Exact solutions for classical few-body systems from the multichannel quantum inverse proble

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几类分数阶微分方程的近似解析解

摘要摘要在近几十年里，分数阶导数越来越引起数学家与物理学家的关注。

分数阶导数的定义有二十种之多，最常被人使用的有：Riemann-Liouville定义，Caputo定义，Jumare’s定义和Conformable定义等。

随着分数阶导数的发展，很多物理工程上的数学模型都可以最终转换成为分数阶微分方程的定解问题，例如：控制论和智能机器人、系统处理和信号识别、热学和光学系统、材料科学及力学和材料系统等。

但是，我们要想找到分数阶微分方程的精确解是相当困难的事情，从而人们转向求分数阶微分方程的近似解析解。

因此，一些逼近方法被应用于求解分数阶微分方程。

目前，在求解分数阶微分方程中比较有效的逼近方法有：同伦摄动法（HPM），同伦分析法（HAM），Adomian分解法(ADM)，变分迭代法（VIM），有限元方法，有限差分方法，线性多步算法和小波分析方法等。

对于上述算法都有其自身的优点与局限性。

在本文中，我们结合了分数阶Sumudu变换和分数阶Elzaki变换，建立了几种新的分数阶微分方程的逼近算法，这些新的算法被成功地应用于求不同类型的分数阶微分方程的近似解析解，通过将新的算法所得逼近解与已有的结果比较，得出我们建立的新的逼近算法具有计算简单、有效、精确度更高等优点。

在本文中我们也成功建立了求解局部分数阶微分方程逼近解的新算法。

本文所建立的四种求分数阶微分方程近似解析解的算法如下：1.分数阶同伦分析变换算法（FHATM）。

分数阶同伦分析变换算法（FHATM）的优点是所求分数阶微分方程的逼近解被辅助参数h所控制，合适的选取h的值将大大加速逼近解的收敛速度，在分数阶同伦分析变换算法中我们加入了分数阶Elzaki变换，使得求解过程简单快捷，通过和传统的经典算法比较可以得出：一些经典的算法可归结为分数阶同伦分析变换算法（FHATM）。

我们使用分数阶同伦分析变换算法（FHATM）成功求解非线性的时间分数阶Fornberg-Whitham方程，二维时间分数阶扩散方程，二维时间分数阶波方程和三维时间分数阶扩散方程。

四世纪城市翻译

The Roman city: from the third into the fourth centuryA number of texts from the late third and early fourth centuries provide us with insights into topography of cities, especially Hermopolis Magna and Oxyrhynchus. This is, therefore, a convenient point at which to obtain something of a snapshot( with an exposure time of fifty years) of the developed cities of the Roman period.Hermopolis magnaRoeder’s (1959) Hermopolis ( excavated between 1929 and 1939) was a vast and ruinous site and even the huge resources that had at his disposal were only sufficient to clear small areas in the centre. He uncovered a series of substantial brick buildings which he associated with the water supply system. Roeder excavated numerous ‘public’buildings and identified these with structures known from the papyrological record, though the accuracy of these identifications is rather difficult to assess and many of the remains were inevitably inidentified.Wace, Megaw and Skeat concentrated on a building to the south of the Great Tetrasty in a prime site in the city. Its location was known as Kom el-Kenissa ( the Kom of the church) and, unsurprisingly, they identified the remains of a fifth-century basilical church, the cathedral of the city. The building that pre-dated thechurch was an enclosure of possibly religious function dedicated by the settlers and cavalry of Ptolemy iii. Surprisingly, it remained largely unaltered perhaps until the church was built. Similar continuities were found to the north of the Great Tetrastylon with the building known as the Bastion, excavated by the British Museum team. This plain hall of uncertain function is dated to the Ptolemaic period.The most significant element of continuity, however, was the temple of Hermes-Thoth. The British Museum team excavated the courtyard of the temple, partly to date the end of occupancy. There, they discovered a wall which closed off access to the temple and a rubbish dump on the temple side of the wall. The latest sherds in this dump were Islamic, but most were of fifth-or sixth-century date. The earliest dated form was African red Slip Form 67 dating from 360-470. although the bottom of the dump was not reached because of high ground-water levels, Bailey saw a connection between the formation of the dump and a closure of the temples by Theodosius in 391. the other closely datable material is lamps. Bailey observed that there were no frog lamps ( a very common type from the second century onwards) at the fourth-century monastic site of Kellis in the Nile Delta, which suggested to him that frog lamps ceased to be used by 390 or even 360. there was asingle frog lamp, of a crude type, in the temple forecourt excavations which Bailey dismissed as a residual. The other lamps mainly date from the early fifth century. Since the bottom of the dump was not excavated and pottery forms and the frog lamp which could date before 390 were present, a slightly lower date for the dump seems possible, though it appears very unlikely that the dump could substantially pre-date 390, which is thus an effective terminus ante quem for the closure of the temple. This would seem to be contemporary with the build-up of rubbish on the dromos of Hermes outside the temple where, although some stray fragments are much earlier (third-fourth century), most fragments of African red slip and Egyptian red slip are form 67 (AD 360-450) or later. It is difficult to push the effective end of the temple any earlier.Hermes-Thoth was an important god in the Roman and late Roman period and was associated with a body of mystical and magical texts that became popular in the Greek and Roman world. Such interest should have meant that the cult was well placed to withstand the threats to paganism in the fourth century. The fourth-century archive of Theophanes emanates from a circle closely involved with the temple and tells us something of cult activity, though the archive cannot be closely dated. We can, I think, assume that the temple continued to operate into the final third ofthe fourth century, though the cult had a long afterlife, as the continued circulation of Hermetic texts attests.South of the temple, by the Great Tetrastylon, the brithsh Museum team excavated an Antonine temple. The temple appears to have been robbed out in the mid-or late fourth century and demolished in the early fifth century when a new pavement was laid over the podium of the temple and wells cut in front of the podium. One of the wells contained a coin of AD 390 and some fifth-century pottery. It looks as if the temple may not have functioned beyond the mid-fourth century.The papyri attest several elements of the topography of the city. The most famous text is the repair papyrus, attesting a sequence of buildings across the centre of Hermopolis. The text costs repairs to the stoai along Antinoe street and sometimes deviates to consider other areas in need of refurbishment. I cannot produce a convincing plan of the site. It lists the Gate of the Moon, the Tetrastylon of Athena, at least two private houses, the Antinoeion, the Hadrianeion, the Sarapeion, the Nilaion, the komasterion, the Nymphaion West and East, the Tuchaion, the Gate of the Sun, the first tetrastylon, the north and south stoai, the Apse, the Aphrodiseion, the Great Tetrastylon, the stoa of Athena and the Gate of the Moon, the city had several other major roads, includingDomitian Street just to the south of Antinoe Street. P. Vind. Sal 11 of AD 142 attests a number of avenues within the city, but the text is lacunose. There was also a Kaisareion by 192, a Sebasteion, a gymnasium, Baths of Hadrian, a city treasury and a mysterious covered area known as the kamara. A women’s baths is attested for the fifth century, though it is plausibly a much earlier construction. A temple of Thotoperios had some connection with the oil-workers’ guild. There was also a temple of Boubastis. In the (late？) fourth century, Bishop Plousianos arbitrated disputes at the gates of the Catholic church. The basic topographical framework of the city was enhanced by colonnades along the main paved road. Any visitor entering the city would have found his or her line sight directed along Antinoe Street towards the three tetrastyla whose columns must have risen considerably above the surrounding buildings, acting as ‘spatial punctuation’, marking the main intersections of the city. Proceeding further into the city, the visitor encountered various Classical-style public buildings and avenues, presumably also decked with stoai running north-south. This elongated civic centre culminated in the Great Tetrastylon, with the mysterious Ptolemaic structure to the south, the Roman komasterion to the north, the Bastion and the Antonine temple. Further to the north lay the temple area dominated by the temple ofHermes Trismegistos. The architecture appears to have been mainly Classical, though there were certainly Egyptian elements.Considerable effort was expended in laying out a city that had elements of Classical urban morphology, driving long straight avenues through a possibly more disordered traditional Egyptian Hermopolis give a rather different impression. The letter-carrier was to enter the city through the Gate of the Moon and walk towards the granaries. He was then to take the first street on the left behind the baths and head to the west, go down and up some steps ( presumably crossing a low-level road), turn right and pass the temple enclosure, before coming to a seven-story house with a statue of Tyche (possibly) on the gatehouse, opposite a basket-weaver’s, and there he is to ask. The instructions do not make obvious use of a grid-plan and suggest that even after such a tortuous journey, he had to rely on local knowledge. There is a contrast between the grandiose ‘front’ of the city with its Classical order and the rather freer ’back’ with more alleys and passageways than avenuees.OxyrhynchusThe best topographical information for Oxyrhynchus comes from the early fourth century. There are two major documents and a number of minor texts combine to produce a deeply confusingpicture of the city. P. oxy I 43 verso is undated, but the recto of the papyrus, a related text, is dated to 296, suggesting a date of c. 300 for the verso. The verso is a list of guards and their stations commencing from the north and moving anti-clockwise around the city, as detailed in table 5.3The last three entries have no guards written in, which shows that the text is incomplete. But it seems unlikely that the topographical register lacks many entries.The second text is a report to the logistes concerning areas of the city in need of restoration in AD 316, as in table 5.4 Reconstructing the city from these attestations is a matter of guesswork and very quickly any attempt makes one’s head spin. Nevertheless, various assumptions listed in n. 104 are used to create figure 5.11, a schematic plan, the inaccuracies and wishfulness of which are obvious(103), after extensive discussion, produced a schematic plan of Oxyrhynchus which did not take account of t he then unpublished P.Oxy. Lxiv 4441. I follow Kruger’s plan in most aspects but would stress the considerable doubt the exact location of buildings within the city.In table 5.3 25 entries cover the west of the city and one would expect that about 50 entries would cover the city. The Sarapeion is The Thoereion is reached at 38, about 75 per cent of the way roundthe city. It would be very attractive therefore to see the Thoereion and the Sarapeion at approximately the same latitude weithin the city. Yet, the gymnasium was In the south, with the Thoereion by and probably someway south of the Sarapeion. Figure 5.11 leaves the north-east short of public buildings and clusters buildings near the Sarapein and between the theatre, the gymnasium and the Thoereion in the central and southern sectors. The traditional temples of Sarapis and Thoeris still dominate the topography of the city, a continued importance reflected in the locating of multiple guards in the two temples. They were rivalled, though not equaled, by the gymnasium, the Sarapeion, with its cluster of associated religious buildings and market was a traditional focal point of the city, probably paralleled by the Thoereion. The new buildings of the Roman period appear to have been located with little account of this traditional bifocal topography, being dispersed, though the major development was towards the south of the city, around the gymnasium and the theatre. As in Hermopolis, it seems that wide avenues were cut through the topography of the city and one presumes that their colonnades drew visitors along the main streets and provided some unity to the urban plan.The archaeological record at Oxyrhynchus is extremely poor but provides some illustrative material. The first excavations werepapyrus hunts and Grenfell and Hunt’s archaeological publications were extremely brief. Grenfell’s 1897 account mentioned a large and completely ruined ‘temple’ discernible only from the mound of stone chippings. He portrays a site with a number of very high mounds of rubbish on which the excavation concentrated so as to rescue as many papyris as possible. The architectural remains were mostly destroyed. Subsequent reports (Grenfell and Hunt 1904; 1905; 1906b; 1907) trace the progress of the excavation across the site. Darbishire drew a sketch of the excavations, published in p.Oxy. L, but this, in fact, adds little of value. The photographs and notebooks of the excavations mention Byzantine coffins, a church uncovered in a clandersting dig conducted by locals, and various granite columns.Subsequent visitors found the architectural remains little more inspiring. Petrie (1925:12-14) paused only briefly at the site to clear part of the theatre(104). He uncovered columns lining the street leading from the thratre towards the centre of the city and other fragments of colonnaded streets, one of which culminated in a pillar dedicated to the emperor phokas (AD 602-10) and, further north, a fifth-century statue base. He also found a fragment of a second-century marble statue and some relief sculpture. He then abandoned the city itself and excavated in the necropolis areawhere he discovered a number of church-like tombs.(105) Breccia led two visits to the site between 1928 and 1932 (Breccia 1932;1934). The mounds still stood several feet high in places and broken columns rose from the debris. Breccia’s team cleared a number of buildings, mostly private, and the photographs show houses(?) with paved courts and columns and bastions for supporting upper storeys. They also show barrel-vaulting (inact). Clearly, the extent of the remains overwhelmed the excavators and they were able to make little headway towards uncovering on obviously extremely confusing and late site. Breccia identified little public architecture but did rescue significant amounts of sculptural material including large numbers of funerary reliefs.(106) These are often of full-scale human figures. The style of these owes little to Egyptian traditions, though some of the figures carry items associated with traditional Egypaian religion(Kessler 1983). Clothing corresponds to Graeco-Roman norms. Such reliefs continued to appear on the antiquities market after 1945 and scholars tend to see parallels between them and Palmyrene sculpture of the second, third and fourth centuries AD (Schneider 1975; 1982; Kitzinger1938; Kessler1983; Parlasca 1978; Breccia 1934-7), which suggests that the elite displayed themselves using a sculptural language current throughout the eastern Mediterraneanbut into which they incorporated distinctive Egyptian elements.A large number of literary papyri have been found at Oxyrhynchus. Of these, over 75 per cent were from the second and third century, as shown in figure5.12 (Kruger 1990: 142-9). A more detailed breakdown of a sample of papyri from P.Oxy. shows that although Homer was by far the most popular single author, there was a tremendous diversity of second-century literary texts. The later literary material is less eclectic and Christian texts gradually achieve dominance.(107) Greek literature appears to have been widely disseminated in Roman Oxyrhynchus, which was, after all, a rather small and unimportant city, which suggests that a significant proportion of the population in pre-modern terms, invested heavily in acquiring a Classical Greek education(108). Greek culture was also disseminated through performance. Homericists were paid to perform in the theatre during festivals, but the number of dramatic texts preserved suggests that the theatre at Oxyrhynchus may have staged Greek tragedies, comedies and satyr plays.The literary papyri, the iconographic material, the archaeology and what can be gleaned from the topography of the city suggest a heavily Hellenized city which preserved some of the traditional aspects of Egyptian religion.Ptolemais EuergetisNo other Egyptian cities are so well attested as Hermopolis and Oxyrhynchus. The topography of Ptolemais Euergetis, for instance, is mysterious. Archaeological exploration of the site has been somewhat desultory. Although Schweinfurth’s map (published in Wicken 1997) points to the survival of a large enclosure to the north of the site, various mounds which one presumes are the remains of ancient structures, and a substantial district of housing immediately south of the large enclose, it was alrerady obvious in 1887 that our understanding of the site would rely very heavily on the papyri (Wilcken 1887)109. the best guide to the topography of the Roman city is a third-century papyrus which lists those paying craft taxes in various streets in the city. Many of the street names are difficult to understand and, unlike those in other tax lists, do not appear in topographical order, as can be seen in table 5.5. there is some relationship。

