by all vertices on which s has a transitiveflow or control dependence.A dependence graph G is a single-point slice iff it is the slice of some dependence graph with respect to some vertex(i.e.,equivalently,iff G=G/v for some vertex v in G).Let G1denote the set of all single-point slices.A partial order ≤is defined on the set G 1as follows:if a and b are single-point slices,b ≤a iff b is a slice of a with respect to some vertex in a (i.e.,iff b =a/v for some vertex v in a ).Thus,≤denotes the relation “is-a-subslice-of”.If p is a program,let S p denote the set of all single-point slices occurring in p (that is,in the program dependence graph of p ).The set S p is downwards-closed with respect to ≤.(A subset A of G 1is said to be a downwards-closed set if for every x ∈A and y ≤x ,y ∈A .)Given a subset A of G 1,DC (A ),the downwards-closed set generated by A ,is defined asDC (A )∆={b ∈G 1|∃a ∈A.(b ≤a )}.Thus,DC (A )is the smallest downwards-closed set containing A ,and hence represents the smallest pro-gram that contains all the slices in A .It can be verified that DC (A ∪B )=DC (A )∪DC (B ),and that a set A is downwards-closed iff DC (A )=A .The concepts of an upwards-closed set ,and UC (A ),the upwards-closed set generated by a set A,are similarly defined.UC (A )∆={b ∈G 1|∃a ∈A.(a ≤b )}.Let DCS denote the set of all downwards-closed sets of single-point slices.Let denote the set G 1,and let ⊥denote the empty set.Let ∪,∩,and −denote the set-theoretic union,intersection and difference operators.DCS is closed with respect to ∪and ∩.DCS is not closed under −,but is closed under the pseudo-difference operator .−defined as follows:X .−Y =DC (X −Y )DCS is also closed under a similar (dual)operator ..−defined as follows,where denotes the set-theoretic complement with respect to :X ..−Y =UC (Y −X ) .Reps [Reps90]shows that (DCS ,∪,∩,.−,..−, )is a double Brouwerian algebra (see below)and that the HPR integration algorithm can be represented by the ternary operator _[_]_defined by:a [base ]b ∆=(a .−base )∪(a ∩base ∩b )∪b .−base ).This permits the use of the properties of Brouwerian algebra in proving various algebraic properties of program-integration.Definition 2.1.A Brouwerian algebra [McKi46]is an algebra (P , , ,.−, )where (i)(P , , )is a lattice (with denoting the join operator,and denoting the meet operator)withgreatest element .The corresponding partial order will be denoted .(ii)P is closed under .−.(iii)For all a ,b ,and c in P ,(a .−b )c iff a (b c ).It can be shown that a Brouwerian algebra has a least element,given by .−,which will be denoted ⊥.Definition 2.2.A double Brouwerian algebra [McKi46]is an algebra (P , , ,.−,..−, )where both (P , , ,.−, )and (P , , ,..−, .− )are Brouwerian algebras.In particular,(i)P is closed under ..−.(ii)For all a ,b ,and c in P ,(a ..−b )c iff a (b c ).Definition 2.3.A ternary operator _[_]_,called the integration operator,of a Brouwerian algebra is defined as follows:a [base ]b ∆=(a .−base ) (a base b ) (b .−base ).In this framework it is convenient to think of the elements of DCS as being programs,though they really are only representations of programs.Further,not every element of DCS represents a program.An element s of DCS is said to be feasible if there exists a program p such that S p =s .Let FDCS denote the set of all feasible elements of DCS .FDCS is not closed with respect to the integration operation:it is pos-sible that a,base and b are feasible elements of DCS,while a[base]b is an infeasible element.The HPR integration algorithm reports interference among the variants exactly under these conditions.The reader is referred to[McKi46,Rasi63,Reps90,Reps91]for a discussion of the algebraic laws that hold in Brouwerian algebras and double Brouwerian algebras.2.2.The fm-algebraic frameworkThe goal of program-integration is to merge or combine changes made to some program.The concept of a “change made to a program”or a program-modification was formalized in[Rama91]and made use of in providing an alternative definition of integration.We briefly review this formalism below.Definition2.4.A modification algebra is a heterogeneous algebra(P,M,∆,+,apply)with two sets P and M and three operations∆,+,and apply,with the following functionalities:∆:P×P→M+:M×M→Mapply:M×P→P.For our purposes,we may interpret the components of a modification algebra as follows.P denotes the set of programs;M denotes a set of allowable program-modification operations;∆(a,base)yields the program-modification performed on base to obtain a1;operation m1+m2combines two modifications m1 and m2to give a new modification;apply(m,base)denotes the program obtained by performing modification m on program base.Definition2.5.The2-variant integration operator_[[_]]_:P×P×P→P of a modification algebra is defined as follows:a[[base]]b∆=apply(∆(a,base)+∆(b,base),base).The U NIX2utility diff yields a simple example of a modification algebra.Here,elements of P are text-files and elements of M are scriptfiles(for an editor).diff(with the option−e)implements∆,while the editor ed implements apply.There is no direct notion of+,butfile concatenation is one obvious choice.This choice,in fact,leads to an integration algorithm that is essentially diff3.We now consider a particular kind of the modification algebra,in which the modifications(elements of M)happen to be certain functions from P to P,and apply is just ordinary function application.Definition2.6.A functional-modification algebra(abbreviated fm-algebra)is an algebra(P,M,∆,+) where M⊆P→P,and∆and+are operations with the following functionalities:∆:P×P→M+:M×M→M.Definition2.7.The2-variant integration operator_[[_]]_:P×P×P→P of an fm-algebra is defined as follows:a[[base]]b∆=(∆(a,base)+∆(b,base))(base).In[Rama91]two different systems of axioms for fm-algebra are introduced,and then used to study the algebraic properties of the_[[_]]_operator.It is shown there how the HPR integration algorithm may be1Of course,there need not be a unique program-modification in M that yields a when applied to base.A more precise interpretation of ∆(a,base),in the fm-algebra we are interested in,will be given in Section5.1.2U NIX is a trademark of AT&T Bell Laboratories.modelled as the integration operation of an appropriately defined fm-algebra,by making use of the Brouwerian-algebraic framework outlined in the Section2.1.The definition of this fm-algebra and the development of its algebraic structure will be the subject of the following sections.3.An fm-algebra from a double Brouwerian algebraThe goal of this work is to study the fm-algebra that models the HPR integration algorithm.In the remain-ing part of the paper,P=(P, , ,.−,..−, )denotes a double Brouwerian algebra.The elements of P will be referred to as programs.The letters x,y,z,a,b,c,and base will typically denote elements of P. The set M of program-modifications of interest is given by the following definition.Definition3.1.M∆={λz.(z y) x|x,y∈P}The elements of M will be referred to as modifications.The symbols m and m i will typically denote ele-ments of M.Let denote function composition.Thus,m1 m2denotes the functionλx.m1(m2(x)).Then,λz.(z y) x=(λz.z x) (λz.z y).Informally,we may think of modificationλz.(z y) x= (λz.z x) (λz.z y)as consisting of two components:λz.z x,which represents an addition of new program components(where x is the set of program components to be added),andλz.z y,which represents deletion of certain program components(where y represents the set of program components to be preserved).We would like to represent the modificationλz.(z y) x by the ordered pair(x,y).However,note thatλz.(z y1) x1andλx.(x y2) x2might be equal even if(x1,y1)≠(x2,y2).But the representation of a modification as an ordered pair may be made unique by restricting attention to ordered pairs(x,y) where x y.Definition3.2.S,a subset of P×P,is defined as follows:S∆={(x,y)∈P×P|x y}Definition3.3.A functionρfrom S to M is defined as follows:ρ((x,y))∆=λz.(z y) xProposition3.4.ρis a bijection from S to M.Proof.Wefirst show thatρis onto;that is,we show that for any modification m∈M,there exists an ordered pair(x,y)∈S such thatρ((x,y))=m.Modification m must be equal toλz.(z y1) x1for some x1and y1,by definition of M.Then,(x1,y1 x1)∈S,andρ((x1,y1 x1))=λz.(z (y1 x1)) x1=λz.(z y1) (z x1) x1=λz.(z y1) x1=m.We now show thatρis1-1;that is,we show that if(x1,y1)and(x2,y2)are two different elements of S,thenρ((x1,y1))≠ρ((x2,y2)).Observe that if(x,y)∈S thenρ((x,y))identifies x and y uniquely as fol-lows:ρ((x,y))(⊥)=x,andρ((x,y))( )=y.It follows thatρis1-1.Hence,ρis a bijection,and we see thatρ−1(m)=(m(⊥),m( )).We use the bijectionρto represent modifications as ordered pairs.We will abbreviateρ((x,y))to <x,y>.In particular,any reference to a modification<x,y>automatically implies that x y. Proposition3.5.(a)M is closed with respect to .In particular,<x1,y1> <x2,y2>=<x1 (x2 y1),y1 (y2 x1)>.(b)Let I denote the modification<⊥, >.Then,(M, ,I)is a monoid:i.e., is associative,and I is the identity with respect to .Proof.(a)<x1,y1> <x2,y2>=(λz.(z y1) x1) (λz.(z y2) x2)=λz.(((z y2) x2) y1) x1=λz.((z y2 y1) (x2 y1)) x1(since a Brouwerian algebra is a distributive lattice)=λz.(z (y2 y1)) ((x2 y1) x1)Hence,<x1,y1> <x2,y2>∈M.Recall that if m∈M,then m=<m(⊥),m( )>.Hence, <x1,y1> <x2,y2>=<x1 (x2 y1),y1 (y2 x1)>.(b)Function composition is associative.<⊥, >denotes the functionλz.(z ) ⊥=λz.z which is the identity function.It follows that(M, ,I)is a monoid.An intuitive explanation of program-integration using pictures has often raised the conjecture that integration was a pushout in an appropriate category.(See Figure1.)Pictures depicting integration typi-cally have programs and arrows between programs,representing modifications to programs.We may for-malize these pictures as diagrams in a category we define below,a category in which the objects denote programs,and arrows denote program-modifications.We will return to the question of whether integration is a pushout in Section8.Definition3.6.Define a category C as follows:The set of objects is P.For every a∈P,and m∈M,there is an arrow labelled[a,m,m(a)]with domain a and codomain m(a).Thus,every arrow[a,m,m(a)]is associated with a program modification m.(If no confusion is likely,the arrow will just be denoted by m.) The operation is defined to be function composition.Thus,[b,m2,c] [a,m1,b]=[a,m2 m1,c].Note that multiple arrows can be associated with the same modification function,provided each of these arrows has a different domain.Every set of arrows has an associated set of modifications.Occasion-ally,we blur the distinction between a set of arrows and the associated set of modifications,in order to give a simple pictorial interpretation for certain sets of modifications.a [ [ base ]] bFigure1.The problem of two variant program-integration.4.A subsumption ordering on program-modificationsIn this section we define a partial ordering≤on the set M of modifications that captures the notion of sub-sumption among modifications.In particular,we may think of“m1≤m2”as a synonym for“modification m1is a part of modification m2”.This provides an intuitive meaning for the meet and join operators with respect to this partial ordering.These operators are closely linked with the function composition operator, and we begin with one of the important properties satisfied by the set M of modification functions.Proposition4.1.(Extended idempotence).m1 m2 m1=m1 m2for all m1,m2∈M.Proof.Let m1be<x1,y1>and m2be<x2,y2>.Then,<x1,y1> <x2,y2> <x1,y1>=<x1 (x2 y1),y1 (y2 x1)> <x1,y1>=<x1 (x2 y1),y1 (y2 x1)>=<x1,y1> <x2,y2>Corollary4.2. is idempotent.Proof.Let m2=I in Proposition4.1.Definition4.3.Define a binary relation≤on M as follows:m1≤m2⇔m1 m2=m2=m2 m1The relation≤is intended to capture the notion of subsumption among modifications.Thus, modification m1is subsumed by(or“is contained in”,“is smaller than”)modification m2if performing m1 and m2in either order does nothing more than performing m2.The following proposition establishes alter-native interpretations of this relation.Proposition4.4.(a)m1≤m2iff m1 m2=m2.(b)≤is a partial order.(c)m1≤m2iff∃m.m2=m1 m.(d)m1 m2≥m1.(e)I is the least element with respect to≤.Proof.(a)If m1≤m2,then m1 m2=m2trivially.Conversely,assume m1 m2=m2.Then,m2 m1= (m1 m2) m1=m1 m2(using Proposition4.1)=m2.Hence,m1≤m2.(b)reflexivity:m m=m,from Corollary4.2.Hence,m≤m,from(a).antisymmetry:Assume m1≤m2and m2≤m1.Hence,m1 m2=m2and m2 m1=m1.We show that m1=m2as follows:m2=m1 m2=(m2 m1) m2=m2 m1((using Proposition4.1)=m1.transitivity:Assume m1≤m2and m2≤m3.Hence,m1 m2=m2and m2 m3=m3.From(a), we can establish that m1≤m3by showing that m1 m3=m3,which,in turn,may be established as fol-lows:m1 m3=m1 (m2 m3)=(m1 m2) m3=m2 m3=m3.(c)If m1≤m2then m2=m1 m2.Hence,∃m.m2=m1 m is true.Conversely,assume m2=m1 m for some m.Then,m1 m2=m1 m1 m=m1 m=m2.Hence,from(a),m1≤m2.(d)Follows from(c).(e)For any m,I m=m=m I.Hence,I≤m.We will often make use of Proposition4.4(a),without specifically referring to it,in showing that m1≤m2for some particular m1and m2.The poset(M,≤)has an interesting structure,as will be apparent soon.。
a rXiv:funct-a n /9724v111F eb1997PSEUDODIFFERENTIAL OPERATORS ON DIFFERENTIAL GROUPOIDS VICTOR NISTOR,ALAN WEINSTEIN,AND PING XU Abstract.We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners.We show that this construction encompasses many examples.The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid,in the sense of non-commutative geometry.Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid.As applications,we give a new proof of the Poincar´e -Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A ∗of a Lie algebroid.Contents Introduction 11.Preliminaries 32.Main definition 63.Differential operators and quantization 134.Examples 185.Distribution kernels 216.The action on sections of E 26References 28Introduction Certain important applications of pseudodifferential operators require variantsof the original definition.Among the many examples one can find in the literature are regular or adiabatic families of pseudodifferential operators [2,40]and pseudo-differential operators along the leaves of foliations [6,8,27,28],on coverings [9,29]or on certain singular spaces [21,22,26,25].Since these classes of operators share many common features,it is natural to ask whether they can be treated in a unified way.In this paper we shall suggest an answer to this question.For any “almost differential”groupoid (a class which allows manifolds with corners),we construct an algebra of pseudodifferential operators.We then show that our construction recovers (almost)all the classes described above (for operators on manifolds with boundary our algebra is slightly smaller thanthe2V.NISTOR,A.WEINSTEIN,AND PING XUone defined in[21]).We expect our results to have applications to analysis on singular spaces,not only manifolds with corners.Our construction and results owe a great deal to the previous work of several authors,especially Connes[6]and Melrose[21,23,20].A hint of the direction we take was given at the end of[37].The basic idea of our construction is to consider families of pseudodifferential operators along thefibers of the domain(or source) map of the groupoid.More precisely,for any almost differentiable groupoid(see Definition3)we consider thefibers G x=d−1(x)of the domain map d consisting of all arrows with domain x.It follows from the definition that thesefibers are smooth manifolds(without corners).The calculus of pseudodifferential operators on smooth manifolds is well understood and by now a classical subject,see for example[14]. We shall consider differentiable families of pseudodifferential operators P x on the smooth manifolds G x.Right translation by g∈G defines an isomorphism G x≡G y where x is the domain of g and y is the range of g.We say that the family P x is invariant if P x transforms to P y under the diffeomorphisms above(for all g). The algebraΨ∞(G)of pseudodifferential operators on G that we shall consider will consist of invariant differentiable families of operators P x as explained above (the actual definition also involves a technical condition on the support of these operators).See Definition7for details.The relation with the work of Melrose relies on an alternative description of our algebra as an algebra of distributions on G with suitable properties(compactly supported,and conormal with singular support contained in the set of units).This is contained in Theorem7.The difference between our theory and Melrose’s lies in the fact that he considers a compactification of G as a manifold with corners,and his distributions are allowed to extend to the compactification,with precise behavior at the boundary.This is useful for the analysis of these operators.In contrast,our work is purely algebraic(or geometric,depending on whether one considers Lie algebroids as part of geometry or algebra).We now review the contents of the sections of this paper.In thefirst section we recall the definitions of a groupoid,Lie algebroid and,the less known definition of a local Lie groupoid.We extend the definition of a Lie groupoid to include manifolds with corners.These groupoids are called almost differentiable groupoids. The second section contains the definition of a pseudodifferential operator on a groupoid(really a family of pseudodifferential operators,as explained above)and the proof that they form an algebra,if a support condition is included.We also extend this definition to include local Lie groupoids.This is useful in the third section where we use this to give a new proof of the Poincar´e-Birkhoff-Witt theorem for Lie algebroids.In the process of proving this theorem we also exemplify our definition of pseudodifferential operators on an almost differentiable groupoid by describing the differential operators in this class.As an application we give an explicit construction of a deformation quantization of the Lie-Poisson structure on A∗,the dual of Lie algebroid A.The section entitled“Examples”contains just what the title suggests:for many particular examples of groupoids G we explicitly describe the algebraΨ∞(G)of pseudodifferential operators on G.This recovers classes of operators that were previously defined using ad hoc constructions.Our definition is often not only more general,but also simpler.This is the case for operators along the leaves of foliations[8,27]or adiabatic families of operators.PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS3 Since one of our main themes is that the Lie algebras of vectorfields which are central in[24]are in fact the spaces of sections of Lie algebroids,we describe these Lie algebroids explicitly in each of our examples.In the sixth section of the paper,we describe the convolution kernels(called reduced kernels)of operators inΨ∞(G).Then we extend to our setting some fun-damental results on principal symbols,by reducing to the classical results.This makes our proofs short(and easy).Finally,the last section treats the action of Ψ∞(G)on functions on the units of G,and a few related topics.Thefirst author would like to thank Richard Melrose for several useful conver-sations.1.PreliminariesIn the following we allow manifolds to have corners.Thus by“manifold”we shall mean a C∞manifold,possibly with corners,and by a“smooth manifold”we shall mean a manifold without corners.By definition,if M is a manifold with corners then every point p∈M has coordinate neighborhoods diffeomorphic to [0,∞)k×R n−k.The transition functions between such coordinate neighborhoods must be smooth everywhere(including on the boundary).We shall use the following definition of submersions between manifolds(with corners).Definition1.A submersion between two manifolds with corners M and N is a differentiable map f:M→N such that d f x:T x M→T f(x)N is onto for any x∈M,and if d f x(v)is an inward pointing tangent vector to N,then v is an inward pointing tangent vector to M.The reason for introducing the definition above is that for any submersion f: M→N,the set M y=f−1(y),y∈N is a smooth manifold,just as for submersions of smooth manifolds.We shall study groupoids endowed with various structures.