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[ "A helicopter circled overhead. ", "Riot police clad in protective gear and balaclavas lined the sidewalks. ", "Columns of police wagons sped past. ", "There was a sense of foreboding in the air in central Moscow on Saturday afternoon as the authorities prepared for the latest demonstration to demand that opposition candidates be allowed to run in the upcoming Moscow City Duma elections. ", "That was especially the case when about a dozen policemen detained Lyubov Sobol, a barred candidate who has become one of the main leaders of the three weeks of protests, as soon as she left her building around 2 p.m. local time to head for what she said would be a walk through central Moscow along the tree-lined Boulevard Ring. “", "Hello to all the brave Muscovites who came out today! ", "Proud of you!” ", "Sobol, who is on the 21st day of a hunger strike, wrote on Twitter several hours later from a police station after being driven around for several hours without access to her phone, she said.", "\n\nAP / TASS\n\nThe protests have roiled Moscow since July 14, when Sobol and other prominent opposition candidates — including fellow allies of opposition leader Alexei Navalny Ivan Zhdanov and Ilya Yashin, as well as Dmitry Gudkov, a prominent critic of Russian President Vladimir Putin — were barred from running. ", "To make it on to the ballot for the Sept. 8 vote, potential candidates had to each collect around 5,000 signatures, depending on the district, from city residents. ", "But over the first days of July, opposition politicians had some of the names they collected invalidated by election officials. ", "They argue that the authorities have barred them from running to stop them from challenging Kremlin-backed incumbents. ", "Over the last 21 days, protesters have mobilized for regular street rallies demanding opposition candidates be allowed to run. ", "At one point, the authorities offered the protesters an outlet to vent their anger by approving a July 20 demonstration on Prospekt Sakharova, a location outside the city center. ", "But that protest drew 22,500 people — making it the largest opposition-led demonstration in recent years. ", "Since then, the authorities have cracked down — hard.", "\n\nhttps://twitter.com/Domozhiroff/status/1157657088941858817\n\nAfter the Investigative Committee opened a criminal investigation into unsanctioned protests, they began night raids on the opposition politicians who were calling on protesters to take part in unauthorized demonstrations. ", "Opposition leader Navalny and ex-state Duma deputy Gudkov have since been handed 30-day prison sentences, while Zhdanov, the director of Navalny’s Anti-Corruption Foundation, has been sentenced to 15 days and municipal deputy Yashin to 12. ", "During last weekend’s protest, police also detained nearly 1,400 people, while the courts — which heard more than 1,000 cases this week — have come down strongly on them. ", "Some 88 people are now facing prison time, while another 330 have been fined. ", "One student faces up to eight years behind bars. ", "This week, Moscow Mayor Sergei Sobyanin had warned the opposition that he would not allow the protest movement to plunge Russia’s capital into chaos. “", "We don’t live in Zimbabwe,” the Putin ally said in an interview on state television, accusing the opposition of having “a desire to usurp power illegitimately.”", "\n\nYet people have continued to come out. ", "Each week, Facebook events advertising the protests have seen increasing interest, and some of Russia’s most prominent celebrities have spoken out in support of the movement. ", "By the time protest kicked off at 2 p.m. local time Saturday, more than 20,000 people had said they were interested in going or would be attending. ", "Another 1,000 gathered in St. Petersburg in support of the Moscow protests, according to local police there. ", "Protesters and journalists, including a Moscow Times reporter, reported having difficulties connecting to the internet along the demonstration route. ", "Last fall, the authorities ordered two mobile operators to cut access to mobile data services in the region of Ingushetia during protests there. ", "Moscow city officials also on Wednesday announced a shish kebab-themed music festival for Saturday, with social media influencers urging people Saturday morning and afternoon to attend. ", "Activists were not deterred — some even travelled to Moscow from other cities, including Volgorad some 1,000 kilometers to the southeast. “", "I needed to see for myself what these protests were about because we don’t see the real thing on television,” said Dmitry Lungu, 22. “", "What I see is a large number of people who believe it’s time for Putin to leave office.”", "\n\nA helicopter circled the protest route in Moscow Saturday afternoon. ", "Valery Sharifulin / Tass\n\nBecause riot police blocked off much of the planned protest route, forcing protesters to turn down side streets, it was difficult to estimate numbers of demonstrators Saturday. ", "Moscow police estimated that 1,500 turned out. ", "For their part, the protesters repeatedly broke out into cheers and claps, accompanied by cars that honked as they passed. ", "At times they chanted familiar slogans like “Russia will be free” and “Sobyanin resign.” ", "Many were detained — 1,001, according to the OVD Info police watchdog. “", "It’s a political crisis for the authorities,” said political analyst Yekaterina Schulmann. “", "So far they are using the methods and instruments they know — we haven’t seen anything new from them, just higher numbers.” “", "But it would also be dumb to disregard the cooling effect that a violent crackdown can have on protests,” she added." ]

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[ "Call Of Cthulhu - Depths of Madness Trailer\n\nCyanide Studioのナラティブデザイナー，マクシミリアン・ルッツ氏\n\nFocus Home Interactiveは，2016年に制作が発表されたホラーRPGのデモプレイを，同社が本拠地を置くパリで開催したイベントで初公開した。開発はフランスのCyanide Studioが担当し，2017年第4四半期の発売も新たに発表された。対応機種は PC および PlayStatioin 4 Xbox One が予定されている。H.P.ラヴクラフトの古典怪奇小説と，それをベースにしたテーブルトップRPGをライセンスした本作は，北米ニューイングランド地方にある架空の孤島を舞台にした物語が描かれる。時は1920年代，女性芸術家のが自分の邸宅で謎の死を遂げ，その調査を依頼された私立探偵がボストンからやってくる。そして，家族や島を覆うミステリーを解き明かしていくことになる。ゲーム画面は主人公の一人称視点で描かれ，プレイヤーは探索やパズル，そしてNPC達との会話などを通してストーリーを進めていくのだ。今回のデモでは，ピアースが島に降り立ったばかりの冒頭のシーンが紹介されており，事故現場となったホーキンス邸に鍵がかけられているために，内部に潜入する方法を考え出さなければならないという設定だった。鍵はという庭師の男が持っているらしいが，主人公を不審に感じたのか，会話をしてもなかなか信用してくれない。会話は，そしてなどから適切だと思うものをダイヤルで選択するシステムだ。波止場から邸宅にやってくる道すがら，サラの墓に花が供えられているのを見たので，これはおそらくウィンチェスターが供えたものと判断。ウィンチェスターはサラに好意を寄せていたらしいことから，「説得」を繰り返して鍵をもらうというステップを踏み，なんとか邸宅内に入った。もっとも，どのような手法で潜入するのかはプレイヤー次第であり，スキルを活かして脅迫してみるといった会話の選択や，庭師の家から鍵を盗む，あるいは裏口を見つけるといったこともできるという。本作は，1981年にリリースされたChaosiumのテーブルトップRPG「Call of Cthulhu」のルールセットを参考にしており，中でも特徴的なのが，と呼ばれるパラメータだ。これは，プレイヤーが恐怖を感じた体験や場所から，特定の恐怖症を引き起こすというもの。暗所，閉所，高所といったいくつかのフォビアが存在し，Sanityゲージが増えていくと，ストーリーの重要なシーンで恐怖症が発生してゲームを進めにくくなる。薬を飲むなど，恐怖感を鎮めるための対処法も存在するが，数あるホラーものの中でもユニークなシステムになりそうだ。今回のデモでも，ホーキンス邸の美術館のような場所で，絵から異様なクリーチャーが抜け出てくるというシーンが紹介されていたが，見つからないよう隠れたピアースの心拍数が上がり，どんどんと恐怖心が増していくのが感じられた。ピアーズはタンスの中にいるのだが，このデモでは閉所恐怖症という設定だったのか，一定の時間以上同じ場所にいると錯乱して声が出るというリスクがある。そのため，時間に追われつつ，クリーチャーがいないスキをうかがって別の場所に移動しなくてはならない。本作の脚本を担当するCyanide Studioのナラティブデザイナー，によれば，主人公は配下の探偵達を使ってを編成し，それぞれの能力に合わせて彼らにミッションを委任するといったシステムも導入される予定だという。失敗するとどうなるのか，すべてのミッションをプレイヤーが直接プレイできるのか，といった不明点も少なくないが，ルッツ氏は，プレイヤーが背景となる物語をより深く知ることができる，さまざまな仕掛けを用意しているとも述べていた。最近はゲーム実況のネタとして人気を博しているホラージャンルだが，「Call of Cthulhu: The Official Video Game」は，独自性の強いさまざまなシステムの導入により，他人がプレイする様子を鑑賞するだけではなく，自分で遊んでみたいと思わせる作品に仕上がりそうだ。新情報の公開に期待したい。" ]

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[ "Our Profile\n\nJyoti Speed Packers and Movers in Indore are well known packers & movers service Provider Company in Indore, They provide excellent packing and moving services. ", "We are the regular, consistent packers movers agency in Indore, India in addition to we'll get out of the packing and moving manufactured goods and equipment with greatest care and transport on time." ]

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[ "\n87 F.Supp.2d 1295 (2000)\nCharles D. MEDALIE, and Janet Medalie, Plaintiffs,\nv.\nFSC SECURITIES CORPORATION, Defendant.", "\nNo. ", "98-3183-CIV.", "\nUnited States District Court, S.D. Florida.", "\nFebruary 1, 2000.", "\n*1296 *1297 Robert B. Miller, Bedzow Korn Brown, Hallendale, FL, for Plaintiffs.", "\nJames Kaplan, Wilson Elser Moskowitz, Miami, FL, for Defendant.", "\n\nORDER GRANTING IN PART AND DENYING IN PART DEFENDANT'S MOTION TO DISMISS AND GRANTING DEFENDANT'S MOTION FOR A MORE DEFINITE STATEMENT\nGOLD, District Judge.", "\nTHIS CAUSE is before the court upon defendant FSC Securities Corporation's motion to dismiss (DE # 16-1, 16-2). ", "Plaintiffs Charles D. Medalie and Janet Medalie were solicited from late 1984 until February 1985 by George Bliss, a representative of FSC, to purchase investments in limited partnerships. ", "Bliss visited George Medalie in school, where he worked, and later both plaintiffs at their home. ", "Plaintiffs allege that Bliss represented the investments would be safe, low risk, have a higher rate of return than money markets or certificates of deposit, and that the investments would be suitable for teachers such as the Medalies. ", "Based on Bliss's representations, the Medalies made five limited partnership purchases on February 5, 1985 and two purchases on February 12, 1987. ", "Plaintiffs' total investment amounted to $157,575.00. ", "Plaintiffs allege that after the purchases, neither Bliss nor FSC reported to them the true value or performance of their investments and represented to them that the investments were \"fine\" and \"doing great.\" ", "Plaintiffs continued to hold these investments \"to their financial detriment.\"", "\nPlaintiffs filed a four count complaint against the defendant alleging violation of § 517.301 of the Florida Securities and Investor Protection Act (Count I), breach of fiduciary duty (Count II), fraud in the inducement (Count III), and breach of contract (Count IV). ", "Plaintiffs' complaint was originally filed in the Circuit Court for the Eleventh Judicial Circuit in and for Miami-Dade County, Florida. ", "The case was timely removed to this court based on the court's diversity jurisdiction. ", "Plaintiffs are citizens and residents of New York and defendant is a Georgia corporation, with its principal place of business in Georgia. ", "The solicitations in this case took place in Waverly, New York. ", "FSC contractually consented to venue, personal jurisdiction, and subject matter jurisdiction in Dade County, Florida. ", "By arguing their motions based on Florida law, the parties have stipulated that Florida law applies in this case.[1] Neither party raised a choice-of-law or conflicts of law problem. ", "Defendant now moves to dismiss all counts, and if count IV for breach of contract, is not *1298 dismissed, defendant moves for a more definite statement on count IV.", "\n\nI. Standard of Review for a Motion to Dismiss under Rule 12(b)(6)\nTo warrant dismissal of a complaint under Rule 12(b)(6) of the Federal Rules of Civil procedure, it must be \"clear that no relief could be granted under any set of facts that could be proved consistent with the allegations.\" ", "Blackston v. Alabama, 30 F.3d 117, 120 (11th Cir.1994) (quoting Hishon v. King & Spalding, 467 U.S. 69, 73, 104 S.Ct. ", "2229, 2232, 81 L.Ed.2d 59 (1984)). ", "Determining the propriety of granting a motion to dismiss requires courts to accept all the factual allegations in the complaint as true and to evaluate all inferences derived from those facts in the light most favorable to the plaintiff. ", "See Hunnings v. Texaco, Inc., 29 F.3d 1480, 1483 (11th Cir.1994). ", "The threshold of sufficiency that a complaint must meet to survive a motion to dismiss is exceedingly low. ", "See Ancata v. Prison Health Servs., ", "Inc., 769 F.2d 700, 703 (11th Cir.1985) (citation omitted); Jackam v. Hospital Corp. of America Mideast, Ltd., 800 F.2d 1577, 1579 (11th Cir.1986). \"[", "U]nless it appears beyond doubt that the plaintiff can prove no set of facts in support of his claim which would entitle him to relief,\" the complaint should not be dismissed on grounds that it fails to state a claim upon which relief can be granted. ", "Sea Vessel, Inc. v. Reyes, 23 F.3d 345, 347 (11th Cir.1994) (citation omitted). ", "Nevertheless, to survive a motion to dismiss, a plaintiff must do more than merely \"label\" his claims. ", "Blumel v. Mylander, 919 F.Supp. ", "423, 425 (M.D.Fla.1996). ", "Moreover, when on the basis of a dispositive issue of law no construction of the factual allegations will support the cause of action, dismissal of the complaint is appropriate. ", "Marshall County Bd. ", "of Educ. ", "v. Marshall County Gas Dist., ", "992 F.2d 1171, 1174 (11th Cir.1993).", "\n\nII. ", "Count I Securities Fraud — Violation of § 517.301, Florida Statutes\nCount I of plaintiff's complaint alleges a violation of § 517.301, Florida Statutes. ", "Defendant moves to dismiss based on the argument that Count I is barred by the statute of limitations. ", "The statute of limitations applicable to Chapter 517 is set forth in § 95.11(4)(e), Florida Statutes, which provides:\nActions other than for recovery of real property shall be commenced as follows:\n...\n4) Within two years. ", "—\n...\ne) An action founded upon a violation of any provision of chapter 517, with the period running from the time the facts giving rise to the cause of action were discovered or should have been discovered with the exercise of due diligence, but not more than 5 years from the date such violation occurred....\nThis statute has been held to bar any action for violation of Chapter 517, Florida Statutes, commenced after the expiration of five years from the date of the violation regardless of whether the facts giving rise to the cause of action were known by the plaintiff. ", "In Wilder v. Meyer, 779 F.Supp. ", "164, 167-68 (S.D.Fla.1991), the court held that fraudulent concealment would not serve to extend the five-year bar provided by the statute of limitations in § 95.11(4)(e) in a § 517.301 action. ", "The Wilder court relied on the Eleventh Circuit's holding in Cook v. Deltona Corp., 753 F.2d 1552, 1562 (11th Cir.1985):\nwhere the statute clearly provides for a tolling period for a fraudulent concealment, and then includes a secondary date which \"in no event\" can be surmounted, there is good basis for belief that the latter date was intended as an absolute barrier to the filing of suit.", "\nIn Cook, the court was interpreting language in the Interstate Land Sales Full Disclosure Act (\"the Act\"), 15 U.S.C. § 1701, et seq. ", "The pre-1979 § 1711 of the Act, providing the limitations period, contains a similar bifurcated limitations scheme as is found in § 95.11(4)(e). ", "The Cook court read the statute as manifesting a legislative intent to avoid equitable tolling *1299 for the secondary limitations period, while retaining the possibility of equitable tolling for the shorter primary limitations period. ", "See id. at 1562.", "\nFirst, looking a the face of the complaint, plaintiffs allege that they did not discover the fraud committed against them until \"mid-1996.\" ", "Amend. ", "Complaint at ¶¶ 25, 41. ", "The original state court action was filed on December 9, 1998, over two years after \"mid-1996.\" ", "See Defendant's Notice of Removal at 1 and attached State Court Complaint. ", "The question is therefore whether the complaint was filed less than \"five years from the date such violation occurred.\" ", "The answer to that question depends on when the violation is deemed to have occurred. ", "If the violation is deemed to have occurred when the securities were actually purchased, then the complaint was filed beyond the five year limitations period and the claim is barred by § 95.11(4)(e). ", "The Amended Complaint indicates that the securities purchases occurred on February 5, 1985 and on February 12, 1987. ", "These purchases occurred more than 11 years before the complaint in this case was filed. ", "See Amend. ", "Complaint at ¶ 20. ", "On the other hand, plaintiffs argue that the violation should be deemed to be a continuous tort that was fraudulently concealed by the defendant. ", "In this case, plaintiffs argue, the defendant's ongoing fraudulent misrepresentations and violations of fiduciary duties would extend the limitations period beyond the date of the original purchase and place it within the five year limitations period of § 95.11(4)(e). ", "The court is unpersuaded by plaintiffs' argument because ample case law suggests that the five year limit is an absolute bar which cannot be extended though equitable tolling, continuous torts, ongoing misrepresentations, or violations of fiduciary duties.[2]See Wilder and Cook, supra.", "\nMoreover, the cases show that the cause of action accrues for purposes of the five year limitations period at the time the securities were purchased. ", "See Byrne v. Gulfstream First Bank & Trust Co. of Boca Raton, 528 F.Supp. ", "692, 695 (S.D.Fla.1981) (determining for purposes of a Section 10(b) and Rule 10b-5 action applying the limitations period in Fla.Stat. § ", "95.11(4)(e) that the cause of action arose when the transaction transferring the securities at issue to the defendant was completed, and the action was therefore barred, even though the fraud was allegedly discovered within 2 years of the filing of the complaint, because 10 years had passed since the transaction was completed); see also Armbrister v. Roland Int'l Corp., 667 F.Supp. ", "802, 823 (M.D.Fla.1987) (\"The right of action provided arises when a sale is made in violation of the Chapter's [517] provisions. ", "The `violation,' then, that triggers the limitations period [of § 95.11(4)(e)] occurs when the purchaser contracts to buy the security.\"); ", "Wilder, supra. ", "Accordingly, because the allegedly fraudulent conduct was discovered more than two years after the complaint was filed and because the securities were purchased more than 11 years before the complaint was filed, Count I is barred by the statute of limitations in § 95.11(4)(e) and is therefore dismissed.[3]\n\nIII. ", "Counts II and III—Breach of Fiduciary Duty and Fraud in the Inducement\n\nA. The Economic Loss Rule\nFlorida's economic loss rule was originally enunciated by the Supreme Court of Florida in the cases of Florida Power & Light Co. v. Westinghouse Elec., ", "510 So.2d 899 (Fla.1987) and AFM Corp. v. Southern Bell, 515 So.2d 180 (Fla.1987). ", "The doctrine provides that contract principles are more appropriate than tort principles for resolving economic loss claims without accompanying physical injury or *1300 injury to property other than that which is the subject of the contract. ", "Consequently, a party to a contract may not pursue a claim in tort for solely economic losses unless the party breaching the contract has committed a tort which is distinguishable from or independent of the breach of contract. ", "Strickland-Collins Constr. ", "v. Barnett Bank of Naples, 545 So.2d 476 (Fla. 2d DCA 1989); Keys Jeep Eagle, Inc. v. Chrysler Corp., 897 F.Supp. ", "1437, 1443 (S.D.Fla.1995).", "\nThe economic loss rule is grounded on the basic difference between contract law—which protects expectations—and tort law—which is determined by the duty owed by all persons to others in society. ", "Casa Clara Condo. ", "Ass'n v. Charley Toppino & Sons, 620 So.2d 1244 (Fla.1993). ", "Economic loss has been defined as \"damages for inadequate value, costs of repair and replacement of the defective product, or consequent loss of profits—without any claim of personal injury or damage to other property.\" ", "Id. Economic losses \"are `disappointed economic expectations,' which are protected by contract, rather than tort law.\" ", "Casa Clara, 620 So.2d at 1246 (quoting Sensenbrenner v. Rust, Orling & Neale Architects, Inc., 236 Va. 419, 374 S.E.2d 55, 58 (1988)). ", "For recovery in tort \"there must be a showing of harm above and beyond disappointed expectations.\" ", "Id. Consequently, when the alleged duty breached is derived from the contractual relationship it cannot form the basis for a separate and distinct tort. ", "Interstate Sec. ", "Corp. v. Hayes Corp., 920 F.2d 769 (11th Cir.1991).", "\n\nB. Breach of Fiduciary Duty\n\n1. ", "Moransais—A New View of the Economic Loss Rule\nDefendant moves to dismiss plaintiffs' breach of fiduciary duty claim based on the argument that the claim is barred by the economic loss rule. ", "In Hayes, the Eleventh Circuit held that a claim for breach of fiduciary duty is barred by the economic loss rule, stating that \"we hold that if a fraud claim is foreclosed as a matter of Florida law under the AFM doctrine, a claim for breach of fiduciary duty must be prohibited as well.\" ", "Hayes, 920 F.2d at 777. ", "Significantly, the Hayes case, just as this case, involved an investor suing a securities broker for fraud. ", "The 1991 Hayes decision would therefore appear to be dispositive of the issue, however, on July 1, 1999, the Florida Supreme Court issued a new pronouncement on the application of the economic loss rule in Moransais v. Heathman, 744 So.2d 973 (Fla.1999). ", "In Moransais, the court held that \"the economic loss rule does not bar a cause of action against a professional for his or her negligence even though the damages are purely economic in nature and the aggrieved party has entered into a contract with the professional's employer.\" ", "744 So.2d at 983-84. ", "The court further held that \"Florida recognizes a common law cause of action against professionals based on their acts of negligence despite the lack of a direct contract between the professional and the aggrieved party.\" ", "Id.\nIn reaching its decision, the Florida Supreme Court engaged in a fundamental reassessment of how the economic loss rule has evolved in Florida and whether it has strayed from its original purpose. ", "See id. at 980 (\"Unfortunately, however, our subsequent holdings have appeared to expand the application of the rule beyond its principled origins and have contributed to application of the rule by trial and appellate courts to situations well beyond our original intent.\") ", "and id. at 980 (\"We must acknowledge that our pronouncements on the rule have not always been clear and, accordingly, have been the subject of legitimate criticism and commentary.\"). ", "The court concluded that the economic loss rule was being applied too broadly and was effectively extinguishing traditional tort causes of action.[4] Therefore, the scope of *1301 the economic loss rule should be narrowed to product liability actions and other similar contexts:\nToday, we again emphasize that by recognizing that the economic loss rule may have some genuine, but limited, value in our damages law, we never intended to bar well-established common law causes of action, such as those for neglect in providing professional services. ", "Rather, the rule was primarily intended to limit actions in the product liability context, and its application should generally be limited to those contexts or situations where the policy considerations are substantially identical to those underlying the product liability-type analysis.", "\nId. at 983.", "\nWhile the holding of this case was limited to professional malpractice, the court implied that other torts would also not be covered by the newly clarified economic loss rule. ", "See id. (\"The rule, in any case, should not be invoked to bar well-established causes of actions in tort, such as professional malpractice.\"). ", "Shortly after the Moransais decision, the Third District Court of Appeal held that \"the Moransais opinion makes it clear that the economic loss rule has not abolished the cause of action for breach of fiduciary duty, even if there is an underlying oral or written judgment on this issue.\" ", "First Equity Corp. of Florida, Inc. v. Watkins, Nos. ", "98-851, 98-589, 1999 WL 542639 (Fla. 3d DCA July 28, 1999). ", "The facts of First Equity are essentially the same as in this case: a securities broker introduced Watkins to an investment opportunity and entered into oral and written contracts. ", "The investment ultimately failed and Watkins brought suit contending that First Equity had misrepresented material facts regarding the investment. ", "See id. at * 1. ", "Noting that breach of fiduciary duty is a \"just such a well-established cause of action in tort\" the Moransais court contemplated, First Equity held the cause of action was not barred by the economic loss rule.", "\nThe First Equity holding therefore appears to be in direct conflict with the Eleventh Circuit's holding in Hayes. ", "The First Equity court recognized the conflict stating:\nWe recognize that the United States Court of Appeals for the Eleventh Circuit has held that the Florida economic loss rule bars a breach of fiduciary claim. ", "See Interstate Securities Corp. v. Hayes Corp., 920 F.2d 769, 776-77 (11th Cir.1991). ", "At the time of that decision, Florida law was not entirely clear. ", "See Moransais, 24 Fla.L. Weekly at S311, [744 So.2d at 980] (\"We must acknowledge that our pronouncements on the rule have not always been clear....\"). ", "After Moransais, Interstate Securities and its progeny cannot be regarded as good law on this point.", "\nId. at * 2 n. ***.", "\nBefore reaching the question of whether the court should apply First Equity instead of Hayes, the court must answer the question of under what circumstances it can depart from Eleventh Circuit precedent.", "\n\n2. ", "Departure from Eleventh Circuit Precedent to Respond to a Change in State Law\nIn cases controlled by state law, federal courts are bound to follow the decisions of the highest court of the state. ", "Huddleston v. Dwyer, 322 U.S. 232, 64 S.Ct. ", "1015, 88 L.Ed. ", "1246 (1944). ", "If the highest state court has not addressed the issue, federal courts should ascertain and apply state law as pronounced by intermediate state appellate courts. ", "Fidelity Union Trust Co. v. Field, 311 U.S. 169, 61 S.Ct. ", "176, 85 L.Ed. ", "109 (1940), reh. ", "den. ", "311 U.S. 730, 61 S.Ct. ", "438, 85 L.Ed. ", "475 and reh. ", "den. ", "314 U.S. 709, 62 S.Ct. ", "118, 86 L.Ed. ", "565 (1941). ", "Federal courts should view an intermediate state court's opinion as \"indicia of the leanings of the state's highest court,\" and follow those \"leanings,\" unless the federal court is convinced *1302 that the state supreme court would reach a different conclusion. ", "Daigle v. Shell Oil Co., 972 F.2d 1527 (10th Cir. ", "1992). ", "See also West v. American Tel. & ", "Tel. ", "Co., 311 U.S. 223, 236-37, 61 S.Ct. ", "179, 183, 85 L.Ed. ", "139 (1940).", "\nLooking to the Eleventh Circuit for guidance, the court appears to be bound to follow state court rulings on issues of state law, particularly where a state court specifically contradicts an earlier interpretation of state law by a federal court. ", "Nussbaum v. Mortgage Service America Co., 913 F.Supp. ", "1548, 1554 (S.D.Fla.1995) (citing Roboserve, Ltd. v. Tom's Foods, Inc., 940 F.2d 1441, 1451 (11th Cir.1991)). ", "The Eleventh Circuit is not alone in considering a more recent interpretation of state law by state courts to justify reconsideration of circuit precedent. ", "See also Jones-Hamilton v. Beazer Materials & Services, 973 F.2d 688, 696 n. 4 (9th Cir.1992); Derflinger v. Ford Motor Co., 866 F.2d 107, 110 (4th Cir.1989); Broussard v. Southern Pac. ", "Transp. ", "Co., 665 F.2d 1387 (5th Cir. ", "1982); Singletary v. Southeastern Freight Lines, Inc., 833 F.Supp. ", "917 (N.D.Ga.1993) (\"a federal court should follow the latest appropriate decision at whatever point in the federal proceedings it comes.\"). ", "The Ninth Circuit requires federal courts to reevaluate the prior federal decision in light of the more recently handed down state decision where a state court has made a subsequent pronouncement of state law. ", "Owen v. United States, 713 F.2d 1461 (9th Cir.1983). ", "In that case, a Ninth Circuit decision was later contradicted by a state court of appeals decision. ", "In evaluating the appropriate course of action, the Court noted that,\n[t]hese recent decisions by the California courts of appeal that have appeared subsequent to our Commercial Union [Insurance Co. v. Ford Motor Co., 640 F.2d 210 (9th Cir.1981)] decision requires us to reconsider the proper interpretation of § 877. ", "Our interpretation in Commercial Union was only binding in the absence of any subsequent indication from the California courts that our interpretation was incorrect.", "\nId. at 1464. ", "In that circuit, if the highest court of the state has not ruled on an issue, federal courts are directed to follow the intermediate appellate courts of the state unless convinced that the highest court would decide differently. ", "Id. Similarly, other circuits have found that \"[w]hen a conflict exists between holdings of the ... Circuit and more recent determinations of state appellate courts, the interpretation of the Circuit is not binding on federal district courts.\" ", "In re New York Asbestos Litig., ", "847 F.Supp. ", "1086, 1111 (S.D.N.Y.1994). ", "In that situation, the federal court is directed to follow the outcome it believes the highest court of the state would follow. ", "In re E. & S. Dists. ", "Asbestos Litig., ", "772 F.Supp. ", "1380, 1391 (E.D.N.Y.1991).", "\n\n3. ", "Which Precedent to Follow—First Equity or Hayes?", "\n\nIn light of Eleventh Circuit authority and the similar practices in other circuits, it appears that this court could follow First Equity instead of Hayes. ", "Defendant, however, argues that Moransais and First Equity are both distinguishable. ", "The holding of Moransais specifically applied only to professionals, stating \"the economic loss rule does not bar a cause of action against a professional for his or her negligence even though the damages are purely economic in nature and the aggrieved party has entered into a contract with the professional's employer.\" ", "Moransais, 744 So.2d at 983-84 (emphasis added). ", "The Moransais court did engage in an inquiry to determine what sort of occupation would be defined as \"profession,\" and it concluded that a \"profession\" is \"any vocation requiring at a minimum a four-year college degree before licensing is possible in Florida.\" ", "Id. at 976 (citing § 95.11(4)(a), Fla. Stat. (", "1997) and Garden v. Frier, 602 So.2d 1273, 1275 (Fla.1992)). ", "Because a securities broker is not required to obtain a four year degree for licensing in Florida, defendant argues that the Moransais holding does not apply in this case.", "\n*1303 The plaintiffs in Moransais sued a professional engineering corporation for breach of contract for failing to discover defects in the home the corporation inspected. ", "The plaintiffs also sued the two engineers that performed the inspection for professional negligence. ", "The Medalies, however, are not suing the defendant for professional negligence, so the holding of Moransais is not directly applicable to their case. ", "However, the Moransais court did make two significant statements in dicta to guide the lower courts in future cases involving the economic loss rule. ", "First, the court directly stated that professional negligence is not the only cause of action that can be maintained despite the economic loss rule — other well-established causes of action in tort also survive. ", "See id. at 983 (\"Today, we again emphasize that by recognizing that the economic loss rule may have some genuine, but limited, value in our damages law, we never intended to bar well-established common law causes of action, such as those for neglect in providing professional services.... The rule, in any case, should not be invoked to bar well-established causes of action in tort, such as professional malpractice.\"). ", "The court also significantly narrowed the application of the economic loss rule to product liability-type cases. ", "See id. (\"[T]he rule was primarily intended to limit actions in the product liability context, and its application should generally be limited to those contexts or situations where the policy considerations are substantially identical to those underlying the product liability-type analysis.\").", "\nThe real problem, which none of the parties address, is the fact that the Supreme Court of Florida did not overturn its decision in AFM. ", "After briefly discussing the factual background and holding of the AFM case, the Florida Supreme Court stated, \"While we continue to believe the outcome of that case [AFM] is sound, we may have been unnecessarily over-expansive in our reliance on the economic loss rule as opposed to fundamental contractual principles.", "\"[5]Id. ", "at 980-81. ", "AFM was heard by the Florida Supreme Court on certification of questions regarding the economic loss rule from the Eleventh Circuit in AFM Corp. v. Southern Bell Tel. ", "and Tel. ", "Co., 796 F.2d 1467 (11th Cir.1986). ", "The Supreme Court of Florida restated the Eleventh Circuit's questions as: \"Does Florida permit a purchaser of services to recover economic losses in tort without a claim for personal injury or property damage?\" ", "AFM, 515 So.2d at 180. ", "The court answered the question in the negative, consistent with its decision in Florida Power & Light Co. v. Westinghouse Electric Corp., 510 So.2d 899 (Fla.1987). ", "See id. The problem therefore is that in holding that a claim for breach of fiduciary duty is barred by Florida's economic loss rule, the Eleventh Circuit relied on AFM, holding that \"If Florida courts dismiss fraud claims between parties to a contract under AFM, it is probable that the Florida courts would also dismiss fiduciary duty claims.\" ", "Hayes, 920 F.2d at 776. ", "The court reasoned that because a fraud claim, which in no way depends on the existence of a contract, is more likely to constitute a separate and independent tort under Florida law than a fiduciary duty claim, which does require existence of a contract, if the fraud claim is barred than the fiduciary duty claim is certainly barred. ", "See id. at 777.", "\nTherefore, because the Florida Supreme Court did not overrule AFM and the holding of Hayes was specifically based on the Florida Supreme Court's answer in AFM, then it is not clear whether Hayes has truly been overruled. ", "First Equity does clearly state that Hayes is no longer good law and that in light of Moransais, claims for breaches of fiduciary duty between security *1304 brokers and clients are not barred by the economic loss rule. ", "As a per curiam opinion, however, First Equity contains no analysis other than a brief citation and quote from Moransais. ", "Additionally, First Equity is in conflict with another post-Moransais case which advocated a narrower approach to the decision. ", "Monroe v. Sarasota County School Board, 746 So.2d 530, 538 (Fla. 2d DCA 1999) (\"We do not believe, however, that Moransais should be read to allow recovery for purely intangible economic losses through negligence in a wider array of cases that do not present the same conflicting issues found in construction law.\").", "\nIn light of this conflict, the court cannot hold that a claim for breach of fiduciary duty is not barred by the economic loss rule in light of Moransais. ", "The continued vitality of AFM calls into doubt exactly how expansive a ruling Moransais intended to be. ", "This question appears to be becoming an issue among the Courts of Appeal in Florida as evidenced by the broader Third District Court of Appeal decision in First Equity and the narrower Second District Court of Appeal in Monroe. ", "Because the Moransais decision is still too new, not specifically on point, and apparently subject to conflicting interpretations in the lower courts in the state, the court declines to depart from the established Eleventh Circuit precedent in Hayes and holds that the economic loss rule bars plaintiffs' breach of fiduciary duty claims.[6]\nIt this court were permitted, it would certify the question of whether the economic loss rule continues to operate as a bar to breach of fiduciary duty claims after Moransais to the Supreme Court of Florida; however, the Florida Constitution does not permit certification from a district court. ", "Fla. Const. ", "Art. ", "5, § 3(b)(6); Fla.R.App.", "P.Rule 9.150(a). ", "Therefore, in light of the lack of clarity in the state law, the court believes this issue is one appropriate for interlocutory review from the Eleventh Circuit Court of Appeals under 28 U.S.C. § 1292(b). ", "The court is of the opinion that this order involves a controlling question of law as to which there is substantial ground for difference of opinion and that an immediate appeal from the order may materially advance the ultimate termination of this litigation. ", "See id. The issues involved in this case are certain to recur, and it would prevent inconsistent decisions if these issues are conclusively resolved. ", "If the parties choose to pursue interlocutory review and the Eleventh Circuit accepts their application for review, the Eleventh Circuit would have the ability to certify the question to the Supreme Court of Florida. ", "Fla. Const. ", "Art. ", "5, § 3(b)(6); Fla.R.App.", "P.Rule 9.150(a). ", "The parties are reminded that in order to invoke the discretionary review by the Eleventh Circuit of this order, the parties must *1305 apply for review within ten days of this order. § ", "1292(b).", "\n\nC. Fraud in the Inducement\n\n1. ", "Economic Loss Rule\nDefendant moves to dismiss count III of plaintiffs' complaint, arguing that the fraud in the inducement claim is barred by the economic loss rule. ", "In HTP, Ltd. v. Lineas Aereas Costarricenses, S.A., 685 So.2d 1238, 1240 (Fla.1996), the Supreme Court of Florida recognized that fraud in the inducement can be an independent tort not barred by the economic loss rule. ", "The court noted that\n[f]raud in the inducement presents a special situation where parties to a contract appear to negotiate freely — which normally would constitute grounds for invoking the economic loss doctrine — but where in fact the ability of one party to negotiate fair terms and make an informed decision is undermined by the other party's fraudulent behavior....\nThe distinction between fraud in the inducement and other kinds of fraud is the same as the distinction drawn by a New Jersey federal district court between fraud extraneous to the contract and fraud interwoven with the breach of contract. ", "With respect to the latter kind of fraud, the misrepresentations relate to the breaching party's performance of the contract and do not give rise to an independent cause of action in tort.", "\nId. (citing Huron Tool & Engineering Co. v. Precision Consulting Services, 209 Mich.App. ", "365, 532 N.W.2d 541, 545 (1995)). ", "The court in Hotels of Key Largo, Inc. v. RHI Hotels, Inc., 694 So.2d 74, 77 (Fla. 3d DCA 1997) added that one cannot avoid the economic loss rule by merely labeling a claim as fraud in the inducement, the fraud must be separate and distinct from the breaching party's performance of the contract.", "\nIn this case, plaintiff's fraud in the inducement claims are sufficient to survive the economic loss rule. ", "Plaintiff alleges that prior to the purchase of the securities in question, defendant made material representations on which the plaintiffs relied to induce them to purchase the securities. ", "See Amend. ", "Complaint at ¶¶ 37-39. ", "Included among the misrepresentations was the amount of risk and return involved with the securities. ", "See id. at ¶ 19. ", "These are the types of allegations that fall under the justification for the fraudulent inducement exception to the economic loss rule described in Hotels. ", "The ability of the plaintiffs to negotiate fair terms and make an informed decision has been undermined by the defendant's fraudulent behavior. ", "For example, had the plaintiffs been told the true risks involved in the investments, they could have protected themselves by investing less money or by contracting for a guaranteed return from the defendant.", "\nDefendant is correct in arguing that some of the material misrepresentations that form the basis of plaintiffs' fraud in the inducement claim are the same allegations that form the basis of the breach of contract claim. ", "It appears, however, that the allegation at ¶ 19(e) which states in part that \"the limited partnerships were long term investments and would generate dividends and thereafter would be sold, with the principal intact and at a profit, such that the rate of return would be substantially above that of Certificates of Deposit and money market funds,\" was not a part of their contract. ", "Count IV makes no allegation that the defendant breached a contractual obligation to provide a rate of return higher than money market funds or CD's. ", "Because plaintiffs' fraudulent inducement claim survives the economic loss rule based on ¶ 19(e), it is not necessary for the court to decide exactly which of plaintiff's allegations for fraud in the inducement mirror allegations that form the basis of the contract. ", "To the extent that defendant induced plaintiff into signing a contract based on the promise of dividends and a high rate of return, plaintiffs' fraud in the inducement claim can be maintained. ", "This allegation meets the Hotels test because it appears that even if defendant had met all the contractual obligations alleged by the plaintiffs including, adequate investigation, *1306 due diligence, meeting reporting requirements, complying with all rules and regulations, and correctly assessing risk tolerance, the plaintiffs would still appear to have a fraud in the inducement claim if their rate of return was not as promised. ", "Accordingly, plaintiffs' fraud in the inducement claim survives the economic loss rule defense and defendant's motion to dismiss count III under the economic loss rule is denied.", "\n\n2. ", "Federal Rule of Civil Procedure 9(b)\nRule 9(b) of the Federal Rules of Civil Procedure provides that: \"[i]n all averments of fraud or mistake, the circumstances constituting fraud or mistake shall be stated with particularity.\" ", "Fed. ", "R.Civ.", "P. 9(b). ", "This rule \"serves an important purpose in fraud actions by alerting defendants to the `precise misconduct with which they are charged' and protecting defendants `against spurious charges of immoral and fraudulent behavior.'\" ", "Durham v. Business Management Associates, 847 F.2d 1505, 1511 (11th Cir.1988) (quoting Seville Indus. ", "Machinery Corp. v. Southmost Machinery Corp., 742 F.2d 786, 791 (3d Cir.1984), cert. ", "denied, 469 U.S. 1211, 105 S.Ct. ", "1179, 84 L.Ed.2d 327 (1985)). ", "Further, \"Rule 9(b) must be read in conjunction with Rule 8(a) [of the Federal Rules of Civil Procedure], which requires a plaintiff to plead only a short, plain statement of the grounds upon which he is entitled to relief.\" ", "O'Brien v. National Property Analysts Partners, 719 F.Supp. ", "222, 225 (S.D.N.Y.1989) (citing Ross v. A.H. Robins Co., 607 F.2d 545, 557 n. 20 (2d Cir.1979), cert. ", "denied, 446 U.S. 946, 100 S.Ct. ", "2175, 64 L.Ed.2d 802 (1980)). ", "See also Durham, 847 F.2d at 1511 (\"The application of [Rule 9(b)] must not abrogate the concept of notice pleading.\"); ", "Berk v. Ascott Investment Corp., 759 F.Supp. ", "245, 254 (E.D.Pa.1991). ", "Rule 9(b) may be satisfied if the complaint sets forth:\n(1) precisely what statements were made in what documents or oral representations or what omissions were made, and\n(2) the time and place of each such statement and the person responsible for making (or, in the case of omissions, not making) same, and\n(3) the content of such statements and the manner in which they misled the plaintiff, and\n(4) what the defendants \"obtained as a consequence of the fraud.\"", "\nFitch v. Radnor Industries, Ltd., No. ", "90-2084, 1990 WL 150110, at *2 (E.D.Pa. ", "Sept.27, 1990); see also Durham v. Business Management Assoc., ", "847 F.2d 1505, 1512 (11th Cir.1988) (\"Allegations of date, time, or place satisfy the Rule 9(b) requirement that circumstances of the alleged fraud must be plead with particularity ...\"); Leonard v. Stuart-James Co., 742 F.Supp. ", "653, 659 (N.D.Ga.1990) (Motion to dismiss granted where complaint failed to allege \"specifically when, where, by whom, or specifically what the representation was.\").", "\nWhen pleading fraud, the plaintiff generally should specifically identify the individuals who made the alleged misrepresentations, the time of the alleged fraud, and the place of the alleged fraud. ", "See Anthony Distributors, Inc. v. Miller Brewing Co., 904 F.Supp. ", "1363, 1365 (M.D.Fla. ", "1995). ", "The plaintiff should also quote or paraphrase the alleged fraudulent misrepresentations made by the defendant. ", "See Metrahealth Insurance Co. v. Anclote Psychiatric Hospital, Ltd., No. ", "96-2547-CIV-T-17C, 1997 WL 728084, at * 2 (M.D.Fla. ", "Oct.23, 1997). ", "However, \"alternative means are also available to satisfy the rule.\" ", "Durham, 847 F.2d at 1512 (citing Seville Indus., ", "742 F.2d at 791 (list containing allegations of fraud describing nature and subject of statements found to be sufficient, even where precise words used were not alleged)). ", "Nevertheless, courts recognize that if the alleged fraud occurred over an extended period of time and the acts were numerous, the specificity requirements are less stringently applied. *", "1307 See Anthony Distributors, Inc., 904 F.Supp. ", "at 1366. ", "This relaxed requirement is applied where \"strict application of Rule 9(b) could result in substantial unfairness to private litigants who could not possibly have detailed knowledge of all the circumstances surrounding the alleged fraud.\" ", "NCR Credit, 155 F.R.D. at 692. ", "This does not negate the plaintiff's duty to adequately plead the contents of the alleged fraudulent representations and the places where the activity was to have occurred. ", "See Anthony Distributors, Inc., 904 F.Supp. ", "at 1366. ", "The court reviews the defendant's' motion against these standards.", "\nPlaintiffs' allegations of fraud are contained in paragraph 19 of their amended complaint. ", "In paragraph 20, plaintiffs identify the seven transactions that are the subject of the lawsuit. ", "Plaintiffs provide the name of the investment purchased, the date it was purchased, and the amount of money they paid for it. ", "Plaintiffs also identified George Bliss as the agent of FSC that made the alleged fraudulent misrepresentations to the plaintiffs. ", "See Amend. ", "Complaint at ¶¶ 12, 19. ", "Plaintiffs state the time frame in which Bliss contacted the plaintiffs to solicit their business —from late 1984 until February 1985. ", "See id. at ¶ 12. ", "Plaintiff identified the place where Bliss contacted them—first in the faculty lounge at a school in the Watkins, N.Y. school system and then at the plaintiffs' home. ", "See id. at ¶ 13-14. ", "Plaintiffs both quote and paraphrase the material representations Bliss made to them at school and at their home. ", "See id. at ¶ 19. ", "The court concludes that the plaintiffs have given defendant sufficient information to satisfy the heightened pleading requirement of Rule 9(b). ", "Plaintiffs have given the defendant the \"who, what, where, and when\" of the alleged fraud that took place. ", "While plaintiffs do not give the exact dates of the meetings, they do provide a sufficiently narrow time frame from which defendant could be on notice as to when the meetings took place. ", "Plaintiffs allege they did not discover the fraud until 1996—more than 10 years after their meetings with Bliss. ", "To require them to list the precise date of their meetings with Bliss would impose a heavy burden not required by Rule 9(b) and Rule 8(a). ", "Plaintiffs cannot be expected to specify the exact time and particular place of each factual omission and misrepresentation. ", "See, e.g., Onesti v. Thomson McKinnon Sec., ", "Inc., 619 F.Supp. ", "1262, 1265 (N.D.Ill.1985) (\"Plaintiffs cannot be expected to specify the exact time and particular place of each factual omission and misrepresentation. ", "The complaint adequately specifies the transactions and the approximate time frame, the contents of the alleged misrepresentations and the essence of omitted information, and the identities of those involved.\"). ", "Accordingly, defendant's motion to dismiss plaintiffs' fraud in the inducement claim for failure to comply with Rule 9(b) is denied.", "\n\nIV. ", "Count IV—Breach of Contract\nDefendant moves to dismiss plaintiffs' breach of contract claim, or in the alternative, moves for a more definite statement under Rule 12(e). ", "It appears that plaintiffs have stated a claim for breach of contract, at the very least, by stating that they entered into an oral contract to receive reports about their investments and defendant failed to provide those reports. ", "See Amend. ", "Complaint at ¶¶ 21, 43, 45. ", "The court agrees with defendant, however, that a more definite statement would be appropriate. ", "In count IV, plaintiffs incorporate by reference paragraphs 1 through 25 of their complaint. ", "Of those paragraphs, paragraph 19 is the only one that appears to lay out contractual terms. ", "The problem is that if paragraph 19 contains the terms of the contract that were entered into between plaintiffs and defendant, then plaintiffs' fraud in the inducement claim would need to be dismissed. ", "As it is, ¶¶ 19(b) and 19(f) clearly appear to be mirrored in ¶¶ 43 and 45, and to the extent plaintiffs believe that those paragraphs form the basis of a fraud in the inducement claim, they do not because they are interwoven with the performance *1308 of the contract. ", "Therefore, the court will require plaintiffs to clearly set forth the terms of the \"oral agreement with FSC for the provision of investment advise [sic] and broker-dealer services\" and the terms of the \"due diligence, management and reporting thereto.\" ", "The plaintiffs will state the parties to which each contract applies, clarify whether more than one contract was formed, state the terms of the contract that were breached, and state which securities laws or rules were breach. ", "Plaintiff may only amend Count IV of the complaint to provide a more definite statement. ", "Should it become apparent that plaintiffs' contract claims are composed of the same terms as plaintiffs' fraud in the inducement claims, the court would be willing to entertain a renewed motion to dismiss plaintiffs' fraud in the inducement claim.", "\nWHEREFORE, IT IS ORDERED THAT:\n1) Defendant's motion to dismiss Count I and Count II of plaintiff's complaint (DE # 16-1) is GRANTED;\n2) Defendant's motion to dismiss Count III of plaintiffs' complaint (DE # 16-1) is DENIED;\n3) Defendant's motion for a more definite statement on Count IV (DE # 16-2) is GRANTED. ", "Plaintiff is ordered to submit a more definite statement to the court within 10 days of this order.", "\n4) Plaintiff's motion for oral argument (DE # 28) is DENIED; and\n5) Defendant's motion to extend time to file joint scheduling report (DE # 11) is now MOOT.", "\nNOTES\n[1] Other than the consent to jurisdiction in Miami-Dade County by the defendant, this case seems to have little connection to the State of Florida or Miami-Dade County. ", "The parties have, however, apparently agreed that Florida law shall govern this case. ", "Because the parties did not raise a conflict of laws issue in the their motions and argued their motions based only on Florida law, the law of the forum will govern, absent any facts justifying the application of some other state's law. ", "Cavic v. Grand Bahama Development Co., 701 F.2d 879, 882 (11th Cir.1983) (\"Because the parties did not raise any conflict of laws issue in the district court and do not raise it on appeal, under applicable conflict of laws principles the law of the forum ([Florida]) would govern the substantive issues due to the absence of facts justifying the application of the law of some other jurisdiction.\"); ", "see also American Fuel Corporation v. Utah Energy Development Co., 122 F.3d 130, 134 (2d Cir. ", "1997) (\"[W]here the parties have agreed to the application of the forum law, their consent concludes the choice of law inquiry.\"). ", "Because the parties have acquiesced to application of Florida law in this case, the parties may not later argue that another jurisdiction's law should apply. ", "See Pulte Home Corp. v. Osmose Wood Preserving, Inc., 60 F.3d 734, 739 n. 15 (11th Cir.1995).", "\n[2] Plaintiffs' cited authorities are inapposite because they do not specifically address the bifurcated limitations period and absolute bar contained in § 95.11(4)(e).", "\n[3] Because the court is dismissing Count I based on the statute of limitations defense, the court need not address defendant's alternative grounds for dismissal.", "\n[4] Florida has been specifically criticized for using the economic loss rule to bar breach of fiduciary duty claims. ", "See, e.g., Amanda K. Esquibel, The Economic Loss Rule and Fiduciary Duty Claims: Nothing Stricter than the Morals of the Marketplace?, ", "42 Vill.", "L.Rev. ", "789, 802-821 (1997).", "\n[5] Justice Wells concurred to specifically state that the Florida Supreme Court should have receded from the AFM decision \"because that opinion erroneously applies the economic loss rule and has given rise to confusion as to the rule's applicability.\" ", "Moransais, 744 So.2d at 984. ", "Justice Wells argues that the economic loss rule \"should be limited to cases involving a product which damages itself by reason of a defect in the product.\" ", "Id.\n[6] If the court were not bound to follow Eleventh Circuit precedent, it would likely allow the breach of fiduciary duty claim to proceed without being barred by the economic loss rule. ", "First, it appears that the thrust of the Supreme Court of Florida's decision in Moransais is to narrow the types of situations in which the economic loss rule applies. ", "Second, valid public policy concerns weigh in favor of allowing breach of fiduciary duty claims to proceed even when the relationship of the parties was formed pursuant to a contract. ", "Consider, for example, the situation where a stockbroker contractually agrees to manage a client's portfolio. \"", "If such a stockbroker misrepresents certain investments, `churns' the investor's account to generate excessive commissions, or engages in acts of self-dealing, the stockbroker should face liability in contract, tort, or both. ", "Allowing the economic loss doctrine to bar tort actions in these situations encourages wrongful conduct because the wrongdoer faces only a minimal risk — the possibility of paying compensatory damages. ", "This application of the doctrine wrongly extends its protections to classes of individuals never intended to reap its benefits.\" ", "James G. Dodrill II, Interstate Securities Corp. v. Hayes Corp.: Should the Economic Loss Doctrine Apply to Actions Against Fiduciaries?, ", "47 U.Miami L.Rev. ", "1193, 1209 (1996) (citing Michael A. Hanzman, Interstate Securities Corporation v. Hayes Corporation: An Unprecedented and Improper Expansion of Florida's \"Economic Loss\" and \"Independent Tort\" Rules, Fla.B.J., Apr. 1992, at 44-45).", "\n" ]

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[ "1. ", "Field of the Invention\nThe present invention relates generally to containers for holding liquid or dry materials, and which are provided with an opening for removal of the material. ", "More specifically, it relates to a bag, in particular a stand-up bag, provided with a frangible piercing point where an extraction device, such as a drinking straw, can be inserted into the container for effecting removal of the contents.", "\n2. ", "Description of the Prior Art\nIn addition to bottles and tin cans, certain types of flexible containers, especially stand-up bags which can be filled with liquid, pasty or dry goods, have recently gained acceptance. ", "Such a stand-up bag is described, for example, in German Pat. ", "No. ", "1,281,140. ", "As opposed to tearing such bags open, cutting off their upper seal or one corner thereof, some of them are designed so that an extraction device, for example a drinking straw, can be pushed through the side wall thereof, the extraction device then being used to remove the contents of the bag.", "\nHowever, since these bags usually consist of laminates, especially laminates including metal foils, e.g., aluminum, the insertion of the straw often presents difficulties, because the straws are relatively thin and tend to break or fold, and thus can no longer be used. ", "On the other hand, if a drinking straw is pushed into the bag too hard, it may happen that both walls of the bag will be pierced, and that the contents will then leak out.", "\nBags of this type have been proposed that have an opening closed by a piece of an adhesive tape, or by a sealing strip on the outside wall of the bag. ", "However, there are disadvantages associated with this manner of closing the opening, particularly if the bag is constructed of laminates. ", "The contents of the bag in such an arrangement are in permanent contact with the different layers of the laminate, which may be foil, at the opening where the bag is to be pierced. ", "Thus, the adhesive binding the layers of the laminate can be attacked by the contents, or, in the case of composite foils having metal layers, these are affected. ", "This may lead to delamination and thus destruction of the bag. ", "Furthermore, the contents may be contaminated by the attack upon the adhesive binding the laminates, or on the metal foil.", "\nThere is thus need for a new bag construction, one that avoids the difficulties associated with known bags of the present type." ]

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[ "Q:\n\nWSO2 IS: Difference between application-authenticator and carbon-authenticator\n\nWhen walking through the code of WSO2 identity server 5.x, I can find a samlsso authenticator in application-authenticator and another one in carbon-authenticator. ", " Same is true for IWA.", "\nWhat is the difference between these? ", "Which one is used when? ", "Or is one of them obsolete?", "\n\nA:\n\nApplication Authenticators are used to authenticate users to the external apps (service providers) using WSO2 products.", "\nCarbon authenticators are used to authenticate users to the admin console of that particular server.", "\n\n" ]

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[ "June 22, 2010\n\nExpectations of Privacy, Revisited\n\nIn City of Ontario v. Quon, the U.S. Supreme Court concluded that, in the particular circumstances of the case, a government employer had a right “to read text messages sent and received on a pager the employer owned and issued to an employee.” ", "Deciding the case in carefully narrow fashion, Justice Anthony Kennedy’s opinion for the Court assumed that the police officer, Jeff Quon, had a reasonable expectation of privacy in his text messages and concluded that the search of those messages was reasonable, because it was conducted for a legitimate, work-related purpose.", "\n\nBefore reaching this conclusion, Kennedy spends some time toying with the question whether Quon had a reasonable expectation of privacy sufficient to trigger Fourth Amendment protections. ", "The Court assumes, for the purposes of this case, that he did, rather than resolving the issue because, as Kennedy puts it, “[t]he Court must proceed with care when considering the whole concept of privacy expectations in communications owned by a government employer. ", "The judiciary risks error by elaborating too fully on the Fourth Amendment implications of emerging technology before its role in society has become clear.”", "\n\nWhen it decided Katz v. United States in 1967, the Court had no such reluctance. ", "In that case, the Court determined that individuals have a reasonable expectation of privacy in a telephone booth. ", "Kennedy suggests that “[i]t is not so clear that courts at present are on so sure a ground,” thus suggesting that, unlike text messaging, the technological issues in Katz could be viewed as within the realm of judicial competence to evaluate. ", "Kennedy goes on to observe that what society regards as private is changing, and that “employer policies concerning communications will of course shape the reasonable expectations of their employees, especially to the extent that such policies are clearly communicated.”", "\n\nThe problem here is that technology has obscured the central issue regarding the privacy the framers designed the Fourth Amendment to protect. ", "One does not have to have an engineering degree to see why Katz had a reasonable expectation of privacy in a phone booth: because the whole point of entering a phone booth and shutting the door was to create a private space in which to have a conversation. ", "Whether Katz’s side of that conversation was actually private was essentially irrelevant; obviously, someone could have read his lips through the glass or perhaps been standing close enough to hear his part of the conversation. ", "Those possibilities did not matter because the conception of reasonable privacy embraced in Katz was normative—the Court assumed some retreat from the world should be sanctioned, even if it could not be perfected; otherwise, we could reach a point at which technology makes nearly every expectation of privacy unreasonable.", "\n\nFor its part, the Quon Court is suggesting that privacy should be judged by such measures as societal standards and employer guidelines. ", "This means that, even if no one save Quon and the person with whom he was texting could see the content of their communications, still there might be no privacy if an employer declared that employees, on employer-owned equipment, had no expectation of privacy. ", "This understanding of privacy, unlike that adopted in Katz, is descriptive: it is defined by what content a person can actually keep from prying eyes, rather than by what a court should regard as necessarily private. ", "Consider that, as in the cases involving information revealed to a bank or a doctor, a court could conclude that, simply because the provider of the text messaging service has access to the content of the user’s texts, there can be no reasonable expectation of privacy.", "\n\nBecause perfect privacy is not possible, this kind of thinking will ultimately forestall arguments about expectations of privacy in respect to new modes of communication. ", "So let’s read Quon in a more positive light. ", "Perhaps the Court’s fussing about the various possibilities for a reasonable expectation of privacy in text messaging and like communications services signals that arguments for the normative conception of privacy articulated in Katz will have some weight in litigation about the privacy of communicative modes that would not exist without third-party intermediaries." ]