北京地区成人本科学位英语重点词汇

词汇表(Vocabulary)Aabandon 放弃，抛弃ability 能力，智能，才能absence 缺席，不在场absolutely 绝对地，极其，完全地academic 学院的，学术的accept 接受，认可accident 事故，意外的事acplish 完成account 账，账目，说明，解释accurate 准确的，精确的accuse 谴责，指控，告发achieve 完成，达到，获得acquaintance 熟人，相识acquire 取得，获得action 行动，动作active 活动的，活跃的，积极的actually 实际上addition 加法，增加adjust 调节，调整admit 允许进入，接纳，承认adopt 收养，采取，通过adult 成人advance 推进，促进，前进advantage 优点，有利条件advertisement 广告advice 忠告，意见affair 事，事情，事件affect 影响afford 担负得起，买得起afraid 担心的，害怕的agent 代理人，代表agreement 同意，一致，协定agriculture 农业aim 志在，旨在，目标alive 活着的，活跃的allow 允许，承认aloud 出声地，大声地alter 改变，变更although 虽然，即使altogether 完全，总之amount 数量，数额，合计analysis 分析，解析analyze/analyse 分析，分解ancient 古代的，古老的anger 愤怒，气愤angle 角，角度，观点announce 宣布，通告anxious 担心的，焦虑的，渴望的apart 分离，隔开，相距apologize/-ise 道歉，认错apparent 明显的appear 出现，出场，仿佛apply 申请，运用，应用appoint 任命，委派，约定approach 接近，途径，方法argue 辩论，争论，主X argument 辩论，论点，论据arise 出现，发生，起源于arouse 引起，激起，唤起，唤醒arrange 整理，布置，安排arrest 逮捕，扣留article 文章，东西，冠词artificial 人工的artist 艺术家，美术家ashamed 惭愧的，害臊的assemble 集合，集会，装配assignment 任务，作业，分配assist 帮助，协助assume 假装，假定，设想，承担，采取assure 保证，使确信astonish 使吃惊，使惊讶atmosphere 空气，大气，气氛attach 贴上，系上，使依附attack 攻击，进攻，抨击attain 获得，达到attempt 试图，努力attend 出席，照顾，注意，留意，专心于attention 注意，注意力，立正attitude 态度，看法，姿势attract 吸引，招引，引诱audience 听众，观众，读者automatic 自动的available 可利用的，可得到的average 平均，平均数，通常的avoid 避免，逃避awake 醒着的，唤醒，醒来award 奖，奖品，授予aware 知道的，意识到的Bbackground 背景，经历badly 坏，差，严重的，非常baggage 行李bake 烤，烘，焙balance 称，平衡，均衡，差额bare 赤裸的，光秃的，空的，仅有的，勉强的bargain 讨价还价，便宜货，契约base 基础，基地，根据地basic 基本的，基础的bathe 洗澡，游泳，浸，冲洗battle 战斗，战役，斗争beach 海滨，海滩bear 忍受，容忍，负担，结果实，生孩子beat 打败，战胜，节拍，跳动，打，敲behave 举动，举止，运转behavio(u)r 行为，举止belief 信仰，信条belong 属，附属，隶属bend 弯曲，曲折处，折弯benefit 利益，恩惠besides 而且，还有，除…之外beyond 在…那边，在远处blame 责备，怨，责任，过失blank 空白表格，无表情的，空着的，茫然的，blind 瞎的，盲目的，使失明bloom 开花，花blow 吹，打气，爆炸，欧打bomb 炸弹，轰炸bother 打扰，麻烦bound 跳，必定brain 大脑，骨髓，智能branch 枝，树枝，分部brand 商标，打烙印于break 打破，折断，违反，破坏，打断，中止，（课间或工间）休息时间breath 呼吸，气息brief 向…作简要的介绍，简短的，简洁的bright 明亮的，聪明的，伶俐的，快活的，美好的brilliant 辉煌的，灿烂的，杰出的，有才华的burden 担子，负担burn 燃烧，烧毁，灼伤burst 破裂，爆炸，突然发作Ccampaign 战役，运动campus （大学）校园cancel 取消，撤消cancer 癌capable 有本领的，有能力的capital 首都，大写，资金，主要的，基本的captain 首领，队长，船长，上校career 生涯，经历，专业carry 搬运，运送，携带case 事实，情况，案件，盒子cast 投，掷，抛casual 偶然的，碰巧的，随便的catch 捕，捉，赶上，感染传法染病cause 原因，缘故，事业celebrate 庆祝central 中心的，中央的ceremony 仪式，典礼，礼节certain 确实的，可靠的，某一，某些，一定，必然的challenge 挑战character 性格，品质，特性characteristic 特有的，独特的，特征charge 指控chase 追求，追逐cheat 欺骗，骗取，行骗check 制止，控制，检查chief 主要的，首要的，首领choice 选择，选择机会choose 选择，挑拣，甘愿circle 圆，圈，圆周，集团，环绕，盘旋，周期，循环civil 公民的，平民的，国内的，文明的，有教养的claim 声称，主X，对…提出要求，索取classical 经典的，古典的coarse 粗的，粗糙，粗劣的，粗俗的，粗鲁的colleague 同事，同僚collect 收集，收（税等），领走bine 结合，联合，化合fortable 舒适的，舒服的，自在的mand 命令，指挥，掌握ment 解说，评论，意见merce 商业mit 犯，干（错事）mon 普通的，通常的，公共的municate 通讯，交流，交际panion 同伴，伴侣pare 比较，相比petent 有能力的，胜任的plain 抱怨plete 完成的，完全的，完成plex 复杂的，复合的pose 写作，作曲，由…组成promise 妥协concentrate 集中，聚集，浓缩concept 概念concern 关心，挂念，涉与conclude 结束，下结论，缔结concrete 混凝土，具体的，实质性的condition 状况，状态，条件conduct 行为，品行conference 会议，讨论会confess 承认，坦白，忏悔confident 确信的，有自信的confine 限制，局限于，禁闭confirm 证实，肯定，批准conflict 战斗，斗争，抵触confuse 混淆，搞乱congratulate 祝贺，贺喜connect 连续，联系conquest 征服conscious 有意识的consent 同意，答应consequence 后果consider 考虑，认为considerable 相当的，可观的considerate 考虑周到的，体谅的consist 由…构成，在于constant 不断的，始终如一的construct 建设，建造consult 商量，请教，咨询consume 消耗，花费contact 接触，联系，交往contain 容纳，含有，装有content 满足，甘愿contest 竞争，比赛continue 连续，继续continuous 连续的，继续的contract 收缩，紧缩，契约，合同contrary 相反的，矛盾的contrast 对比contribute 捐助，投稿control 控制，抑制convenient 方便的convey 传达，输送convince 使信服，使相信cope 对付，应付correct 正确的，合适的，修正correspond 相当于，符合corresponding 相应的corridor 走廊cost 成本，费用，值cover 盖，包括，涉与create 创造，创作，建立creative 有创造力的crime 罪，罪行criticize 批评，评论cure 治愈，矫正Ddamage 毁坏dangerous 危险的，不安全的deal 处理，交易，契约，做买卖debt 债，欠债deceive 欺骗，蒙蔽declare 宣布，表明decorate 装饰，布置defeat 战胜，挫败defend 保卫，辩护definite 明确的，限定的degree 度，程度，学位delay 推迟，耽搁deliver 投递，送交，发表，接生demand 要求，需要，质问demonstrate 证实，表明deny 否认，拒绝dependent （on,upon）依赖的describe 描述，形容deserve 应受，值得design 计划，企图，设计desire 愿望，要求despair 失望，绝望despite 不管，尽管destination 目的地destroy 破坏，毁灭detail 细节，详细说明detect 察觉，发现determine 决定，测定develop 发展，进展，发扬differ 不同，分歧direct 径直，直接，指引disaster 灾害，灾难discover 发现，显示，暴露discuss 讨论dismiss 驳回，对…不予受理display 陈列，展览distribute 分配，散布disturb 扰乱，妨碍，使不安divide 分，划分，分担domestic 家庭的，国内的doubt 怀疑，疑问，不相信dramatic 戏剧性的，引人注目的drive 驾驶，开动，迫使due 预定，应得的，到期的duty 职务，义务，税Eearn 赚得，羸得，获得economic 经济的edge 刃，边缘，棱，挤进educate 教育，培养effect 效果，作用effective 有效的，生效的efficient 效率高的，有能力的effort 努力，尽力elect 推选，选举emerge 出现，冒出，（事情）暴露emotion 情感，情绪emphasis 强调employ 雇用，使用encourage 鼓励，助长，促进endure 忍受，忍耐，持续energy 精力，气力，能量entertain 招待，使欢乐environment 环境，周围状况，自然环境equal 同等的，相等的，匹敌者escape 逃跑，逃避especially 特别，格外essential 必不可少的，必要的establish 建立，确立estimate 估计，评价evaluate 估价，评价event 事件，大事，比赛项目eventually 最后，终于exact 确切的，精确的exaggerate 夸大examine 检验，审查，考试exceed 超过excellent 优秀的except 除…之外excessive 过多的，极度的exchange 交换，调换excite 激动，唤起，刺激exhaust 用尽，竭力exhibit 展览，显示，展品exist 在，存在exit 出口，太平门，离去expect 期待，盼望experience 经验，经历explain 解释explode 爆炸，破裂explore 探险，探索expose 揭露export 出口，输出express 表示Ffactor 因素，要素failure 失败，失败者fair 公平的，相当的，晴朗的,定期集市，交易会fairly 公平地，相当，完全faith 信任，信用，信仰familiar 熟悉的，交情好的famous 著名的fashion 样子，方式，风尚fatal 致命的fate 命运fault 缺点，过失favo(u)r 好感，恩惠，支持fear 恐惧，担心feature 面貌，特征festival 音乐节，节日fetch （去）拿来，请来few 少数的，几平没有fight 打仗，斗争figure 外形，图形，人物finish 完毕，完成firm 坚固的，坚强的，公司fit 合适的，恰当的，健壮的fix 固定，安装，修理，确定flexible 柔软的，易弯曲的flood 洪水，水灾，淹没flourish 繁荣，兴旺focus 使聚集，使集中follow 跟随，追求，遵循，理解forbid 禁止force 力，力气，军队，强迫forecast 预测forever 永远，总是forget 忘记，遗忘forgive 宽恕formal 正式的，礼仪上的former 在前的fortunate 幸运的fortune 命运，财富frame 框架，体格frank 坦白的frequently 时常，往往fresh 新的，有生气的，淡水的fuel 燃料fulfil(l) 完成，履行function 职责，功能fund 资金，专款fundamental 基本的furniture 家具furthermore 而且Ggain 获得，增加，获利gaze 凝视general 普通的，总的，大体的，将军generation 产生，代，时代generous 慷慨的，宽厚的genius 天才gentle 和蔼的，有礼貌的genuine 真正的，真诚的gesture 手势，姿态glance 看一眼glimpse 一瞥globe 地球，地球仪govern 统治graceful 优美的gradual 逐步的graduate 毕业生，研究生的grant 同意，给予grasp 掌握，抓紧grateful 感激的gratitude 感激growth 生长，增长guarantee 保证guilty 有罪的Hhabit 习惯handle 手柄，触，处理harmony 和谐，和睦harvest 收获，后果haste 匆忙，赶快hazard 危险，危害healthy 健康的hesitate 犹豫，踌躇hide 隐藏，躲藏hint 暗示hire 雇用，租借hobby 业余爱好hollow 空的，中空的honest 诚实的，正直的hono(u)r 荣誉，敬意hostile 敌对的hotel 旅馆however 然而，无论，不管human 人的，人类的humo(u)rous 幽默的hunt 打猎，寻找hurt 伤害Iideal 理想，称心的identify 认出idle 闲着的，无效的，空虚igore 