([32]is a general reference for some of what follows.)We recallfirst that a small category is a category whose class of morphisms is a set.The class of objects of a small category is then a set as well.Definition2.A groupoid is a small category G in which every morphism is in-vertible.This is the shortest but least explicit definition.We are going to make this definition more explicit in cases of interest.The set of objects,or units,of G will be denoted byM=G(0)=Ob(G).The set of morphisms,or arrows,of G will be denoted byG(1)=Mor(G).We shall sometimes write G instead of G(1)by abuse of notation.For example, when we consider a space of functions on G,we actually mean a space of functions on G(1).We will denote by d(g)[respectively r(g)]the domain[respectively,the range]of the morphism g:d(g)→r(g).We thus obtain functions(1)d,r:G(1)−→G(0)4V.NISTOR,A.WEINSTEIN,AND PING XUthat will play an important role bellow.The multiplication operator µ:(g,h )→µ(g,h )=gh is defined on the set of composable pairs of arrows G (2):µ:G (2)=G (1)×M G (1):={(g,h ):d (g )=r (h )}−→G (1).(2)The inversion operation is a bijection ι:g →g −1of G (1).Denoting by u (x )the identity morphism of the object x ∈M =G (0),we obtain an inclusion of G (0)into G (1).We see that a groupoid G is completely determined by the spaces G (0)and G (1)and the structural morphisms d,r,µ,u,ι.We sometimes write G =(G (0),G (1),d,r,µ,u,ι).The structural maps satisfy the following properties:(i)r (gh )=r (g ),d (gh )=d (h )for any pair (g,h )∈G (2),and the partially defined multiplication µis associative.(ii)d (u (x ))=r (u (x ))=x ,∀x ∈G (0),u (r (g ))g =g and gu (d (g ))=g ,∀g ∈G (1)and u :G (0)→G (1)is one-to-one.(iii)r (g −1)=d (g ),d (g −1)=r (g ),gg −1=u (r (g ))and g −1g =u (d (g )).Definition 3.An almost differentiable groupoid G =(G (0),G (1),d,r,µ,u,ι)is a groupoid such that G (0)and G (1)are manifolds with corners,the structural maps d,r,µ,u,ιare differentiable,and the domain map d is a submersion.We observe that ιis a diffeomorphism and hence d is a submersion if and only if r =d ◦ιis a submersion.Also,it follows from the definition that each fiber G x =d −1(x )⊂G (1)is a smooth manifold whose dimension n is constant on each connected component of G (0).The ´e tale groupoids considered in [5]are extreme examples of differentiable groupoids (corresponding to dim G x =0).If G (0)is smooth (i.e.if it has no corners)then G (1)is also smooth and G becomes what is known as a differentiable,or Lie groupoid.1We now introduce a few important geometric objects associated to an almost differentiable groupoid.The vertical tangent bundle (along the fibers of d )of an almost differentiable groupoid G isT d G =ker d ∗=x ∈G (0)T G x ⊂T G (1).(3)Its restriction A (G )=T d GG (0)to the set of units is the Lie algebroid of G [19,30].We denote by T ∗d G the dual of T d G and by A ∗(G )the dual of A (G ).In addition tothese bundles we shall also consider the bundle Ωλd of λ-densities along the fibers of d .If the fibers of d have dimension n then Ωλd =|Λn T ∗d G|λ=∪x Ωλ(G x ).By invariance these bundles can be obtained as pull-backs of bundles on G (0).For example T d G =r ∗(A (G ))and Ωλd =r ∗(D λ)where D λdenotes Ωλd |G (0).If E is a (smooth complex)vector bundle on the set of units G (0)then the pull-back bundle r ∗(E )on G will have right invariant connections obtained as follows.A connection ∇on E lifts to a connection on r ∗(E ).Its restriction to any fiber G x defines a linear connection in the usual sense,which is denoted by ∇x .It is easy to see that these connections are right invariant in the sense thatR ∗g ∇x =∇y ,∀g ∈G such that r (g )=x and d (g )=y.(4)The bundles considered above will thus have invariant connections.PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS5 The bundle A(G),called the Lie algebroid of G,plays in the theory of almost differentiable groupoids the rˆo le Lie algebras play in the theory of Lie groups.We recall for the benefit of the reader the definition of a Lie algebroid[30].Definition4.A Lie algebroid A over a manifold M is a vector bundle A over M together with a Lie algebra structure on the spaceΓ(A)of smooth sections of A,and a bundle mapρ:A→T P,extended to a map between sections of these bundles, such that(i)ρ([X,Y])=[ρ(X),ρ(Y)];and(ii)[X,fY]=f[X,Y]+(ρ(X)f)Yfor any smooth sections X and Y of A and any smooth function f on M.Note that we allow the base M in the definition above to be a manifold with corners.If G is an almost differentiable groupoid then A(G)will naturally have the struc-ture of a Lie algebroid[19].Let us recall how this structure is defined(the original definition easily extends to include manifolds with corners).Clearly A(G)is a vec-tor bundle.The right translation by an arrow g∈G defines a diffeomorphism R g:G r(g)∋g′→g′g∈G d(g).This allows us to talk about right invariant dif-ferential geometric quantities as long as they are completely determined by their restriction to all submanifolds G x.This is true of functions and d–vertical vector fields,and this is all that is needed in order to define the Lie algebroid structure on A(G).The sections of A(G)are in one-to-one correspondence with vectorfields X on G that are d–vertical,in the sense that d∗(X(g))=0,and right invariant.The condition d∗(X(g))=0means that X is tangent to the submanifolds G x,thefibers of d.The Lie bracket[X,Y]of two d–vertical right–invariant vectorfields X and Y will also be d–vertical and right–invariant,and hence the Lie bracket induces a Lie algebra structure on the sections of A(G).To define the action of the sections of A(G)on functions on G(0),observe that the right invariance property makes sense also for functions on G and that C∞(G(0))may be identified with the subspace of right–invariant functions on G.If X is a right–invariant vectorfield on G and f is a right–invariant function on G then X(f)will still be a right invariant function. This identifies the action ofΓ(A(G))on functions on G(0).Not every Lie algebroid is the Lie algebroid of a Lie groupoid(see[1]for an example).However,every Lie algebroid is associated to a local Lie groupoid[31]. The definition of a local Lie(or more generally,almost differentiable)groupoid [10]is obtained by relaxing the condition that the multiplicationµbe everywhere defined on G(2)(see Equation(2)),and replacing it by the condition thatµbe defined in a neighborhood U of the set of units.Definition5(van Est).An almost differentiable local groupoid L=(L(0),L(1))is a pair of manifolds with corners together with structural morphisms d,r:L(1)→L(0),ι:L(1)→L(1),u:L(0)→L(1)andµ:U→L(1),where U is a neighborhood of(u×u)(L(0))={(u(x),u(x))}in L(2)={(g,h),d(g)=r(h)}⊂L(1)×L(1). The structural morphisms are required to be differentiable maps such that d is a submersion,u is an embedding,and to satisfy the following properties:(i)The products u(d(g))g,gu(r(g)),gg−1and g−1g are defined and coincide with,respectively,g,g,u(r(g))and u(d(g));where we denoted g−1=ι(g)as usual.(ii)If gh is defined,then h−1g−1is defined and equal to(gh)−1.(iii)(Local associativity)If gg′,g′g′′and(gg′)g′′are defined then g(g′g′′)is also defined and equal to(gg′)g′′.6V.NISTOR,A.WEINSTEIN,AND PING XUThe set U is the set of arrows for which the product gh=µ(g,h)is defined.We see that the only difference between a groupoid and a local groupoid L is the fact that the condition d(g)=r(h)is necessary for the product gh=µ(g,h) to be defined,but not sufficient in general.The product is defined as soon as the arrows g and h are“small enough”.A consequence of this definition is that the right multiplication by an arrow g∈L(1)defines only a diffeomorphismU g−1∋g′→g′g∈U g(5)of an open(and possibly empty)subset U g−1of L y,y=r(g)to an open subset U g⊂L x,x=d(g).This will not affect the considerations above,however,so we can associate a Lie algebroid A(L)to any almost differentiable local groupoid L.In the following,when considering groupoids,we shall sometimes refer to them as global groupoids,in order to stress the difference between groupoids and local groupoids.2.Main definitionConsider a complex vector bundle E on the space of units G(0)of an almost differentiable groupoid G.Denote by r∗(E)its pull-back to G(1).Right translations on G define linear isomorphismsU g:C∞(G d(g),r∗(E))→C∞(G r(g),r∗(E))(6)(U g f)(g′)=f(g′g)∈(r∗E)g′which makes sense because(r∗E)g′=(r∗E)g′g=E r(g′).If G is merely a local groupoid then(6)is replaced by the isomorphismsU g:C∞(U g,r∗(E))→C∞(U g−1,r∗(E))(7)defined for the open subsets U g⊂G d(g)and U g−1⊂G r(g)defined in(5).Let B⊂R n be an open subset.Define the space S m(B×R n)of symbols on the bundle B×R n→B as in[14]to be the set of smooth functions a:B×R n→C such that|∂αy∂βξa(y,ξ)|≤C K,α,β(1+|ξ|)m−|β|(8)for any compact set K⊂B and any multiindicesαandβ.An element of one of our spaces S m should properly be said to have“order less than or equal to m”; however,by abuse of language we will say that it has“order m”.A symbol a∈S m(B×R n)is called classical if it has an asymptotic expansion as an infinite sum of homogeneous symbols a∼ ∞k=0a m−k,a l homogeneous of degree l:a l(y,tξ)=t l a l(y,ξ)if ξ ≥1and t≥1.(“Asymptotic expansion”is used here in the sense that a− N−1k=0a m−k belongs to S m−N(B×R n).)The space of classical symbols will be denoted by S m cl(B×R n).We shall be working exclusively with classical symbols in this paper.This definition immediately extends to give spaces S m cl(E;F)of symbols on E with values in F,whereπ:E→B and F→B are smooth euclidian vector bundles. These spaces,which are independent of the metrics used in their definition,are sometimes denoted S m cl(E;π∗(F)).Taking E=B×R n and F=C one recovers S m cl(B×R n)=S m cl(B×R n;C).A pseudodifferential operator P onB is a linear map P:C∞c(B)→C∞(B) that is locally of the form P=a(y,D y)plus a regularizing operator,where for anyPSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS7 complex valued symbol a on T∗W=W×R n,W an open subset of R n,one defines a(y,D y):C∞c(W)→C∞(W)bya(y,D y)u(y)=(2π)−n R n e iy·ξa(y,ξ)ˆu(ξ)dξ.(9)Recall that an operator T:C∞c(U)→C∞(V)is called regularizing if and only if it has a smooth distribution(or Schwartz)kernel.This happens if and only if T is pseudodifferential of order−∞.The class of a in S m cl(T∗W)/S m−1cl (T∗W)does not depend on any choices;thecollection of all these classes,for all coordinate neighborhoods W,patches togetherto define a classσm(P)∈S m cl(T∗W)/S m−1cl (T∗W)which is called the principal sym-bol of P.If the operator P acts on sections of a vector bundle E,then the principalsymbolσm(P)will belong to S m cl(T∗B;End(E))/S m−1cl (T∗B;End(E)).See[14]formore details on all these constructions.We shall sometimes refer to pseudodifferential operators acting on a smooth manifold as ordinary pseudodifferential operators,in order to distinguish them from pseudodifferential operators on groupoids,a class of operators which we now define (and which are really families of ordinary pseudodifferential operators).Throughout this paper,we shall denote by(P x,x∈G(0))a family of order m pseudodifferential operators P x,acting on the spaces C∞c(G x,r∗(E))for some vector bundle E over G(0).Operators between sections of two different vector bundles E1 and E2are obtained by considering E=E1⊕E2.Definition6.A family(P x,x∈G(0))as above is called differentiable if for any open set V⊂G,diffeomorphic through afiber preserving diffeomorphism to d(V)×W,for some open subset W⊂R n,and anyφ∈C∞c(V),we canfind a∈S m cl(d(V)×T∗V;End(E))such thatφP xφcorresponds to a(x,y,D y)under the diffeomorphism G x∩V≃W,for each x∈d(V).Afiber preserving diffeomorphism is a diffeomorphismψ:d(V)×W→V satisfying d(ψ(x,w))=x.Thus we require that the operators P x be given in local coordinates by symbols a x that depend smoothly on all variables,in particular on x∈G(0).Definition7.An order m invariant pseudodifferential operator P on an almost differentiable groupoid G,acting on sections of the vector bundle E,is a differ-entiable family(P x,x∈G(0))of order m classical pseudodifferential operators P x acting on C∞c(G x,r∗(E))and satisfyingP r(g)U g=U g P d(g)(invariance)(10)for any g∈G(1),where U g is as in(6).Replacing the coefficient bundle E by E⊗Dλand using the isomorphismΩλd≃r∗(Dλ),we obtain operators acting on sections of density bundles.Note that P can generally not be considered as a single pseudodifferential operator on G(1). This is because a family of pseudodifferential operators on a smooth manifold M, parametrized by a smooth manifold B,is not a pseudodifferential operator on the product M×B,although it acts naturally on C∞c(M×B).(See[2]or[14],page 94.)Recall[13]that distributions on a manifold Y with coefficients in the bundle E0 are continuous linear maps C∞c(Y,E′0⊗Ω)→C,where E′0is the dual bundle to E08V.NISTOR,A.WEINSTEIN,AND PING XUandΩ=Ω(Y)is the space of1-densities on Y.The collection of all distributions on Y with coefficients in the(finite dimensional complex vector)bundle E0is denoted C−∞(Y;E0).If P=(P x,x∈G(0))is a family of pseudodifferential operators acting on G x denote by k x the distribution kernel of P x(11)k x∈C−∞(G x×G x;r∗1(E)⊗r∗2(E)′⊗Ω2).HereΩ2is the pull-back of the bundle of vertical densitiesΩd on G x to G x×G x via the second projection.These distribution kernels are obtained using Schwartz’kernel theorem.We define the support of the operator P to besupp(P)=PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS9 Proof.If P consists of regularizing operators thenP f(g)= G x k x(g,h)f(h),where x=d(g).Lemma1implies that the formula above for P f involves only the integration of smooth(uniformly in g)compactly supported sections,and hence we can exchange integration and derivation to obtain the smoothness of P f.This proves(i)in case P consists of regularizing operators.The proof of(ii)if both P and Q consist of regularizing operators follows the same reasoning.We prove now(i)for P arbitrary.Fix g∈G x and V a neighborhood of gfiber preserving diffeomorphic to d(V)×W for some open convex subset W in R n,0∈W, such that(x,0)maps to g.Replacing P x by P x−R x for a smooth regularizing family R x we can assume that the distribution kernels k x of P x satisfyp−11(d(V)×W/4)∩∪supp(k′x)⊂(d(V)×W/2)×(d(V)×3W/4)where k′x are the distribution kernels of Q x.The support estimates above for P and Q show that the P y Q y for y∈d(V)are the compositions of smooth families of pseudodifferential operators acting on W⊂R n.The result is then known.The smaller class of uniformly supported operators is also closed under compo-sition.Lemma3.The composition P Q=(P x Q x,x∈G(0))of two uniformly supported families of operators P=(P x,x∈G(0))and Q=(Q x,x∈G(0))is uniformly supported.Proof.The reduced support suppµ(P Q)(see(12))of the composition P Q satisfiessuppµ(P Q)⊂µ suppµ(P)×suppµ(Q)whereµis the composition of arrows.Since suppµ(P)and suppµ(Q)are compact, the equation above completes the proof of the lemma.Let G be an almost differentiable groupoid.The space of order m,invariant,uni-formly supported pseudodifferential operators on G,acting on sections of the vector bundle E will be denoted byΨm(G;E).We denoteΨ∞(G;E)=∪m∈ZΨm(G;E) andΨ−∞(G;E)=∩m∈ZΨm(G;E).Thus an operator P∈Ψm(G;E)is actually a differentiable family P=(P x,x∈G(0))of ordinary pseudodifferential operators. Theorem1.The setΨ∞(G;E)of uniformly supported invariant pseudodifferential operators on an almost differentiable groupoid G is afiltered algebra,i.e.