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[ "‘Rise Of The Tomb Raider’ Sales Tank But Dev Team Is ‘Very Happy’\n\n'Rise of the Tomb Raider' faced tough competition when it launched as an Xbox exclusive on the same day as 'Fallout 4.' ", "However, Microsoft and Square Enix as well as Crystal Dynamics and Eidos Montreal are 'very happy,' according to director Brian Horton.", "\n( Square Enix )\n\nAdvertisement\n\nWhen Rise of the Tomb Raider was launched, the title didn't meet the expected sales figure, which implied a disheartened publisher and development team. ", "However, Brian Horton, game director at Crystal Dynamics, says otherwise, noting that both Microsoft and Square Enix are \"very happy\" along with the rest of the team at Crystal Dynamics and Eidos Montreal.", "\n\nThe latest Tomb Raider installment sold about 330,000 in its first week. ", "Some say that the game's poor sales can be attributed to its Xbox exclusivity, whereas some blame its launch date, as the title went official at the same time as Fallout 4.", "\n\nFar from being in low morale because of the numbers, Horton's reaction suggests that sales aren't the most important thing in making a successful game.", "\n\n\"Microsoft and Square are very happy with Rise and the dev team, thanks for the concerns. ", "Crystal and Eidos Montreal made a game we are proud of and we appreciate the overwheling [sic] number of positive reviews and fan feedback,\" Horton replies to concerned fans on an unofficial Tomb Raider forum.", "\n\nHe then continues that the team is looking forward to the Game Awards 2015, where Rise of the Tomb Raider was nominated for Best Action/Adventure game. ", "Actress Camilla Luddington also earned a nomination for Best Performance.", "\n\nHorton adds that they don't assess the game's success based on sales count, saying that they are not responsible for that segment. ", "Instead, the devs look at the reviews and release time to determine their performance, both of which Rise of the Tomb Raider accomplished swimmingly.", "\n\nAccording to the game director's criteria, Rise of the Tomb Raider is a huge success, as it has been receiving plenty of positive reviews.", "\n\nThe title initially launched on the Xbox 360 and Xbox One a few weeks ago, and it will arrive on the PlayStation 4 some time in the fourth quarter of 2016 and on the PC by early 2016." ]

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[ "697 F.2d 287\nBakerv.", "McCall\n82-2039\nUNITED STATES COURT OF APPEALS Second Circuit\n6/16/82\nS.D.N.Y., 543 F.Supp. ", "498\nAFFIRMED\n" ]

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[ "Q:\n\nRescue given anonymous module instead of exception class\n\nWe can put a class or module after a rescue statement, but in the code below, I see a method following rescue, which does not fit into this pattern. ", "How is it working and how is it producing the output it has been designed to show?", "\ndef errors_with_message(pattern)\n # Generate an anonymous \"matcher module\" with a custom threequals\n m = Module.new\n (class << m; self; end).instance_eval do\n define_method(:===) do |e|\n pattern === e.message\n end\n end\n m\nend\nputs \"About to raise\"\nbegin\n raise \"Timeout while reading from socket\"\nrescue errors_with_message(/socket/)\n puts \"Ignoring socket error\"\nend\nputs \"Continuing...\"\n\nOutput\nAbout to raise\nIgnoring socket error\nContinuing...\n\nA:\n\nRescue requires a class or a module, true. ", "So, that method creates an anonymous module with special behaviour. ", "You see, when rescue searches for a handler, it applies === operator to exception classes/modules you provided, passing as an argument the actual exception.", "\nbegin\n # do something\nrescue MyCustomError\n # process your error\nrescue StandardError\n # process standard error\nend\n\nSo, if StandardError (or one of its descendants) was raised, the first handler will be skipped and second handler will be matched.", "\nNow, the module from errors_with_message is special. ", "It redefines threequals operator to match on exception message. ", "So, if an error was raised and its message contains word \"socket\", this handler will match. ", "Cool trick, huh?", "\n\n" ]

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[ "Alfred John North\n\nAlfred John North (11 June 1855 – 6 May 1917) was an Australian ornithologist.", "\n\nNorth was born in Melbourne and was educated at Melbourne Grammar School. ", "He was appointed to the Australian Museum, Sydney in 1886 and was given a permanent position there five years later.", "\nHe wrote a List of the Insectivorous Birds of New South Wales (1897) and a Descriptive Catalogue of the Nests and Eggs of Birds Found Breeding in Australia and Tasmania (1889) with George Barnard as co-author. ", "He described a number of birds for the first time, many in the Victorian Naturalist, the magazine of the Field Naturalists Club of Victoria of which he was a founding member.", "\n\nReferences\nNorth, Alfred John (1855 - 1917) at Bright Sparcs, University of Melbourne\n\nCategory:1855 births\nCategory:1917 deaths\nCategory:People educated at Melbourne Grammar School\nCategory:Australian ornithologists\nCategory:Australian curators" ]

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[ "You are here\n\nHome » Bulldogs track and field compete in Laurel, Colstrip and Glendive\n\nBulldogs track and field compete in Laurel, Colstrip and Glendive\n\nBy Jim Eshleman / Big Horn County News\n\nIt was a busy week for Bulldog track and field, starting with a JV meet in Laurel on Tuesday, a varsity meet in Colstrip on Thursday and Glendive Saturday for the Elks Invitational.", "\n\nIn Colstrip last Thursday, both the boys’ and girls’ teams took second place overall with four Bulldogs taking first place: Sydney Little Light in the 3,200, Ezekiel Coyote Runs in the 1,600, Trajan Hill in the 3,200 and Malik Touré in the 300-meter hurdles.", "\n\nThe boys’ 1,600-meter relay team also placed first. ", "The team consisted of Hunter Bear Cloud, Paul Little Light, Coyote Runs and Touré.", "\n\nForty-four athletes ran, jumped or threw for their best personal marks in Colstrip.", "\n\nOn Saturday, the Bulldogs traveled to Glendive for the Elks Invite.", "\n\nIn Glendive, both teams placed sixth overall with Hill taking the top spot in the 800-meter race. ", "Coyote Runs placed second in the same race.", "\n\nLibby Nedens took second place in both the 1,600 and 3,200-meter runs. ", "Madison Harmer placed third in the 3,200.", "\n\nMarie Five took third in the shot put and the boys’ 1,600 relay team placed third behind the running of Bear Cloud, Little Light, Hill and Coyote Runs.", "\n\nThere were 18 athletes who set personal records.", "\n\nAt the JV event in Laurel on Tuesday, three athletes placed first overall: Bear Cloud in the 200-meter, Ronald Anderson in the 100-meter hurdles and Harmer won the 3,200-meter run. ", "Bear Cloud also took third in the 100-meter and Anderson placed second in the 300-meter hurdles.", "\n\nThe girls’ 1,600 meter relay team placed first. ", "Running in the relay were Harmer, Marion Hugs, Khylah Two Leggins and Hannah Mark.", "\n\nSaturday, the varsity team will travel to Miles City for their invite." ]

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[ "package mocktracer\n\nimport (\n\t\"fmt\"\n\t\"reflect\"\n\t\"time\"\n\n\t\"github.com/opentracing/opentracing-go/log\"\n)\n\n// MockLogRecord represents data logged to a Span via Span.", "LogFields or\n// Span.", "LogKV.", "\ntype MockLogRecord struct {\n\tTimestamp time.", "Time\n\tFields []MockKeyValue\n}\n\n// MockKeyValue represents a single key:value pair.", "\ntype MockKeyValue struct {\n\tKey string\n\n\t// All MockLogRecord values are coerced to strings via fmt.", "Sprint(), though\n\t// we retain their type separately.", "\n\tValueKind reflect.", "Kind\n\tValueString string\n}\n\n// EmitString belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitString(key, value string) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitBool belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitBool(key string, value bool) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitInt belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitInt(key string, value int) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitInt32 belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitInt32(key string, value int32) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitInt64 belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitInt64(key string, value int64) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitUint32 belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitUint32(key string, value uint32) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitUint64 belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitUint64(key string, value uint64) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitFloat32 belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitFloat32(key string, value float32) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitFloat64 belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitFloat64(key string, value float64) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitObject belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitObject(key string, value interface{}) {\n\tm.Key = key\n\tm.ValueKind = reflect.", "TypeOf(value).Kind()\n\tm.ValueString = fmt.", "Sprint(value)\n}\n\n// EmitLazyLogger belongs to the log.", "Encoder interface\nfunc (m *MockKeyValue) EmitLazyLogger(value log.", "LazyLogger) {\n\tvar meta MockKeyValue\n\tvalue(&meta)\n\tm.Key = meta.", "Key\n\tm.ValueKind = meta.", "ValueKind\n\tm.ValueString = meta.", "ValueString\n}\n" ]

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[ "Short-term effects of TiO2, CeO2, and ZnO nanoparticles on metabolic activities and gene expression of Nitrosomonas europaea.", "\nNanosized TiO2 (n-TiO2), CeO2 (n-CeO2), and ZnO (n-ZnO) and bulk ZnO were chosen for a 4-h exposure study on a model ammonia oxidizing bacterium, Nitrosomonas europaea. ", "n-ZnO displayed the most serious cytotoxicity while n-TiO2 was the least toxic one. ", "The change of cell morphologies, the retardance of specific oxygen uptake rates and ammonia oxidation rates, and the depression of amoA gene expressions under NP stresses were generally observed when the cell densities and membrane integrities were not significantly impaired yet. ", "The TEM imaging and the synchrotron X-ray fluorescence microscopy of the NPs impacted cells revealed the increase of the corresponding intracellular Ti, Ce or Zn contents and suggested the intracellular NP accumulation. ", "The elevation of intracellular S contents accompanied with higher K contents implied the possible activation of thiol-containing glutathione and thioredoxin production for NP stress alleviation. ", "The NP cytotoxicity was not always a function of NP concentration. ", "The 200 mg L(-1) n-TiO2 or n-CeO2 impacted cells displayed the similar ammonia oxidation activities but higher amoA gene expression levels than the 20 mg L(-1) NPs impacted ones. ", "Such phenomenon further indicated the possible establishment of an anti-toxicity mechanism in N. europaea at the genetic level to redeem the weakened AMO activities along with the NP aggregation effects." ]

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[ "Allseasons in Fowey, Cornwall\n\nAllseasons description\n\nSet within the conservation area of the historic port of Fowey (the home of Daphne du Maurier), this contemporary airy property has patio doors leading out to its own small south facing garden within larger shared grounds. ", "A comfortable, warm holiday cottage, Allseasons is equally suitable for winter breaks and visiting the many winter attractions Cornwall has to offer. ", "A popular yachting centre, Fowey has a special blend of pubs, restaurants and shops, boat hire and a sailing school. ", "The beauty of the estuary may be appreciated by walking the famous Hall Walk, taking a boat trip or perhaps the ferry to Bodinnick, Polruan or Mevagissey. ", "Swimmers and children will enjoy Ready Money Cove, and ramblers the beautiful Cornish Coast Path and Saints Way. ", "Pub 400 yards.", "Ground floor: Living/dining room/kitchen with patio doors. ", "First floor: 2 bedrooms: 1 double, 1 twin. ", "Bathroom with shower over bath and toilet. ", "Gas CH, Elec, bed linen and towels inc. ", "TV. ", "DVD. ", "M/wave. ", "W/machine. ", "D/washer. ", "Freezer. ", "Small garden with patio and furniture. ", "Shared grounds. ", "On-site parking (1 car). ", "No smoking. ", "No children under 2 years.", "\n\nReviews\n\nBe the first one to tell us about your rental experience while staying at Allseasons. ", "Let us know if the listing matched your expectations. ", "You can share your comments about the location, property owner/manager and all the unique features that made your stay a memorable one." ]

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[ "Q:\n\nBuilt in colormaps in Matlab\n\nI want a lighter version of the \"cyan\" color, using the function colormap('cyan'). ", "How do you do this?", "\n\nA:\n\nCheck out the function BRIGHTEN:\nX = spiral(8);\nimage(X)\ncolormap(winter), colorbar\nbrighten(0.6)\n\nAnother trick is to right click on the colorbar and select Interactive Colormap Shift, this allows to shift the color-to-data mapping using mouse dragging.", "\n\n" ]

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[ "Justin working on musical project\n\nAccording to a recent blog post from progressive metal band Intronaut, apparently Dave contributed drums two two tracks on a musical project with Justin:\n\nDave contributed drums to a song or two in a project with JustinChancellor from Tool and a Polish guy named Peter. ", "I’m a little afraid to try and describe the music, but it rules and kind of sounds like African techno with Polish lyrics. ", "That’s actually not at all what it sounds like. ", "Forget I said anything. ", "It rules though! ", "We’ve also been talking with these two characters about collaborating on a new Intronaut song.", "\n\nSounds intriguing! ", "Anyone have any idea who this Polish guy could be?", "\n\nAnd here we were thinking that Justin’s done nothing but sit on his arse during the break. ", "Check out Intronaut’s music on their Myspace page.", "\n\n15 thoughts on “Justin working on musical project”\n\nJustin has been working on stuff in a project called “One White Onion” for awhile. ", "I think this might be something different from that, though. ", "He’s had a few other low-key projects over the past few years, it’s just that nobody in the Tool fan community ever hears about them.", "\n\nThe reason most people dont hear about his side projects is probably because he wants it that way. ", "He probably doesnt want a bunch of people just going to the show to see him and take attention off the the rest of his band, he also most likely doesnt want to hear TOOL shouted out between songs. ", "If he wanted his side projects made more public all he would have to do is talk to blair and he would be given the same treatment volto, puscifer and adams bartending gigs get on toolarmy.", "\n\nWhat if Justin just wants to make a name for himself and a low-key band without the ungodly network that Tool has at its disposal? ", "There’s nothing wrong with that.", "\n\nJustin, Adam, and to a certain degree, Danny all have the amazing gift of being world-class rockstars that can walk around cities, play unknown gigs, and eat dinner without being hounded by fans or the media. ", "This type of fame comes few and far between if you know what I mean…\n\npeter mohamad is one badass music superstar on the international scene especially in west and east europe and russia and africa . ", "he was the leader of a band called Sweet Noise that was the headlining act at polish woodstock in 2003 with drew over 3million people. ", "peter wrote all the music for sweet noise including the electronica samples used he is quite a brilliant musician and it is no surprise to me that Justin and Peter teamed up on a projecthttp://www.myspace.com/sweetnoisehttp://www.youtube.com/user/sweetnoise\n\nM.t.void sounds cool. ", "I like the vocals, and it has a really neat dark, surreal vibe.", "\nI definitely will keep an eye on their myspace.", "\nI believe Justin is extremely talented (as are all the Tool band members) and I’d like to hear more of whatever he sinks his teeth into." ]

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[ "Q:\n\nIs the set $\\{y\\in Y\\mid f^{-1}(y)\\subset H\\}$ closed?", "\n\nSuppose $f\\colon X\\to Y$ is a surjective morphism of projective varieties, $H\\subseteq X$ is a closed subset in $X$, does this hold: $\\{y\\in Y \\mid f^{-1}(y)\\subset H\\}$ is a closed subset of $Y$? ", "\n\nA:\n\nThis looks like it should be true, but it isn't. ", "Here's an example. ", "\nLet $Y=\\mathbf P^2$, let $X$ be the blowup of $\\mathbf P^2$ in a point $p$, and let $f: X \\rightarrow Y$ be the blowdown morphism.", "\nNow take $H$ to the proper transform on $X$ of a smooth curve $C$ through $p$. Then the set you are asking about will be exactly $C - \\{p\\}$.\n\n" ]

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[ "Design is of paramount importance when you are creating a news based website. ", "Your readers expect a website that is easy to navigate and a disturbance-free zone. ", "Having unnecessary over the top interface would drive them away. ", "This is why you have to be really careful while choosing the news WordPress themes for your website. ", "You wouldn’t want your website to drive away traffic due to poor designing.", "\n\nYou cannot look for a specific feature for your news theme as it differs with your site’s purpose. ", "However, you’ll unquestionably choose a theme that will be readable enough and has easy navigation. ", "Also, make sure the theme is search engine optimizable because readers want relevant and fresh news content. ", "You must have easy support for ads if you have plans to earn revenue from it. ", "To help you decide the right theme, we have compiled 10 best WordPress news themes.", "\n\nList Of Best News WordPress Themes\n\nLet us discuss the best news themes of WordPress.", "\n\nNewspaper\n\nThe Newspaper is a news and magazine based theme which takes the needs of a certain type of user into account. ", "The interface is really easy to use for inexperienced users but it also provides ample of professional-level options. ", "The theme is mobile optimized and therefore is fast and hassle-free prioritizing performance and functionality. ", "Additionally, the theme allows an abundance of content and high-resolution images and does not compromise with the loading time.", "\n\nThere are over 50 pre-built demos you can choose from. ", "Also, you can create your own layout. ", "You can do it either with two bundled page builder plugins – tagDiv Composer or Visual Composer – and the front end CSS editor. ", "You can create articles and author, category, or search pages using tagDiv Cloud Library. ", "Apart from all of these, there are news based widgets for weather, social media, exchange markets, and more. ", "You can also embed videos from the web. ", "The theme is so optimized that will it detect ads and make them immediately responsive.", "\n\nIf you want the theme, click below:\n\nDownload\n\nPenNews\n\nPenNews allows you to have visually striking articles on your homepage but at the same time, it remains minimalistic. ", "The WPBakery Page Builder plugin is already bundled with the theme. ", "It arranges your site elements with ease via a drag-and-drop interface. ", "Another bundled Covert Plug Plugin helps to grow and manage your email subscriber list.", "\n\nWhat sets this theme apart is the huge scope of layout customization. ", "There are more than 40 content block styles and a broad array of custom video playlist and gallery styles. ", "PenNews is great value for money, you should shortlist it for all the right reasons.", "\n\nDownload\n\nMH Magazine\n\nMH Magazine theme offers you an uncluttered design with flexible layout options. ", "This ultimately makes your website layout look clean and inviting. ", "There are more than 25 widget locations that help create flexible layouts, and prominent sections to display advertising. ", "For category-based content, there is a number of custom widgets. ", "With bundled FlexSlider 2 plugin, you can display all the prominent news stories.", "\n\nYou can match your website’s design with your branding with a custom logo uploader, an extensive color palette, and a wide selection of Google Fonts. ", "You can do more customization with the child theme files. ", "The theme is ideal for a clean and clear website layout. ", "If you wish to know its working then click the button below:\n\nDownload\n\nThe Issue\n\nIf you publish your content regularly then The Issue is flexible and extremely easy to use. ", "It has a wide range of demos and templates which will provide great momentum to your website. ", "For publishing and displaying news articles, the demo content is ideal. ", "If you want to change the way your content appears then there are various post format, gallery display, and review style options to pick from.", "\n\nThe Issue helps your site to integrate with social media. ", "With this theme, you can display the number of share counts of the article. ", "You can also include links to your social media handles which will provide an easier way for readers to share your content. ", "Want to try out The Issue? ", "Click below:\n\nDownload\n\nJannah\n\nJannah helps bring the best out of your website highlighting the journalistic effort and website’s visuals. ", "The theme also offers a wide variety of demo templates which includes a wide variety of niches. ", "If you want to further customize your layouts you can turn to the built-in-drag-and-drop page builder.", "\n\nJannah offers you an option to automatically remove duplicate content which otherwise would impact your SEO. ", "It also has desktop notification so that your loyal and regular readers will directly land on the news. ", "It has some really useful features in a stylish design. ", "If we want to avail the theme then here is the button:\n\nDownload\n\nHerald\n\nHerald is not fancy like other themes but it has an abundance of content on its page which is easily navigable. ", "But it also offers design customization. ", "The theme has models that enable more than 500 different layout combinations, as well as a customizable header. ", "You also get a related post to feature and an option to feature audio, video, images, and galleries.", "\n\nHerald has predefined locations for ads on home pages, archive pages, and posts. ", "It also ships with Theme translator which helps you create translations of all parts of your website in the Theme Options panel. ", "If you want to use the theme click the button below:\n\nDownload\n\nPrime News\n\nPrime News successfully capitalizes on your reader’s interest. ", "In the time when minimalism has taken a back seat, Prime News encompasses plenty of features without cluttering the screen. ", "Prime News has the traditional navigation option, ticker bars, ad space, image sliders, and much more but it doesn’t compromise the home page display. ", "The elements can be additionally customized with Live Customizer or theme Options Panel.", "\n\nYou might think it is one-dimensional seeing its initial appearance but it is not like that. ", "It doesn’t matter what type of news site you are building you can use this theme. ", "It will help you build an engaging, captivating, and engrossing final product. ", "If you want this theme for your news site then click the button below.", "\n\nDownload\n\nNewsTime\n\nNewsTime has flat designs that are catching popularity. ", "It has a combination of prominent photography and easy navigation which is rare. ", "Having said that, the theme does not compromise with the design of the individual article page. ", "It has parallax scrolling on a featured image and functional content/sidebar layout. ", "It has all the necessary features that a news site requires.", "\n\nIn the Theme Options Panel and drag-and-drop Homepage, you can customize the theme as you want. ", "It is one of the strongest contenders in the race yet it has a cheap price. ", "If you want to download NewsTime then click the button below.", "\n\nDownload\n\nGrand News\n\nGrand News offers you an array of design combinations and layouts keeping the unifying principle new and clean. ", "The pre-built demo layouts are one of the best features of the themes. ", "That includes General News, Technology News, Lifestyle and Magazine, Fashion News, Sports News, and Design Magazine, each being functional. ", "You can load these with a single click. ", "If you are new in the business this should be the first theme you should have.", "\n\nGrand News offers you a lot of customization options. ", "With the drag and drop content builder and a shortcode generator plugin, you can modify your website design in as many ways as possible. ", "Also, you can customize the site with the help of 500 Google fonts and a variety of gallery display options. ", "To buy this theme, click the button below.", "\n\nDownload\n\nGoodLife\n\nGoodLife is a clean and simple theme. ", "It offers you straightforward typography and minimal background textures that keep your content front and center. ", "GoodLife offers you Music and Game configuration and if you are publishing about either of it, it would be a good option.", "\n\nGoodLife offers every feature which any other theme offers. ", "It can be built-in locations for ads, a variety of display templates, and support for gallery and video posts. ", "With the Visual Composer and Essential Grid plugins, you can complete drag and drop control over your site’s design. ", "If you want to avail the theme, click below.", "\n\nDownload\n\nSo, these were some beautiful and best News WordPress themes for you. ", "We hope you’ll definitely make use of these themes for making your news website.", "\n\nimage source- Athemes" ]

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[ "The former Inter midfielder penned a new deal with Atletico last week.", "\n\nHowever, Italian sources say there is a clause in the contract allowing Simeone to walk away after two years without paying a penalty. ", "The option is in place allowing him to make his dream move to Inter Milan without delay." ]