不理，忽视illegal 不合法的imaginary 想像的imagination 想像imagine 想像，料想imitate 模仿immediate 立即的，最接近的impact 影响impatient 不耐烦的imply 意指import 输入，进口物资importance 重要性impossible 不可能的impress 给…深刻的印象improve 改善，进步improvement 改进incident 小事件，事变include 包含，计入ine 收入increase 增长indicate 指示，暗示individual 个人，个别的inevitable 必然的infer 推断inferior 次的influence 影响，权势inform 通知，告发injure 损害，伤害innocent 清白的insist 坚持inspect 检查inspire 使产生灵感install 装置instance 例instant 立刻，瞬间instead 代替instinct 本能，天性instruct 教，指示insult 侮辱insurance 保险intelligent 聪明的intend 想要，企图intensive 加强的，集中的intention 意图interfere 干涉intermediate 中间的interrupt 打断，断绝interval 间隔interview 接见，面试intimate 亲密的introduce 介绍，引进invent 发明invest 投资invite 邀请，招待involve 卷入，连累，含有isolate 隔离item 条，条款Jjealous 妒忌的journey 旅行judge 审判员，评价，审判junior 年少的，后进的，下级的Ikeen 锋利的，尖锐的，敏捷的，热心的，渴望的kick 踢kid 小孩，戏弄knock 敲打，碰撞knot 结，节疤knowledge 知识，知道Llabo(u)r 工作，劳动，劳力largely 大部分，基本上lately 近来latter 后面的，后者launch 发射，发起，开始lawyer 律师lay 放，搁，下（蛋），铺设lead 领导，引导，铅leading 指导的，最主要的leak 渗漏lecture 演讲，讲课legal 法律（上）的，合法的leisure 空闲lest 惟恐，以免level 水平，水准，铺平liberate 解放license/licence 准许，认可，执照lie 躺，平放，说慌，慌言likely 可能的，大概likewise 同样，也limit 界限，X围，限定link 环，联系list 表，，列入living 活的，现存的living-room 起居室local 地方的，局部的locate 找出，位于lonely 孤独的，荒凉的loyal 忠诚的luxury 奢侈，奢侈品M magic 戏法，魔法，有魔力的mainly 主要地，大体上maintain 保养，维修，继续major 较大的，主要的，主修manage 管理，处理，设法manner 方式，态度，规矩mark 记号，痕迹，分数，作标记mask 面具，遮盖master 主人，能手，掌握match 比赛，敌手，相配material 材料，物质的，具体的meanwhile 其时，同时measure 量度，办法，量medical 医学的，医药的medium 中等的，中间，传导体memory 记忆力，回忆mental 思想的，脑力的mention 提与，讲述merely 仅仅，只不过metal 金属，金属制品method 方法miracle 奇迹，令人惊奇的人（或事）miserable 悲惨的miss 小姐，没达到，没赶上，省去，惦记mission 使节，代表团，天职，任务model 样式，模特，典型moment 片刻，瞬间mood 心境，语气moral 道德的，寓意，教育意义murder 谋杀mutual 相互的，共同的mysterious 神秘的，可疑的，难理解的Nnative 土著，当地人，土生的natural 自然界的，自然的，固有的necessarily 必定，当然negative 否定的，反面的neglect 忽视，忽略neutral 中立的，中性的nevertheless 然而，不过nuclear 核心的，原子核的numerous 众多的，无数的Oobey 服从，听由object 物体，对象，宾语，目的objective 目标，客观的observe 遵守，观察，注意到obtain 获得obvious 明显的occasion 场合，时节occupy 占，占据occur 发生，想起offend 冒犯，使不快offer 提供，提议operate 操作，动手术opinion 意见，主X opportunity 机会oppose 反对optimistic 乐观（主义）的organize 组织，编组origin 起源output 产量overe 克服overlook 俯瞰，忽略，宽容overtake 赶上，超过Ppace 步子，步速package 包，盒pardon 原谅，请再说一遍participate 参与particularly 特别地，尤其地partly 在一定程度上，不完全地partner 伙伴，舞伴passage 通过，走廊，段落passenger 旅客passion 激情passive 被动的，消极的patient 有耐心的，病人payment 支付peace 和平，平静peculiar 特殊的perfect 完美的，十足的perform 做，完成，表演performance 表演，执行permanent 永久的permit 允许，执照persist 坚持persuade 说服plain 平原，平易的，简单的，朴素的plastic 塑料pleasant 令人愉快的plentiful 丰富poisonous 有毒的policy 政策，方针polish 擦亮，抛光polite 有礼貌的，客气的pollute 污染，玷染popular 广受欢迎的，有名的，流行的position 位置，职位，姿态positive 确定的，正面的，积极的possess 拥有，拥有（物）possibility 可能性postpone 推迟potential 潜力poverty 贫困powerful 强大的，有权的practical 实际的practice 实践，练习，开业precious 珍贵的precise 精确的predict 预言prefer 更喜欢preserve 维护prescribe （医生）开（药），嘱咐（疗法），指定present 出席的，目前的，礼品pressure 压力，紧Xpretend 假装prevent 预防previous 先的，在…以前的price 价格，代价pride 骄傲，自满primarily 主要地，首先principal 主要的，负责人principle 原理，主义private 私人的，秘密的privilege 优惠，特权prize 奖，珍视probable 有希望的，大概的procedure 程序，手续proceed 继续进行process 过程，处理produce 生产，显示，演出profit 收益，益处program(me) 计划，节目progress 前进，进步project 计划，工程，设计，放映promise 答应，承诺，出息promote 促进，发扬，提升proper 适当的，特有的proposal 提议，求婚prospect 展望protect 保护protest 抗议proud 骄傲的，引以自豪的prove 证明provide 提供psychological 心理的publish 发表，出版punish 惩罚，处罚purchase 买，购买purpose 目的，用途，效果pursue 从事，追求Qqualify 取得资格quality 质量，特性quantity 量，数量quiet 安静，平定quit 离开，停止，辞职quite 完全，相当quote 引用，援引Rracial 种的，种族的raise 举起，提升，建立range X围，排列，（山）脉rank 排，等级，评价rapid 快的reach 伸手，够到，到达reaction 反应reasonable 合理的，公道的receive 收到，受到recently 最近recognize 认出，承认reduce 缩小，使成为，还原refer 参考，查询，提交reflection 映像reform 改革regard 考虑，看作，敬意，注重region 地区，X围regret 遗憾，抱歉regular 定时的，正规的，整齐的relax （使）松弛，放松release 释放，放出relieve 缓解，换班，减少religion XX，信仰rely 依靠，信赖remind 使想起，提醒replace 放回，取代report 报告，传说represent 表示，说明，代表request 请求require 需要resist 抵抗，忍住resource 资源，谋略respect 尊敬，问候responsible 应负责的restrict 限制review 复习revise 修订，复习risk 风险rough 粗糙的，粗鲁的rouse 唤醒，惊起route 航线，路线ruin 毁坏，崩溃rule 统治，规章，惯例rural 农村的Ssafety 安全salary 薪金satisfactory 令人满意的scale 标度，天平，比例尺，音阶，鱼鳞scarce 稀少的，不足的scare 惊恐score 得分，记分，二十secret 秘密的，奥秘section 章节，部分，地区security 安全select 选择，精选的senior 资格老的，地位高的，年长的sense 感觉到，感官，辨别力，意思sensible 懂事的，明智的sensitive 有感觉的，敏感的separate 分离的，分开serve 服务，招待，适用severe 严厉的，严重的，凛冽的，严格的shade 阴影，遮蔽，遮光物，罩shadow 阴影，阴暗shake 震动，颤抖shallow 肤浅的，浅滩shame 羞愧，耻辱shape 形状，形成share 分配，分享，股份sharp 尖锐的，突然的，线条分明的，刺耳的，敏锐的shelter 掩蔽，庇护shift 转移，转变shock 震动，电击sight 视觉，瞥见sign 符号，招牌，签signal 信号，发信号significance 意义silent 沉默的similar 类似的simple 简单的，朴素的single 单身的，单个的site 场所situation 形势，局面skill 熟练，技能slight 轻微的，瘦弱的social 交际的，社会的society 社会，协会，上流社会solution 解决办法，溶液sort 种类，整理source 源，根源spare 多余的，抽出（时间）special 特殊的，附加的specific 明确的，具体的speed 速度，速率，飞驰spend 消耗，消磨，花费spirit 精神，情绪，气魄，酒精，勇气spoil 损坏，溺爱staff 杆，棒，参谋部，工作人员，全体职员stand 站立，坐落，忍受，台standard 标准，规格stare 盯，凝视state 国家，情况，陈述status 地位，身份steady 稳定的，使稳定steal 窃取，巧取stimulate 刺激，鼓励straight 直接，直率，平直的strange 奇怪的，陌生的，生疏的strategy 战略，策略stress 压力，紧迫，强调strict 严格的，精确的structure 结构，建筑物，建造style 风格，时髦substitute 代替品，代替success 成功，成就suffer 受苦，经历，忍受sufficient 充分的suggest 建议，使想起，暗示summary 摘要superior 上级，更好的，优秀的supply 供应，补充support 支持，拥护，赡养suppose 猜想，假定survive 比…寿命长，幸存suspect 嫌疑犯，怀疑symbol 象征，标志system 系统，体系，制度Ttalent 才能，天资target 耙子，目标task 任务，工作temper 情绪，脾气tendency 倾向，趋势tense 时态，拉紧的，紧X的term 学期，期限，措辞，词，条款threaten 威胁，快要来临，有…征兆title 书名，题目，称号topic 论题torture 拷问，折磨total 总数，完全的tourist 旅游者tradition 传统，惯例traffic 交通tragedy 悲剧，惨事transform 转变，变革transport 运输travel 旅行treat 对待，医疗，请客trend 趋势，倾向trial 审讯，实验trust 信任，委托truth 真实，真理typical 典型的，有代表性的Uunderline 强调，在…下面划线undertake 接受，承担unfortunate 不幸的，令人遗憾的unless 如果不，除非unusual 稀有的，例外的upset 弄翻，打乱，使不安urban 城市的urge 鼓励urgent 紧迫的used to 过去经常useful 有用的usually 通常Vvacation 休假valuable 有价值的vanish 消散variety 多样性，品种various 各种各样的vehicle 车辆victim 牺牲品，受害者victory 胜利view 观察，眼界，观点，看法，风景，考虑violence 强暴，暴行，猛烈violent 猛烈的，强烈的vital 重要的vocabulary 词汇（量）volume （一）卷，（一）册，体积，音量voluntary 志愿的vote 选票，选举Wwage 工资，进行warn 警告waste 荒废的，浪费，废物wave 波浪，飘扬，挥舞wealth 财富，丰富weapon 武器wear 穿戴，磨损weave 编weigh 称，量，考虑weight 重量welfare 福利wide 宽阔的，完全的，充分的wisdom 智慧wise 聪明的withdraw 收回，撤消witness 目击者，见证人wonder 惊奇，奇迹，纳闷worm 虫，蠕虫worn 破烂的，损坏的worry 发愁，烦恼worship 礼拜worth 值…钱，价值worthless 无价值的worthwhile 值得的wound 创伤，伤害wreck 失事，海滩，沉船，遇难，残骸11 / 11。