Ψm(G;E)Ψm′(G;E)⊂Ψm+m′(G;E).In particularΨ−∞(G;E)is a two-sided ideal.10V.NISTOR,A.WEINSTEIN,AND PING XUProof.Let P=(P x,x∈G(0))and Q=(Q x,x∈G(0))be two invariant uniformly supported pseudodifferential operators on G,of order m and m′respectively.Their composition P Q=(P x Q x),is a uniformly supported operator of order m+m′,in view of Lemma3.It is also a differentiable family due to Lemma2.We now check the invariance condition.Let g be an arbitrary arrow and U g:C∞c(G x,r∗(E))→C∞c(G y,r∗(E)),x=d(g)and y=r(g),be as in the definition above.Then(P Q)y U g=P y Q y U g=P y U g Q x=U g P x Q x=U g(P Q)x.This proves the theorem.Properly supported invariant differentiable families of pseudodifferential opera-tors also form afiltered algebra,denotedΨ∞prop(G;E).While it is clear that in order for our class of pseudodifferential operators to form an algebra we need some con-dition on the support of their distribution kernels,exactly what support condition to impose is a matter of choice.We prefer the uniform support condition because it leads to a better control at infinity of the family of operators P=(P x,x∈G(0)) and allows us to identify the regularizing ideal(i.e.the ideal of order−∞operators) with the groupoid convolution algebra of G.The choice of uniform support will also ensure thatΨm(G;E)behaves functorially with respect to open embeddings.The compact support condition enjoys the same properties but is usually too restrictive. The issue of support will be discussed again in examples.The definition of the principal symbol extends easily toΨm(G;E).Denote by π:A∗(G)→M,(M=G(0))the projection.If P=(P x,x∈G(0))∈Ψm(G;E)is an order m pseudodifferential differential operator on G,then the principal symbol σm(P)of P will be represented by sections of the bundle End(π∗E)and will be defined to satisfyσm(P)(ξ)=σm(P x)(ξ)∈End(E x)ifξ∈A∗x(G)=T∗x G x(13)(the equation above is mod S m−1cl (A∗x(G);End(E))).This equation will obviouslyuniquely determine a linear mapσm:Ψm(G)→S m cl(A∗(G);End(E))/S m−1cl(A∗(G);End(E)).provided we can show that for any P=(P x,x∈G(0))there exists a symbol a∈S m cl(A∗(G);End(E))whose restriction to A∗x(G)is a representative of the prin-cipal symbol of P x in thatfiber for each x.We thus need to choose for each P x a representative a x∈S m cl(A∗x(G);End(E))ofσm(P x)such that the family a x is smooth and invariant.Assumefirst that E is the trivial line bundle and proceed as in[14]Section18.1,especially Equation(18.1.27)and below.Choose a connection∇on the vector bundle A(G)→G(0)and consider the pull-back vector bundle r∗(A)→G of A(G)→G(0)endowed with the pull-back connection ∇=r∗∇.Its restriction on anyfiber G x defines a linear connection in the usual sense,which is denoted by∇x.These connections are right invariant in the sense thatR∗g∇x=∇y,∀g∈G such that r(g)=x and d(g)=y.(14)Using such an invariant connection,we may define the exponential map of a Lie algebroid,which generalizes the usual exponential map of a manifold with a connection and the exponential map of a Lie algebra as follows.For any x∈G(0),PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS11 define a map exp x:A x→G as the composition of the maps:A x i−→T x G x˜exp x−−−−→Gwhere i is the natural inclusion and˜exp x=exp∇x is the usual exponential mapat x∈G x on the manifold G x.By varying the point x,we obtain a map exp∇defined in a neighborhood of the zero section,called the exponential map of the Lie algebroid2.Clearly,exp∇is a local diffeomorphismA(G)⊃V0∋v−→exp∇(v)=y∈V⊂G(15)mapping an open neighborhood V0of the zero section in A(G)diffeomorphically to a neighborhood V of G(0)in G,and sending the zero section onto the set of units. Choose a cut-offfunctionφ∈C∞(G)with support in V and equal to1in a smaller neighborhood of G(0)in G.If y∈V,x=d(y)andξ∈A∗x(G)let v∈V0be the unique vector v∈A x(G)such that y=exp∇(v)and denote eξ(y)=φ(y)e iv·ξwhich extends then to all y∈G due to the cut-offfunctionφ.Define the(∇,φ)–complete symbolσ∇,φ(P)byσ∇,φ(P)(ξ)=(P x eξ)(x),∀ξ∈T∗x G x=A∗x(G).(16)Lemma4.If P=(P x,x∈G(0))is an operator inΨm(G)then the function σ∇,φ(P)defined above is differentiable and defines a symbol in S m cl(A∗(G)).More-over if(∇1,φ1)is another pair consisting of an invariant connection∇1and a cut-offfunctionφ1thenσ∇,φ(P)−σ∇,φ1(P)is in S−∞cl(A∗(G))andσ∇,φ(P)−σ∇1,φ1(P)is in S m−1cl(A∗(G)).Proof.For eachξ∈A∗x the function eξis smooth with compact support on G x so P x eξis defined.Equation(18.1.27)of[14]shows that a(ξ)=σ∇,φ(P)(ξ)is the restriction of the complete symbol of P xφto T∗x G x if the complete symbol is defined in the normal coordinate system at x∈G x(given by the exponential map). The normal coordinate system defines,using a local trivialization of A(G),afiber preserving diffeomorphismψ:d(V)×W→V for some open subset W of R n(i.e. satisfying d(ψ(x,w))=x).From the definition of the smoothness of the family P x (Definition6)it follows that the complete symbol of Pφis in S m cl(d(V)×T∗W)if the support ofφis chosen to be in V.This proves thatσ∇,φ(P)is in S m cl(A∗(G)).The rest follows in exactly the same way.The lemma above justifies the following definition of the principal symbol as the class ofσ∇,φ(P)modulo terms of lower order(for the trivial line bundle E=C). This definition will be,in view of the same lemma,independent on the choice of ∇orφand will satisfy Equation(13).If E is not trivial one can still define a complete symbolσ∇,∇′,φ(P),depending also on a second connection∇′on the bundle E,which is used to trivialize r∗(E)on V⊂G(assuming also that V0is convex).Alternatively,we can use Proposition3below.Proposition1.Let∇andφbe as above.The choice of a connection∇′on E defines a complete symbol mapσ∇,∇′,φ:Ψm(G;E)→S m cl(A∗(G)).The principalsymbolσm:Ψm(G;E)→S m cl(A∗(G))/S m−1cl(A∗(G)),defined byσm(P)=σ∇,∇′,φ(P)+S m−1cl (A∗(G))(17)。
【单选】 All oil spills must be reported to the ______. --------------------------------------------------------------------------------A.MOCB.MSA OF CHINAC.local fire departmentD.local police【单选】 Many of the lights on this coast are placed so high as to be frequently obscured by ______.A.cover B.shower C.tower D.power【单选】 The collision bulkhead is located ______.A.on the bridge deck B.at the stern of the ship C.as the first watertight bulkhead ahead D.between the passenger and cargo areas【单选】 The charts sold are of ______.A.the current edition and incorporate the last Notices to Mariners correction B.brand-new one with up to date correction and clean writing C.newly edition with up to date correction and in reasonable prices D.the current edition and incorporate the latest Notices to Mariners correction at the time of sale【单选】 ______ is not contained in the NM Weekly.A.Amendments to Admiralty List of Radio Signals B.Supplement to Guide to Port Entry C.Amendments to Admiralty Sailing Directions D.Amendments to Admiralty List of Lights and Fog Signals 单选】 Which document will describe lifesaving equipment located aboard your vessel?A.Muster List B.Certificate of Inspection C.Forecastle Card D.Clearance Papers 【单选】 While proceeding towards a distress site you hear the message PRU-DONCE over the radiotelephone. Which action should you take?A.Advise the sender of your course,speed,position,and ETA at the distress site e that frequency only for restricted working communicationsC.Shift your radio guard to the working frequency that will be indicated in the message D.Resume base course and speed because the distress is terminated 【单选】 Which statement about stowing spare hose is TRUE?A.Fold the hose so that the male coupling is about 4 feet from the female coupling,then roll it up B.Roll the hose starting at the female end C.Roll the hose starting at the male end D.Fold the hose into lengths about 6 feet long and then lash the folds together 【单选】 What is the primary purpose for Digital Selective Calling (DSC)?A.DSC is to be used for transmitting and receiving distress alerts to and from other ships or coast radio stations via radio B.This aids SAR authorities in tracking a vessel's position by satellite C.DSC provides reception of weather and navigational warnings plus search and rescue informationD.DSC provides low-cost,routine communications for the vessel operator 【单选】 The fire-protected lifeboats are found ______.A.satisfied B.being satisfied C.satisfactory D.to satisfy 【单选】 After jacking down your liftboat you have an unexpected list. You find that the only cause of this list must be a flooded leg. The list caused by a flooded leg means your vessel has a(n) ______.A.decrease in the GZ (righting arm) B.less chance of deck edge immersion C.negative GM (metacentric height) D.increase in the waterplane and the metacentric height 【单选】 In practice，it is usual for the ship to be loaded ______ to improve the vessel's movement through the water．A.a balance between two sides B.at the same draught between fore and aft C.a little deeper forwardD.a little deeper aft 单选】 What is NOT a form used by ice support services to disseminate information?A.Ice Bulletins B.Ice Outlooks C.Ice Analyses D.Ice Forecasts【单选】 In order to pay out or slack a mooring line which is under strain，you should ______.A.stopper the line B.surge the line C.sluice the line D.slip the line 【单选】 What is NOT a form used by ice support services to disseminate information?A.Ice Bulletins B.Ice Outlooks C.Ice Analyses D.Ice Forecasts【单选】 In order to pay out or slack a mooring line which is under strain，you should ______.A.stopper the line B.surge the line C.sluice the line D.slip the line 【单选】 What is NOT a form used by ice support services to disseminate information?A.Ice Bulletins B.Ice Outlooks C.Ice Analyses D.Ice Forecasts【单选】 In order to pay out or slack a mooring line which is under strain，you should ______.A.stopper the line B.surge the line C.sluice the line D.slip the line 单选】 Which radiotelephone signal indicates receipt of a distress message?A.SOS acknowledged B.Romeo，romeo，romeo C.Mayday roger D.Roger wilco 【单选】 When fighting a fire in a space containing an IMO class 1 hazardous cargo,the most effective fire fighting procedure is to ______.e high-expansion foam B.activate the fixed CO2 firefighting system e water from fire hoses or a sprinkler system D.shut down the ventilation and exclude all air to smother the fire【单选】 In the Northern Hemisphere a wind is said to veer when the wind ______.A.changes direction violently and erratically B.changes direction counterclockwise,as from south to east,etc. C.remains constant in direction and speed D.changes direction clockwise,as from north to east,etc. 单选】 When using an ARPA,what should you consider in order to evaluate the information displayed?A.The target vessel's generated course and speed are based solely on radar inputs B.Navigational constraints may require a target vessel to change course C.You cannot determine if a small target has been lost due to sea return D.The trial maneuver feature will automatically determine a course that will clear all targets 【单选】 The movement of water away from the shore or downstream is called a(n) ______.A.reversing current B.ebb current C.flood current D.slack current【单选】 How many GMDSS radio operators are needed／required on board according to the provisions of the International Convention for the Safety of Life at Sea，1974？A.2 operators B.1 operator C.4 operators D.3 operators 【单选】 The ship's tanks most effective for trimming are the ______.A.settlers B.peaks C.domesticsD.deeps【单选】 How many GMDSS radio operators are needed／required on board according to the provisions of the International Convention for the Safety of Life at Sea，1974？A.2 operators B.1 operator C.4 operators D.3 operators 【单选】 The ship's tanks most effective for trimming are the ______.A.settlers B.peaks C.domestics D.deeps 【单选】 First aid means ______.A.setting of broken bones B.emergency treatment at the scene of the injury C.medical treatment of accident D.dosage of medications单选】 The ship's tanks most effective for trimming are the ______.A.settlers B.peaks C.domestics D.deeps 单选】 First aid means ______.A.setting of broken bones B.emergency treatment at the scene of the injury C.medical treatment of accident D.dosage of medications 【单选】 An orange flag showing a black circle and square is a ______.A.signal indicating a course change B.distress signal C.signal of asking to communicate with another vessel D.signal indicating danger 单选】 All electrical appliances aboard a vessel should be grounded to ______.A.prevent them from falling when the vessel rolls B.increase their operating efficiency C.protect personnel from electrical shock D.prevent unauthorized personnel from operating them 【单选】 It would be possible that part of the cargo got ______ as great change in the weather during the voyage caused heavy ______ in the hold．A.dampness／sweater B.wetness／dew C.damp／water D.wet／condensation 【单选】 Do you need to measure oxygen levels before entering an enclosed space?A.No B.Yes, always C.Yes, but not if you ventilate properly first for 24 hours D.Not if you measure for toxic gases 【单选】 When is a stand-on vessel FIRST allowed by the Rules to take action in order to avoid collision?A.When the give-way vessel is not taking appropriate action to avoid collision B.When collision is imminent C.The stand-on vessel is never allowed to take action D.When the two vessels are less than half a mile from each other 【单选】 Which of the following is the most useful factor for predicting weather?A.The present reading of the barometer B.The difference in the barometric readings within the past 24 hours C.The rate and direction of change of barometric readings D.The previous reading of the barometer Communications over relatively short distances can be made by visual or sound signals. Visual signals can be sent byusing flags or an Aldis lamp. An Aldis lamp is an electric lamp used for flashing messages in Morse code. The traditional method of signaling from one ship to another is by using flags. There are different colored flags for each letter of the alphabet. There are also pennant-shaped flags for numbers, and a long pennant, known as an answering or code pennant. Three other flags, which are burgee-shaped, are known as substitutes. These show that the flat or pennant is being repeated. Besides standing for a letter of the alphabet, each flag, when hoisted along, has another meaning. For example, the 'W' flag also means: 'I require medical assistance'. Flags can also be hoisted in combinations of two, three or four. Siren, whistle, bell or other sound signals can be used in fog and similar circumstances when visual signals can not be seen.(1)An Aldis lamp is used for ______.A.sending flag signals B.sending sound signalsC.transmitting Morse code D.flashing flags(2)Burgee-shaped flags are used as substitutes to show ______.A.'I requiring medical assistance' B.'repeating' C.'answering' D.'code' pennant(3)Communications over relatively short distances may be made by ______.A.Morse Code B.Either visual or sound signals C.sound signals D.visual signals(4)______ are used in fog and similar circumstances when visual signals can not be seen.A.Substitutes B.The ship's siren, whistle or bell C.Visual signalsD.Pennant-shaped flagsThe official original data (S57 data) of the electronic nautical chart is usually supplied on CD-ROM or, in case of updates, via digital telephone or satellite communication system. This original data is also called electronic nautical chart (ENC).The chat database is organized in cells that cover the entire earth's surface without overlapping. The cells store all nautical chat objects as well as objects created only during the operation of the system, such as waypoints and leg lines, notes, positions of own ship and of other vessels, etc.The data in the System Electronic Nautical Chart (SENC) is generated from the original data of the ENC. The Enc has to be kept unaltered in order to be able to reconstruct the SENC data if this is unintentionally damaged or destroyed. In SENC, the chart data is stored in a proprietary file format designed by the ECDIS manufacturer for speed and reliability.The S57 data represents a specific kind of attributed vector data. The kind of data (object description with geometry and geographical position) requires an efficient kind of storage.For compactness and speed, vector data is the optimal solution in contrast to the voluminous raster data.(1)Which of the following is correct statement?A.ENC is the raw data of the Electronic Nautical Charts and it may be changed by ECDIS manufacturer for speed and reliability B.The database of an Electronic Nautical Chart System is compacter and the speed of the system is faster if those use voluminous data C.ENC is a kind of voluminous raster data while the SENC is a kind of vector data D.SENC is the raw data of the Electronic Nautical Charts and it may be changed by ECDIS manufacturer for speed and reliability(2)The passage implies ______.A.ENC is the raw data of the Electronic Nautical Chart B.The vector data is the optimal solution for ECDIS C.SENC is the raw data of the Electronic Nautical Chart D.SENC may be designed with different representation purposes by different ECDIS manufacturers(3)Which of the following is Not Correct as to the ENC and SENC?A.The ENC can be reconstructed from the SENC B.The ENC is the raw data of the Electronic Nautical Chart C.The SENC is generated form the S-57 D.The SENC can be reconstructed from the S-57(4)The passage indicates that the update of the S-57 can be done via ______.① CD-ROM②Digital telephone③Satellite communication system④Floppy DiskA.②③ B.①～④ C.②～④ D.①～③ 【单选】 You are offloading garbage to another ship. Your records must identify that ship by name and show her ______.A.official number B.Master C.next port-of-call D.home port 【单选】 The master shall be ______ the marine environment when taking collision-avoiding action．A.aware of B.in charge of C.interested in D.clear of 【单选】 What is important to remember when using AIS for collision avoidance?A.AIS is not as accurate as ARPA B.AIS is not allowed to be used for collision avoidance C.AIS may not give a complete picture of the traffic situation D.AIS is more accurate than ARPA 【单选】 While proceeding to a distress site,you hear the words “Seelonce mayday” on the radiotelephone.Which action should you take?A.Monitor the radiotelephone but do not transmit B.Relay the original distress message as no other vessel has acknowledged it C.Resume base course and speed as your assistance is no longer required D.Acknowledge receipt and advise your course,speed,and ETA 【单选】 The end of the joint with the exterior threads is called the ______.A.stem B.box C.stand D.pin【单选】 When a vessel is entering or leaving a port，a record of engine speeds is kept in the ______.A.Official Logbook B.engine rough log C.bell book D.deck rough logbook 【单选】 What is used to prevent twisting of a towing bridle?A.A fishplate B.A V-spring C.A bitt D.A bulkhead 【单选】 All marine low-speed diesels are of what design?A.Two-stroke B.Electronic ignition C.Forced exhaust D.Four-stroke 单选】 A single heavy wire made up for the topping lift is called a ______.A.working guy B.bale C.spanner wire D.bull line By paying out more anchor cable，you ______.A.decrease the holding power of your anchor B.decrease the swing of your vessel while at anchor C.increase the possibility that your vessel will drag anchor D.increase the holding power of your anchor According to the Rules，a vessel's length is her ______.A.registered length B.length overall C.length between the perpendiculars D.length along the waterline 【单选】 After using a Halon extinguisher,it should be ______.A.repainted B.recharged C.put back in service if more than 50% of the charge remains D.discarded【单选】 According to the Rules，a vessel's length is her ______.A.registered length B.length overall C.length between the perpendiculars D.length along the waterline单选】 After using a Halon extinguisher,it should be ______.A.repainted B.recharged C.put back in service if more than 50% of the charge remains D.discarded【单选】 MAINLY VARIABLE 3 to 4 VEERING NELY 5 TOMORROW MORNING．