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[ "---\nabstract: 'We study Liouville fillable contact manifolds with non-zero Rabinowitz Floer homology and assign spectral invariants to paths of contactomorphisms. ", "As a consequence we prove all such contact manifolds are orderable in the sense of Eliashberg and Polterovich. ", "If the contact manifold is in addition periodic or a prequantization space $M\\times S^{1}$ for $M$ a Liouville manifold, then we construct a contact capacity and define a bi-invariant metric on the group of contactomorphisms. ", "This can be used to obtain elements of infinite order in fundamental group of the contactomorphism group. ", "We also deduce the existence of infinitely many iterated translated points in the sense of Sandon [@Sandon2010] for negative contactomorphisms whose Hofer norm is smaller than the minimal period of a contractible (in the filling) Reeb orbit. ", "Finally we prove a general non-squeezing result, which amongst other examples in particular recovers the beautiful non-squeezing results from [@EliashbergKimPolterovich2006].'", "\naddress:\n- |\n Peter Albers\\\n Institut\\\n Westfälische Wilhelms – Universität Münster\\\n Germany\n- |\n Will J. Merry\\\n Department of Mathematics\\\n ETH Zürich\\\n Switzerland\nauthor:\n- 'Peter Albers and Will J. Merry'\nbibliography:\n- 'willmacbibtex.bib'\ntitle: 'Orderability, contact non-squeezing, and Rabinowitz Floer homology'\n---\n\nIntroduction and results\n========================\n\nSuppose $(\\Sigma,\\xi)$ is a closed coorientable contact manifold. ", "Denote by $\\mbox{Cont}_{0}(\\Sigma,\\xi)$ the identity component of the group of contactomorphisms, and denote by $\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$ the set of smooth paths of contactomorphisms starting at the identity. ", "The universal cover $\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ is then $\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)/\\sim$, where $\\sim$ denotes the equivalence relation of being homotopic with fixed endpoints. ", "Suppose $\\alpha\\in\\Omega^{1}(\\Sigma)$ is a contact form defining $\\xi$, and $\\theta^{t}$ its Reeb flow. ", "To a path $\\{\\varphi_{t}\\}_{0\\leq t\\leq1}\\in\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$ we can associate its contact Hamiltonian $h_{t}$ $$h_{t}\\circ\\varphi_{t}:=\\alpha\\left(\\frac{d}{dt}\\varphi_{t}\\right):\\Sigma\\rightarrow\\mathbb{R},$$ which uniquely determines the path $\\varphi$. In this article we are be interested in three classes of contact manifolds, labelled **(A)**, **(B)**, and **(C)**. ", "See Sections \\[sub:Periodicity\\] and \\[sub:Prequantisation-spaces\\] below for the other two, and Section \\[sec:Preliminaries\\] for precise definitions of the terms involved.", "\n\n[(A)]{}\n\n: $(\\Sigma,\\xi)$ admits a Liouville filling $W$ such that the Rabinowitz Floer homology $\\mbox{RFH}_{*}(\\Sigma,W)$ is non-zero.", "\n\n\\[Theorem-A\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)***. ", "Then there is map $c:\\mathcal{P}\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{R}$ with the following properties\n\n1. ", " If $\\varphi\\sim\\psi$ then $c(\\varphi)=c(\\psi)$. Thus $c$ descends to define a map $c:\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{R}$.\n\n2. ", " For any $T\\in\\mathbb{R}$, $c(t\\mapsto\\theta^{tT})=-T$ and thus $c(\\mbox{\\emph{id}}_{\\Sigma})=0$.\n\n3. ", " The map $c$ is Lipschitz continuous with respect to the $C^{0}$-norm on $\\mathcal{P}\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$, and the following triangle inequality is satisfied: $$c(\\varphi\\psi)\\leq c(\\varphi)+c(\\psi),$$ where $\\varphi\\psi$ denotes the path $t\\mapsto\\varphi_{t}\\psi_{t}$.\n\n4. ", " If $\\varphi$ is generated by a contact Hamiltonian $h_{t}$ with $h_{t}<0$ then $c(\\varphi)>0$. Similarly if $h_{t}>0$ then $c(\\varphi)<0$.\n\n5. ", " If one only assumes that $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$ then one still has $c(\\varphi)>0$, and moreover $c(\\varphi^{\\nu+1})>c(\\varphi^{\\nu})$ for all $\\nu\\in\\mathbb{N}$. However if $h_{t}(\\varphi_{t}(x))>0$ whenever $\\varphi_{t}(x)\\ne x$ then in general one only has $c(\\varphi)\\leq0$.\n\nNote that in general replacing $\\alpha$ with $-\\alpha$ changes the sign of the contact Hamiltonian $h_{t}$. However in general if $(\\Sigma,\\alpha)$ is Liouville fillable then there is no canonical Liouville filling of $(\\Sigma,-\\alpha)$, and thus for our purposes the distinction between positive and negative is not arbitrary. ", "A good non-trivial example to bear in mind is $\\Sigma=\\mathbb{R}P^{3}$, with $\\xi$ the contact structure induced from from the standard contact structure on $S^{3}$. Then $\\Sigma$ is Liouville fillable with $W=T^{*}S^{2}$, and $\\mbox{RFH}_{*}(\\mathbb{R}P^{3},T^{*}S^{2})$ is infinite dimensional (cf. [", "@AbbondandoloSchwarz2009; @CieliebakFrauenfelderOancea2010]). ", "However we can also view $\\mathbb{R}P^{3}$ as a prequantization space over $S^{2}$, which corresponds to replacing $\\alpha$ with $-\\alpha$. Indeed, in this case the completion $W$ is the total space of $\\mathcal{O}(2)\\rightarrow S^{2}$, as opposed to $T^{*}S^{2}$ which is the total space of $\\mathcal{O}(-2)\\rightarrow S^{2}$.\n\nWe state now several immediate corollaries of Theorem \\[Theorem-A\\] and Proposition \\[prop:orderability criterion-1\\]. ", "Recall that Eliashberg and Polterovich in [@EliashbergPolterovich2000] introduced the concept of *orderability* of $(\\Sigma,\\xi)$ (more precisely, of $\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$) and proved the following characterization [@EliashbergPolterovich2000 Section 2.1]:\n\n\\[prop:orderability criterion-1\\]$(\\Sigma,\\xi)$ is non-orderable if and only if there exists a contractible loop $\\varphi$ whose contact Hamiltonian $h_{t}$ is strictly positive.", "\n\nThe following result includes the orderability results of [@EliashbergKimPolterovich2006] as special cases.", "\n\n***\\[cor:(Orderability)\\]***Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)***. ", "Then $(\\Sigma,\\xi)$ is orderable.", "\n\nBy Proposition \\[prop:orderability criterion-1\\] it suffices to show that there does not exist a contractible loop $\\varphi=\\{\\varphi_{t}\\}_{t\\in S^{1}}\\subseteq\\mbox{Cont}_{0}(\\Sigma,\\xi)$ with $\\varphi_{0}=\\varphi_{1}=\\mbox{id}_{\\Sigma}$ whose contact Hamiltonian $h_{t}$ is strictly positive. ", "Suppose such a loop $\\varphi$ exists. ", "Then part (3) of Theorem \\[Theorem-A\\] asserts that $c(\\varphi)<0$. However $c(\\varphi)=c(\\mbox{id}_{\\Sigma})$, and by part (1) and (2) of Theorem \\[Theorem-A\\].", "\n\n\\[cor:converse orderabler\\]If $(\\Sigma,\\xi)$ is non-orderable and Liouville fillable then the Rabinowitz Floer homology or the symplectic homology of any its Liouville fillings vanishes.", "\n\n**The last statement follows as the Rabinowitz Floer homology $\\mbox{RFH}_{*}(\\Sigma,W)$ vanishes if and only if the symplectic homology $\\mbox{SH}_{*}(W)$ vanishes see [@CieliebakFrauenfelderOancea2010; @Ritter2010]. ", "Moreover the non-vanishing of the Rabinowitz Floer homology is these days now easy to verify in many cases. ", "As an example, take** $(S^{2n-1},\\xi_{\\textrm{st}})$ which is non-orderable [@EliashbergKimPolterovich2006 Theorem 1.10]. ", "Corollary \\[cor:converse orderabler\\] for $\\Sigma=S^{2n-1}$ has been proved before by Smith see [@Seidel2006].", "\n\n\\[cor:infinite order\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)***. ", "Then if $\\varphi$ is a loop satisfying either $c(\\varphi)<0$ (e.g. if the contact Hamiltonian $h_{t}$ of $\\varphi$ is strictly positive[^1]), or if the contact Hamiltonian $h_{t}$ satisfies $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$ , then $\\varphi$ is of infinite order in $\\pi_{1}(\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi))$.\n\nIf $c(\\varphi)<0$ then by the triangle inequality one has $c(\\varphi^{\\nu})\\leq\\nu c(\\varphi)<0$. Thus $\\varphi^{\\nu}$ cannot be contractible for any $\\nu\\in\\mathbb{N}$ . ", "Similarly if $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$ then part (5) of Theorem \\[Theorem-A\\] shows that $\\varphi^{n}$ can never be contractible for any $\\nu\\in\\mathbb{N}$.\n\n\\[def cU\\]Suppose $U\\subset\\Sigma$ is open. ", "We define $$c(U):=\\sup\\left\\{ c(\\varphi)\\mid\\mathfrak{S}(\\varphi)\\subset U\\right\\} ,$$ where $\\mathfrak{S}(\\varphi):=\\bigcup_{0\\leq t\\leq1}\\overline{\\left\\{ x\\in\\Sigma\\mid\\varphi_{t}(x)\\ne x\\right\\} }$ denotes the *support* of $\\varphi$.\n\nThe quantity $c(U)$ is *not* a contact invariant, that is, it is not necessarily the case that $c(\\psi(U))=c(U)$ for every $\\psi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$. We will see in Section \\[sub:Periodicity\\] below that if the Reeb flow is periodic then it is possible to define a related number $\\bar{c}(U)$ which *is* a contact invariant, and this is where many of applications will lie. ", "Nevertheless, it is still possible to extract some interesting information in this setting, as the following result and Proposition \\[cor:infinitely many-1\\] below shows.", "\n\n\\[preliminary result of the cU\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)***. ", "Then:\n\n1. ", " $c(U)>0$ for any non-empty open set $U$, and $c(\\Sigma)=\\infty$.\n\n2. ", " If $U\\subset V$ are open sets then $$c(U)\\leq c(V).$$\n\n3. ", " If $U\\subset\\Sigma$ is an open set with the property that there exists $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ with the property that $$\\psi(U)\\cap\\left(\\bigcup_{t\\in\\mathbb{R}}\\theta^{t}(U)\\right)=\\emptyset$$ then for any $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ satisfying $\\mathfrak{S}(\\varphi)\\subset U$ one has $$0\\leq c(\\varphi)\\leq c(\\psi)+c(\\psi^{-1}),$$ and thus in particular $c(U)<\\infty$.\n\n\\[smallest period\\]Let us denote by $$\\wp_{W}(\\Sigma,\\alpha):=\\inf\\left\\{ T>0\\mid\\exists\\ \\mbox{a closed contractible (in }W\\mbox{) Reeb orbit of }\\alpha\\mbox{ of period }T>0\\right\\} .$$ If there are no contractible closed Reeb orbits we set $\\wp_{W}(\\Sigma,\\alpha):=\\infty$. Following Sandon [@Sandon2010 Theorem 3.1], we can prove the following result (see Definitions \\[def trans poin\\], \\[def contractible translated point\\], and \\[hofer norm def\\] for the relevent terms).", "\n\n\\[cor:infinitely many-1\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)*** and $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ satisfies $\\mathfrak{S}(\\varphi)\\subset U$ and assume that the contact Hamiltonian $h_{t}$ of $\\Sigma$ satisfies $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$. Assume that either:\n\n1. ", " One has $\\left\\Vert h\\right\\Vert _{\\textrm{\\emph{osc}}}<\\wp_{W}(\\Sigma,\\alpha)$ and $c(U)<\\infty$.\n\n2. ", " There exists $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ with $\\psi(U)\\cap\\left(\\bigcup_{t\\in\\mathbb{R}}\\theta^{t}(U)\\right)=\\emptyset$ and $\\left\\Vert \\psi\\right\\Vert <\\wp_{W}(\\Sigma,\\alpha)$.\n\nThen $\\varphi$ has infinitely many geometrically distinct iterated translated points which are ** contractible in the Liouville filling $(W,d\\lambda)$.\n\nSandon [@Sandon2010] proved this result for $\\Sigma=\\mathbb{R}^{2n+1}$ and $\\Sigma=\\mathbb{R}^{2n}\\times S^{1}$ for open sets $U$ with compact closure. ", "Neither of these examples satisfy Assumption **(A)**. ", "However our proof also works (cf. ", "Corollary \\[cor:translated point in MxS1\\]) when $(\\Sigma,\\xi)$ is a prequantization space $M\\times S^{1}$ . ", "Taking $M=\\mathbb{R}^{2n}$ recovers Sandon’s result.", "\n\nIf there are no closed contractible Reeb orbits then both Conditions (1) and (2) in Proposition \\[cor:infinitely many-1\\] are vacuous, and in this case one has that if $\\varphi$ satisfies $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$, and $\\mathfrak{S}(\\varphi)$ is contained in an open set $U$ with $c(U)<\\infty$ , then $\\varphi$ has infinitely many geometrically distinct iterated contractible translated points. ", "In fact, if there are no contractible closed Reeb orbits then we can prove the following additional result.", "\n\n\\[prop:depends only on endpoint\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)*** and that there are no closed contractible Reeb orbits. ", "Then the value of $c(\\varphi)$ depends only on $\\varphi_{1}$. Thus the map $c:\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{R}$ descends to $\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$.\n\n\\[question 1\\]Let us note that in the presence of contractible Reeb orbits Proposition \\[prop:depends only on endpoint\\] is clearly not true. ", "For instance, in the next section we study ** contact manifolds with periodic Reeb flow $\\theta^{t}$ (see Assumption **(B)** below). ", "In this case one has $\\theta^{\\nu}=\\mbox{id}_{\\Sigma}$ for each $\\nu\\in\\mathbb{Z}$, but $c(t\\mapsto\\theta^{t\\nu})=\\nu$.\n\n\\[sub:Periodicity\\]Periodic contact manifolds\n---------------------------------------------\n\n\\[def - periodic\\]A closed coorientable contact manifold $(\\Sigma,\\xi)$ is called *periodic* if there exists a contact form $\\alpha$ for $\\xi$ such that the Reeb flow $\\theta^{t}$ of $\\alpha$ is $1$-periodic $$\\theta^{1}=\\theta^{0}=\\mbox{id}_{\\Sigma}.$$\n\nIn this Section the standing Assumption **(B)** that we will make is the following strengthening of Assumption **(A)**.", "\n\n[(B)]{}\n\n: $(\\Sigma,\\xi)$ admits a Liouville filling $(W,d\\lambda)$ such that the Rabinowitz Floer homology $\\mbox{RFH}_{*}(\\Sigma,W)$ is non-zero and such that $\\alpha:=\\lambda|_{\\Sigma}$ is periodic.", "\n\n\\[Examples-of-periodic \\]Examples of periodic contact manifolds are unit cotangent bundles $S_{g}^{*}B$ where $g$ is a Riemannian metric on $B$ all of whose geodesics are closed (e.g.. $B=S^{n}$, $g$ the round metric). ", "Another class of examples arise by taking *prequantization spaces* of Liouville manifolds. ", "For this, one starts with a Liouville manifold $(M,d\\gamma)$. Then we equip $\\Sigma:=M\\times S^{1}$ with the contact form $\\alpha:=\\gamma+d\\tau$, where $\\tau$ is the coordinate on $S^{1}$. We will study this latter setting in depth see Assumption **(C)** below.", "\n\nAn immediate consequence of Corollary \\[cor:infinite order\\] is the following.", "\n\nSuppose $(\\Sigma,\\xi)$ satisfies Assumption ***(B)***. ", "Then the Reeb flow $\\theta^{1}$ is of infinite order in $\\pi_{1}(\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi))$.\n\nThe key extra piece of information one gains by working with periodic contact manifolds **** is that in this case the spectral numbers are (almost) *conjugation invariant*. ", "More precisely, following Sandon [@Sandon2010a] we define a function $\\bar{c}:\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}$ by $$\\overline{c}(\\varphi):=\\left\\lceil c(\\varphi)\\right\\rceil .$$\n\n\\[prop:intro conju invariance\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(B)***. ", "Then $\\overline{c}$ is conjugation invariant under the natural action of $\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ on its universal cover. ", "Moreover $\\overline{c}$ continues to satisfy the triangle inequality $\\overline{c}(\\varphi\\psi)\\leq\\overline{c}(\\varphi)+\\overline{c}(\\psi)$.\n\nWe define $\\gamma:\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$ by $$\\gamma(\\varphi):=\\left|\\overline{c}(\\varphi)\\right|+\\left|\\overline{c}(\\varphi^{-1})\\right|,$$ and $\\hat{\\gamma}:\\mbox{Cont}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$ by $$\\hat{\\gamma}(\\vartheta):=\\inf\\left\\{ \\gamma(\\varphi)\\mid\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\,\\varphi_{1}=\\vartheta\\right\\} .$$\n\nSince $\\mbox{Cont}_{0}(\\Sigma,\\xi)$ is a *simple* group [@Rybicki2010] and $\\hat{\\gamma}^{-1}(0)$ is a normal subgroup (by conjugation invariance of $\\gamma$), the function $\\hat{\\gamma}$ induces a bi-invariant metric on $\\mbox{Cont}_{0}(\\Sigma,\\xi)$.\n\n\\[thm:metric\\]Define $$\\mbox{\\emph{d}}_{\\hat{\\gamma}}:\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)\\times\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$$ by $$\\mbox{\\emph{d}}_{\\hat{\\gamma}}(\\varphi,\\psi)=\\hat{\\gamma}(\\varphi\\psi^{-1}).$$ Then $\\mbox{\\emph{d}}_{\\hat{\\gamma}}$ is a bi-invariant metric on $\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$. Similarly if one defines $$\\mbox{\\emph{d}}_{\\gamma}:\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\times\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$$ in the same way, but using $\\gamma$ instead of $\\hat{\\gamma}$, then $\\mbox{\\emph{d}}_{\\gamma}$ is a bi-invariant pseudo-metric on $\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ whose degeneracy locus is the (possibly empty) set of non-contractible loops $\\varphi=\\{\\varphi_{t}\\}_{t\\in S^{1}}$ in $\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ with $c(\\varphi)=c(\\varphi^{-1})=0$.\n\nBi-invariants metrics have been defined before by Sandon [@Sandon2010a] on $\\mathbb{R}^{2n}\\times S^{1}$, by Zapolsky [@Zapolsky2012] on $T^{*}B\\times S^{1}$ and Fraser-Polterovich-Rosen [@FraserPolterovichRosen2012] for a certain class of periodic contact manifolds and Colin-Sandon [@ColinSandon2012] for all contact manifolds. ", "It is an interesting question what the precise relationship is between their metrics and ours. ", "It is still possible to define $\\gamma$ and $\\hat{\\gamma}$ under only Assumption **(A)**. ", "Nevertheless in this case $\\mbox{d}_{\\hat{\\gamma}}$ is not bi-invariant, and $\\hat{\\gamma}^{-1}(0)$ is no longer a normal subgroup of $\\mbox{Cont}_{0}(\\Sigma,\\xi)$.\n\nNote that the metric is always unbounded as $\\mbox{d}_{\\hat{\\gamma}}(\\mbox{id},\\{t\\mapsto\\theta^{\\nu t}\\})=\\nu$ for any $\\nu\\in\\mathbb{Z}$.\n\nHere is an example where the metric $\\mbox{d}_{\\gamma}$ is degenerate. ", "Take $\\Sigma=S^{*}\\mathbb{T}^{2}=\\mathbb{T}^{3}$. Let $\\varphi_{t}:\\Sigma\\rightarrow\\Sigma$ denote the lift of a full rotation of $\\mathbb{T}^{2}$. This is a non-contractible loop. ", "Since in this case $\\Sigma$ has no contractible Reeb orbits, Proposition \\[prop:depends only on endpoint\\] implies that $c(\\varphi)=c(\\varphi^{-1})=0$, and hence $\\mbox{d}_{\\gamma}(\\varphi,\\mbox{id}_{\\Sigma})=0$.\n\nAs in Definition \\[def cU\\], we define the quantity $\\overline{c}(U)\\in\\mathbb{Z}_{\\geq0}$ for $U\\subset\\Sigma$ open by $$\\bar{c}(U):=\\sup\\left\\{ \\overline{c}(\\varphi)\\mid\\mathfrak{S}(\\varphi)\\subset U\\right\\} .$$ Clearly Proposition \\[preliminary result of the cU\\] continues to hold for $\\bar{c}$, and in particular $\\bar{c}(U)\\geq1$ for any non-empty open set $U\\subset\\Sigma$. However the key point is that in this setting we can prove rather more. ", "Namely, $\\overline{c}$ is a *contact capacity*, that is (\\[eq:inv under cont\\]), a notion that was first introduced by Sandon in [@Sandon2011b].", "\n\n\\[Theorem-B\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(B)***. ", "Let $U\\subset\\Sigma$ be open:\n\n1. ", " $\\bar{c}$ is a *contact capacity*: that is, for all $\\psi\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$, one has $$\\bar{c}(\\psi(U))=\\bar{c}(U).\\label{eq:inv under cont}$$\n\n2. ", " If there exists $\\psi\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ with $\\psi(\\overline{U})\\cap\\overline{U}=\\emptyset$ then $\\overline{c}(U)\\leq\\overline{c}(\\psi)+\\overline{c}(\\psi^{-1})<\\infty$.\n\nNote that part (2) of Theorem \\[Theorem-B\\] allows us to slightly improve Proposition \\[cor:infinitely many-1\\]. ", "Namely, if in Proposition \\[cor:infinitely many-1\\] if we assume Assumption **(B)** holds then condition (2) can be replaced with the assertion that $\\psi(\\overline{U})\\cap\\overline{U}=\\emptyset$ and $\\left\\Vert \\psi\\right\\Vert <1=\\wp(\\Sigma,\\alpha)$.\n\nAn application of Theorem \\[Theorem-B\\] is the following *abstract non-squeezing* result.", "\n\n\\[cor:non squeezing\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(B)***. ", "Let $U\\subset V\\subset\\Sigma$ be open sets and assume that there exists $\\varphi\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ with $\\varphi(V)\\subset U$. Then $$\\overline{c}(U)=\\overline{c}(V).$$ In particular, if $\\overline{c}(U)<\\overline{c}(V)$ then there exists no contact isotopy mapping $V$ into $U$.\n\nSuppose $\\varphi$ is as in the statement of the Corollary. ", "Then by part (2) of Proposition \\[preliminary result of the cU\\] we have $\\overline{c}(U)\\leq\\overline{c}(V)$ and $\\overline{c}(\\varphi(V))\\leq\\overline{c}(U)$. By part (2) of Theorem \\[Theorem-B\\] we also have $\\overline{c}(\\varphi(V))=\\overline{c}(V)$, and hence $\\overline{c}(U)=\\overline{c}(V)$ as claimed.", "\n\n\\[sub:Prequantisation-spaces\\]Prequantization spaces \n-----------------------------------------------------\n\nFix a Liouville manifold $(M,d\\gamma)$ **(i.e. the completion of Liouville domain, cf. ", "Definition \\[A-Liouville-manifold\\]). ", "The** *prequantization space* **of $M$ is the contact manifold $\\Sigma:=M\\times S^{1}$, equipped with the contact structure $\\xi:=\\ker\\,\\alpha$, where** $$\\alpha:=\\gamma+d\\tau,$$ where $\\tau$ is the coordinate on $S^{1}=\\mathbb{R}/\\mathbb{Z}$. These contact manifolds are the third type we study in this paper:\n\n[(C)]{}\n\n: $(\\Sigma,\\xi=\\ker\\,\\zeta)$ is a prequantization space $\\Sigma=M\\times S^{1}$, where $(M,d\\gamma)$ is a Liouville manifold, and $\\alpha=\\gamma+d\\tau$.\n\nLet $P_{0}$ denote a torus with a small disc removed, so that $\\partial P_{0}=S^{1}$. Equip $P_{0}$ with an exact symplectic form $d\\beta_{0}$ such that $\\beta_{0}|_{\\partial P_{0}}=d\\tau$. Let $(P,d\\beta)$ denote the completion of the Liouville domain $(P_{0},d\\beta_{0})$, and consider $$W:=M\\times P,$$ equipped with the symplectic form $d\\lambda$ where $\\lambda:=\\gamma+\\beta$. Even though $\\Sigma$ is periodic, $W$ is *not* a Liouville filling of $\\Sigma$, and in fact $\\Sigma$ does not satisfy either Assumptions **(A)** or **(B)** - for instance, $\\Sigma$ is non-compact. ", "Nevertheless, it is still possible to define the Rabinowitz Floer homology $\\mbox{RFH}_{*}(\\Sigma,W)$, and we prove that $$\\mbox{RFH}_{*}(\\Sigma,W)\\cong\\mbox{HF}_{*}(M)\\otimes\\mbox{H}_{*}(S^{1};\\mathbb{Z}_{2}),\\label{eq:what rfh is}$$ where $\\mbox{HF}_{*}(M)\\cong\\mbox{H}^{n-*}(M;\\mathbb{Z}_{2})$ denotes the Hamiltonian Floer homology of $M$, defined using *compactly supported* Hamiltonians (see Frauenfelder-Schlenk [@FrauenfelderSchlenk2007]).", "\n\nSince the Hamiltonian Floer homology is non-zero, one can associate a spectral number $c_{M}(f)$ to any $f\\in\\widetilde{\\mbox{Ham}}_{c}(M,d\\gamma)$, the universal cover of the group of compactly supported Hamiltonian diffeomorphisms (see eg. ", "Schwarz [@Schwarz2000] or Frauenfelder-Schlenk [@FrauenfelderSchlenk2007]) . ", "As in the contact case described above, $c_{M}$ can then be used to define a *symplectic capacity* $c_{M}(\\mathcal{O})$ for $\\mathcal{O}\\subset M$ open, by setting $$c_{M}(\\mathcal{O}):=\\sup\\left\\{ c_{M}(f)\\mid\\mathfrak{S}(f)\\subset\\mathcal{O}\\right\\} .$$\n\nThe proof that $c_{M}(f)$ is invariant under conjugation, that is, $c_{M}(hfh^{-1})=c_{M}(f)$ for $f\\in\\widetilde{\\mbox{Ham}}_{c}(M,d\\gamma)$ and $h\\in\\mbox{Symp}_{c}(M,d\\gamma)$ is immediate, since in this case the action spectrum of $hfh^{-1}$ is the same as the action spectrum of $f$ (see for instance [@HoferZehnder1994 Chapter 5, Proposition 7]). ", "This in turn immediately implies that $c_{M}$ is a symplectic capacity, that is, $c_{M}(f(\\mathcal{O}))=c_{M}(\\mathcal{O})$ for any symplectomorphism $f$ and any open set $\\mathcal{O}\\subset M$.\n\nGoing back to $\\Sigma=M\\times S^{1}$, let us denote by $\\mbox{Cont}_{0,c}(\\Sigma,\\xi)$ those contactomorphisms $\\varphi$ with compact support. ", "There is a natural way to lift an element $f\\in\\widetilde{\\mbox{Ham}}_{c}(M,d\\gamma)$ to obtain an element $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$, as we now explain. ", "The equation $$f_{t}^{*}\\gamma-\\gamma=da_{t},\\ \\ \\ a_{0}\\equiv0,$$ determines a smooth compactly supported function $a_{t}:M\\to\\mathbb{R}$. Define $\\varphi_{t}:\\Sigma\\rightarrow\\Sigma$ by $$\\varphi_{t}(y,\\tau):=\\bigl(f_{t}(y),\\underset{\\textrm{mod 1}}{\\underbrace{\\tau-a_{t}(y)}}\\bigr).\\label{eq:phi from f}$$ One can define the spectral numbers $c(\\varphi)$ for $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$ in much the same way as before, and the triangle inequality $c(\\varphi\\psi)\\leq c(\\varphi)+c(\\psi)$ continues to hold, see Section \\[sub:Hamiltonian-Floer-homology\\] or [@FrauenfelderSchlenk2007]. ", "If $U\\subset\\Sigma$ is an open set with *compact closur*e then both Proposition \\[preliminary result of the cU\\] and Theorem \\[Theorem-B\\] also go through in this setting. ", "Note that in this case the metric is again always unbounded; this follows from Theorem \\[thm:equality of capacities\\] below, since the Hofer norm on $\\widetilde{\\mbox{Ham}}_{c}(M,d\\gamma)$ is always unbounded for $(M,d\\gamma)$ a Liouville manifold. ", "Proposition \\[cor:infinitely many-1\\] goes through without changes in fact since there are no closed contractible Reeb orbits, a stronger result holds.", "\n\n\\[cor:translated point in MxS1\\]Suppose $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0,c}(\\Sigma,\\xi)$ and assume that $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$. Assume that $\\mathfrak{S}(\\varphi)\\subset U$ for some open set $U$ with compact closure and finite capacity. ", "Then $\\varphi$ has infinitely many geometrically distinct iterated contractible translated points.", "\n\nIn the same vein Proposition \\[prop:depends only on endpoint\\] also holds in this setting. ", "Finally, Theorem \\[thm:metric\\] also holds, provided one works only with elements of $\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$. See Section \\[sub:Applications\\] for more information. ", "It is important to note that there is no analogue to part (2) of Theorem \\[Theorem-A\\], since the Reeb flow $\\theta^{t}$ is not compactly supported and hence its spectral number is not defined. ", "In Appendix A we show how to construct a ‘compactly supported Reeb flow’ $\\vartheta^{t}$ (from the point of view of translated points.) ", "However, as proved in Theorem \\[prop:compactly supported reeb\\], this compactly supported Reeb flow has $c(\\vartheta^{T})=0$ whenever $T>0$, which suggests that there is no sensible definition of $c(\\varphi)$ for non-compactly supported contactomorphisms $\\varphi$. The following corollary is immediate from part (2) of Proposition \\[preliminary result of the cU\\].", "\n\nIf $U\\subset\\Sigma$ is an open set with compact closure such that the projection $\\overline{\\mathcal{O}}:=\\mbox{\\emph{pr}}_{1}(\\overline{U})$ is displaceable in $M$, that is, there exists $f\\in\\widetilde{\\mbox{\\emph{Ham}}}_{c}(M,d\\gamma)$ such that $f_{1}(\\overline{\\mathcal{O}})\\cap\\overline{\\mathcal{O}}=\\emptyset$. Then for any $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0,c}(\\Sigma,\\xi)$ with $\\mathfrak{S}(\\psi)\\subset U$ one has $$0\\leq c(\\psi)\\leq c(\\varphi)+c(\\varphi^{-1}),$$ where $\\varphi$ denotes the lift of $f$ defined in (\\[eq:phi from f\\]). ", "In particular, if $M=\\mathbb{R}^{2n}$ (where every compact set is displaceable) then $c(\\psi)\\geq0$ for all $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0,c}(\\mathbb{R}^{2n}\\times S^{1},\\xi)$.\n\nIt is natural to ask the question: if $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$ is the lift of $f\\in\\widetilde{\\mbox{Ham}}_{c}(M,d\\gamma)$, how is $c(\\varphi)$ related to $c_{M}(f)$? ", "Note that $\\mathfrak{S}(\\varphi)=\\mathfrak{S}(f)\\times S^{1}$, and hence another question is how the $c_{M}$ capacity of $\\mathcal{O}\\subset M$ is related to the $c$ capacity of $\\mathcal{O}\\times S^{1}$. The following result answers these questions in the nicest possible way.", "\n\n\\[thm:equality of capacities\\]Suppose $f\\in\\widetilde{\\mbox{\\emph{Ham}}}_{c}(M,d\\gamma)$, and let $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$ denote the lift of $f$. Then $$c_{M}(f)=c(\\varphi).$$ Moreover, if $\\mathcal{O}\\subset M$ is open and has compact closure then $$c_{M}(\\mathcal{O})=c(\\mathcal{O}\\times S^{1}).$$\n\nTheorem \\[thm:equality of capacities\\] allows us to prove non-squeezing results on $\\Sigma$ by making use of the known results on $M$. Let $l^{t}:M\\rightarrow M$ denote the flow of the Liouville vector field on $M$ and set $\\zeta^{t}:=l^{\\log\\, t}$. We will prove the following general result.", "\n\n\\[thm:liouville flow\\]Suppose $\\mathcal{O}\\subset M$ is a non-empty open set with compact closure and finite capacity: $c_{M}(\\mathcal{O})<\\infty$. Suppose there exists a contact isotopy $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0,c}(\\Sigma,\\xi)$ such that $$\\varphi_{1}\\left(\\zeta^{r_{2}}(\\mathcal{O})\\times S^{1}\\right)\\subset\\zeta^{r_{1}}(\\mathcal{O})\\times S^{1}$$ for $r_{1},r_{2}\\in\\mathbb{R}$. Then $\\left\\lceil r_{2}\\right\\rceil \\leq\\left\\lceil r_{1}\\right\\rceil $. ", "More generally if $\\mathcal{O}\\subset\\mathcal{Q}\\subset M$ are open sets with the property that there exists $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ with $\\varphi_{1}(\\mathcal{Q}\\times S^{1})\\subset\\mathcal{O}\\times S^{1}$ then $\\left\\lceil c_{M}(\\mathcal{Q})\\right\\rceil \\leq\\left\\lceil c_{M}(\\mathcal{O})\\right\\rceil $.", "\n\nNote that for any $r\\in\\mathbb{R}$, $$c_{M}(\\zeta^{r}(\\mathcal{O}))=rc_{M}(\\mathcal{O})\\ne0,$$ since $c_{M}(\\mathcal{O})>0$ as $\\mathcal{O}$ is non-empty. ", "Thus without loss of generality we may assume that $c_{M}(\\mathcal{O})=1$. Then $$\\overline{c}(\\zeta^{r}(\\mathcal{O})\\times S^{1})=\\left\\lceil c_{M}(\\zeta^{r}(\\mathcal{O}))\\right\\rceil =\\left\\lceil r\\right\\rceil .$$ Since as remarked above, Theorems \\[Theorem-A\\] and \\[Theorem-B\\] extend to this setting, the result follows from the analogue of Corollary \\[cor:non squeezing\\]. ", "The last statement follows similarly.", "\n\nTheorem \\[thm:liouville flow\\] recovers the beautiful non-squeezing result of [@EliashbergKimPolterovich2006 Theorem 1.2]. ", "In this case one takes $M=\\mathbb{R}^{2n}$ and $U$ the unit ball. ", "They prove that if $\\left\\lceil r_{1}\\right\\rceil <\\left\\lceil r_{2}\\right\\rceil $ then it is not possible to squeeze $B(r_{2})\\times S^{1}$ into the *cylinder*[^2] $C(r_{1})\\times S^{1}$. This result was also recovered by Sandon [@Sandon2011b] using generating functions.", "\n\nOne can play a similar game using the *symplectic homology* $\\mbox{SH}_{*}(M)$ rather than the Hamiltonian Floer homology $ $$\\mbox{HF}_{*}(M)$. This corresponds to using a different class of Hamiltonians instead of taking compactly supported Hamiltonians one uses Hamiltonians that grow superlinearly. ", "This leads to a truly infinite dimensional theory: for instance taking $M=T^{*}B$, one has $\\mbox{SH}_{*}(T^{*}B)\\cong\\mbox{H}_{*}(\\Lambda(B);\\mathbb{Z}_{2})$ [@Viterbo1996; @SalamonWeber2006; @AbbondandoloSchwarz2006], as opposed to $\\mbox{HF}_{*}(T^{*}B)\\cong\\mbox{H}^{n-*}(B;\\mathbb{Z}_{2})$. It is possible to define a different Rabinowitz Floer homology $\\mbox{RFH}_{*}^{\\textrm{SH}}(M\\times S^{1},M\\times P)$ which satisfies $$\\mbox{RFH}_{*}^{\\textrm{SH}}(M\\times S^{1},M\\times P)\\cong\\mbox{SH}_{*}(M)\\otimes\\mbox{H}_{*}(S^{1};\\mathbb{Z}_{2})$$ instead of (\\[eq:what rfh is\\]). ", "This leads to non-squeezing results on $T^{*}B\\times S^{1}$, which we will investigate in a sequel to this paper.", "\n\nFurther applications of Theorem \\[thm:equality of capacities\\] are given in Section \\[sub:Applications\\].", "\n\nFinally, following [@EliashbergKimPolterovich2006 Section 1.7] we investigate a rigidity phenomenon of positive contractible loops of contactomorphisms. ", "Suppose now that $(M,d\\gamma)$ is the completion of a Liouville domain $(M_{0},d\\gamma_{0})$. Set $S:=\\partial M_{0}$ and $\\kappa:=\\gamma|_{S}$, so that $(S,\\kappa)$ is a contact manifold. ", "Abbreviate $$M(r):=\\begin{cases}\nM_{0}\\backslash(S\\times[r,1), & 0<r<1,\\\\\nM_{0}\\cup_{S}(S\\times[1,r), & r\\geq1.", "\n\\end{cases}$$ We can prove the following result, which roughly speaking says that if $(S,\\kappa)$ is *non-orderable*, so there exists a positive contractible loop $\\chi=\\{\\chi_{t}\\}_{t\\in S^{1}}\\subset\\mbox{Cont}_{0}(S,\\kappa)$ of contactomorphisms, then it is *not* possible to homotope $\\zeta$ through *positive* loops to $\\mbox{id}_{S}$. In [@EliashbergKimPolterovich2006 Theorem 1.11] this was proved for $S=S^{2n-1}$.\n\nSet $c:=c_{M}(M_{0})$ and assume that $c<\\infty$. Suppose that $(S,\\kappa)$ is non-orderable, with $\\chi=\\{\\chi_{t}\\}_{t\\in S^{1}}\\subset\\mbox{\\emph{Cont}}_{0}(S,\\kappa)$ a positive contractible loop of contactomorphisms. ", "Let $g_{t}:S\\rightarrow(0,\\infty)$ denote the contact Hamiltonian of $\\chi$, and set $$\\varepsilon:=\\min_{(t,y)\\in S^{1}\\times S}g_{t}(y)>0.$$ Then if $\\{\\chi_{s,t}\\}_{0\\leq s\\leq1}$ is any homotopy of loops of contactomorphisms such that $\\chi_{1,t}=\\chi_{t}$ and $\\chi_{0,t}=\\mbox{\\emph{id}}_{S}$ with corresponding contact Hamiltonian $g_{s,t}:S\\rightarrow\\mathbb{R}$ then there exists $(s,t,y)\\in[0,1]\\times S^{1}\\times S$ such that $g_{s,t}(y)=-(1-\\varepsilon)c$.\n\nThis follows directly from Theorem \\[thm:liouville flow\\] and the material from [@EliashbergKimPolterovich2006 Section 2.1]. ", "Indeed, suppose there exists $\\delta>0$ such that $g_{s,t}(y)>-(1-\\varepsilon)(c-\\delta)$ for all $(s,t,y)\\in[0,1]\\times S^{1}\\times S$. Set $a:=\\min\\{\\varepsilon,\\varepsilon c\\}$. Then as proved in [@EliashbergKimPolterovich2006 Section 2.1] for any $r<\\frac{1}{c-\\delta}$ it is possible to squeeze $M(r)\\times S^{1}$ into $M(\\tfrac{r}{1+ar})\\times S^{1}$. Fix $0<\\lambda<\\min\\{a,\\delta\\}$ and take $r=\\tfrac{1}{c-\\lambda}$. Then $$\\begin{aligned}\n\\overline{c}(M(r)\\times S^{1}) & =\\left\\lceil c_{M}(M(r))\\right\\rceil \\nonumber \\\\\n & =\\left\\lceil rc_{M}(M(1))\\right\\rceil \\nonumber \\\\\n & =\\left\\lceil \\tfrac{c}{c-\\lambda}\\right\\rceil =2.\\end{aligned}$$ But $$\\begin{aligned}\n\\overline{c}(M(\\tfrac{r}{1+ar})\\times S^{1}) & =\\left\\lceil c_{M}(M(\\tfrac{r}{1+ar}))\\right\\rceil \\nonumber \\\\\n & =\\left\\lceil \\frac{cr}{1+ar}\\right\\rceil \\nonumber \\\\\n & =\\left\\lceil \\frac{c}{c-\\lambda+a}\\right\\rceil =1.\\end{aligned}$$ This contradicts Theorem \\[thm:liouville flow\\].", "\n\nWe are very grateful to Leonid Polterovich and Frol Zapolsky for their helpful comments and discussions. ", "We are also grateful to Daniel Rosen and Sheila Sandon for their useful remarks. ", "The first author is supported by the SFB 878 - Groups, Geometry and Actions. ", "The second author is supported by an ETH Postdoctoral Fellowship.", "\n\n\\[sec:Preliminaries\\]Preliminaries\n==================================\n\n\\[sub:Introductory-definitions\\]Introductory definitions\n--------------------------------------------------------\n\nSuppose $(\\Sigma,\\xi)$ is a connected closed coorientable contact manifold. ", "We denote by $\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$ the set of all smoothly parametrized paths $\\{\\varphi_{t}\\}_{0\\leq t\\leq1}$ with $\\varphi_{0}=\\mbox{id}_{\\Sigma}$. We introduce an equivalence relation $\\sim$ on $\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$ by saying that two paths $\\varphi$ and $\\psi$ are equivalent if $\\varphi_{1}=\\psi_{1}$ and we can connect $\\varphi$ and $\\psi$ via a smooth family $\\varphi^{s}=\\{\\varphi_{t}^{s}\\}_{0\\leq s,t\\leq1}$ of paths such that $\\varphi^{0}=\\varphi$, $\\varphi^{1}=\\psi$ and such that $\\varphi_{1}^{s}$ is independent of $s$. The universal cover $\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ of $\\mbox{Cont}_{0}(\\Sigma,\\xi)$ is then $\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)/\\sim$. We now give the precise definition of a Liouville manifold, and what it means for $\\Sigma$ to be Liouville fillable.", "\n\n\\[A-Liouville-manifold\\]A *Liouville* *domain* $(W_{0},d\\lambda_{0})$ is a compact exact symplectic manifold such that $\\lambda_{0}|_{\\partial W_{0}}$ is a positive contact form on $\\partial W_{0}$. Equivalently the vector field $Z_{0}$ on $W_{0}$ defined by $i_{Z_{0}}\\lambda_{0}=d\\lambda_{0}$ should be transverse to $\\partial W_{0}$ and point outwards. ", "$Z_{0}$ is called the *Liouville vector field*, and we denote by $l^{t}$ the flow of $Z_{0}$, which is defined for all $t\\leq0$, and thus induces an embedding $\\partial W_{0}\\times(0,1]\\rightarrow W_{0}$ defined by $(x,r)\\mapsto l^{\\log\\, r}(x)$. Thus we can form the *completion* $(W,d\\lambda)$ of $(W_{0},d\\lambda_{0})$ by by attaching $\\partial W_{0}\\times[1,\\infty)$ onto $\\partial W_{0}$: $$W:=W_{0}\\cup_{\\partial W_{0}}(\\partial W_{0}\\times[1,\\infty)).$$ We extend $\\lambda_{0}$ to a 1-form $\\lambda$ on $W$ by setting $\\lambda=r\\lambda|_{\\partial W_{0}}$ on $\\partial W_{0}\\times[1,\\infty)$. Thus $d\\lambda$ is a symplectic form on $W$. Similarly we extend $Z_{0}$ to a vector field $Z$ on $W$ by setting $Z=r\\partial_{r}$ on $\\partial W_{0}\\times[1,\\infty)$. One calls $(W,d\\lambda)$ a *Liouville manifold* - thus Liouville manifolds are exact non-compact symplectic manifolds obtained by completing a Liouville domain.", "\n\n\\[def - liouville fillable\\]We say that a closed connected coorientable contact manifold $(\\Sigma,\\xi)$ is *Liouville fillable* if there exists a Liouville domain $(W_{0},d\\lambda_{0})$ such that $\\Sigma=\\partial W_{0}$ and such that if $\\alpha:=\\lambda|_{\\Sigma}$ then $\\alpha$ is a positive contact form on $\\Sigma$ with $\\ker\\,\\alpha=\\xi$. By a slight abuse of notation we will generally refer to the Liouville manifold $(W,d\\lambda)$ obtained from completing $(W_{0},d\\lambda_{0})$ as ‘the’ filling of $\\Sigma$.\n\nThe *symplectization* $S\\Sigma$ of a contact manifold $(\\Sigma,\\xi=\\ker\\,\\alpha)$ is the symplectic manifold $\\Sigma\\times(0,\\infty)$ equipped with the symplectic form $d(r\\alpha)$. If $\\Sigma$ is Liouville fillable with filling $(W,d\\lambda)$ then one can embed $S\\Sigma\\hookrightarrow W$ by using the flow $l^{t}$ of the Liouville vector field $Z$ of $V$. Next we recall how to lift a path $\\varphi=\\{\\varphi_{t}\\}_{0\\leq t\\leq1}$ to a symplectic isotopy $\\Phi=\\{\\Phi_{t}\\}_{0\\leq t\\leq1}$ on the symplectization $S\\Sigma$. Write $\\varphi_{t}^{*}\\alpha=\\rho_{t}\\varphi_{t}$. Then define $\\Phi_{t}:S\\Sigma\\rightarrow S\\Sigma$ by $$\\Phi_{t}(x,r):=\\left(\\varphi_{t}(x),\\frac{r}{\\rho_{t}(x)}\\right).\\label{eq:lift path to symplectization}$$ The path $\\Phi_{t}$ is Hamiltonian (in fact it preserves $\\lambda$) with Hamiltonian function $$H_{t}(x,r):=rh_{t}(x):S\\Sigma\\rightarrow\\mathbb{R}.\\label{eq:Ht}$$ For later use we note the following lemma, whose proof is a simple computation which we omit.", "\n\n\\[lem:computations\\]Suppose that $\\varphi=\\{\\varphi_{t}\\}_{0\\leq t\\leq1}$ and $\\psi=\\{\\psi_{t}\\}_{0\\leq t\\leq1}$ belong to $\\mathcal{P}\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$, and let $\\vartheta\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$. Let $X_{t}$ and $Y_{t}$ denote the infinitesimal generators of $\\varphi$ and $\\psi$ respectively, so that $$X_{t}\\circ\\varphi_{t}=\\frac{\\partial}{\\partial t}\\varphi_{t},\\ \\ \\ Y_{t}\\circ\\psi_{t}=\\frac{\\partial}{\\partial t}\\psi_{t}.$$ We define smooth positive functions $\\rho_{t}$, $\\sigma_{t}$ and $\\kappa$ via $$\\varphi_{t}^{*}\\alpha=\\rho_{t}\\alpha,\\ \\ \\ \\psi_{t}^{*}\\alpha=\\sigma_{t}\\alpha,\\ \\ \\ \\vartheta^{*}\\alpha=\\kappa\\alpha.$$ Let $h_{t}$ denote the contact Hamiltonian of $\\varphi$ and $k_{t}$ denote the contact Hamiltonian of $\\psi$. Then:\n\n1. ", " One has $(\\varphi_{t}\\psi_{t})^{*}\\alpha=\\tau_{t}\\alpha$ where $$\\tau_{t}(x)=\\rho_{t}(\\psi_{t}(x))\\sigma_{t}(x),$$ and the contact Hamiltonian $l_{t}$ of $\\varphi\\psi$ is given by $$l_{t}(x)=h_{t}(x)+\\rho_{t}(\\varphi_{t}^{-1}(x))k_{t}(\\varphi_{t}^{-1}(x)).$$\n\n2. ", " One has $(\\varphi_{t}\\psi_{t}^{-1})^{*}\\alpha=\\upsilon_{t}\\alpha$ where $$\\upsilon_{t}(x)=\\frac{\\rho_{t}(\\psi_{t}^{-1}(x))}{\\sigma_{t}(\\psi_{t}^{-1}(x))},$$ and the contact Hamiltonian $m_{t}$ of $\\varphi\\psi^{-1}$ is given by $$m_{t}(x)=h_{t}(x)-\\frac{\\rho_{t}(\\varphi_{t}^{-1}(x))k_{t}(\\mu_{t}^{-1}(x))}{\\sigma_{t}(\\varphi_{t}^{-1}(x))}.$$\n\n3. ", " Let $\\vartheta\\varphi\\vartheta^{-1}\\in\\mathcal{P}\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ denote the path $t\\mapsto\\vartheta\\varphi_{t}\\vartheta^{-1}$. Then one has $(\\vartheta\\varphi_{t}\\vartheta^{-1})^{*}\\alpha=\\chi_{t}\\alpha$ where $$\\chi_{t}(x)=\\frac{\\kappa(\\varphi_{t}(\\vartheta^{-1}(x))\\rho_{t}(\\vartheta^{-1}(x))}{\\kappa(\\vartheta^{-1}(x))},$$ and the contact Hamiltonian $q_{t}$ of $\\vartheta\\varphi\\vartheta^{-1}$ is given by $$q_{t}(x):=\\kappa(\\vartheta^{-1}(x))h_{t}(\\vartheta^{-1}(x)).$$\n\nWe next define precisely what it means for a contact form $\\alpha$ generating $\\xi$ to be of Morse-Bott type.", "\n\n\\[def morse bott type\\]A contact 1-form $\\alpha\\in\\Omega^{1}(\\Sigma)$ generating $\\xi$ is said to be of *Morse-Bott type* if for each $T>0$, the set $P_{T}\\subset\\Sigma$ of points $x\\in\\Sigma$ satisfying $\\theta^{T}(x)=x$ is a closed submanifold of $\\Sigma$, with the property that $\\mbox{rank}\\, d\\alpha|_{P_{T}}$ is locally constant and $$T_{x}P_{T}=\\ker\\left(D\\theta^{T}(x)-\\mbox{id}_{T_{x}\\Sigma}\\right)\\ \\ \\ \\mbox{for all }x\\in P_{T}.$$ A Liouville fillable contact manifold $(\\Sigma,\\xi)$ is said to admit a *Morse-Bott Liouville filling* if there exists a filling $(W,d\\lambda)$ such that $\\alpha:=\\lambda|_{\\Sigma}$ is of Morse-Bott type.", "\n\nLet us now recall the definition of a translated point of a contactomorphism. ", "This notion was introduced by Sandon in [@Sandon2010].", "\n\n\\[def trans poin\\]Let $(\\Sigma,\\xi)$ denote a closed connected coorientable contact manifold, and fix a contact form $\\alpha\\in\\Omega^{1}(\\Sigma)$ generating $\\xi$. Fix $\\varphi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$. We can write $\\varphi^{*}\\alpha=\\rho\\alpha$ for a smooth positive function $\\rho$ on $\\Sigma$. A *translated point* of $\\varphi$ is a point $x\\in$ with the property that there exists $\\eta\\in\\mathbb{R}$ such that $$\\varphi(x)=\\theta^{\\eta}(x),\\ \\ \\ \\mbox{and}\\ \\ \\ \\rho(x)=1.\\label{eq:TP}$$ We call $\\eta$ the *time-shift* of $x$. Note that if the leaf $\\{\\theta^{t}(x)\\}_{t\\in\\mathbb{R}}$ is closed (which is always the case when $\\alpha$ is periodic) then the time-shift is not unique. ", "Indeed, if the leaf $\\{\\theta^{t}(x)\\}_{t\\in\\mathbb{R}}$ has period $T>0$ then $\\varphi(x)=\\theta^{\\eta+\\nu T}(x)$ for all $\\nu\\in\\mathbb{Z}$. In this case one needs to keep track of the $\\eta$. Finally we say a point $x\\in\\Sigma$ is an *iterated translated point* of $\\varphi$ if it is a translated point for some iteration $\\varphi^{\\nu}$.\n\nNow let us define what it means for a translated point $x$ of an element $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ to be *contractible* with respect to a Liouville filling $(W,d\\lambda)$.\n\n\\[def contractible translated point\\]Let $(W,d\\lambda)$ denote a Liouville filling of $(\\Sigma,\\xi)$, with $\\alpha=\\lambda|_{\\Sigma}$. Suppose $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ and $x$ is a translated point of $\\varphi$ of time-shift $\\eta$. We say that the pair $(x,\\eta)$ is a *contractible* translated point if the continuous loop $l:\\mathbb{R}/2\\mathbb{Z}\\rightarrow\\Sigma$ obtained from concatenating the path $\\{\\varphi_{t}(x)\\}_{0\\leq t\\leq1}$ with the path $\\{\\theta^{-\\eta t}(x)\\}_{0\\leq t\\leq1}$ is contractible in $W$. It is easy to see that this does not depend on path $\\{\\varphi_{t}\\}_{0\\leq t\\leq1}\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$ representing $\\varphi$. A point $(x,\\eta)$ is a *contractible iterated translated point* of $\\varphi$ if $(x,\\eta)$ is a contractible translated point of some iterate $\\varphi^{\\nu}$.\n\nNote that if the leaf $\\{\\theta^{t}(x)\\}_{t\\in\\mathbb{R}}$ is closed then it is important to take into account which time-shift we are working with. ", "For instance, suppose $\\varphi=\\theta^{\\tau}$ for some $\\tau>0$ and the leaf $\\{\\theta^{t}(x)\\}_{t\\in\\mathbb{R}}$ is closed and non-contractible in $W$. Then $(x,\\tau)$ is a contractible translated point of $\\varphi$ but for all $\\nu\\in\\mathbb{Z}\\backslash\\{0\\}$, $(x,\\tau+\\nu T)$ is not contractible.", "\n\nFor us, the usefulness of translated points stems from the fact that the translated points of $\\varphi$ are the generators of the Rabinowitz Floer homology associated to $\\varphi$, when the Rabinowitz Floer homology is well defined; see Lemma \\[lem:critical points\\] or [@AlbersMerry2011] for more information.", "\n\nThe Rabinowitz action functional $\\mathcal{A}_{\\varphi}$ on $\\Lambda(S\\Sigma)\\times\\mathbb{R}$\n----------------------------------------------------------------------------------------------\n\nWrite $\\Lambda(S\\Sigma):=C_{\\textrm{contr}}^{\\infty}(S^{1},S\\Sigma)$ for the component of the free loop space containing the contractible loops.", "\n\nFix a path $\\varphi\\in\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$ as above, and let $H_{t}$ denote the Hamiltonian (\\[eq:Ht\\]). ", "We define the *perturbed Rabinowitz action functional* $$\\mathcal{A}_{\\varphi}:\\Lambda(S\\Sigma)\\times\\mathbb{R}\\rightarrow\\mathbb{R}$$ by $$\\mathcal{A}_{\\varphi}(u,\\eta):=\\int_{0}^{1}u^{*}\\lambda-\\eta\\int_{0}^{1}\\beta(t)(r(t)-1)dt-\\int_{0}^{1}\\dot{\\chi}(t)H_{\\chi(t)}(u(t))dt,\\label{eq:Rab functional}$$ where $\\beta:\\mathbb{R}/\\mathbb{Z}\\rightarrow\\mathbb{R}$ is a smooth function with $$\\beta(t)=0\\ \\forall t\\in[\\tfrac{1}{2},1],\\ \\ \\ \\mbox{and}\\ \\ \\ \\int_{0}^{1}\\beta(t)dt=1,$$ and $\\chi:[0,1]\\rightarrow[0,1]$ is a smooth monotone map with $\\chi(\\tfrac{1}{2})=0$, $\\chi(1)=1$, and $r(t)$ is the $\\mathbb{R}$-component of the map $u:S^{1}\\to S\\Sigma=\\Sigma\\times\\mathbb{R}$. Denote by $\\mbox{Crit}(\\mathcal{A}_{\\varphi})$ the set of critical points of $\\mathcal{A}_{\\varphi}$, and denote by $\\mbox{Spec}(\\mathcal{A}_{\\varphi}):=\\mathcal{A}_{\\varphi}(\\mbox{Crit}(\\mathcal{A}_{\\varphi}))$.\n\nIn this paper we define the Rabinowitz action functional only on the contractible component of the free loop space, as all the applications we have in mind here pertain only to the contractible component. ", "Nevertheless, it is possible to carry out all of our constructions on the full loop space without any changes. ", "This is because the symplectic form (on $S\\Sigma$ or on the Liouville filling $W$) is exact, and so there are no ambiguities in the definition of $\\mathcal{A}_{\\varphi}$ on non-contractible loops.", "\n\nThe following lemma explains why the perturbed Rabinowitz action functional is useful in detecting translated points. ", "It is a minor variant of an argument originally due to the first author and Frauenfelder [@AlbersFrauenfelder2010c Proposition 2.4]. ", "For the convenience of the reader we recall the proof again here.", "\n\n\\[lem:critical points\\][@AlbersMerry2011]A pair $(u,\\eta)$ is a critical point of $\\mathcal{A}_{\\varphi}$ if and only if, writing $u(t)=(x(t),r(t))\\in\\Sigma\\times(0,\\infty)$, $p:=x(\\tfrac{1}{2})$ is a translated point of $\\varphi$, with time-shift $-\\eta$ (taken modulo the period of the leaf $\\{\\theta^{t}(p)\\}_{t\\in\\mathbb{R}}$ if $\\{\\theta^{t}(p)\\}_{t\\in\\mathbb{R}}$ is a closed leaf). ", "Moreover if $(u,\\eta)$ is a critical point of $\\mathcal{A}_{\\varphi}$ then $$\\mathcal{A}_{\\varphi}(u,\\eta)=\\eta.\\label{eq:value on crit}$$ If $\\varphi$ is an exact path of contactomorphisms then $r(t)\\equiv1$ for every critical point $(u=(x,r),\\eta)$.\n\nDenote by $\\Phi_{t}:S\\Sigma\\rightarrow S\\Sigma$ the symplectic isotopy (\\[eq:lift path to symplectization\\]). ", "A pair $(u,\\eta)$ with $u=(x,r):S^{1}\\rightarrow\\Sigma\\times(0,\\infty)$ belongs to $\\mbox{Crit}(\\mathcal{A}_{\\varphi})$ if and only if $$\\begin{cases}\n\\dot{u}(t)=\\eta\\beta(t)R(x(t))+\\dot{\\chi}(t)X_{H_{\\chi(t)}}(u(t)),\\\\\n\\int_{0}^{1}\\beta(t)(r(t)-1)dt=0.", "\n\\end{cases}\\label{eq:critical points}$$ Thus for $t\\in[0,\\tfrac{1}{2}]$, we have $r(t)=1$ and $\\dot{x}(t)=-\\eta R(x(t))$, and $x(1)=\\Phi_{1}(x(\\tfrac{1}{2}))$. Suppose $(u,\\eta)\\in\\mbox{Crit}(\\mathcal{A}_{\\varphi})$. Thus $u(\\tfrac{1}{2})=(\\theta^{-\\eta}(x(0)),1)$. Next, for $t\\in[\\tfrac{1}{2},1]$ we have $\\dot{u}(t)=\\dot{\\chi}(t)X_{H_{\\chi(t)}}(u(t))$. In particular, $\\varphi(x(\\tfrac{1}{2}))=\\theta^{-\\eta}(x(\\tfrac{1}{2}))$, and thus $x(\\tfrac{1}{2})$ is a translated point of $\\varphi$. Moreover the time shift of $x$ is given by $-\\eta$ mod 1.", "\n\nNext, we note that $$\\lambda(X_{H}(x,r))=dH(x,r)\\left(r\\frac{\\partial}{\\partial r}\\right)=H(x,r),$$ and hence $$\\begin{aligned}\n\\mathcal{A}_{\\varphi}(u,\\eta) & =\\int_{0}^{\\tfrac{1}{2}}(r\\alpha)(\\eta\\beta(u)R(x))dt+\\int_{\\tfrac{1}{2}}^{1}\\left[\\lambda(\\dot{\\chi}X_{H_{\\chi}}(u))-\\dot{\\chi}H_{\\chi}(u)\\right]dt\\nonumber \\\\\n & =\\eta+0.\\end{aligned}$$ Finally if $\\varphi_{t}$ is exact for each $t$ then the path $\\Phi_{t}$ of symplectomorphisms defined in (\\[eq:lift path to symplectization\\]) is simply given by $\\Phi_{t}(x,r)=(\\varphi_{t}(x),r)$, from which the last statement immediately follows.", "\n\n\\[the ubiquitous minus sign\\]We emphasize again that if $(u=(x,r),\\eta)$ is a critical point of $\\mathcal{A}_{\\varphi}$ then the time-shift of the translated point $x(\\tfrac{1}{2})$ is the *negative* of the action value. ", "This explains the Reeb flow will turn out to have a *negative* spectral number (cf. ", "part (1) of Theorem \\[Theorem-A\\]).", "\n\n\\[rem: spec only depends on the end\\]Note that $\\mbox{Spec}(\\mathcal{A}_{\\varphi})$ depends only on the terminal map $\\varphi_{1}$.\n\nWe now define what if means for $\\varphi$ to be non-degenerate. ", "In the periodic case we also introduce the notion of being non-resonant.", "\n\n\\[def: nondegen\\]A path $\\varphi$ is *non-degenerate* if $\\mathcal{A}_{\\varphi}:\\Lambda(S\\Sigma)\\times\\mathbb{R}\\rightarrow\\mathbb{R}$ is a Morse-Bott function. ", "In the periodic case we say that $\\varphi$ is *non-resonant* if $\\mbox{Spec}(\\mathcal{A}_{\\varphi})\\cap\\mathbb{Z}=\\emptyset$.\n\nThe identity $\\mbox{id}_{\\Sigma}$ is non-degenerate if and only if $\\alpha$ is of Morse-Bott type (see [@CieliebakFrauenfelder2009 Appendix B]). ", "In [@AlbersMerry2011 p7] we explained why a generic path $\\varphi$ is non-degenerate (actually a stronger result is true: for generic $\\varphi$ the functional $\\mathcal{A}_{\\varphi}$ is even Morse). ", "It is also easy to see that a generic $\\varphi$ is non-resonant.", "\n\nThe next lemma explains in why we pay particular attention to periodic contact manifolds. ", "It will prove crucial in the construction of the contact capacity (cf. ", "Section \\[sec:The-periodic-case - capacities\\], in particular Proposition \\[lem the use of periodicity\\]). ", "Fix $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ and fix a contactomorphism $\\psi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$.\n\n[\\[lemma:integral\\_critical\\_points\\_are\\_conjugation\\_invariant\\]]{}Assume $\\alpha$ is periodic. ", "If $(u=(x,r),\\eta)\\in\\mbox{\\emph{Crit}}(\\mathcal{A}_{\\varphi})$ with $\\eta\\in\\mathbb{Z}$ then there exists a critical point $(u_{1}=(x_{1},r_{1}),\\eta)$ of $\\mathcal{A}_{\\psi\\varphi\\psi^{-1}}$ with $x_{1}(\\tfrac{1}{2})=\\psi(x(\\tfrac{1}{2}))$. In particular, $$\\mbox{\\emph{Spec}}(\\mathcal{A}_{\\varphi})\\cap\\mathbb{Z}=\\emptyset\\ \\ \\ \\Leftrightarrow\\ \\ \\ \\mbox{\\emph{Spec}}(\\mathcal{A}_{\\psi\\varphi\\psi^{-1}})\\cap\\mathbb{Z}=\\emptyset.\\label{eq:key eq}$$ Moreover $(u,\\eta)$ is non-degenerate if and only if $(u_{1},\\eta)$ is non-degenerate.", "\n\nIf $(u,\\eta)\\in\\mbox{Crit}(\\mathcal{A}_{\\varphi})$ with $\\eta\\in\\mathbb{Z}$ then since $\\theta^{t}$ is 1-periodic, this means that if we write $u(t)=(x(t),r(t))$ then $x(\\tfrac{1}{2})$ is a ** fixed point of $\\varphi$. Thus $\\psi(x(\\tfrac{1}{2}))$ is a fixed point of $\\psi\\varphi\\psi^{-1}$. Thus by Lemma \\[lem:critical points\\] for each $\\nu\\in\\mathbb{Z}$ there exists a critical point $(u_{\\nu}=(x_{\\nu},r_{\\nu}),\\nu)$ of $\\mathcal{A}_{\\psi\\varphi\\psi^{-1}}$ with $x_{\\nu}(\\tfrac{1}{2})=\\psi(x_{\\nu}(\\tfrac{1}{2}))$. In particular, this is true for $\\nu=\\eta$.\n\nThe final statement follows from the fact that $(u,\\eta)$ is non-degenerate if and only if the linear map $D\\varphi((x(\\tfrac{1}{2}))-\\mathbb{1}$ is invertible, and similarly $(u_{1},\\eta)$ is non-degenerate if and only if the linear map $D(\\psi\\varphi\\psi^{-1})(\\psi(x((\\tfrac{1}{2})))-\\mathbb{1}$ is invertible.", "\n\nThe following lemma can be proved by arguing as in [@Schwarz2000 Lemma 3.8]. ", "The important thing to note is that in the lemma one does *not* necessarily assume that $\\varphi$ is non-degenerate.", "\n\n\\[lem:spectrum is discrete\\]For any path $\\varphi$ and any $T>0$ the set $$\\mbox{\\emph{Spec}}^{[-T,T]}(\\mathcal{A}_{\\varphi}):=\\mbox{\\emph{Spec}}(\\mathcal{A}_{\\varphi})\\cap[-T,T]$$ is a discrete subset of $\\mathbb{R}$. In particular, if $\\alpha$ is periodic then $\\mbox{\\emph{Spec}}(\\mathcal{A}_{\\varphi})$ is itself discrete for any path $\\varphi$.\n\n\\[sec:Rabinowitz-Floer-homology\\]Rabinowitz Floer homology\n----------------------------------------------------------\n\nLet us now assume that $(\\Sigma,\\xi)$ is Liouville fillable with a Morse-Bott Liouville filling $(W,d\\lambda)$. We would like to extend $\\mathcal{A}_{\\varphi}$ to a functional defined on all of $\\Lambda(W)\\times\\mathbb{R}$, where $\\Lambda(W):=C_{\\textrm{contr}}^{\\infty}(S^{1},W)$ as before. ", "In order to do this we must extend the function $(x,r)\\mapsto r-1$ and the Hamiltonian $H_{t}$ to functions defined on all of $W$. At the same time, it is convenient to truncate them. ", "As in [@AlbersFrauenfelder2010c], we proceed as follows. ", "Define $m:W\\rightarrow\\mathbb{R}$ so that $$m(x,r):=r-1\\ \\ \\ \\mbox{on }\\Sigma\\times(\\tfrac{1}{2},\\infty),$$ $$\\frac{\\partial m}{\\partial r}(x,r)\\geq0\\ \\ \\ \\mbox{for all }(x,r)\\in S\\Sigma,$$ $$m|_{W\\backslash S\\Sigma}:=-\\tfrac{3}{4}.$$ Next, for $\\kappa>0$ let $\\varepsilon_{\\kappa}\\in C^{\\infty}([0,\\infty),[0,1])$ denote a smooth function such that $$\\varepsilon_{\\kappa}(r)=\\begin{cases}\n1, & r\\in[e^{-\\kappa},e^{\\kappa}],\\\\\n0, & r\\in[0,e^{-\\kappa}]\\cup[e^{\\kappa}+1,\\infty),\n\\end{cases}$$ and such that $$0\\leq\\varepsilon_{\\kappa}'(r)\\leq2e^{2\\kappa}\\ \\ \\ \\mbox{for }r\\in[e^{-2\\kappa},e^{-\\kappa}],$$ $$-2\\leq\\varepsilon_{\\kappa}'(r)\\leq0\\ \\ \\ \\mbox{for }r\\in[e^{\\kappa},e^{\\kappa}+1].$$ Then define $H_{t}^{\\kappa}:W\\rightarrow\\mathbb{R}$ by setting $H_{t}^{\\kappa}|_{W\\backslash S\\Sigma}=-\\tfrac{3}{4}$ and $$H_{t}^{\\kappa}(x,r):=\\varepsilon_{\\kappa}(r)H_{t}(x,r)\\ \\ \\ \\mbox{for }(x,r)\\in S\\Sigma.\\label{eq:truncating H}$$ We denote by $\\mathcal{A}_{\\varphi}^{\\kappa}:\\Lambda(W)\\times\\mathbb{R}\\rightarrow\\mathbb{R}$ the Rabinowitz action functional defined using the Hamiltonians $m$ and $H_{t}^{\\kappa}$: $$\\mathcal{A}_{\\varphi}^{\\kappa}(u,\\eta):=\\int_{0}^{1}u^{*}\\lambda-\\eta\\int_{0}^{1}\\beta(t)m(u(t))dt-\\int_{0}^{1}\\dot{\\chi}(t)H_{\\chi(t)}^{\\kappa}(u(t))dt.$$\n\n\\[hofer norm def\\]Now let us recall the definition of the *oscillation ‘norm’*[^3] ** on $\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$. Firstly, suppose $\\{\\varphi_{t}\\}_{0\\leq t\\leq1}\\in\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$. Let $h_{t}:\\Sigma\\rightarrow\\mathbb{R}$ denote the contact Hamiltonian. ", "The *oscillation ‘norm’* $\\left\\Vert h\\right\\Vert _{\\textrm{osc}}$ is defined by $$\\left\\Vert h\\right\\Vert _{\\textrm{osc}}:=\\left\\Vert h\\right\\Vert _{+}+\\left\\Vert h\\right\\Vert _{-},\\label{eq:osc 1}$$ $$\\left\\Vert h\\right\\Vert _{+}:=\\int_{0}^{1}\\max_{x\\in\\Sigma}h_{t}(x)dt,\\ \\ \\ \\left\\Vert h\\right\\Vert _{-}:=-\\int_{0}^{1}\\min_{x\\in\\Sigma}h_{t}(x)dt.\\label{eq:osc 2}$$ We then define the oscillation norm $\\left\\Vert \\varphi\\right\\Vert $ for $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ by taking the infimum of the oscillation norms $\\left\\Vert h_{t}\\right\\Vert _{\\textrm{osc}}$ over all possible paths $\\{\\varphi_{t}\\}_{0\\leq t\\leq1}$ representing $\\varphi$ (with corresponding contact Hamiltonians $h_{t}$).", "\n\nDenote by $\\Phi^{\\kappa}\\in\\mbox{Ham}_{c}(W,d\\lambda)$ the Hamiltonian diffeomorphism generated by $H_{t}^{\\kappa}$. The ****** *Hofer norm* $\\left\\Vert \\Phi^{\\kappa}\\right\\Vert $ of $H_{t}^{\\kappa}$ is related to $\\left\\Vert \\varphi\\right\\Vert $ by $$\\left\\Vert \\Phi^{\\kappa}\\right\\Vert \\leq e^{\\kappa}\\left\\Vert \\varphi\\right\\Vert .$$ Here by definition the Hoer norm $\\left\\Vert \\Phi^{\\kappa}\\right\\Vert $ is the infimum of the oscillation norms of all possible Hamiltonians generating $\\Phi^{\\kappa}$. One such Hamiltonian is $H_{t}^{\\kappa}$, and it is clear that $\\left\\Vert H^{\\kappa}\\right\\Vert _{\\textrm{osc}}\\leq e^{\\kappa}\\left\\Vert h\\right\\Vert _{\\textrm{osc}}$.\n\nSuppose $\\varphi\\in\\mathcal{P}\\mbox{Cont}_{0}(\\Sigma,\\xi)$. Let $\\rho_{t}:\\Sigma\\rightarrow(0,\\infty)$ is defined by $\\varphi_{t}^{*}\\alpha=\\rho_{t}\\alpha$. Define a constant $\\kappa(\\varphi)>0$ by $$\\kappa(\\varphi):=8\\max\\left\\{ \\left|\\frac{\\dot{\\rho}_{t}(x)}{\\rho_{t}^{2}(x)}\\right|\\mid(t,x)\\in[0,1]\\times\\Sigma\\right\\} .\\label{eq:c phi}$$ Note that if $\\varphi$ is exact then $\\kappa(\\varphi)=0$.\n\nIn [@AlbersMerry2011 Proposition 2.5] we proved:\n\n\\[prop:linfinity-1\\]If $\\kappa>\\kappa(\\varphi)$ then if $(u,\\eta)\\in\\mbox{\\emph{Crit}}(\\mathcal{A}_{\\varphi}^{\\kappa})$ then $u(S^{1})\\subseteq S\\Sigma$, and moreover if we write $u(t)=(x(t),r(t))$ then $r(S^{1})\\subseteq(e^{-\\kappa/2},e^{\\kappa/2})$.\n\nFix $w>0$. Suppose we rescale $\\alpha$ to $w\\alpha$. This corresponds to identifying $\\Sigma$ with $\\Sigma\\times\\{w\\}$ instead of $\\Sigma\\times\\{1\\}$ inside $S\\Sigma$. Note that the contact Hamiltonian of $w\\alpha$ is $wh_{t}$, and $\\varphi_{t}^{*}(w\\alpha)=w\\rho_{t}\\alpha$. In particular, the oscillation norm $\\left\\Vert \\{\\varphi_{t}\\}\\right\\Vert _{\\textrm{osc};w}$ with respect to $w\\alpha$ is given simply by $\\left\\Vert \\{\\varphi_{t}\\}\\right\\Vert _{\\textrm{osc};w}=w\\left\\Vert \\{\\varphi_{t}\\}\\right\\Vert _{\\textrm{osc}}$. Moreover, if $\\wp_{W}(\\Sigma,\\alpha)$ denote the smallest period of a Reeb orbit of $\\alpha$ which is contractible in $W$ (see Definition \\[smallest period\\]) one clearly has $\\wp_{W}(\\Sigma,w\\alpha)=w\\wp_{W}(\\Sigma,\\alpha).