关于柜子种类的英语作文

关于柜子种类的英语作文Title: A Comprehensive Exploration of Different Types of Cabinets。

Cabinets play a crucial role in our daily lives, providing storage solutions for various items and enhancing the organization of our living spaces. From kitchens to bedrooms, offices to bathrooms, cabinets come in diverse types, each serving specific functions and catering to different aesthetic preferences. In this essay, we will delve into the world of cabinets, exploring their types, functionalities, and design variations.1. Kitchen Cabinets:Kitchen cabinets are perhaps the most widely recognized type of cabinets, essential for storing utensils, cookware, and pantry items. They are typically installed above and below countertops, maximizing storage space in the kitchen. Kitchen cabinets come in various styles such as basecabinets, wall cabinets, tall cabinets, and corner cabinets, each optimized for different storage needs.2. Bathroom Cabinets:Similar to kitchen cabinets, bathroom cabinets servethe purpose of storing toiletries, towels, and other essentials. They are available in freestanding or wall-mounted designs, with options for single or multiple shelves, drawers, and compartments. Bathroom cabinets often prioritize moisture resistance and durability due to the humid environment of bathrooms.3. Storage Cabinets:Storage cabinets are versatile pieces of furniture used in different areas of the home, including living rooms, bedrooms, and offices. They come in a wide range of sizes, shapes, and materials, offering ample storage for books, clothing, documents, and decorative items. Storage cabinets may feature doors, drawers, or open shelves, allowing for customizable organization solutions.4. Display Cabinets:Display cabinets, also known as showcase cabinets or curio cabinets, are designed to showcase and protect valuable or decorative items such as china, collectibles, and memorabilia. They often feature glass doors and shelves, providing visibility while keeping the items safe from dust and damage. Display cabinets can be standalone units or integrated into larger furniture pieces like entertainment centers or buffets.5. Media Cabinets:With the proliferation of electronic devices, media cabinets have become essential for organizing and storing media equipment such as televisions, gaming consoles, and stereo systems. These cabinets are equipped with features like cable management systems, adjustable shelves, and ventilation slots to accommodate various electronic devices while keeping cables organized and accessible.6. Wardrobes:Wardrobes are specialized cabinets designedspecifically for storing clothing and accessories. They come in different configurations, including single-door, double-door, and sliding-door designs, with options for hanging rods, shelves, drawers, and shoe racks. Wardrobes are often customizable to fit the specific storage needs and spatial constraints of bedrooms.7. Pantry Cabinets:Pantry cabinets are dedicated storage solutions for food items and kitchen supplies, typically installed in or near the kitchen. They come in various forms, including walk-in pantries, pantry closets, and pull-out pantry cabinets. Pantry cabinets are designed to maximize vertical space and facilitate easy access to stored items, with adjustable shelves and organizers for efficient organization.8. Custom Cabinets:For those with unique storage requirements or specific design preferences, custom cabinets offer a tailored solution. Custom cabinets are built to fit the exact dimensions and specifications of a space, allowing for maximum utilization of available space and integration with the overall design aesthetic. Whether it's a built-in bookcase, a bespoke kitchen island, or a customized wardrobe, custom cabinets offer limitless possibilities for personalization.In conclusion, cabinets come in a myriad of types, each serving distinct functions and catering to diverse storage needs. Whether it's organizing kitchen essentials, displaying cherished collectibles, or storing media equipment, there's a cabinet type suited for every purpose and space. Understanding the different types of cabinets empowers homeowners to make informed decisions when planning and designing their living spaces, ensuring optimal functionality and aesthetic appeal.。