This forecast refers to ______ in the designated area．A.fog B.visibility C.winds D.seas 【单选】 Which statement concerning GPS is TRUE?A.There are 12 functioning GPS satellites at present B.It may be suspended without warning C.It cannot be used in all parts of the world D.Two position lines are used to give a 2D fix单选】 The necessity of the segregation of cargoes is determined by ______.A.various types of cargoes B.different types of shipsC.personal abilities D.experience from practice 【单选】 The error in the measurement of the altitude of a celestial body，caused by refraction，increases as the ______.A.altitude of the body decreases B.observer's height above sea level increases C.horizontal parallax decreases D.humidity of the atmosphere decreases【单选】 Clouds that form as small white flakes or scaly globular masses covering either small or large portions of the sky are ______.A.altostratus B.cirrocumulus C.dirrostratus D.cirrus【单选】 Panting frames are located in the ______.A.centerline tanks on tankers B.after double bottoms C.fore and after peaks D.forward double bottoms【单选】 If a person gets something in his or her eye and you see that it is not embedded,you can ______.A.get them to rub their eye until the object is gone B.remove it with a match or toothpick C.remove it with a piece of dry sterile cotton D.remove it with a moist,cotton-tipped applicator 【单选】 Contour elevations on this chart refer to heights in feet above mean ______.A.high waterB.low water C.lower low water D.sea level 【单选】 If a person gets something in his or her eye and you see that it is not embedded,you can ______.A.get them to rub their eye until the object is gone B.remove it with a match or toothpick C.remove it with a piece of dry sterile cotton D.remove it with a moist,cotton-tipped applicator 【单选】 Contour elevations on this chart refer to heights in feet above mean ______.A.high water B.low water C.lower low water D.sea level 【单选】 For any given pedestal crane，when the boom is lengthened，the lifting capacity is ______.A.unchanged B.decreasedC.eliminated D.increased 单选】 When reversing, the tidal stream will have a period with little or no effect. This is called the ______.A.Range B.Spring C.Rise D.Slack 【单选】 A high-velocity fog stream can be used in fire fighting situations to drive heat and smoke ahead of the fire fighters in a passageway. This technique should only be used when ______.ing a 2-1/2 inch hose B.at least two fog streams can be used C.there is an outlet for the smoke and heat D.the fire is totally contained by the ship's structure 单选】 When reversing, the tidal stream will have a period with little or no effect. This is called the ______.A.Range B.SpringC.Rise D.Slack 【单选】 A high-velocity fog stream can be used in fire fighting situations to drive heat and smoke ahead of the fire fighters in a passageway. This technique should only be used when ______.ing a 2-1/2 inch hose B.at least two fog streams can be used C.there is an outlet for the smoke and heat D.the fire is totally contained by the ship's structure【单选】 The correct method of switching off a marine radar is to turn power switch to ______ position first，then to ______ position．A.close／standby B.standby／close C.standby／off D.off／standby 【单选】 Magnetic compass deviation ______.A.is the angular difference between magnetic north and compass north B.varies with the bearing used C.is the angular difference between geographic and magnetic meridians D.is published on the compass rose on most nautical charts 单选】 When a sea anchor for a survival craft is properly rigged,it will ______.A.help to prevent broaching pletely stop the survival craft from drifting C.prevent the survival craft from rolling D.prevent the survival craft from pitching 【单选】 All entries in Logbook，______ made，must not be erased or amended．A.just B.while C.once D.whether【单选】 Protection of cargo against tainting damage can best be obtained by ______.A.not ventilating the space B.segregation of cargo by using different hatches C.ventilating the space D.proper use of paper separation and dunnage【单选】 Which aid is NOT marked on a chart with a magenta circle?A.Radiobeacon B.Aero light C.Radar transponder beacon D.Radar station单选】 MIDSHIPS refers to rudder to be held ______.A.in position to port B.in position fore and aft C.in position to anywhere D.in position to starboard【单选】 The datum from which the predicted heights of tides are reckoned in the tide tables is ______.A.the highest possible level B.the same as that used for the charts of the locality C.mean low water D.given in table three of the tide tables 单选】 The circular steel structure installed around the propeller of a towboat is the ______.A.hood B.strut C.nozzleD.shroud 【单选】 The characteristic of a lighted cardinal mark may be ______.A.occulting B.very quick flashing C.fixed D.flashing 【单选】 Jettisoning weight from topside ______.A.lowers the center of gravity B.raises the center of buoyancy C.reduces free surface effect D.returns the vessel to an even keel 【单选】 The characteristic of a lighted cardinal mark may be ______.A.occulting B.very quick flashing C.fixed D.flashing【单选】 Jettisoning weight from topside ______.A.lowers the center of gravity B.raises the center of buoyancy C.reduces free surface effect D.returns the vessel to an even keel【单选】 After having been pulled aloft in a bosun's chair on a mast,you must now make yourself fast in the chair prior to painting the mast. You should first ______.A.make the tail of the line leading from the becket bend fast to a padeye on the mast B.frap yourself to the mast to take the strain off the hauling part C.seize the hauling part and the standing part firmly in one hand to support your weight D.have the sailor on deck make the hauling part fast to a cleat on the mast 单选】 The annual change in ______ is 0．2 degree．A.Marine Insurance B.Maritime Accident C.Magnetic Variation D.Mean High Water Spring 【单选】 Buoyancy is a measure of the ship's ______.A.freeboard B.deadweight C.ability to float D.midships strength 【单选】 A design modification of an anchor chain which prevents kinking is the ______.A.Kenter link B.detachable link C.stud link D.connecting link 【单选】 A look-out should report objects sighted using ______.A.gyro bearings B.relative bearings C.true bearingsD.magnetic bearings 【单选】 Heave is motion along the ______.A.longitudinal axis B.centerline axis C.vertical axis D.transverse axis【单选】 You are using VHF channel 16 (156.8 mHz) or 2182 kHz. You need help but are not in danger. You should use the urgent signal ______.A.ASSISTANCE NEEDED B.MAYDAY C.SECURITE D.PAN-PAN 【单选】 You are on watch and the Pilot has the conn. The Master has temporarily gone below. The Pilot orders a course change, which you are certain, will put the vessel intoimminent danger. Your first action should be to ______.A.immediately sound a short ring on the general alarm B.countermand the order and immediately notify the Master C.immediately call the Master and await further orders from him D.make an appropriate entry in the deck log concerning the Pilot's order【单选】 If the compass heading and the magnetic heading are the same then ______.A.there is no deviation on that heading B.the deviation has been offset by the variation C.the compass is being influenced by nearby metals D.there is something wrong with the compass 【单选】 Why should you report accidents to the Designated Person?A.So the company can calculate your safety bonusesB.To find some one to blame C.To be able to receive an insurance claim D.To prevent it from happening again 单选】 What information is NOT found in the chart title?A.Projection B.Survey information C.Scale D.Date of first edition 【单选】 A vessel is ______ when she is not at anchor，made fast to the shore，or aground．A.dead in the water B.a power driven vessel C.making way D.underway【单选】 You are docking a vessel. Wind and current are most favorable when they are ______.A.crossing your course in the same direction B.setting you on the pier C.parallel to the pier from ahead D.crossing your course in opposite directions 单选】 ______ will be paid by shipowners after tallyman doing the tally work．A.Tally fees B.Tally money C.Cargo-tallying dues D.Cargo-handling expenses 【单选】 The shortest distance between any two points on earth defines a ______.A.rhumb line B.hyperbola C.small circle D.great circle 【单选】 Why is 6X19 class wire rope more commonly used for cargo runners than the more flexible 6× 37 wire rope?A.It is less expensiveB.It hugs the winch drum better C.It resists abrasion better D.It is longer 【单选】 When own ships position input to ECDIS wrong, what is the result?A.ECDIS will give warning B.ECDIS will automatically be switched off C.Positions, range and bearings taken on the ECDIS will be wrong D.Nothing 【单选】 The exact and complete identification of all cargo on board must be found on the ______.A.Loading List B.Hatch Report C.Mate's Receipt D.Cargo Manifest【单选】 The term SHORT BLAST means ______.A.a blast of about two seconds' duration B.a blast of about one second's duration C.a blast of about three seconds' duration D.a blast of about four seconds' duration 【单选】 From ______ mariners can know the data of tide．A.the Sea Pilot B.the Port List C.the Cargo Plan D.the Tide Table 【单选】 The following information may not be required to be communicated to a distressed craft?A.Own vessel speed and ETA to distressed craft B.Own vessels identity, call sign, name and position C.Distressed crafts true bearing and distance from ship D.Rescue award if successful 单选】 As matter of fact，the damage to the winches was due to ______.A.insufficiency of packagingB.rough handling C.inherent vice of the cargo D.improper stowage 【单选】 You are using a radar in which your own ship is shown at the center，and the heading flash always points to 0° .If bearings are measured in relation to the flash，______ bearings are produced.A.magnetic pass C.true D.relative 【单选】 The length of a wave is the length ______.A.measured from crest to trough B.of the wave's trough C.measured from crest to crest D.of the wave's crest 【单选】 Fire protection regulations apply to those towing vessels ______.ed only on inland waters ed only for pollution responseC.owned and operated by the US government ed only within a barge fleeting area 【单选】 A large navigational buoy (LNB) is painted ______.A.yellow B.with red and white vertical stripes C.with a distinct color and pattern unique to each buoy D.red。
a r Xiv:809.3434v1[mat h.SP ]19Se p28A F AMILY OF SCHR ¨ODINGER OPERATORS WHOSE SPECTRUM IS AN INTER V AL HELGE KR ¨UGER Abstract.By approximation,I show that the spectrum of the Schr¨o dinger operator with potential V (n )=f (n ρ(mod 1))for f continuous and ρ>0,ρ/∈N is an interval. 1.Introduction In this short note,I wish to describe a family of Schr¨o dinger operators on l 2(N )whose spectrum is an interval.To set the stage introduce for a bounded sequence V :N →R and u ∈l 2(N )the Schr¨o dinger operator H V defined by (H V u )(n )=u (n +1)+u (n −1)+V (n )u (n )n ≥2(1.1)(H V u )(1)=u (2)+V (1)u (1).We will denote by σ(V )the spectrum of the operator H V .It is well known that if V (n )is a sequence of independent identically distributed random variables with distribution µsatisfying supp(µ)=[a,b ],we have that the spectrum σ(V )is [a −2,b +2].For the Almost–Mathieu Operator with potential (1.2)V λ,α,θ(n )=2λcos(2π(nα+θ)),where λ>0,α/∈Q ,the set σ(V λ,α,θ)is a Cantor set [2].Bourgain conjectured in [4],that if one considers the potential (1.3)V λ,α,x,y (n )=λcos n (n −1)Date :September 19,2008.2000Mathematics Subject Classification.Primary 81Q10;Secondary 47B36.Key words and phrases.Schr¨o dinger Operators,Spectrum.H.K.was supported by NSF grant DMS–0800100.12H.KR¨UGERLemma1.2.Let r≥0be an integer and r<ρ<r+1.Given anyα,θ,K≥1,ǫ>0,a0,...,a r,there exists an integer n≥1such that(1.6)sup|k|≤K |α(n+k)ρ+θ−rj=0a j k j|≤ǫ.We will prove this lemma in the next section.Proof of Theorem1.1.By Lemma1.2,we canfind for any x∈[0,1)a sequence n l such thatsup|k|≤l|α(n l+k)ρ+θ−x|≤1/l.Hence,the sequence V l(n)=V(n−n l)converges pointwise to f(x).The claim now follows from strong convergence.It is remarkable that combined with the Last–Simon semicontinuity of the abso-lutely continuous spectrum[6],one also obtains the following resultTheorem1.3.For r≥0an integer,r<ρ<r+1,and f a continuous function on T,introduce the set B r(f)as(1.7)B r(f)= a0,...,a rσac(f(r j=0a j k j)).Then(1.8)σac(f(αnρ+θ))⊆B r(f).Hereσac(V)denotes the absolutely continuous spectrum of H V.We note that for r=0,we have that(1.9)B0(f)=[−2+max(f),2−min(f)].Under additional regularity assumptions on f,Stolz has shown in[7]that we have equality in(1.8).Furthermore note that(1.10)B r+1(f)⊆B r(f).In[6],Last and Simon have stated the following conjecture(1.11)B1(2λcos(2π.))=∅forλ>0.They stated this in poetic terms as Does Hofstadter’s Butterfly have wings?.The best positive result in this direction as far as I know,is by Bourgain [3]showing(1.12)|B1(2λcos(2π.))|→0,λ→0.It is an interesting question if Theorem1.1holds forρ≥2an integer.In the particular case of f(x)=2λcos(2πx),ρ=2,this would follow from Bourgain’s conjecture.However,there is also the following negative evidence.Consider the skew-shift T:T2→T2given byT(x,y)=(x+α,x+y),(1.13)T n(x,y)=(x+nα,n(n−1)A FAMILY OF SCHR¨ODINGER OPERATORS WHOSE SPECTRUM IS AN INTER V AL3 whereα/∈Q.Then Avila,Bochi,and Damanik have shown that for generic continuous f:T2→R,the spectrumσ(f(T n(x,y)))is a Cantor set.So,it is not clear what to expect in this case.As afinal remark,let me comment on a slight extension.If one replaces V by the following family of potentialsV(n)=f(αnρ+Kk=1αk nβk),whereβk<ρandαk are any numbers,then Theorem1.1and1.3remain still valid.2.Proof of Lemma1.2Let in the following r be an integer such that r<ρ<r+1.By Taylor expansion, we have that(2.1)α(n+k)ρ=rj=0x j(n)k j+αρ...(ρ−r)j!nρ−j.We nowfirst note the following lemmaLemma2.1.For any K≥1andǫ>0,there exists a N0(K,ǫ)such that(2.3)|α(n+k)ρ−rj=0x j(n)k j|≤ǫfor|k|≤K and n≥N0(K,ǫ).Proof.This follows from(2.1)and thatρ−r−1<0.We furthermore need the following consequence of Theorem1.8in[5].Recall for this,that a sequence x(n)is uniformly distributed in T r+1if for any0≤a j<b j≤1, j=0,...,r we have that(2.4)limn→∞14H.KR¨UGERProof of Lemma1.2.By Lemma2.1,there exists a N0such that|α(n+k)ρ−rj=0x j(n)k j|≤ǫ2rK l ,ǫ。
小学下册英语第五单元测验试卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The ______ (海豹) is known for its playful behavior.2.The ancient Greeks are known for their ________ and art.3.Did you hear the _____ (小狗) barking playfully?4.The dog is _____ with a ball. (playing)5.I have a _____ (滑梯) in my backyard. I love to slide down it! 我在后院有一个滑梯。