$ However the constant $\\kappa(\\varphi)$ rescales to $$\\kappa_{w}(\\varphi)=\\frac{1}{w}\\kappa(\\varphi).$$ Thus we conclude the following result, which will be useful in the proof of Proposition \\[rem strict inequalities\\] below.", "\n\n\\[cor:needed for translated point\\]Suppose $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ satisfies $\\left\\Vert \\varphi\\right\\Vert <\\wp_{W}(\\Sigma,\\alpha)$. Suppose $\\varphi$ has contact Hamiltonian $h_{t}$. Given $\\kappa,w>0$ let $H_{t}^{\\kappa;w}:W\\rightarrow W$ be defined by $$H_{t}^{\\kappa;w}(x,r):=\\varepsilon_{\\kappa}(wr)wrh_{t}(x),\\ \\ \\ H_{t}^{\\kappa;w}|_{W\\backslash S\\Sigma}=-\\tfrac{3}{4}.$$ Then if $\\Phi^{\\kappa;w}\\in\\mbox{\\emph{Ham}}_{c}(W,d\\lambda)$ is generated by $H_{t}^{\\kappa;w}$ then for $w\\gg0$ one can choose $\\kappa>\\kappa_{w}(\\varphi)$ such that $$\\left\\Vert \\Phi^{\\kappa;w}\\right\\Vert <\\wp_{W}(\\Sigma,w\\alpha).$$\n\nIf $\\varphi$ is non-degenerate in the sense of Definition \\[def: nondegen\\], then as explained in [@AlbersMerry2011], Lemma \\[prop:linfinity-1\\] allows to define for $a,b\\in[-\\infty,\\infty]\\backslash\\mbox{Spec}(\\mathcal{A}_{\\varphi})$ the *Rabinowitz Floer homology* $$\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W).$$ This is a semi-infinite dimensional Morse theory associated to the functional $\\mathcal{A}_{\\varphi}^{\\kappa}$ (for some $\\kappa>\\kappa_{0}(\\varphi)$), and we sketch the definition here and refer to e.g. [@AlbersFrauenfelder2010c; @AlbersMerry2011] for more information.", "\n\nLet $\\mathcal{J}_{\\textrm{conv}}(S\\Sigma)$ denote the set of time dependent almost complex structures $J=\\{J_{t}\\}_{t\\in S^{1}}$ on $S\\Sigma$ that are $d(r\\alpha)$-compatible and that are *convex*. ", "Here we use the sign convention that $J$ is $d\\lambda$-compatible if $d(r\\alpha)(J\\cdot,\\cdot)$ defines a Riemannian metric, and to say that $J$ is convex is to ask that there exists $S_{0}>0$ such that $$dr\\circ J_{t}=d(r\\alpha)\\ \\ \\ \\mbox{on }\\Sigma\\times[S_{0},\\infty)\\label{eq:convex type}$$ (in particular, $J$ is independent of $t$ on $\\Sigma\\times[S_{0},\\infty)$). ", "We denote by $\\mathcal{J}_{\\textrm{conv}}(W)$ the set of time-dependent almost complex structures $J=\\{J_{t}\\}_{t\\in S^{1}}$ with the property that $J|_{S\\Sigma}\\in\\mathcal{J}_{\\textrm{conv}}(S\\Sigma)$.\n\nGiven $J\\in\\mathcal{J}_{\\textrm{conv}}(W)$ we can define an $L^{2}$-inner product $\\left\\langle \\left\\langle \\cdot,\\cdot\\right\\rangle \\right\\rangle _{J}$ on $\\Lambda(W)\\times\\mathbb{R}$: for $(u,\\eta)\\in\\Lambda(W)\\times\\mathbb{R}$, $\\zeta,\\zeta'\\in\\Gamma(u^{*}TW)$ and $b,b'\\in\\mathbb{R}$, set $$\\left\\langle \\left\\langle (\\zeta,b),(\\zeta',b')\\right\\rangle \\right\\rangle _{J}:=\\int_{0}^{1}d\\lambda(J_{t}\\zeta,\\zeta')dt+bb'.\\label{eq:l2 norm}$$ We denote by $\\nabla_{J}\\mathcal{A}_{\\varphi}^{\\kappa}$ the gradient of $\\mathcal{A}_{\\varphi}^{\\kappa}$ with respect to $\\left\\langle \\left\\langle \\cdot,\\cdot\\right\\rangle \\right\\rangle _{J}$.\n\nIn this paper all sign conventions are the same as in [@AbbondandoloSchwarz2009; @AlbersMerry2011].", "\n\nAssume that $\\varphi$ is non-degenerate and fix $\\kappa>\\kappa_{0}(\\varphi)$ and $J\\in\\mathcal{J}_{\\textrm{conv}}(W)$. By assumption $\\mathcal{A}_{\\varphi}^{\\kappa}$ is a Morse-Bott function. ** ", "Pick a Morse function $g:\\mbox{Crit}(\\mathcal{A}_{\\varphi}^{\\kappa})\\rightarrow\\mathbb{R}$, and choose a Riemannian metric $\\varrho$ on $\\mbox{Crit}(\\mathcal{A}_{\\varphi}^{\\kappa})$ such that the negative gradient flow of $\\nabla_{\\varrho}g$ is *Morse-Smale*. ", "Given two critical points $w^{-},w^{+}\\in\\mbox{Crit}(g)$, with $w^{\\pm}=(u^{\\pm},\\eta^{\\pm})$, we denote by $\\mathcal{M}_{w^{-},w^{+}}(\\mathcal{A}_{\\varphi}^{\\kappa},g,J,\\varrho)$ the moduli space of *gradient flow lines with cascades* of $-\\nabla_{J}\\mathcal{A}_{\\varphi}^{\\kappa}$ and $-\\nabla_{\\varrho}g$ running from $w^{-}$ to $w^{+}$. See [@Frauenfelder2004 Appendix A] or [@CieliebakFrauenfelder2009 Appendix A] for the precise definition.", "\n\nIntroduce a grading on $\\mbox{Crit}(g)$ by setting $$\\mu(u,\\eta):=\\begin{cases}\n\\mu_{\\textrm{CZ}}(u)-\\frac{1}{2}\\dim_{(u,\\eta)}\\mbox{Crit}(\\mathcal{A}_{\\varphi}^{\\kappa})+\\mbox{ind}_{g}(u,\\eta), & \\eta>0,\\\\\n\\mu_{\\textrm{CZ}}(u)-\\frac{1}{2}\\dim_{(u,\\eta)}\\mbox{Crit}(\\mathcal{A}_{\\varphi}^{\\kappa})+\\mbox{ind}_{g}(u,\\eta)+1, & \\eta<0,\\\\\n1-n+\\mbox{ind}_{g}(u,\\eta), & \\eta=0.", "\n\\end{cases}\\label{eq:grading}$$ Here $\\mu_{\\textrm{CZ}}(u)$ denotes the Conley-Zehnder index of the loop $t\\mapsto u(t/\\eta)$ and $\\dim_{(u,\\eta)}\\mbox{Crit}(\\mathcal{A}_{\\varphi}^{\\kappa})$ denotes the local dimension of $\\mbox{Crit}(\\mathcal{A}_{\\varphi}^{\\kappa})$ at $(u,\\eta)$. Actually in most cases of interest in this paper, we may assume that $\\mathcal{A}_{\\varphi}$ is actually Morse. ", "In this case the Morse function $g$ is taken to be identically zero, and (\\[eq:grading\\]) continues to hold.", "\n\n\\[Our-grading-convention\\] Our normalization convention for the Conley-Zehnder index is that if $H$ is a $C^{2}$-small Morse function on $W$ and $x$ is a critical point of $W$ then $$\\mu_{\\textrm{CZ}}(x)=n-\\mbox{ind}_{H}(x),$$ where $\\mbox{ind}_{H}(x)$ denotes the Morse index of $x$.\n\nGiven $-\\infty<a<b<\\infty$ denote by $\\mbox{RFC}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi}^{\\kappa},g):=\\mbox{Crit}_{*}^{(a,b)}(g)\\otimes\\mathbb{Z}_{2}$, where $\\mbox{Crit}_{*}^{(a,b)}(g)$ denotes the set of critical points $w$ of $g$ with $a<\\mathcal{A}_{\\varphi}(w)<b$. We only do this when $a,b\\notin\\mbox{Spec}(\\mathcal{A}_{\\varphi})$, even if this is not explicitly stated. ", "Generically the moduli spaces $\\mathcal{M}_{w^{-},w^{+}}(\\mathcal{A}_{\\varphi},g,J,\\varrho)$ carry the structure of finite dimensional smooth manifolds, whose components of dimension zero are compact. ", "One defines a boundary operator $\\partial$ on $\\mbox{RFC}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi}^{\\kappa},g)$ by counting the elements of the zero-dimensional parts of the moduli spaces $\\mathcal{M}_{w^{-},w^{+}}(\\mathcal{A}_{\\varphi},g,J,\\varrho)$.\n\nThe homology $\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W):=\\mbox{H}_{*}(\\mbox{RFC}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi}^{\\kappa},g),\\partial)$ does not depend on any of the auxiliary choices we made. ", "We emphasize though that $\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W)$ depends on the choice of filling $(W,d\\lambda)$. Finally we define $$\\mbox{RFH}_{*}^{b}(\\mathcal{A}_{\\varphi},W):=\\underset{a\\downarrow-\\infty}{\\underrightarrow{\\lim}}\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W),$$ $$\\mbox{RFH}_{*}^{(a,\\infty)}(\\mathcal{A}_{\\varphi},W):=\\underset{b\\uparrow\\infty}{\\underleftarrow{\\lim}}\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W),$$ $$\\mbox{RFH}_{*}(\\mathcal{A}_{\\varphi},W):=\\underset{a\\downarrow-\\infty}{\\underrightarrow{\\lim}}\\underset{b\\uparrow\\infty}{\\underleftarrow{\\lim}}\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W)\\label{eq:inverse and direct limit}$$ (the order of the limits in (\\[eq:inverse and direct limit\\]) matters). ", "As pointed out by Ritter [@Ritter2010], it follows from work of Cieliebak-Frauenfelder-Oancea [@CieliebakFrauenfelderOancea2010] that the Rabinowitz Floer homology $\\mbox{RFH}_{*}(\\Sigma,W)$ vanishes if and only if the symplectic homology $\\mbox{SH}_{*}(W)$ vanishes.", "\n\nWe briefly summarize now the key properties that we will need about the Rabinowitz Floer homology $\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W)$:\n\n1. ", " \\[enu:The-Rabinowitz-Floer1\\]The Rabinowitz Floer homology is independent of $\\varphi$ in the following strong sense. ", "There is a universal object $\\mbox{RFH}_{*}(\\Sigma,W)$ (which may be thought as corresponding to the case $\\varphi=\\mbox{id}_{\\Sigma}$, together with canonical isomorphisms $$\\zeta_{\\varphi}:\\mbox{RFH}_{*}(\\Sigma,W)\\rightarrow\\mbox{RFH}_{*}(\\mathcal{A}_{\\varphi},W).$$ Given two paths $\\varphi$ and $\\psi$, there is a map $\\zeta_{\\varphi,\\psi}:\\mbox{RFH}_{*}(\\mathcal{A}_{\\varphi},W)\\rightarrow\\mbox{RFH}_{*}(\\mathcal{A}_{\\psi},W)$ with the property that $$\\zeta_{\\psi}=\\zeta_{\\varphi,\\psi}\\circ\\zeta_{\\varphi}.\\label{eq:continuation homomorphisms}$$\n\n2. ", " If $a\\leq b\\leq\\infty$ there is a natural map $$j_{\\varphi}^{a,b}:\\mbox{RFH}_{*}^{a}(\\mathcal{A}_{\\varphi},W)\\rightarrow\\mbox{RFH}_{*}^{b}(\\mathcal{A}_{\\varphi},W).$$ Similarly there is a natural map $$p_{\\varphi}^{a,b}:\\mbox{RFH}_{*}^{b}(\\mathcal{A}_{\\varphi},W)\\rightarrow\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W).$$ If $b=\\infty$ we abbreviate $j_{\\varphi}^{a,\\infty}=j_{\\varphi}^{a}$, and we write $j^{a}$ for the map $\\mbox{RFH}_{*}^{a}(\\Sigma,W)\\rightarrow\\mbox{RFH}_{*}(\\Sigma,W)$, with similar conventions for the maps $p_{\\varphi}^{a,b}$. If $\\mbox{Spec}(\\mathcal{A}_{\\varphi})\\cap[a,b]=\\emptyset$ then the map $j_{\\varphi}^{a,b}:\\mbox{RFH}_{*}^{a}(\\mathcal{A}_{\\varphi},W)\\rightarrow\\mbox{RFH}_{*}^{b}(\\mathcal{A}_{\\varphi},W)$ is an isomorphism and $p_{\\varphi}^{a,b}:\\mbox{RFH}_{*}^{b}(\\mathcal{A}_{\\varphi},W)\\rightarrow\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W)$ is the zero map (as $\\mbox{RFH}_{*}^{(a,b)}(\\mathcal{A}_{\\varphi},W)=0$).", "\n\n3. ", " Moreover there is a filtered version of (\\[eq:continuation homomorphisms\\]), which gives the existence of a maps $$\\zeta_{\\varphi,\\psi}^{a}:\\mbox{RFH}_{*}^{a}(\\mathcal{A}_{\\varphi},W)\\rightarrow\\mbox{RFH}_{*}^{a+K(\\varphi,\\psi)}(\\mathcal{A}_{\\psi},W)\\label{eq:fil cont hom}$$ for some constant $K(\\varphi,\\psi)\\ge0$. The maps (\\[eq:fil cont hom\\]) are a special case of [@AlbersFrauenfelder2010c Lemma 2.7]. ", "It will be important however to note that if the paths $\\varphi,\\psi$ have contact Hamiltonians $h_{t}$ and $k_{t}$ then then the constant $K(\\varphi,\\psi)$ satisfies $$K(\\varphi,\\psi)\\leq e^{\\max\\{\\kappa(\\varphi),\\kappa(\\psi)\\}}\\left\\Vert h^{\\varphi}-h^{\\psi}\\right\\Vert _{+},\\label{eq:the constant K}$$ where we are using the notation from (\\[eq:osc 1\\])-(\\[eq:osc 2\\]). ", "Finally one has for all $A\\in\\mbox{RFH}_{*}^{a}(\\mathcal{A}_{\\varphi},W)$ that $$\\zeta_{\\varphi,\\psi}\\left(j_{\\varphi}^{a}(A)\\right)=j_{\\psi}^{a+K(\\varphi,\\psi)}\\left(\\zeta_{\\varphi,\\psi}^{a}(A)\\right).\\label{eq:commutativity}$$\n\n4. ", " The Rabinowitz Floer homology $\\mbox{RFH}(\\Sigma,W)$ carries the structure of a unital ring (see [@AbbondandoloMerry2012]). ", "That is, given $a,b\\in\\mathbb{R}\\cup\\{+\\infty\\}$, there is a map $$\\pi^{a,b}:\\mbox{RFH}_{i}^{a}(\\Sigma,W)\\otimes\\mbox{RFH}_{j}^{b}(\\Sigma,W)\\rightarrow\\mbox{RFH}_{i+j-n}^{a+b}(\\Sigma,W)$$ with the property that $\\pi:=\\pi^{\\infty,\\infty}$ gives $\\mbox{RFH}_{*}(\\Sigma,W)$ the structure of a unital ring. ", "Thus there exists an element $\\mathbf{1}\\in\\mbox{RFH}_{*}(\\Sigma,W)$ such that $\\pi(A,\\mathbf{1})=\\pi(\\mathbf{1},A)=A$ for all $A\\in\\mbox{RFH}_{*}(\\Sigma,W)$. In particular, $\\mbox{RFH}_{*}(\\Sigma,W)=0$ if and only if $\\mathbf{1}=0$. The unit lives in degree $n=\\frac{1}{2}\\dim\\, V$. The maps $\\pi^{a,b}$ are compatible with the action filtration in the sense that for all $A\\in\\mbox{RFH}_{*}^{a}(\\Sigma,W)$ and $B\\in\\mbox{RFH}_{*}^{b}(\\Sigma,W)$ one has $$j^{a+b}(\\pi^{a,b}(A,B))=\\pi(j^{a}(A),j^{b}(A)).$$ The unit $\\mathbf{1}$ satisfies $$\\mathbf{1}\\in j^{a}(\\mbox{RFH}^{a}(\\Sigma,W))\\ \\ \\ \\mbox{if and only if }a>0.\\label{eq:c id equals zero}$$\n\n5. ", " \\[5\\]Consider again the case of the identity $\\mbox{id}_{\\Sigma}$. In this case we don’t need to worry about the cutoff function $\\beta$, and we simply study the unperturbed functional $\\mathcal{A}$ defined by $$\\mathcal{A}(u,\\eta):=\\int_{0}^{1}u^{*}\\lambda-\\eta\\int_{0}^{1}m(u(t))dt.\\label{unperturbed}$$ Since $\\Sigma\\times\\{0\\}\\subseteq\\mbox{Crit}(\\mathcal{A})$, if as above we choose a Morse function $g$ on $\\mbox{Crit}(\\mathcal{A})$, we may think of $g_{\\Sigma}:=g|_{\\Sigma\\times\\{0\\}}$ as a Morse function on $\\Sigma$. If we choose $\\varepsilon>0$ sufficiently small then $$\\mbox{RFH}_{*}^{(-\\varepsilon,\\varepsilon)}(\\mathcal{A},g)\\cong\\mbox{HM}_{*+n-1}(g_{\\Sigma}),$$ since in this case the boundary operator $\\partial:\\mbox{RFC}_{*}^{(-\\varepsilon,\\varepsilon)}(\\mathcal{A},g)\\rightarrow\\mbox{RFC}_{*-1}^{(-\\varepsilon,\\varepsilon)}(\\mathcal{A},g)$ coincides with the Morse differential $\\partial^{\\textrm{Morse}}:\\mbox{CM}_{*+n-1}(g_{\\Sigma})\\rightarrow\\mbox{CM}_{*+n-2}(g_{\\Sigma})$. Suppose $g_{\\Sigma}$ has a unique maximum $x_{\\max}\\in\\Sigma$. Then $x_{\\max}$ is necessarily a cycle in the Morse complex $\\mbox{CM}_{2n-1}(g_{\\Sigma})$, and under the isomorphism $\\mbox{HM}_{*}(g_{\\Sigma})\\cong\\mbox{H}_{*}(\\Sigma;\\mathbb{Z}_{2})$ the class $[x_{\\max}]$ represents the fundamental class $[\\Sigma]\\in\\mbox{H}_{2n-1}(\\Sigma;\\mathbb{Z}_{2})$. It turns out that $(x_{\\max},0)$ is also a cycle in $\\mbox{RFC}_{*}(\\mathcal{A},g)$, and the class $[(x_{\\max},0)]$ is the unit $\\mathbf{1}\\in\\mbox{RFH}_{n}(\\Sigma,W)$. In particular, one has $$p^{-\\varepsilon}(\\mathbf{1})=j^{\\varepsilon}([\\Sigma]).$$\n\n6. ", " The product structure carries over to the groups $\\mbox{RFH}_{*}(\\mathcal{A}_{\\varphi},W)$. Given $\\varphi$ and $\\psi$ one can define a product $$\\pi_{\\varphi,\\psi}^{a,b}:\\mbox{RFH}_{i}^{a}(\\mathcal{A}_{\\varphi},W)\\otimes\\mbox{RFH}_{j}^{b}(\\mathcal{A}_{\\psi},W)\\rightarrow\\mbox{RFH}_{i+j-n}^{a+b}(\\mathcal{A}_{\\varphi\\psi},W).$$ The product $\\pi_{\\varphi,\\psi}$ is compatible with the action filtration in the sense that for all $A\\in\\mbox{RFH}_{*}^{a}(\\mathcal{A}_{\\varphi},W)$ and $B\\in\\mbox{RFH}_{*}^{b}(\\mathcal{A}_{\\psi},W)$ one has $$j_{\\varphi\\psi}^{a+b}(\\pi_{\\varphi,\\psi}^{a,b}(A,B))=\\pi_{\\varphi,\\psi}(j_{\\varphi}^{a}(A),j_{\\psi}^{b}(B)),\\label{eq:key equation for triangle inequality-1}$$ where as before we abbreviate $\\pi_{\\varphi,\\psi}:=\\pi_{\\varphi,\\psi}^{\\infty,\\infty}$. Moreover the product $\\pi_{\\varphi,\\psi}$ is related to the continuation homomorphisms $\\zeta_{\\varphi},\\zeta_{\\psi}$ via $$\\pi_{\\varphi,\\psi}(\\zeta_{\\varphi}(A),\\zeta_{\\psi}(B))=\\zeta_{\\varphi\\psi}(\\pi(A,B))\\ \\ \\ A,B\\in\\mbox{RFH}_{*}(\\Sigma,W)$$ We define the ‘unit’ $\\boldsymbol{1}_{\\varphi}\\in\\mbox{RFH}_{n}(\\mathcal{A}_{\\varphi},W)$ by $$\\mathbf{1}_{\\varphi}=\\zeta_{\\varphi}(\\mathbf{1}),\\label{eq:continuation map does what it should to unit}$$ (note though that for $\\varphi\\ne\\mbox{id}_{\\Sigma}$, $\\mathbf{1}_{\\varphi}$ is not strictly a unit in any sense of the word!) ", "Note that $$\\zeta_{\\varphi,\\psi}(\\mathbf{1}_{\\varphi})=\\mathbf{1}_{\\psi},\\label{eq:unit goes to unit}$$ and $$\\pi_{\\varphi,\\psi}(\\mathbf{1}_{\\varphi},\\mathbf{1}_{\\psi})=\\mathbf{1}_{\\varphi\\psi}.\\label{eq:unit times unit equals unit}$$\n\n\\[sec:Spectral-invariants\\]Spectral invariants and orderability\n===============================================================\n\nThroughout this section we require Assumption **(A)** from the Introduction to hold. ", "More precisely, recall we say that a closed connected coorientable contact manifold $(\\Sigma,\\xi)$ satisfies Assumption **(A)** if:\n\n[(A)]{}\n\n: $(\\Sigma,\\xi)$ admits a Liouville filling $(W,d\\lambda)$ such that $\\alpha:=\\lambda|_{\\Sigma}$ is Morse-Bott and the Rabinowitz Floer homology $\\mbox{RFH}_{*}(\\Sigma,W)$ is non-zero.", "\n\n\\[def of c\\]Let $\\varphi$ denote a non-degenerate path. ", "We define its *spectral number* by $$c(\\varphi):=\\inf\\left\\{ a\\in\\mathbb{R\\mbox{\\,:\\,}}\\mathbf{1}_{\\varphi}\\in j_{\\varphi}^{a}(\\mbox{RFH}_{*}^{a}(\\mathcal{A}_{\\varphi},W))\\right\\} .\\label{eq:def of c}$$\n\nThen we have:\n\n\\[lem: c is positive\\]$c(\\varphi)$ is a finite real number, and moreover $$c(\\varphi)\\in\\mbox{\\emph{Spec}}(\\mathcal{A}_{\\varphi}).$$\n\nAssumption **(A)** implies that $c(\\varphi)>-\\infty$. The fact that $c(\\varphi)$ is a critical value of $\\mathcal{A}_{\\varphi}$ follows immediately from the fact (stated in (2) above) that the map $j_{a}^{b}:\\mbox{RFH}_{*}^{a}(\\mathcal{A}_{\\varphi},W)\\rightarrow\\mbox{RFH}_{*}^{b}(\\mathcal{A}_{\\varphi},W)$ is an isomorphism if $\\mbox{Spec}(\\mathcal{A}_{\\varphi})\\cap[a,b]=\\emptyset$.\n\nWe now have the following *Lipschitz continuity property* for the spectral invariant $c(\\varphi)$:\n\n\\[lemma:c(phi)\\_is\\_Lipschitz\\] For two non-degenerate paths $\\varphi,\\psi$, one has $$c(\\psi)\\le c(\\varphi)+K(\\varphi,\\psi),$$ and thus in particular $\\varphi\\mapsto c(\\varphi)$ is Lipschitz continuous with respect to the $C^{0}$-norm on $\\mathcal{P}\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$.\n\nIf $j_{\\varphi}^{a}(A)=\\mathbf{1}_{\\varphi}$ then by (\\[eq:commutativity\\]) $$j_{\\psi}^{a+K(\\varphi,\\psi)}\\left(\\zeta_{\\varphi,\\psi}^{a}(A)\\right)=\\zeta_{\\varphi,\\psi}\\left(j_{\\varphi}^{a}(A)\\right)=\\zeta_{\\varphi,\\psi}(\\mathbf{1}_{\\varphi})=\\mathbf{1}_{\\psi}.$$ This proves that $c(\\psi)\\leq c(\\varphi)+K(\\varphi,\\psi)$.\n\nLemma \\[lemma:c(phi)\\_is\\_Lipschitz\\], together with the fact that a generic path $\\varphi$ is non-degenerate, allows us to extend the definition of $c$ to degenerate paths $\\varphi$:\n\n\\[degen def\\]Let $\\varphi$ denote a degenerate path. ", "Pick a sequence $\\varphi^{\\nu}$ of non-degenerate paths $\\varphi^{\\nu}$ with $\\varphi^{\\nu}\\rightarrow\\varphi$. Then we define $$c(\\varphi):=\\lim_{\\nu}c(\\varphi^{\\nu}).\\label{eq:how defined in degen case}$$ Note that in the degenerate case we still have $c(\\varphi)\\in\\mbox{Spec}(\\mathcal{A}_{\\varphi})$.\n\n\\[c id equals zero\\]It follows from (\\[eq:c id equals zero\\]) that $$c(\\mbox{id}_{\\Sigma})=0.$$\n\nMore generally, we have:\n\n\\[thereebflow\\]For any $T\\in\\mathbb{R}$ one has $$c(\\theta^{T})=-T.$$\n\nOne has $\\mbox{Spec}(\\mathcal{A}_{\\theta^{T}})=-T+\\mbox{Spec}(\\mathcal{A}_{\\theta^{0}})$. Since by assumption $\\mbox{Spec}(\\mathcal{A}_{\\theta^{0}})$ is nowhere dense, the result follows from continuity of $c$ and the fact that $c(\\theta^{0})=c(\\mbox{id}_{\\Sigma})=0$.\n\nRecall that the universal cover $\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ of $\\mbox{Cont}_{0}(\\Sigma,\\xi)$ consists of equivalence classes of paths where two paths $\\{\\varphi_{t}\\}_{0\\leq t\\leq1}$ and $\\{\\psi_{t}\\}_{0\\leq t\\leq1}$ with $\\varphi_{1}=\\psi_{1}$are equivalent if there exists a smooth family $\\varphi^{s}=\\{\\varphi_{t}^{s}\\}_{0\\leq s,t\\leq1}$ such that $\\varphi^{0}=\\varphi$, $\\varphi^{1}=\\psi$, and $\\varphi_{1}^{s}$ is independent of $s$.\n\n\\[cor:descends to universal cover-1\\]If $\\varphi$ and $\\psi$ are equivalent then $c(\\varphi)=c(\\psi)$. Thus $c$ descends to define a map (still denoted by) $$c:\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{R}.$$\n\nLet $\\{\\varphi^{s}\\}_{0\\leq s\\leq1}=\\{\\varphi_{t}^{s}\\}_{0\\leq s,t\\leq1}\\subseteq\\mbox{Cont}_{0}(\\Sigma,\\xi)$ denote a smooth family such that $\\varphi^{0}=\\varphi$, $\\varphi^{1}=\\psi$, and $\\varphi_{1}^{s}$ is independent of $s$. Note that $c(\\varphi^{s})\\in\\mbox{Spec}(\\mathcal{A}_{\\varphi^{s}})$, which depends only on $\\varphi_{1}^{s}$ by Remark \\[rem: spec only depends on the end\\]. ", "Moreover Lemma \\[lemma:c(phi)\\_is\\_Lipschitz\\] implies that $s\\mapsto c(\\varphi^{s})$ is continuous. ", "There exists $T>0$ such that $c(\\varphi^{s})\\in[-T,T]$ for all $s\\in[0,1]$, and hence in the discrete set $\\mbox{Spec}^{[-T,T]}(\\mathcal{A}_{\\varphi})$ (cf. ", "Lemma \\[lem:spectrum is discrete\\]). ", "Thus $c(\\varphi^{s})$ does not depend on $s$.\n\nThe following ‘triangle inequality’ is what makes $c$ a useful function.", "\n\n\\[lem:triangle inequality\\]Given $\\varphi,\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ one has $$c(\\varphi\\psi)\\leq c(\\varphi)+c(\\psi).$$\n\nIn the non-degenerate case we argue as follows: if $j_{\\varphi}^{a}(A)=\\mathbf{1}_{\\varphi}$ and $j_{\\psi}^{b}(B)=\\mathbf{1}_{\\psi}$ then $C:=\\pi_{\\varphi,\\psi}^{a,b}(A,B)$ belongs to $\\mbox{RFH}_{*}^{a+b}(\\mathcal{A}_{\\varphi\\psi},W)$, and by (\\[eq:key equation for triangle inequality-1\\]) and (\\[eq:unit times unit equals unit\\]) we have $$j_{\\varphi\\psi}^{a+b}(C)=\\pi_{\\varphi,\\psi}(j_{\\varphi}^{a}(A),j_{\\psi}^{b}(B))=\\pi_{\\varphi,\\psi}(\\mathbf{1}_{\\varphi},\\mathbf{1}_{\\psi})=\\mathbf{1}_{\\varphi\\psi}.$$ The general case now follows from an approximation argument.", "\n\n\\[prop:the comparison prop\\]Suppose $\\varphi,\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$. Suppose the contact Hamiltonian of $\\varphi$ is $h_{t}$, and the contact Hamiltonian of $\\psi$ is $k_{t}$. Define smooth positive functions $\\rho_{t}$ and $\\sigma_{t}$ via $\\varphi_{t}^{*}\\alpha=\\rho_{t}\\alpha$ and $\\psi_{t}^{*}\\alpha=\\sigma_{t}\\alpha$. If for all $t\\in[0,1]$ one has $$\\frac{h_{t}\\circ\\varphi_{t}}{\\rho_{t}}\\geq\\frac{k_{t}\\circ\\psi_{t}}{\\sigma_{t}}\\label{eq:key inequality}$$ then $$c(\\varphi)\\leq c(\\psi).\\label{eq:key inequality-1}$$ Moreover if the inequality (\\[eq:key inequality\\]) is strict then so is (\\[eq:key inequality-1\\]).", "\n\nSet $\\mu_{t}:=\\varphi_{t}\\psi_{t}^{-1}$. It suffices to prove the result under the assumption that $\\varphi,\\psi$ and $\\mu$ are all non-degenerate. ", "and let $l_{t}$ denote the contact Hamiltonian of $\\mu$. By part (2) of Lemma \\[lem:computations\\], one has $$l_{t}(x):=h_{t}(x)-\\rho_{t}(\\varphi_{t}^{-1}(x))\\frac{k_{t}(\\psi_{t}(\\varphi_{t}^{-1}(x)))}{\\sigma_{t}(\\varphi_{t}^{-1}(x))}.$$ Lemma \\[lem:triangle inequality\\] implies that $$c(\\varphi)=c(\\mu\\psi)\\leq c(\\mu)+c(\\psi),$$ and (\\[lemma:c(phi)\\_is\\_Lipschitz\\]) implies that $$c(\\mu)\\leq c(\\mbox{id}_{\\Sigma})+K(\\mbox{id}_{\\Sigma},\\mu),$$ and hence $c(\\mu)\\leq0$ if $K(\\mbox{id}_{\\Sigma},\\mu)=0$. Finally $K(\\mbox{id}_{\\Sigma},\\mu)=0$ if $l_{t}\\geq0$ (cf. (", "\\[eq:the constant K\\])), that is, $$h_{t}(x)\\sigma_{t}(\\varphi_{t}^{-1}(x))\\geq\\rho_{t}(\\varphi_{t}^{-1}(x))k_{t}(\\psi_{t}(\\varphi_{t}^{-1}(x))),$$ which is equivalent to (\\[eq:key inequality\\]). ", "Now suppose the inequality (\\[eq:key inequality\\]) is strict. ", "Choose $$0<\\varepsilon_{0}<\\min_{(t,x)\\in S^{1}\\times\\Sigma}\\left(\\frac{h_{t}(\\varphi_{t}(x))}{\\rho_{t}(x)}-\\frac{k_{t}(\\psi_{t}(x))}{\\sigma_{t}(x)}\\right),$$ and choose $$0<\\varepsilon_{1}<\\min_{(t,x)\\in S^{1}\\times\\Sigma}\\sigma_{t}(x).$$ Set $$\\varepsilon:=\\varepsilon_{0}\\varepsilon_{1},$$ and consider the exact contactomorphism $t\\mapsto\\theta^{\\varepsilon t}$, which has contact Hamiltonian $\\varepsilon$ and satisfies $c(\\theta^{\\varepsilon})=-\\varepsilon$ by Lemma \\[thereebflow\\]. ", "If we write $\\mu_{t}^{*}\\alpha=\\tau_{t}\\alpha$ then from part (2) of Lemma \\[lem:computations\\], we have $$\\tau_{t}(x)=\\frac{\\rho_{t}\\left(\\psi_{t}^{-1}(x)\\right)}{\\sigma_{t}\\left(\\psi_{t}^{-1}(x))\\right)},$$ and hence $$\\begin{aligned}\n\\frac{l_{t}(\\mu_{t}(x))}{\\tau_{t}(x)} & =\\frac{h_{t}(\\mu_{t}(x))-\\rho_{t}(\\psi_{t}^{-1}(x))\\frac{k_{t}(x)}{\\sigma_{t}(\\psi_{t}^{-1}(x))}}{\\frac{\\rho_{t}\\left(\\psi_{t}^{-1}(x)\\right)}{\\sigma_{t}\\left(\\psi_{t}^{-1}(x))\\right)},}\\nonumber \\\\\n & =\\frac{h_{t}(\\mu_{t}(x))\\sigma_{t}(\\psi_{t}^{-1}(x))-\\rho_{t}(\\psi_{t}^{-1}(x))k_{t}(x)}{\\rho_{t}(\\psi_{t}^{-1}(x))}\\nonumber \\\\\n & \\geq\\left(\\min_{(t,y)\\in S^{1}\\times\\Sigma}\\sigma_{t}(y)\\right)\\left(\\min_{(s,z)\\in S^{1}\\times\\Sigma}\\left(\\frac{h_{s}(\\varphi_{s}(z))}{\\rho_{s}(z)}-\\frac{k_{s}(\\psi_{s}(z))}{\\sigma_{s}(z)}\\right)\\right)\\nonumber \\\\\n & >\\varepsilon_{0}\\varepsilon_{1}=\\varepsilon\\end{aligned}$$ Thus from what we have already proved, $$c(\\mu)\\leq c(\\vartheta)=-\\varepsilon,$$ and hence $$c(\\varphi)=c(\\varphi\\psi^{-1}\\psi)\\leq c(\\mu)+c(\\psi)\\leq-\\varepsilon+c(\\psi),\\label{eq:this equation}$$ which completes the proof.", "\n\nBy taking either $\\varphi=\\mbox{id}_{\\Sigma}$ or $\\psi=\\mbox{id}_{\\Sigma}$ in Proposition \\[prop:the comparison prop\\] we obtain the following immediate corollary.", "\n\n\\[cor:posi and negi\\]Suppose that $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ is generated by a contact Hamiltonian $h_{t}$ with $h_{t}>0$. Then $c(\\varphi)<0$. Similarly if $h_{t}<0$ then $c(\\varphi)>0$.\n\nWe have now completed the proof of Theorem \\[Theorem-A\\] from the Introduction. ", "Given a path $\\varphi$ of contactomorphisms, we define the *support* of $\\varphi$, $$\\mathfrak{S}(\\varphi):=\\bigcup_{0\\leq t\\leq1}\\mbox{supp}(\\varphi_{t}),$$ where $\\mbox{supp}(\\varphi_{t}):=\\overline{\\left\\{ x\\in\\Sigma\\mid\\varphi_{t}(x)\\ne x\\right\\} }$.", "\n\n\\[def of cU\\]For an open set $U\\subset\\Sigma$ we set $$c(U):=\\sup\\left\\{ c(\\varphi)\\mid\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\ \\mathfrak{S}(\\varphi)\\subset U\\right\\} \\in(-\\infty,\\infty].$$\n\n\\[By-Example-infinit\\]By Example \\[c bar reeb\\] one has immediately that $c(\\Sigma)=\\infty$.\n\nLet us show that $c(U)>0$ for any non-empty set $U\\subset\\Sigma$.\n\n\\[lem:co is nontrivial-1\\]Given any open set $U\\subset\\Sigma$, there exists $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ such that $\\mathfrak{S}(\\psi)\\subset U$ and $c(\\psi)>0$.\n\nWe prove the proposition in three steps.", "\n\n**Step 1.** ", "We use an idea from Sandon [@Sandon2011 p2]. ", "Fix a $C^{2}$-small function $b:\\Sigma\\rightarrow\\mathbb{R}$. We use $b$ to build a contactomorphism $\\Psi:T^{*}\\Sigma\\times\\mathbb{R}\\rightarrow T^{*}\\Sigma\\times\\mathbb{R}$, where the 1-jet bundle $T^{*}\\Sigma\\times\\mathbb{R}$ is equipped with the standard contact form $\\lambda_{0}+d\\tau$ and $\\lambda_{0}=pdx$ in local coordinates. ", "Namely, we set $$\\Psi(x,p,\\tau)=(x,p-db(x),\\tau+b(x)).$$ Note that critical points of $b$ are in 1-1 correspondence with Reeb chords between the two Legendrians $\\Sigma\\times\\{0\\}$ and $\\Psi(\\Sigma\\times\\{0\\})$ (where $\\Sigma\\subset T^{*}\\Sigma$ is the zero section). ", "Since $b$ is assumed to be $C^{2}$-small, $\\Psi$ determines a contactomorphism of $\\psi$ of $(\\Sigma,\\alpha)$, defined as follows. ", "Firstly, Weinstein’s neighborhood theorem for Legendrian submanifolds (see [@AbbasHofer Theorem 2.2.4]) implies that there is an exact contactomorphism $$\\Xi:N\\times(-\\delta,\\delta)\\rightarrow Q\\times(-\\varepsilon,\\varepsilon)\\label{eq:the contacto big xi-1}$$ between a neighborhood $N\\times(-\\delta,\\delta)$ of $\\Sigma\\times\\{0\\}$ inside $T^{*}\\Sigma\\times\\mathbb{R}$ and a neighborhood $Q\\times(-\\varepsilon,\\varepsilon)$ of $\\Delta\\times\\{0\\}$ inside $\\Sigma\\times\\Sigma\\times\\mathbb{R}$, where $\\Delta$ is the diagonal in $\\Sigma\\times\\Sigma$. Here $\\Sigma\\times\\Sigma\\times\\mathbb{R}$ is equipped with the contact form $e^{r}\\mbox{pr}_{1}^{*}\\alpha-\\mbox{pr}_{2}^{*}\\alpha$, where $\\mbox{pr}_{j}$ is the projection onto the $j$th factor. ", "The contactomorphism $\\psi$ is then defined by looking at the restriction of $\\Xi\\circ\\Psi\\circ\\Xi^{-1}$ to $\\Delta\\times\\{0\\}$ inside $Q\\times(-\\varepsilon,\\varepsilon)$; we can write\n\n$$\\Xi\\circ\\Psi\\circ\\Xi^{-1}(x,x,0)=:(x,\\psi(x),\\log\\rho(x)),$$\n\nfor $\\psi:\\Sigma\\rightarrow\\Sigma$ and $\\rho:\\Sigma\\rightarrow(0,\\infty)$. One readily checks that $\\psi^{*}\\alpha=\\rho\\alpha$, and hence $\\psi$ is a contactomorphism.", "\n\nSimilarly, if we start with an isotopy $\\{b_{t}\\}_{0\\leq t\\leq1}$ with $b_{0}=0$ then we obtain a path $\\psi=\\{\\psi_{t}\\}_{0\\leq t\\leq1}$ of contactomorphisms with $\\psi_{0}=\\mbox{id}_{\\Sigma}$. In this case one can check that the contact Hamiltonian of $\\psi$ is $-b_{t}$: $$\\alpha\\left(\\frac{\\partial}{\\partial t}\\psi_{t}\\right)=-b_{t}\\circ\\psi_{t}.$$ The key point now is that the translated points $x\\in\\Sigma$ of $\\psi_{1}$ with time-shift $\\eta\\in(-\\varepsilon,\\varepsilon)$ are in 1-1 correspondence with the critical points of $b_{1}$: if $x\\in\\mbox{crit}(b_{1})$ then $$\\psi(x)=\\theta^{-b_{1}(x)}(x).$$ Thus for each $x\\in\\mbox{crit}(b_{1})$ there is a critical point $(u_{x},b_{1}(x))\\in\\mbox{crit}(\\mathcal{A}_{\\psi})$, and any critical point $(u,\\eta)$ of $\\mathcal{A}_{\\psi}$ not of this form necessarily satisfies $\\left|\\eta\\right|>\\varepsilon$.\n\n**Step 2.** ", "Suppose now that we start with a $C^{2}$-small Morse-Bott function $b$ on $\\Sigma$. Define $b_{t}:=tb$ for $t\\in[0,1]$, and let $\\psi=\\{\\psi_{t}\\}_{0\\leq t\\leq1}$ denote the corresponding path of contactomorphisms. ", "Then if $x\\in\\mbox{crit}(b)$ then the critical point $(u_{x},b(x))$ belongs to a Morse-Bott component of $\\mathcal{A}_{\\psi}$, and moreover we claim that $$\\mu(u_{x},b(x))=1-n+\\mbox{ind}_{b}(x),$$ where $\\mbox{ind}_{b}(x)$ denotes the maximal dimension of a subspace on which the Hessian $\\mbox{Hess}_{b}(x)$ of $b$ at $x$ is *strictly* negative definite.", "\n\nTo see this, we consider the Hamiltonian diffeomorphism $\\Phi$ of $T^{*}\\Sigma\\times\\mathbb{R}\\times\\mathbb{R}$ obtained by lifting $\\Psi$, which as $\\Psi$ is exact, is given simply by $$\\Phi(q,p,\\tau,\\sigma)=\\left(\\Psi(q,p,\\tau),\\sigma\\right).$$ A translated point $x$ of $\\psi_{1}$ gives rise to the following path of Lagrangian subspaces: $$L_{t}:=\\left\\{ (\\hat{x},-t\\mbox{Hess}_{b}(x)(\\hat{x}),0,\\hat{\\sigma})\\mid\\hat{x}\\in T_{x}\\Sigma,\\,\\hat{\\sigma}\\in\\mathbb{R}\\right\\} \\subset T_{(x,0,0,\\sigma)}(T^{*}\\Sigma\\times\\mathbb{R}\\times\\mathbb{R}),$$ The desired index is then given by $$\\mu(u_{x},b(x))=1-n+\\mu_{\\textrm{RS}}(L_{0},L_{1}),$$ which in this case is just $1-n+\\mbox{ind}_{b}(x)$ as claimed; note that the $1-n$ summand comes from the normalization used in the definition of the Rabinowitz index (\\[eq:grading\\]) above, and we are using the grading convention from Remark \\[Our-grading-convention\\].", "\n\n**Step 3.** ", "We now prove the theorem. ", "Suppose $U\\subset\\Sigma$ is open. ", "Choose a function $b:\\Sigma\\rightarrow[0,\\infty)$ such that $\\mbox{supp}(b)\\subset U$ and such that $b$ is Morse on the interior of its support. ", "Moreover we insist that $b$ has a unique maximum $x_{0}\\in\\Sigma$, with $0<b(x_{\\max})<\\varepsilon$, where $\\varepsilon$ is as in (\\[eq:the contacto big xi-1\\]). ", "Finally, we insist that $b$ is sufficiently $C^{2}$-small so that the corresponding contactomorphism $\\psi=\\{\\psi_{t}\\}_{0\\leq t\\leq1}$ is defined and satisfies $$K(\\mbox{id}_{\\Sigma},\\psi)<\\varepsilon.$$ From Lemma \\[lemma:c(phi)\\_is\\_Lipschitz\\] we obtain $$c(\\psi)\\leq c(\\mbox{id}_{\\Sigma})+K(\\mbox{id}_{\\Sigma},\\psi)<\\varepsilon.$$ Since the contact Hamiltonian of $\\psi$ is $-b$ which is non-positive, one has $c(\\psi)\\geq0$ by Corollary \\[cor:posi and negi\\]. ", "Thus $c(\\psi)$ is necessarily a critical value of $b$. Since $\\mathbf{1}_{\\psi}$ has index $n$, and $x_{\\max}$ is the only critical point of $b$ of index $2n-1$ (so that the corresponding critical point $(u_{x_{\\max}},b(x_{\\max}))$ has index $1-n+2n-1=n$), we see that $$c(\\psi)=\\mathcal{A}_{\\psi}(u_{x_{\\max}},b(x_{\\max}))=b(x_{\\max})>0.$$ The proof is complete.", "\n\nIt is *not* true that one can always find $\\psi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ such that $\\mathfrak{S}(\\psi)\\subset U$ and $c(\\psi)<0$ - see Lemma \\[lem:always positive\\] below. ", "One can also see directly from the preceding proof that the argument would completely break down: if we tried replaced $x_{\\max}$ with a minimum $x_{\\min}$ of $b$ then the corresponding critical point $(u_{x_{\\min}},b(x_{\\min}))$ would no longer have the correct index.", "\n\n\\[lem:always positive\\]Suppose $U\\subset\\Sigma$ is an open set with the property that there exists $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ satisfying $$\\psi_{1}(U)\\cap\\left(\\bigcup_{t\\in\\mathbb{R}}\\theta^{t}(U)\\right)=\\emptyset.\\label{eq:reeb displaceable-1}$$ Then $0\\leq c(\\varphi)\\leq c(\\psi)+c(\\psi^{-1})$ for every $\\varphi$ with support in $U$.\n\nSuppose $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ satisfies $\\mathfrak{S}(\\varphi)\\subset U$. We claim that $$c(\\psi\\varphi)=c(\\psi).\\label{eq:c the same]}$$ Indeed, any translated point $x\\in\\Sigma$ of $\\psi$ must necessarily lie outside of $U$ (due to (\\[eq:reeb displaceable-1\\])), and since $\\mathfrak{S}(\\varphi)\\subset U$ the same is true of $\\psi\\varphi$. Thus $\\mbox{Spec}(\\mathcal{A}_{\\psi\\varphi})=\\mbox{Spec}(\\mathcal{A}_{\\psi})$. But now if we choose a path $\\varphi^{s}=\\{\\varphi_{t}^{s}\\}_{0\\leq s,t\\leq1}$ such that $\\varphi^{0}=\\mbox{id}_{\\Sigma}$ and $\\varphi^{1}=\\varphi$ with $\\mathfrak{S}(\\varphi^{s})\\subset U$ then the same argument shows that $\\mbox{Spec}(\\mathcal{A}_{\\psi\\varphi^{s}})=\\mbox{Spec}(\\mathcal{A}_{\\psi})$ for all $s$. Thus $s\\mapsto c(\\psi\\varphi^{s})$ is a continuous function on the discrete set $\\mbox{Spec}(\\mathcal{A}_{\\psi})$ (cf. ", "Lemma \\[lem:spectrum is discrete\\]), and thus $c(\\psi\\varphi^{s})$ is constant. ", "This proves (\\[eq:c the same\\]\\]).", "\n\nTo see that $c(U)\\leq c(\\psi)+c(\\psi^{-1})$, we note that for $\\varphi$ with $\\mathfrak{S}(\\varphi)\\subset U$ one has $$c(\\varphi)=c(\\psi^{-1}\\psi\\varphi)\\leq c(\\psi^{-1})+c(\\psi\\varphi)=c(\\psi^{-1})+c(\\psi).$$ Finally to see that $c(\\varphi)\\geq0$ for all $\\varphi$ with $\\mathfrak{S}(\\varphi)\\subset U$. We apply (\\[eq:c the same\\]\\]) to $\\varphi^{-1}$, which also satisfies $\\mathfrak{S}(\\varphi^{-1})\\subset U$. Then we have $$c(\\psi)=c(\\psi\\varphi^{-1}\\varphi)\\leq c(\\psi\\varphi^{-1})+c(\\varphi)=c(\\psi)+c(\\varphi),$$ which implies that $c(\\varphi)\\geq0$ as claimed.", "\n\nWe now present a version of Corollary \\[cor:posi and negi\\] for contactomorphisms $\\mathfrak{S}(\\varphi)$ contained in some open set.", "\n\n\\[cor:posi but not negi\\]Suppose $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ has contact Hamiltonian $h_{t}$. If $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$ then $c(\\varphi)>0$.\n\nWe need a slight extension of the argument used in the proof of Proposition \\[lem:co is nontrivial-1\\]. ", "Suppose $\\{b_{t}\\}_{0\\leq t\\leq1}$ is a family of $C^{2}$-small functions $b_{t}:\\Sigma\\rightarrow\\mathbb{R}$ such that $b_{0}=\\mbox{id}_{\\Sigma}$ and such that $b_{1}$ is Morse on the interior of its support, with a unique maximum $x_{\\max}$. We denote by $\\psi^{b}=\\{\\psi_{t}^{b}\\}_{0\\leq t\\leq1}$ the contact isotopy with contact Hamiltonian $-b_{t}$. Similarly we denote by $\\psi^{b_{1}}=\\{\\psi_{t}^{b_{1}}\\}_{0\\leq t\\leq1}$ the contact isotopy with contact Hamiltonian $-tb_{1}$. In the proof of Proposition \\[lem:co is nontrivial-1\\] we showed that $c(\\psi^{b_{1}})=b_{1}(x_{\\max})$. Now let us note that as the family $\\{b_{t}\\}_{0\\leq t\\leq1}$ is homotopic to $\\{tb_{1}\\}_{0\\leq t\\leq1}$, the usual invariance argument (cf. ", "Corollary \\[cor:descends to universal cover-1\\]) shows that $c(\\psi^{b})=c(\\psi^{b_{1}}),$ and hence we still have $$c(\\psi^{b})=b_{1}(x_{\\max})>0.\\label{eq:extension}$$ Now we prove the Corollary. ", "Let $\\rho_{t}:\\Sigma\\rightarrow(0,\\infty)$ denote the smooth positive function defined by $\\varphi_{t}^{*}\\alpha=\\rho_{t}\\alpha$. Choose as above a family $\\{b_{t}\\}_{0\\leq t\\leq1}$ of $C^{2}$-small functions $b_{t}:\\Sigma\\rightarrow\\mathbb{R}$ such that $b_{0}=\\mbox{id}_{\\Sigma}$ and such that $b_{1}$ is Morse on the interior of its support, with a unique maximum $x_{\\max}$, together with the additional property that if $\\sigma_{t}^{b}:\\Sigma\\rightarrow(0,\\infty)$ is defined by $(\\psi_{t}^{b})^{*}\\alpha=\\sigma_{t}^{b}\\alpha$, where $\\psi^{b}$ is as above, then $$\\frac{-b_{t}\\circ\\psi_{t}^{b}}{\\sigma_{t}^{b}}\\geq\\frac{h_{t}\\circ\\varphi_{t}}{\\rho_{t}}.$$ This is certainly possible as $h_{t}<0$ on the interior of the support of $\\mbox{supp}(\\varphi_{t})$. Then by Corollary \\[prop:the comparison prop\\] and (\\[eq:extension\\]) we have $c(\\varphi)\\geq b_{1}(x_{\\max})>0$.\n\nWe have now completed the proof of Proposition (\\[preliminary result of the cU\\]) from the Introduction.", "Unfortunately this is as far as one can take the argument of Proposition \\[prop:the comparison prop\\] for the case of contactomorphisms supported in a set $U$. The point is that the proof of Proposition \\[prop:the comparison prop\\] used the Reeb flow $\\theta^{\\varepsilon t}$ for small $\\varepsilon$, which has $c(\\theta^{\\varepsilon})<0$. In order to extend Proposition \\[prop:the comparison prop\\] to this setting we would need to be able to construct (for any $\\varepsilon>0$) a contactomorphism $\\varphi$ with support in $U$ such that, writing $\\varphi_{t}^{*}\\alpha=\\rho_{t}\\alpha$ and denoting the contact Hamiltonian of $\\varphi$ by $h_{t}$, that $$0\\leq\\sup_{(t,x)\\in S^{1}\\times\\Sigma}\\frac{h_{t}(\\varphi_{t}(x))}{\\rho_{t}(x)}<\\varepsilon,\\ \\ \\ \\mbox{and}\\ \\ \\ c(\\varphi)<0.$$ In general there is no way to do this, and indeed Lemma \\[lem:always positive\\] shows that for some open sets $U$ no such contactomorphisms $\\varphi$ exist. ", "See also Remark \\[making a real cptly sptd reeb flow\\] below.", "\n\n\\[question 2\\]In general we do not know whether the homology class $\\mathbf{1}_{\\varphi}\\in\\mbox{RFH}_{n}(\\mathcal{A}_{\\varphi},W)$ is necessarily represented by critical points of $\\mathcal{A}_{\\varphi}$ with image contained in $\\mathfrak{S}(\\varphi)$. In some circumstances this cannot happen however. ", "Since from Lemma \\[lemma:c(phi)\\_is\\_Lipschitz\\] one always has $c(\\varphi)\\leq K(\\mbox{id}_{\\Sigma},\\varphi)$, if $K(\\mbox{id}_{\\Sigma},\\varphi)<T$ then $(x,T)$ can definitely not contribute to $\\mathbf{1}_{\\varphi}$. In particular, if $K(\\mbox{id}_{\\Sigma},\\varphi)<\\wp_{W}(\\Sigma,\\alpha)$, where $\\wp_{W}(\\Sigma,\\alpha)$ was defined in Definition \\[smallest period\\], then no closed Reeb orbits outside the support of $\\varphi$ can contribute to $\\mathbf{1}_{\\varphi}$. In fact, this statement can be improved by making use of Corollary \\[cor:needed for translated point\\]. ", "Since $K(\\mbox{id}_{\\Sigma},\\varphi)\\leq e^{\\kappa(\\varphi)}\\left\\Vert \\varphi\\right\\Vert $ (see (\\[eq:the constant K\\])), by identifying $\\Sigma$ with $\\Sigma\\times\\{w\\}$ inside $S\\Sigma$ instead of $\\Sigma\\times\\{1\\}$, then for $w\\gg0$ we may assume that the corresponding constant $K_{w}(\\mbox{id}_{\\Sigma},\\varphi)$ satisfies $K_{w}(\\mbox{id}_{\\Sigma},\\varphi)<\\wp_{W}(\\Sigma,w\\alpha)$. We summarize this in the following lemma.", "\n\n\\[lem:rescaling lemma\\]Suppose $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ satisfies $\\left\\Vert \\varphi\\right\\Vert <\\wp_{W}(\\Sigma,\\alpha)$ and is supported in an open set $U\\subset\\Sigma$. Then after identifying $\\Sigma$ with $\\Sigma\\times\\{w\\}$ instead of $\\Sigma\\times\\{1\\}$ in $W$ for some $w\\gg0$, the unit $\\mathbf{1}_{\\varphi}$ is necessarily represented by critical points of $\\mathcal{A}_{\\varphi}$ with image contained in $U$.\n\n\\[rem strict inequalities\\]Suppose $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ has support in an open set $U\\subset\\Sigma$. Denote the contact Hamiltonian of $\\varphi_{t}$ by $h_{t}$, and assume that $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$. Assume either that\n\n1. ", " One has $\\left\\Vert h\\right\\Vert _{\\textrm{\\emph{osc}}}<\\wp_{W}(\\Sigma,\\alpha)$ and $c(U)<\\infty$.\n\n2. ", " There exists $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ with $\\psi(U)\\cap\\left(\\bigcup_{t\\in\\mathbb{R}}\\theta^{t}(U)\\right)=\\emptyset$ and $\\left\\Vert \\psi\\right\\Vert <\\wp_{W}(\\Sigma,\\alpha)$.\n\nThen $$c(\\varphi^{\\nu-1})<c(\\varphi^{\\nu})\\ \\ \\ \\mbox{for all }\\nu\\in\\mathbb{N}.$$\n\nLet us first assume that case (1) holds. ", "We then perform a rescaling so that Lemma \\[lem:rescaling lemma\\] holds for every iterate $\\varphi^{\\nu}$. For clarity however we will suppress this rescaling for the remainder of the proof. ", "Write $\\varphi_{t}^{*}\\alpha=\\rho_{t}\\alpha$ and write $(\\varphi_{t}^{\\nu})^{*}\\alpha=\\rho_{t}^{\\nu}\\alpha$. Let $h_{t}^{\\nu}$ denote the contact Hamiltonian of $\\varphi^{\\nu}$. Then using part (1) of Lemma \\[lem:computations\\], one easily checks that $$\\rho_{t}^{\\nu}(x)=\\rho_{t}(\\varphi_{t}^{\\nu-1}(x))\\rho_{t}^{\\nu-1}(x)\\label{eq:pho nus}$$ and $$h_{t}^{\\nu}(x)=h_{t}^{\\nu-1}(x)+\\rho_{t}(\\varphi_{t}^{-1}(x))h_{t}^{\\nu-1}(\\varphi_{t}^{-1}(x)).$$ We now prove by induction that $$h_{t}^{\\nu}(\\varphi_{t}^{\\nu}(x))<0\\ \\ \\ \\mbox{whenever}\\ \\ \\ \\varphi_{t}(x)\\ne x.\\label{eq:induction step}$$ By assumption this is true for $\\nu=1$. Since $$h_{t}^{\\nu}(\\varphi_{t}^{\\nu}(x))=h_{t}^{\\nu-1}(\\varphi_{t}^{\\nu}(x))+\\rho_{t}(\\varphi_{t}^{\\nu-1}(x))h_{t}^{\\nu-1}(\\varphi_{t}^{\\nu-1}(x)).\\label{eq:inductive step 2}$$ The first term on the right-hand side in (\\[eq:inductive step 2\\]) is certainly non-negative, and if $\\varphi_{t}(x)\\ne x$ then by induction then second term on the right-hand side in (\\[eq:inductive step 2\\]) is strictly negative. ", "This completes the inductive step, and hence (\\[eq:induction step\\]) is proved. ", "Proposition \\[prop:the comparison prop\\] tells us that $c(\\varphi^{\\nu-1})\\leq c(\\varphi^{\\nu})$ for any $\\nu\\in\\mathbb{N}$, since $$\\frac{h_{t}^{\\nu}(\\varphi_{t}^{\\nu}(x))}{\\rho_{t}^{\\nu}(x)}=\\frac{h_{t}^{\\nu-1}(\\varphi_{t}^{\\nu-1}(x))}{\\rho_{t}^{\\nu-1}(x)}+\\frac{h_{t}(\\varphi_{t}^{\\nu}(x))}{\\rho_{t}^{\\nu}(x)},$$ and thus $$\\frac{h_{t}^{\\nu-1}(\\varphi_{t}^{\\nu-1}(x))}{\\rho_{t}^{\\nu-1}(x)}-\\frac{h_{t}^{\\nu}(\\varphi_{t}^{\\nu}(x))}{\\rho_{t}^{\\nu}(x)}=-\\frac{h_{t}(\\varphi_{t}^{\\nu}(x))}{\\rho_{t}^{\\nu}(x)}\\geq0.\\label{eq:nonstrict inequality}$$ It remains to prove that the inequality is strict. ", "Fix $\\delta>0$ and define $$U_{\\delta}(\\varphi_{t}):=\\left\\{ x\\in\\mbox{supp}(\\varphi_{t})\\mid\\mbox{dist}(x,\\partial\\mbox{supp}(\\varphi_{t}))>\\delta\\right\\} ,$$ where $\\mbox{dist}$ is taken with respect to some metric on $\\Sigma$. For $\\delta>0$ sufficiently small the set $U_{\\delta}(\\varphi_{t})$ is an open subset of $\\Sigma$ varying smoothly in $t$ which is non-empty for $t>0$. Given $\\varepsilon>0$, let $k_{t}^{\\delta,\\varepsilon}:\\Sigma\\rightarrow[0,\\varepsilon]$ denote a smooth function such that $k_{t}^{\\delta,\\varepsilon}(x)=\\varepsilon$ for all $x\\in U_{\\delta}(\\varphi_{t})$ and such that $\\mbox{supp}(k_{t}^{\\delta,\\varepsilon})\\subset\\mbox{supp}(\\varphi_{t})$. We may assume that $k_{t}^{\\delta,\\varepsilon}$ is also smooth in $t$, and that as $\\varepsilon\\rightarrow0$ one has $k_{t}^{\\delta,\\varepsilon}\\rightarrow0$ in the $C^{\\infty}$-topology. ", "Thus from (\\[eq:induction step\\]), if $\\varepsilon>0$ is chosen sufficiently small, we have $$k_{t}^{\\delta,\\varepsilon}(x)+2h_{t}^{\\nu}(\\varphi_{t}^{\\nu}(x))\\leq0\\ \\ \\ \\mbox{for all }(t,x)\\in[0,1]\\times\\Sigma.\\label{eq:to hold 1}$$ Moreover if $\\psi^{\\delta,\\varepsilon}\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ denotes the contactomorphism with contact Hamiltonian $k_{t}^{\\delta,\\varepsilon}$ then for $\\varepsilon>0$ sufficiently small the function $\\sigma_{t}^{\\delta,\\varepsilon}:\\Sigma\\rightarrow(0,\\infty)$, defined by $(\\psi_{t}^{\\delta,\\varepsilon})^{*}\\alpha=\\sigma_{t}^{\\delta,\\varepsilon}\\alpha$, satisfies $$\\sigma_{t}^{\\delta,\\varepsilon}(x)\\leq2\\ \\ \\ \\mbox{for all }(t,x)\\in[0,1]\\times\\Sigma.\\label{eq:to hold 2}$$ We now claim that choosing $\\delta>0$ and then $\\varepsilon>0$ sufficiently small, so that both (\\[eq:to hold 1\\]) and (\\[eq:to hold 2\\]) hold then, $$c(\\varphi^{\\nu-1})\\leq c(\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}).$$ Using part (1) of Lemma \\[lem:computations\\] again, we see that the contact Hamiltonian $l_{t}(x)$ of $\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}$ is given by $$l_{t}(x)=k^{\\delta,\\varepsilon}(x)+\\sigma_{t}^{\\delta,\\varepsilon}\\left((\\psi^{\\delta,\\varepsilon})^{-1}(x)\\right)h_{t}\\left((\\psi^{\\delta,\\varepsilon})^{-1}(x)\\right),$$ and that $$(\\psi^{\\delta,\\varepsilon}\\varphi_{t}^{\\nu})^{*}\\alpha=\\tau_{t}\\alpha,$$ where $$\\tau_{t}(x)=\\sigma_{t}^{\\delta,\\varepsilon}(\\varphi_{t}^{\\nu}(x))\\rho_{t}^{\\nu}(x).$$ Thus $$\\begin{aligned}\n\\frac{l_{t}(\\psi_{t}^{\\delta,\\varepsilon}\\varphi_{t}^{\\nu}(x))}{\\tau_{t}(x)} & =\\frac{k^{\\delta,\\varepsilon}(\\psi_{t}^{\\delta,\\varepsilon}\\varphi_{t}^{\\nu}(x))}{\\sigma_{t}^{\\delta,\\varepsilon}(\\varphi_{t}^{\\nu}(x))\\rho_{t}^{\\nu}(x)}+\\frac{h_{t}^{\\nu}(\\varphi_{t}^{\\nu}(x))}{\\rho_{t}^{\\nu}(x)}.\\nonumber \\\\\n & =\\frac{1}{\\rho_{t}(x)\\sigma_{t}^{\\delta,\\varepsilon}(\\varphi_{t}^{\\nu}(x))}\\left(k_{t}^{\\delta,\\varepsilon}(x)+\\sigma_{t}^{\\delta,\\varepsilon}(\\varphi_{t}^{\\nu}(x))h_{t}^{\\nu}(\\varphi_{t}^{\\nu}(x))\\right).\\\\\n & \\leq0,\\end{aligned}$$ where we are using both (\\[eq:to hold 1\\]) and (\\[eq:to hold 2\\]). ", "Thus from (\\[eq:nonstrict inequality\\]), we see that $$c(\\varphi^{\\nu-1})\\leq c(\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu})$$ as required. ", "Next, $$c(\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu})\\leq c(\\psi^{\\delta,\\varepsilon})+c(\\varphi^{\\nu})\\leq c(\\varphi^{\\nu}),$$ since $c(\\psi^{\\delta,\\varepsilon})\\leq0$ as $k^{\\delta,\\varepsilon}\\geq0$. Now write $$\\mbox{Spec}(\\mathcal{A}_{\\varphi^{\\nu}})=\\mbox{Spec}'(\\mathcal{A}_{\\varphi^{\\nu}})\\cup\\mbox{Spec}''(\\mathcal{A}_{\\varphi^{\\nu}}),$$ where $$\\mbox{Spec}'(\\mathcal{A}_{\\varphi^{\\nu}}):=\\left\\{ \\mathcal{A}_{\\varphi^{\\nu}}(u,\\eta)\\mid(u,\\eta)\\in\\mbox{Crit}(\\mathcal{A}_{\\varphi^{\\nu}}),\\, u(0),u(1/2)\\in U_{\\delta}(\\varphi_{1})\\right\\} ,$$ $$\\mbox{Spec}''(\\mathcal{A}_{\\varphi^{\\nu}}):=\\mbox{Spec}(\\mathcal{A}_{\\varphi^{\\nu}})\\backslash\\mbox{Spec}'(\\mathcal{A}_{\\varphi^{\\nu}}).$$ Since $c(\\varphi^{\\nu-1})>0$ by Proposition \\[prop:the comparison prop\\], we see that for $\\delta>0$ small enough one has $c(\\varphi^{\\nu})\\in\\mbox{Spec}'(\\mathcal{A}_{\\varphi^{\\nu}})$ and $\\left|c(\\varphi^{\\nu})-c\\right|>\\delta$ for all $c\\in\\mbox{Spec}''(\\mathcal{A}_{\\varphi^{\\nu}})$. Note here we are using the fact that Lemma \\[lem:rescaling lemma\\] holds for $\\varphi^{\\nu}$ (as established at the beginning of the proof). ", "Now for the punchline: on $U_{\\delta}(\\varphi_{t})$, $\\psi_{t}^{\\delta,\\varepsilon}$ is just the Reeb flow $\\theta^{\\varepsilon t}$, and for $\\delta>0$ sufficiently small, if $\\mbox{Spec}'(\\mathcal{A}_{\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}})$ is defined in the same way then $$\\mbox{Spec}'(\\mathcal{A}_{\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}})=\\left\\{ c-\\varepsilon\\mid c\\in\\mbox{Spec}'(\\mathcal{A}_{\\varphi^{\\nu}})\\right\\} .\\label{eq:reeb flow changes spec}$$ We can’t really say much about $\\mbox{Spec}''(\\mathcal{A}_{\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}})$, apart from that it must be very close (in the Hausdorff topology) to $\\mbox{Spec}''(\\mathcal{A}_{\\varphi^{\\nu}})$. However, it follows that provided we choose both $\\delta$ and $\\varepsilon$ small enough, one has $c(\\varphi^{\\nu})\\in\\mbox{Spec}'(\\mathcal{A}_{\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}})$ and $\\mbox{Spec}'(\\mathcal{A}_{\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}})\\cap\\mbox{Spec}'(\\mathcal{A}_{\\varphi^{\\nu}})=\\emptyset$, and hence $$c(\\varphi^{\\nu})\\notin\\mbox{Spec}(\\mathcal{A}_{\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu}}).$$ Thus we see that for $\\delta,\\varepsilon>0$ sufficiently small, $$c(\\varphi^{\\nu-1})\\leq c(\\psi^{\\delta,\\varepsilon}\\varphi^{\\nu})<c(\\varphi^{\\nu}),$$ which completes the proof in case (1). ", "The proof in case (2) is very similar, and makes use of Lemma \\[lem:always positive\\] and (\\[eq:the constant K\\]).", "\n\n\\[making a real cptly sptd reeb flow\\]One should note that it is not at all clear what the translated points of $\\psi^{\\delta,\\varepsilon}$ are, since we have no control over the behaviour of $k^{\\delta,\\varepsilon}$ on $\\mathfrak{S}(\\varphi)\\backslash U_{\\delta}$. In Appendix \\[app:A-’compactly-supported\\] we show how, in the periodic case, one can construct a variant $\\vartheta^{\\varepsilon}$ of the contactomorphism $\\psi^{\\delta,\\varepsilon}$ which from the point of view of translated points, really is a ‘compactly supported Reeb flow’. ", "In this case it is possible to compute the spectral value precisely, and one gets $c(\\vartheta^{\\varepsilon})=-\\varepsilon$ if $-1<\\varepsilon<0$ and $c(\\vartheta^{\\varepsilon})=0$ if $\\varepsilon\\geq0$ see Theorem \\[prop:compactly supported reeb\\]. ", "This highlights a distinct contrast between the compactly supported Reeb flow and the genuine Reeb flow.", "\n\nWe now can now prove the following result, which was stated as Corollary \\[cor:infinitely many-1\\] in the Introduction. ", "This argument is due to Sandon [@Sandon2010 Theorem 3.1].", "\n\n\\[cor:infinitely many\\]Suppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)***. ", "Suppose $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ satisfies $\\mathfrak{S}(\\varphi)\\subset U$ and assume that the contact Hamiltonian $h_{t}$ of $\\Sigma$ satisfies $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$. Assume that either:\n\n1. ", " One has $\\left\\Vert h\\right\\Vert _{\\textrm{\\emph{osc}}}<\\wp_{W}(\\Sigma,\\alpha)$ and $c(U)<\\infty$.\n\n2. ", " There exists $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ with $\\psi(U)\\cap\\left(\\bigcup_{t\\in\\mathbb{R}}\\theta^{t}(U)\\right)=\\emptyset$ and $\\left\\Vert \\psi\\right\\Vert <\\wp_{W}(\\Sigma,\\alpha)$.\n\nThen $\\varphi$ has infinitely many distinct iterated translated points.", "\n\nProposition \\[rem strict inequalities\\] implies that there exists a sequence $(x_{\\nu},\\eta_{\\nu})\\in\\Sigma\\times(0,\\infty)$ such that $\\varphi^{\\nu}(x_{\\nu})=\\theta^{-\\eta_{\\nu}}(x_{\\nu})$, with the $\\eta_{\\nu}$ pairwise disjoint. ", "Either infinitely many of the points $x_{\\nu}$ are disjoint, in which case we are done, or there exists a point $x\\in\\Sigma$ such that $x_{\\nu}=x$ for infinitely many $\\nu$. In the latter case, set $y_{\\nu}:=\\varphi^{\\nu-1}(x)$. Then the $y_{\\nu}$ are pairwise disjoint for infinitely many $\\nu$, and for infinitely many $\\nu$, $y_{\\nu}$ is a translated point of $\\varphi$. Here we are using the fact that if $(\\varphi^{\\nu})^{*}\\alpha=\\rho^{\\nu}\\alpha$ then using (\\[eq:pho nus\\]) we see that $\\rho^{\\nu}(x)=1\\Rightarrow\\rho(y_{\\nu})=1$. This completes the proof.", "\n\nFinally let us prove the following result, which was given as Proposition \\[prop:depends only on endpoint\\] in the Introduction.", "\n\nSuppose $(\\Sigma,\\xi)$ satisfies Assumption ***(A)*** and that there are no closed contractible Reeb orbits. ", "Then the value of $c(\\varphi)$ depends only on $\\varphi_{1}$. Thus the map $c:\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{R}$ descends to $\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$.\n\nSuppose $\\varphi,\\psi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ satisfy $\\varphi_{1}=\\psi_{1}$. Set $\\mu:=\\varphi\\psi^{-1}$. We claim that $c(\\mu)=0$. Indeed, since $\\mu_{1}=\\mbox{id}_{\\Sigma}$ and there are no contractible Reeb orbits, $\\mbox{Spec}(\\mathcal{A}_{\\mu_{1}})=\\{0\\}$, and thus necessarily $c(\\mu)=0$. Thus $$c(\\varphi)=c(\\mu\\psi)\\leq c(\\psi).$$ Interchanging the roles of $\\varphi$ and $\\psi$ proves the result.", "\n\n\\[sec:The-periodic-case - capacities\\]The periodic case - capacities \n=====================================================================\n\nLet us now assume that $(\\Sigma,\\xi)$ satisfies Assumption **(B)** from the Introduction:\n\n[(B)]{}\n\n: $(\\Sigma,\\xi)$ admits a Liouville filling $(W,d\\lambda)$ such that the Rabinowitz Floer homology $\\mbox{RFH}_{*}(\\Sigma,W)$ is non-zero and such that $\\alpha:=\\lambda|_{\\Sigma}$ is periodic.", "\n\n\\[Def of c bar when resonant\\]We define for $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ an integer $\\overline{c}(\\varphi)$ by $$\\overline{c}(\\varphi):=\\left\\lceil c(\\varphi)\\right\\rceil .$$\n\nThe reason periodicity is helpful is this function $\\overline{c}$ is *conjugation invariant*. ", "We will prove this shortly in Proposition \\[lem the use of periodicity\\] below, but to begin with we present the following lemma. ", "Recall from Definition \\[def: nondegen\\] that we say $\\varphi$ is *non-resonant* if $\\mbox{Spec}(\\mathcal{A}_{\\varphi})\\cap\\mathbb{Z}=\\emptyset$.\n\n\\[lem:resonant and nondegen\\]Suppose $\\varphi$ is both resonant and degenerate with $c(\\varphi)\\in\\mathbb{Z}$. Then there exists $\\varphi^{\\nu}\\rightarrow\\varphi$ such that $\\varphi^{\\nu}$ is resonant and non-degenerate such that for all $\\nu$ sufficiently large one has $c(\\varphi^{\\nu})=c(\\varphi)$.\n\nStart with any sequence $(\\varphi^{\\nu})$ of non-degenerate paths such that $\\varphi^{\\nu}\\rightarrow\\varphi$. Since $\\varphi$ is resonant, for each $\\nu\\in\\mathbb{N}$ there exists a translated point $x^{\\nu}\\in\\Sigma$ for $\\varphi^{\\nu}$ with time-shift $\\eta^{\\nu}$ such that $\\eta^{\\nu}\\rightarrow0$ and such that for all $\\nu$ sufficiently large one has $$c(\\varphi^{\\nu})=c(\\varphi)+\\eta^{\\nu}.$$ The sequence $\\theta^{-\\eta^{\\nu}}\\circ\\varphi^{\\nu}$ still converges to $\\varphi$, and it is easy to check that $\\theta^{-\\eta^{\\nu}}\\circ\\varphi^{\\nu}$ is still non-degenerate, and for all $\\nu$ sufficiently large one has that $$c(\\theta^{-\\eta^{\\nu}}\\circ\\varphi^{\\nu})=c(\\varphi^{\\nu})-\\eta^{\\nu}=c(\\varphi)$$ (see (\\[eq:reeb flow changes spec\\])\n\n\\[lem the use of periodicity\\]The function $\\overline{c}:\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}$ is conjugation invariant: if $\\psi\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ and $\\varphi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ then $$\\bar{c}(\\psi\\varphi\\psi^{-1})=\\bar{c}(\\varphi).$$ The triangle inequality $\\overline{c}(\\varphi\\vartheta)\\leq\\overline{c}(\\varphi)+\\overline{c}(\\vartheta)$ continues to hold for $\\overline{c}$.\n\nAssume that $\\varphi$ is non-resonant, that is, $\\mbox{Spec}(\\mathcal{A}_{\\varphi})\\cap\\mathbb{Z}=\\emptyset$ (see Definition \\[def: nondegen\\]). ", "Fix $\\psi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$. We show that $\\left\\lceil c(\\psi\\varphi\\psi^{-1})\\right\\rceil =\\left\\lceil c(\\varphi)\\right\\rceil $. ", "Let $\\psi_{s}\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$ be a path connecting $\\mbox{id}_{\\Sigma}$ to $\\psi$. Then we consider the map $$s\\mapsto c(\\psi_{s}\\varphi\\psi_{s}^{-1}).$$ Lemma \\[lemma:c(phi)\\_is\\_Lipschitz\\] implies that this map is continuous. ", "Lemma \\[lemma:integral\\_critical\\_points\\_are\\_conjugation\\_invariant\\] implies that $\\mbox{Spec}(\\mathcal{A}_{\\psi_{s}\\varphi\\psi_{s}^{-1}})\\cap\\mathbb{Z}=\\emptyset$ for all $s\\in[0,1]$, and hence $\\left\\lceil c(\\psi\\varphi\\psi^{-1})\\right\\rceil =\\left\\lceil c(\\varphi)\\right\\rceil $ as required.", "\n\nThere are now three cases to consider. ", "Firstly, suppose that $\\varphi$ is resonant but that $c(\\varphi)\\notin\\mathbb{Z}$. Suppose $\\psi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$. Then for $\\varphi'$ non-resonant and sufficiently close to $\\varphi$ we have $$\\overline{c}(\\psi\\varphi\\psi^{-1})=\\overline{c}(\\psi\\varphi'\\psi^{-1})=\\overline{c}(\\varphi')=\\overline{c}(\\varphi),$$ where the second equality used the step above. ", "The second case is when $\\varphi$ is resonant and non-degenerate, with $c(\\varphi)\\in\\mathbb{Z}$. As before, given $\\psi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$ we choose a path $\\psi_{s}$ connecting $\\mbox{id}_{\\Sigma}$ to $\\psi$. The key point now is that for any $s_{0}\\in[0,1]$, if $(u_{s_{0}},\\eta_{s_{0}})$ is a critical point of $\\mathcal{A}_{\\psi_{s_{0}}\\varphi\\psi_{s_{0}}^{-1}}$ with $\\eta_{s_{0}}\\in\\mathbb{Z}$ then $(u_{s_{0}},\\eta_{s_{0}})$ is automatically non-degenerate by the last statement of Lemma \\[lemma:integral\\_critical\\_points\\_are\\_conjugation\\_invariant\\]. ", "It follows that there exists $\\varepsilon>0$ such that $$\\mbox{Spec}(\\mathcal{A}_{\\psi_{s}\\varphi\\varphi_{s}^{-1}})\\cap[c(\\varphi)-\\varepsilon,c(\\varphi)+\\varepsilon]=\\{c(\\varphi)\\},$$ and the result follows as above. ", "The final case is when $\\varphi$ is both resonant and degenerate and $c(\\varphi)\\in\\mathbb{Z}$. In this case we employ Lemma \\[lem:resonant and nondegen\\] to find a sequence $\\varphi^{\\nu}\\rightarrow\\varphi$ such that $\\varphi^{\\nu}$ is both resonant, non-degenerate, and such that for all large $\\nu$ one has $c(\\varphi^{\\nu})=c(\\varphi)$. The argument above then implies that for any $\\psi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$ and for all $\\nu$ sufficiently large, $c(\\psi\\varphi^{\\nu}\\psi^{-1})=c(\\varphi^{\\nu})$ is an integer. ", "Since $c(\\psi\\varphi^{\\nu}\\psi^{-1})\\rightarrow c(\\psi\\varphi\\psi^{-1})$ the result follows.", "\n\nThe triangle inequality follows from the triangle inequality for $c$, and the fact that $\\left\\lceil a+b\\right\\rceil \\leq\\left\\lceil a\\right\\rceil +\\left\\lceil b\\right\\rceil $ for any $a,b\\in\\mathbb{R}$.\n\n\\[c bar reeb\\]One has $\\bar{c}(t\\mapsto\\theta^{tT})=\\left\\lceil T\\right\\rceil $ for any $T\\in\\mathbb{R}$.\n\nLemma \\[thereebflow\\].", "\n\nWe now define $\\overline{c}(U)$ in the same way as $c(U)$ was defined in Definition \\[def of cU\\].", "\n\nFor an open set $U\\subset\\Sigma$ we define the *contact capacity* $$\\overline{c}(U):=\\sup\\left\\{ \\overline{c}(\\varphi)\\mid\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\ \\mathfrak{S}(\\varphi)\\subset U\\right\\} \\in\\mathbb{Z}_{\\geq0}\\cup\\{\\infty\\}.$$ Note that $\\overline{c}(U)\\geq0$ by Proposition \\[lem:co is nontrivial-1\\].", "\n\n\\[prop:cu\\_co\\_are\\_invariant\\_under\\_contactos\\_and\\_monotone\\_wrt\\_inclusions\\] For all $\\psi\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$, one has $$\\overline{c}(\\psi(U))=\\overline{c}(U).