Bethe Ansatz Solutions and Excitation Gap of the Attractive Bose-Hubbard Model

a r X i v :c o n d -m a t /0108314v 1 [c o n d -m a t .s t a t -m e c h ] 20 A u g 2001Bethe Ansatz Solutions and Excitation Gap of the Attractive Bose-Hubbard ModelDeok-Sun Lee and Doochul KimSchool of Physics,Seoul National University,Seoul 151-747,KoreaThe energy gap between the ground state and the ﬁrst excited state of the one-dimensional attractive Bose-Hubbard Hamiltonian is investigated in connection with directed polymers in random media.The excitation gap ∆is obtained by exact diagonalization of the Hamiltonian in the two-and three-particle sectors and also by an exact Bethe Ansatz solution in the two-particle sector.The dynamic exponent z is found to be 2.However,in the intermediate range of the size L where UL ∼O (1),U being the attractive interaction,the eﬀective dynamic exponent shows an anomalous peak reaching high values of 2.4and 2.7for the two-and the three-particle sectors,respectively.The anomalous behavior is related to a change in the sign of the ﬁrst excited-state energy.In the two-particle sector,we use the Bethe Ansatz solution to obtain the eﬀective dynamic exponent as a function of the scaling variable UL/π.The continuum version,the attractive delta-function Bose-gas Hamiltonian,is integrable by the Bethe Ansatz with suitable quantum numbers,the distributions of which are not known in general.Quantum numbers are proposed for the ﬁrst excited state and are conﬁrmed numerically for an arbitrary number of particles.I.INTRODUCTIONThe dynamics of many simple non-equilibrium sys-tems are often studied through corresponding quantum Hamiltonians.Examples are the asymmetric XXZ chain Hamiltonian and the attractive Bose-Hubbard Hamilto-nian for the single-step growth model [1]and the directed polymers in random media (DPRM)[2],respectively.The single-step growth model is a Kardar-Parisi-Zhang (KPZ)universality class growth model where the inter-face height h (x,t )grows in a stochastic manner under the condition that h (x ±1,t )−h (x,t )=±1.The process is also called the asymmetric exclusion process (ASEP)in a diﬀerent context.The evolution of the probability distri-bution for h (x,t )is generated by the asymmetric XXZ chain Hamiltonian [3].The entire information about the dynamics is coded in the generating function e αh (x,t ) .Its time evolution,in turn,is given by the modiﬁed asym-metric XXZ chain Hamiltonian [4–6],H XXZ (α)=−L i =1e 2α/L σ−i σ+i +1+12L i =1(b i b †i +1+b †i b i +1−2)−UL i =1b †ib i (b †ib i −1)4Lρ(1−ρ)and −n√4Lρ(1−ρ)≫1and the density of particles is ﬁ-nite in the limit L →∞,∆(α)behaves as ∆(α)∼L −1.However,when α∆(α)∼L−3/2[3,11].The dynamic exponent z=3/2is a characteristic of the dynamic universality class of the KPZ-type surface growth.When the number of par-ticles isﬁnite and the density of particles is very low, it is known that∆(α)∼L−2[12].However,whenα<0,which corresponds to the ferromagnetic phase, most Bethe Ansatz solutions are not available althoughthe Bethe Ansatz equations continue to hold.Asαbe-comes negative,the quasi-particle momenta appearing inthe Bethe Ansatz equations become complex,so solutions are diﬃcult to obtain analytically.The attractive Bose-Hubbard Hamiltonian is expected to have some resemblance to the ferromagnetic phaseof the asymmetric XXZ chain Hamiltonian consider-ing the equivalence ofαand−n.The equivalence isidentiﬁed indirectly by comparing the two scaling vari-ablesα LU under the relation U= 4ρ(1−ρ)or the two generating functions exp(αh(x,t) and Z(x,t)n under the relation Z(x,t)=e−h(x,t).In contrast to the asymmetric XXZ chain Hamiltonian,theBose-Hubbard Hamiltonian does not satisfy the Bethe Ansatz except in the two-particle sector[13].Instead, the attractive delta-function Bose-gas Hamiltonian,H D(n)=−1∂x2i−Ui<jδ(x i−x j),(4)which is the continuum version of the attractive Bose-Hubbard Hamiltonian,is known to be integrable by the Bethe Ansatz.The attractive delta-function Bose gas has been studied in Refs.[14]and[15].The ground-state energy is obtained from the Bethe Ansatz solution by us-ing the symmetric distribution of the purely imaginary quasi-particle momenta.However,the structure of the energy spectra is not well known for the same reason as in the asymmetric XXZ chain Hamiltonian withα<0. The unknown energy spectra itself prevents one from un-derstanding the dynamics of DPRM near the stationary state.In this paper,we discuss in Section II the distribu-tion of the quantum numbers appearing in the Bethe Ansatz equation for theﬁrst excited state of the attrac-tive delta-function Bose-gas Hamiltonian,the knowledge of which is essential for solving the Bethe Ansatz equa-tion.In Section III,the excitation gap of the attractive Bose-Hubbard Hamiltonian with a small number of par-ticles is investigated through the exact diagonalization method.We show that the gap decays as∆∼L−2,i.e., z=2,but that the exponent becomes anomalous when U∼L−1.The emergence of the anomalous exponent is explained in connection with the transition of theﬁrst excited state from a positive energy state to a negative energy state.The Bethe Ansatz solutions in the two-particle sector show how the behavior of the gap varies with the interaction.We give a summary and discussion in Section IV.II.QUANTUM NUMBER DISTRIBUTION FOR THE FIRST EXCITED STATEIn this section,we study the Bethe Ansatz solutions for the ground state and theﬁrst excited state of the attrac-tive delta-function Bose-gas Hamiltonian.The eigenstate of H D(n),Eq.(4),is of the formφ(x1,x2,...,x n)= P A(P)exp(ik P1x1+ik P2x2+···+ik P n x n),(5)where P is a permutation of1,2,...,n and x1≤x2≤...≤x n with no three x’s being equal.The quasi-particle momenta k j’s are determined by solving the Bethe Ansatz equations,k j L=2πI j+ l=jθ(k j−k l2+j,(j=1,2,...,n),(7)and the quasi-particle momenta are distributed symmet-rically on the imaginary axis in the complex-k plane. Care should be taken when dealing with theﬁrst excited state.For the repulsive delta-function Bose-gas Hamilto-nian,where U is replaced by−U in Eq.(4),the quantum numbers for one of theﬁrst excited states areI j=−n+12.(8)However,for the attractive case,by following the move-ment of the momenta as U changes sign,weﬁnd that the quantum numbers for theﬁrst excited state should be given byI j=−n−12(=I1).(9)That is,the two quantum numbers I1and I n become the same.Such a peculiar distribution of I j’s does not ap-pear in other Bethe Ansatz solutions such as those for the XXZ chain Hamiltonian or the repulsive delta-function Bose-gas Hamiltonian.We remark that even though the two I j’s are the same,all k j’s are distinct;otherwise,the wavefunction vanishes.Such a distribution of quan-tum numbers is conﬁrmed by the consistency between the energies obtained by diagonalizing the Bose-HubbardHamiltonian exactly and those obtained by solving the Bethe Ansatz equations with the above quantum num-bers for very weak interactions,for which the two Hamil-tonians possess almost the same energy spectra.When there is no interaction(U=0),all quasi-particlemomenta,k j’s,are zero for the ground state while for theﬁrst excited state,all the k j’s are zero except thelast one,k n=2π/L.In the complex-k plane,as the very weak repulsive interaction is turned on,the n−1momenta are shifted inﬁnitesimally from k=0withk1<k2<···<k n−1,and the n th momentum is shifted inﬁnitesimally to the left from k=2π/L.All the mo-menta remain on the real axis.When the interaction is weakly attractive,the n−1momenta become complexwith Im k1<Im k2<···<Im k n−1and Re k j≃0for j=1,2,...,n−1,and the n th momentum remains on thereal axis,but is shifted to the left.Figure1shows the dis-tribution of the quantum numbers and the quasi-particlemomenta in the presence of a very weak attractive in-teraction.The quasi-particle momenta are obtained by solving Eq.(6).Knowledge of the distribution of the quantum num-bers is essential for solving the Bethe Ansatz equations of the attractive delta-function Bose-gas Hamiltonian.For the original attractive Bose-Hubbard Hamiltonian,the Bethe Ansatz solutions are the exact solutions for the two-particle sector only,but are good approximate so-lutions in other sectors provided the density is very low and the interaction is very weak.This is because the Bethe Ansatz for the Bose-Hubbard Hamiltonian fails once states with sites occupied by more than three parti-cles are included.Thus,for the sectors with three or more particles,the Bethe Ansatz solutions may be regarded as approximate eigenstates provided states with more than three particles at a site do not play an important role in the eigenfunctions.In Ref.[13],it is shown that the error in the Bethe Ansatz due to multiply-occupied sites (occupied by more than three particles)is proportional to U2,where U(>0)in Ref.[13]corresponds to−U in Eq.(2).This applies to the attractive interaction case also.For the repulsive Bose-Hubbard Hamiltonian,the Bethe Ansatz is a good approximation when the density is low and the interaction is strong because the strong re-pulsion prevents many particles from occupying the same site[16].For the attractive Bose-Hubbard Hamiltonian, the Bethe Ansatz is good when the density is low and the interaction is weak because a weak attraction is better for preventing many particles from occupying the same site and because the error is proportional to U2.III.POWER-LA W DEPENDENCE ANDANOMALOUS EXPONENTWe are interested in the scaling limit L→∞with the scaling variable n√byE 0=−4sinh 2 κ2Lcosh2q 2L sinh2q2sinh κ−U,(12)andqL =logU +2cos(πU −2cos(π2s κ−s U,(14)which gives s κ≃1.151.When the size of the system L is increased by δL with U =U∗,the changes of κand q ,δκand δq ,are,from Eqs.(12)and (13),δκ=−πs κ(4s κ2−s U 2)L 2≡−πΓδL(4/π)s U −s U 2+4δLL 2.(15)The perturbative expansion ∆(L +δL )≃∆(L )(1−z (δL/L )),under the assumption that ∆(L )∼L −z ,gives the value of z eﬀat U ∗:z eﬀ=21+s κΓ+Σlog((L −1)/(L +1))(17)by using the solutions of Eqs.(12)and (13)for suﬃciently large L .As discussed above,the exponent z eﬀshows an anomalous peak near U =U ∗or UL/π=s U and ap-proaches 2.0as UL/π→0or ∞.Figure 6shows a plot of z eﬀversus the scaling variable UL/πat L =10000.IV.SUMMARY AND DISCUSSIONAs the asymmetric XXZ chain generates the dy-namics of the single-step growth model,the attractive Bose-Hubbard Hamiltonian governs the dynamics of the DPRM.We studied the attractive Bose-Hubbard Hamil-tonian and its continuum version,the attractive delta-function Bose-gas Hamiltonian concentrating on the be-havior of the excitation gap,which is related to the char-acteristics of DPRM relaxing into the stationary state.For the attractive delta-function Bose gas Hamiltonian,The quantum numbers for the ﬁrst excited state in the Bethe Ansatz equation are found for the attractive delta-function Bose gas Hamiltonian,and the distribution of the quasi-particle momenta is discussed in the presence of a very weak attractive interaction.Our result is the start-ing point for a further elucidation of the Bethe Ansatz solutions.We show that the excitation gap depends on the size of the system as a power law,∆∼L −z ,and that the exponent z can be calculated by using an exact diag-onalization of the attractive Bose-Hubbard Hamiltonian in the two-and the three-particle sectors and by using the Bethe Ansatz solution in the two-particle sector.The exponent z is 2.0.However,for the intermediate region where UL ∼O (1),the eﬀective exponent z eﬀshows a peak.The equivalence of the diﬀerential equations govern-ing the single-step growth model and DPRM implies some inherent equivalence in the corresponding Hamil-tonians.The power-law behavior of the excitation gap,∆∼L −2,for the attractive Bose-Hubbard Hamiltonian with a very weak interaction is the same as that for the asymmetric XXZ chain Hamiltonian with a small num-ber of particles,which is expected considering the rela-tion U =4ρ(1−ρ).The fact that the excitation gap behaves anomalously for U ∼L −1implies the possibility of an anomalous dynamic exponent z for a ﬁnite scaling variable n√[1]M.Plischke,Z.Racz,and D.Liu,Phys.Rev.B 35,3485(1987).[2]M.Kardar,Nucl.Phys.B 290[FS20],582(1987).[3]L.H.Gwa and H.Spohn,Phys.Rev.A 46,844(1992).[4]B.Derrida and J.L.Lebowitz,Phys.Rev.Lett.80,209(1998).[5]D.-S.Lee and D.Kim,Phys.Rev.E 59,6476(1999).[6]B.Derrida and C.Appert,J.Stat.Phys.94,1(1999).[7]J.Krug and H.Spohn,in Solids Far from Equilibrium ,edited by C.Godr´e che (Cambridge University Press,Cambridge,1991),p.412.[8]B.Derrida and K.Mallick,J.Phys.A 30,1031(1997).[9]S.-C.Park,J.-M.Park,and D.Kim,unpublished.[10]E.Brunet and B.Derrida,Phys.Rev.E 61,6789(2000).[11]D.Kim,Phys.Rev.E 52,3512(1995).[12]M.Henkel and G.Sch¨u tz,Physica A 206,187(1994).[13]T.C.Choy and F.D.M.Haldane,Phys.Lett.90A ,83(1982).[14]E.H.Lieb and W.Liniger,Phys.Rev.130,1605(1963).[15]J.G.Muga and R.F.Snider,Phys.Rev.A 57,3317(1998).[16]W.Krauth,Phys.Rev.B 44,9772(1991).(a)ω-0.10.1-0.50.5(b)Re k Im k 0FIG.1.For the ﬁrst excited state,(a)the quantum num-bers I j ’s are depicted in the complex-ωplane with ω=e 2πiI/L and (b)the quasi-particle momenta k j ’s are shown in the complex-k plane.Here,the size of the system L is 20,the number of particles n is 10,and the attractive interaction U is 0.0025.The ﬁlled circle in (a)is where the two quantum numbers overlap.0.5 1102030EL n =2 U =0.05ground state first excited state0.5 1102030EL n =2 U =0.5ground state first excited state-3.4-3.3-3.2 102030EL n =2 U =5ground state first excited state0.5 11020 30EL n =3 U =0.05ground state first excited state-0.5 0 0.5 1020 30ELn =3 U =0.5ground state first excited state-12.17-12.16-12.15 1020 30ELn =3 U =5ground state first excited stateFIG.2.Ground-state energies and ﬁrst excited-state ener-gies are plotted versus the size of the system L (4≤L ≤30)for U =0.05,0.5,and 5in the two-and the three-particle sectors.The dotted line represents E =0.For all values of U and L ,the ground-state energy is negative.On the other hand,when U =0.5,the excited-state energy becomes nega-tive near L ≃14in the two-particle sector and L ≃6in the three-particle sector.The signs of the excited-state energies for U =0.05and 5do not change in the range of L shown here.0.0010.010.1110102030∆LU=0.05U=0.5U=5FIG.3.Log-log plot of the excitation gaps (∆)versus the size of the system (L )in the two-particle sector.Data for U =0.05and 5approach straight lines with slope z =2.0,but those for U =0.5show a strong crossover before approach-ing the asymptotic behavior.The solid line for U =0.5is that ﬁtted in the range 14≤L ≤18,and shows an eﬀective z ≃2.4.0.00010.001 0.010.1110102030∆LU=0.05U=0.5U=5FIG.4.Same as in Fig.3,but for the three-particle sec-tor.The ﬁtted solid line used the data for 8≤L ≤12,and has a slope of approximately 2.7.(a)k (b)k FIG.5.Distributions of the quasi-particle momenta,k j ’s,for the ground state (ﬁlled circles)and the ﬁrst excited state (open circles)are shown in the complex-k plane for n =2.The size of the system L is 100and the interaction U is (a)0.001and (b)0.1.22.22.4s U510z e f fU L/πFIG.6.Eﬀective exponent z eﬀin the two-particle sector versus the scaling variable UL/πat L =10000.The interac-tion U varies from 0.0001to 0.001.At UL/π=s U ≃2.181,z eﬀ≃2.401.。