我喜欢从上面滑下来！6.What is the term for a young hen?A. ChickB. DucklingC. GoslingD. Pullet答案:D Pullet7. A ______ is a representation of a scientific relationship.8.How many letters are in the English alphabet?A. 24B. 25C. 26D. 27答案:C 269.My favorite toy is my ________.10.The dog is ________ and friendly.11.I like _____ (chocolate/vanilla) ice cream.12.The Kuiper Belt is located beyond the orbit of _______.13.The ancient Romans are known for their contributions to ________ (法律).14.What is the term for the study of stars and planets?A. BiologyB. ChemistryC. AstronomyD. Geology答案:C15.What is the main ingredient in pizza?A. RiceB. BreadC. CheeseD. Meat答案:B16. A spider's web is an engineering marvel for catching ________________ (猎物).17.My friend enjoys playing __________ (桌游) with family.18.The deer can easily navigate through ______ (森林).19.I can’t wait to have a __________ (形容词) __________ (玩具名) party.20.I help my parents _______ (做饭) on special occasions.21.The puppy loves to play with a ______.22.The lizard sunbathes on a _______ (热的) rock.23. A __________ (化学职业) offers diverse opportunities in various fields.24.My _____ (亲戚) are visiting this weekend.25.The _____ can have many moons orbiting it.26.When we push or pull something, we are using _______.27. A hamster stores food in its ________________ (脸颊).28.What do we call the animal known for its long trunk?A. RhinoB. GiraffeC. ElephantD. Hippo29.What is the largest organ inside the human body?A. LiverB. HeartC. BrainD. Lung答案:A30. A catalyst is not consumed in a chemical ______.31.The sun sets in the ______. (west)32.What is the capital of Mexico?A. CancunB. GuadalajaraC. Mexico CityD. Tijuana答案:C33.The ________ was a crucial period in the history of global teamwork.34.The ________ was a key treaty that marked the end of hostilities.35. A ________ (清真寺) is a place of worship in Islam.36.Which festival is known as the Festival of Lights?A. ChristmasB. HanukkahC. DiwaliD. Eid答案:C Diwali37.She wears _____ (裙子) to the party.38.The chemical symbol for nickel is _____.39.I will _____ (帮助) you with homework.40.We should _______ (关心)我们的环境.41.Which month has Halloween?A. JulyB. OctoberC. DecemberD. January答案:B42.What is the color of the sun?A. YellowB. GreenC. BlueD. Red答案:A43.The ________ (果实) is sweet and juicy.44.I can ______ (表达) my feelings clearly.45.How many legs does an octopus have?A. SixB. EightC. TenD. Twelve答案:B46.I enjoy ___ (going) on adventures.47. A _______ is a chemical process that produces gas.48.What is the name of the famous scientist known for the theory of relativity?A. Isaac NewtonB. Galileo GalileiC. Albert EinsteinD. Nikola Tesla答案:C49.What instrument do you play to make music?A. BrushB. GuitarC. PencilD. Paper50.What is the name of the famous waterfall in North America?A. Niagara FallsB. Angel FallsC. Victoria FallsD. Iguazu Falls答案:A51.________ (植物多样性改善) benefits ecosystems.52.What is the most common pet?A. CatB. DogC. FishD. Bird53.The _____ (猴子) is very intelligent and clever.54.What do you call the process of planting seeds?A. GrowingB. HarvestingC. SowingD. Watering答案:C55.The chemical equation shows the reactants on the ______ side.56.The chemical formula for oleic acid is ______.57.The playground is ________.58.My dad is a ______. He enjoys gardening.59. A frog's skin is often ______ (湿润).60.Plant cells have ______ that capture sunlight.61.Which of these is a winter sport?A. SwimmingB. SkiingC. SoccerD. Tennis62. A _____ (seedbed) is where seedlings start growing.63.What do you call a person who studies rocks?A. GeologistB. BiologistC. MeteorologistD. Archaeologist答案:A64.The teacher is _____ the lesson. (starting)65. A monkey uses its tail for ______.66. A chemical reaction can involve the absorption of _____.67.My ________ (玩具名称) is made of soft material.68.My favorite _____ is a fluffy bunny.69.She paints a _____ (画).70.Which of these is a fruit?A. CarrotB. PotatoC. TomatoD. Onion答案:C71. A lion is a brave _______ that lives in the jungle.72.The _____ (chard) is a colorful leafy green.73.What do we call the place where we watch movies?A. TheaterB. MuseumC. LibraryD. Park74.What do we call the bright display of lights seen in the northern sky?A. Aurora BorealisB. Northern LightsC. Southern LightsD. Star Shower答案:A Aurora Borealis75.I like to _______ (参加) dance classes on Fridays.76.What instrument has black and white keys?A. GuitarB. ViolinC. PianoD. Flute77.I like to play ______ (策略游戏). It helps improve my critical thinking and decision-making skills.78.What is the main gas in the air we breathe?A. OxygenB. NitrogenC. Carbon dioxideD. Helium答案:B79.The chemical formula for ammonia sulfate is _______.80.I enjoy playing ______ (视频游戏) with my friends on weekends.81.The sunflower is bright and ______.82. A chick pecks its way out of the ______ (蛋).83.What do we call the main character in a story?A. HeroB. VillainC. ProtagonistD. Antagonist答案:C Protagonist84.I have a _____ (计算器) for math class.85.What do you call the outer layer of the Earth?A. MantleB. CoreC. CrustD. Lithosphere答案:C86. A ____ is a small creature that spins webs in corners.87.What is 100 50?A. 40B. 50C. 60D. 70答案:B88.The rabbit has long ______ (耳朵). It can hear very ______ (好).89.Which continent is known for its deserts?A. AsiaB. AfricaC. EuropeD. Antarctica答案:B90.The ______ has a strong family bond.91.There are five ______ in my family. (people)92.The kitten is ___ (purring) softly.93.The hamster runs on its _________. (轮子)94.We can paint a ________ together.95.My dad is a __________ (企业家).96.I can ________ (imagine) a better world.97.I have a plant that needs lots of _____.98.The __________ (火山口) is fascinating to visit.99.She is ______ (聪明) and kind.100.I have a soft ________ (玩具熊) that I hug every night before I sleep.。
Gallager博士论文LDPCON THE THEORY OF GENERAL PARTIAL DIFFERENTIALOPERATORSBYLARS HORMANDERin LuridCONTENTSPagePREFACE 162CHAPTER I. Differential operators from an abstract point of view1.0. Introduction1631.1. Definitions and results from the abstract theory of operators 1641.2. The definition of differential operators1671.3Cauchy data and boundary problems 171CHAPTERII. Minimal differential operators with constant coefficients2.0. Introduction1742.1. Notations and formal properties of differential operators withconstant coef-ficients. 1762.2Estimates by Laplace transforms1772.3. The differential operators weaker than a given one1782.4. The algebra of energy integrals1802.5Analytical properties of energy integrals. 1822.6. ]~stimates by energy integrals 1832.7. Some special cases of Theorem 2.2. 1852.8The structure of the minimal domain 1892.9. Some theorems on complete continuity2012.10. On some sets of polynomials. 2072.11Remarks on the case of non-bounded domains. 208CHAPTER III. imal differential operators with constant cofficients3.0. Introduction2103.1. Comparison of the domains of imal differential operators 2113.2. The existence of null solutions 2163.3. Differential operators of local type. 2183.4. Construction of a fundamental solution of a complete operator of local type2223.5. Proof of Theorem 3.3. 22911- 553810. Acta Mathematica. 94. Imprim~t le 26 septembre 1955. 162 LARS HORMANDER3.6. The differentiability of the solutions of a complete operator of local type. 2303.7. Spectral theory of complete self-adjoint operators of local type2333.8. Examples of operators of local type3.9. An approximation theoremCHAPTERIV. Differential operators with variable coefficients4.0. Introduction2424.1. Preliminaries2424.2. Estimates of the minimal operator. 244REFERENCES. 247PREFACE0.1. The main interest in the theory of partial differential equations has always beenconcentrated on elliptic and normally hyperbolic equations. During the last few yearsthe theory of these equations has attained a very satisfactory form, at least where Dirich-let's and Cauchy's problems are concerned. There is also a vivid interest in other differentialequations of physical importance, particularly in the mixed elliptic-hyperbolic equationsof the second order. Very little, however, has been written concerning differential equationsof a general type. Petrowsky [25], p. 7, pp. 38-39 stated in 1946 that "it is unknown, evenfor most of the very simplest non-analytical equations, whether even one solution exists",and "there is, in addition, a sizable class of equations for which we do not know any correctlyposed boundary problems. The so-called ultra-hyperbolic equation U ~2 U U U+ ' + - + "'" +with p ~ 2 appears, for example, to be one of these." Some important papers have appearedsince then. In particular, we wish to mention the proof by Malgrange [19] that any differen-tial equation with constant coefficients has a fundamental solution. Explicit constructionsof distinguished fundamental solutions have been performed for the ultra-hyperbolicequations by de Rham [27] and others. Apart from this result, however, no efforts toexplore the properties of general differential operators seem to havebeen made. Theprincipal aim of this paper is to make an approach to such a study. The general point ofview may perhaps illuminate the theory of elliptic and hyperbolic equations also0.2. A pervading characteristic of the modern theory of differential equations is the useof the abstract theory of operators in Hilbert space. Our point of view here is also purelyoperator theoretical. To facilitate the reading of this paper we have included an exposition~2 ~2 ~2241238 GENERAL PARTIAL DIFFERENTIAL OPERATORS 163of the necessary abstract theory in the first chapter, where we introduce our main problems3Using the abstract methods we prove that the answer to our questions depends on theexistence of so-called a priori inequalities. The later, chapters are to a great extent devotedto the proof of such inequalities. In Chapters II and IV the proofs are based on the energyintegral method in a general form, i.e. on the study of the integralsof certain quadraticforms in the derivatives of a function. For the wave equation, where it has a physicalinterpretation as the conservation of energy, this method was introduced by Friedrichs andLewy [6]. Recently Leray [19] has found a generalization which applies to normally hyper-bolic equations of higher order. In Chapter II we study systematically the algebraic aspectsof the energy integral method. This chapter deals only with equations with constant coef-ficients. The extension to a rather wide class of equations with variable coefficients isdiscussed in Chapter IVIn Chapter III we chiefly study a class of differential operators with constant coefficients,which in several respects appears to be the natural class for the study of problems usuallytreated only for elliptic operators. For example, Weyl's lemma holds true in this class, i.eall weak solutions are infinitely differentiable. Our main arguments use a fundamentalsolution which is constructed there. The results do not seem to be accessible by energyintegral arguments in the general case, although many important examples can be treatedby a method due to Friedrichs [5]0.3. A detailed exposition of the results would not be possible without the use of theconcepts introduced in Chapter I. However, this chapter, combined with the introductions ofeach of the following ones, gives a summary of the contents of the whole paper0.4. It is a pleasure for me to acknowledge the invaluable help which professor B. Lvan der Waerden has given me in connection with the problems of section 3.1. I also wantto thank professor A. Seidenberg, who called my attention to one of his papers, which isvery useful in section 3.4CHAPTER IDifferential Operators from an Abstract Point of View1.0. IntroductionIn the preface we have pointed out that the present chapter has the character of anintroduction to the whole paper. Accordingly we do not sum up the contents here, but1 Chapter I, particularly section 1.3, overlaps on several points witha part of an important paperby VIw [34] on general boundary problems for elliptic equations ofthe second order. 164 LARS HORMANDERmerely present the general plan. First, in section 1.1, we recall some well-known theoremsand definitions from functional analysis. Then in section 1.2 we define differential operatorsin Hilbert space and specialize the theorems of section 1.1 to the case of differential opera-tors. A discussion of the meaning of boundary data and boundary problems is given insection 1.3. This study has many ideas in common with Vi~ik [34]. It is not logically in-dispensable for the rest of the paper but it serves as a general background1.1. Definitions and results from the abstract theory of operatorsLet B o and Bj be two complex Banach spaces, i.e. two normed and complete complexvector spaces. A linear transformation operator T from B 0 to B 1 is a function defined ina linear set Dr in B 0 with values in B 1 such that1.1.1 Tocx +fly ocTx + fl Tyfor x, y E OT and complex x, ft. It follows from 1.1.1 that the range of values ~T is a linearset in B1;The set B 0B I of all pairs x [Xo, Xl] with x, E B, i 0, 1, where we introducethe natural vector operations and the norm 11.1.2 Ixl ix01 +is also a Banach space, called the direct sum of B 0 and B r If T is a linear transformationfrom B o to B1, the set in B 0B 1 defined by1.1.3 Gr [x Txo], XOfiOTis linear and contains no element of the form [0, xl] with x 1 # 0. The set GT is called thegraph of T. A linear set G in B oB1, containing no element of the form [0, xl] with x I # 0,is the graph of one and only one linear transformation TA linear transformation T is said to be closed, if the graph Gr is closed. We shah alsosay that a linear transformation T is pre-closed, if the closure Gr of the graph Gr is a graph,i.e. does not contain any element of the form [0, x~] with xl # 0. The transformation withthe graph Gr is then called the closure of T. Thus T is pre-closed if and only if, wheneverx n-+ 0 in B 0 and Tx n-+ y in B1, we have y 0. We also note that any hnear restriction ofa linear pre-closed operator is pre-closedThe following theorem gives a useful form of the theorem on the closed graph, whichstates that a closed transformation from B 0 to B 1 must be continuous, if O r B CfBourbaki, Espaces vectoriels topologiques, Chap. I, w 3 Paris 1953.i Any equivalent norm in B oB x can be used, but this choice has the advantage of giving a Hilbertnorm, if B 0 and B 1 have Hilbert normsoo, GENERAL pARTIAL DIFFERENTIAL OPERATORS165THEORE~ 1.1. Let i 0, 1, 2 Banach spaces and i 1, 2 linear trans/ormations /tom B o to Then, i/T 1 is closed, Tz pre-closed and ~T, ~T,, there exists aconstant C such that1.1.4 I C[ TlUl2 lull, ueO~,PROOF. The graph of T x is by assumption closed. Hence the mapping1.1.5 ~ [u, Tlu]~ T2uE B~is defined in a Banach space. We shall prove that the mapping is closed. Thus supposethat [un, Tlun] converges in and that T~u n converges in B~. SinceT 1 is closed, thereis an element nEtT, such that un-+u and virtue of the assumptions, u is in ~0T, and, since T~ is pre-closed, the existing limit of T~un can only be T2u. Hencethe mapping 1.1.5 is closed and defined in the whole of a Banach space, so that it iscontinuous in virtue of the theorem on the closed graph. This proves the theoremTheorem 1.1 is the only result we need for other spaces than Hilbert spaces; it will alsobe used when some of the spaces are spaces of continuous functions with uniform normIn the rest of this section we shall only consider transformations from a Hilbert space Hto itself. In that case the graph is situated in HH, which is also a Hilbert space, the innerproduct of x [x0, xl] and y [Y0, Yl] being given byx,yxo, yo+xl,ylFor the definition of adjoints, products of operators and so on, we refer thereader to Nagy[23], p. 27 ff.L E M M A 1.1. The range ~ r o/a closed densely de/ined linear operator T is equal to H i/and only i/ T *-1 exists and is continuous, and consequently is de/ined in a closed subspacePROOF. We first establish the necessity of the condition. Thus suppose that ~T HSince T* u 0 implies that Tv, u v,T* u 0 for every v E Dr, it follows that T* u 0only if u 0. Hence T *-1 is defined. Now for any element v in H we can find an elementw such that Tw v. Hence we have, if near.,u,v u, Tw T* u,w,so that for fixed vIu,vlLet u, be a sequence of elements in such that T* u nil is bounded. Since Iun, v[ isthen bounded for every fixed v, it follows from Banach-Steinhaus' theorem cf, Nagy [23],II lOT*CHT*uH,B~:In TlU. Tlun-GT~GT,GT,A- T~ulz-_B~be T~ be B~ 166 LARS HSRMANDERp. 9 that Ilun[[ must be bounded. Hence T *-1 is continuous, and since it is obviouslyclosed, we conclude that T*-I is defined in a closed subspaceThe sufficiency of the condition is easily proved directly but follows also as a corollaryof the next lemmaL ~ M M A 1.2. The densely de/ined closed operator T has a bounded right inverse S i/andonly i/ T *-1 exists and is continuous. 1PR O O F. Since TS I implies that ~T H, it follows from the part of Lemma 1.1,which we have proved, that a bounded right inverse can only exist if T *-1 is continuousThe remaining part of Lcmma 1.1 will also follow when we have constructed the rightinverse in Lemma 1.2In virtue of a well-known theorem of von Neumann [24], the operator TT* is self-adjoint and positive. Under the conditions of the lemma we have TT* u, u T'u, T'u C2u, u,where C is a positive constant. Hence TT* ~ C2I. Let A be the positive square root ofTT*. Since A2~ C ~ I, it follows from the spectral theorem that 0 A-l C-1I. Theoperator A -1 is bounded and self-adjoint, IIA-1]IC -1. Furthermore, the operatorT*A -1 is isometric according to von Neumann's theorem. Now we define1.1.6 S T*TT* -1 T*A-1A -1Since S is the product of an isometric operator and A -1, it must be bounded, and wehave S C-I- Finally, it is obvious that TS IL ~ M, 1.3. The densely de/ined closed operator T has a completely continuous right in- verse S i/ and only i/ T *-1 exists and is completely continuousP R o O F. We first note that the operator S given by 1.1.6 is completely continuousif T *-1 and consequently A -1 is completely continuous. This proves one half of the lemmaNow suppose that there exists a completely continuous right inverse S. If UE~T., wehave for any v E Hu, v u, TS v S* T* u, v,and therefore u S* T*u. Hence, if v E ~r., we have T*-I v S* v, which proves thatT *-1 is completely continuous, since it is a restriction of a completely continuous operator1 This means that S is continuous and defined in the whole of H, and satisfies the equality TS I, where I is the identity operator]~II IIUE~TT*, GENERAL PARTIAL DIFFERENTIAL OPERATORS 1671.2. The definition of differential operatorsLet be a v-dimensional infinitely differentiable manifold. We shalldenote byC ~ ~ the set of infinitely differentiable functions defined in ~, and by C~ ~ the set ofthose functions in C ~ ~ which vanish outside a compact set in ~. When no confusionseems to be possible, we also write simply C ~ and C~A transformation to from C a t:l to itself is called a differential operator, if, in localcoordinate systems x., x~, it has the form1.2.1 tou~a. kx 1 0 1i ~ x ~' i ~ x u,where the sum contains only a finite number of terms 40, and the coefficients a s~are infinitely differentiable functions of x which do not change if we permute the indices~j.1 We shall denote the sequence ~k of indices between 1 and v by ~ and its lengthk by l a[. Furthermore, we set1D~ i ~x" D~D~D~ kFormula 1.2.1 then takes a simplified form, which will be used throughout:1.2.2 to u ~ a ~ x D~ uHere the summation shall be performed overall sequences ~We shall say that we have a differential operator with constant coefficients, if ~ is adomain in the v-dimensional real vector space R and the coefficients in 1.2.2 are constant,when the coordinates are linearLet Q be a fixed density in i.e. e x is a positive function, defined in every local coor-dinate system, such that ~ xdxl dx ~ is an invariant measure, which will be denoted dxWe require that ~ x shall be infinitely differentiable, and, in cases where has constantcoefficients, we always take Q x constantThe differential operators shall be studied in the ttilbert space L 2 of all equivalence classes of square integrable functions with respect to the measure dx, the scalar productin this space being1.2.3 u, v f u x v x dxWith respect to this scalar product we define the algebraic adjoint p of as follows1 We restrict ourselves to the infinitely differentiable case for simplicity in statements; most argu- ments and results are, however, more general and will later, in Chapter IV, be used under the weakercondition of a sufficient degree of differentiabilitytOtOg2,~,~1~k1,r~2 168 LARS HORMANDERLet vEC and let u be any function in C~. Integrating pu, v by parts, we find thatthere is a unique differential operator ~ such that1.2.4 p u, ~ u, ~ ~In fact, we obtainWhen the coefficients are constant we thus obtain ]0 by conjugating the coefficients,which motivates our notationL]~MMA 1.4. The operator p, de/ined /or those/unctions u in C ~ /or which u and Duare square integrable, is pre-closed in LP R O O F. Let u n be a sequence of functions in this domain such that u n ~ 0 and ]: un ~ vwith L2-convergence. Then we have for any /EC~v, / lim Pun,/ lim Un, p/ 0Hence v 0, which proves the lemmaRE~ARK. It follows from the trivial proof that Lemma 1.4 would also hold if, forexample, we consider ]0 as an operator from L ~ to C, the space ofcontinuous functionswith the uniform normLemma 1.4 justifies the following important definitionD F I N I T I 0 ~ 1.1. The closure Po o/the operator in L ~ with domain C~, defined by p,is called the minimal operator de/ined by p. The adjoint P o/ the minimal operator Po, definedby ~, is called the imal operator de/ined by ]:The definition of the imal operator means that u is in O~ and Pu / if and only if u and / are in L and for any v E C~ r we have/, v u, p vOperators defined in this way are often called weak extensions. In terms of the more generalconcept of distributions see Schwartz [28], we might also say that the domain consistsof those functions u in L 2 for which pu in the sense of the theory of distributions is afunction in LIf u E C and u and ]: u are square integrable, it follows from 1.2.4 that Pu exists andequals u. This is of course the idea underlying the definition. Since P is an adjoint operator,it is closed and therefore an extension of P0~D~~~,]~~yD~qa~v. ]gvq-lr162 GENERAL PARTIAL DIFFERENTIAL OPERATORS 169It is unknown to the author whether in general P is the closure ofits restriction toN C ~. For elliptic second order equations in domains with a smoothboundary thisfollows from the results of Birman [1]. If p is a homogeneous operatorwith constantcoefficients and ~ is starshaped with respect to every point in anopen set, it is also easilyproved by regularization. In section 3.9 we shall prove an affirmativeresult for a classof differential operators with constant coefficients, when ~ is anydomainWe now illustrate Definition 1.1 by an elementary example. Let ~be the finite intervala, b of the real axis, and let p be the differential operator d~/dxIt is immediately veri-fied that the domain of P consists of those n-。