$$\n\nSince $$\\mathfrak{S}(\\psi\\varphi\\psi^{-1})=\\psi(\\mathfrak{S}(\\varphi)),$$ we conclude from Proposition \\[lem the use of periodicity\\] that $$\\begin{aligned}\n\\overline{c}(U) & =\\sup\\left\\{ \\overline{c}(\\varphi)\\mid\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\ \\mathfrak{S}(\\varphi)\\subset U\\right\\} \\nonumber \\\\\n & =\\sup\\left\\{ \\overline{c}(\\varphi)\\mid\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\ \\psi(\\mathfrak{S}(\\varphi))\\subset\\psi(U)\\right\\} \\nonumber \\\\\n & =\\sup\\left\\{ \\overline{c}(\\varphi)\\mid\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\ \\mathfrak{S}(\\psi\\varphi\\psi^{-1})\\subset\\psi(U)\\right\\} \\nonumber \\\\\n & =\\sup\\left\\{ \\overline{c}(\\psi\\varphi\\psi^{-1})\\mid\\varphi\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\ \\mathfrak{S}(\\psi\\varphi\\psi^{-1})\\subset\\psi(U)\\right\\} \\nonumber \\\\\n & =\\sup\\left\\{ \\overline{c}(\\mu)\\mid\\mu\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\ \\mathfrak{S}(\\mu)\\subset\\psi(U)\\right\\} \\nonumber \\\\\n & =\\overline{c}(\\psi(U)).\\end{aligned}$$\n\nThe following result will prove very useful.", "\n\n\\[lem:displacement gives bound\\]Suppose $\\varphi,\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ are such that $\\psi$ is non-resonant and $$\\psi_{1}(\\mathfrak{S}(\\varphi))\\cap\\mathfrak{S}(\\varphi)=\\emptyset.\\label{eq:displaces}$$ Then $$\\overline{c}(\\psi\\varphi)=\\overline{c}(\\psi).\\label{eq:equality}$$\n\nLet $\\mu^{s}=\\{\\psi_{t}\\varphi_{st}\\}_{0\\leq t\\leq1}$ so that $\\psi\\varphi=\\mu^{1}$ and $\\psi=\\mu^{0}$. Suppose $(u,\\eta)\\in\\mbox{Crit}(\\mathcal{A}_{\\mu^{s}})$, and let $x:=u(\\tfrac{1}{2})$. Then $$\\psi\\varphi_{s}(x)=\\theta^{-\\eta}(x)$$ (see Lemma \\[lem:critical points\\]). ", "If $x\\in\\mbox{supp}(\\varphi_{s})$ then $\\varphi_{s}(x)\\in\\mbox{supp}(\\varphi_{s})$, and hence $\\psi(\\varphi_{s}(x))\\notin\\mbox{supp}(\\varphi_{s})$. Thus $\\varphi_{s}^{-1}\\psi\\varphi_{s}(x)=\\psi\\varphi_{s}(x)=\\theta^{-\\eta}(x)$, and so $\\eta\\in\\mbox{Spec}(\\mathcal{A}_{\\varphi_{s}^{-1}\\psi\\varphi_{s}})$. If $x\\notin\\mbox{supp}(\\varphi_{s})$ then $\\psi(x)=\\theta^{-\\eta}(x)$, and hence $\\eta\\in\\mbox{Spec}(\\mathcal{A}_{\\psi})$. Thus $$\\mbox{Spec}(\\mathcal{A}_{\\mu^{s}})\\subseteq\\mbox{Spec}(\\mathcal{A}_{\\varphi_{s}^{-1}\\psi\\varphi_{s}})\\cup\\mbox{Spec}(\\mathcal{A}_{\\psi}).\\label{eq:spec cont}$$ Thus from Lemma \\[lemma:integral\\_critical\\_points\\_are\\_conjugation\\_invariant\\], we see that $\\mbox{Spec}(\\mathcal{A}_{\\mu^{s}})\\cap\\mathbb{Z}=\\emptyset$ for all $s\\in[0,1]$. Now (\\[eq:equality\\]) follows by continuity of $c$.\n\nWe now give a criterion for $\\overline{c}(U)$ to be finite.", "\n\n\\[prop:upper bound on cU\\]Let $U\\subseteq\\Sigma$ be open. ", "If there exists a non-resonant $\\psi_{1}\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ such that $$\\psi_{1}(U)\\cap U=\\emptyset$$ then $\\overline{c}(U)\\leq\\overline{c}(\\psi)+\\overline{c}(\\psi^{-1})$ for any path $\\psi=\\{\\psi_{t}\\}_{0\\leq t\\leq1}$ connecting with endpoint $\\psi_{1}$. Thus $\\overline{c}(U)<\\infty$ whenever there exists $\\psi_{1}\\in\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ such that $\\psi_{1}(\\overline{U})\\cap\\overline{U}=\\emptyset$.\n\nLet $\\psi$ be a path connecting $\\mbox{id}_{\\Sigma}$ with $\\psi_{1}$. If $\\varphi$ satisfies $\\mathfrak{S}(\\varphi)\\subset U$ then by Lemma \\[lem:displacement gives bound\\] we have $\\overline{c}(\\psi\\varphi)=\\overline{c}(\\psi)$, and hence by Proposition \\[lem the use of periodicity\\] $$\\overline{c}(\\varphi)=\\overline{c}(\\psi\\varphi\\psi^{-1})\\leq\\overline{c}(\\psi)+\\overline{c}(\\psi^{-1}).$$ The last statement follows from the fact that if non-resonance is a generic condition, and hence if $\\psi_{1}(\\overline{U})\\cap\\overline{U}=\\emptyset$ then one can find another $\\psi_{1}'$ which is non-resonant and satisfies $\\psi_{1}'(U)\\cap U\\ne\\emptyset$.\n\nWe have now completed the proof of Theorem \\[Theorem-B\\] from the Introduction.", "\n\nWe define another function $\\gamma:\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$ by $$\\gamma(\\varphi):=\\left|\\overline{c}(\\varphi)\\right|+\\left|\\overline{c}(\\varphi^{-1})\\right|.$$\n\n\\[prop:Suppose–satisfies\\]Suppose $\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ satisfies $\\gamma(\\psi)=0$. Then $\\psi_{1}=\\mbox{\\emph{id}}_{\\Sigma}$.\n\nIf $\\psi_{1}\\ne\\mbox{id}_{\\Sigma}$ then since $c(\\psi)=0$ we must have $\\psi_{1}\\ne\\theta^{T}$ for all $T\\in\\mathbb{R}$. Thus there exists an open set $U\\subset\\Sigma$ with the property that $\\psi_{1}(U)\\cap\\left(\\bigcup_{t\\in\\mathbb{R}}\\theta^{t}(U)\\right)=\\emptyset$. Thus Lemma \\[lem:always positive\\] implies that $c(\\psi)+c(\\psi^{-1})\\geq c(U)>0$, which contradicts the fact that $\\gamma(\\psi)=0$.\n\nDefine $\\hat{\\gamma}:\\mbox{Cont}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$ by setting $$\\hat{\\gamma}(\\psi):=\\inf\\left\\{ \\gamma(\\varphi)\\mid\\varphi=\\{\\varphi_{t}\\}_{0\\leq t\\leq1}\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi),\\,\\varphi_{1}=\\psi\\right\\} .$$\n\n\\[cor:we get a metric on cont\\]For any $\\psi\\ne\\mbox{\\emph{id}}_{\\Sigma}$ one has $\\hat{\\gamma}(\\psi)>0$.\n\nBi-invariance implies that $\\hat{\\gamma}^{-1}(0)$ is a normal subgroup of $\\mbox{Cont}_{0}(\\Sigma,\\xi)$. A theorem due to Rybicki [@Rybicki2010] tells us that $\\mbox{Cont}_{0}(\\Sigma,\\xi)$ is a *simple* group, and thus it suffices to show that there exists *some* $\\psi\\in\\mbox{Cont}_{0}(\\Sigma,\\xi)$ with $\\hat{\\gamma}(\\psi)>0$. This is clear by Proposition \\[prop:Suppose–satisfies\\] and the fact that $\\gamma$ is integer valued.", "\n\nWe conclude this section by using $\\hat{\\gamma}$ to define a bi-invariant metric $\\mbox{d}_{\\hat{\\gamma}}$ on $\\mbox{Cont}_{0}(\\Sigma,\\xi)$. The following result was stated as Theorem \\[thm:metric\\] in the Introduction.", "\n\nDefine $$\\mbox{\\emph{d}}_{\\hat{\\gamma}}:\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)\\times\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$$ by $$\\mbox{\\emph{d}}_{\\hat{\\gamma}}(\\varphi,\\psi)=\\hat{\\gamma}(\\varphi\\psi^{-1}).$$ Then $\\mbox{\\emph{d}}_{\\hat{\\gamma}}$ is a well defined bi-invariant metric on $\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$. Similarly if one defines $$\\mbox{\\emph{d}}_{\\gamma}:\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\times\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)\\rightarrow\\mathbb{Z}_{\\geq0}$$ in the same way, but using $\\gamma$ instead of $\\hat{\\gamma}$, then $\\mbox{\\emph{d}}_{\\gamma}$ is a bi-invariant pseudo-metric on $\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ whose degeneracy locus is the (possibly empty) set of non-contractible loops $\\varphi=\\{\\varphi_{t}\\}_{t\\in S^{1}}$ in $\\mbox{\\emph{Cont}}_{0}(\\Sigma,\\xi)$ with $c(\\varphi)=c(\\varphi^{-1})=0$.\n\nThis is by now straightforward. ", "The triangle inequality for $\\mbox{d}_{\\gamma}$ and $\\mbox{d}_{\\hat{\\gamma}}$ follows directly from the triangle inequality for $\\overline{c}$ - given $\\varphi,\\psi,\\vartheta\\in\\widetilde{\\mbox{Cont}}_{0}(\\Sigma,\\xi)$ we compute: $$\\begin{aligned}\n\\mbox{d}_{\\gamma}(\\varphi,\\psi) & =\\gamma(\\varphi\\psi^{-1})\\nonumber \\\\\n & =\\left|\\overline{c}(\\varphi\\psi^{-1})\\right|+\\left|\\overline{c}(\\psi\\varphi^{-1})\\right|\\nonumber \\\\\n & =\\left|\\overline{c}(\\varphi\\vartheta^{-1}\\vartheta\\psi^{-1})\\right|+\\left|\\overline{c}(\\psi\\vartheta^{-1}\\vartheta\\varphi^{-1})\\right|\\nonumber \\\\\n & \\leq\\left|\\overline{c}(\\varphi\\vartheta^{-1})\\right|+\\left|\\overline{c}(\\vartheta\\psi^{-1})\\right|+\\left|\\overline{c}(\\psi\\vartheta^{-1})\\right|+\\left|\\overline{c}(\\vartheta\\varphi^{-1})\\right|\\nonumber \\\\\n & =\\gamma(\\varphi\\vartheta^{-1})+\\gamma(\\vartheta\\psi^{-1})\\nonumber \\\\\n & =\\mbox{d}_{\\gamma}(\\varphi,\\vartheta)+\\mbox{d}_{\\gamma}(\\vartheta,\\psi^{-1}).\\end{aligned}$$ Next, bi-invariance follows as $\\overline{c}$ is conjugation invariant. ", "That $\\mbox{d}_{\\hat{\\gamma}}$ is non-degenerate is precisely Corollary \\[cor:we get a metric on cont\\], and the statement about when $\\mbox{d}_{\\gamma}$ is non-degenerate follows from Proposition \\[prop:Suppose–satisfies\\].", "\n\n\\[sec:Relations-to-symplectic\\]Prequantization spaces\n=====================================================\n\n\\[sub:Hamiltonian-Floer-homology\\]Hamiltonian Floer homology \n-------------------------------------------------------------\n\nFix a Liouville domain $(M_{0},d\\gamma_{0})$. Let $S:=\\partial M_{0}$ and $\\kappa:=\\gamma_{0}|_{S}$, so that $(S,\\kappa)$ is a contact manifold. ", "Let $(M,d\\gamma)$ denote the completion of $M_{0}$, so that $M=M_{0}\\cup_{S}(S\\times[1,\\infty))$. It is convenient in this section to introduce the notation $$M(\\sigma):=\\begin{cases}\nM_{0}\\backslash(S\\times[\\sigma,1), & 0<\\sigma<1,\\\\\nM_{0}\\cup_{S}(S\\times[1,\\sigma), & \\sigma\\geq1.", "\n\\end{cases}$$ Thus $M_{0}=\\overline{M(1)}$. Note here we are using $\\sigma$ to denote the $\\mathbb{R}$-coordinate on the end of $M$ - this is so as to avoid confusion in Section \\[sub:Rabinowitz-Floer-homology on Mxs1\\], when a second Liouville domain will come into play.", "\n\nDenote by $\\mbox{Ham}_{c}(M,d\\gamma)$ the group of Hamiltonian diffeomorphisms $f$ on $M$ with compact support. ", "As before, a *path* $f=\\{f_{t}\\}_{0\\leq t\\leq1}$ of compactly supported Hamiltonian diffeomorphisms is assumed to be smoothly parametrized and begin at the identity: $f_{0}=\\mbox{id}_{M}$. Given such a path $f=\\{f_{t}\\}_{0\\leq t\\leq1}$, let $X_{f}$ denote the time-dependent vector field on $M$ defined by $$\\frac{\\partial}{\\partial t}f_{t}=X_{f_{t}}\\circ f_{t}.$$ The equation $$f_{t}^{*}\\gamma-\\gamma=da_{t},\\ \\ \\ a_{0}\\equiv0\\label{at-1}$$ determines a smooth compactly supported function $a_{t}:M\\to\\mathbb{R}$. If we define $$F_{t}=i_{X_{f_{t}}}\\gamma-\\left(\\frac{\\partial}{\\partial t}a_{t}\\right)\\circ f_{t}^{-1},\\label{eq:Ft from at-1}$$ then $F_{t}$ generates $f_{t}$: $f_{t}=f_{F}^{t}$. We can recover $a_{t}$ from $F_{t}$ via $$a_{t}=\\int_{0}^{t}\\left(i_{X_{f_{t}}}\\gamma-F_{s}\\right)\\circ f_{s}ds\\label{eq:relate a and F-1}$$ (see for instance [@McDuffSalamon1998 p294]).", "\n\nWe briefly explain the construction of the Hamiltonian Floer homology of $f$ in this section. ", "The setting we consider here is a special case of the one considered by Frauenfelder and Schlenk in [@FrauenfelderSchlenk2007], to which we refer to for more details. ", "However it will be convenient for us to use Frauenfelder’s Morse-Bott framework [@Frauenfelder2004], in order to make the link with the Rabinowitz Floer homology of $\\Sigma:=M\\times S^{1}$ clearer in the next section.", "\n\nGiven $F\\in C_{c}^{\\infty}(S^{1}\\times M,\\mathbb{R})$, the flow $f_{F}^{t}$ has many 1-periodic orbits, since $f_{F}^{t}$ is compactly supported. ", "Denote by $$\\mathfrak{r}_{F}:=\\inf\\left\\{ \\sigma>0\\mid\\mathfrak{S}(f_{F})\\subseteq M(\\sigma)\\right\\} .$$ Given a path $f=\\{f_{t}\\}_{0\\leq t\\leq1}$ in $\\mbox{Ham}_{c}(M,d\\gamma)$, we set $\\mathfrak{r}_{f}:=\\mathfrak{r}_{F}$, where $F$ is given by (\\[eq:Ft from at-1\\]). ", "Next, we set $$\\mathcal{P}_{F}:=\\left\\{ y\\in M(\\mathfrak{r}_{F})\\mid f_{F}^{1}(y)=y\\right\\} .$$ Note that $\\mathcal{P}_{F}$ is *not* necessarily the same as $\\{y\\in\\mathfrak{S}(f_{F})\\mid f_{F}^{1}(y)=y\\}$ - see (\\[eq:def of G\\]) for an example of a function $G$ for an example where $\\mathcal{P}_{G}\\ne\\{y\\in\\mathfrak{S}(f_{G})\\mid f_{G}^{1}(y)=y\\}$.\n\nDefine a subset $\\mathcal{H}_{c}^{\\textrm{mb}}\\subseteq C_{c}^{\\infty}(S^{1}\\times M,\\mathbb{R})$ (here the ‘mb’ stands for Morse-Bott) to consist of those functions $F$ with the property that $\\mathcal{P}_{F}$ is either a closed submanifold of $M$ or an open domain whose closure is a compact manifold, and for which $$T_{y}\\mathcal{P}_{F}=\\ker(Df_{F}^{1}(y)-\\mathbb{1})\\ \\ \\ \\mbox{for all }y\\in\\mathcal{P}_{F}.\\label{eq:morse bott}$$ It is well known that the subset $\\mathcal{H}_{c}^{\\textrm{mb}}$ is generic in $C_{c}^{\\infty}(S^{1}\\times M,\\mathbb{R})$. We say that a path $f=\\{f_{t}\\}_{0\\leq t\\leq1}$ is *non-degenerate* if the function $F$ defined in (\\[eq:Ft from at-1\\]) belongs to $\\mathcal{H}_{c}^{\\textrm{mb}}$.\n\nWe denote by $R_{S}$ the Reeb vector field of $\\kappa$. Denote by $\\widehat{\\mathcal{H}}$ the set of time-dependent smooth functions $\\widehat{F}$ on $M$ with the property that there exists $C>0$ such that $\\widehat{F}_{t}|_{S\\times[C,\\infty)}$ is of the form $\\widehat{F}_{t}(y,\\sigma)=e(\\sigma)$ for some smooth function $e:[C,\\infty)\\rightarrow\\mathbb{R}$ satisfying $$0\\leq e'(\\sigma)<\\wp(S,\\kappa).\\label{eq:asymptotic m behaviour-2}$$ Here $$\\wp(S,\\kappa):=\\inf\\left\\{ T>0\\mid\\exists\\ \\mbox{a closed Reeb orbit of }R_{S}\\mbox{ of period }T>0\\right\\} .$$ This ensures that if $ $$\\varphi_{\\widehat{F}}^{1}$ denotes the flow of $\\widehat{F}$ then $\\varphi_{\\widehat{F}}^{1}$ has no non-constant 1-periodic orbits on $S\\times(C,\\infty)$. Note that if $F\\in C_{c}^{\\infty}(S^{1}\\times M,\\mathbb{R})$ then one can find $\\widehat{F}\\in\\widehat{\\mathcal{H}}$ such that $\\widehat{F}|_{S^{1}\\times M(\\mathfrak{r}_{F})}=F$.\n\nFix a path $f=\\{f_{t}\\}_{0\\leq t\\leq1}$, and let $F$ denote the function defined in (\\[eq:Ft from at-1\\]), and fix an extension $\\widehat{F}\\in\\widehat{\\mathcal{H}}$ such that $\\widehat{F}|_{S^{1}\\times M_{\\mathfrak{r}_{F}}^{\\textrm{int}}}=F$. Recall that $\\Lambda(M):=C_{\\textrm{contr}}^{\\infty}(S^{1},M)$. Define the *Hamiltonian action functional* $\\mathcal{A}_{f}:\\Lambda(M)\\rightarrow\\mathbb{R}$ by $$\\mathcal{A}_{f}(v):=\\int_{0}^{1}v^{*}\\gamma-\\widehat{F}_{t}(v)dt.\\label{A f}$$\n\nDenote by $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{f})$ the set of critical points $v$ of $\\mathcal{A}_{f}$ with $v(S^{1})\\subseteq M(\\mathfrak{r}_{F})$. Then $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{f})$ doesn’t depend on the extension $\\widehat{F}$ - in fact $$\\mbox{Crit}^{\\circ}(\\mathcal{A}_{f})=\\mathcal{P}_{F},$$ and hence the assumption (\\[eq:morse bott\\]) implies that each component of $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{f})$ is a Morse-Bott component for $\\mathcal{A}_{f}$.\n\nFix $J\\in\\mathcal{J}_{\\textrm{conv}}(M)$ (cf. (", "\\[eq:convex type\\])). ", "We define an $L^{2}$-inner product $\\left\\langle \\left\\langle \\cdot,\\cdot\\right\\rangle \\right\\rangle _{J}$ on $\\Lambda(M)$ as before (cf. (", "\\[eq:l2 norm\\]), only this time there is no $bb'$ term). ", "We denote by $\\nabla_{J}\\mathcal{A}_{F}$ the gradient of $\\mathcal{A}_{F}$ with respect to $\\left\\langle \\left\\langle \\cdot,\\cdot\\right\\rangle \\right\\rangle _{J}$. ** Pick a Morse function $g:\\mbox{Crit}^{\\circ}(\\mathcal{A}_{f})\\rightarrow\\mathbb{R}$ and a Riemannian metric $\\varrho$ on $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{f})$ such that $(g,\\varrho)$ is a Morse-Smale pair. ", "In the case where $\\mathcal{P}_{F}$ is an open domain in $M$ whose boundary is a compact manifold, $g$ must be chosen so that $\\left\\langle dg,\\mathbf{n}\\right\\rangle <0$ on the boundary, where $\\mathbf{n}$ is an outward pointing normal. ", "As before we define moduli spaces $\\mathcal{M}_{v^{-},v^{+}}(\\mathcal{A}_{f},g,J,\\varrho)$ of gradient flow lines with cascades for critical points $v^{\\pm}\\in\\mbox{Crit}(g)$. This time we grade $v\\in\\mbox{Crit}(g)$ simply by $\\mu(W):=\\mu_{\\textrm{CZ}}(W)+\\mbox{ind}_{g}(W)$, where $\\mu_{\\textrm{CZ}}(W)$ is the Conley-Zehnder index. ", "A standard convexity argument gives the necessary compactness needed to define Floer homology - see Frauenfelder-Schlenk [@FrauenfelderSchlenk2007].", "\n\nGiven $-\\infty<a<b<\\infty$ denote by $\\mbox{CF}_{*}^{(a,b)}(\\mathcal{A}_{f},g):=\\mbox{Crit}_{*}^{(a,b)}(g)\\otimes\\mathbb{Z}_{2}$, where $\\mbox{Crit}_{*}^{(a,b)}(g)$ denotes the set of critical points $v$ of $g$ with $a<\\mathcal{A}_{f}(v)<b$. As before one defines a boundary operator $\\partial$ on $\\mbox{CF}_{*}^{(a,b)}(\\mathcal{A}_{f},g)$ by counting the elements of the zero-dimensional parts of the moduli spaces $\\mathcal{M}_{v^{-},v^{+}}(\\mathcal{A}_{f},g,J,\\varrho)$ for $v^{-}\\ne v^{+}$. We denote by $\\mbox{HF}_{*}^{(a,b)}(\\mathcal{A}_{f})$ the associated homology, which as the notation suggests, is independent of the auxiliary data $(g,J,\\varrho)$. In fact, one can also show it is also independent of the choice of path $f$. We abbreviate $\\mbox{HF}_{*}^{a}(\\mathcal{A}_{f}):=\\mbox{HF}_{*}^{(-\\infty,a)}(\\mathcal{A}_{f})$ and $\\mbox{HF}_{*}(\\mathcal{A}_{f}):=\\mbox{HF}_{*}^{(-\\infty,\\infty)}(\\mathcal{A}_{f})$. We denote the natural maps $\\mbox{HF}_{*}^{a}(\\mathcal{A}_{f})\\rightarrow\\mbox{HF}_{*}(\\mathcal{A}_{f})$ by $j_{f}^{a}$ in the same way as before. ", "Under our grading convention explained in Remark \\[Our-grading-convention\\], there is a canonical isomorphism $$\\mbox{HF}_{*}(\\mathcal{A}_{f})\\cong\\mbox{H}_{n+*}(M_{0},\\partial M_{0})\\cong\\mbox{H}^{n-*}(M_{0}).\\label{eq:computng HF}$$ See the proof of Lemma \\[ex the unit for cM\\] below for one way to see this.", "\n\nNext, the Floer homology $\\mbox{HF}_{*}(\\mathcal{A}_{f})$ carries the structure of a unital ring with unit $\\mathbf{1}_{f}\\in\\mbox{HF}_{n}(\\mathcal{A}_{f})$, and as a result, as since $\\mbox{HF}_{*}(\\mathcal{A}_{f})$ is necessarily non-zero, as before we can define the *spectral number* $$c_{M}(f):=\\inf\\left\\{ a\\in\\mathbb{R}\\mid\\mathbf{1}_{f}\\in j_{f}^{a}(\\mbox{HF}_{*}^{a}(\\mathcal{A}_{f}))\\right\\} .$$ As before, $c_{M}$ is a well defined function $$c_{M}:\\widetilde{\\mbox{Ham}}_{c}(M,d\\gamma)\\rightarrow\\mathbb{R}.$$ We can use $c_{M}$ to define a capacity on open subsets $\\mathcal{O}\\subset M$, $$c_{M}(\\mathcal{O}):=\\sup\\left\\{ c_{M}(f)\\mid\\mathfrak{S}(f)\\subset\\mathcal{O}\\right\\} ,$$ in the same way as before. ", "We use the subscript $c_{M}$ to differentiate it from the function $c$ associated to $\\Sigma:=M\\times S^{1}$ that we will define shortly.", "\n\nWe now introduce a special function $G$ on $M$ such that $G=0$ on $M(1)=\\mbox{int}(M_{0})$ and $G=0$ on $S\\times[1+\\varepsilon,\\infty)$ for some $\\varepsilon>0$, and such that $$G(y,r)=e(\\sigma)\\label{eq:def of G}$$ on $S\\times[1,1+\\varepsilon]$ with $e(1)=e(1+\\varepsilon)=0$ and $e(\\sigma)>0$ on $(1,1+\\varepsilon)$, with $\\left|e'(\\sigma)\\right|<\\wp(S,\\kappa)$. Note that $$\\mathcal{P}_{G}=M(1),\\label{eq:int M0}$$ where points in $M_{0}$ are thought of as constant loops. ", "In particular, $f_{G}^{t}|_{M(1)}=\\mbox{id}_{M(1)}$. By a slight abuse of notation we thus think of $G$ as an extension to $\\widehat{\\mathcal{H}}$ of the ‘Hamiltonian diffeomorphism’ $\\mbox{id}_{M(1)}$.\n\n\\[ex the unit for cM\\]In the case $\\mbox{\\emph{id}}_{M(1)}$ one has $c_{M}(\\mbox{\\emph{id}}_{M(1)})=0$. Moreover in this case the unit $\\mathbf{1}$ is simply given by the fundamental class $[M_{0}]$.\n\nWe define $\\mathcal{A}_{\\textrm{id}_{M(1)}}$ using the function $G$ defined in (\\[eq:def of G\\]). ", "Thus $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{\\textrm{id}_{M(1)}})=M(1)$, and every element of $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{\\textrm{id}_{M(1)}})$ has action value zero. ", "Thus there are no gradient flow lines of $\\mathcal{A}_{\\textrm{id}_{M(1)}}$, and hence the Floer complex $\\mbox{CF}_{*}(\\mathcal{A}_{\\textrm{id}_{M(1)}},g)$ reduces to the Morse complex of a Morse function $g$ on $M(1)$. Such a Morse function $g$ can be chosen so that $g>1$ on $M(1)$ and such that $g$ is the restriction of a Morse function $\\widehat{g}:M\\rightarrow\\mathbb{R}$ such that $\\widehat{g}(y,\\sigma)=\\frac{1}{\\sigma}$ on $S\\times[1,\\infty)$. This shows that $$\\mbox{HF}_{*}(\\mathcal{A}_{\\textrm{id}_{M(1)}})\\cong\\mbox{HM}_{n+*}(g)\\cong\\mbox{H}_{n+*}(M_{0},\\partial M_{0}),$$ which proves (\\[eq:computng HF\\]). ", "Moreover if the Morse function $g$ has a unique maximum $y_{\\max}$ then $[y_{\\max}]$ is a cycle in $\\mbox{HF}_{n}(\\mathcal{A}_{\\textrm{id}_{M(1)}})$, and in fact $\\mbox{HF}_{n}(\\mathcal{A}_{\\textrm{id}_{M(1)}})=\\mathbb{Z}_{2}[y_{\\max}]$. Note that under the isomorphism of the Morse homology of $g$ with the relative homology of $(M_{0},\\partial M_{0})$, $[y_{\\max}]$ represents the fundamental class $[M_{0}]$. This shows that $\\boldsymbol{1}_{\\textrm{id}_{M_{0}}}=[M_{0}]$, and $c_{M}(\\mbox{id}_{M(1)})=0$ as claimed.", "\n\nThe prequantization space $\\Sigma=M\\times S^{1}$ \n-------------------------------------------------\n\n**The** *prequantization space* **of $M$ is the contact manifold $\\Sigma:=M\\times S^{1}$, equipped with the contact structure $\\xi:=\\ker\\,\\alpha$, where** $$\\alpha:=\\gamma+d\\tau,$$ and $\\tau$ is the coordinate on $S^{1}\\cong\\mathbb{R}/\\mathbb{Z}$. The third class of contact manifolds we study in this paper are these prequantization spaces, which for convenience we refer to as Assumption **(C)**:\n\n[(C)]{}\n\n: $(\\Sigma,\\xi=\\ker\\,\\zeta)$ is a prequantization space $\\Sigma=M\\times S^{1}$, where $(M,d\\gamma)$ is a Liouville manifold, and $\\alpha=\\gamma+d\\tau$.\n\nIn this case $\\Sigma$ is obviously periodic, but it is *not* Liouville fillable in the previous sense. ", "Aside from anything else, $\\Sigma$ is necessarily *non-compact*. ", "However $\\Sigma$ does still retain enough of the properties needed above in order to define a Rabinowitz Floer homology, as will explain in the next section.", "\n\nLet us denote by $\\mbox{Cont}_{0,c}(\\Sigma,\\xi)$ those contactomorphisms $\\varphi$ with compact support. ", "There is a natural way to obtain a path $\\varphi=\\{\\varphi_{t}\\}_{0\\leq t\\leq1}$ of compactly supported contactomorphisms on $\\Sigma$ from a path $f=\\{f_{t}\\}_{0\\leq t\\leq1}$ of compactly supported Hamiltonian diffeomorphisms on $M$. Indeed, given such a path $f$, define $\\varphi_{t}:\\Sigma\\rightarrow\\Sigma$ by $$\\varphi_{t}(y,\\tau):=\\bigl(f_{t}(y),\\underset{\\textrm{mod }1}{\\underbrace{\\tau-a_{t}(y)}}\\bigr),\\label{eq:phi from f-1}$$ where $a_{t}$ was defined in (\\[at-1\\]). ", "One easily checks that $\\varphi_{t}$ is an *exact* contactomorphism. ", "We say that the contact isotopy $\\varphi$ is the *lift* of the Hamiltonian isotopy $f$. In this case the contact Hamiltonian $h_{t}$ associated to $\\varphi_{t}$ is simply $F_{t}$: $$h_{t}\\circ\\varphi_{t}=\\alpha\\left(\\frac{\\partial}{\\partial t}\\varphi_{t}\\right)=F_{t}\\circ\\varphi_{t},\\label{eq:h equals F}$$ where $F_{t}$ was defined in (\\[eq:Ft from at-1\\]). ", "Fix a function $\\widehat{F}\\in\\widehat{\\mathcal{H}}$ such that $\\widehat{F}=F$ on $S^{1}\\times M(\\mathfrak{r}_{F})$, and define $\\widehat{H}_{t}:S\\Sigma\\rightarrow\\mathbb{R}$ by $\\widehat{H}_{t}:=r\\widehat{F}_{t}$.\n\nConsider again the Rabinowitz action functional $\\mathcal{A}_{\\varphi}:\\Lambda(S\\Sigma)\\times\\mathbb{R}\\rightarrow\\mathbb{R}$ defined as in (\\[eq:Rab functional\\]), using $\\widehat{H}_{t}$. Suppose $(u,\\eta)\\in\\mbox{Crit}(\\mathcal{A}_{\\varphi})$. Write $u(t)=(v(t),\\tau(t),r(t))\\in M\\times S^{1}\\times(0,\\infty)$. Then from (\\[eq:critical points\\]) we have $$\\begin{aligned}\n(f_{1}(v\\left(\\tfrac{1}{2}\\right),\\underset{\\textrm{mod 1}}{\\underbrace{\\tau\\left(\\tfrac{1}{2}\\right)-a_{1}\\left(v\\left(\\tfrac{1}{2}\\right)\\right)}}\\bigr) & =\\varphi_{1}\\left(u\\left(\\tfrac{1}{2}\\right)\\right),\\nonumber \\\\\n & =\\theta^{-\\eta}\\left(v\\left(\\tfrac{1}{2}\\right),\\tau\\left(\\tfrac{1}{2}\\right)\\right)\\nonumber \\\\\n & =\\bigl(v\\left(\\tfrac{1}{2}\\right),\\underset{\\textrm{mod 1}}{\\underbrace{\\tau\\left(\\tfrac{1}{2}\\right)-\\eta}}\\bigr)\\end{aligned}$$ and hence if $y:=v(\\tfrac{1}{2})$ then $f_{1}(y)=y$ and $a_{1}(y)=\\eta$ mod 1. ", "Moreover since $\\varphi_{t}$ is exact one has $r(t)\\equiv1$ for all $t$ (cf. ", "the last statement of Lemma \\[lem:critical points\\]). ", "Since we only consider *contractible* critical points of $\\mathcal{A}_{\\varphi}$, we deduce:\n\n\\[lem:relating the two functionals\\]There exists a bijective map $$\\pi:\\mbox{\\emph{Crit}}(\\mathcal{A}_{\\varphi})\\rightarrow\\mbox{\\emph{Crit}}(\\mathcal{A}_{f})$$ given by $$\\pi(u=(v,\\tau,r),\\eta):=\\left(t\\mapsto f_{t}\\left(v\\left(\\tfrac{1}{2}\\right)\\right)\\right).$$ Moreover $$\\mathcal{A}_{\\varphi}(u,\\eta)=\\mathcal{A}_{f}(\\pi(u,\\eta)).$$ In particular, every critical point $(u,\\eta)$ of $\\mathcal{A}_{\\varphi}$ has $$u(S^{1})\\subseteq M(\\mathfrak{r}_{f})\\times S^{1}\\times\\{1\\}.$$\n\nGiven a contactomorphism $\\varphi\\in\\mbox{Cont}_{0,c}(\\Sigma,\\xi)$, we denote by $$\\mathfrak{r}_{\\varphi}=\\inf\\left\\{ \\sigma>0\\mid\\mathfrak{S}(\\varphi)\\subseteq M(\\sigma)\\times S^{1}\\right\\} .$$ Thus if $\\varphi$ is the lift of $f$ then $$\\mathfrak{r}_{\\varphi}=\\mathfrak{r}_{f}.$$\n\n\\[sub:Rabinowitz-Floer-homology on Mxs1\\]Rabinowitz Floer homology on $\\Sigma$\n------------------------------------------------------------------------------\n\nLet $P_{0}$ denote a torus with a disc removed, so that $\\partial P_{0}=S^{1}$. Equip $P_{0}$ with an exact symplectic form $d\\beta_{0}$ such that $\\beta_{0}|_{\\partial P_{0}}=d\\tau$. Denote by $(P,d\\beta)$ the completion of $P$, so that $$\\beta=rd\\tau\\ \\ \\ \\mbox{on}\\ \\ \\ \\partial P_{0}\\times[1,\\infty).$$ Consider $$W:=M\\times P,$$ equipped with the symplectic form $d\\lambda$ where $\\lambda:=\\gamma+\\beta$. Note that $W$ is *not* a Liouville filling of $\\Sigma$. In particular, when equipped with the symplectic form $d\\lambda$, there is no embedding $(S\\Sigma,d(r\\alpha))\\hookrightarrow(W,d\\lambda)$ that we can use in order to extend the Rabinowitz action functional $\\mathcal{A}_{\\varphi}$ to a functional defined on all on $\\Lambda(W)\\times\\mathbb{R}$. Ideally we would like to work with a symplectic form $\\omega'$ such that $\\omega'|_{M\\times\\partial P_{0}\\times(0,\\infty)}=d(r\\alpha)$. Sadly no such symplectic form exists, and even if one did it would not have the ‘right’ properties at infinity. ", "To circumvent this problem we will work with a family $\\{\\lambda^{s}\\}_{s\\geq1}$ of 1-forms as specified by the following lemma.", "\n\nThere exists a family $\\{\\lambda^{s}\\}_{s\\geq1}$ such that\n\n1. ", " For all $s\\geq1$, $d\\lambda^{s}$ is a symplectic form on $W$.\n\n2. ", " For all $s\\geq1$, $d\\lambda^{s}$ is split-convex at infinity ** - that is, on $\\left(M\\backslash M(2s-1)\\times P\\right)\\cup\\left(M\\times P\\backslash P(2s-1)\\right)$, the 1-form $\\lambda^{s}$ is just the split form $(2s-1)\\gamma+\\beta$. Thus $(W,d\\lambda^{s})$ is geometrically bounded for each $s\\geq1$.\n\n3. ", " For all $s\\geq1$, on $M(s)\\times P(\\tfrac{1}{2s-1})$, $\\lambda^{s}$ is the split form $\\frac{1}{2s-1}\\gamma+\\beta$.\n\n4. ", " For $s=1$, $\\lambda^{1}=\\lambda$ on all of $W$.\n\n5. ", " For $s>1$, define $\\Omega_{s}\\subset S\\Sigma$ to be the subset $$\\Omega_{s}:=M(s)\\times S^{1}\\times(1/s,s).$$ Then for each $s>1$, the ‘inclusion’ $$\\iota_{s}:\\Omega_{s}\\hookrightarrow W\\label{eq:embedding i}$$ satisfies $\\iota^{*}\\lambda^{s}=r\\alpha$.\n\nDefine a smooth function $$k:[1,\\infty)\\times[0,\\infty)\\times[0,\\infty)\\rightarrow(0,\\infty)$$ such that $$k(1,\\sigma,r)=1\\ \\ \\ \\mbox{for all }\\sigma,r\\in[0,\\infty)$$ and if $s>1$ then: $$k(s,\\sigma,r)=r\\ \\ \\ \\mbox{if }\\sigma\\in[0,s)\\mbox{ and }r\\in(1/s,s),$$ $$k(s,\\sigma,r)=2s-1\\ \\ \\ \\mbox{if either }\\sigma>2s-1\\mbox{ or }r>2s-1,$$ $$k(s,\\sigma,r)=\\frac{1}{2s-1}\\ \\ \\ \\mbox{if }\\sigma\\in[0,s)\\mbox{ and }r\\in\\left(0,\\frac{1}{2s-1}\\right),$$ and finally such that $$\\frac{\\partial k}{\\partial\\sigma}(s,\\sigma,r)\\geq0\\ \\ \\ \\mbox{for all }(s,\\sigma,r)\\in[1,\\infty)\\times[0,\\infty)\\times[0,\\infty).$$ Now define a function $\\tilde{k}:[1,\\infty)\\times V\\rightarrow\\mathbb{R}$ by setting $$\\tilde{k}(s,(y,\\sigma),(r,\\tau)):=k(s,\\sigma,r)\\ \\ \\ \\mbox{for }(y,\\sigma)\\in\\partial M_{0}\\times\\mathbb{R}^{+},\\ (r,\\tau)\\in\\partial P_{0}\\times\\mathbb{R}^{+},$$ and such that for $z\\in M(0)$ and $(r,\\tau)\\in\\partial P_{0}\\times(0,\\infty)$ one has $$\\tilde{k}(s,z,(r,\\tau))=k(s,0,r),$$ and if $(y,\\sigma)\\in\\partial M_{0}\\times(0,\\infty)$ and $w\\in P(0)$ then $$\\tilde{k}(s,(y,\\sigma),w)=k(s,\\sigma,0).$$ and finally such that for $z\\in M(0)$ and $w\\in P(0)$ one has $$\\tilde{k}(s,z,w):=k(s,0,0)$$ Then finally we define $$\\lambda^{s}:=\\tilde{k}(s,\\cdot,\\cdot)\\gamma+\\beta.$$ It is clear that properties (1)-(5) are satisfied for $\\lambda^{s}$ defined in this way.", "\n\nWe now prove the following result.", "\n\n\\[thm:rfh doesnt depend on s\\]For any non-degenerate path $\\varphi$ there exists a constant $s_{0}(\\varphi)\\geq1$ such that if $s\\geq s_{0}(\\varphi)$ then it is possible to define the Rabinowitz Floer homology $\\mbox{\\emph{RFH}}_{*}(\\mathcal{A}_{\\varphi},W,d\\lambda^{s})$ (here the notation indicates that we are working with the symplectic structure $d\\lambda^{s}$ on $W$). ", "Moreover the Rabinowitz Floer homology is independent of the choice of $s\\geq s_{0}(\\varphi)$.\n\nLet $\\mathcal{J}_{s}(W)$ denote the set of time-dependent almost complex structures $J=\\{J_{t}\\}_{t\\in S^{1}}$ on $W$ that are $d\\lambda^{s}$ compatible, and satisfy:\n\n1. ", " If $\\iota_{s}$ is the embedding (\\[eq:embedding i\\]) then $\\iota_{s}^{*}J\\in\\mathcal{J}_{\\textrm{conv}}(\\Omega_{s}\\subset S\\Sigma)$.\n\n2. ", " The restriction of $J$ to $\\left(M\\backslash M(2s-1)\\times P\\right)\\cup\\left(M\\times P\\backslash P(2s-1)\\right)$ is *split* - that is, there exist almost complex structures $J'\\in\\mathcal{J}_{\\textrm{conv}}(M)$ and $J''\\in\\mathcal{J}_{\\textrm{conv}}(P)$ such that $J=(J',J'')$ on this set.", "\n\nExtend $\\mathcal{A}_{\\varphi}$ to a functional $\\mathcal{A}_{\\varphi}^{\\kappa}$ defined on all of $\\Lambda(W)\\times\\mathbb{R}$ in the same way as before, by replacing $\\widehat{H}_{t}$ with a truncated function $\\widehat{H}_{t}^{\\kappa}$ as in (\\[eq:truncating H\\]). ", "As with the Hamiltonian Floer homology, we are now only interested in the set $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{\\varphi}^{\\kappa})$ of critical points $(u=(v,\\tau,r),\\eta)$ of critical points of $\\mathcal{A}_{\\varphi}^{\\kappa}$ with $v(S^{1})\\subset M(\\mathfrak{r}_{\\varphi})$. For $s$ large enough, all elements of $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{\\varphi}^{\\kappa})$ are contained in $\\Omega_{s}$ and $\\widehat{H}_{t}^{\\kappa}$ is constant outside of $\\Omega_{2s-1+\\varepsilon}$ for some small $\\varepsilon>0$. Thus if we work with an almost complex structure $J\\in\\mathcal{J}_{s}(W)$, no flow lines of $-\\nabla_{J}\\mathcal{A}_{\\varphi}^{\\kappa}$ can ever escape $\\Omega_{2s-1+\\varepsilon}$. Thus the Rabinowitz Floer homology is well defined for this $s$.\n\nIn order to prove independence of $s$, first note that for $s\\geq\\max\\{s_{0}(\\varphi),s_{0}(\\psi)\\}$ the continuation maps from points (1)-(3) on the properties of Rabinowitz Floer homology go through to show that $$\\mbox{RFH}_{*}(\\mathcal{A}_{\\varphi},W,d\\lambda^{s})\\cong\\mbox{RFH}_{*}(\\mathcal{A}_{\\psi},W,d\\lambda^{s}).\\label{eq:indep of s 1}$$ Next we note that if $\\mbox{id}_{M(1)\\times S^{1}}$ is the contactomorphism with contact Hamiltonian $G$ as defined in (\\[eq:def of G\\]) then $s_{0}(\\mbox{id}_{M(1)\\times S^{1}})=1$. More generally, this is true for any *exact* path $\\varphi$ of contactomorphisms, since in this case for any $\\varepsilon>0$, every critical point of $\\mathcal{A}_{\\varphi}^{\\varepsilon}$ is contained in $\\Sigma\\times\\{0\\}$ - see Lemma \\[prop:linfinity-1\\]. ", "By (\\[eq:indep of s 1\\]) it suffices to show that $\\mbox{RFH}_{*}(\\mathcal{A}_{\\textrm{id}_{M(1)\\times S^{1}}},W,d\\lambda^{s})$ is independent of $s$. But this is clear, since every critical point of the Rabinowitz action functional $\\mathcal{A}_{\\textrm{id}_{M(1)\\times S^{1}}}$ has action value zero, as we are only looking at contractible critical points and we have filled $S^{1}$ with a punctured torus $P_{0}$ rather than a disc $D^{2}$. Thus $\\mbox{Crit}^{\\circ}(\\mathcal{A}_{\\textrm{id}_{M(1)\\times S^{1}}})=M(1)\\times S^{1}\\times\\{0\\}$, and hence regardless of which symplectic structure we use, as in Lemma \\[ex the unit for cM\\], the Rabinowitz complex reduces to the Morse complex of a Morse function $\\tilde{g}:M(1)\\times S^{1}\\rightarrow\\mathbb{R}$ . ", "In particular, it does not depend on $s$.\n\nWe denote by $\\mbox{RFH}_{*}(\\mathcal{A}_{\\varphi},W)$ the groups $\\mbox{RFH}_{*}(\\mathcal{A}_{\\varphi},W,d\\lambda^{s})$ for any $s\\geq s_{0}(\\varphi)$.\n\n\\[thm computation\\]If $\\varphi=\\{\\varphi_{t}\\}_{0\\leq t\\leq1}$ is the lift of $f=\\{f_{t}\\}_{0\\leq t\\leq1}$ then there exists a natural isomorphism $$\\mbox{\\emph{RFH}}_{*}(\\mathcal{A}_{\\varphi},W)\\cong\\mbox{\\emph{HF}}_{*}(\\mathcal{A}_{f})\\otimes\\mbox{\\emph{H}}_{*}(S^{1}),$$ under which the unit $\\mathbf{1}_{\\varphi}$ maps to $\\boldsymbol{1}_{f}\\otimes[S^{1}]$.\n\nBy naturality it suffices to prove the theorem in the case $f=\\mbox{id}_{M(1)}$, $\\varphi=\\mbox{id}_{M(1)\\times S^{1}}$. In this case as in the proof of the last part of Theorem \\[thm:rfh doesnt depend on s\\], one has$\\mbox{}$ $$\\mbox{RFH}_{*}(\\mathcal{A}_{\\textrm{id}_{M(1)\\times S^{1}}},W)\\cong\\mbox{HM}_{*+n}(\\tilde{g}),$$ where $\\tilde{g}$ is a Morse function on $M(1)\\times S^{1}$. We choose $\\widetilde{g}=(g,g')$, where $g$ is the Morse function considered in the proof of Lemma \\[ex the unit for cM\\], and $g':S^{1}\\rightarrow\\mathbb{R}$ is a Morse function with two critical points $\\tau_{\\min}$ and $\\tau_{\\max}$. This gives $$\\mbox{RFH}_{*}(\\mathcal{A}_{\\textrm{id}_{M(1)\\times S^{1}}},W)\\cong\\mbox{HM}_{n+*}(\\tilde{g})\\cong\\mbox{HM}_{*}(g)\\otimes\\mbox{HM}_{*}(g')\\cong\\mbox{H}_{n+*}(M_{0},\\partial M_{0})\\otimes\\mbox{H}_{*}(S^{1}).$$ To complete the proof we check that the unit in $\\mbox{RFH}_{*}(\\mathcal{A}_{\\textrm{id}_{M(1)\\times S^{1}}},V)$ is given by $[((y_{\\max},\\tau_{\\max}),0]$. This is clear since $((y_{\\max},\\tau_{\\max}),0)$ is the only critical point of index $n$ (cf. ", "point (5) from the properties of Rabinowitz Floer homology on page ).", "\n\n\\[sub:Applications\\]Relating the capacities\n-------------------------------------------\n\nWe define $c(\\varphi)$ in the same way as before for $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$. It is by now clear that almost all the results from Sections \\[sec:Spectral-invariants\\]-\\[sec:The-periodic-case - capacities\\] go through in this setting, provided we only work with elements of $\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$. There are two points where things are not quite the same. ", "Firstly, there is no analogue of Lemma \\[thereebflow\\]. ", "This is because the Reeb flow is not compactly supported. ", "In fact, in the case where $M=\\mathbb{R}^{2n}$ one has $c(\\varphi)\\geq0$ for all $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\mathbb{R}^{2n}\\times S^{1},\\xi)$. This follows from Lemma \\[lem:always positive\\]. ", "As a consequence it is no longer possible to prove the strict inequality in Proposition \\[prop:the comparison prop\\], and in fact this is simply not true in this setting (cf. ", "Remark \\[making a real cptly sptd reeb flow\\] and Appendix \\[app:A-’compactly-supported\\]). ", "Thus in this setting Proposition \\[prop:the comparison prop\\] is reduced to the following weaker statement:\n\n\\[prop:the comparison prop-1\\]Suppose $\\varphi,\\psi\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0,c}(\\Sigma,\\xi)$. Suppose the contact Hamiltonian of $\\varphi$ is $h_{t}$, and the contact Hamiltonian of $\\psi$ is $k_{t}$. Write $\\varphi_{t}^{*}\\alpha=\\rho_{t}\\alpha$ and $\\psi_{t}^{*}\\alpha=\\sigma_{t}\\alpha$. Then if for all $t\\in[0,1]$ one has $$\\frac{h_{t}\\circ\\varphi_{t}}{\\rho_{t}}\\geq\\frac{k_{t}\\circ\\psi_{t}}{\\sigma_{t}},\\label{eq:key inequality-2}$$ then $$c(\\varphi)\\leq c(\\psi).\\label{eq:key inequality-1-1}$$\n\nNevertheless we never actually made use of the strict inequality anywhere in the paper except in the proof of Corollary \\[cor:posi and negi\\]. ", "Corollary \\[cor:posi but not negi\\] shows that one direction continues to hold, namely that if $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$ has contact Hamiltonian $h_{t}$ with $h_{t}(\\varphi_{t}(x))<0$ whenever $\\varphi_{t}(x)\\ne x$ then $c(\\varphi)>0$. However the converse is false (see Appendix \\[app:A-’compactly-supported\\]). ", "Note that the proof of Proposition \\[rem strict inequalities\\] still goes through in this setting (provided the set $U$ has compact closure), and hence so does the proof of Corollary \\[cor:infinitely many\\] (again, assuming $U$ has compact closure). ", "Of course, in this setting there are no contractible closed Reeb orbits. ", "The key extra piece of information we have here is the following statement, which was stated as Theorem \\[thm:equality of capacities\\] in the Introduction.", "\n\n\\[thm:equality of capacities-1\\]Suppose $f\\in\\widetilde{\\mbox{\\emph{Ham}}}_{c}(M,d\\gamma)$, and let $\\varphi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$ denote the lift of $f$. Then $$c_{M}(f)=c(\\varphi).$$ Moreover, if $\\mathcal{O}\\subset M$ is open then $$c_{M}(\\mathcal{O})=c(\\mathcal{O}\\times S^{1}).$$\n\nThe first statement follows from Theorem \\[thm computation\\] and Lemma \\[lem:relating the two functionals\\]. ", "Thus clearly $c_{M}(\\mathcal{O})\\leq c(\\mathcal{O}\\times S^{1})$. In order to complete the proof, we must show that given any $\\psi\\in\\widetilde{\\mbox{Cont}}_{0,c}(\\Sigma,\\xi)$ with $\\mathfrak{S}(\\psi)\\subset\\mathcal{O}\\times S^{1}$ there exists $f\\in\\widetilde{\\mbox{Ham}}_{c}(M,d\\gamma)$ with $\\mathfrak{S}(f)\\subset\\mathcal{O}$ and such that lifted contactomorphism $\\varphi$ satisfies $$c(\\psi)\\leq c(\\varphi).\\label{eq:penultimate}$$ This follows from Proposition \\[prop:the comparison prop\\]. ", "Writing $\\psi_{t}^{*}\\alpha=\\rho_{t}\\alpha$, and letting $h_{t}$ denote the contact Hamiltonian of $\\psi$, let us choose $f$ with $\\mathfrak{S}(f)\\subset\\mathcal{O}$ and such that the corresponding function $F$ from (\\[eq:Ft from at-1\\]) satisfies $$F_{t}\\circ\\varphi_{t}\\leq\\frac{h_{t}\\circ\\psi_{t}}{\\rho_{t}},\\label{eq:final eq}$$ where $\\varphi$ denotes the lift of $f$. Since $\\varphi$ is exact and has contact Hamiltonian $F_{t}$, (\\[eq:penultimate\\]) follows from (\\[eq:final eq\\]) and Proposition \\[prop:the comparison prop-1\\].", "\n\nIn order to present an application of Theorem \\[thm:equality of capacities-1\\], let us quickly recall the definition of the *Hofer-Zehnder capacity*. ", "See for instance [@HoferZehnder1994] for an in depth treatment.", "\n\nLet $\\mathcal{O}$ be an open subset of $M$. We define the *Hofer-Zehnder capacity* $c_{\\textrm{HZ}}(\\mathcal{O},M)$ of $\\mathcal{O}$ to $$c_{\\textrm{HZ}}(\\mathcal{O},M):=\\sup\\left\\{ \\left\\Vert H\\right\\Vert \\mid H\\mbox{ is admissible}\\right\\} ,$$ where $H\\in C_{c}^{\\infty}(\\mathcal{O},\\mathbb{R})$ is *admissible* if there exists an open set $O\\subset\\mathcal{O}$ such that $H|_{O}=\\max H$, and if the flow $\\varphi_{H}^{t}$ has no non-constant periodic orbits of period $\\leq1$.\n\nWe also define the *displacement energy* by $$e(\\mathcal{O},M):=\\inf\\left\\{ \\left\\Vert H\\right\\Vert \\mid\\varphi_{H}^{1}(\\mathcal{O})\\cap\\mathcal{O}=\\emptyset\\right\\} .$$ The following result is due to Frauenfelder and Schlenk [@FrauenfelderSchlenk2007 Corollary 8.3], see also [@FrauenfelderGinzburgSchlenk2005; @Schwarz2000].", "\n\n\\[thm:HZ cap thm\\]If $(M_{0},\\gamma_{0})$ is a Liouville domain then $$c_{\\textrm{\\emph{HZ}}}(\\mathcal{O},M)\\leq c_{M}(\\mathcal{O})\\leq e(\\mathcal{O},M).$$\n\nDenote by $B(r)$ the open ball of radius $r$ in $\\mathbb{R}^{2m}$. Then $c_{\\textrm{HZ}}(B(r),\\mathbb{R}^{2m})=\\pi r^{2}$. We can now prove:\n\n\\[prop:our HZ result\\]Let $(M,d\\gamma)$ denote a Liouville manifold. ", "Equip $\\mathbb{R}^{2m}$ with the standard symplectic form $d\\lambda_{0}$, and consider the contact manifold $(\\widetilde{\\Sigma},\\alpha+\\lambda_{0})$, where $\\widetilde{\\Sigma}:=M\\times\\mathbb{R}^{2m}\\times S^{1}$. Suppose $\\mathcal{O}\\subseteq M$ is open and $c_{\\textrm{\\emph{HZ}}}(\\mathcal{O},M)<\\infty$. Choose $r_{0}>0$ such that $$\\left\\lceil \\pi r_{0}^{2}\\right\\rceil <\\left\\lceil c_{\\textrm{\\emph{HZ}}}(\\mathcal{O},M)\\right\\rceil$$ and set $$r_{1}:=\\sqrt{\\tfrac{1}{\\pi}c_{\\textrm{\\emph{HZ}}}(\\mathcal{O},M)}+1$$ Then there does not exist $\\varphi\\in\\mbox{\\emph{Cont}}_{0,c}(\\widetilde{\\Sigma},\\alpha+\\lambda_{0})$ such that $$\\varphi(\\mathcal{O}\\times B(r_{1})\\times S^{1})\\subset\\mathcal{O}\\times B(r_{0})\\times S^{1}.$$\n\nWe first prove that for $r>r_{1}$,\n\n$$c_{\\textrm{HZ}}(\\mathcal{O}\\times B(r),M\\times\\mathbb{R}^{2m})\\geq c_{\\textrm{HZ}}(\\mathcal{O},M).\\label{eq:going one way}$$\n\nFix $\\varepsilon>0$. We consider a cutoff function $\\beta:[0,\\infty)\\rightarrow[0,1]$ such that $\\beta(s)=1$ for $ $$s\\in[0,r-1-\\varepsilon]$ and $\\beta(s)=0$ for $s>r$, and such that $-1\\leq\\beta'(s)\\leq0$ for all $s\\in[0,\\infty)$. Now suppose $H$ is any admissible function on $\\mathcal{O}$. Define $H_{\\beta}:M\\times\\mathbb{R}^{2m}\\rightarrow\\mathbb{R}$ by $$H_{\\beta}(x,y):=\\beta(\\left|y\\right|)H(x).$$ The symplectic gradient of $H_{\\beta}$ with respect to $d\\gamma\\oplus d\\lambda_{0}$ is $$X_{H_{\\beta}}(x,y)=\\left(\\beta(\\left|y\\right|)X_{H}(x),H(x)X_{\\beta}(y)\\right).$$ Suppose $\\gamma:\\mathbb{R}\\rightarrow M\\times\\mathbb{R}^{2m}$ is a non-constant periodic orbit of $X_{H_{\\beta}}$, with $\\gamma(t+T)=\\gamma(t)$ for all $t\\in\\mathbb{R}$. We shall show that $T>1$, so that $H_{\\beta}$ is admissible. ", "Write $\\gamma(t)=(\\gamma_{x}(t),\\gamma_{y}(t))$. Then $$\\dot{\\gamma}_{x}=\\beta(\\left|\\gamma_{y}\\right|)X_{H}(\\gamma_{x}),\\ \\ \\ \\dot{\\gamma}_{y}=H(\\gamma_{x})X_{\\beta}(\\gamma_{y}).$$ Since $\\left|\\beta'\\right|\\leq1$ we see that if $\\gamma_{x}$ is non-constant then $T>1$. But if $\\gamma_{x}$ is constant, say $\\gamma_{x}(t)=x_{0}$, then we must have $H(x_{0})\\ne0$. Since $\\beta'$ is non-zero only for $\\left|\\gamma_{y}\\right|\\in(r-1-\\varepsilon,r)$ we necessarily have $$T\\geq\\frac{1}{H(x_{0})}\\pi(r-1-\\varepsilon)^{2}\\geq\\frac{1}{c_{\\textrm{HZ}}(\\mathcal{O},M)}\\pi(r-1-\\varepsilon)^{2}.$$ Thus as long as $$\\pi(r-1-\\varepsilon)^{2}>c_{\\textrm{HZ}}(\\mathcal{O},M),\\label{eq:with epsilon}$$ $H_{\\beta}$ is indeed admissible. ", "Since clearly $\\max\\, H_{\\beta}=\\max\\, H$, we see that $$c_{\\textrm{HZ}}(\\mathcal{O}\\times B(r),M\\times\\mathbb{R}^{2n})\\geq c_{\\textrm{HZ}}(U,M)$$ provided that (\\[eq:with epsilon\\]) holds. ", "Since $\\varepsilon$ was arbitrary we obtain (\\[eq:going one way\\]). ", "Moreover for any $r>0$ one always has $$e(\\mathcal{O}\\times B(r),M\\times\\mathbb{R}^{2n})\\leq\\pi r^{2},$$ as can be checked directly. ", "The remainder of the proof is an easy application of Theorem \\[thm:equality of capacities-1\\], Theorem \\[thm:HZ cap thm\\] and Corollary \\[cor:non squeezing\\]. ", "Indeed, we have $$\\begin{aligned}\n\\overline{c}(\\mathcal{O}\\times B(r_{0})\\times S^{1}) & =\\left\\lceil e(\\mathcal{O}\\times B(r_{0}),M\\times\\mathbb{R}^{2m})\\right\\rceil \\nonumber \\\\\n & \\leq\\left\\lceil \\pi r_{0}^{2}\\right\\rceil \\nonumber \\\\\n & <\\left\\lceil c_{\\textrm{HZ}}(\\mathcal{O},M)\\right\\rceil \\nonumber \\\\\n & \\leq\\left\\lceil c_{\\textrm{HZ}}(\\mathcal{O}\\times B(r_{1}),M\\times\\mathbb{R}^{2m})\\right\\rceil \\nonumber \\\\\n & \\leq\\overline{c}_{M\\times\\mathbb{R}^{2m}}(\\mathcal{O}\\times B(r_{1}))\\nonumber \\\\\n & =\\overline{c}(\\mathcal{O}\\times B(r_{1})\\times S^{1}).\\end{aligned}$$\n\nHere is an application of Proposition \\[prop:our HZ result\\], which can be seen as a more quantitive (albeit weaker, and with more hypotheses) version of the infinitesimal result of [@EliashbergKimPolterovich2006 Theorem 1.18].", "\n\nSuppose $X$ is a closed connected oriented Riemannian manifold which admits a circle action $S^{1}\\times X\\rightarrow X$ such that the loop $t\\mapsto t\\cdot p$ is not contractible for some $p\\in X$. Then if $\\mathcal{O}\\subset T^{*}X$ is any neighborhood of the zero section then the conclusion of Proposition \\[prop:our HZ result\\] holds.", "\n\nA result of Kei Irie [@Irie2011] proves that in this setting the Hofer-Zehnder capacity of the unit disc bundle $D^{*}X\\subset T^{*}X$ is finite. ", "Thus the same is true of any neighborhood $\\mathcal{O}\\subset T^{*}X$ of the zero section, and hence the hypotheses of Proposition \\[prop:our HZ result\\] are satisfied.", "\n\n\\[app:A-’compactly-supported\\]The ‘compactly supported Reeb flow’\n=================================================================\n\nIn this Appendix we construct a ‘compactly supported Reeb flow’ whose support is contained in a tubular neighborhood of a non-contractible orbit, and explicitly compute the spectral value. ", "This result has been alluded to in Remark \\[making a real cptly sptd reeb flow\\] and also in the discussion before and after Proposition \\[prop:the comparison prop-1\\].", "\n\n\\[prop:compactly supported reeb\\]Suppose $(\\Sigma,\\xi)$ satisfies either Assumption ***(B)*** or ** ***(C)***, and suppose $\\gamma:S^{1}\\rightarrow\\Sigma$ is a closed *non-contractible* Reeb orbit. ", "Then there exists $\\rho_{0}>0$ (which under Assumption ***(B)*** additionally satisfies $\\rho_{0}<1$) and a tubular neighborhood $N$ of $\\gamma$ with the following significance: For all $\\rho\\in\\mathbb{R}$ with $\\left|\\rho\\right|<\\rho_{0}$, there exists an exact contactomorphism $\\vartheta^{\\rho}\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0}(\\Sigma,\\xi)$ with $\\mathfrak{S}(\\vartheta^{\\rho})\\subset N$ with the property that if $x\\in\\mathfrak{S}(\\vartheta^{\\rho})$ is a translated point with of $\\vartheta^{\\rho}$ then $$\\vartheta^{\\rho}(x)=\\theta^{\\rho}(x).$$ In other words, from the point of view of translated points, $\\vartheta^{\\rho}$ is ‘the Reeb flow supported on $\\gamma$’. ", "Moreover if $N'\\subset N$ is any tubular neighborhood of $\\gamma$ then for $\\left|\\rho\\right|$ sufficiently small we have $\\mathfrak{S}(\\vartheta^{\\rho})\\subset N'$.\n\nThe spectral value $c(\\vartheta^{\\rho})$ is given by $$c(\\vartheta^{\\rho})=\\begin{cases}\n0, & 0\\leq\\rho<\\rho_{0},\\\\\n-\\rho, & -\\rho_{0}<\\rho\\leq0.", "\n\\end{cases}$$\n\nWe will prove Theorem \\[prop:compactly supported reeb\\] using Proposition \\[prop:compactly supported Reeb flow on R2n\\] below, which deals with the special case $\\Sigma=\\mathbb{R}^{2n}\\times S^{1}$.\n\n**** In this appendix we equip $\\mathbb{R}^{2n}\\backslash\\{0\\}$ with polar coordinates $(s,\\phi)$ where $s\\in(0,\\infty)$ and $\\phi=(\\phi_{1},\\dots,\\phi_{2n-1})$ with $\\phi_{j}\\in\\mathbb{R}/2\\pi\\mathbb{Z}$. In these coordinates we can have $$\\alpha_{0}=\\sum_{j}\\tfrac{1}{2}s^{2}d\\phi_{j}+d\\tau.$$ Thus $\\tau$ is 1-periodic but the $\\phi_{j}$ are $2\\pi$-periodic! ", "This is so that $c_{\\mathbb{R}^{2m}}(B(r))=\\pi r^{2}$ and not $\\frac{1}{2}r^{2}$.\n\n\\[prop:compactly supported Reeb flow on R2n\\]Denote by $B(r)\\subseteq\\mathbb{R}^{2n}$ the open ball of radius $r$ about the origin. ", "Then for any $r>0$ and any $\\rho\\in\\mathbb{R}$ with $\\left|\\rho\\right|<\\pi r^{2}$, there exists an exact contactomorphism $\\vartheta^{\\rho}\\in\\widetilde{\\mbox{\\emph{Cont}}}_{0,c}(\\mathbb{R}^{2n}\\times S^{1},\\xi_{0})$ with $\\mathfrak{S}(\\vartheta^{\\rho})\\subset B(r)\\times S^{1}$ with the property that if $x\\in\\mathfrak{S}(\\vartheta^{\\rho})$ is a translated point with of $\\vartheta^{\\rho}$ then $$\\vartheta^{\\rho}(x)=\\theta^{\\rho}(x).$$ The spectral value is given by\n\n$$c(\\vartheta^{\\rho})=\\begin{cases}\n0, & 0\\leq\\rho<\\pi r^{2},\\\\\n-\\rho, & -\\pi r^{2}<\\rho\\leq0.", "\n\\end{cases}$$\n\nNote that this result gives a direct proof of the fact that $c(B(r)\\times S^{1})\\geq\\pi r^{2}$ (cf. ", "Theorem \\[thm:equality of capacities-1\\]).", "\n\nBefore getting started on the proofs we need two preliminary results. ", "The following lemma is proved for instance in [@HryniewiczMacarini2012 Lemma 5.2].", "\n\n\\[lem:HM\\]Suppose $(\\Sigma^{2n+1},\\alpha)$ is any contact manifold and $\\gamma:\\mathbb{R}/T\\mathbb{Z}\\rightarrow\\Sigma$ is a closed orbit of the Reeb vector field $R$ of $\\alpha$. Then there exists a tubular neighborhood $N$ of $\\gamma(\\mathbb{R}/T\\mathbb{Z})$ in $\\Sigma$ and and $r>0$ together with a chart $$\\mathtt{x}:B(r)\\times S^{1}\\rightarrow N$$ with the following two properties:\n\n1. ", " $\\mathtt{x}(\\{0\\}\\times S^{1})=\\gamma(\\mathbb{R}/T\\mathbb{Z})$,\n\n2. ", " $\\mathtt{x}^{*}\\alpha=\\sum_{j}\\tfrac{1}{2}s^{2}d\\phi_{j}+Kd\\tau$, where $K:B(r)\\rightarrow(0,\\infty)$ is a positive function satisfying $K_{\\tau}(0)=T$ and $dK_{\\tau}(0)=0$ (here $K_{\\tau}:=K(\\cdot,\\tau)$).", "\n\nWe will use the following simple corollary of this result.", "\n\n\\[cor:nice coordinates\\]Suppose $(\\Sigma^{2n+1},\\alpha)$ is a periodic contact manifold and $\\gamma:S^{1}\\rightarrow\\Sigma$ is a closed orbit of the Reeb vector field $R$ of $\\alpha$. Then there exists a tubular neighborhood $N$ of $\\gamma(S^{1})$ in $\\Sigma$ and $r>0$, together with a chart $$\\mathtt{x}:B(r)\\times S^{1}\\rightarrow N$$ such that $\\mathbf{x}(\\{0\\}\\times S^{1})=\\gamma(S^{1})$, and $$\\mathtt{x}^{*}\\alpha=\\alpha_{0}.$$\n\nIt suffices to show that the periodic condition forces $K\\equiv1$ in Lemma \\[lem:HM\\]. ", "Let $\\beta:=\\mathtt{x}^{*}\\alpha$, so that $\\beta$ is a contact form on $\\mathbb{R}^{2n}\\times S^{1}$. We wish to show that $\\beta=\\alpha_{0}$. The Reeb vector field $X$ of $\\beta$ is given by $$X=\\frac{1}{K_{\\tau}-\\lambda(X_{K_{\\tau}})}\\left(\\frac{\\partial}{\\partial\\tau}-X_{K_{\\tau}}\\right),$$ as is easily checked, where $X_{K_{\\tau}}$ is the symplectic gradient of $K_{\\tau}$ with respect to $\\sum_{j}\\tfrac{1}{2\\pi}sds\\wedge d\\phi_{j}$. Thus denoting by $$F_{\\tau}(s,\\phi):=\\frac{1}{K_{\\tau}(s,\\phi)-\\lambda_{(s,\\phi)}(X_{K_{\\tau}}(s,\\phi))},$$ the Reeb flow $\\theta_{\\beta}^{t}$ of $\\beta$ is given by $$\\theta_{\\beta}^{t}(s,\\phi,\\tau)=\\left(\\varphi_{K}^{\\int_{0}^{t}F_{\\sigma}(s,\\phi)d\\sigma}(s,\\phi),\\tau+\\int_{0}^{t}F_{\\sigma}(s,\\phi)d\\sigma\\right),\\label{eq:reeb flow of beta}$$ where $\\varphi_{K}^{t}$ is the flow of $X_{K_{t}}$. In particular, if $\\theta_{\\beta}^{1}(s,\\phi,\\tau)=(s,\\phi,\\tau)$ for every point $(s,\\phi,\\tau)$ then examining the second coordinate we see that one must have $$\\int_{0}^{1}F_{\\sigma}(s,\\phi)d\\sigma\\in\\mathbb{Z}\\ \\ \\ \\mbox{for all }(s,\\phi)\\in\\mathbb{R}^{2n}.$$ Since the function $(s,\\phi)\\mapsto\\int_{0}^{1}F_{\\sigma}(s,\\phi)d\\sigma$ is continuous and for $(s,\\phi)=(0,0)$ one has $F_{\\sigma}(0)=1$ for all $\\sigma$, it follows that we must have $\\int_{0}^{1}F_{\\sigma}(s,\\phi)d\\sigma=1$ for all $(s,\\phi)$. Now examining the first coordinate we see that $\\varphi_{K}^{1}(s,\\phi)=(s,\\phi)$ for all $(s,\\phi)$, and hence $K_{\\tau}(s,\\phi)=1$ for every $(s,\\phi,\\tau)$.\n\nObserve that Theorem \\[prop:compactly supported reeb\\] is an immediate consequence of Corollary \\[cor:nice coordinates\\] and Proposition \\[prop:compactly supported Reeb flow on R2n\\], where under Assumption **(B)** we require $\\rho_{0}<1=\\wp_{W}(\\Sigma,\\alpha)$ so as to ensure that Lemma \\[lem:rescaling lemma\\] holds. ", "It thus remains to show Proposition \\[prop:compactly supported Reeb flow on R2n\\].", "\n\nThe Reeb vector field $R$ of $\\alpha_{0}$ is just $\\frac{\\partial}{\\partial\\tau}$, and the Reeb flow $\\theta^{t}$ is given by $$\\theta^{t}(s,\\phi,\\tau)=(s,\\phi,\\underset{\\textrm{mod 1}}{\\underbrace{\\tau+t}}).$$ Fix $\\rho\\in\\mathbb{R}$ such that $0<\\left|\\rho\\right|<\\pi r^{2}$. Let $f:[0,\\infty)\\times[0,\\infty)\\rightarrow\\mathbb{R}$ denote a smooth function with the following properties:\n\n1. ", " There exists $\\varepsilon>0$ such that $f(s)=\\rho$ for $0\\leq s\\leq\\varepsilon$ and $f(s)=0$ for $r-\\varepsilon\\leq s\\leq r$.\n\n2. ", " If $\\rho<0$ then $f'(s)\\geq0$ for all $s$. If $\\rho>0$ then $f'(s)\\leq0$ for all $s$.\n\n3. ", " If $\\rho<0$ then $2\\pi s-f'(s)>0$ for all $s>0$. If $\\rho<0$ then $2\\pi s+f'(s)<0$ for all $s>0$.\n\nNote that such a function only exists because $\\left|\\rho\\right|<\\pi r^{2}$. Indeed, if $\\rho<0$ then since $2\\pi s-f'(s)>0$ one has $$-\\rho=\\int_{0}^{r}f'(s)ds<\\int_{0}^{r}2\\pi sds=\\pi r^{2}.\\label{eq:its empty}$$ Conversely it is easy to see that when $\\left|\\rho\\right|<\\pi r^{2}$ such functions really do exist. ", "Now consider the contactomorphism $\\vartheta^{\\rho}$ of $\\mathbb{R}^{2n}\\times S^{1}$ whose contact Hamiltonian $h_{t}:\\mathbb{R}^{2n}\\times S^{1}$ is given by $$h_{t}(s,\\zeta,\\tau)=f(r).$$ The contact vector field $X_{t}$ of $h_{t}$ is defined by the equations $$\\alpha(X_{t})=h_{t},\\ \\ \\ i_{X_{t}}d\\alpha=dh_{t}(R)\\alpha-dh_{t}.$$ This gives $$X_{t}(s,\\phi,\\tau)=\\sum_{j}\\frac{f'(s)}{s}\\frac{\\partial}{\\partial\\phi_{j}}+\\left(f(s)-\\frac{sf'(s)}{2}\\right)\\frac{\\partial}{\\partial\\tau}.$$ We can integrate this to obtain $$\\vartheta_{t}^{\\rho}(s,\\phi,\\tau)=\\left(s,\\phi_{1}+\\frac{f'(s)}{s}t,\\dots,\\phi_{2n-1}+\\frac{f'(s)}{2}t,\\tau+\\left(f(s)-\\frac{sf'(s)}{2}\\right)t\\right),$$ and hence translated points of $\\vartheta_{1}^{\\rho}$ are tuples $(s,\\phi,\\tau)$ with $$\\frac{f'(s)}{s}\\in2\\pi\\mathbb{Z},\\label{eq:tp1}$$ and the time-shift is given by $$\\eta=f(s)-\\frac{sf'(s)}{2}.\\label{eq:tp2}$$ By assumption one never has $f'(s)/2\\pi s\\in\\mathbb{Z}$ unless $f'(s)=0$. In other words, translated points only occur when $0\\leq s\\leq\\varepsilon$ or when $ $$r-\\varepsilon\\leq s\\leq\\infty$. In particular, the only translated points of $\\vartheta^{\\rho}$ that lie in the interior of the support of $\\vartheta^{\\rho}$ are the points in $B(\\varepsilon)\\times S^{1}$. Since $\\vartheta^{\\rho}=\\theta^{\\rho}$ on $B(\\varepsilon)\\times S^{1}$, this justifies our claim that ‘from the point of view of translated points’, $\\vartheta^{\\rho}$ is the Reeb flow.", "\n\nTo complete the proof let us compute the spectral value of $\\vartheta^{\\rho}$. Note that the *contractible* action spectrum of $\\mathcal{A}_{\\vartheta^{\\rho}}$ is just $\\{0,-\\rho\\}$, and hence we certainly have $c(\\vartheta^{\\rho})\\in\\{0,-\\rho\\}$. For $\\rho<0$, one has $h_{t}<0$ on the interior of its support and hence by Corollary \\[cor:posi but not negi\\] one has $c(\\vartheta^{\\rho})>0$, which implies $c(\\vartheta^{\\rho})=-\\rho$. Next, Lemma \\[lem:always positive\\] implies that for any $\\rho,$ $c(\\vartheta^{\\rho})\\geq0$ since one can find a contactomorphism $\\psi\\in\\mbox{Cont}_{0,c}(\\mathbb{R}^{2n}\\times S^{1},\\alpha_{0})$ such that $\\psi(B(r)\\times S^{1})\\cap\\left(B(r)\\times S^{1}\\right)=\\emptyset$ (take $\\psi$ to be the lift of a translation of $\\mathbb{R}^{2n}$ which is suitably truncated). ", "Thus for $\\rho>0$ we must have $c(\\vartheta^{\\rho})=0$. This completes the proof.", "\n\nFix $r=1$ for convenience. ", "Let us denote by $\\mathcal{F}_{\\rho}$ the set of functions satisfying (1), (2) and (3) from the proof of Proposition \\[prop:compactly supported Reeb flow on R2n\\]. ", "Denote also by $\\widetilde{\\mathcal{F}}_{\\rho}$ the larger subset of functions that satisfy (1) and (2) but instead of (3) we impose the weaker criterion that $$4\\pi s-f'(s)>0\\ \\ \\ \\mbox{for all }s\\in[0,1].$$ It follows from (\\[eq:its empty\\]) that $\\mathcal{F}_{\\rho}$ is empty if $\\left|\\rho\\right|>\\pi$ and $\\widetilde{\\mathcal{F}}_{\\rho}$ is empty if $\\left|\\rho\\right|>2\\pi$.\n\nSuppose $f\\in\\widetilde{\\mathcal{F}}_{\\rho}$ for some $-2\\pi<\\rho<\\pi$. Then the (non-trivial) translated points of the corresponding contactomorphism $\\varphi_{f}$ come in two flavours: firstly there are the points in $B(\\varepsilon)\\times S^{1}$ as before, but there are also now the points $(s,\\zeta,\\tau)$ where $2\\pi s=f'(s)$. In this case the time-shift is $f(s)-\\frac{sf'(s)}{2}$.\n\nNow suppose we are given a path $$[0,3\\pi/2]\\ni\\rho\\mapsto f_{\\rho}\\in\\widetilde{\\mathcal{F}}_{-\\rho},$$ with the property that there exists $0<\\rho_{0}<\\pi$ such that $f_{\\rho}\\in\\mathcal{F}_{-\\rho}$ for $0\\leq\\rho\\leq\\rho_{0}$. We know that $c(\\varphi_{f_{\\rho}})=\\rho$ for $0\\leq\\rho\\leq\\rho_{0}$, but for $\\rho\\geq\\pi$ this is no longer the case as $c(B(1)\\times S^{1})=\\pi$. We also know that the function $\\rho\\mapsto c(\\varphi_{f_{\\rho}})$ is continuous. ", "From (\\[eq:tp1\\]) and (\\[eq:tp2\\]) this implies that there must exist some point $\\rho_{1}$ such that $\\rho_{0}\\leq\\rho_{1}<\\pi$ and a point $s\\in[0,1]$ such that $f_{\\rho_{1}}'(s)=2\\pi s$ and such that $f_{\\rho_{1}}(s)-\\frac{1}{2}s^{2}=-\\rho_{1}$. Here is a direct proof of that assertion.", "\n\nConsider the function $l:[\\rho_{0},3\\pi/2]\\rightarrow[0,1]$ defined by $$l(\\rho):=\\sup\\left\\{ s\\in[0,1]\\mid f_{\\rho}'(s)\\geq2\\pi s\\right\\} .$$ Note that the function $l$ is not necessarily continuous (it may have jump discontinuities). ", "Define $$g(\\rho):=f_{\\rho}(l(\\rho))-\\pi l(\\rho)^{2}+\\rho.$$ The function $g$ is also not necessarily continuous. ", "However $g$ is lower semicontinuous in the sense that if $\\rho_{*}$ is a point of discontinuity of $g$ (and hence of $l$) then $$\\lim_{\\rho\\downarrow\\rho_{*}}g(\\rho)\\leq\\lim_{\\rho\\uparrow\\rho_{*}}g(\\rho).$$ Indeed, the point is that if $l^{+}(\\rho_{*}):=\\lim_{\\rho\\downarrow\\rho_{*}}l(\\rho)$ and $l^{-}(\\rho_{*}):=\\lim_{\\rho\\uparrow\\rho_{*}}l(\\rho)$ then $f_{\\rho}'(s)<2\\pi s$ for $s\\in(l^{-}(\\rho_{*}),l^{+}(\\rho_{*}))$, and hence $$\\int_{l^{-}(\\rho_{*})}^{l^{+}(\\rho_{*})}f_{\\rho}'(s)ds\\leq\\int_{l^{-}(\\rho_{*})}^{l^{+}(\\rho_{*})}2\\pi sds$$ which implies $$\\lim_{\\rho\\downarrow\\rho_{*}}g(\\rho)\\leq\\lim_{\\rho\\uparrow\\rho_{*}}g(\\rho).$$ Next note that $$\\rho=\\int_{0}^{1}f_{\\rho}'(s)ds\\leq\\int_{0}^{l(\\rho)}f_{\\rho}'(s)ds+\\int_{l(\\rho)}^{1}2\\pi sds=f_{\\rho}(l(\\rho))+\\rho+\\pi-\\pi l(\\rho)^{2},$$ and thus for $\\rho>\\pi$ one has $g(\\rho)>0$. For $\\rho=\\rho_{0}$ one has the strict inequality $$\\rho_{0}=\\int_{0}^{1}f_{\\rho_{0}}'(s)ds<\\int_{0}^{l(\\rho_{0})}2\\pi sds+\\int_{l(\\rho_{0})}^{1}f_{\\rho_{0}}'(s)ds=\\pi l(\\rho_{0})^{2}-f_{\\rho_{0}}(l(\\rho_{0})),$$ and hence $g(\\rho_{0})<0$. Thus there exists $\\rho_{1}\\in(\\rho_{0},\\pi)$ such that $g(\\rho_{1})=0$.\n\n[^1]: Although note if $h_{t}>0$ then the assertion that $\\varphi$ is of infinite order follows directly from orderability.", "\n\n[^2]: Note that whilst $C(r_{1}):=B^{2}(r_{1})\\times\\mathbb{R}^{2n-2}$ does not have compact closure in $\\mathbb{R}^{2n}$, and thus $\\overline{c}(C(r_{1})\\times S^{1})$ is not defined, since we only work with compactly supported contactomorphisms we can deduce this from the second statement of Theorem \\[thm:liouville flow\\] by taking $\\mathcal{O}=B(r_{2})$ and $\\mathcal{Q}$ a sufficiently large ellipse contained in $C(r_{1})$.\n\n[^3]: Note that this is not really a norm! ", "Any exact contactomorphism $\\varphi$ has $\\left\\Vert \\varphi\\right\\Vert =0$.\n" ]