Journal of Mathematical Analysis and Applications

J. Math. Anal. Appl. 409 (2014) 760-776ELSEVIERContents lists available at ScienceDirectJournal of Mathematical Analysis andApplicationsjournal homepage: /locate/jmaaBlow-up solutions for the inhomogeneous Schrodinger equation with L 2 supercritical nonlinearityShihui ZhuCollege ofMathematics and Software Science, Sichuan Normal University, Chengdu, 610066, ChinaA R T I C L E I N F OA B S T R A C TArticle history:Received 16 April 2013 Available online 29July 2013 Submitted by Xu ZhangKeywords:Nonlinear inhomogeneousSchrodinger equation Blow-upsolution Generalized Gagliardo-Nirenberg inequality Weak concentrationThis paper studies blow-up solutions for the inhomogeneous Schrodinger equation with L 2 supercritical nonlinearity. In terms of Strauss ，arguments in Strauss (1977) [22], we find a new compactness lemma for radial symmetric functions. Thus, we use it to derive the best constants of two generalized Gagliardo-Nirenberg type inequalities. Moreover, we obtain a more precisely sharp criteria ofblow-up and global existence, and derive the weak concentration phenomenon of blow-up solutions by the variational methods.© 2013 Elsevier Inc. All rights reserved.S. Zhu /J. Math. Anal. Appl^ 409 (2014) 760^776 761u x(t, x)=入^u(X2t, kx)is also a solution if u(t, x)is a solution. The scale-invariant Lebesgue norm for Eq. (1.1) is L pc-norm, where pc = U), and s c = N—22+2 is the invariant Sobolev index for Eq. (1.1) as lH sc4L p c. Hence, the assumption p > 2N+f+4 implies pc= N2+2) 2, and we may call Eq. (1.1) as a class of inhomogeneous Schrodinger equations with L2supercritical nonlinearity when 2N+2&+4 < p < 2N+P.Forthe Cauchy problem (1.1)-(1.2), the local well-posedness is established by Chen and Guo [4] in = H) (R N), where H) (R n) is the set of radial symmetric functions in H1(R N). Meanwhile, Chen and Guo [4] studied the sharp conditions of blow-up and global existence for the Cauchy problem (1.1)-(1.2) by the cross-constrained variational arguments. We remark that the sharp conditions ofblow-up and global existence obtained in [4] are not determined in some sense, because the precisely upper bound value d of the energy functional E (u)is not given. When p> 2N+f+4, Eq. (1.1) is a class of nonlinear Schrodinger equations with L2supercritical nonlinearity. Whether there is a concentration phenomenon ofblow- up solutions is still unknown. Motivated by these problems, we further study the blow-up solutions of the inhomogeneous Schrodinger equation (1.1)with L2 supercritical nonlinearity (p > 2N+N&+4).In fact, if one takes b = 0 in Eq. (1.1), then Eq. (1.1) is degenerated to the classical nonlinear homogeneous Schrodinger equation. Now, we recall some known results ofblow-up solutions for the classical nonlinear homogeneous Schrodinger equation in the following form:i^t + + |少|CT~2f = 0, 少(0，x)=少o ^ H1, (1.4) where f = f(t，x): [0，r) x R N— C. The local well-posedness of the Cauchy problem (1.4) is established by Ginibre and Velo [7] in H1 (see [2] for a review). The existence of finite time blow-up solutions was firstly studied by Glassey in [8].Thus, the blow-up solutions have been extensively studied (see [9,18,14-17,19,24-26]). In particularly, forthe critical nonlinearity a = 2 + 4, the structure and formation of singularity is studied by Weinstein in [25]. Based on this property, Merle and Tsutsumi [18] found a new phenomenon for the blow-up solution which is called the L2-concentration. The exact blow-up solution with critical mass is constructed by Merle in [12]. Recently, using the variational methods, Merle and Raphael [15] improved the Weinstein’s results in [25] and obtained the sharp decomposition ofblow-up solutions for small supercritical mass. Thus, using this sharp decomposition, Merle and Raphael [14-16] obtained a large body ofbreakthrough work by the Spectral Properties (see [15]), such as sharp blow-up rates, profiles of blow-up solutions, etc. But there are few results on the dynamics ofblow-up solutions for Eq. (1.4) with L2 supercritical nonlinearity a > 2 + n4. Indeed, it is easy to check that a > 2 + N implies —2)N > 2, and one cannot use the classical Gagliardo-Nirenberg inequality (see [24]):(2—N)a +2N (a—2)NIH < C||vy L T^一 ||vvy L7^，v v e H1, (1.5) to control the nonlinear term in the energy by ||Vf ||^. Recently, using a new Gagliardo-Nirenberg type inequality, Holmer and Roudenko [9] obtained the L3-concentration phenomena of the radial symmetric blow-up solution u(t，x)for a —1 = N= 3, which is called a type of weak concentration compared with the L2-concentration.In the present paper, firstly, we consider the sharp criteria of blow-up and global existence for the inhomogeneous Schrodinger equation (1.1) with L2supercritical nonlinearity. In terms of Weinstein’s arguments in [24], we find the best constant of the following generalized Gagliardo-Nirenberg type inequality: for N > 3，b > 0，2 + n—1 < p < 2N+P and v v e H1, 疒p 2N+2b+2p—Np Np—2N—2b|x|b|v|p dx <^V^ l|v|L r^^ IIVv|T^，(1.6) J R N2||R"p-2 L Lwhere R is the ground state solution of—(Np—2N—2b)AR + (2N + 2b + 2p—Np)R— 4|x|b|R|p—2R = 0，R e H;.(1.7) Moreover, in terms of Holmer and Roudenko’s argument in [9], we use (1.6) to obtain a precisely sharp criteria ofblow-up and global existence of solutions by constructing two invariant evolution flows generated by the Cauchy problem (1.1)-(1.2). Thus, we can give a precisely bound value of the energy functional E (u), which corresponds to d in [4]. Secondly, we consider the concentration phenomenon ofblow-up solutions for the inhomogeneous Schrodinger equation (1.1) with L2supercritical nonlinearity. Because of the nonlinearity |x|b|u|p—2u, Holmer and Roudenko’s argument in [9] cannot be applied to show the weak concentration of blow-up solutions. Our strategy is to find a new sharp generalized Gagliardo-Nirenberg type inequality, which can control the nonlinear potential energy in the energy functional E(u) by ||Vulg. More precisely, we find a new compactness result for the homogeneous Sobolev space Hf「| Hi 1: s 1q2N Hf f 】H r—L q compactly for pc < q < ———-.This is similar to Strauss’ compactness lemma in [22]. Moreover, using the Hardy inequality and the above compactness lemma, we derive the best constant of a generalized Gagliardo-Nirenberg type inequality: for N > 3，b > 0，pc + 芒where Q is the minimal action solution of762 S. Zhu /J. Math. Anal. Appl. 409 (2014) 760-776-AQ + p^—2(-A)sc Q -|x|b|Q|p-2Q = 0, Q g 电 Q Hi 1 (1.9)There are two remarks needed to be pointed out. One is that Chen [3] showed the generalized Gagliardo-Nirenberg inequality (1.6) for2 + < p < N-N2 + , and we improve the range of p to 2 + N— < p < NN + n2^ . The other one is that we derive a new sharp generalized Gagliardo-Nirenberg type inequality (1.8), which gives a new v-riational structure for the nonlinear inhomogeneous Schrodinger equation (1.1).Finally, using the sharp generalized Gagliardo-Nirenberg type inequality (1.8), we obtain the weak concentration of blow-up solutions for the Cauchy problem (1.1)-(1.2). It reads that let N > 3, b > 0 and max [pc + , 2N+Nf+4} < p < min{6, be such that(N— 1 —Sc)(p—pc) > 2b and (Np—2N—2b)(p— 2) < 16.(Indeed, there exist N, b and p satisfyingthe above assumptions, for example N= 3, b= 2, p= 4) If u(t, x) G C([0, T); H1) is the radial symmetric blow-up solution of the Cauchy problem (1.1)-(1.2), then for all M > 0, u(t, x) satisfies liminf ||u(t,x)^H sc(w<m) > IIQ 11柄>0, (1.10) t^Twhere T is the finite blow-up time and Q is the solution of Eq. (1.9).There are two major difficulties to study the blow-up solutions of the nonlinear inhomogeneous Schrodinger equation (1.1). The first one is the nonlinearity |x|b|u|p—2u containing an unbounded potential |x|b. The second one is that the nonlinearity is L2 supercritical for p >2N b+4. Whether there is a concentration prolegomenon for the blow-up solutions of the Cauchy problem (1.1)-(1.2) is open, and using inequalities (1.5) and (1.6), one cannot give a positive reply. In this paper, we give a partial reply in (1.10). Indeed, by establishing the sharp generalized Gagliardo-Nirenberg inequality (1.8), we obtain the Hi Sc -concentration of radial symmetric blow-up solutions, which is another type of weak concentration. Here, the lowerbound value of ||u(t, x)||h sc (|x卜m) is determined, which is different from the L3-concentration obtained by Holmer and Roudenko (see [9]).2.Notations and preliminariesIn this paper, pc and Sc are defined by pc = U, Sc = f—p+b. We denote L q(R N), || • ||[q(rn), H!(R N),句(R N) and /rN •dx by L q,|| • ||q, H1, H and f •dx respectively. We also denote S(R N) the Schwartz space and S(R N) is the topological dual of S(R n). The various positive constants will be simply denoted by C.For the Cauchy problem (1.1)-(1.2), the work space is defined byH] := {v g L2| V v g L2, v(x) = v(r)},wherer = |x| x2 + x\ + …•+ xf,which is a Hilbert space. In order to study the best constants of generalized Gagliardo-Nirenberg inequalities, we introduce two auxiliary Hilbert spaces:jH 1 := j v g S/(R n) is radial symmetric \ j吃 |2pVg)|2d会 < J ,jH S c：= j v g S;(R n) is radial symmetric \ j|会 |2Sc tW(^)|2d与 < J ,where^denotes the Fourier transform. Moreover, we define the energy functional £(v) on H)E(v) := - J |Vv|2dx ------ J |x|b|v|p dx.The functional E(v)is well-defined according to the Sobolev embedding theorem (see [2]). Chen and Guo [4] established the local well-posedness for the Cauchy problem (1.1)-(1.2) in H1.S. Zhu /J. Math. Anal. Appl^ 409 (2014) 760^776 763(2.2)Proposition 2.1 (Chen and Guo [4]). Let N > 3, 2 < p < N+f + and u 〇 g H/. There exists a unique solution u(t, x) ofthe Cauchy problem (1.1)-(1.2) on the maximal time interval [0, T) such that u(t, x) g C ([0, T ); H^) and either T = (global existence), or else 0 < T < and lim t —t ||u(t)||忖1 = (bl^'w-up). Furthermore, for all t g [0, T), u(t, x) satisfies thefollowing conservation laws: (i) Conservation of mass_ )|丨2 =||叱|丨2.(ii) C onservation ofenergyE(u(t)) = E(u 。