航海及海运专业英语词汇(L2)lean bow 尖进流段lean bow 尖形船首lean a. 贫的偏倾依靠lean 贫的lean-to skylight 单坡的天窗leaning lettering 斜体字形leaning 倾斜的leap month 闰月leap year 闰年leaper 海面)小浪花leaper 小浪花;船首溅沫lease agreement 租赁合同lease contract with interchange system 互换使用契约lease purchase 先租后买lease rate 租费率lease termination charge 提前终止费lease 租契lease 租约;租借leased channel 租用信道leased circuit 租用线路leased circuit 租用电路leased line channel 租用线路信道leaseholder 承租人leasing company depot 租箱公司堆场leasing trade 租赁贸易least common multiple 最小公倍数least depth 最小深度least draft 最小吃水least draught 最小吃水least freeboard 最小干舷least motion 摆动最小least moulded depth 最小型深least of water 最小水深least radius of gyration 最小回转半径least reading 最小读数least significant bit 最低有效位least significant digits 最低有效位least square method 最小二乘法least 最小least-cost estimating and scheduling system 最低成本估算与调度法leather belt 皮带leather blower 皮囊吹风器leather boat-body-rest pads 牛皮靠把leather bowl 皮碗leather cloth 漆布leather cloth 人造革leather diaphragm 皮膜leather gasket 皮垫leather gloves 皮手套leather hose 皮革软管leather hose 皮软管leather measuring tape 皮卷尺leather packing ring 皮密封圈leather packing 密封环leather packing 皮衬leather packing 皮垫leather palm working gloves 皮工作手套leather strap 皮带leather tool bag 皮工具袋leather washer 皮垫圈leather 桨颈包皮皮leather 皮leather 皮革leather 皮革桨颈包皮leatheret 人造皮leatheroid 人造革leave about 乱扔leave break 超过归船期leave breaker 逾期未归船者leave draft 出港吃水leave for enforcement 裁判执行许可leave for 起程去leave over 剩下leave port 驶离港leave quarantine 留检leave with pay 工资照付的假期leave without pay 留职停薪leave 离开leave 离开;遗留;委托;离船一天以上的假期leave 离开留下leave 离启程leave-joint 离合leaving draft 离港时吃水leaving loss 排出损失leaving loss 排出损失余速损失leaving the land 离开陆地leaving 离开lebanon 黎巴嫩leblanc connection 二相变三相的接线法leblanc connection 二相变三相的接线法勒布朗克接头leblanc connection 勒布朗克接头leclanche cell 勒克朗谢电池ledeburite 莱氏体ledge bar 舱口盖承铁ledge bar 舱口型材ledge 环向凸起痕舱口横梁ledge 礁脉礁脉ledge 礁脉石脊ledge 石脊ledger 分类帐ledger 总帐ledgment 展开图lee anchor 惰锚lee anchor 下风锚lee board 防横漂板lee board 下风板lee cable 惰链lee cable 惰链惰链lee chain 惰链lee current 顺风潮流lee depression 背风坡低压lee helm 上风舵(推舵柄向下风lee helm 下风舵lee of the shore 岸的下风侧lee port 避风港lee rudder 下风舵lee shore 下风岸lee tide 顺风潮lee tide 顺风潮流lee tide 顺风潮顺风潮流lee wave 顺风波浪lee 背风面lee 下风lee-bowing 利用潮流以避免船首吹向下风lee-helm 下风舵leeboard 披水板leech line hole 起帆索眼leech lining 帆缘贴边leech rope 帆缘绳leech rope 帆缘绳;天幕边绳leech rope 帆缘绳天幕边绳leech 纵帆下风缘leech 纵帆下风缘;横帆两侧边缘;帆缘leech 纵帆下风缘横帆两侧边缘帆缘leech-line 起帆索leechrope 帆缘索leeflange 甲板门形驶帆架leeside 下风舷leeward line 背风缆leeward side 背风面leeward squall 迎面大风雨leeward tidal current 顺风潮流leeward tide 顺风潮leeward tide 顺风潮流leeward tide 顺风潮顺风潮流leeward 顺风的leeward 向下风leewardliness 偏向下风leewardly 向下风leeway angle 风流压差leeway angle 风压偏位角leeway drift 风流压差leeway indicator 风压差指示器leeway 风流压差leeway 风压差leeway 风压漂移;风压差left bank 左岸(面向河流下游left elevation 左视图left hand lay 左搓绳left hand lay 左搓绳(绳纹成left hand lay 左搓绳(绳纹成s形left hand propeller 左旋螺旋桨left hand rule 左手定律left hand rule 左手定则left hand rule 左手定则左手定律left hand 左转左侧left handed lang's lay 股丝同向的左搓left handed rope 左搓绳left handed screw 左旋螺旋桨left handed 左旋的;左手的;左撇子left handrope 反搓绳left laid rope 反搓绳left laid rope 左搓绳left rudder 左舵left screw 左侧螺旋桨left screw 左侧螺旋桨left semicircle 左半圆left side 左侧left turning test 左回转试验left 左left 左边left 左左边left 左左边的左舷left-hand helix 左向螺旋线left-hand nut 左转螺母left-hand quartz 左旋石英left-hand revolving engine unit 左转机组left-hand rotation diesel engine 左旋柴油机left-hand rotation 左向旋转left-hand rule 左手定则left-hand thread tap 左旋螺纹丝锥left-hand thread 左旋螺纹left-hand turn 左向旋转left-hand winding 左向绕组left-hand 左手的左转的left-handed screw 左旋螺杆左旋螺钉左旋螺旋桨left-handed lang's lay 股丝同向的左搓left-handed nut 左旋螺母left-handed propeller 左旋螺旋桨left-handed propeller 左旋推进器left-handed rope 左搓绳left-handed rotation 左向旋转left-handed screw 左旋螺杆left-handed thread 左旋螺纹left-handed turning 左旋left-handed vessel 左旋推进器船left-handed 左侧的left-handed 左转的leftward heeling 左倾leftwardheeling 左倾leg along 排绳leg length 焊脚长度leg member 支柱leg node 桩腿节点leg of a bridge 电桥臂leg of a circuit 相线leg of angle 角材边leg of angle 角钢的边leg of fillet weld 填角焊焊足leg of mutton 三角软帆leg of test rig 实验装置支架leg row boat 脚划船leg up 撑顶起来leg 脚leg 腿;支柱;某一索股;一段航程;某一边;脚;引线;相leg 支脚legal act 法律行为legal action 诉讼legal address 合法地址legal advice 法律意见legal adviser 法律顾问legal affairs department 法律事务处legal aid 法律顾问的协助legal ampere 法定安培legal analysis 法律分析legal application of maritime lien 船舶优先权法律适用legal basis 法律基础legal branch 法律部门legal capacity 权利能力legal charges 法定费用legal committee 法律委员会legal consciousness 法律意识legal consultancy service 法律咨询服务legal costs 诉讼费用legal discharge 合法排放legal document 法定单证legal documents 法律文件legal duty reduction or exemption 法定减免legal effect of ship arrest 船舶扣押法律效力legal effect 法律效力legal entity 法律实体legal entity 法人legal expert 法律专家legal fact 法律事实legal field 法律领域legal framework 法律框架legal holiday 法定假日legal inspection 法定检验legal instrument 法律文件legal investigation 法律调查legal issue 法律问题legal liability 法律责任legal measure 法律措施legal merchandise 合法商品legal metering unit 法定计量单位legal nature 法律性legal net weight 法定净重legal norm 法律规范legal obligation 法律义务legal office 法律办公室legal ohm 法定欧姆legal order 法律命令legal order 法律命令法律秩序legal order 法律秩序legal papers 法律文件legal person of profit making 营利法人legal person 法人legal person 法人法人资格legal person 法人资格legal personal representative 法定代理人legal personality 法人身分legal procedure 法律程序legal proceeding 法律程度legal proceedings 法律诉讼legal proceedings 诉讼legal quay 法定码头legal quay 海关码头legal quay 特许码头legal regime 法律机制legal relation of property 财产法律关系legal relations arising from maritime transport 海上运输法律关系legal relations pertaining to ships 船舶法律关系legal relationship 法律关系legal remedy for administrative action 对行政行为的司法救济legal remedy 法律方面的补救legal representative 法律代表legal review 法律评价legal sanction 法律制裁legal service 法律服务legal situation 法律形势legal statues of an international freight forwarder 国际货运代理的法律地位legal status 法律地位legal system of security of maritime claim 海事请求保全法律制度legal system of ship arrestamerica 船舶扣押制度legal system of ship arrestaustralia 船舶扣押制度legal system of ship arrestbelgium 船舶扣押制度比利时legal system of ship arrestcanada 船舶扣押制度legal system of ship arrestengland 船舶扣押制度legal system of ship arrestformer western germany 船舶扣押制度legal system of ship arrestfrance 船舶扣押制度legal system of ship arrestgreece 船舶扣押制度legal system of ship arrestnetherland 船舶扣押制度legal system of ship arrestnorway 船舶扣押制度legal system of ship arrestsingapore 船舶扣押制度legal system of ship arresttaiwan 船舶扣押制度legal system 法律体系legal tare 法定皮重legal unit 法定单位legal weight 法定重量legal 法定的legal 法律上的legalization 使合法化批准legation 公使馆legend 标题legend 图例legend 图示符号legiered steel 合金钢legislation clause 当与立法规定岐视外国船时legislation clause 立法条款(定期租船合同规定legislation 法律legislation 立法legislative assistant 法律助理legislative 立法的legitimate procedure 合法手续legitimate 合法的legitimum 合法权利leguce co. v. ward co. 兰杠斯公司诉沃德公司案leibnitzchen method 赖布津司法lemniscate 双纽线lemonade 柠檬汽水lend a hand 协助lend 贷与lend 借给lender 出借人length breadth ratio 长宽比length draft ratio 长度吃水比length aft of amidships 船中后长度length along path 线段长length at waterline 水线长度length between draft marks 吃水标志间船长length between load-waterline perpendiculars 载重水线垂线间长length between perpendiculars 垂线间长length between perpendiculars 垂线间长度length between perpendiculars 垂线间长度垂线间长两柱间长首尾垂线间长length between perpendiculars 垂线间距length between perpendiculars 首尾垂线间长length between perpends 水线间长度length breadth ratio 长宽比length deformation 长度变形length depth ratio 长深比length for displacement 排水量船长length for flooded stability 破舱稳性船长length forward of amidships 船中前长度length in freeboard rule 干舷法规船长length machined 切削长度length of a knot 节长length of a wave 波长length of action 啮合长度length of aft non-watertight hull 后部非水密船体长length of cable 链长length of common normal 公法线长度length of cut 切削长度length of damage 破损长度length of designed waterplane 设计水线长length of entrance 进流段长度length of fit 配合长度旋入长度length of flooding 浸水长度length of fracture 断裂长度length of hatchcover section 舱盖分块长度length of hull 船体长length of life 使用寿命length of line 线路长度length of load waterline 满载水线长length of painter 首缆长度length of port stay 停港时间length of pressure hull 耐压船体长length of registry 登记长length of registry 登记长度length of run per electrode 每根焊条可焊长度length of run 船尾端长length of service 服务年限length of ship 船长length of stroke 行程长度length of superstructure 上层建筑长度length of swell 涌长length of tank 液舱长度length of the berth 泊位长度length of the parallel portion of the designed waterline 设计水线平行部分长度length of thread 螺纹长度length of travel 行程length of vitality 使用期限length of water line 水线长度length of watertight hull 水密船体长length on designed waterline 设计水线长度length on load waterline 载重水线长度length on summer load water line 夏季满载水线长度length on water line 水线长length on water line 水线长度length on water line 水线上长length over all 船舶总长度length over all 全长length over fender 带碰垫船长length over keel blocks 跨龙骨墩长length overall submerged 浸体总长length overall 全长length overall 总长length rod 测杆length stop 纵向制动器length to depth ratio 船长与船深比length ×width×height长×宽×高length ×width×height长×宽×高×width×height长度×宽度×高度长度持续时间船长段节length 长度length 长度;距离;持续时间length 长度持续时间length-balance ratio 舵平衡部分长度比length-balance ratio 舵平衡部分长度与舵总长度之比length-beamratio 长宽比length-depth ratio 长深比length-diameter 长度直径比length-draft ratio 长度吃水比length-gauge 长度计length-hull height ratio 船长型深比length-width ratio 长宽比lengthbreadth ratio 长宽比lengthdraft ratio 长度吃水比lengthen 使延长lengthened 船身加长法lengthening piece 活钩链段lengthening 船身加长法lengthening 加长lengthwise movement 纵向运动lengthwise oscillations 纵向振荡lengthwise a.lengthwise 纵长地;纵向的;纵排lengthwise 纵向的纵长地lengthwise 纵向地lengthy cargo 长件货lengthy cargo 长件货物lengthy charge 长货附加运费lengthy charge 长件货加价额lengthy 超长的length×width×height 长度×宽度×高度lenox's anchor 无档锚的一种lens frame 舷窗玻璃窗框lens 玻璃罩lens 透镜lenticular 夹状云;透镜状的lenticular 夹状云透镜状的lenz's law 楞次定律leo minor 小狮星座leo 狮子星座leonard control 发电机电动机组控制leonard dynamo 变速用直流发电机leonard system steering 发电机电动机系统操舵leonard 伦纳德leonis minoris 小狮座的leonis 狮子座的lepas anatifera 一种船底附生物leporis 天兔座的lepth cable 铅包聚乙烯电缆lepth 铅聚乙烯包皮lepus 天免星座lepus 天兔星座天免星座les ran out of processingcommunication capacity to process your message 陆上地面站已离开你的作用区less than a full container load cargo 拼箱货less than a full container load 拼箱货less than a full container loadfull container load 拼箱货但整箱交货责任less than carload 零担less than carload 零担less than carload 零担拼车货less than container load cargo 拼箱货less than container load cargo 集装箱拼箱货less than container load cargo 拼箱货less than container load cargo 拼箱货集装箱拼箱货拼箱货less than normal refraction 小于正常折射less than or equal to 小于或等于less than trailer load cargo 不够一拖车装载的小宗货物less than truckload 不足一车零担less-than-trailer load cargo 不够—拖车的小宗货lessee 承租人lessen 减少lesser angle of heel 较小的横倾角lesser draught 较小吃水lesser ebb 较弱落潮流lesser extent of damage 较小破损范围lesser flood 较弱涨潮流lesser hazard 较小危险lesser high water 低高潮lesser high water 较)低高潮lesser 较小lessor 出租人leste 立士德风let fall 放下let fly! 迅速和完全放松!let go anchor! 抛锚let go anchor 抛锚let go … line! 解掉……缆let go! 放松!解掉let go 放松let her go off! 再离开些!let out! 放走!(对绳索let out 伸出let up 停止let 令let 让let 使let-go current 容许电流let-go-through current 故障时通过的电流leteorological map 气象图letgo both anchors! 抛双锚letgo port anchor! 抛左锚letgo sarboard anchor! 抛右锚lethal action 致命作用lethal concentration 50 半致死浓度lethal dose 50 半致死量lethal stand-off distance 安全避开距离lethal voltage 致命电压letter board 船名牌letter mark 文字标志letter of abandonment 委付书letter of advice 建议书通知书letter of advice 起运通知letter of affirmation of ship auction 拍卖船舶成交确认书letter of agreement 协定书letter of appointment 委任书letter of assurance 航运许可证letter of attorney 授权书letter of attorney 诉讼授权委托书letter of attorney 委任状letter of authority 授权证书letter of authority 授权证书委托证书letter of cancellation 解约书letter of credit terms 信用证条款letter of credit without recourse 不可追索的信用证letter of credit 信用证letter of credit 信用证信用证信用证件letter of delegation 代理委托收款委托书letter of deposit 抵押证书letter of guarantee 保函letter of guarantee 保证书letter of guarantee 保证书保函letter of hypothecation 抵押证书letter of hypothecation 押汇负责书letter of hypothecation 押汇负责书抵押证书letter of indemnity 担保函letter of indemnity 赔偿保证书letter of indemnity 赔偿保证书保函letter of information 通知书letter of inquiry 询价信letter of instruction 信用指示函letter of intent 意向书letter of introduction 介绍信letter of lien 留置权书letter of readiness 准备就绪书letter of recommendation 推荐信letter of subrogation 索赔代理证书letter of transmittal 送文函letter of trust 信托书letter of undertaking 担保书letter patent 专利证letter punch 打印字母letter report 通信报告letter shift 字母转换letter stamps 字母印模letter telegram 书信电报letter telegrams 书信电报letter 文字;信;证书letter 信无线电报字母letter 字letter-board 船名牌lettered rules 字母规则levant 立凡德风levante 立凡德风levanter 立凡脱风levanto 立凡多东南风leveche 立维琪风levee revetment 堤防护岸levee 堤levee 防洪堤;堤防;天然冲积堤;码头level adjust 电平调整level attitude 水平位置level constant 水准仪常数level control float 液位控制浮子level control valve 水位调节阀level control 电平控制水平控制level controller 水位控制器level controller 液面控制器level controller 液位调节器level crossing 平面交叉level error 水平差level float 水位浮标level gauge 水准仪level gauge 水准仪;液面指示器level gauge 水准仪液位表level gauge 液位表level ice 平坦海冰level ice 平坦海冰平坦海水level indicator 水平指示器电平指示器level indicator 液面指示器level line 水准线level lines 水平线level luffing crane 平行运送旋臂起重机level luffing crane 水平俯仰起重机level meter 电平计level meter 电平计水位表level meter 水位表level meter 水准表level of automation 自动化程度level of cap 封顶水平level of communication priority 通信优先等级level of enforcement 执行程度level of fund 基金水平level of lowest possible low water 最低低潮面level of noise 噪声度level of protection 保水平level of response 反应程度level of safety 安全程度level of sea 海平面level of the sea 海平面level on boundary line 限界级level plane 水准面level recorder 电平记录仪level recorder 水平记录仪level regulator 液位调节器level regulator 液位调节器电平调节器level sensing device 液位传感器level sight glass 玻璃液位表level surface 水平面level surface 水准面level switch 电位开关level switch 水平开关level switch 水平开关电位开关level translator 电平转换器level trier 水准检定器level trim 水平纵倾level tube 水准测管level 电平水平等级平舱电平水位level 电平水位level 水平level 水平线leveler 矮平机leveler 矫平机leveler 矫平机电平器leveling block 矫正平台leveling block 水平压块leveling by level gauge 液面计法找平leveling charges 平舱费leveling equipment 调平装置leveling gear 重心自动找正装置leveling instrument 水准仪leveling lines 平衡索leveling machine 整直机leveling of engine bed 机座找平leveling of foundation 基座找平leveling of plates 钢板矫平leveling pipe 调平连通管leveling point 水准点leveling rod 水准尺leveling surveying 水准测量leveling the bedplate 机座找平leveling 使平直leveling 水准测量leveling 水准测量校平levelling charge 平舱费levelness 水平度lever arm of stability 稳性力臂lever arm 杆臂lever arm 横倾力臂lever chain stopper 闸刀止链器lever controller 活砧式制链器lever engines 控制杆杠杆lever for adjusting oil 调节控制杆lever handle 手柄lever handle 手柄操纵杆lever latch block 杆形刷扣lever light 操纵杆指示灯lever of form stability 形状稳性臂lever of stability by weights 移重法稳性臂lever of stability 稳性力臂lever of statical stability 静稳性臂lever of wind pressure 风压力臂lever pin 杆销lever principle 杠杆原理lever ratio 杠杆比lever safety valve 杠杆式安全阀lever spring 杠杆式弹簧lever steerer 操舵杆lever stopper 活砧式制链器lever switch 杠杆开关lever type hand grease gun 杠杆式牛油枪lever type starter 杠杆式起动器lever 动稳性力臂lever 杆lever 杆杠杆操纵杆lever 控制杆lever-operated valve 杠杆操纵阀lever-type dial indicator 杠杆式千分表leverage 杠杆传动杠杆系统leverage 杠杆作用杠杆率levy on 扣押levy on 扣押提取levy 征收levy 征税lewis bolt 带眼螺栓lewis 起重爪lewis-bolt 带眼螺栓lex loci actus 行为地法lex loci contracts 合同缔结地法lex loci delicti 侵权行为地法lex voluntatis 当事人自主选择的法律liabilities of tugowner and tow party 拖带责任拖航责任liabilities 负债liability and compensation regime 责任和赔偿制度liability and compensation system 责任及赔偿制度liability convention 责任公约liability for breach of contract 违约责任liability for contribution 分摊责任liability for damage 损害责任liability for delay 延迟责任liability for fault 过错责任liability for fault 过失责任liability for fault 过失责任过错责任liability for loss 损失责任liability forms of container carrier 集装箱运输承运人责任形式liability in tort 侵权责任liability insurance of marine oil exploration development 海上石油开发责任保险liability insurance 对第三者负责的保险liability insurance 责任保险liability insurer 责任保险承保人liability not yet determined 责任尚未确定liability of an international freight forwarder 国际货运代理的责任liability of container shipper 集装箱运输发货人责任liability of leaser 租箱人责任liability of legal person 法人责任liability of operator 经营人责任liability of the assured 被保险人的责任liability of the carrier 承运人的责任liability of the insurer 承保人责任liability regime 责任机制liability retroact 责任追溯liability risk 责任风险liability salvage 责任救助liability scheme 责任机制liability system 责任制liability to prime 易于汽水共腾liability without fault 无过错责任liability without fault 无过失责任liability without fault 无过失责任无过错责任liability 倾向性liability 责任liability 责任有……的倾向性不利条件liability-sharing 责任分摊liable for 对……负责liable for 对…负责liable to duty 应付税liable to 易于……liable to 易于发生的liable 有责任的liaison office 联络处liaison 联络liblibera 利比里亚liberation of heat 放热liberation 释放liberation 释放游离释出逸出liberia 利比里亚liberty day 放假日期liberty man 放假上岸的船员liberty ship 低标准船自由轮liberty to touch and stay 驻泊条款liberty to tranship 随意转船liberty 上岸许可liberty 自由liberty 自由;随意;短假期liberty 自由上岸许可liberum 公海libra 磅libra 磅镑磅天平星座libra 常衡磅libra 天平星座librae 天平座的library of congress classification 图书馆图书分类法library 图书馆库藏书library 图书馆书库藏书libration 天平动licence =license执照licence 执照licenced pilot 有执照的引水员licenced =licensed得到许可的licenced 得到许可的licencee 许可证接收方licencer =licenser发许可证书licencer 发许可证书licencer 发许可证者license licensed licenser licence 许可证license trade 许可证贸易license 许可证license 执照licensed pilot 有执照的引水员licensed 得到许可的licensee 被许可的人licensee 被许可者licenser licensorlicenser 发许可证者licensor 同licenserlid off 盖失落lid 盖lid 盖罩顶帽lidar 激光雷达lido deck 游泳池甲板lie along the land 沿岸航行lie always safely afloat 始终安全漂浮lie at anchor 锚泊lie athwart the wind 横风停泊lie athwart 横风或横流停泊lie by 靠近lie in berth 停泊lie off 稍离lie off 稍离(陆地或其他船舶lie on the oars! 