{ "pile_set_name": "ArXiv" }

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[ "Delayed reduction of tissue water diffusion after myocardial ischemia.", "\nThe apparent diffusion coefficient (ADC) of water after regional myocardial ischemia was measured in isolated, perfused rabbit hearts by using magnetic resonance imaging (MRI) techniques. ", "After ligation of the left anterior descending coronary artery, the ADC of the nonperfused region showed a gradual but significant decreasing trend over time, whereas that of the normally perfused myocardium remained constant. ", "Morphological analysis revealed that the ADC decrease reflected the expansion of a subregion of reduced ADC within the nonperfused myocardium. ", "The dynamics of the diffusion change and the morphological progression of the affected tissue suggest that the ADC decrease may be linked to the onset of myocardial infarction, which is known to involve myocyte swelling. ", "The ADC reduction provides a potentially valuable MRI tissue-contrast mechanism for noninvasively determining the viability of the ischemic myocardium and assessing the dynamics of acute myocardial infarction." ]

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[ "A total audience of more than 240,000 Milwaukee households watched television coverage of last night's presidential and local election results.", "\n\nNBC affiliate WTMJ-TV (Channel 4) led all four local stations with a 9.9 average rating, followed by ABC affiliate WISN-TV (Channel 12) with a 7.7, Fox affiliate WITI-TV (Channel 6) with a 6.4 and CBS affiliate WDJT-TV (Channel 58) with a 2.7.", "\n\nAll but WITI showed network coverage between 7 and 10 p.m.\n\nAt 9 p.m., WITI aired its regular local newscast and saw its ratings jump more than two points, or about 20,000 households. ", "About 57,000 additional households tuned in for all four 10 p.m. local newscasts.", "\n\nWTMJ-TV also won at 10 p.m. with an 11.8 ratings average, followed by WITI with a 10.7, WISN-TV with a 9.2 and WDJT with a 4." ]

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[ "Stimulating effects of quercetin and phenamil on differentiation of human dental pulp cells.", "\nDentin formation is preferred in the healing response of the pulp to pulp-capping agents during vital pulp therapy. ", "Enhancement of the dentinogenic differentiation of dental pulp cells is thought to accelerate pulp repair. ", "The aim of this study was to evaluate the dentinogenic activity of small molecules (three flavonoids and phenamil) that have been shown previously to induce osteoblast differentiation. ", "Among the flavonoids (quercetin, genistein and baicalin), quercetin induced the highest alkaline phosphatase (ALP) activity of human dental pulp (HDP) cells. ", "Phenamil, an amiloride derivative, elicited higher ALP activity than quercetin. ", "However, increased expression of dentin sialophosphoprotein (DSPP) mRNA and mineral deposition were seen in cultures treated with quercetin compared with phenamil. ", "This would seem to suggest that quercetin is the most dentinogenic agent among the tested chemicals. ", "The increase in ALP activity in the quercetin-treated cells was not affected by ICI 182,780, an estrogen receptor inhibitor, and was partially blocked by PD98059, an extracellular signal-regulated kinase 1/2 (ERK1/2) inhibitor. ", "This suggests that ERK1/2 is activated in the quercetin-induced differentiation of HDP cells without the mediation of estrogen receptors, which are known to be involved in osteoblast differentiation induced by quercetin." ]

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[ "Media playback is unsupported on your device Media caption Trump's nominee for judge can't answer basic legal questions\n\nUS President Donald Trump's latest judiciary nominee has withdrawn his name after failing to answer basic legal questions at his Senate confirmation hearing.", "\n\nVideo of Matthew Petersen's embarrassing encounter went viral.", "\n\n\"It has become clear to me over the last few days that my nomination has become a distraction,\" Mr Petersen wrote in a letter.", "\n\nMr Petersen was nominated as a federal judge in the District of Columbia.", "\n\nMr Trump had accepted his offer to withdraw, the White House said.", "\n\nMr Peterson, a Republican member of the Federal Election Commission, is the latest of Mr Trump's conservative judicial nominations to fail.", "\n\nAt the hearing, Mr Peterson stumbled over questions asked by Republican Senator John Kennedy.", "\n\nThe exchange\n\nSenator Kennedy starts by asking Mr Petersen and the four other nominees who appeared with him: \"Have any of you not tried a case to verdict in a courtroom?\"", "\n\nOnly Mr Petersen raises his hand.", "\n\nSenator Kennedy: Have you tried a jury trial?", "\n\nMatthew Petersen: I have not.", "\n\nSenator Kennedy: Civil?", "\n\nMatthew Petersen: No.", "\n\nSenator Kennedy: Criminal?", "\n\nMatthew Petersen: No.", "\n\nSenator Kennedy: Bench?", "\n\nMatthew Petersen: No.", "\n\nSenator Kennedy: State or federal court?", "\n\nMatthew Petersen: I have not.", "\n\nSenator Kennedy: Have you ever taken a deposition by yourself?", "\n\nMatthew Petersen: Uh, I believe no.", "\n\nSenator Kennedy: Just for the record, do you know what a motion in limine is?", "\n\nMatthew Petersen: I would probably not be able to give you a good definition right here at the table.", "\n\nThe Louisiana Senator and former law professor, who still fully supports Mr Trump, openly criticised his nominee in a later TV interview.", "\n\n\"Just because you've seen My Cousin Vinny doesn't qualify you to be a federal judge,\" Mr Kennedy told local station WWL-TV, referring to the 1992 comedy film about a novice lawyer.", "\n\nMr Kennedy told the station that Mr Trump called him after learning of the exchange and agreed that Mr Peterson was too inexperienced.", "\n\nMr Peterson, who became the third of Mr Trump's judicial picks to flounder in just a week, said: \"I had hoped that my nearly two decades of public service might carry more weight than my two worst minutes on television.\"", "\n\nDespite recent setbacks, Mr Trump has made progress filling other judicial vacancies with conservative judges and notably restored the Supreme Court's conservative majority with the appointment of Justice Neil Gorsuch." ]

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[ "Privatanwender und kleine Unternehmen Unmanaged: Plug-and-Play-Lösung, mit der Sie Ihre gesamten Netzwerkgeräte ohne komplizierte Konfigurations- oder Managementsoftware verbinden können – und gleichzeitig ein zuverlässiges und sicheres Netzwerk schaffen. ", "Smart Managed Plus: Bietet wachsenden kleinen Unternehmen grundlegende Funktionen zur Konfiguration, Sicherung und Überwachung ihres Netzwerks, die über einfache Konnektivität hinausgehen, zu einem erschwinglichen Preis – ein ideales Upgrade von Unmanaged Switches.", "\n\nKleine und mittelständische Unternehmen Smart Managed Pro: Für größere Unternehmen, die eine erweiterte Konfiguration und eine verbesserte Kontrolle für die sichere Konfiguration und Verwaltung komplexer Unternehmensnetzwerke und einen geschützten Zugriff auf diese benötigen. ", "Unterstützt die Anforderungen von Edge-Networking und der automatisierten Verwaltung bezüglich der Priorisierung von Sprach-, Video- und anderem Datenverkehr. ", "Smart Cloud: Bietet vollständig integrierte Verwaltung, Transparenz und Kontrolle von jedem Gerät und Ort aus. ", "Erweiterte Konfiguration und Remote-Zugriff mit der NETGEAR Insight App und dem Cloud-Portal für beispiellose Flexibilität." ]

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[ "41 F.3d 665\nHernandezv.", "Shalala*\nNO. ", "94-50135United States Court of Appeals,Fifth Circuit.", "\nNov 25, 1994\nAppeal From: W.D.Tex., ", "No. ", "A-92-CA-512\n\n1\nAFFIRMED.", "\n\n" ]

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[ "There's Even Hope for Pro-McCain Oklahoma\n\nKirk Johnson: \"Not a single county in Oklahoma stirred from the orderly phalanx marching behind Mr. McCain, the senator from Arizona who was the Republican nominee....But that staunchly Republican, conservative Oklahoma is harder to find now.\"", "\n\nRepublican John McCain made his best electoral showing in Oklahoma, winning the state by 66%-34%. ", "Reporter Kirk Johnson, who finds right-wingers wherever he goes on his Midwest beat, can't seem to get his mind around the fact that some places didn't fall in line behind Barack Obama in 2008, and implicitly chides the \"staunchly Republican, conservative\" state for marching in an \"orderly phalanx\" behind McCain. ", "But for Johnson there are signs of hope even in that benighted place, as he documented in his Inaugration Day story, \"McCain Country Warms to Its New President.\"", "\n\n\"I voted for John McCain and still would,\" said Tim Driskill, in a flatly drawled declaration of certainty that still speaks for many in this place underwhelmed last November by the charms of Barack Obama, then the Democratic nominee for president.", "\n\nNot a single county in Oklahoma stirred from the orderly phalanx marching behind Mr. McCain, the senator from Arizona who was the Republican nominee, and Mr. Driskill, the owner of an insurance agency in downtown Tulsa, said he was proud to be in those ranks. ", "Statewide, two out of three voters supported Mr. McCain, the highest percentage in the nation.", "\n\nBut that staunchly Republican, conservative Oklahoma is harder to find now. ", "While there are countless Mr. Driskills here - and hardly anyone doubts that Mr. McCain would easily win again in a redo of the vote - there are also new fractures and fault lines as some voters have shifted toward accepting what the rest of the country wrought in giving Mr. Obama a lopsided victory.", "\n\nIn interviews in the week leading up to Mr. Obama's inauguration, many people here said a tolerant spirit toward his presidency has been hastened, paradoxically, by some of the same groups that voted mostly Republican in the election. ", "Those include active or former military personnel, and people who identify themselves as evangelical Christians, two groups with traditions of respecting hierarchical order and strong leadership.", "\n\n....\n\nBut some people have, in fact, changed their minds. ", "Leonard Nelson, 63, a 23-year veteran of both the Army and the Navy, said he had voted for Mr. McCain mainly through military fealty, believing that Mr. McCain's own military record would make him a better commander in chief.", "\n\n\"But I've come to think the better man won,\" said Mr. Nelson, owner of the Humidor Cigar Shop, an aromatic haven of pipes, blended tobaccos and customers on a first-name basis. ", "Mr. Nelson said that Mr. Obama, through his cabinet selections, sent a signal of centrist government intention that feels all right to him.", "\n\nMr. Nelson's customers like Cliff A. Stark, a lawyer and pipe smoker, were more representative of the spirit of pained resignation that is common here. \"", "It's just something you can't do anything about,\" Mr. Stark said.", "\n\nJohnson concluded by hinting at racism, while describing white-owned businesses who refused to ante up money for an inauguration party:\n\nPrincetta Rudd-Newman is living through that mix of hope and anxiety. ", "She exults one minute over Mr. Obama's election, she said, and frets the next over the future of the city she loves.", "\n\nHer family has a long history here - an uncle began one of Tulsa's oldest black-owned businesses, a funeral home, in 1917 - and Ms. Rudd-Newman has been trying this month to organize an inauguration party in the city's historically black north end. ", "But the money has not been coming in, especially at the $150-a-ticket Patriot level, pitched to local white-dominated corporations.", "\n\nMs. Rudd-Newman said she did not think it was about race. \"", "It's financial, in my perception,\" she said. \"", "It's hard times.\"", "\n\nFederal employees and military personnel can donate to the Media Research Center through the Combined Federal Campaign or CFC. ", "To donate to the MRC, use CFC #12489. ", "Visit the CFC website for more information about giving opportunities in your workplace." ]

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[ "NASCAR Hall of Fame\n\nThe NASCAR Hall of Fame, located in Charlotte, North Carolina, honors drivers who have shown exceptional skill at NASCAR driving, all-time great crew chiefs and owners, broadcasters and other major contributors to competition within the sanctioning body.", "\n\nHistory and construction\nNASCAR committed to building a Hall of Fame and on March 6, 2006, the City of Charlotte was selected as the location. ", "Ground was broken for the $160 million facility on January 26, 2007, and it officially opened on May 11, 2010, with the inaugural class inducted the day following the 2010 NASCAR Sprint All-Star Race. ", "The new Hall of Fame brings hundreds of jobs and an increase in tourism to Charlotte. ", "In addition to the Hall of Fame, the NASCAR Plaza, a 20-story office building, opened in May 2009. ", "The structure serves as the home of Hall of Fame-related offices, NASCAR Digital Media, NASCAR's licensing division, as well as NASCAR video game licensee Dusenberry Martin Racing (now known as 704Games). ", "Other tenants include the Charlotte Regional Partnership and Lauth Property Group. ", "Richard Petty and Dale Inman helped unveil the first artifact at the Hall of Fame — the Plymouth Belvedere that Petty drove to 27 wins in 1967.", "\n\nThe City of Charlotte was responsible for the construction of the building and is the owner of the NASCAR Hall of Fame. ", "However, it is operated by the Charlotte Regional Visitors Authority. ", "Winston Kelley is the NASCAR Hall of Fame executive director. ", "Internationally renowned Pei Cobb Freed & Partners led the design effort, and Leslie E. Robertson Associates were the structural engineers. ", "Little Diversified Architectural Consulting based in Charlotte is the local architectural firm overseeing many aspects of design and construction of the project. ", "LS3P Associates, Ltd. was the associate architect for the office tower. ", "Tobin Starr + Partners served as site architect, providing full-time representation for Pei Cobb Freed & Partners during construction. ", "Engineering and fabrication of the stainless steel möbius that wraps around the structure was completed by Zahner, of Kansas City. ", "Exhibition design is by Ralph Appelbaum Associates, and exhibition lighting by Technical Artistry. ", "Tobin Starr + Partners is architect-of-record for exhibit and auditorium spaces. ", "Jaros, Baum & Bolles (JB&B) was the mechanical, electrical and plumbing engineer. ", "Site excavation and grading services started on May 21, 2007. ", "The facility features a Hall of Fame and a 19-story office tower. ", "The NASCAR Hall of Fame is set on a 150,000 square feet surface. ", "In June 2008, NASCAR announced that the inauguration is planned in May 11, 2010. ", "In 2009, Nascar Hall of Fame established a partnership with Buffalo Wild Wings to be its exclusive restaurant partner in Charlotte.", "\n\nSite selection\n\nBecause of stock car racing's roots in and wealth of famous drivers from North Carolina, Charlotte was considered the favorite by many fans and commentators. ", "There are many NASCAR offices in the area and many teams in the three major NASCAR series (Cup, Xfinity and Truck Series) totaling over 73% of motorsports employees in the United States, in what the committee called \"NASCAR Valley.\" ", "The Hall of Fame is in Uptown Charlotte, about 25 minutes south of Charlotte Motor Speedway. ", "The bid was led by NASCAR car owner Rick Hendrick, then Mayor Pat McCrory, and business leaders in Charlotte. ", "Pei Cobb Freed & Partners were enlisted to design the complex, which is near the Charlotte Convention Center.", "\n\nHall of Fame building\nThe building contains the following:\n\nFirst Floor:\nBelk High Octane Theater – A screening room below ground level which shows videos to guests, including a primer video for first-time visitors.", "\nSecond Floor:\nCeremonial Plaza – An outdoor \"patio\" with a video screen.", "\nGlory Road – A 33-degree banked ramp (matching that of Talladega Superspeedway) featuring 18 different cars and saluting 46 past and current tracks.", "\nThe Great Hall – Dubbed as the Times Square of the hall, a -by- video screen and rotating exhibits will be staged here.", "\n\"Studio 43\" – Named in honor of Richard Petty's car number – used for television production.", "\nThird Floor:\nHall of Honor – A 360-degree wall with the honorees enshrined serves as the centerpiece of the building with each enshrinee with their own exhibit.", "\nTransporter and Racecar Simulators – Simulators provided by iRacing.com.", "\nRace Week Experience – Simulates an actual week in a NASCAR team, from race prep through inspection, practice, time trials and the race.", "\nFourth Floor:\nHeritage Speedway – The six decade history of NASCAR is focused here, including a glass-enclosed section with historic artifacts from the history of stock car racing.", "\n\nThere is a gift shop, the Hall of Fame Café and a Buffalo Wild Wings restaurant on site. ", "An expansion, which includes a new ballroom, is part of the project.", "\n\nWhile most information on the Charlotte bid has been released voluntarily, the Charlotte Observer has asked the state Attorney General for an opinion requiring full disclosure of the financial details.", "\n\nThe self-proclaimed slogan used by Charlotte for the Hall of Fame was \"Racing Was Built Here. ", "Racing Belongs Here.\"", "\n\nOther final candidates\nThe other two cities at the time of the announcement that were in the running were the cities of Atlanta and Daytona Beach.", "\n\nOther bids\nThe state of Alabama had been mentioned as a potential candidate location, and was no longer seen as a contender, possibly because Lincoln, Alabama is home to the International Motorsports Hall of Fame, which is not affiliated with NASCAR. ", "The only northern area that considered bidding was in the state of Michigan. ", "Detroit prepared bids, but state officials decided not to submit the proposals. ", "The cities of Richmond, Virginia and Kansas City, Kansas, were among the five finalists, but on January 5, 2006, NASCAR announced they had been eliminated from the running, leaving just Atlanta, Charlotte and Daytona Beach as the remaining cities.", "\n\nEligibility and selection process\n\nEligibility\nFormer drivers must have been active in NASCAR for at least 10 years and retired for at least three. ", "Starting with the 2015 Hall of Fame nominations that were voted in the 2014 nomination process, the three-year rule is waived for drivers who compete in 30 or more years in NASCAR-sanctioned competition or turn 55 years of age. ", "The rule applies to all NASCAR-sanctioned competitions; some drivers in the Hall of Fame did not participate in the Cup Series.", "\n\nNon-drivers must have been involved in the industry at least ten years. ", "Some candidates with shorter careers will be considered if there were special circumstances.", "\n\nSelection process\n\nNomination\nA 20-member nominating committee chooses nominees from those who are eligible. ", "The committee consists of:\nSeven NASCAR representatives;\nNASCAR Hall of Fame Executive Director Winston Kelley;\nNASCAR Hall of Fame Historian;\nTrack owners (Two each from International Speedway Corporation and Speedway Motorsports Inc.), a representative of Hulman & Company (Indianapolis Motor Speedway), a representative of Mattco Inc. (Pocono Raceway) and a representative of Dover Motorsports Inc. (Dover International Speedway)\nFour track owners from historic short tracks: Bowman Gray Stadium, Rockford Speedway, the Holland Motorsports Complex, and West Coast track operator Ken Clapp.", "\n\nInduction\nAfter the nomination committee selects the list of candidates, a total of 48 votes are cast by a voting committee consisting of the nominating committee and the following:\n14 media representatives: Three each from the National Motorsports Press Association, the Associated Press Sports Editors and the Eastern Motorsports Press Association; one each from current media rights holders Fox Broadcasting Company, Comcast Corporation, Motor Racing Network, Performance Racing Network and Sirius XM NASCAR Radio.", "\nOne representative each from the current manufacturers – Chevrolet, Ford and Toyota;\nThree retired drivers;\nThree retired owners;\nThree retired crew chiefs;\nThe reigning NASCAR Cup Series champion;\nOne ballot which represents the results of a fan vote on NASCAR.com.", "\n\nInductees\n\nA total of 50 individuals have been inducted into the NASCAR Hall of Fame. ", "35 were inducted as drivers, 24 of whom were inducted solely as drivers. ", "The other 15 were inducted for their accomplishments as drivers, owners and/or broadcasters. ", "Among non-drivers, seven were inducted for being owners, four as promoters, and three for being crew chiefs.", "\n\nReferences\n\nExternal links\n\nNASCAR Hall of Fame official website\nCollection of links to articles for all prospective locations\n USA Today article from May 25\nCharlotte basks, boasts at Hall groundbreaking\n\nHall\nCategory:Auto racing museums and halls of fame\nCategory:Sports in Charlotte, North Carolina\nCategory:Museums in Charlotte, North Carolina\nCategory:Sports museums in North Carolina\nCategory:Halls of fame in North Carolina\nCategory:Awards established in 2010\nCategory:Buildings and structures completed in 2010\nCategory:2010 establishments in North Carolina\nCategory:Skyscrapers in Charlotte, North Carolina" ]

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[ "When it comes to feeding the world, most of us support the idea. ", "We are taught from a young age that if someone is hungry it’s our moral duty to feed them, whether they live down the street or in another country. ", "For decades, agriculture companies have used the noble goal of “feeding the world” to increase yields by any means possible, from genetic modification to the use of chemical fertilizers and pesticides. ", "This logic has justified ecological destruction from prairies to rainforests. ", "It has wreaked havoc on indigenous and small-farming communities. ", "And with 870 million chronically undernourished people on earth right now, it has failed to get food to the people who need it most.", "\n\nInstead of a fed planet, we have monoculture farms, poisons on food, and toxic runoff in our land and water. ", "Into our air, the global agriculture industry emits about 14 percent of total greenhouse gases, according to the Consultative Group on International Agricultural Research (CGIAR). ", "If we include agricultural deforestation, that number jumps to 27.5 percent. “[", "I]t’s impossible,” writes CGIAR, “to address climate issues without including agriculture—and vice versa.”", "\n\nFortunately, real solutions aren’t difficult to imagine. ", "Raj Patel interviewed one Wisconsin farmer, Jim Goodman, who seems to have a lot of this figured out.", "\n\nBrain Merchant writes at Motherboard: What’s the number one reason we riot? ", "The plausible, justifiable motivations of trampled-upon humanfolk to fight back are many—poverty, oppression, disenfranchisement, etc—but the big one is more…\n\nA senior economist at the worldwide bank HSBC has warned of civil unrest in Britain if food prices continue to soar.", "\n\nSpeaking on Jeff Randall Live, senior global economist Karen Ward cautioned that the UK could experience the kind of food riots seen in other countries.", "\n\n“Even in the developed world I think we have very, very low wage growth, so people aren’t getting more in their pay packet to compensate them for food and energy, and I think we could see social unrest certainly in parts of the developed world and the UK as well.”…" ]

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[ "In 2016 the United States gambled its 70-year supremacy in the Asia-Pacific region and—with Donald Trump’s victory—lost. ", "The risks were already growing. ", "Barack Obama argued for American leadership, but never got Congress behind its economic dimension, the Trans-Pacific Partnership. ", "On the security front, American reticence helped turn the South China Sea into a Chinese lake. ", "China won over the chippy new Philippines president with flattery and an export market for bananas. ", "The North Korean nuclear threat engendered no new ideas (as ever); Japan was left hanging; and Mr Trump sold out Taiwan by suggesting that it might be a bargaining chip, wholly misunderstanding the mainland authorities’ sentiments towards the island democracy. ", "The risks of American-Chinese confrontation have grown. ", "Growing faster are the odds that a China-centred hierarchy will replace a largely rules-based American order. ", "Few neighbours want to submit to such new arrangements—or to be forced to make the choice." ]

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[ "Click here: Latest News Article - FreeRealTime.com \nJeff, \nKnowing that ENE is making market in the paper industry I thought it would be \nof interest to you to possibly work a deal with DOMTAR (DTC) who just \nannounced earnings were cut in half due to energy costs. ", " Sounds like an \nopportunity there. ", "\nRegards, \nCharlie Farrell \n281 395 5908" ]