custom和标准

custom和标准Custom and Standard。

In today's competitive market, businesses are constantly seeking ways to stand out from their competitors and attract customers. One way to achieve this is by offering custom products and services tailored to the specific needs and preferences of individual customers. On the other hand, standard products and services are also important for businesses to maintain efficiency and consistency in their operations. In this document, we will explore the benefits and considerations of custom and standard offerings in business.Custom products and services are designed to meet the unique requirements of individual customers. By offering customization, businesses can create a personalized experience for their customers, leading to higher satisfaction and loyalty. Customization can also allow businesses to charge premium prices for their products and services, as customers are willing to pay more for personalized offerings that meet their exact needs.Furthermore, custom products and services can be a source of competitive advantage for businesses. By offering unique and personalized solutions, businesses can differentiate themselves from their competitors and attract a niche market of customers who value customization. This can lead to increased market share and higher profitability for the business.However, offering custom products and services also comes with its challenges. Customization can be time-consuming and costly, as it requires businesses to invest in the development of unique solutions for each customer. Additionally, managing a large number of custom orders can be complex and may require a high level of coordination and communication with customers.On the other hand, standard products and services are essential for businesses to maintain efficiency and consistency in their operations. Standardization allows businesses to streamline their processes, reduce costs, and ensure quality and reliability in theirofferings. Standard products and services also make it easier for customers to compare and evaluate different options, leading to more informed purchasing decisions.Standardization can also enable businesses to scale their operations and reach a larger customer base. By offering standardized products and services, businesses can replicate their offerings across different locations and markets, leading to increased brand recognition and market penetration. Standardization also allows businesses to benefit from economies of scale, as they can produce and deliver large volumes of standardized offerings at lower costs.However, relying solely on standard products and services may limit businesses' ability to meet the diverse needs and preferences of individual customers. In today's market, customers are increasingly seeking personalized experiences and solutions, and businesses that fail to offer customization may risk losing out to competitors who do.In conclusion, both custom and standard offerings have their own benefits and considerations for businesses. While customization can lead to higher customer satisfaction and competitive advantage, it also comes with challenges such as increased costs and complexity. On the other hand, standardization can help businesses maintain efficiency and reach a larger customer base, but may limit their ability to meet the unique needs of individual customers. Ultimately, the key for businesses is to find the right balance between custom and standard offerings, taking into account the needs of their target market and the resources available to them. By doing so, businesses can maximize their competitive advantage and achieve long-term success in the market.。

7选5强化练习有详细解答

七选五(满分50 2.5*5*4)How to Remember What You ReadReading is important. But the next step is making sure that you remember what you've read! 1．you may have just read the text. But the ideas, concepts and images（形象）may fly right out of your head. Here are a few tricks for remembering what you read.● 2．If the plot, characters, or word usage is confusing for you, you likely won't be able to remember what you read. It's a bit like reading a foreign language. If you don't understand what you're reading, how would you remember it? But there area few things you can do... Use a dictionary; look up the difficult words.● Are you connected?Does a character remind you of a friend? Does the setting make you want to visit the place? Does the book inspire you, and make you want to read more? With some books, you may feel a connection right away. 3． How willing are you to make the connections happen?● Read it; hear it; be it!Read the lines. Then, speak them out loud. And, put some character into the words. When he was writing his novels, Charles Dickens would act out the parts of the characters. He'd make faces in the mirror, and change his voice for each character. 4．● How often do you read?If you read frequently, you'll likely have an easier time with remembe ring what you’re reading and what you've read. 5．As you make reading a regular part of your life, you'll make more connections, stay more focused and understand the text better. You’ll learn to enjoy literature- as you remember what you read!A．Are you confused?B．Practice makes perfect.C．What's your motivation?D．Memory is sometimes a tricky thing.E. Marking helps you remember what you read.F. But other books require a bit more work on your part.G. You can do the same thing when you are reading the text!Being scared is not always a treat. When a balloon pops, we may jump. 6．How does the body triggered（激发）these reactions? When we encounter something we're not familiar with, our brain and body start making changes to help prepare us for the unknown. These changes are part of the body's " threat," "stress," or "fight — or — flight" response. This response is triggered by the five senses, which are always sending information to the brain. 7．, the brain sounds adanger alarm. It releases chemicals into the bloodstream, and those chemicals act on different parts of the body. Once the body is alert（警觉的），the brain decides whether or not the danger is real.8．It urges a person to hide, run away, or defend oneself. Faster heartbeat and breathing send more blood and oxygen to the muscles so they're ready when they're needed. Stress chemicals can make a person more alert. Our memories also help protect us. We keep a close eye on babies because they don't understand what might hurt them. As babies grow, they learn what's unsafe, and their brains store those memories 9．Experiences can lead to long-lasting fears that may stop a person from trying new things. We're born with the ability to feel fear, but a lot of what we're afraid of is learned. A long-lasting fear can mean you’re had a bad experience, and your brain doesn't want you to have it again. Many people have beaten their own fears. Here's one method. 10．, they learn the facts about anything that seems scary. Knowledge can help a person feel less afraid.A．Instead of listening to their imaginationB．When they signal something unexpectedC．Our heart beats harder, and we breathe fasterD．Then their brains use fear to warn them of dangersE. It's good to have a healthy sense of what's dangerousF. Fear is a tool that the brain uses to keep the body safeG. Memories are an important factor in fear-lasting experiencesSymptoms of Dehydration(脱水)You’re bad- tempered.Researchers tested the mood and concentration of 25 women who drank healthy amounts of water one day, and then didn’t over the next two days. When slightl y dehydrated, the women reported tiredness, bad temper, headaches, and difficulty focusing. In a separate test, men with mild dehydration also had trouble with mental tasks. 11．Scientists are still trying to figure out why.You have a bad workout.12．It impacts how much you can push yourself. Even a 2 to 3 percent fluid loss affects your ability to get a good workout and more than 5 percent dehydration decreases exercise capacity by about 30 percent. A no-brainer way to avoid dehydration and dehydration symptoms is to aim for this much water per day.13．Driving while dehydrated may be just as dangerous as getting behind the wheel drunk, in terms of how many mistakes you could make on the road. British researchers had participants take two-hour drives [using a simulator (模拟器). When they drank enough water, there were 47 driving errors. 14．You feel dizzy when you stand up too fast.One of the signs of dehydration is when your blood volume and pressure drops, it can leave you feeling dizzy or faint, or bring on that rush of light-headedness after you quickly get up from sitting or lying down.The exact treatment for dehydration symptoms depends on age and how severely dehydrated someone is. 15．Most of the time, however, people use some over-the-counter solutions for kids, and adults can drink more waterA．You drive like you’re drunk.B．Dehydration can be life-threatening.C．To get rid of dehydration you have to replace lost water.D．In extreme cases, people might go to the hospital for a treatment.E. Dehydration reduces blood pressure and makes the heart work harder.F. But when it came to mood changes, women changed much more than men’s.G. But when they are short of water, there were more than doubled driving errors to 101.The Changing Diet in North AmericaWhat do most Americans and Canadians usually eat? Many people think that the typical North American diet consists of fast food—hamburgers, hot dogs, French fries, pizza, fried chicken, and so on. They think Americans and Canadians also eat a lot of convenience foods, usually frozen or canned, and junk food—candy, cookies, potato chips, and other things without much nutritional value. 16．The American diet is generally high in sugar, salt, fat, and cholesterol(胆固醇) and these can cause health problems.However, some people’s eating habits are changing. 17．North Americans are eating less red meat and fewer eggs, and they are eating more chicken and fish. Chicken and fish contain less fat than meat and eggs. Many people are also buying more fresh vegetables and eating them raw or cooked quickly in very little water in order to keep vitamins.18．The “typical” North American diet now includes food from many different countries. More restaurants are opening in big cities in the United States and Canada. Foods from China, Japan, Korea, Thailand, India, and the Middle East are very popular. Even fast food places now offer “lean” (low-fat) hamburgers, roasted (instead of fried)chicken, and salad bars with a wide variety of fresh fruits and vegetables.19．Because we now know about the importance of nutrition, we will probably continue to eat more fish and vegetables and less meat. We will still buy convenience foods in supermarkets, but frozen foods may be more nutritious and canned foods may have less salt and sugar. 20．Instead of candy bars we will eat “nutrition bars” with a lot of vitamins and protein. In the future, our diet will probably be even more interesting and healthier than it is now.A．Our junk food will not be “junk” at all.B．How are we going to eat in the future?C．Unfortunately, this description is not totally incorrect.D．The nature of American diet has changed a lot in this century.E. Restaurant menus are also changing to reflect people’s growing concern with good nutrition.F. We can choose foods from a variety of sources, control the quantities that we eat, and limit fats.G. They are becoming more interested in good health, and nutrition is an important part of health.参考答案1．D2．A3．F4．G5．B本文为说明文，给出了几种方法来帮助读者记忆阅读过的内容，包括扫清生词障碍，将所读到的内容与已有知识建立联系，把自己置身于情节之中和经常性阅读。