平桨lie on the oars 平桨lie over 倾侧lie over 倾侧;延期lie over 倾斜lie the course 在预定的航线上lie to 受阻lie up 进坞lie 躺lie 位于lien clause 留置权条款lien clause 留置权条款留置权条款lien for freight 运费留置权lien of broker 经纪人留置权lien on cargo for unpaid freight 为欠付的运费而留置货物lien on cargo 对货物的留置权lien on cargo 货物留置权lien on the accessories for the vessel 对船舶附属物留置权lien on vessel insurance 船舶留置保险lien on vessel 对船舶的留置权lien 留置权lien 留置权扣押权lienee 留置权人lienon cargo 对货物的留置权lienon vessel 对船舶的留置权lieu of hold cleaning 扫舱包干费life afloat 海上生活life arrow 救生抛缆的引头life assurance 人寿保险life at sea 海上人命life belt 救生带life bench 救生凳life boat compass 救生艇罗经life boat davit 救生艇吊柱life boat equipment 救生艇属具life boat hook 救生艇钩篙life boat hook 救生艇钩篙;救生艇挽钩life boat hook 救生艇钩篙救生艇挽钩life boat signal mirror 救生艇日光信号镜life boat station 救生站life boat station 救生站救生站life boat tackle 救生艇吊杆绞辘life boat 救生艇life boat 救生艇救生舢板life buoy flare 救生火焰life buoy flare 救生圈烟火life buoy flare 救生圈烟火信号life buoy flare 烟火信号life buoy life line lifting device 起重装置life buoy light marker 救生圈浮灯life buoy light 救生火焰life buoy light 救生烟火life buoy light 烟火信号life buoy self-ighiting light 救生圈自亮灯life buoy self-lighiting light 救生圈自亮灯life buoy signal 救生圈信号life buoy signal 烟火信号life buoy 救生圈life combination suit 救生服life craft 救生艇life cycle cost 使用周期费life cycle 寿命周期life expectancy 预计使用期限life float 救生筏life float 救生浮具life float 救生浮具救生筏life grab 救生抓索life guard 救生员life insurance policy 人寿保险单life insurance policy 人寿保险单人身保险单life insurance 人寿保险life jacket light marker 救生衣标志灯life jacket light 救生衣灯life jacket 救生衣life jacket=lifejacket 救生衣life kite 救生发报风筝life kite 救生风筝life light 救生灯life line mortar 救生索投索器life line pistol 抛绳枪life line throwing gun 救生抛绳枪life line throwing gun 救生索发射枪life line 安全索life line 救生绳life line 救生索life line-throwing appliance 救生抛绳设备life meter 寿命测定计life net 救生网life of ship 船舶使用年限life predictability 寿命可预计性life preserver 救生具life preserver 救生用具life raft autoreleasing 救生筏自动释放life raft 救生筏life raft=liferaft 救生筏life rail 安全拦杆life rail 安全拦杆安全栏杆船舷扶手life rail 船舷扶手life ring 救生圈life ring 救生圈;救生艇舷握索环life ring 救生索环life rocket 救生火箭life rocket 遇难信号火箭life rope 防风暴绳life saving apparatus 救生器具life saving apparatus 救生设备life saving appliance 救生设备life saving appliances certificate 《救生设备证书》life saving appliances certificate 救生设备证书life saving biscuit 救生饼干life saving equipment 救生设备life saving float 救生浮具life saving gun 抛绳枪life saving gun 救生)抛绳枪life saving gun 抛绳枪life saving gun 抛绳枪抛绳枪life saving load 救生容量life saving net 救生网life saving rocket 救生火箭life saving service 救生服务组life saving service 救生机构life saving ship 救生船life saving station 救生站life signal 救生圈信号life span 使用年限life suit 救生服life support system 生命支持系统life test 使用期限试验life test 使用寿命试验life test 寿命试验life thermal protective aids 救生保温用具life throwing appliance 救生索发射器life time 使用期限life time 使用寿命life time=lifetime 使用寿命life vest 救生衣life waistcoat 救生背心life 救生衣耐用度life 耐用度期限life 设计寿命life 生命life 寿命life-belt 救生带life-boat receiver 救生艇接收机life-jacket light 救生衣灯life-jacket locker 救生衣柜life-jacket rack 救生衣架life-line throwing gun 救生索发射炮life-line 救生索life-saving apparatus 救生设备life-saving appliance 救生设备life-saving arrangement 救生装置life-saving capsule 救生舱life-saving equipment 救生设备life-saving gear 救生设备life-saving gun 救生信号枪life-saving service 救生服务组织life-saving station 救生站life-saving suit 救生服life-saving system 救生系统life-saving 救生life-support system 生命保障系统lifeboat accident 救生艇事故lifeboat ax 救生艇手斧lifeboat buoyancy 救生艇浮力lifeboat capacity 救生艇搭载量lifeboat cutter 救生艇lifeboat davit 救生艇吊柱lifeboat deck 救生艇甲板lifeboat drag 救生艇浮锚lifeboat engine 救生艇发动机lifeboat fittings 救生艇舾装件lifeboat floor surface 救生艇底表面lifeboat launching gear 救生艇降落装置lifeboat marking 救生艇标记lifeboat overload test 救生艇超载试验lifeboat painter 救生艇索lifeboat propulsion 救生艇推进lifeboat station 救生艇站救生站lifeboat station 救生站lifeboat with self-contained air support system 自供空气救生艇lifeboat 救生艇lifeboatman certificate 救生艇员证书lifeboatman 救生艇水手lifeboatman 救生艇员lifebuoy flare 救生火焰lifebuoy self-activating smoke signal 救生圈自发烟雾信号lifebuoy self-igniting light 救生圈自亮灯lifebuoy specifications 救生圈规格lifebuoy 救生圈lifebuoy 救生圈@n.救生圈救生浮标lifebuoy 救生圈救生浮标lifeguard 救生员救生器lifejacket box 救生衣箱lifejacket for children 儿童救生衣lifejacket light test 救生衣试验lifejacket light 救生衣灯lifejacket locker 救生衣柜lifejacket rack 救生衣架lifejacket stowage 救生衣灯存放lifejacket 救生衣lifeline mortar 救生索发射炮lifeline mortar 救生索投索器lifeline pistol 抛绳枪lifeline pistol 投索器lifeline 救生索lifelinegun 救生索投掷器lifenet 救生网liferaft canopy 救生筏顶篷liferaft container 救生筏箱liferaft fittings 救生筏舾装件liferaft light test 救生筏灯试验liferaft 救生筏liferaft's light waterline 救生筏最轻水线liferocket 救生火箭lifesaving equipment 救生设备lifesignal 救生浮标信号lifetime 使用寿命lift bridge 升降桥lift by the stern 船尾浮扬lift by the stern 尾部起浮lift car 升降车lift check valve 提升止回阀lift coefficient 升力系数lift component 升力分量lift counter 上下往复计数器lift cradle 集装箱升降机lift crane 起重机lift curve 升力曲线lift cylinder 升降汽缸lift cylinder 升降汽缸提升油缸lift cylinder 提升油缸lift effect damping 升力阻尼lift engine 主机lift fan 垫升风机lift fan 升力风扇lift force 升力lift gate 提升式闸门lift hatch cover 升降式舱口盖lift line man 潜水员助手lift line man 信号绳员lift lock arm 提升式舱口盖扳手lift lock type hatchcover 吊锁式舱口盖lift lock 提升式闸门lift magnetic disc 起重电磁盘lift nozzle 腾起喷口lift of gas 气体升力lift of hydrofoil 水翼升力lift of propeller 推进器升力lift of pump 泵吸高lift of pump 泵扬程lift of rudder 舵升力lift of static suction 静吸高lift off 升起lift onlift off charge 吊箱费lift onlift off ship 吊上吊下船lift onlift off system 吊上吊下lift onlift off system 吊上吊下集装箱装卸系统lift onlift off system 吊上吊下装卸系统lift onlift off vessels 吊进吊出集装箱装卸船舶lift onlift off 吊式装卸lift onlift-off type 吊装式lift out piston 吊缸lift pump 抽水泵lift system 垫升系统lift tackle 升降绞辘lift test 使用寿命试验lift transporter 用拖头牵引的起重运输机lift truck satellite 叉车附属装卸机lift truck 铲车lift trunk 升降机通道lift trunk 升降机围壁lift valve 提升阀lift vessel 起重船lift 举lift 升降机lift-away hatchcover 起吊式舱盖lift-bolt 起重螺杆lift-bolt 起重螺杆提升螺杆lift-jigger 顶索复滑车lift-off cover 拼装舱盖lift-off cover 拼装舱口盖lift-off hatchcover 拼装舱盖lift-onlife-off full container ship 吊装式全集装箱船lift-onlife-off ship 吊装式货船lift-onlift-off container-ship 吊装集装箱船lift-rolling hatch cover 层叠型舱口盖lift-rolling hatchcover 背载型舱口盖lift-rolling hatchcover 层叠式舱盖liftcapacity 起重能力liftdrag 升阻比lifter 升降机起重机lifter 升降机起重机电磁铁衔铁lifting and moving equipment 升降和平移设备lifting apparatus 起重装置lifting appliance annual certificate 吊货设备年检证lifting appliance quadrenial certificate 吊杆年一次检验证lifting appliance 起吊装置lifting appliance 起重设备lifting appliances inspection and testing certificate 《起重设备检验和试验证书》lifting appliances test record 《舶起重和起货设备试验证书》lifting bar 吊货梁lifting batten 起吊撑条lifting beam 吊重梁lifting beam 起吊梁lifting beam 提板起重梁lifting bolt 起重螺旋lifting bracket 起重支架lifting bridge 升降桥lifting cable 起重索lifting camel 捞船浮筒lifting capacity of floating dock 浮船坞举力lifting capacity of vessel's derrick 船舶吊杆负荷单lifting capacity 负荷量lifting capacity 起重能力lifting capacity 起重能力;浮力lifting chain 起重链lifting controlling link 起升联动杆lifting craft 负重船lifting device 起重装置lifting device 升降装置lifting dipping 船首突然下倾现象。
a rX iv:mat h /1895v1[mat h.AP]13Aug212000]Primary 35J70;Secondary 47A05,58J05,58J32ADJOINTS OF ELLIPTIC CONE OPERATORS JUAN B.GIL AND GERARDO A.MENDOZA Abstract.We study the adjointness problem for the closed extensions of a general b -elliptic operator A ∈x −νDiffm b (M ;E ),ν>0,initially defined as an unbounded operator A :C ∞c (M ;E )⊂x µL 2b (M ;E )→x µL 2b (M ;E ),µ∈R .The case where A is a symmetric semibounded operator is of particular interest,and we give a complete description of the domain of the Friedrichs extension of such an operator. 1.Introduction Let M 0be a smooth paracompact manifold,m a smooth positive measure on M 0.Suppose A :C ∞c (M 0)→C ∞c (M 0)is a scalar linear partial differential operator with smooth coefficients.Among all possible domains D ⊂L 2(M 0,m )for A as an unbounded operator on L 2(M 0)there are two that stand out:D max (A )={u ∈L 2(M 0,m )|Au ∈L 2(M 0,m )}where Au is computed in the distributional sense,and D min (A )=completion of C ∞c (M 0)with respect to the norm u + Au ,which can be regarded as a subspace of L 2(M 0,m ).Both domains are dense in L 2(M 0,m ),since C ∞c (M 0)⊂D min (A )⊂D max (A ),and with each domain A is a closed operator.Clearly D min (A )is the smallest domain containing C ∞c (M 0)with respect to which A is closed,and D max (A )contains any domain on which the action of the operator coincides with the action of A in the distributional sense.Also clearly,A with domain D max (A )is an extension of A with domain D min (A ).If M 0is compact without boundary and A is elliptic then D max (A )=D min (A );the interesting situations occur when A is non-elliptic or M 0is noncompact.In this paper we shall analyze the latter problem,assuming that M 0is the interior of a smooth compact manifold M with boundary and that A ∈x −νDiffm b (M ;E ),ν>0,is a b -elliptic ‘cone’operator acting on sections of a smooth vector bundle E →M ;here x :M →R is a smooth defining function for ∂M ,positive in M 0.The elements of Diffm b (M ;E )are the totally characteristic differential opera-tors introduced and analyzed systematically by Melrose [8].These are linear op-erators with smooth coefficients whichnear the boundary can be written in lo-cal coordinates (x,y 1,...,y n )as P = k +|α|≤m a kα(xD x )k D αy .Such an opera-tor is b -elliptic if it is elliptic in the interior in the usual sense and in addition k +|α|=m a kα(0,y )ξk ηαis invertible for (ξ,η)=0;this is expressed more concisely by saying that the principal symbol of P ,as an object on the compressed cotan-gent bundle (see Melrose,op.cit.),is an isomorphism.It follows from the definition of b -ellipiticity that the family of differential operators on ∂M given locally byˆP 0(σ)= j +|α|≤m a α,j (0,y )σj D αy ,called the indicial operator or conormal symbol2JUAN B.GIL AND GERARDO A.MENDOZAof P,is elliptic of order m for anyσ∈C.For the general theory of these oper-ators and the associated pseudodifferential calculus the reader is referred to the paper of Melrose cited above,his book[9],as well as Schulze[14,15].An oper-ator A∈x−νDiffm b(M;E)is b-elliptic if P=xνA is b-elliptic.By definition the conormal symbol of A is that of P.For more details see Section2.The measure used to define the L2spaces will be of the form xµm for some real µ,where m is a b-density,that is,x m is a smooth positive density.The spaces L2(M;E;xµm)are defined in the usual way with the aid of a smooth but otherwise arbitrary hermitian metric on E.These spaces are related among themselves by canonical isometries with which L2(M;E;xµm)=x−µ/2L2b(M,E),where L2b(M,E) is the space defined by the measure m itself.This said,the general problem we are concerned with is the description of the ad-joints of the closed extensions of a general b-elliptic operator A∈x−νDiffm b(M;E) initially defined as an unbounded operator(1.1)A:C∞c(M;E)⊂xµL2b(M;E)→xµL2b(M;E).The case where A is a symmetric semibounded operator is of particular interest,and we give,in Theorem8.12,a complete description of the domain of the Friedrichs extension of such an operator.Differential operators in x−νDiffm b(M;E)arise in the study of manifolds with conical singularities.The study of such manifolds from the geometric point of view began with Cheeger[3],and by now there is an extensive literature on the subject. In the specific context of our work,probably the most relevant references,aside from those cited above,are the book by Lesch[7],and the papers by Br¨u ning and Seeley[1,2],and Mooers[11].See also Coriasco,Schrohe,and Seiler[13].As already mentioned,the domains D min(A)and D max(A)need not be the same. The object determining the closed extensions of A is its conormal symbolˆP0(σ): C∞(∂M)→C∞(∂M).Because of the b-ellipticity,this operator is invertible for allσ∈C except a discrete set spec b(A),the boundary spectrum of A(or P),a set which(again due to the b-ellipticity)intersects any strip a<ℑσ<b in afinite set. It was noted by Lesch[7]that D min(A)=D max(A)if and only if spec b(A)∩{−µ−ν<ℑσ<−µ}=∅.Also proved by Lesch[op.cit.,Proposition1.3.16]was the fact that D max(A)/D min(A)isfinite dimensional.This provides a simple description of the closed extensions of A:they are given by the operator A acting in the distributional sense on subspaces D⊂xµL2b(M;E)with D min(A)⊂D⊂D max(A). These results and others,also due to Lesch(op.cit.)are reproved in Section3for the sake of completeness,with a slightly different approach emphasizing the use of the pseudodifferential calculus,for totally characteristic operators[10],or for the cone algebra[14].In that section we also prove a relative index theorem for the closed extensions of A.From the point of view of closed extensions there is nothing more to understand than that they are in one to one correspondence with the subspaces of E(A)= D max(A)/D min(A).The problem offinding the domain of the adjoint is more delicate,and forces us to pass to the Mellin transforms of representatives of elements of E.Our approach entails rewriting the pairing[u,v]A=(Au,v)−(u,A⋆v),defined for u∈D max(A)and v∈D max(A⋆),where A⋆is the formal adjoint of A,in terms of the Mellin transforms of u and v,and proving certain specific nondegeneracy properties of the pairing.It is generally true(and well known)that in a general abstract setting,[·,·]A induces a nonsingular pairing of E(A)and E(A⋆).In theADJOINTS OF ELLIPTIC CONE OPERATORS3 case at hand,there is an essentially well defined notion of an element u∈D max(A) with pole‘only’atσ0∈spec b(A)∩{−µ−ν<ℑσ<−µ}.Ifσ0∈spec b(A)thenσ0−i(ν+2µ)is nonsingular(modulo the respective minimal domains).Our analysis of the pairing begins in Section5with a careful description of the Mellin transforms of elements in D max(A).The main result in that section is Proposition5.9,a result along the lines of part of the work of Gohberg and Sigal[4]. In Section6we prove in a somewhat abstract setting that the pairing alluded to above for solutions with poles at conjugate points is nonsingular(Theorem6.4).In Section7we link the pairing[·,·]A with the pairing of Section6.The main results are Theorems7.11and7.17.Thefirst of these gives a formula for the pairing which in particular shows that the pairing is null when the poles in question are not conjugate,and the second,which is based on the formula in Theorem7.11and Theorem6.4,shows that the pairing of elements associated to conjugate poles is nonsingular.The formulas are explicit enough that in simple cases it is easy to determine the domain of the adjoint of a given extension of A.We undertake the study of the domain of the Friedrichs extension of b-elliptic semibounded operators in Section8.The main result there is Theorem8.12,which is a complete description of the domain of the Friedrichs extension.Loosely speak-ing,the domain consists of the sum of those elements u∈D max(A)with poles‘only’in−µ−ν<ℑσ<−µ−ν/2and those with pole‘only’atℑσ=−µ−ν/2and‘half’of the order of the pole.Finally,in Section9we collect a number of examples that illustrate the use of Theorems7.11,7.17,and8.12.Any closed extension of a b-elliptic operator A∈x−νDiffm b(M;E)as an operator D⊂xµL2b(M;E)→xµL2b(M;E)has the space xµ+νH m b(M;E)in its domain.The space H m b(M;E)is the subspace of L2b(M;E)whose elements u are such that for any smooth vectorfields V1,...,V k,k≤m,on M tangent to the boundary,V1...V k u∈L2b(M;E).Other than this,the set of domains of the closed extensions of different b-elliptic operators in x−νDiffm b(M;E)are generally not equal.In Section4we provide simple sufficient conditions for the domains of two such operators to be the same.Not unexpectedly,these conditions are on the equality of Taylor expansion at the boundary of the operators involved,up to an order depending onν.We prove in particular that under the appropriate condition the domains of the Friedrichs extensions of two different symmetric semibounded operators are the same.This is used in Section8as an intermediate step to determine the domain of the Friedrichs extension of such an operator,and a refinement of the condition is obtained as a consequence.Operators of the kind we investigate arise naturally as geometric operators.In such applicationsµis determined by the actual situation.It is convenient for us, however,to work with the normalizationx−µ−ν/2Axµ+ν/2:C∞c(M;E)⊂x−ν/2L2b(M;E)→x−ν/2L2b(M;E). rather than(1.1).Since the mappings x s:xµL2b(M;E)→xµ+s L2b(M;E)are surjective isometries,this represents no loss.In particular,adjoints and symmetry properties of operators are preserved.Also to be noted is that these transformations represent translations on the Mellin transform side,so it is a simple exercise to4JUAN B.GIL AND GERARDO A.MENDOZArecast information presented in terms of Mellin transforms of the modified operator as information on the original operator.2.Geometric preliminariesThroughout the paper M is a compact manifold with(nonempty)boundary with afixed positive b-density m,that is,a smooth density m such that for some (hence any)defining function x,x m is a smooth positive density.We will alsofix a hermitian vector bundle E→M.Fix a collar neighborhood U Y for each boundary component Y of M,so we have a trivialfiber bundleπY:U Y→Y withfiber[0,1).We can then canonically identify the bundle of1-densities