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[ "Psychological well-being in advanced NSCLC patients in Serbia: impact of sociodemographic and clinical factors.", "\nPatients with advanced non-small cell lung cancer (NSCLC) usually undergo toxic treatment (chemotherapy and/or radiotherapy). ", "They can experience devastating effects of illness and therapies on their psychological and emotional well-being. ", "On the other hand, untreated psychological distress is associated with reduced quality of life and inadequate palliation of physical symptoms.<br />In order to estimate frequency of anxiety and depressive symptoms and influence of demographic, socioeconomic and clinical factors on psychological well-being, we performed this cross-sectional study in group of 100 patients with advanced stage of disease. ", "Symptoms of anxiety and depression were assessed using the Hamilton Anxiety Rating Scale (HARS) and Hamilton Depression Rating scale (HDRS). ", "Health-related quality of life data are obtained by EORTC QLC C30 and SF 36.<br />Patients with poor performance status (PS) experienced significantly more anxiety and depressive symptoms (p=0.001) and worse emotional (p=0.001) and mental functioning (p=0.001). ", "Treated patients had significantly better mental (p=0.011) and emotional (p=0.001) functioning in compared to newly diagnosed ones. ", "Somewhat unusual, unemployed participants reported significantly less anxiety (p=0.029) and depressive (p=0.002) symptoms, better mental (p=0.030) and emotional functioning (p=0.007). ", "Additionally, nausea and vomiting adversely affected mental health and emotional functioning and correlated significantly positively with HARS and HDRS scores.<br />Our findings suggest significant impact of some disease-related factors (PS, active treatment) and treatment-related factors (chemotherapy -induced nausea and vomiting) on psychological well-being of patients with advanced NSCLC. ", "This should be taking an account when appropriate interventions are planned. ", "lung cancer, anxiety, depression, quality of life, chemotherapy, chemotherapy-induced nausea and vomiting." ]

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[ "Eva and Sagar Hindu Wedding @ Brampton\n\nEva and Sagar’s Summer Wedding was on a rainy summer day. ", "They had a 2hr hindu ceremony in their backyard – tastefully setup in the comfort of their own home. ", "Luckily only the last few minutes had to be moved to a tent due to rain. ", "We then moved on to Chincagousy park in Brampton for their Bridal party photos and finally ended the day with their reception held at Bombay Palace in Brampton where both Eva and Sagar were dressed up in a stunning blue!", "\n\nI got the opportunity to second shoot the entire wedding alongside Photography by Shivani. ", "Amazing vendor to work with! ", "Do check out her website for more of their work." ]

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[ "Finding Voice: Telling Our Own Stories\n\nI seem to have been spending quite a bit of\ntime lately urging other activist friends to write more.", "\nMany already are. ", "And I regularly ask myself why I write.", "\n\nWe need more people who write for social movements in\nstruggle, are actively engaged in organising in those\nmovements, and who take the same political risks as others.", "\nWe also need more quality research explicitly directed by\nand accountable to people in struggle.", "\n\nWe should not\nunderestimate the role of research and writing in our\ncampaigns. ", "But these resources must meet the needs of\ncommunities, activists and organisers daily engaged in\nspecific struggles. ", "We should not confuse the production of\nyet more information with education for empowerment and\nliberation.", "\n\nFinding space and time to reflect upon and\nevaluate our tactics, strategies, processes, goals and\nvisions, our successes and failures, victories and setbacks\ncan be hard in day-to-day organising that is often\nall-consuming. ", "But it is so vital.", "\n\nWho else can better\narticulate the complexities, contradictions and challenges\nthat lie within our organisations and movements which\narguably need as much attention as the institutions and\nforces which we mobilise against? ", "Who better to articulate\nthe alternatives to the neoliberal agenda which are already\nbeing built. ", "And who better to make the links and\nconnections between global and local struggles and\ninjustices?", "\n\nMy motivation to start writing more came\nequally from frustrations with much “alternative” media and\nNGO literature - and seeing whose voices and which views are\nheard and valued and which silenced - as from the slop\nserved by corporate media and official propagandists. ", "Often\nwhat was out there did not reflect the realities of the work\nI was doing or what other people were telling me about their\nstruggles. ", "I started writing as one other way to contribute\nto movements for change.", "\n\nIn an earlier ZNet commentary\n(All this Civil Society talk takes us nowhere, 13/01/02) I\nwrote about the dangerous separation of the “brain and\nbrawn” of the “anti-globalisation movement”.", "\n\nUnfortunately, within many movements and networks\ncommitted to social change, it has often been implicitly\n(and rather arrogantly) assumed that people organising\nmilitant actions have little or no analysis, theoretical\nunderstanding or worthwhile things to say, unlike more\n“respectable” NGO policy people, lobbyists, and\ncommentators. ", "Such elitism and the cults of personality and\npower which develop around “experts” have no place in\nmovements for social justice.", "\n\nMoreover, many NGOs which\nhave the material resources and organisational\ninfrastructure to regularly produce research documents and\nliterature studiously avoid confronting fundamental issues\nlike imperialism, capitalism or colonisation in any\nsubstantive way. ", "Perhaps that is hardly surprising – there\nis little funding available for organisations which take\nexplicitly anti-colonial, anti-capitalist and\nanti-imperialist standpoints by comparison with those\nadvancing more reformist – or at least, ambiguous\n-positions.", "\n\nIn their tiresome territorial turf wars to\nbrand themselves as the authoritative ‘alternative’ voice,\nthey can be as condescending and patronising to the rest of\nus as those institutions and actors which they are pitted\nagainst. ", "Consciously or inadvertently, they often reproduce\nor reinforce the same imperial, patriarchal, racist and\nclass relations which underpin the world we live in. ", "We\ntalk, you listen. ", "You organise, and we’ll write about it.", "\n\nCul-de-sacs of convenience and comfort are frequently\nconstructed which exclude any focus on the issues of\nimperialism and colonialism, often couched in the language\nof “reality” and pragmatism. ", "They invite people to question\nthings up to a certain point, but fail to grapple with, and\noften obscure, the root causes of injustice.", "\n\nIn his\ncritique of international NGOs (in Globalisation Unmasked:\nImperialism in the 21st Century), James Petras writes that\nthey frequently act as “intellectual policemen who define\n“acceptable” research, distribute research funds and filter\nout topics and perspectives that project a class analysis\nand struggle perspective.” ", "When their essentially reformist\nagendas threaten to set the parameters of accepted debate in\nthe “alternative” world about a particular issue, how do we\nengage and confront them?", "\n\nHow do we challenge the subtle\nand not-so-subtle ways in which many more powerful NGOs seek\nto dominate discourse on issues like globalisation? ", "How do\nwe move the analysis forward? ", "What responsibility and what\ntools do we have, drawn from our popular education and\nmobilization work, to challenge this situation?", "\n\nWhat\noften strikes me is that ideas, concerns and arguments which\nseem basic and hardly original, are not being articulated\nand shared outside of a small circle of friends and\ncomrades. ", "Putting these into words can be a powerful way to\nmove discussion forward and build bridges with others with\nsimilar concerns.", "\n\nI have been amazed at how well ZNet\narticles like the one I wrote on civil society, and last\nAugust’s “Bringing It All Back Home: Anti Globalisation\nActivism cannot ignore Colonial Realities” have been\nreceived, even though it seems that quite a few of us have\nbeen saying these kinds of things for many years.", "\n\nThe\nwork of identifying linkages, articulating and building a\nlarger vision linking local and global issues, for example,\nbuilding campaigns which expose the connections between\nneoliberalism, racist immigration policies, colonial laws\nand culture which continue to subjugate Indigenous Peoples,\nis an important project. ", "But by comparison with the analyses\nof corporate power and the demise of “democracy”, few are\nwriting about these issues. ", "The organising in our\ncommunities needs to go hand in hand with a conscious focus\non framing the debate and the issues at a more conceptual\nlevel.", "\n\nWe cannot deal with every issue at once in our\ncampaigns and mobilizations – but we can link our actions\nand campaign work to an analysis based on broader\nunderstandings. ", "There is a mutually-reinforcing relationship\nbetween mobilising on local concerns and contextualising\nwhat we do in light of broader national or global questions.", "\n\nWhy should it be left up to detached academics,\nprogressive social commentators, professional NGO policy\nanalysts and investigative journalists who occupy positions\nof relative privilege, to frame debates and analysis? ", "They\nmay well be fine people, with important insights. ", "And we\nneed them. ", "But is what they have to say and write inherently\nany more legitimate, credible and valid than people who put\nthemselves on the line on a daily basis? ", "Or people who have\nno choice in being on the frontlines of struggles?", "\n\nThe\nhigh value placed on the written word, and the exalted place\nwe give to those who write is highly problematic. ", "Not\neveryone is a writer, and the way in which “alternative”\norganisations, especially in the North, value and validate\nthose who write over those who do not is highly problematic.", "\nSome of the sharpest critical thinking and analyses come\nfrom people who have never written and some who cannot\nwrite. ", "Where and how are their voices and ideas valued? ", "We\nneed to examine and confront the tendency of alternative\nmovements to attribute higher values and status to those of\nus who write than those who don’t, won’t or can’t.", "\n\nAnother problem is the continued dominance of the English\nlanguage which often serves as a filter. ", "It largely\ndetermines whose voices we hear, articles and reports we\nread, and who from movements in the non-English-speaking\nmajority of the world gets exposed to debates beyond their\ncountry or region.", "\n\nWe are drowning in information,\noverwhelmed by the enormity of the injustices that we seek\nto overcome, and the urgent need to act immediately. ", "Having\nthe opportunity to step back, talk with comrades, assess\nwhere we are at, and share our experiences often gets put\nfarther down the priority list than organising the next\ndemonstration or public meeting, or getting the next\nleaflet, poster or flyer out.", "\n\nNonetheless, writing can be\nan essential tool in crystallizing arguments and concerns\nthat are not likely to be otherwise addressed and can play a\nrole as a catalyst for change.", "\n\nCombining organising,\nintense campaign and mobilization work with writing is not\neasy. ", "It is often a frustrating juggling act. ", "Nonetheless I\nthink it is very important that more of us try.", "\n\nInternational internet and digital technology commentator Cory Doctorow talked about machine learning and big data at the Privacy Commissioner’s PrivacyLive event on 13 March 2018 in Wellington. ", "More>>\n\nThe #MeToo campaign has been approached by Sweden in an impressively systematic fashion. ", "As Bloomberg News reports, #MeToo subsets have emerged in almost every imaginable sector of Swedish society.. More>>\n\nGun ownership in the US is a mystery to New Zealanders, and so is the constitutional fetish that surrounds it. ", "However, the attitudes involved are not static and unchanging, even if it can feel that way in the wake of each new gun atrocity. ", "More>>\n\nIn itself, the indictment of 13 Russian nationals and three Russian companies for interfering in the 2016 US wlll do little to change pre-existing views about the Robert Mueller investigation into Russia’s meddling in US presidential politics... More>>" ]

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[ "By Joy Press June 14, 2005\n\nThat aggressive, chain-smoking swagger and fuck-you rasp, the toxic provocations and limitless well of bile: Denis Leary's stand-up act rarely translated any of his blue-collar male rage into actual laughter as far as I was concerned. ", "But all the things that made me hate Leary as a comic have been reconfigured as effective components of Leary as a Method... More >>>" ]

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[ "Dave Skingley, Elder\n\nI’m married to Davina. ", "We have 3 children … Sarah-Beth, Gemma & Nathan. ", "We moved to Farnham in 1993 and now live in Badshot Lea. ", "I work as an IT Manager with British Car Auctions.", "\n\nAlthough I was brought up by Christian parents, I made a decision to follow Jesus for myself when I was 10 years old. ", "We joined this church in 2006, and I have been part of the leadership team since 2010. ", "My main passion in the church is men’s outreach. ", "Besides this, I lead a Community Group, do some preaching and service leading, play bass with the music group and support some of the youth work.", "\n\nI enjoy riding my road bike and running. ", "I don’t generally enjoy following Sunderland FC or playing golf, but I still do both of them." ]

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[ "Just about everyone has heard of Einstein's Theory of Relativity. ", "But how many know that he worked to develop hearing aids? ", "Or that a student spotted a mistake in one of his papers? ", "Or that Einstein's brain endured a bizarre odyssey after he died? ", "He was even offered the presidency of Israel—twice—and humbly declined. ", "Filled with such tidbits and anecdotes in 101 entries, this browsable book from a physicist and a technical writer takes a novel approach to the life and work of an iconic scientific and cultural figure." ]

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[ "Q:\n\nStabilizer of doubly transitive group action\n\nI am reading about doubly transitive group actions. ", "I am trying to prove that if $G$ acts doubly transitively on $X$, then $G_x$ is maximal for all $x\\in X$ ($G_x$ is the stabilizer of $x$). ", "\nA part of the argument goes like this: Let $G_x\\subsetneq H\\subsetneq G$ for some subgroup $H$ of $G$. Let $Y=\\lbrace hx \\text{ }\\mid\\text{ }h\\in H\\rbrace$. Then $Y\\neq X$.\nMy question is why $Y\\neq X$ ?", "\n\nA:\n\nPick $g\\in G\\setminus H$. Then $gx\\notin Y$. Indeed suppose $gx=hx$ for some $h\\in H$. Then $h^{-1}gx=x$, so $h^{-1}g\\in G_x\\subseteq H$, so $g=h(h^{-1}g)\\in H$, a contradiction.", "\n\n" ]

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[ "In the conventional Oldham coupling, projections formed in the end surfaces of shafts, and grooves formed in the intermediate disk are machined to a high accuracy, and the grooves of the intermediate disk are formed so that the bottom walls of the grooves are extremely rigid, to ensure an accurate transmission of a rotation of one shaft to the other.", "\nHowever, the projections of the shafts and the side surfaces of the grooves of the intermediate disk are abraded in an extended period of operation due to the sliding movement of the projections and the intermediate disk relative to each other, and consequently, the clearances between the projections and the grooves increase to cause a loss of motion, which deteriorates the accuracy of the rotation." ]

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[ "Pao Ham Phan\n\nPao Ham Phan () was the governor of Ratanakiri Province, Cambodia, until his retirement in May 2014.", "\n\nIn 1997, he was deputy governor. ", "He is a member of the central committee of the Cambodian People's Party, the ruling party of Cambodia.", "\n\nReferences\n\nCategory:Living people\nCategory:Cambodian People's Party politicians\nCategory:Year of birth missing (living people)" ]

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[ "Thursday, April 25, 2013\n\nOnion tomato thokku preparation is very instant and Easy recipe. ", "It goes very well with chapati, idli,dosa chapathi,paratha's and roti varieties.", "Onions are good sources of several vitamins, and in particular vitamin A, vitamin C and vitamin E.onion has folic content in it, which regulates some system in body. ", "Tomatoes are a rich source of potassium; Tomatoes work against impotency and help increase the sperm count. ", "Tomatoes are powerful blood purifiers and clear up urinary tract infections.", "Now we prepare simple onion tomato thokku." ]

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[ "TNA Impact Wrestling Results – January 30, 2014\n\nTNA Impact opened with Mike Tenay solemnly narrating a video on Sting’s contract being ripped up by Dixie Carter last week. ", "Tenay said Sting is gone from TNA.", "\n\nBackstage: The show started with agents amateurishly attempting to separate Samoa Joe & Kurt Angle from Magnus, Rockstar Spud, and Ethan Carter.", "\n\nTo the ring, Magnus led out ECIII and Spud as Mike Tenay and Jeremy Borash introduced the show. ", "JB recapped some business about Scotland not wanting to be part of the U.K. anymore, so the crowd does not recognize Englishman Magnus as their champion. ", "Once in the ring, Magnus insulted Scotland before claiming victory over Sting.", "\n\nMagnus went on to discuss the business of restructuring or removing dead wood, like the fans in the crowd losing their jobs. ", "Magnus referenced Dixie Carter to boos, then noted they have decided to rid TNA of the dinosaurs lumbering around the company. ", "He said they are making way for the new stars of the business. ", "Notably, himself. ", "And, singlehandedly.", "\n\nEthan Carter III interrupted, asking Magnus if he really thinks he did this by himself. ", "Magnus mocked him to deflect ECIII’s claims, then Kurt Angle’s music played. ", "Angle and Samoa Joe hit the ring to loud cheers as JB noted that Angle and Joe are not happy with how Dixie Carter has been running the company. ", "After the heels cleared out, Angle took the mic and said he can’t believe Sting is gone. ", "He called Magnus a paper champion, which Magnus warned him about saying.", "\n\nAngle said he can’t believe what Magnus turned into after they welcomed him into their Main Event Mafia family. ", "Magnus replied that what’s really killing Angle and Joe is that he is World champion, not them. ", "Joe said the only thing killing them is that Magnus is walking around breathing. ", "He said they came to Glasgow to kick his ass. ", "Magnus took that as a threat, then called for a tag match. ", "ECIII didn’t like the sound of that.", "\n\nMagnus raised the stakes that if they win the match, then Angle and Joe are gone from TNA for good. ", "Angle and Joe thought it over, then Joe raised the stakes – if either one of them scores a fall, then they become #1 contender to the TNA World Title. ", "Joe told Magnus to really think it over because – he understands – this might be too intense for a paper champion. ", "Magnus flipped out, then told Joe that he’s got the match.", "\n\nSuddenly, Dixie Carter walked out on-stage to boos as Joe’s music stopped playing. ", "Carter said this is not happening, then asked for the music to be cut. ", "Magnus and Carter argued about whether this tag match will be happening, then Dixie fessed up that she’s worried about Magnus losing the title to Kurt or Joe. ", "Magnus ignored her, then re-affirmed that they have a deal. ", "Tenay said the TNA champ has over-ruled the TNA president.", "\n\nBackstage: A vehicle pulled up backstage. ", "JB excitedly said they will find out the “new investor” later tonight.", "\n\n[Q2] [Commercial Break at 9:15]\n\nBackstage: A cameraman talked to himself about needing to get a close-up on the mini-van backstage. ", "Out came the Wolves (Eddie Edwards and Davey Richards), who told the cameraman to back off. ", "He said anyone who wants to find out the investor has to go through them.", "\n\n1 — KO champion MADISON RAYNE & VELVET SKY vs. GAIL KIM & LEI’D TAPA\n\nBefore the match started, TNA flashed back through the Beautiful People Era that included Rayne and Sky. ", "The heels then jumped the re-united KO tandem to begin the match. ", "But, the faces bought back and Sky dropped Gail with a Pedigree. ", "It was good for the win.", "\n\nWINNERS: Madison and Sky at 1:54.", "\n\nPost-match, Chris Sabin stormed down to the ring. ", "Sabin told “tramp” Madison to get out of the ring because he needs to talk to his girlfriend. ", "Sabin said he hasn’t heard from Sky for a week and she is still his girlfriend. ", "Sky sold dejection as Sabin talked down to her about it being her time of the month. ", "So, he’s willing to give her one chance to say she’s sorry, put it behind them, and they go backstage, where Sky will give him a massage. “", "One chance. ", "Say you’re sorry,” Sabin said. ", "Sky pretended to think it over, then quietly said she has one thing to say to him. “", "We’re Done!” ", "she shouted, then shoved the mic into Sabin’s chest and left the ring. ", "Sabin freaked out in the ring as Sky headed to the back.", "\n\nBackstage: Samoa Joe and Kurt Angle talked about needing to take out Magnus and ECIII tonight. ", "Angle then stormed into the locker room, where he found Bobby Roode and aggressively approached him. ", "But, the feed cut out before any physicality was documented.", "\n\n[Commercial Break at 9:25]\n\nBackstage: Angle was mid-beating on Roode, but Joe pulled him off. ", "Angle complained about Roode costing Sting and A.J. Styles their jobs, then Roode talked back to Angle, so Joe grabbed him around the throat and threatened to choke the life out of him. “", "I will end you, Bobby,” Joe shouted in Roode’s ear as he gasped for air. ", "Angle pulled off Joe, then the duo left. ", "Roode was then shown smirking after getting to his two rivals.", "\n\n[Q3] In-ring: James Storm’s music played to bring out Storm dressed to wrestle. ", "Once in the ring, Storm referenced Gunner to boos. ", "Storm said something during their partnership made things fall apart. ", "He then called down Gunner to put an end to this. ", "Loud crashing music played to bring out Gunner with possession of his World Title MITB briefcase.", "\n\nOnce in the ring, Gunner told a story about them coming together to do great things and pick up a couple of bar fights. ", "But, once he obtained this briefcase, things kind of blew up. ", "Actually, the problems were there before the briefcase, which brought their issues to the surface. ", "In any event, Gunner talked about his family, James’s family, and Gunner’s military experience. ", "Gunner said one thing he learned in the Marines was not fighting for the government, but for the man next to him.", "\n\nGunner continued with a story about some of his friends dying out in the battlefield, but they always fought side-by-side. ", "Storm said he knows Gunner was a great Marine and a great man, then called Gunner a great wrestler. ", "Storm then talked up his alcohol consumption before noting his kids are most important. ", "He said her daughter asked why he’s mad at Gunner, then he thought it over and said he had to look at himself. ", "Storm said he’s proud of Gunner and has his back. ", "He said he wants to be the bigger man, so he will shake his hand and say he’s sorry. “", "I got your back,” Storm said.", "\n\nProviding the interruption was Bad Influence’s music, bringing out Christopher Daniels and Kazarian. ", "Kaz asked Storm if he honestly believes anyone should believe what he says. ", "He then told Gunner that he’s all beard and no brains, but he shouldn’t trust what Storm says.", "\n\nDaniels spoke next that they should have a tag match tonight. ", "And, if Gunner thinks Storm is trustworthy, he should put the briefcase on the line so that when they win, one of them will get the TNA Title shot. ", "Gunner answered with a punch and a fight was on. ", "A ref showed up to ring the bell heading to break.", "\n\nBack from break, Storm and Gunner showed tag team cohesion by wearing down Kaz, who was still dressed in his pre-match t-shirt. ", "Kaz then tagged in Daniels, who helped Kaz take control of the match. ", "Gunner fought back on Kaz, then tried to tag in Storm, but Storm was occupied with Daniels. ", "So, the heels resumed control on Gunner.", "\n\n[Q4] Gunner got a second wind and climbed to the top rope, fought off Daniels, and splashed Kaz with a diving headbutt. ", "It was good for the pin and the win. ", "GunStorm then celebrated their victory in the ring.", "\n\nWINNERS: GunStorm at 8:01. ", "And, they’re back on the same page. ", "For now. ", "That was a lot of time invested in a storyline that is more than miss than hit.", "\n\nBackstage: Dixie Carter was freaking out. ", "About the limo, not the events in the ring. ", "Rockstar Spud tried to calm down Carter, who demanded to know who this new investor is now. ", "Spud left, then Impact cut to break with Dixie selling that she was on the verge of a nervous break down.", "\n\nBackstage: Dixie Carter was on the phone with her attorney. ", "He didn’t have any answers on the investors. ", "He’s right outside – why not ask him? ", "Bobby Roode then walked in and Dixie said she doesn’t have time for his issues. ", "Roode demanded a TNA Title shot at Lockdown after all he did for her helping Magnus remain TNA champ. ", "Dixie told him to calm down and appreciate that she has a full plate. ", "Dixie pulled out her pocketbook to give Roode money, but Roode said he doesn’t want any money. ", "He told her to figure it out by next week. ", "Until then, no more favors.", "\n\nBackstage: Eric Young was shown talking to his “wife,” ODB, who asked EY about his master plan with Abyss/Joseph Park. ", "ODB said she’s worried about him because Abyss is a monster for a reason. ", "ODB left, then Abyss walked in with a bag of thumbtacks in his right hand. ", "Abyss breathed heavily, then walked toward the ring.", "\n\nBackstage: Rockstar Spud found a camera and demanded the operator follow him. ", "Spud said he’s going to find the Wolves and find answers on the investor. ", "Spud then walked through a back door.", "\n\n[Commercial Break at 9:56]\n\n[Q5 -- second hour] The second hour started with a review of events earlier tonight when Joe & Angle vs. Magnus & ECIII was booked for the main event.", "\n\nIn-ring: Another round of Rockstar Spud, which prompted Tenay to blame Borash for Spud being in TNA since JB was an executive producer of “British Bootcamp.” ", "Once in the ring, Spud ran down Scotland again. ", "Spud then demanded to know who the new investor is.", "\n\nInstead, the Wolves came out on-stage dressed in all-black suits. ", "Davey Richards and Eddie Edwards marched down to the ring as Spud stood by annoyed. ", "Once the Wolves entered the ring, Spud knocked their “very 2005″ wardrobe, then demanded to know who the investor is. ", "Spud then pulled out a flashlight to investigate them like on one of those U.S. cop shows like “Magnum T.A.” No answers. ", "Spud removed his jacket, then noted who he is. ", "Spud heelishly claimed that he pushed Jeff Hardy off a 500 pound ladder and beat Samoa Joe within an inch of his life. “", "And, Mr. Carter told me that I’m fierce!”", "\n\nSpud slapped Davey, who no-sold, then both Wolves removed their jackets. ", "Spud demanded to know who the investor is one more time, but the Wolves kicked him in the gut before giving him a double-team move. ", "Edwards then told Spud that Dixie will find out who the investor is at the same time everyone else will later tonight. ", "He added that if anyone interferes in the main event later tonight, that person will be fired. ", "Eddie and Davey left the ring as Spud coughed up a lung in the ring.", "\n\nVideo Package: Eric Young and Joseph Park/Abyss. ", "EY was shown talking in a new piece of video that it’s okay that Park is Abyss and Abyss is Park. ", "But, the video showed Park not being so sure.", "\n\nIn-ring: Abyss’s music played and Christy Hemme introduced a Tag Title match up next. ", "Eric Young and Abyss came down to the ring and TNA accidentally put up an X Division Title match graphic on the screen. ", "Combined with audio issues in the opening segment, it’s been a challenging night for the production team.", "\n\nD.J. Zema Ion then paved the way for the Bro-Mans to come out to defend their titles. ", "Zema tried talking on the mic as Bro-Mans made their entrance, but he couldn’t be heard. ", "The Tag Title match is next.", "\n\nThe heels dominated EY early on as Tenay referenced a U.K. referee assigned to this match. ", "Abyss then tagged himself in, cleaned house, spotted D.J. Zema trying to get involved, and choked him. ", "The ref reprimanded Abyss, who then put the referee on his shoulders for Shock Treatment, but EY tried to calm him down. ", "Instead, Abyss hit Shock Treatment on the ref, presumably causing a DQ.", "\n\nPost-match, EY shook his head as JB noted their Tag Title aspirations are out the door. ", "Abyss then chokeslammed Young for good measure. ", "Abyss posed, then stomped around the ring to no music. ", "Abyss left the ring, then headed to the back as EY grabbed a mic. ", "Young said they only have one experiment left. ", "EY suggested they get crazy with a Monster’s Ball match next week. ", "From the stage, Abyss smiled, which counted as an acceptance.", "\n\nWINNER: No Decision around 6:00. ", "Will Abyss/Park/a distant cousin be attacked $10,000 for taking out a referee? ", "Also, the ship has sailed on TNA protecting wrestling matches, so the match ended without a result and no one batted an eye, just moved on to the next thing.", "\n\nBackstage: The roving cameraman was hounding Eddie Edwards about the investor. ", "Edwards told the guy to relax while Davey talked to him.", "\n\nBackstage: A coffin. ", "Bully Ray was shown slowly wheeling the coffin down the hallway like a zombie.", "\n\n[Commercial Break at 10:24]\n\nEarlier This Week: Christy Hemme and Sam Shaw. ", "They were drinking wine at Shaw’s place. ", "Hemme excitedly said he’s going to check out his place as Shaw walked behind her too close for comfort. ", "Shaw then took a chunk of Hemme’s hair and put it in his pocket. ", "TNA cut to a camera shot of another room, which was adorned with Hemme memorabilia. ", "The unsub kissed a photo of Hemme, who called out off-camera if he was going upstairs with her.", "\n\n[Q7] In-ring: Bully Ray wheeled a coffin down to ringside, then entered the ring. ", "Ray, his face hidden by a dark hoodie and dark sunglasses, took the mic to ask the crowd if they know who he is. ", "Ray introduced himself to boos, then noted he used to be president of Aces & Eights. ", "And, he used to be the World Hvt. ", "champion. ", "But, one man took that away from him. ", "All of it. ", "Ray said this man is loved and adored by the people – Ken Anderson.", "\n\nRay said he has tried to break him on multiple occasions, but he keeps coming back. ", "So, he’s challenging him to one final match. ", "And, the only way to win is not by pinfall or submission. ", "But, by placing your opponent in this coffin.", "\n\nMr. Anderson’s music interrupted to address the challenge. ", "Anderson walked out on-stage, then slowly walked down toward the coffin to meet Ray on the floor. ", "Anderson asked Ray if he really intends to put him in this coffin. ", "Then, what? ", "Ray said maybe he’ll piledrive Anderson’s wife and kids. ", "Anderson didn’t flinch, then asked Ray if he’s really going to do that. ", "Anderson leaned back, then aggressively told Ray that they’re all tired of Ray running his mouth. ", "He vowed to shut Ray’s mouth, then slammed the coffin lid into Ray’s face.", "\n\nOn the floor, Anderson grabbed a chair, slid it into the ring, and knocked Ray into the ring. ", "Anderson crotched Ray, then dropped him with the Green Bay Plunge. ", "Anderson then jabbed Ray with the chair, but Ray escaped the ring before he could take more punishment. ", "Ray recovered to the stage as Anderson stood tall in the ring.", "\n\nVideo: It’s “Terminator” Kurt Angle. ", "The video hyped Angle, similar to Joe, ahead of the main event.", "\n\nBackstage: Ethan Carter III walked into Magnus’s holding area to ask if he’s ready for the main event. ", "Magnus talked down to ECIII, calling him a green rookie. ", "ECIII said it sounds like he’s making excuses in-case they lose tonight. ", "Dixie Carter then walked in and her boys cheered up. ", "Magnus said he is excited to team with ECIII, then Dixie whispered to Carter not to screw up. ", "Dixie walked off with her prize pupil as ECIII mocked them from behind, selling frustration with Magnus taking his place of favor.", "\n\n[Commercial Break at 10:40]\n\n[Q8]\n\nIn-ring: Christy Hemme was standing by in the ring to introduce the main event. ", "Has anyone alerted Christy to the Sam Shaw footage? ", "Why didn’t the cameraman filming Shaw’s creepiness help out?", "\n\nAngle and Joe dominated early on as JB announced the 2015 European Tour with initial dates in Glasgow, Manchester, and London in January 2015. ", "The heels took control, then TNA cut to a split-screen of the Wolves and the investor’s feet in the parking area. ", "The trio left their holding place as events continued in the ring.", "\n\n[Commercial Break at 10:50]\n\nBack from break, TNA went to a variety of wide shots to try to keep the audience’s attention as the heels isolated Angle. ", "ECIII then missed a corner attack, allowing Angle to spring on Carter with a German Suplex. ", "Angle then reached his corner to tag in Joe, who cleaned house. ", "Joe lit up Magnus with a chop, then delivered a snap powerslam.", "\n\nECIII entered the ring, so Angle yanked him to the floor, where Angle delivered a belly-to-belly suplex. “", "What are you doing?” ", "Magnus shouted down to Carter. ", "Joe then grabbed Magnus and put on the rear-naked choke. ", "Magnus faded, then tapped out, making Joe the #1 contender. ", "JB said the investor made it two-on-two, creating an even playing field.", "\n\nWINNERS: Joe & Angle at 11:04.", "\n\nPost-match: Dixie Carter marched down to the ring shouting she told you so. ", "Dixie said she knew this was going to happen. ", "From the ring, Joe taunted Dixie and Magnus that he is the #1 contender to the TNA World Title. ", "Joe then told Magnus not to blame ECIII because he tapped out tonight. ", "And, soon, there will be no excuses. ", "He told Magnus to shine his belt because he’s going to kill him.", "\n\n[Q9 -- over-run] Angle said it’s night like this that make him proud to be in this business. ", "After thanking Glasgow, Angle noted he’s done it all in TNA, but the last two weeks have been pretty great. “", "I’m happy and glad to say that Kurt Angle is back,” he said. ", "Angle then told Dixie Carter to bring her paycheck because the time has come for him to be inducted into the TNA Hall of Fame. ", "He told Dixie to invite the new investor, too.", "\n\nDixie responded that she’s tired of hearing about the new investor. ", "She shouted that she wants this person to show his or her face right now. ", "Carter broke the fourth wall shouting into the camera to show up right now.", "\n\nSuddenly, a familiar clock ticked. ", "The lights went down, then Edwards and Richards came out on-stage. ", "Out came MVP dressed in a suit with “investor” glasses. ", "Tenay’s voice cracked as he shouted: “The mystery is solved. ", "The investor is M-Veee-Peeeee!” ", "Impact signed off two minutes past the top of the hour with MVP standing on the stage with his fist in the air." ]

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[ "315 F.3d 383\nPG&E GAS TRANSMISSION, NORTHWEST CORPORATION, Petitioner,v.FEDERAL ENERGY REGULATORY COMMISSION, Respondent.", "Avista Corporation, et al., ", "Intervenors.", "\nNo. ", "01-1167.", "\nUnited States Court of Appeals, District of Columbia Circuit.", "\nArgued October 11, 2002.", "\nDecided January 21, 2003.", "\n\nOn Petition for Review of Orders of the Federal Energy Regulatory Commission.", "\nStefan M. Krantz argued the cause for petitioner. ", "With him on the briefs were Lee A. Alexander and Debra H. Rednik. ", "Carl M. Fink entered an appearance.", "\nLaura J. Vallance, Attorney, Federal Energy Regulatory Commission, argued the cause for respondent. ", "With her on the brief were Cynthia A. Marlette, General Counsel, and Dennis Lane, Solicitor.", "\nBefore: SENTELLE, RANDOLPH and TATEL, Circuit Judges.", "\nOpinion for the Court filed by Circuit Judge SENTELLE.", "\nSENTELLE, Circuit Judge:\n\n\n1\nPG&E Gas Transmission, Northwest Corporation (PG&E) petitions for review of Federal Energy Regulatory Commission (FERC) orders suspending, PG&E Gas Transmission, Northwest Corp., 90 F.E.R.C. ¶ 61,349, 2000 WL 337605 (2000), and then rejecting, 92 F.E.R.C. ¶ 61,202, 2000 WL 1310693 (2000), reh'g denied, 94 F.E.R.C. ¶ 61,119, 2001 WL 112330 (2001), PG&E's tariff filing, in which it sought to change its method for allocating natural gas transportation capacity. ", "FERC claims that PG&E's petition is moot because FERC later approved an alternative method of allocating capacity. ", "We disagree. ", "Reaching the merits of the petition, we hold that FERC failed to adequately address relevant Commission precedent and thus acted arbitrarily in denying PG&E's filing. ", "Therefore, we grant the petition for review and vacate FERC's orders and remand for further consideration in light of this opinion.", "\n\nI. Background\n\n2\nPG&E operates a natural gas pipeline running 612 miles from the Washington/Canada border to the border between Oregon and California. ", "On its pipeline, PG&E sells two primary types of natural gas transportation capacity — firm and interruptible. ", "Firm capacity is purchased on a monthly basis and cannot be interrupted or curtailed except in limited circumstances. ", "Interruptible transportation (IT) capacity can be interrupted when necessary to provide service to higher priority customers, such as firm customers. ", "IT capacity is bid for as needed, rather than purchased monthly. ", "PG&E's gas tariff sets the maximum per-mile rates PG&E can charge for its IT services. ", "The total amount a shipper pays for service, and thus the revenue generated, is derived by multiplying the per-mile bid by the number of miles the gas is to be transported.", "\n\n\n3\nPrior to the proceedings under review, PG&E allocated IT capacity first to shippers bidding the maximum per-mile rate, regardless of distance, and hence regardless of revenue. ", "PG&E then allocated any remaining capacity to shippers bidding less than the maximum per-mile tariff rate by ranking bids based on total revenue. ", "Ties between bidders, at both the maximum and sub-maximum rates, were broken according to a shipper's position in the IT queue. ", "Thus, if two shippers' bids were tied, the shipper with the higher position in the queue would be allocated the IT capacity. ", "Queue positions were determined by a lottery held by PG&E in 1987. ", "See Pacific Gas Transmission Co., 40 F.E.R.C. ¶ 61,193, at 61,615, 1987 WL 117714 (1987).", "\n\n\n4\nOn March 1, 2000, PG&E submitted a tariff filing pursuant to Section 4 of the Natural Gas Act, 15 U.S.C. § 717c (2000) (NGA), seeking to change its IT capacity allocation method. ", "PG&E proposed to use the system it employed to rank sub-maximum rate bidders to rank bids from maximum rate bidders as well. ", "Under this \"revenue-based\" or \"distance-based\" proposal, allocation would be based on net revenue generated per dekatherm, with net revenue being determined by multiplying the distance in pipeline miles from the receipt point to the delivery point by the rate bid plus surcharges. ", "Consequently, a long-haul maximum rate bidder would always defeat a shorter-haul maximum rate bidder, because the long-haul shipper's total bid would always generate greater revenue. ", "If any ties remained between bids generating the same net revenue, capacity would be allocated pro rata — that is, each tied bidder would receive a proportionate share of the remaining capacity. ", "In sum, under PG&E's filing, the IT queue would be replaced with revenue-based allocation followed by a pro rata tiebreaker. ", "On March 31, 2000, the Commission accepted and suspended FERC's tariff filing, and asked PG&E to provide further \"justification as to the benefits gained by the pipeline and its shippers if such a change is implemented.\" ", "PG&E Gas Transmission, Northwest Corp., 90 F.E.R.C. ¶ 61,349, at 62,154 (Suspension Order). ", "PG&E filed additional supporting evidence for its proposal and sought rehearing of the Suspension Order, claiming that the Suspension Order improperly required PG&E to submit evidence of the unreasonableness of the IT queue and the superiority of its proposed distance-based allocation mechanism. ", "Under NGA § 4, PG&E contended, a pipeline proposing a rate change need only prove that its proposed rate is \"just and reasonable.\"", "\n\n\n5\nOn September 14, 2000, FERC rejected PG&E's revenue-based allocation proposal. ", "PG&E Gas Transmission, Northwest Corp., 92 F.E.R.C. ¶ 61,202, 2000 WL 1310693 (2000) (PG&E I). ", "The Commission held that the revenue-based mechanism would unduly discriminate against maximum rate short-haul shippers because longer-haul maximum rate shippers could always outbid shorter-haul shippers for capacity, even though both would be bidding the same per-mile rate. ", "Id. at 61,677. ", "Cf. ", "18 C.F.R. § 284.9(b) (2002) (banning undue discrimination in the allocation of IT capacity by reference to 18 C.F.R. § 284.7(b)). ", "FERC explained that this result would contradict the Commission's policy of allocating capacity to those who value it most, since, at the maximum rate, both short-haul and long-haul shippers value the capacity equally. ", "PG&E I, 92 F.E.R.C. at 61,677. ", "Nonetheless, FERC realized that the IT queue may be complex, inefficient, and administratively burdensome. ", "Id. at 61,676. ", "Thus, FERC did not preclude PG&E from submitting a later proposal to replace the queue, id. at 61,677, and FERC noted that it had \"accepted other methods of allocating capacity when shippers all bid the maximum rate, such as pro rata,\" id. at 61,676. ", "Finally, FERC rejected PG&E's request for rehearing of the Suspension Order. ", "Id. at 61,677-78. ", "The Commission held that it was justified in seeking further evidentiary support for the relative benefits of PG&E's distance-based mechanism in light of protesters' concerns about discrimination against short-haul shippers. ", "Id. at 61,678.", "\n\n\n6\nShortly after FERC's ruling in PG&E I, PG&E submitted a new tariff filing that replaced the queue with simple pro rata allocation among all maximum rate bidders. ", "Under that tariff, each maximum rate bidder receives a proportionate share of capacity regardless of revenue generated by its total bid, and thus regardless of distance. ", "On October 25, 2000, FERC approved PG&E's filing over the protest of some of PG&E's IT customers, relying on Commission precedents accepting pro rata allocation and reasoning that pro rata allocation would eliminate the need for a complex queue and improve efficiency along the pipeline. ", "See PG&E Gas Transmission, Northwest Corp., 93 F.E.R.C. ¶ 61,072, at 61,187, 2000 WL 1597087 (2000), reh'g denied, 94 F.E.R.C. ¶ 61,114, 2001 WL 112329 (2001) (PG&E II).1 PG&E subsequently implemented its pro rata allocation mechanism, and it remains in effect today.", "\n\n\n7\nMeanwhile, PG&E requested rehearing of FERC's PG&E I ruling, claiming that FERC failed to address several Commission precedents that allowed distance-based allocation. ", "On February 8, 2001, FERC denied PG&E's request for rehearing. ", "PG&E Gas Transmission, Northwest Corp., 94 F.E.R.C. ¶ 61,119, 2001 WL 112330 (2001) (Rehearing Order). ", "In the Rehearing Order, FERC attempted to distinguish the cases on which PG&E relied on the grounds that the cases \"do not address the issue raised here, but merely find that breaking ties at the maximum rate should be done in a nondiscriminatory manner.\" ", "Id. at 61,451.", "\n\n\n8\nPG&E timely filed this petition for review.", "\n\nII. ", "Analysis\n\n9\nUnder NGA § 4, a pipeline proposing a rate change has the burden of showing that the proposed rate is just and reasonable. ", "Exxon Corp. v. FERC, 206 F.3d 47, 51 (D.C.Cir.2000). ", "If the pipeline meets that burden, \"FERC approves the rate regardless of whether there may be other rates that would also be just and reasonable.\" ", "Id. We will uphold FERC's decision to reject a tariff filing unless it is \"arbitrary, capricious, an abuse of discretion, or otherwise not in accordance with law.\" ", "5 U.S.C. § 706(2)(A) (2000). ", "Under this deferential standard, \"the Commission must be able to demonstrate that it has made a reasoned decision based upon substantial evidence in the record.\" ", "Northern States Power Co. v. FERC, 30 F.3d 177, 180 (D.C.Cir.1994) (quotation omitted). ", "In addition, the Commission must have \"examine[d] the relevant data and articulate[d] a satisfactory explanation for its action including a rational connection between the facts found and the choice made.\" ", "Motor Vehicle Mfrs. ", "Ass'n of United States, Inc. v. State Farm Mut. ", "Auto. ", "Ins. ", "Co., 463 U.S. 29, 43, 103 S.Ct. ", "2856, 2866, 77 L.Ed.2d 443 (1983) (quotation omitted). ", "PG&E's central claim is that FERC did not exercise reasoned decisionmaking when it rejected PG&E's filing. ", "Before we reach the merits of PG&E's contention, however, we first address whether this case is properly before us.", "\n\nA. Mootness\n\n10\nFERC claims that we should dismiss this petition as moot because FERC approved, and PG&E implemented, its pro rata allocation proposal after FERC rejected PG&E's revenue-based proposal. ", "Thus, FERC argues that allowing the revenue-based mechanism would not provide PG&E with any relief because the pro rata mechanism has already replaced the queue and performs the same capacity allocation function the revenue-based mechanism would perform. ", "We disagree. ", "A controversy persists over whether PG&E should be allowed to implement its preferred method of IT capacity allocation.", "\n\n\n11\nArticle III of the Constitution limits federal courts to resolving \"actual, ongoing controversies.\" ", "Honig v. Doe, 484 U.S. 305, 317, 108 S.Ct. ", "592, 601, 98 L.Ed.2d 686 (1988). ", "To satisfy Article III's case or controversy requirement, \"a litigant must have suffered some actual injury that can be redressed by a favorable judicial decision.\" ", "Iron Arrow Honor Soc'y v. Heckler, 464 U.S. 67, 70, 104 S.Ct. ", "373, 375, 78 L.Ed.2d 58 (1983). ", "FERC does not dispute that PG&E suffered an injury when the Commission initially rejected its revenue-based mechanism. ", "Rather, it contends that PG&E's injury has been negated by the Commission's later acceptance of a different proposal for allocation of IT capacity, and thus this Court can offer no effective relief.", "\n\n\n12\nWhile FERC is certainly correct that both the pro rata and revenue-based proposals replace the queue and allocate IT capacity between maximum rate bidders, the revenue-based mechanism is undisputedly PG&E's preferred allocation method. ", "Indeed, PG&E proposed pro rata allocation only after FERC rejected its revenue-based proposal. ", "Therefore, while approval of the \"second-best\" pro rata mechanism may have lessened PG&E's injury, the injury persists in that PG&E has been precluded from implementing its preferred method of allocation.", "\n\n\n13\nUnquestionably, a favorable ruling of this Court would redress PG&E's injury. ", "If this Court grants PG&E's petition for review, we will remand the case to FERC for it to reconsider the revenue-based mechanism in light of this Court's opinion. ", "If, on remand, the Commission approves the filing, PG&E will implement the revenue-based mechanism in place of the existing pro rata system. ", "Thus, a favorable decision would provide PG&E with its desired relief: a fair consideration of the revenue-based mechanism, and possibly, an opportunity to implement its preferred method for allocating IT capacity.", "\n\n\n14\nThe fact that both the existing pro rata mechanism and the proposed revenue-based mechanism replace the queue and perform the same function — allocating IT capacity — carries no weight in the mootness analysis. ", "FERC has cited no authority for its proposition that merely because an agency approves a litigant's second-best option to perform a given function, the litigant may not continue to appeal the rejection of its most favored option. ", "Our decision in Rio Grande Pipeline Co. v. FERC, 178 F.3d 533 (D.C.Cir.1999) is instructive. ", "In Rio Grande, a pipeline company sought approval of its rates under 18 C.F.R. § 342.2(a) and submitted evidence of reasonableness of the rates. ", "178 F.3d at 536. ", "Commission approval under § 342.2(a) would have protected the pipeline from paying reparations to customers if the rate was later found unreasonable. ", "Id. In the alternative, the pipeline sought approval under § 342.2(b), which allowed approval if at least one pipeline customer had agreed to the rate, but required reparations if the rate was successfully protested. ", "Id. FERC approved the rate under § 342.2(b), but not under the more favorable provisions of § 342.2(a). ", "Id. at 537. ", "The pipeline petitioned for review of FERC's rejection of its § 342.2(a) claim. ", "Id. On appeal, FERC argued that the pipeline had suffered no injury because its rate had been approved. ", "Id. at 540. ", "We disagreed, holding that the pipeline was aggrieved because approval under § 342.2(b) was less desirable than approval under § 342.2(a), in that it subjected the pipeline to greater economic risks. ", "Id. Similarly, in the present case FERC has approved one method for allocating capacity among PG&E's maximum bidders, but it is a less desirable option than PG&E's preferred method, which FERC rejected. ", "Consequently, PG&E has been aggrieved by FERC's ruling despite subsequent approval of the pro rata tiebreaker. ", "Since this Court's ruling can remedy that injury, the case is not moot. ", "Therefore, we reach the merits of PG&E's petition.", "\n\nB. Merits\n\n15\nOn the merits, PG&E argues that FERC acted arbitrarily and capriciously by failing to adequately address Commission precedents that approved distance-based mechanisms for allocating transportation capacity. ", "We agree.", "\n\n\n16\nIn its orders below, the Commission held that PG&E's proposal violated 18 C.F.R. § 284.9(b)'s prohibition against \"undue discrimination,\" by making it more difficult for maximum rate short-haul shippers to obtain IT capacity. ", "PG&E I, 92 F.E.R.C. at 61,677. ", "In examining the reasoning that undergirds FERC's ruling, we note that discrimination is undue only if \"a pipeline's rate schedule creates a preference without a reasonable basis.\" ", "Algonquin Gas Transmission Co. v. FERC, 948 F.2d 1305, 1316 (D.C.Cir.1991) (quotation omitted). ", "By contrast, when differences in treatment are \"based on relevant, significant facts which are explained,\" the disparate treatment does not run afoul of the NGA. ", "TransCanada Pipelines Ltd. v. FERC, 878 F.2d 401, 413 (D.C.Cir.1989).", "\n\n\n17\nThroughout the course of this litigation, PG&E has consistently pointed to several cases in which FERC approved distance-based allocation methods and held that such methods did not constitute undue discrimination against short-haul shippers. ", "In PG&E I, FERC utterly failed to confront these cases. ", "In its Rehearing Order, FERC's attempt to distinguish these precedents was confusing at best, if not outright disingenuous. ", "Finally, on review before us, FERC counsel concocted a harmonization of Commission precedent with the Commission ruling. ", "Of course, this Court cannot consider such post hoc justifications, but may only consider the grounds on which the Commission actually relied in making its decision. ", "Algonquin Gas Transmission, 948 F.2d at 1312 n. 12 (citing SEC v. Chenery Corp., 332 U.S. 194, 196, 67 S.Ct. ", "1575, 1577, 91 L.Ed. ", "1995 (1947)). ", "We now briefly address the relevant Commission precedents and FERC's attempts to distinguish them in its orders.", "\n\n\n18\nIn Northern Natural Gas Co., 82 F.E.R.C. ¶ 61,077, 1998 WL 32696, reh'g, 84 F.E.R.C. ¶ 61,154, 1998 WL 765463 (1998), FERC approved the pipeline's proposal to allocate capacity based on net present value (NPV), with NPV determined in part by whether the gas was to be transported over two zones or wholly within one zone. ", "82 F.E.R.C. at 61,287. ", "The Commission recognized that under the pipeline's proposal, \"continuous service that includes both [Zone A] and [Zone B] should be awarded the service if they have a higher NPV that [sic] purely [Zone A] transportation at the maximum rate.\" ", "Id. Even so, FERC approved the proposal over protests that the allocation system would unduly discriminate against single zone shippers. ", "Id. Despite Northern Natural's obvious relevance to the present case, FERC failed even to mention Northern Natural in its orders below, much less attempt to distinguish it.", "\n\n\n19\nSimilarly, in Tennessee Gas Pipeline Co., 65 F.E.R.C. ¶ 61,224, at 62,111, 1993 WL 467762 (1993), the Commission approved a zone-based system with allocation based on price, wherein price was based on the route traveled and quantity of gas transported. ", "Id. Tennessee's distance-based allocation proposal applied to both maximum and sub-maximum rate bids. ", "Id. Moreover, while the Commission ordered Tennessee to delete its consideration of quantity, the Commission allowed Tennessee to consider distance, specifically rejecting arguments that the distance-based mechanism would unduly discriminate against short-haul shippers. ", "Id. In a later proceeding, the Commission again rejected an attempt to eliminate Tennessee's consideration of distance in allocating IT capacity, despite complaints that it discriminated against maximum rate short-haul shippers. ", "See Tennessee Gas Pipeline Co., 71 F.E.R.C. ¶ 61,399, at 62,582, 1995 WL 384147 (1995).", "\n\n\n20\nIn its Rehearing Order in this proceeding, FERC misleadingly tried to distinguish Tennessee. ", "See 94 F.E.R.C. at 61,452. ", "The Commission only mentioned Tennessee's holding that the pipeline had to delete any consideration of quantity from its allocation mechanism so as not to discriminate against small or short-haul shippers. ", "Id. However, FERC completely ignored the next paragraph in Tennessee, in which the Commission explicitly upheld consideration of distance against charges of undue discrimination. ", "Tennessee, 65 F.E.R.C. at 62,111. ", "Thus, FERC utterly failed to distinguish the relevant holding of Tennessee and explain why its reasoning should not be applied to the present case.", "\n\n\n21\nFinally, in Trunkline Gas Co., 64 F.E.R.C. ¶ 61,141, at 62,126, 1993 WL 349643 (1993), reh'g denied in relevant part, 65 F.E.R.C. ¶ 61,355, 1993 WL 594424 (1994), FERC again approved a zone-based system that allocated IT capacity based, in part, on number of zones traveled. ", "The distance-based mechanism applied to both maximum and sub-maximum rate bids. ", "Id. Furthermore, FERC rejected a protestor's proposal that would have allocated capacity based on percentage of maximum rate paid, regardless of distance traveled. ", "Id. The Commission explicitly acknowledged that under Trunkline's filing a maximum rate short-haul shipper could be trumped by a lower rate longer-haul shipper. ", "Id. Nonetheless, the Commission concluded that the proposal did not unduly discriminate against short-haul shippers and promoted allocative efficiency. ", "Id. at 62,126-27.", "\n\n\n22\nThe Commission's attempt to distinguish Trunkline in its Rehearing Order is confusing at best. ", "First, FERC seems to claim that Trunkline applied only to below maximum rate bids. ", "See Rehearing Order, 94 F.E.R.C. at 61,452. ", "This is flatly wrong, as even FERC counsel admitted at oral argument. ", "See Tr. ", "of Oral Arg. ", "at 28-29. ", "Second, FERC claimed \"[t]he Trunkline issue is different because it did not discuss allocation methodology for resolving ties when the bids are equal.\" ", "Rehearing Order, 94 F.E.R.C. at 61,452. ", "It is true that Trunkline's revenue-based proposal did not involve breaking ties between equal bidders, but neither does PG&E's proposal in any relevant sense. ", "That is, in Trunkline, bids were ranked based on adding the rates for each zone traveled. ", "Similarly, in PG&E's proposal, bids are ranked based on multiplying the per-mile rate by the number of miles traveled. ", "The fact that PG&E's proposal breaks ties between bidders who bid the same per-mile rate is not a relevant difference from the proposal in Trunkline when, under both systems, a maximum rate short-haul shipper can always be outbid by a longer-haul shipper. ", "Thus, the distinction to which FERC alluded between the zone-based system in Trunkline and the distance-based system in the PG&E tariff is a distinction without a difference.", "\n\n\n23\nIn sum, FERC's attempts to distinguish its precedents approving distance-based allocation were alternately nonexistent, misleading, and irrelevant. ", "On brief to this Court, counsel argues that these precedents are distinguishable because they involved zone-based systems with rates based on several factors, of which distance is only one. ", "This contrasts, counsel contends, with PG&E's proposal, in which distance is the sole determinant of capacity allocation at the maximum rate. ", "Thus, PG&E's proposed allocation mechanism is unduly discriminatory while those in Northern Natural, Tennessee, and Trunkline were not. ", "We cannot consider this argument. ", "FERC's order \"must stand or fall on the grounds articulated by the agency in that order,\" not the reasoning proffered by its appellate counsel. ", "NorAm Gas Transmission Co. v. FERC, 148 F.3d 1158, 1165 (D.C.Cir. ", "1998) (quotation omitted). ", "We therefore make no judgment about the validity of counsel's argument because it is a purely post hoc justification which cannot sustain the Commission's orders. ", "See Chenery, 332 U.S. at 196, 67 S.Ct. ", "at 1577.", "\n\n\n24\nAs outlined above, FERC's orders in this case do not adequately explain why the Commission's precedent in favor of distance-based allocation does not compel approval of PG&E's filing. ", "In \"gloss[ing] over\" these precedents, FERC \"cross[ed] the line from the tolerably terse to the intolerably mute.\" ", "Greater Boston Television Corp. v. FCC, 444 F.2d 841, 852 (D.C.Cir.1970). ", "Indeed, FERC has given no explanation whatsoever for this apparent shift in Commission policy. ", "FERC's failure to come to terms with its own precedent reflects the absence of a reasoned decisionmaking process. ", "See North Carolina Utils. ", "Comm'n v. FERC, 42 F.3d 659, 666 (D.C.Cir.1994) (rejecting a FERC order because the Commission did not \"sufficiently explain[] its departure from its prior cases\"); Hatch v. FERC, 654 F.2d 825, 834 (D.C.Cir.1981) (\"[A]n agency must provide a reasoned explanation for any failure to adhere to its own precedents.\"). ", "Consequently, we vacate FERC's orders and remand, so that FERC may reconsider PG&E's proposal in light of this opinion and Commission precedent.", "\n\nIII. ", "Conclusion\n\n25\nWe hold that this case did not become moot when FERC approved PG&E's pro rata allocation proposal after it rejected PG&E's revenue-based mechanism. ", "A live controversy persists regarding whether PG&E should be able to implement revenue-based allocation of IT capacity. ", "On the merits, we hold that FERC failed to adequately address Commission precedent allowing pipelines to consider distance when allocating transportation capacity. ", "Accordingly, we grant the petition for review and vacate FERC's orders and remand for further consideration in light of this opinion.", "\n\n\n26\n\nSo ordered.", "\n\n\n\n\nNotes:\n\n\n1\n We address the petition for review of PG&E's customers in a companion case,Duke Energy Trading & Mktg., ", "L.L.C. v. FERC, 315 F.3d 377 (D.C.Cir.2003), issued by this Court today.", "\n\n\n" ]