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a r X i v :q uant-ph/01234v17D ec2Exact solutions for classical few-body systems from the multichannel quantum inverse problem Zakhariev B.N.,Chabanov boratory of the Theoretical Physics,Joint Institute for Nuclear Research Dubna,141980,Russia;e-mail:zakharev@thsun1.jinr.ru;chabanov@thsun1.jinr.ru;homepage:http://thsun1.jinr.ru ∼zakharev Abstract A surprising ”duality”of the Newton equation with time-dependent forces and the stationary Schr¨o dinger equation is discussed.Wide classes of exact solutions not known before for few-body Newton equations are generated directly from exactly solvable multichannel models discovered in quantum mechanics due to the inverse problem and the supersymmetry (SUSYQ)approach.The application of this duality to the control of the stability (bifurcations)of classical motions is suggested.1Introduction In this paper the results related to the control of classical few-body systems are presented.We use the duality of corresponding solutions with the known exact solutions of quantum problems.The complete sets of such exactly solvable models have been recently intensively discussed [1,2].The transition between quantum and classical solutions is possible through simply renaming variables and functions.Physics is permanently developing in a tight interplay with mathematics (”the laws of nature are written in the mathematical language”).There are cases when the same mathematical equation can describe physical systems of diﬀerent nature,which broadens and deepens their understanding.We shall demonstrate this for the example of the multichannel formalism that is a powerful and universal tool in quantum inverse and direct problems of few-body physics [1,2,3].We start with brieﬂy reminding the one-dimensional and one-channel case.The corresponding stationary Schr¨o dinger equation−ψ′′(x )+v (x )ψ(x )=Eψ(x )(1)can be rewritten by changing the notationψ→z ;x →t ;d 2/dx 2→d 2/dt 2;[v (x )−E ]ψ→F (t,z )so that it will look like an equation of motion of a classical particle under the action of the force F (t,z )dependent on time t and coordinate z :¨z (t )=[v (t )−E ]z ≡F (t,z ).(2)It is a Newton equation:the acceleration of a particle moving along the z axis is determined by the force F (t,z )linearly dependent on z (an oscillator-type po-tential ∼z 2)with a time-dependent strength.In a classical equation,the former quantum energy E and the potential v (x )become,respectively,a parameter and a function characterizing the external force.This information was obtained at the School on Quantum Mechanics in Mex-ico,1998,from Prof.Bogdan Mielnik and Dr.David Fernandez and paper[4], see also references therein.It was surprising for us and particularly interesting because it means that complete sets of exactly solvable quantum models[2,5] simultaneously provide exact solutions to classical problems of quite a diﬀerent nature.It is also a”paradox”that the eigenvalues E n for stable bound states in quan-tum mechanics correspond to the points of bifurcations(absolute instability)of solutions for classical motion.Let us comment on this statement.At an energy slightly lower than a ground state energy level E n,the nonphysical solution of the Schr¨o dinger equation decreasing asymptotically to the left grows exponentially to the right.And somewhat above the energy level,the solution with the same asymptotic behavior on the left bends more strongly,acquires an additional node and grows exponentially to the right with another sign.In the classical case it corresponds to an instantaneous change of the regime for a particle motion when the parameter E goes through the point E n:a particle being initially at the origin in theﬁrst case goes to the right with increasing accelerations and in the second case it goes to the left.At the point E=E n the particle localized previ-ously at the origin after the inﬂuence of the time-depending externalﬁeld returns to the origin.So,the possibility of shifting,creating and removing arbitrary energy levels in quantum systems[1,2,5]corresponds in the classical case to the ability to control the bifurcation points which are the parameters of principal importance characterizing the instability of motion.Of course,at the same value E,a classical scattering with another initial(t→−∞)condition is possible corresponding to another dual quantum asymptotic (x→−∞)condition.In the quantum inverse problem,the potential is uniquely determined by a complete set of spectral parameters:energy level values,normalizing parame-ters(spectral weights)and phase shifts(or resonance positions if S-matrix has fractional-rational form[7])for scattering.Therefore,classical counterparts of these parameters(e.g.bifurcations)are control levers of the corresponding clas-sical system.This interesting fact was not so evident from the point of view of Newton equations.The intuition developed in exactly solvable quantum models[1,5]allows one to qualitatively predict also classical motion without computation.2Multichannel,multi-dimensional and few-body systemsThe multichannel systems[6,7,8,9,10]for vector-valued wave functions with partial channel componentsψi(x)are simply matrix generalizations of the one-dimensional Schr¨o dinger equation(1)−ψ′′i(x)+ j v ij(x)ψj(x)=E iψi(x),E i≡E−ǫi(3)where the interaction matrix v ij(x)replaces the ordinary potential andǫi are threshold energies that can be diﬀerent in partial channels.This system can be rewritten as multi-dimensional or few-body equations of classical mechanics with special forces.Let us replace the partial channel wave functionψi(x)and its space coordinate variable x in the system(3)by the coordinate z i of the i th classical particle and time:z i(t).Then the second derivativeψi(x)is substituted by the acceleration ¨z i and all other terms can be considered as the forces F i(t,z)dependent on time and space coordinates acting on the i th particle.So,elements of the potential matrix v ij(x)and channel energy terms E iψi(x)become constituents of the forcesF ij(t)= j v ij(t)z j(t)−E i z i(t).The whole system(3)becomes a system of equations for several classical particles,where the particle accelerations are determined by functions of the time-dependent forces¨z i= j v ij(t)z j(t)−E i z i(t)≡F i(t).(4) Again as in(2)the parameters E i are no longer the channel energies,but simply enter as parameters into the deﬁnition of the forces acting in the system.There-fore,every exact solution of the multichannel quantum problem(and complete sets of them)corresponds to an exact solution of the few-body classical problem with time-dependent forces.The asymptotic and boundary conditions of quan-tum problems determine initial andﬁnal conditions of the corresponding classical solutions.For the classical scattering solutions it is possible to use nonphysical quantum solutions growing linearly in asymptotic regions:ψ(x)=ax+b(for E i=0).The constants a and bﬁxing the inclination of the straight line and the position point of the node of the wave function determine in the classical problem the position and velocity of a particle in some moment of time(for instance,t=0): z=at+b.As an example of the prediction of a peculiar behavior of few-body systems, one can choose the dual model of multichannel exact solution when one of the partial channel wave functions is concentrated at the origin.This corresponds to increasing the spectral weight parameter of this partial channel C i≡dψi(x)dxM i M j e−(κi+κj)x2κme−2κm x,(5)κi=where M i stand for partial channel spectral parameters which are pre-exponential factors in the decreasing asymptotic tails of the bound state wave functions.An interested reader canﬁnd other formulae for a complete set of inverse problem and SUSYQ transformations in[2].The transformation of the three-channel system of type(3)by renaming:ψ1(x)→x(t),ψ2→y(t),ψ3→z(t)gives us the classical equations for the three-dimensional motion of a particle.As interesting instructive examples can be considered the classical counter-parts of quantum systems with”paradoxical”coexistence of bound and scattering states at the same energy[5],1999or combinations of absolute transparency and strong reﬂection for diﬀerent linearly independent solutions of multichannel equa-tions.It is possible to signiﬁcantly extend the class of exactly solvable models if we introduce,into the quantum equations nonhomogeneous terms which can be treated as sources with an arbitrary dependence on coordinates.From the clas-sical point of view,they correspond to forces F n(t)which are independent of coordinates.Having the sets of independent exact solutions of the homogeneous equations we get the expressions for Green functions.They give explicit analytic expressions for the solutions of the initial nonhomogeneous equations.The quantum multichannel systems obtained by adiabatic expansions have derivatives ofﬁrst order which can be interpreted as friction forces depending on the velocity.But exact solutions of these systems remain an open problem yet. 3ConclusionThe authors consider it a pleasure and an honor to participate in this issue of the Few-Body Systems dedicated to Prof.W.Gl¨o ckle,the scientist having record results in this fundamentalﬁeld of quantum physics.Already hisﬁrst work[11] produced a strong impression on us.It was a signiﬁcant contribution to the multichannel formalism.Prof.W.Gl¨o ckle also stimulated our investigations on qualitative theory of quantum design reviewed later in the books”Lessons on Quantum Intuition”,1”New ABC of Quantum Mechanics(in pictures)”2and in a series of review articles[1,2].We appreciate very much that Prof.W.Gl¨o ckle was one of theﬁrst physicists in the Western Europe who organized,at Ruhr University at his seminar consideration of our theory.4AcknowledgmentsThe authors are thankful to Prof.Bogdan Mielnik and Dr.David Fernandez for the information about the previous works on classical interpretation of the quantum equations and to Prof.Man’ko for discussions of these problems.Thework of Chabanov V.M.was supported in part by the INTAS grant96-0457 within the research program of the International Center for Fundamental Physics in Moscow.References[1]Zakhariev,B.N.:Few-Body Systems4,25(1988);Lessons on Quantum Intuition,Dubna,JINR,1996,(in Russian);New ABC of Quantum Mechanics.Izhevsk,USU,1998,(in Russian). 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