over U Y with| |[0,1)⊗| |Y(the tensor product of the pullback to[0,1)×Y of the respective density bundles).Let m be a b-density on| |[0,1)⊗| |Y.Then there is a smooth defining function x:Y×[0,1)→R vanishing at Y×0,and a smooth density m Y on Y,such thatm=dxξ⊗m Y with h smooth,positive,h(0,y)=1.Let x=gξwhere g is determined modulo constant factorby the requirement that dξxξ(dξmeans differential in the variableξ;thismakes sense since we are dealing with a product manifold).Thus g(ξ,y)should satisfy the equation∂gξg=0Since h=1whenξ=0,the solutions g are smooth acrossξ=0.Pick the one which is1whenξ=0.Wefix the choice of x for each boundary component.When working near the boundary we will always assume that the defining function was chosen above,and that the coordinates,if at all necessary,are consistent with a choice of product structure as above.By∂x we mean the vectorfield tangent to thefibers of U Y→Y such that∂x x=1.If E,F→M are(smooth)vector bundles and P∈Diffm b(M;E,F)is a b-elliptic differential operator,then E and F are isomorphic.This follows from the fact that the principal symbol of P is an isomorphismπ∗E→π∗F whereπ:b T∗M\0→M is the projection,and the fact that the compressed cotangent bundle b T∗M admits a global nonvanishing section(since it is isomorphic to T∗M and M is a manifold with boundary).Thus when analyzing b-elliptic operators in Diffm b(M;E,F)we may assume F=E(for more on this see[6]).The Hilbert space structure of the space of sections of E→M is the usual one, namely integration with respect to m of the pointwise inner product in E:(u,v)L2b (M;E)= (u,v)E m if u,v∈L2b(M;E).ADJOINTS OF ELLIPTIC CONE OPERATORS5 Fix a hermitian connection∇on E.If P∈Diffm b(M;E),then near a boundary component one can writeP=mℓ=0Pℓ◦(∇xD x)ℓwhere the Pℓare differential operators of order m−ℓ(defined on U Y)such that for any smooth functionφ(x)and section u of E over U Y,Pℓ(φ(x)u)=φ(x)Pℓ(u),inother words,of order zero in∇xDx .P is said to have coefficients independent of xnear Y if∇∂x P k(u)=P k(∇∂xu)for any smooth section u of E supported in U Y.By means of parallel transport along thefibers of U Y→Y one can show that if P∈Diffm b(M;E),then for any N there are operators P k,˜P N∈Diffm b(M;E)such thatP=Nk=0P k x k+˜P N x Nwhere P k has coefficients independent of x near Y.If P k has coefficients independent of x near Y then so does its formal adjoint P⋆k.To see this recall that since theconnection is hermitian,∂x(u,v)E=(∇∂x u,v)+(u,∇∂xv)if u and v are supportednear Y,so if they vanish on Y then(∇∂x u,v)L2b(M;E)=−(u,∇∂xv)L2b(M;E).One derives the assertion easily from this.Fixω∈C∞c(−1,1)real valued,nonnegative and such thatω=1in a neigh-borhood of0.The Mellin transform of a section of C∞c(◦M;E)is defined to be the entire functionˆu:C→C∞(Y;E)such that for any v∈C∞(Y;E|Y)(x−iσω(x)u,π∗Y v)L2b (M;E)=1σ))∗.3.Closed extensionsRecallfirst the abstract situation(cf.[12]),where A:D max⊂H→H is a densely defined closed operator in a Hilbert space H.Thus D max is complete with the graph norm · A induced by the inner product(u,v)+(Au,Av),and if D⊂D max is a subspace,then A:D⊂H→H is a closed operator if and only if D6JUAN B.GIL AND GERARDO A.MENDOZAis closed with respect to · A.Fix D min⊂D max,suppose D min is dense in H and closed with respect to · A.Let(3.1)D={D⊂D max|D min⊂D and D is closed w.r.t. · A}Thus D is in one to one correspondence with the set of closed operators A:D⊂H→H such that D min⊂D⊂D max.For our purposes the following restatement is more appropriate.Proposition3.2.The set D is in one to one correspondence with the set of closed subspaces of the quotient(3.3)E=D max/D minIn our concrete case A∈x−νDiffm b(M;E),ν>0,is a b-elliptic cone operator, considered initially as a densely defined unbounded operator(3.4)A:C∞c(M;E)⊂x−ν/2L2b(M;E)→x−ν/2L2b(M;E).We take D min(A)as the closure of(3.4)with respect to the graph norm,andD max(A)={u∈x−ν/2L2b(M;E)|Au∈x−ν/2L2b(M;E)},which is also the domain of the Hilbert space adjoint ofA⋆:D min(A⋆)⊂x−ν/2L2b(M;E)→x−ν/2L2b(M;E).Thus D max is the largest subspace of x−ν/2L2b(M;E)on which A acts in the dis-tributional sense and produces an element of x−ν/2L2b(M;E);one can define A on any subspace of D max by restriction.These definitions have nothing to do with ellipticity.The following almost tautological lemma is based on the continuity of A:xν/2H m b(M;E)→x−ν/2L2b(M;E)and the fact that C∞c(◦M;E)is dense in xν/2H m b(M;E).Lemma3.5.Suppose A∈x−ν/2L2b(M;E),let D⊂D max(A)be such that A:D⊂x−ν/2L2b(M;E)→x−ν/2L2b(M;E)is closed.If D contains xν/2H m b(M;E)then D contains D min(A).In particular,xν/2H m b(M;E)⊂D min(A).Adding the b-ellipticity of A as a hypothesis provides the following precise char-acterization of D min(A):Proposition3.6.If A∈x−ν/2L2b(M;E)is b-elliptic,then1.D min(A)=D max(A)∩ ε>0xν/2−εH m b(M;E)2.D min(A)=xν/2H m b(M;E)if and only if spec b(A)∩{ℑσ=−ν/2}=∅.The proof requires a number of ingredients,beginning with the following funda-mental result[10],[14]:Theorem3.7.Let A∈x−νDiffm b(M;E)be b-elliptic.For every real s andγ,(M;E)A:xγH s b(M;E)→xγ−νH s−mbis Fredholm if and only if spec b(A)∩{ℑσ=−γ}=∅.In this case,one canfind a bounded pseudodifferential parametrixQ:xγ−νH s−m(M;E)→xγH s b(M;E)bsuch that R=QA−1and˜R=AQ−1are smoothing cone operators.ADJOINTS OF ELLIPTIC CONE OPERATORS7Note that if A is b-elliptic we always canfind an operator Q such thatQA−1:xγH s b(M;E)→xγH∞b(M;E)andAQ−1:xγ−νH s b(M;E)→xγ−νH∞b(M;E)are bounded for every s∈R,even if the boundary spectrum intersects the line {ℑσ=−γ}.Also in this case ker A and ker A⋆arefinite dimensional spaces. Moreover,for every u∈xγH s b(M;E),u xγH sb ≤ (QA−1)u xγH sb+ QAu xγH sb,which impliesu xγH sb ≤C s,γ u xγH s b+ Au xγ−νH s−m b(3.8)for some constant C s,γ>0.As noted above,there is a bounded operator Q:xγ−νH s−mb →xγH s b such thatR=QA−1:xγH s−mb→xγH∞b is bounded.Hence for u∈xγH s bu xγH sb ≤ Ru xγH sb+ QAu xγH sb,which implies the estimate(3.8).Lemma3.9.There existsε>0such thatD max(A)֒→x−ν/2+εH m b(M;E).Proof.The inclusion follows from(3.8)and the fact that,if u∈D max(A)thenˆu has no poles on{ℑσ=ν/2}.Chooseε>0smaller than the distance between spec b(A)∩{ℑσ<ν/2}and the line{ℑσ=ν/2}.The continuity of the embedding is a consequence of the closed graph theorem since D max(A)and x−ν/2+εH m b are both continuously embedded in x−ν/2L2b.Recall that x−ν/2+εH m b(M;E)is compactly embedded in x−ν/2L2b(M;E).Hence if A with domain D is closed,then(D, · A)֒→x−ν/2L2b(M;E)compactly.(3.10)That this embedding is compact is a fundamental difference between the situation at hand and b-elliptic totally characteristic operators and is due to the presence of the factor x−νin A.Lemma3.11.Letγ∈[−ν/2,ν/2]be such that spec b(A)∩{ℑσ=−γ}=∅.Then A with domain D max(A)∩xγH m b(M;E)is a closed operator on x−ν/2L2b(M;E). Proof.Let{u n}n∈N⊂D max(A)∩xγH m b with u n→u and Au n→f in x−ν/2L2b. Further,let Q be a parametrix of A as in Theorem3.7.In particular,QA=1+R:xγH m b(M;E)→xγH m b(M;E)is Fredholm.So,Au n→f implies(1+R)u n→Qf in xγH m b,and Qf=(1+R)˜u for some ˜u∈xγH m b.Now,since dim ker(1+R)<∞,there is a closed subspace H⊂x−ν/2L2b such that x−ν/2L2b(M;E)=H⊕ker(1+R).IfπH denotes the orthogonal projection onto H,thenπH u n→πH u in x−ν/2L2b and,as above,(1+R)πH u n→(1+R)πH˜u in xγH m b.ThusπH u n→πH˜u in xγH m b֒→x−ν/2L2b which implies thatπH u=πH˜u∈xγH m b.8JUAN B.GIL AND GERARDO A.MENDOZAOn the other hand,(1−πH)u n→(1−πH)u∈ker(1+R)⊂xγH m b.Therefore, u∈xγH m b and Au=f.In other words,A with the given domain is closed.Momentarily returning to the abstract situation of a densely defined closed op-erator A:D max⊂H→H,let D⋆min be the domain of its adjoint.Further,let D⋆max be the domain of the adjoint of A:D min⊂H→H,and denote this adjoint by A⋆.Then we have D⋆min⊂D⋆max.Let D⋆and E⋆be the analogues of(3.1)and(3.3)for A⋆.Define[·,·]A:D max×D⋆max→Cby[u,v]A=(Au,v)−(u,A⋆v).(3.12)Then[u,v]A=0if either u∈D min or v∈D⋆min,and[·,·]A induces a nondegenerate pairing[·,·]♭A:E×E⋆→C.Indeed,suppose u∈D max is such that[u,v]A=0for all v∈D⋆max.Then(Au,v)= (u,A⋆v)for all v∈D⋆max,which implies that u belongs to the domain of the adjoint of A⋆:D∗max⊂H→H,that is,u∈D min.Thus the class of u in E is zero. Likewise,if v∈D⋆max and[u,v]A=0for all u∈D max then v∈D⋆min.Given D∈D,let D⊥⊂D⋆max be the orthogonal of D with respect to[·,·]A. Proposition3.13.Let D∈D.The adjoint A∗of A:D⊂H→H is precisely the operator A⋆restricted to D⊥.Consequently,A is selfadjoint if and only if D=D⊥. Proof.Let A∗:D∗⊂H→H be the adjoint of A|D.We have(Au,v)=(u,A⋆v) for all u∈D,v∈D⊥.Thus D⊥⊂D∗and A∗v=A⋆v if v∈D⊥.On the other hand,since D∗⊂D⋆max(because D⊃D min),it makes sense to compute[u,v]A for u∈D and v∈D∗.For such u,v we then have[u,v]A=0since A∗v=A⋆v for every v∈D∗.Thus D∗⊂D⊥which completes the proof that D∗=D⊥.Proof of Proposition3.6.To prove part1we will show inclusion in both directions. Ifε>0is such that spec b(A)∩{ℑσ=−ν/2+ε}=∅,thenxν/2H m b(M;E)⊂D max∩xν/2−εH m b(M;E),so from Lemmas3.5and3.11we deduce D min(A)⊂D max(A)∩xν/2−εH m b(M;E). Thus,D min(A)⊂D max(A)∩ ε>0xν/2−εH m b(M;E).To prove the reverse inclusion let u∈D max∩ ε>0xν/2−εH m b and set u n=x1/n u for n∈N.Then{u n}n∈N is a sequence in xν/2H m b and as n→∞u n→u in xν/2−εH m b,and Au n→Au in x−ν/2−εL2bfor everyε>0.In particular,xεAu n→xεAu in x−ν/2L2b.Chooseεsufficiently small such that D max(A⋆)⊂x−ν/2+εH m b(Lemma3.9).Then for v∈D max(A⋆) (Au n,v)=(xεAu n,x−εv)→(xεAu,x−εv)=(Au,v)as n→∞.On the other hand,(u n,A⋆v)→(u,A⋆v)and(Au n,v)=(u n,A⋆v)since u n∈D min(A).Hence(Au,v)=(u,A⋆v)for all v∈D max(A⋆),that is,[u,v]A=0for all v∈D max(A⋆)which implies u∈D min(A)since D min(A)=D max(A⋆)⊥.ADJOINTS OF ELLIPTIC CONE OPERATORS9 To prove part2,supposefirst that spec b(A)∩{ℑσ=−ν/2}=∅.Then Lemma3.11gives that A with domain D max(A)∩xν/2H m b(M;E)is closed.Since xν/2H m b(M;E)⊂D max(A),Lemma3.5implies D min(A)=xν/2H m b(M;E).On the other hand,if D min(A)=xν/2H m b(M;E),then A with domain xν/2H m b(M;E) is Fredholm,so by Theorem3.7spec b(A)∩{ℑσ=−ν/2}=∅We will now prove that D max/D min isfinite dimensional.This is a consequence of the following proposition,which is interesting on its own,see Lesch[7,Lemma 1.3.15,Prop.1.3.16].Proposition3.14.If A∈x−νDiffm b(M;E)is b-elliptic,every closed extensionA:D⊂x−ν/2L2b(M;E)→x−ν/2L2b(M;E)is a Fredholm operator.Moreover,dim D(A)/D min(A)isfinite,andind A|D=ind A|D+dim D(A)/D min(A).minIn particular,E(A)=D max(A)/D min(A)isfinite dimensional.Below we will give a proof different from that of Lesch.That the dimension of D max(A)/D min(A)isfinite can also be proved by observing that if A=x−νP with P∈Diffm b(M;E)and Au∈x−ν/2L2b(M;E),then P u∈xν/2L2b(M;E),so the Mellin transform of P u is holomorphic inℑσ>−ν/2,from which it follows thatˆu(σ)is meromorphic inℑσ>−ν/2,and holomorphic inℑσ>ν/2since u∈x−ν/2L2b(M;E)(see[5],also[10]).On the other hand,if u∈D min,thenˆu(σ) is holomorphic inℑσ>−ν/2.This type of argument leads toCorollary3.15.If A∈x−νDiffm b(M;E)is b-elliptic thenD min(A)=D max(A)if and only if spec b(A)∩{ℑσ∈(−ν/2,ν/2)}=∅.We will analyze E(A)more carefully in the next sections.This will entail some repetition of work done by Gohberg and Sigal[4].Our proof of Proposition3.14requires the following classical result[16]:Lemma3.16.Let X,Y and Z be Banach spaces such that X֒→Y is compact. Further let T∈L(X,Z).Then the following conditions are equivalent:1)dim ker T<∞and rg T is closed,2)there exists C>0such that for every u∈Xu X≤C( u Y+ T u Z)(a-priori estimate).Proof of ing(3.10)and Lemma3.16with X=(D, · A)and Y=Z=x−ν/2L2b(M;E),we obtain dim ker A|D<∞and rg A|D closed.On the other hand,the adjoint A∗of a cone operator A is just a closed extension of its b-elliptic formal adjoint A⋆,cf.Proposition3.13.Applying again Lemma3.16we get dim ker(A|D)∗<∞.To verify the index formula consider the inclusionι:D min→D which is clearly Fredholm(use the same argument but now with X=(D min, · A),Y= x−ν/2L2b(M;E)and Z=(D, · A)).Then indι=−dim D/D min,hence=ind(A|D◦ι)=ind A|D+indι=ind A|D−dim D/D min.ind A|Dmin10JUAN B.GIL AND GERARDO A.MENDOZAAs a consequence of Propositions3.2and3.14,for any closed extension ofA:C∞c(◦M;E)⊂x−ν/2L2b(M;E)→x−ν/2L2b(M;E)with domain D(A)there is afinite dimensional space E′⊂D max(A)such thatD(A)=D min(A)⊕E′(algebraic direct sum).From Proposition3.14we also get the following two corollariesCorollary3.17.Let A:D1→x−ν/2L2b(M;E)and A:D2→x−ν/2L2b(M;E)beclosed extensions of A such that D1⊂D2,ind A|D1<0and ind A|D2>0.Thenthere exists a domain D with D1⊂D⊂D2such that ind A|D=0.The interest of this corollary lies in the fact that the vanishing of the index is necessary for the existence of the resolvent and of selfadjoint extensions. Corollary3.18.Let A:D1→x−ν/2L2b(M;E)and A:D2→x−ν/2L2b(M;E)be closed extensions of A.Thenind A|D2−ind A|D1=dim D2/D min−dim D1/D min.In particular,if D1⊂D2thenind A|D2−ind A|D1=dim D2/D1.This corollary implies the well-known relative index theorems for operators acting on the weighted Sobolev spaces xγH m b(M;E),cf.[10],[14].4.Equality of domainsIn this section we give sufficient conditions for the domains of different operators to be equal.Of these results,only the one concerning Friedrichs extensions will be used later on.An operator A∈x−νDiffm b(M;E)is said to vanish on∂M to order k(k∈N)if for any u∈C∞(M;E),xνAu vanishes to order k on∂M.Let⌈s⌉=min{k∈N|s≤k}.Proposition4.1.Let A0,A1∈x−νDiffm b(M;E)be b-elliptic.1.If A0−A1vanishes on∂M,then D min(A0)=D min(A1).2.If A0−A1vanishes to orderℓ≤ν−1on∂M,ℓ∈N,thenD max(A0)∩xν2−ℓ−1H m b(M;E).3.If A0−A1vanishes to order⌈ν−1⌉on∂M,then D max(A0)=D max(A1).4.If A0and A1are symmetric and bounded from below and A0−A1vanishes toorder⌈ν−1⌉on∂M,then the domains of their Friedrichs extensions coincide, that is,D F(A0)=D F(A1).Proof.First of all,observe that in all cases it is enough to prove only one inclusion; the equality of the sets follows then by exchanging the roles of A0and A1.To prove part1,writeA1=A0+(A1−A0)=A0+x−νP xwith P∈Diffm b(M;E)and suppose u∈D min(A0).There is then a sequence {u n}n∈N⊂C∞c(◦M;E)such that u n→u and A0u n→A0u,in x−ν/2L2b.Con-sequently,xu n→xu in xν/2H m b and x−νP xu n→x−νP xu in x−ν/2L2b.Thus A1u n→A0u+x−νP xu which implies D min(A0)⊂D min(A1).ADJOINTS OF ELLIPTIC CONE OPERATORS11 Now,letℓ∈N,ℓ≤ν−1,and let u∈D max(A0)∩xν/2−ℓ−1H m b.This means u∈xν/2−ℓ−1H m b and A0u∈x−ν/2L2b.To prove that u∈D max(A1)∩xν/2−ℓ−1H m b we only need to show that A1u belongs to x−ν/2L2b.Let P∈Diffm b(M;E)be such that A1=A0+x−νP xℓ+1.Since u∈xν/2−ℓ−1H m b then x−νP xℓ+1u∈x−ν/2L2b. Hence A1u∈x−ν/2L2b which proves the second statement.To prove the third statement,let u∈D max(A0),ℓ=⌈ν−1⌉,and P as above. Then u∈x−ν/2H m b and x−νP xℓ+1u∈x−ν/2L2b.Thus A1u∈x−ν/2L2b and D max(A0)⊂D max(A1).To prove part4wefirst prove the rather useful and well known abstract char-acterization of the domain of the Friedrichs extension of a symmetric semibounded operator given in Lemma4.3below.Suppose A:D min⊂H→H is a densely defined closed operator which is symmetric and bounded from below.Let A⋆: D max⊂H→H be its adjoint and let A F:D F⊂H→H be the Friedrichs extension of A.Define(u,v)A⋆=c(u,v)+(A⋆u,v)for u,v∈D max,(4.2)where c=1−c0and c0≤0is a lower bound of A.Lemma4.3.u∈D max belongs to D F if and only if there exists a sequence{u n}n∈N in D min such that(u−u n,u−u n)A⋆→0as n→∞.Proof.Because of the fact that D min is dense in D F with respect to the norm · A⋆induced by(4.2),every u∈D F can be approximated as claimed.Let now u∈D max and let{u n}n∈N⊂D min be such that(u−u n,u−u n)A⋆→0, so u n→u in H.Let K⊂H be the domain of the positive square root R of A F+cI. Recall that D F=D max∩K.Hence u∈D max belongs to D F if u∈K,and the identityu n−uℓ A⋆= R(u n−uℓ)implies that{Ru n}n∈N also converges in H.Thus u∈K since R is closed.We now prove part4of Proposition4.1.Suppose that A0and A1satisfy the hypotheses there.Then by parts1and3,D min=D min(A0)=D min(A1)and D max=D max(A0)=D max(A1),and from the fact that A0+A1−2A0vanishes to order⌈ν−1⌉we also get that D min(A0+A1)=D min and D max(A0+A1)=D max. Since A0and A1are symmetric and bounded from below,so is A0+A1.We will show that these three operators share the same Friedrichs domain by showing thatD F(A0+A1)⊂D F(A0)∩D F(A1).(4.4)Suppose this has been shown.Since[u,v]A0=[u,v]A1=112JUAN B.GIL AND GERARDO A.MENDOZABut then also(1−c i )(u −u n ,u −u n )x −ν/2L 2b +(A i (u −u n ),u −u n )x −ν/2L 2bas n →∞,i =0,1,so u ∈D F (A 0)∩D F (A 1).5.Spaces of Meromorphic SolutionsIf K is a finite dimensional complex vector space,we let M σ0(K )be the space of germs of K -valued meromorphic functions with pole at σ0and Hol σ0(K )be the subspace of holomorphic germs.These are naturally modules over the ring Hol σ0(C ).Let R ⊥be another finite dimensional complex vector space.If P (σ):K →R ⊥is a linear map depending holomorphically on σin a neighborhood of σ0,then P defines a map M σ0(K )→M σ0(R ⊥),which we also denote by P .Lemma 5.1.Suppose that P (σ)is defined near σ=σ0,is invertible for σ=σ0but P (σ0)=0.Then there are ψ1,...,ψd ∈M σ0(K )be such that βj (σ)=P (σ)(ψj (σ))is holomorphic and β1(σ0),...,βd (σ0)is a basis of R ⊥.For any such ψj ,ifu ∈M σ0(K )and P u ∈Hol σ0(R ⊥),then there are f j ∈Hol σ0(C )such that u = dj =1f j ψj .Proof.Let {b j }d j =1be a basis of R ⊥and define ψj =P −1(b j ).Then the ψj are meromorphic with pole at σ0,P ψj =βj =b j is holomorphic,and the βj (σ0)form a basis of R ⊥.Let now ψ1,...,ψd ∈M σ0(K )be such that βj (σ)=P (σ)(ψj (σ))is holomorphic and β1(σ0),...,βd (σ0)is a basis of R ⊥.If f ∈Hol σ0(R ⊥)then f = j f j βj for some f j ∈Hol σ0(C ),because the βj (σ0)form a basis,and each βj (σ0)can be written as a linear combination (over Hol σ0(C ))of β1,...,βd .If P u =f then P (u − dj =0f j ψj )=0,so u = dj =0f j ψj for σ=σ0,which is the equality ofmeromorphic functions.The lemma asserts that P −1(Hol σ0(R ⊥))is finitelygenerated as a submodule of M σ0(K )over Hol σ0(C ).We will be interested in ˆE σ0=P −1(Hol σ0(R ⊥))/Hol σ0(K )as a vector space overC .The following fundamental lemma paves the way todescribing a basis of ˆE σ0.Lemma 5.2.Suppose that P (σ)is defined near σ=σ0,is invertible for σ=σ0but P (σ0)=0.There are ψ1,...,ψd ∈M σ0(K )such that each βj (σ)=P (σ)(ψj (σ))is holomorphic,β1(σ0),...,βd (σ0)form a basis of R ⊥,and ifψj =µj −1ℓ=01。
