Adjoints of elliptic cone operators

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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10 marks)Leave blank 6*P48241A0628*DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 2. 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 hours.(4)________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Leaveblank7*P48241A0728*Turn 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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 D.Sedighi,Phys.Rev.A60(1999)3410.[10]R.Matloob,Phys.Rev.A60(1999)3421.[11]R.Matloob and H.Falinejad,Phys.Rev.A64(2001)042102.[12]M.S.Tomas,Phys.Rev.A66(2002)052103.[13]G.L.Klimchitskaya,U.Mohideen,and V.M.Mostepa-nenko,Phys.Rev.A61(2000)062107.[14] D.Kupiszewska and J.Mostowski,Phys.Rev.A41(1990)4636.[15] D.Kupiszewska,Phys.Rev.A46(1992)2286.[16]R.Matloob,Phys.Rev.A60(1999)50.[17] C.Raabe,L.Knoll,and D.G.Welsch,Phys.Rev.A68(2003)033810.[18]R.E.Sirvent, C.Villarreal,and G.H.Cocoletzi,Phys.Rev.A64(2001)02108.[19]onni,R.J.Cook,and M.E.Goggin,Phys.Rev.A38(1988)1621.[20]onnim,The Quantum Vacuum,Academic Press,San Diego(1994)p.97.[21]M.Khanbekyan,L.Knoll,and D.G.Welsh,Phys.Rev.A67(2003)063812.[22]T.Gruner and D.G.Welsh,Phys.Rev.A54(1996)1661.[23]L.H.Ford,Phys.Rev.A48(1993)2962.[24] F.Zhou and L.Spruch,Phys.Rev.A52(1995)297.[25] B.Geyer,G.L.Klimchitskaya,and V.M.Mostepanenko,Phys.Rev.A67(2003)062102.[26]V.B.Bezerra,G.L.Klimchitskaya,and V.M.Mostepa-nenko,Phys.Rev.A66(2002)062112.。

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.。