{ "pile_set_name": "FreeLaw" }

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[ "// Copyright (c) 2007-2009 Nokia Corporation and/or its subsidiary(-ies).", "\n// All rights reserved.", "\n// This component and the accompanying materials are made available\n// under the terms of the License \"Eclipse Public License v1.0\"\n// which accompanies this distribution, and is available\n// at the URL \"http://www.eclipse.org/legal/epl-v10.html\".", "\n//\n// Initial Contributors:\n// Nokia Corporation - initial contribution.", "\n//\n// Contributors:\n//\n// Description:\n// This is a test application for tutorial reference driver. ", "This purpose of \n// this application is to illustrate using and calling the driver API. ", "Here \n// this application will load the driver, open and call the driver API to \n// configure the uart, transmit and receive some test data over the uart. ", "\n// This application can be built and run for armv5 platform\n// This application tests the driver for its asynchronous request handling and\n// and the Tx,Rx functionality with driver functioning in DMA mode\n// API:\n// Symbian Device driver framework API\n// Platforms:\n// H4,ARMV5\n// Capability:\n// CommDD\n// Assumptions:\n// 1) Tests On COM0,KUnit1 uses internal loopback and external loopback\n// External loopback is used on same port COM0 of H4\n// 2) Changes the debugport to 3 and reverts back\n// 3) If driver enables debug messages, the timeout values used here\n// will have to be increased\n// \n//\n\n// include files\n//\n#include <e32test.h>\t\t\t// for test framework\n#include \"exdma.h\"\t\t\t\t// driver interface\n#include <hal.h>\t\t\t\t// for HAL API\n#include <hal_data.h>\t\t\t// for HAL attributes\n#include \"t_exdma_testdata.h\" \t// test data\n\n// Literal string descriptors for LDD and PDD names of the driver\n//\n_LIT(KExDriverLdd, \"d_exdma_ldd\");\t// Ldd name\n_LIT(KExDriverPdd, \"d_exdma_pdd\");\t// Pdd name\n\n// Constants\n//\n// Use debug port 3 to output data. ", "This is done as we shall be testing\n// this application for UART1 and UART2. ", "To avoid clashing of the tutorial\n// driver with default driver, we want the default driver use UART3\n//\nconst TInt KDebugPort=3;\nconst TInt KUnit1=1;\t\t\t// Unit number for UART1\nconst TInt KUnit2=2;\t\t\t// Unit number for UART2\nconst TInt KUnit3=3;\t\t\t// Invalid unit number, UART3\nconst TInt KTestRxSize = 1000; // Rx request data size in bytes\n\nconst TUint KTimeOut=1000000; \t// 1 sec, interms of microseconds\n\n// 40 sec, interms of microseconds. ", "This is more, as large buffers\n// of data is being used. ", "However, this timeout value is only used\n// to provide an exit method from wait on asynch requests in some\n// error conditions.", "\n//\nconst TUint KTimeOutTxRx=40000000; \n\n/**\n Class for grouping the test cases\n */\nclass TestExDriver\n\t{\npublic:\n\t// Constructor\n\tTestExDriver();\n\t// Destructor\t\t\n\t~TestExDriver();\t\n\t/* \n\t Called from E32Main() \n\t */\n\t // Load the driver\n\tvoid TestLoadDriver();\n\t// Unload the driver\t\t \n\tvoid TestUnloadDriver(); \n\t// Test the logical device API of the driver\n\tvoid TestDevice();\n\t// Test the driver for channels and units support\n\tvoid TestChannelsAndUnits();\n\t// Test synchronous requests - caps, configure, loopback\n\tvoid TestSynchRequests();\n\t// Test asynchronous requests, cancelling, concurrent requests etc\n\tvoid TestAsynchRequests();\t\t\n\t// Test data flow Tx&Rx on multiple units using loopback\n\tvoid TestDataflowMultipleUnitsLoopback();\n\t// Change the debug port (required for H4)\t\n\tTInt ChangeDebugPort(TInt aPort);\nprivate: \n\t// Note:: These are not independent tests. ", "Their preconditions\n\t// have to be satisfied, like loading the driver\n\t//\t\n\t/* \n\t Channel & Unit Tests \n\t */\t\n\t// Test channel on a single valid unit, invalid unit\n\tvoid TestChannelOnMultipleUnits();\n\t// Test multiple channels on a unit\n\tvoid TestMultipleChannelsOnUnit();\n\t// Test dulication of channel handle\n\tvoid TestHandleDuplication();\n\t// Test multi thread access\n\tvoid TestMultipleThreadAccess();\t\t\n\t/* \n\t Synchronous Tests \n\t */\n\t// Test channel capabilities\n\tvoid TestChannelCaps();\n\t// Configure the device\n\tvoid TestConfigure();\t\t\n\t/* \n\t Asynchronous Tests \n\t */\t\n\t// Test Transmit Data, complete\n\tvoid TestTransmit();\n\t// Test cancel asynchronous request\n\tvoid TestCancelRequests();\n\t// Test multiple same requests\n\tvoid TestMultipleSameRequests();\t\n\t// Test Tx and Rx duplex mode, concurrent requests in loopback mode\n\tvoid TestConcurrentTxRx(TInt aLoopback, const TDesC8& aTxData, TInt aRxSize);\t\n\t/* \n\t Data flow tests \n\t */\t\n\tvoid TestDataFlow(TInt aUnit, TInt aLoopback);\t\npublic:\n\tRTest iTest;\t\t\t\t\t// RTest reference object\n\tRTimer iTimer;\t\t\t\t\t// Timer reference object\t\n\tRExDriverChannel iLdd;\t\t\t// Logical channel\n\tTRequestStatus\tiTimeStatus;\t// Asynch request object\n\t};\n\n/**\n E32Main - Application, exe main entry point.", "\n \n @return KErrNone\n */\nGLDEF_C TInt E32Main()\n\t{\t\n\tTInt debugport;\n\t\n\t// Create a TestExDriver object\n\tTestExDriver exDrv;\t\n\t\n\t// Change the debugport. ", "This function is optional and specific \n\t// to variant. ", "Incase of H4 variant, default debugport is 1. ", "However\n\t// since this tutorial driver tests both unit1(port1) and unit2(port2),\n\t// default port is being changed to port3 to avoid conflict\n\t//\n\tdebugport=exDrv.", "ChangeDebugPort(KDebugPort);\n\t\t\n\t// Print the title of the test\t\n\texDrv.iTest.", "Title();\t\n\t\n\t// Start the test sets\n\texDrv.iTest.", "Start(_L(\"Tutorial Driver:Phase-III\"));\t\n\t\n\t// Load the driver\n\texDrv.", "TestLoadDriver();\n\t\n\t// Test the device functionality\n\texDrv.", "TestDevice();\n\n\t// Test Channels and units\n\texDrv.", "TestChannelsAndUnits();\n\n\t// Test Synchronous requests\n\texDrv.", "TestSynchRequests();\n\n\t// Test Asynchronous Requests, Cancel and handle Concurrent Requests\n\texDrv.", "TestAsynchRequests();\n\t\n\t// Test Dataflow on multiple units with loopback\n\texDrv.", "TestDataflowMultipleUnitsLoopback();\n\t \t\n \t// Unload the driver\n\texDrv.", "TestUnloadDriver();\t\n \t\n\t// If debugport is changed then, revert it back to the default debug port\n \tif (debugport>=0)\n \t\t{\n \t\texDrv.iTest.", "Printf(_L(\"\\nReverting the debugport to Port (%d)\\n\"),debugport); \t\t\n \t\t// Revert debugport?", "\n \t\texDrv.", "ChangeDebugPort(debugport);\n \t\t}\n \t\t \t\t \t\n\t// return success\n \treturn KErrNone;\n\t}\n\t\n/*\n TestExDriver functions\n */\n\t\n/**\n Constructor\n */\nTestExDriver::TestExDriver()\n// Initialize RTest with title string\n:iTest(_L(\"Tutorial Driver:Phase-III\"))\n\t{\t\n\t}\n\n/**\n Destructor\n */\t\nTestExDriver::~TestExDriver()\n\t{\n\t// Ends the current sets of test\n\tiTest.", "End();\n\t// Closes the console and frees any resources acquired\n\tiTest.", "Close();\t\n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0623\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver (LDD and PDD) installing\n//! ", "@SYMTestActions 1.", "\tLoad the PDD by name, d_exdma.pdd\n//! ", "\t\t\t\t\t2.", "\tLoad the LDD by name, d_exdma.ldd\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tPDD loads successfully\n//! ", "\t\t\t\t\t\t\ta.\treturns KErrNone if not existing already\n//! ", "\t\t\t\t\t\t\tb.\treturn KErrAlreadyExists when the driver is already loaded.", "\n//! ", "\t\t\t\t\t\t2.", "\tLDD loads successfully\n//! ", "\t\t\t\t\t\t\ta.\treturns KErrNone if not existing already\n//! ", "\t\t\t\t\t\t\tb.\treturn KErrAlreadyExists when the driver is already loaded.", "\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Load the driver, LDD and PDD and verify if successful\n */\nvoid TestExDriver::TestLoadDriver()\n\t{\n\tTInt r;\n\t\n\t// [RTest::Next() starts new set of tests]\t\n\tiTest.", "Next(_L(\"Test driver Loading\"));\n\t\n\t// Load the PDD\n \tiTest.", "Printf(_L(\"Load Physical Device\\n\"));\n\n\t// Load the PDD. ", "User API will load the PDD dll by name and\n\t// also enable the loader to search for the PDD object by name.", "\n\t//\n \tr = User::LoadPhysicalDevice(KExDriverPdd);\n\n\t// PDD loading is considered successful, if either it is loaded now\n\t// or it is already loaded and is existing\n\t//\n \tiTest((r==KErrNone)||(r==KErrAlreadyExists));\n \t\n\t// Load the LDD\n \tiTest.", "Printf(_L(\"Load Logical Device\\n\"));\n\n\t// Load the LDD. ", "User API will load the LDD dll by name and\n\t// also enable the loader to search for the LDD object by name.", "\n\t//\n \tr = User::LoadLogicalDevice(KExDriverLdd);\n\n\t// LDD loading is considered successful, if either it is loaded now\n\t// or it is already loaded and is existing\n \tiTest((r==KErrNone)||(r==KErrAlreadyExists));\n\t}\n\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0624\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver (LDD and PDD) un-installing\n//! ", "@SYMTestActions 1.", "\tUn-Load the PDD by object name\n//! ", "\t\t\t\t\t2.", "\tUn-Load the LDD by object name\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tPDD un-loads successfully, returns KErrNone\n//! ", "\t\t\t\t\t\t2.", "\tLDD unloads successfully, returns KErrNone\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Unload the driver, LDD and PDD and verify if successful\n @pre TestLoadDriver() called\n */\t\nvoid TestExDriver::TestUnloadDriver()\t\n\t{\n\tTInt r;\n\t\n\tiTest.", "Next(_L(\"Test driver Un-Loading\"));\n\t\n\t// Free the logical device / ldd\n\tiTest.", "Printf(_L(\"Free Logical Device\\n\"));\n \tr=User::FreeLogicalDevice(KDriverName);\n \tiTest(r==KErrNone);\n \t\n \t// Free the physical device / pdd\n \tiTest.", "Printf(_L(\"Free Physical Device\\n\")); \t\n \t\n \t// Instead of directly using the pdd name, get the PDD factory object name\n \t// and append with extension name, to unload the PDD.", "\n \t//\n\tTName pddName(KDriverName);\n\t_LIT(KVariantExtension,\".pdd\");\n pddName.", "Append(KVariantExtension); \n \t\n // Free the PDD, resulting in freeing pdd factory object. ", "Name passed here\n // should match with the one used by driver to register itself in DoCreate().", "\n //\n \tr=User::FreePhysicalDevice(pddName);\n \tiTest(r==KErrNone);\n\t}\n\n/*\n Testing Logical device functionality/API \n */\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0625\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes logical device API/functionality\n//! ", "@SYMTestActions 1.", "\tOpen the device\n//! ", "\t\t\t\t\t2.", "\tGet the device capabilities\n//! ", "\t\t\t\t\t3.", "\tClose the device\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tLogical Device is opened successfully\n//! ", "\t\t\t\t\t\t2.", "\tReads the device capabilities\n//! ", "\t\t\t\t\t\t3.", "\tCloses the logical device\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Tests the operations on the logical device\n\t- Opens the device\n\t- Reads the device capabilities\n\t- Closes the device \n @pre \tTestLoadDriver() called\n */\nvoid TestExDriver::TestDevice()\n\t{\n\tiTest.", "Next(_L(\"Testing Logical Device API\"));\n\n\tTInt r;\n\n\t// RDevice is the user side handle of LDD factory object.", "\n \tRDevice device;\n\n\tiTest.", "Printf(_L(\"Open the Device\\n\")); \n\t// Open the device with reference to the driver name.", "\n \tr = device.", "Open(KDriverName);\n \tif (r==KErrNone)\n \t\t{\n \t\tiTest.", "Printf(_L(\"Get Device Capabilities\\n\")); \t\t\n\t\t// Package the device capabilities structure to a descriptor\n\t\t// suitable to be passed to RDevice::GetCaps()\n\t\t//\n \t\tTPckgBuf<TCapsDevCommV01> caps;\n\n\t\t// Get the device capabilities. ", "Driver fills in the capabilities\n\t\t// in caps and returns to user.", "\n\t\t//\n \t\tdevice.", "GetCaps(caps); \t\t\n\t\n\t\t// Retrieve the information from caps and validate name and version\n \t\tTVersion expectedVer(RExDriverChannel::VersionRequired()); \t\t\n \t iTest(caps().version.iMajor==expectedVer.iMajor);\n \tiTest(caps().version.iMinor==expectedVer.iMinor);\n \tiTest(caps().version.iBuild==expectedVer.iBuild);\n\n\t\tiTest.", "Printf(_L(\"Close the device\\n\"));\t\t\n\t\t// Close the device. ", "This handle is required only to get any device \n\t\t// related information from LDD factory object.", "\n\t\t//\n \t\tdevice.", "Close();\n \t\t}\n\t}\n\n\n/**\n Test the driver for various scenarios of using channels and units\n @pre\tTestLoadDriver() called\n */\nvoid TestExDriver::TestChannelsAndUnits()\n\t{\n\tiTest.", "Next(_L(\"Testing channel access on multiple units\"));\n\t// Test channel on a single valid unit, invalid unit\n\tTestChannelOnMultipleUnits();\n\n\tiTest.", "Next(_L(\"Testing multiple channel access on units\"));\n\t// Test multiple channels on a unit\n\tTestMultipleChannelsOnUnit();\n\n\tiTest.", "Next(_L(\"Testing multiple client access\"));\n\t// Test dulication of channel handle\n\tTestHandleDuplication();\n\n\tiTest.", "Next(_L(\"Testing multiple thread access\"));\n\t// Test multi thread access\n\tTestMultipleThreadAccess();\t\n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0626\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver support for channels on multiple units\n//! ", "@SYMTestActions 1.", "\tOpen channel on unit1, valid unit\n//! ", "\t\t\t\t\t2.", "\tOpen channel on unit2, valid unit\n//! ", "\t\t\t\t\t3.", "\tOpen channel on unit3, invalid unit\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tOpen on unit1 returns KErrNone, success case\n//! ", "\t\t\t\t\t\t2.", "\tOpen on unit2 returns KErrNone, success case\n//! ", "\t\t\t\t\t\t3.", "\tOpen on unit3 returns KErrNotSupported, success case\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Test opening channel on multiple units\n @pre\tTestLoadDriver() called\n */\t\nvoid TestExDriver::TestChannelOnMultipleUnits()\n\t{\n\tTInt r;\t\n\t\n \tiTest.", "Printf(_L(\"Open Channel on Unit-1\\n\")); \t\n\t// Open thelogical channel, driver for unit 1. ", "This is a user-side wrapper\n\t// function for RBusLogicalChannel::DoCreate() API. ", "Hence DoCreate() is \n\t// called in Open() for unit 1.", "\n\t//\n \tr=iLdd.", "Open(KUnit1); \n \tiTest(r==KErrNone); \t\n \t// Close the channel opened\n \tiLdd.", "Close();\n \t\n \tiTest.", "Printf(_L(\"Open Channel on Unit-2\\n\"));\n \t// Valid unit, hence return success\n \tr=iLdd.", "Open(KUnit2);\n \tiTest(r==KErrNone);\n \t// Close the channel opened\n \tiLdd.", "Close();\n \t\n \tiTest.", "Printf(_L(\"Open Channel on Unit-3\\n\"));\n \t// Invalid unit, hence return error\n \tr=iLdd.", "Open(KUnit3);\n \tiTest(r==KErrNotSupported);\n \tiTest.", "Printf(_L(\"Open Channel on Unit-3: Failed as Expected\\n\"));\n \t// Channel opened fail, hence no need to close\n\t}\n\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0627\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver support for multiple channels on a unit \n//! ", "@SYMTestActions 1.", "\tOpen channel-1 on unit1\n//! ", "\t\t\t\t\t2.", "\tOpen channel-2 on unit1\n//! ", "\t\t\t\t\t3.", "\tOpen channel-1 on unit2\n//! ", "\t\t\t\t\t4.", "\tClose Channel-1 on unit1\n//! ", "\t\t\t\t\t5.", "\tClose Channel-1 on unit2\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tOpen channel-1 on unit1 returns KErrNone\n//! ", "\t\t\t\t\t\t2.", "\tOpen channel-2 on unit1 returns KErrInUse\n//! ", "\t\t\t\t\t\t3.", "\tOpen channel-1 on unit2 returns KErrNone\n//! ", "\t\t\t\t\t\t4.", "\tClose completes and exits successfully\n//! ", "\t\t\t\t\t\t5.", "\tClose completes and exits successfully\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Test opening multiple channels on units\n @pre\tTestLoadDriver() called\n */\nvoid TestExDriver::TestMultipleChannelsOnUnit()\n\t{\n\tTInt r;\t\n\tRExDriverChannel ldd1;\n\tRExDriverChannel ldd2;\n\t\t\n\tiTest.", "Printf(_L(\"Open Channel-1 on Unit1\\n\"));\n\t// Open a channel on Unit1\n\tr=iLdd.", "Open(KUnit1);\n\tiTest(r==KErrNone);\n\tiTest.", "Printf(_L(\"Open Channel-1 on Unit1: Success\\n\"));\n\t\t\n\tiTest.", "Printf(_L(\"Open Channel-2 on Unit1\\n\"));\n\t//Open another channel on Unit1\n\tr=ldd1.Open(KUnit1);\n\tiTest(r==KErrInUse);\n\tiTest.", "Printf(_L(\"Open Channel-2 on Unit1: Failed as Expected\\n\"));\n\t\n\tiTest.", "Printf(_L(\"Open Channel-1 on Unit2\\n\"));\n\t//Open another channel on Unit1\n\tr=ldd2.Open(KUnit2);\n\tiTest(r==KErrNone);\n\tiTest.", "Printf(_L(\"Open Channel-1 on Unit2: Success\\n\"));\n\t\n\t// Close the channel1 that is opened successfuly\n\tiTest.", "Printf(_L(\"Close Channel-1 on Unit1\\n\"));\n\tiLdd.", "Close();\n\t\n\t// Close the channel on unit2\n\tiTest.", "Printf(_L(\"Close Channel-1 on Unit2\\n\"));\n\tldd2.Close();\n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0628\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver support for sharing handles \n//! ", "@SYMTestActions 1.", "\tDuplicate handle with EOwnerProcess\n//! ", "\t\t\t\t\t2.", "\tDuplicate handle with EOwnerThread\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tDuplicate handle with EOwnerProcess returns KErrAccessDenied, success case\n//! ", "\t\t\t\t\t\t2.", "\tDuplicate handle with EOwnerThread returns KErrNone, success case\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Tests if there is a duplication in client access and handle\n @pre\tTestLoadDriver() called\n */\nvoid TestExDriver::TestHandleDuplication()\n\t{\n\tTInt r;\n\t\n\t// Test set 5 - Check if the driver can verify and deny access to wrong clients \t\n \tiTest.", "Printf(_L(\"Check Wrong client access\\n\"));\n \t\n\t// Open a channel on Unit1\n\tr=iLdd.", "Open(KUnit1);\n\tiTest(r==KErrNone);\t\n\t\n\t// Create another user side handle for the logical channel\n RExDriverChannel ldd2=iLdd;\n\n\t// RBusLogicalChannel::Duplicate() creates a valid handle to the kernel \n\t// object for which the specified thread already has a handle. ", "Check with \n\t// ownership as process.", "\n\t//\n r=ldd2.Duplicate(RThread(),EOwnerProcess);\n\n\t// Above API should return KErrAccessDenied error to verify that driver \n\t// could deny wrong client access\n\t//\n iTest(r==KErrAccessDenied);\n\n\t// Test set 6 - Handle duplication\n iTest.", "Printf(_L(\"Check handle duplication\\n\"));\n ldd2=iLdd;\n \n\t// Check handle duplication, with ownership as this Thread.", "\n r=ldd2.Duplicate(RThread(),EOwnerThread);\n iTest(r==KErrNone);\n ldd2.Close();\n \n iLdd.", "Close();\n\t}\n\n/**\n Thread that is created from main application. ", "This gets executed\n once RThread::Resume() is called. ", "Tests to validate the access\n from multiple threads are done here\n */\t \n static TInt ThreadFunc(TAny* aPtr)\n {\n TInt r;\n RTest test(_L(\"Test Thread\"));\n \n test.", "Printf(_L(\"Test Thread scheduled\\n\"));\n \n test.", "Printf(_L(\"Open channel from Test thread\\n\")); \n // Test1: Creating channel from other thread\n RExDriverChannel ldd1;\n r=ldd1.Open(KUnit1);\n test((r==KErrNone)||(r==KErrInUse));\n if (r==KErrInUse)\n \t{\n\t\ttest.", "Printf(_L(\"Channel already opened in another thread, hence failed:Expected result\\n\"));\n\t\t}\n \n // Test2: Using the handle created in other thread \n \tRExDriverChannel *ldd2=(RExDriverChannel*)aPtr; \t\t\n\tTCommCaps3 chCaps;\t\n\tr=ldd2->Caps(chCaps);\t\n\ttest(r==KErrAccessDenied);\n test.", "Printf(_L(\"Access denied from other thread:Expected result\\n\"));\n \n test.", "Close();\n \treturn KErrNone;\n }\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0629\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver support for multiple thread access \n//! ", "@SYMTestActions 1.", "\tOpen channels in different threads at a time \n//! ", "\t\t\t\t\t2.", "\tShare the handle between threads by passing handle to new thread and access driver API\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tOpen channel in new thread returns KErrInUse, success case\n//! ", "\t\t\t\t\t\t2.", "\tDriver API access in new thread, using handle from original thread panics, success case\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Test access from a different thread\n @pre\tTestLoadDriver() called\n */ \nvoid TestExDriver::TestMultipleThreadAccess()\n\t{\n\tTInt r;\n\tRThread thrd;\t\t \n TRequestStatus tS;\n\t_LIT(KTestThreadName,\"TestThread\");\n\t\n\tiTest.", "Printf(_L(\"Test multiple thread access\\n\"));\n\t\n\t// Create the iTimer that is relative to the thread \n \tr = iTimer.", "CreateLocal();\n \tiTest(r==KErrNone);\n \t\n \t// Open the channel \t \n\tr=iLdd.", "Open(KUnit1);\n\tiTest(r==KErrNone);\n\t \n\t// Create a new thread, where we can test access to the driver\n iTest (thrd.", "Create(KTestThreadName,&ThreadFunc,KDefaultStackSize,NULL,&iLdd)==KErrNone);\n \n // Logon() will request asynchronous notification on thread termination\n thrd.", "Logon(tS);\n \n // Trigger iTimer expiry after KTimeOut. ", "The status should be pending\n\tiTimer.", "After(iTimeStatus,KTimeOut);\n\tiTest(iTimeStatus==KRequestPending);\n\t\n\t// Resume() schedules the thread created\n thrd.", "Resume();\n \n // Wait for the thread termination notification\n //User::WaitForRequest(tS,iTimeStatus);\n User::WaitForRequest(tS);\n \n // Incase thread is not scheduled and timeout occurs\n if ((iTimeStatus==KErrNone))\n \t{\n \tiTest.", "Printf(_L(\"Timeout for TestThread schedule and exit\\n\"));\n \t}\n \n // Cancel the timer and close\t\n iTimer.", "Cancel();\t\n iTimer.", "Close();\n \n // Close the thread created\t\n thrd.", "Close(); \n // Close the channel opened \n iLdd.", "Close(); \n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0630\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes all the synchrnous requests supported by the driver \n//! ", "@SYMTestActions Call all synchronous API of the driver:\n//! ", "\t\t\t\t\t\t1.", "\tGet the channel capabilities\n//! ", "\t\t\t\t\t\t2.", "\tConfigure the device\n//! ", "\t\t\t\t\t\t3.", "\tSet internal loopback\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tGet channel capabilities reads the channel capabilities to user buffer, success case\n//! ", "\t\t\t\t\t\t2.", "\tSetConfig() API returns KErrNone\n//! ", "\t\t\t\t\t\t3.", "\tSetIntLoopback() returns KErrNone\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Test Synchronous requests\n \t- 1) Get Channel capabilities\n \t- 2) Set Configuration\n \t- 3) Set loopback\n @pre TestLoadDriver() called\t\n */\t\nvoid TestExDriver::TestSynchRequests()\n\t{\n\tTInt r;\n\t\n\tiTest.", "Next(_L(\"Test Synchronous Requests\"));\n\t\n\t// Open Channel\n\tr=iLdd.", "Open(KUnit1);\n\tiTest(r==KErrNone);\n\n\t// Test channel capabilities\n\tTestChannelCaps();\n\n\t// Configure the device\n\tTestConfigure();\n\n\t// Set the device's internal loopback mode, i.e enable or disable\n \tr = iLdd.", "SetIntLoopback(KIntLoopbackEnable);\n \tiTest(r==KErrNone); \t\n\n\t// Close channel\n\tiLdd.", "Close();\n\t}\n\n\n/**\n Get the channel capabilities and verify against the expected capabilities\n @pre \tLoadDriver() called with its pre conditions\n */\t\nvoid TestExDriver::TestChannelCaps()\n\t{\n\tTInt r;\n\t\n\t// Test - Get the channel capabilities\n\tiTest.", "Printf(_L(\"Get the channel capabilities\\n\"));\n\t\n\t// Package buffer of TCommCapsV03, which holds the uart capabilities\n\tTCommCaps3 chCaps;\n\t// Get the channel's capabilities to be configured\t\n\tr=iLdd.", "Caps(chCaps);\t\n\tiTest(r==KErrNone);\n\t\n\t// Check if 9600 baud rate is supported,(want to configure to this later)\n\tiTest(chCaps().iRate&KCapsBps9600);\t\n\t}\n\n\t\n/**\n Configure the device\n @pre \tLoadDriver(), TestOpenChannel() called with their pre conditions\n */\nvoid TestExDriver::TestConfigure()\n\t{\n\tTInt r;\n\t\n\t// Configure the device (uart)\n \tiTest.", "Printf(_L(\"Configure UART\\n\"));\n \t\n \t// Package the comm config structure to descriptor and initialize the data\n \tTPckgBuf<TCommConfigV01> config;\n \tconfig().iRate = EBps9600;\n \t\n \t// Configure the channel\n \tr = iLdd.", "SetConfig(config);\n \tiTest(r==KErrNone); \t \t\n\t}\n\t\n/**\n Test Asynchronous requests\n \t- 1) Transmit\n \t- 2) Receive \n */\nvoid TestExDriver::TestAsynchRequests()\n\t{\n\tTInt r;\n\t\n\tiTest.", "Next(_L(\"Test Asynchronous Requests\"));\n\t\n\t// Open channel & configure\t\n\tr=iLdd.", "Open(KUnit1);\n\tiTest(r==KErrNone);\n\t\n\t// Run the device @ default configuration with no loopback\n\n\t// Test Transmit data, complete\n\tTestTransmit();\n\t\t\n\t// Test cancel asynchronous request\n\tTestCancelRequests();\n\n\t// Test multiple same requests\n\tTestMultipleSameRequests();\n\t\t\n\t// Close channel\n\tiLdd.", "Close();\n\t\t\n\t// Test Tx and Rx duplex mode, concurrent requests in loopback mode\n\tTestConcurrentTxRx(KIntLoopbackEnable,KTestTxDataSmall,KTestRxSize);\n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0631\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver transmit functionality, as a part of\n//! ", "\t\t\t\t\ttesting the asynchrnous request implementation\t\n//! ", "@SYMTestActions 1.", "\tTransmit data over the device\n//! ", "\t\t\t\t\t2.", "\tWait for transmit request completion\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tTransmit returns KErrNone\n//! ", "\t\t\t\t\t\t2.", "\tWait for request completes with KErrNone\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Transmits data over the device\n @pre\tTestLoadDriver(), channel opened\n */\t\nvoid TestExDriver::TestTransmit()\n\t{\n\tTInt r;\n\t\n\t// Transmit data to the device (uart)\n \tiTest.", "Printf(_L(\"Test Transmit Asynchronous Request\\n\"));\n\t\t \t \t\n \t// Request status object for transmit. ", "This object will be used to read the \n \t// status of asynchronous requests after the notification of request completion \n \t//\n \tTRequestStatus txStatus;\n \t\n \t// Create the iTimer that is relative to the thread \n \tr = iTimer.", "CreateLocal();\n \tiTest(r==KErrNone);\n \t \t \t\n \t// Trigger iTimer expiry after KTimeOut. ", "The status should be pending\n\tiTimer.", "After(iTimeStatus,KTimeOutTxRx);\n\tiTest(iTimeStatus==KRequestPending);\n \t\n \t// Call ldd interface TransmitData() API test data descriptor as parameter\n \tr = iLdd.", "TransmitData(txStatus, KTestTxDataMedium);\n \tiTest(r==KErrNone);\n \t \t\n \t// Wait till the request is complete on txStatus or iTimeStatus. ", "User thread is \n \t// blocked with this call, till it is notified about the request completion or \n \t// iTimer expiry\n \t//\n \tUser::WaitForRequest(txStatus, iTimeStatus);\n \t\n \t// if transmit has occured correctly, the iTimeStatus will not be KErrNone, else\n \t// no transmit complete has occured and iTimer has expired\n \tiTest (iTimeStatus!=KErrNone);\n \t\n \t// Cancel the iTimer request\n \tiTimer.", "Cancel();\n \t\n \t// txStatus holds the request completion. ", "TRequestStatus::Int() returns the\n \t// completion code. ", "It will be KErrNone in case of successful completion\n \t//\n \tr = txStatus.", "Int();\n \tiTest(r==KErrNone);\n \t \t\n \t// Cancel the iTimer request\n \tiTimer.", "Cancel();\n \t\n \t// Close the handle to the iTimer\n \tiTimer.", "Close(); \t\t\n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0632\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver's asynchronous request cancelling\n//! ", "\t\t\t\t\tfunctionality as a part of testing the asynchrnous request implementation\n//! ", "@SYMTestActions 1.", "\tIssue a receive request by calling ReceiveData()\n//! ", "\t\t\t\t\t2.", "\tCancel the pending receive request\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tReceive request is pending with KRequestPending\n//! ", "\t\t\t\t\t\t2.", "\tRequest status object status is KErrCancel after cancelling the request\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Test cancelling asynchronous requests. ", "This function cancels a pending \n asynchronous receive request.", "\n \n @pre\tTestLoadDriver(), channel opened\n */\nvoid TestExDriver::TestCancelRequests()\n\t{\n\tTInt r;\n\t\n\t// Cancel Asynchronous Receive Request\n \tiTest.", "Printf(_L(\"Test Cancelling Asynchronous Receive Request\\n\"));\n \t\n \t// Create the iTimer that is relative to the thread \n \tr = iTimer.", "CreateLocal();\n \tiTest(r==KErrNone);\n \t\t\t\n\t// Create a buffer that has to be filled and returned by the driver\t\n \tTBuf8<KTestRxSize> rxBuf;\n \t\n\t// Request status object for receive. ", "This object will be used to read the \n \t// status of asynchronous requests after the notification of request completion \n \t//\n \tTRequestStatus rxStatus;\n \t \t \t \t\n \t// Trigger iTimer expiry after KTimeOutTxRx. ", "The status should be pending\n\tiTimer.", "After(iTimeStatus,KTimeOutTxRx);\n\tiTest(iTimeStatus==KRequestPending);\n\t \t \n \t// Call ldd interface ReceiveData() API to get data to RxBuf\n \tr = iLdd.", "ReceiveData(rxStatus, rxBuf);\n \tiTest(r==KErrNone);\n \t\n \t// Cancel the Receive Request, This invokes a DoCancel()\n \tiLdd.", "CancelReceive();\n \t\n \t// Wait till the request is complete on rxStatus. ", "User thread is blocked \n \t// with this call, till it is notified about the request completion.", "\n \t// \t\n \tUser::WaitForRequest(rxStatus,iTimeStatus); \t \t\n \n \t// If receive has occured correctly, the iTimeStatus will not be KErrNone,\n \t// else no receive complete has occured and iTimer has expired\n \tiTest (iTimeStatus!=KErrNone);\n \t\n \t// rxStatus holds the request completion. ", "TRequestStatus::Int() returns\n \t// the completion code. ", "It should be KErrCancel in case of successful\n \t// cancellation of request\n \t//\n \tr=rxStatus.", "Int();\n iTest(r==KErrCancel); \t\n \t\n \t// Cancel the iTimer\n \tiTimer.", "Cancel();\n \t\n // Close the handle to the iTimer\n \tiTimer.", "Close();\n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0633\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver's support for multiple asynchronous requests\n//! ", "@SYMTestActions 1.", "\tIssue a receive request by calling ReceiveData()\n//! ", "\t\t\t\t\t2.", "\tIssue another receive request by calling ReceiveData() again\n//! ", "\t\t\t\t\t3.", "\tCancel the pending requests\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tFirst receive request returns KErrNone\n//! ", "\t\t\t\t\t\t2.", "\tSecond receive request returns KErrNone\n//! ", "\t\t\t\t\t\t3.", "\tA cancel on receive should cancel both the pending requests\n//! ", "\t\t\t\t\t\t\ti.e request status should be cancel for both receive requests\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n\n/**\n Test multiple same requests\n */\t\nvoid TestExDriver::TestMultipleSameRequests()\n\t{\n\tTInt r;\n\t \n\tiTest.", "Printf(_L(\"Test Multiple same type Asynchronous Requests\\n\"));\n\t\t \t\n\t/* Test multiple Receive */\n\t\n\t// Create a buffer that has to be filled and returned by the driver\t\n \tTBuf8<KTestRxSize> rxBuf;\n \t\n\t// Request status object for receive. ", "This object will be used to read\n\t// the status of asynchronous requests after the notification of \n\t// request completion \n \t//\n \tTRequestStatus rxStatus1;\n \t\n \t// Call ldd interface ReceiveData() API to get data to rxBuf\n \tr = iLdd.", "ReceiveData(rxStatus1, rxBuf);\n \tiTest((r==KErrNone)&&(rxStatus1.Int()==KRequestPending));\n \t\n \t// Second asynchronous request object for receive\n \tTRequestStatus rxStatus2;\n \t \t\n \t// Call ldd interface ReceiveData() API to get data to rxBuf\n \tr = iLdd.", "ReceiveData(rxStatus2, rxBuf);\n \tiTest((r==KErrNone)&&(rxStatus2.Int()==KRequestPending)); \n \t\n \t// Cancel the pending requests, to cleanup this test\n \tiLdd.", "CancelReceive();\n \tiTest(rxStatus1.Int()==KErrCancel);\n \tiTest(rxStatus2.Int()==KErrCancel);\n\t}\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0634\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver's support for supporting concurrent \n//! ", "\t\t\t\t\tasynchronous requests, Tx and Rx full duplex mode\n//! ", "@SYMTestActions 1.", "\tTransmit and Receive data over device concurrently (in Full Duplex mode) with loopback\n//! ", "\t\t\t\t\t\ta.\tEnable loopback of transmit and receive\n//! ", "\t\t\t\t\t\tb.\tIssue a receive request \n//! ", "\t\t\t\t\t\tc.\tIssue a transmit request\n//! ", "\t\t\t\t\t\td.\tWait on Transmit request to complete\n//! ", "\t\t\t\t\t\te.\tWait on Receive request to complete\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tData received is same as data transmitted\n//! ", "\t\t\t\t\t\t\ta.\tInternal loopback enabled successfully\n//! ", "\t\t\t\t\t\t\tb.\tReceive completes successfully\n//! ", "\t\t\t\t\t\t\tc.\tTransmit completes successfully\n//! ", "\t\t\t\t\t\t\td.\tCompletes the wait on Transmit with KErrNone\n//! ", "\t\t\t\t\t\t\te.\tCompletes the wait on Receive with KErrNone\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Test the driver data flow path by loopback, \n i.e Transmit the data and receive same data back. ", " \n It runs the device @ default configuration\n \n @param\t\taLoopback\n \t\t\t\tLoopback mode as internal or external\n \t\t\taTxData\n \t\t\t\tTransmit Data buffer\n \t\t\taRxSize\n \t\t\t\tReceive data size \n */\t\nvoid TestExDriver::TestConcurrentTxRx(TInt aLoopback, const TDesC8& aTxData, TInt aRxSize)\n\t{\n\tTInt r;\t\n\t\n \t// Request status object for transmit and receive. ", "These object will be \n \t// used to read the status of asynchronous requests after the notification\n \t// of request completion \n \t//\n \tTRequestStatus txStatus;\n \tTRequestStatus rxStatus;\n \t\n \t// Timers and their respective status objects for Tx and Rx\n \tRTimer timerTx;\n\tRTimer timerRx;\n\tTRequestStatus timeStatusTx;\n\tTRequestStatus timeStatusRx;\n\t\n\tiTest.", "Printf(_L(\"Test Concurrent Asynchronous Requests - Tx/Rx\\n\"));\n\t\n\t// Open channel \t\n\tr=iLdd.", "Open(KUnit1);\n\tiTest(r==KErrNone);\n\t\n \t// Create a buffer that has to be filled and returned by the driver\t\t\n \tRBuf8 rxBuf; \t\n \tr=rxBuf.", "Create(aRxSize); \t\n \tiTest(r==KErrNone);\t\n\t \t\n\tr=iLdd.", "SetIntLoopback(aLoopback);\n\tiTest(r==KErrNone);\n\t \t\n \t// Create the timer that is relative to the thread \n \tr = timerTx.", "CreateLocal();\n \tiTest(r==KErrNone);\n \tr = timerRx.", "CreateLocal();\n \tiTest(r==KErrNone);\n \t \t \t \t\n \t// Trigger timerRx expiry after KTimeOutTxRx. ", "The status should be pending\n\ttimerRx.", "After(timeStatusRx,KTimeOutTxRx);\n\tiTest(timeStatusRx==KRequestPending);\n\t \t \t\t\t\n \t// Call ldd interface ReceiveData() API to get data to rxBuf\n \tr = iLdd.", "ReceiveData(rxStatus, rxBuf);\n \t// In case of zero length request\n \tif (aRxSize==0)\n \t\t{\n \t\t // Driver should return error immediately\n \t\tiTest(r!=KErrNone); \t\t\n \t\t// Close the RBuf\n \t\trxBuf.", "Close(); \t\t\n \t\t// Close channel\n\t\tiLdd.", "Close();\n \t\treturn;\n \t\t}\n \telse\n \t\t{\n \t\t// Asynchronous request should be pending, with request message\n \t\t// posted to driver successfully\n \t\t//\n \t\tiTest((r==KErrNone)&&(rxStatus.", "Int()==KRequestPending));\n \t\t}\n \t\t\n \t// Trigger timerTx expiry after KTimeOutTxRx. ", "The status should be pending \t\n\ttimerTx.", "After(timeStatusTx,KTimeOutTxRx);\n\tiTest(timeStatusTx==KRequestPending);\n\t\t\n \t// Call ldd interface TransmitData() API test data descriptor as parameter\n \tr = iLdd.", "TransmitData(txStatus, aTxData);\n \t// In case of zero length request\n \tif (aTxData.", "Size()==0)\n \t\t{\n \t\t// Driver should return error immediately\n \t\tiTest(r!=KErrNone);\n \t\t// Close the RBuf\n\t\trxBuf.", "Close();\n\t\t// Close channel\n\t\tiLdd.", "Close(); \t\n\t\treturn;\n \t\t}\n \telse\n \t\t{\n \t\t// Asynchronous request should be pending, with request message\n \t\t// posted to driver successfully\n \t\t//\n \t\tiTest(r==KErrNone);\n \t\t}\n \t\t \t \t \t\n \t// Wait till the request is complete on rxStatus and txStatus. ", "User thread\n \t// is blocked with this call, till it is notified about the request \n \t// completion.", "\n \t//\n \tif(txStatus.", "Int()==KRequestPending)\n \t\tUser::WaitForRequest(txStatus,timeStatusTx);\n \tif(rxStatus.", "Int()==KRequestPending)\n \t\tUser::WaitForRequest(rxStatus,timeStatusRx);\n \t\n \t// if transmit has occured correctly, the iTimeStatus will not be KErrNone, else\n \t// no transmit complete has occured and iTimer has expired\n \tiTest (timeStatusTx!=KErrNone);\n \tiTest (timeStatusRx!=KErrNone);\n \t\n \t// Cancel the iTimer request\n \ttimerTx.", "Cancel();\n \ttimerRx.", "Cancel();\n \t\n \t// txStatus holds the request completion. ", "TRequestStatus::Int() returns the\n \t// completion code. ", "It will be KErrNone in case of successful completion\n \t//\n \tr = txStatus.", "Int();\n \tiTest((r==KErrNone)||(r==KErrTimedOut));\n \tr = rxStatus.", "Int();\n \tiTest((r==KErrNone)||(r==KErrTimedOut));\n \t\n \t// Close the handle to the timerTx and timerRx\n \ttimerTx.", "Close();\n \ttimerRx.", "Close();\n \t\n \t// Print the recieve data to display and verify the data received manually.", "\n \tTInt i;\n \tiTest.", "Printf(_L(\"Received Data of size (%d):\"),rxBuf.", "Size());\n \tfor (i=0;i<rxBuf.", "Size();i++)\n\t \t{\n\t \tiTest.", "Printf(_L(\"%c\"),(TUint8*)(rxBuf.", "Ptr())[i]);\t\n\t \tif ((rxBuf.", "Ptr())[i] !", "= aTxData[i])\n \t\t\t{\n \t\t\tiTest.", "Printf(_L(\"Transmit and Receive data do not match\\n\"));\t\t\n \t\t\tiTest(EFalse);\n \t\t\t}\n\t \t}\n\t \t\n \tiTest.", "Printf(_L(\"\\n\"));\n \t\n \t// Free the receive buffer\n \trxBuf.", "Close();\n \t\n\t// Close channel\n\tiLdd.", "Close();\n\t}\n\n\n//----------------------------------------------------------------------------------------------\n//! ", "@SYMTestCaseID PBASE-T_EXDMA-0635\n//! ", "@SYMTestType UT\n//! ", "@SYMPREQ PREQ1212\n//! ", "@SYMTestCaseDesc This test case executes driver's Tx and Rx path for various buffer sizes\n//! ", "\t\t\t\t\tand loopback modes on all supported units\n//! ", "\t\t\t\t\tasynchronous requests, Tx and Rx full duplex mode\n//! ", "@SYMTestActions 1.", "\tTransmit and Receive Data in full duplex mode on unit1 and internal loopback with buffers of\n//! ", "\t\t\t\t\t\ta.\tTx size 0, Rx size 0 \n//! ", "\t\t\t\t\t\tb.\tTx size less than Rx size\n//! ", "\t\t\t\t\t\tc.\tTx size greater than Rx size\n//! ", "\t\t\t\t\t\td.\tTx and Rx size less than LDD frame size (256 bytes) but greater than PDD frame (64 bytes)\n//! ", "\t\t\t\t\t\te.\tTx and Rx large buffers (>3000 bytes) same size\n//! ", "\t\t\t\t\t2.", "\tRepeat step (1) on unit1 with external loopback\n//! ", "\t\t\t\t\t3.", "\tRepeat step (1) on unit2 with internal loopback\n//! ", "\t\t\t\t\t4.", "\tRepeat step (1) on unit2 with external loopback\n//!", "\n//! ", "@SYMTestExpectedResults 1.", "\tTransmit and Receive complete successfully in full duplex mode on unit1, internal loopback and\n//! ", "\t\t\t\t\t\t\ta.\tTx and Rx requests returns KErrArgument\n//! ", "\t\t\t\t\t\t\tb.\tTx and Rx are successful, returning KErrNone and complete the wait on the requests with KErrNone\n//! ", "\t\t\t\t\t\t\tc.\tTx and Rx are successful, returning KErrNone and Tx completes the wait on the requests with KErrNone and RX complete the wait on request with KErrTimedOut\n//! ", "\t\t\t\t\t\t\td.\tTx and Rx are successful, returning KErrNone and complete the wait on the requests with KErrNone\n//! ", "\t\t\t\t\t\t\te.\tTx and Rx are successful, returning KErrNone and complete the wait on the requests with KErrNone\n//! ", "\t\t\t\t\t\t2.", "\tGets results as in expected result (1) on unit1 with external loopback\n//! ", "\t\t\t\t\t\t3.", "\tGets results as in expected result in (1) on unit2 with internal loopback\n//! ", "\t\t\t\t\t\t4.", "\tDoes not continue with Tx and Rx and exit as unit2 doesn't support external loopback\n//!", "\t\t\t\t\t\t\t\n//! ", "@SYMTestPriority High\n//! ", "@SYMTestStatus Implemented\n//----------------------------------------------------------------------------------------------\n/**\n Test data flow on multiple units with loopback enable and disable\n @pre TestLoadDriver() called\n */\t\nvoid TestExDriver::TestDataflowMultipleUnitsLoopback()\n\t{\n\tiTest.", "Printf(_L(\"TestDataflow1:Unit1,Internal Loopback\\n\"));\n\t// Test data flow on unit1, internal loopback\n\tTestDataFlow(KUnit1, KIntLoopbackEnable);\n\t\n\t// Test data flow on unit1, external loopback\n\tiTest.", "Printf(_L(\"\\nTestDataflow2:Unit1,External Loopback\\n\"));\n\tiTest.", "Printf(_L(\"COM0 needs to be loopbed back externally\\n\\n\"));\n\tUser::After(2000000); // 2 secs, in units of microseconds\n\tTestDataFlow(KUnit1, KIntLoopbackDisable);\t\n\t\n\t// Test data flow on unit2, internal loopback\n\tiTest.", "Printf(_L(\"\\nTestDataflow3:Unit2,Internal Loopback\\n\"));\n\tTestDataFlow(KUnit2, KIntLoopbackEnable);\n\t\n\t// Test data flow on unit2, external loopback\n\t// external loopback is not possible on unit2 and external \n\tiTest.", "Printf(_L(\"\\nTestDataflow4:Unit2,External Loopback\\n\\n\"));\n\tTestDataFlow(KUnit2, KIntLoopbackDisable); \n\t}\n\t\t\n/**\n Test data flow for various buffer sizes\n Loopback Tx data to Rx\n \n @param\t\taUnit\n \t\t\tUnit in which this test is being done\n \t\t\taLoopback\n \t\t\tLoopback mode - internal or external\n */ \t\nvoid TestExDriver::TestDataFlow(TInt aUnit, TInt aLoopback)\n\t{\n\t// External loopback is possible only on unit1\n\tif ((aUnit!=KUnit1)&(aLoopback==KIntLoopbackDisable))\n\t\treturn;\n\t\t\n\t// case1: Tx size 0, Rx size 0\t\n\tTestConcurrentTxRx(aLoopback,KTestTxDataZero,0);\n\t\n\t// case2: Tx size less than Rx size, will result in Rx timeout \n\tTestConcurrentTxRx(aLoopback,KTestTxDataMedium,KTestTxDataMedium.iTypeLength);\n\t\n\t// case3: Tx size greater than Rx size\n\tTestConcurrentTxRx(aLoopback,KTestTxDataLarge,KTestRxSize);\n\t\t\n\t// case4: Tx and Rx size less than LDD frame size but > than PDD frame\t\n\tTestConcurrentTxRx(aLoopback,KTestTxDataSmall,KTestRxSize/10);\n\t\n\t// case5: Tx and Rx large buffers same size\n\tTestConcurrentTxRx(aLoopback,KTestTxDataLarge,KTestTxDataLarge.iTypeLength);\n\t}\n\t\n/**\n Change the debugport to port KDebugPort(3)\n \n This function is optional and specific to variant. ", "Incase of H4 variant,\n default debugport is 1. ", "However since this tutorial driver tests both \n unit1(port1) and unit2(port2), default port is being changed to port3\n to avoid conflict\n \n @post\tDebugport changed to COM port3\n */\nTInt TestExDriver::ChangeDebugPort(TInt aNewPort)\n\t{\n\tTInt muid;\n\tTInt val=-1;\n\tTInt newval;\t\n\n\t// HAL API provides interface to read and set some system attributes\n\t// specified by HALData. ", "Here, HAL::EMachineUid gives the machine uid\n\t// on which this test is being executed.", "\n\t// \t\n\tconst TInt r = HAL::Get(HAL::EMachineUid, muid);\n\tif (r==KErrNone)\n\t\t{\n\t\t// If the board is H4, we would like to change the debugport to 3\n\t\t// to enable testing unit1. ", "If unit1 is not being tested, then\n\t\t// no need to change the debugport\n\t\t//\n\t \tif (muid == HAL::EMachineUid_OmapH4)\t\n\t \t\t{\n\t \t\t// Get the deugport that is set currently using HAL API\n\t \t\tHAL::Get(HALData::EDebugPort,val);\t\n\t \t\tif (val!=aNewPort)\n\t \t\t\t{\n\t \t\t\t// Set the debugport to KDebugPort\n\t \t\t\tHAL::Set(HALData::EDebugPort,aNewPort);\t\n\t \t\t\t// Read again and verify\n\t \t\t\tHAL::Get(HALData::EDebugPort,newval);\t\n\t \t\t\t// Here, test() macro is not used to check as we dont want\n\t \t\t\t// to panic or exit if the check fails. ", "test() macro check\n\t \t\t\t// panics and exits on the condition cheked being false\n\t \t\t\t//\n\t \t\t\tif(newval!=aNewPort)\n\t \t\t\t\t{\n\t \t\t\t\t// Failed changing debugport, we'll not be able to view\n\t \t\t\t\t// log messages on the debug port, but can see on display\n\t \t\t\t\t//\n\t \t\t\t\tiTest.", "Printf(_L(\"\\nChanging DebugPort Failed\\n\"));\n\t \t\t\t\t}\t\t\t\n\t \t\t\t\t\t\n\t \t\t\t// Debugport changed to KDebugPort\n\t \t\t\tiTest.", "Printf(_L(\"\\nDebugPort Changed to (%d)\\n\"),newval);\n\t \t\t\t\t\n\t\t\t\t// Let the user know the current debug port\t\n\t\t\t\tiTest.", "Printf(_L(\"\\nPlease use COM port%d to view the debug messages on PC's hyperterminal\\n\"), newval);\n\t\n\t\t\t\t// User::After() suspends the current thread until a specified time\n\t\t\t\t// interval has expired. ", "Time is given in microseconds. ", "\n\t\t\t\t// This is added here to give the user view the above messages before\n\t\t\t\t// continuing\n\t\t\t\t//\n\t\t\t\tUser::After(2000000); // 2 secs, in units of microseconds\t\n\t \t\t\t}\n\t \t\t}\n\t\t}\n\t\n\t// return the initial debug port\n\treturn val;\n\t}\t\n\t\n//\n// End of t_exdma_user.cpp\n" ]

{ "pile_set_name": "Github" }

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[ "Q:\n\nmethod calling in C++ / number parameters not matching\n\nIn reading other programmer's C++ code I am confused when the method takes 3 parameters, but the call to the method only passes one.", "\nFor example \n.", "\n.", "\nCarList myCarList;\nread_next(myCarList);\n.", "\n.", "\nsize_t CarListReader::read_next(CarList &cl, bool theBool, size_t skip)\n\nIn java I know of method overloading based on the method signature, but I am unclear what is going on in this case in C++.", "\n\nA:\n\nThat's C++ code. ", " If you find the function's prototype (maybe in a header file), you should see something like this:\nsize_t CarListReader::read_next(CarList &cl, bool theBool = true, size_t skip = 4);\n\nThose are default arguments, and will be used if they are not supplied in the function call.", "\nYour confusion probably comes from the fact that the default values are specified in the function prototype (which may be buried in a header file somewhere), not in the function definition. ", " They will only be in the function definition if there is no prototype*).", "\n*) in which case the function definition is the prototype alias declaration.", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "The role of Glu39 in MnII binding and oxidation by manganese peroxidase from Phanerochaete chrysoporium.", "\nManganese peroxidase (MnP) is a heme-containing enzyme produced by white-rot fungi and is part of the extracellular lignin degrading system in these organisms. ", "MnP is unique among Mn binding enzymes in its ability to bind and oxidize Mn(II) and efficiently release Mn(III). ", "Initial site-directed mutagenesis studies identified the residues E35, E39, and D179 as the Mn binding ligands. ", "However, an E39D variant was recently reported to display wild-type Mn binding and rate of oxidation, calling into question the role of E39 as an Mn ligand. ", "To investigate this hypothesis, we performed computer modeling studies which indicated metal-ligand bond distances in the E39D variant and in an E35D--E39D--D179E triple variant which might allow Mn binding and oxidation. ", "To test the model, we reconstructed the E35D and E39D variants used in the previous study, as well as an E39A single variant and the E35D--E39D--D179E triple variant of MnP isozyme 1 from Phanerochaete chrysosporium. ", "We find that all of the variant proteins are impaired for Mn(II) binding (K(m) increases 20--30-fold) and Mn(II) oxidation (k(cat) decreases 50--400-fold) in both the steady state and the transient state. ", "In particular, mutation of the E39 residue in MnP decreases both Mn binding and oxidation. ", "The catalytic efficiency of the E39A variants decreased approximately 10(4)-fold, while that of the E39D variant decreased approximately 10(3)-fold. ", "Contrary to initial modeling results, the triple variant performed only as well as any of the single Mn ligand variants. ", "Interestingly, the catalytic efficiency of the triple variant decreased only 10(4)-fold, which is approximately 10(2)-fold better than that reported for the E35Q--D179N double variant. ", "These combined studies indicate that precise geometry of the Mn ligands within the Mn binding site of MnP is essential for the efficient binding, oxidation, and release of Mn by this enzyme. ", "The results clearly indicate that E39 is a Mn ligand and that mutation of this ligand decreases both Mn binding and the rate of Mn oxidation." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ " FILED\n NOT FOR PUBLICATION DEC 11 2009\n\n MOLLY C. DWYER, CLERK\n UNITED STATES COURT OF APPEALS U .S. C O U R T OF APPE ALS\n\n\n\n\n FOR THE NINTH CIRCUIT\n\n\n\nREESE P. HUGHES, No. ", "09-35032\n\n Plaintiff - Appellant, D.C. No. ", "2:08-cv-00077-JPH\n\n v.\n MEMORANDUM *\nMICHAEL J. ASTRUE, Commissioner of\nSocial Security,\n\n Defendant - Appellee.", "\n\n\n\n Appeal from the United States District Court\n for the Eastern District of Washington\n James P. Hutton, Magistrate Judge, Presiding\n\n Submitted December 9, 2009 **\n Seattle, Washington\n\nBefore: BEEZER, GOULD, and TALLMAN, Circuit Judges.", "\n\n Reese Hughes appeals the district court’s judgment affirming the\n\nCommissioner of Social Security’s final decision to deny his application for\n\nDisability Insurance Benefits and Supplemental Security Income under Titles II\n\n *\n This disposition is not appropriate for publication and is not precedent\nexcept as provided by Ninth Circuit Rule 36-3.", "\n **\n The panel unanimously finds this case suitable for decision without\noral argument. ", "See Fed. ", "R. App. ", "P. 34(a)(2).", "\n\fand XVI of the Social Security Act. ", "Because the parties are familiar with the facts\n\nand procedural history of this case, we will discuss them only as necessary to\n\nexplain our decision. ", "We have jurisdiction pursuant to 28 U.S.C. § 1291, and we\n\naffirm the judgment of the district court.", "\n\n We may set aside the Administrative Law Judge’s (“ALJ”) decision only\n\nwhen the decision is not supported by substantial evidence or is premised on legal\n\nerror. ", "Bayliss v. Barnhart, 427 F.3d 1211, 1214 n.1 (9th Cir. ", "2005). ", "The ALJ\n\nfound that, without Hughes’s drug and alcohol abuse, Hughes retained the residual\n\nfunctional capacity (“RFC”) to perform a modified range of light work. ", "At step\n\nfive, the ALJ determined that, while Hughes could not perform his past relevant\n\nwork, he could perform other jobs existing in significant numbers in the national\n\neconomy. ", "Hughes argues that (1) the ALJ impermissibly rejected opinion\n\ntestimony, (2) the hypothetical posed to the vocational expert failed to consider the\n\ntotality of Hughes’s limitations, and (3) substantial evidence does not support the\n\nALJ’s determination.", "\n\n First, after finding Hughes not entirely credible, the ALJ properly rejected\n\nopinion testimony based on Hughes’s own self-reports. ", "Hughes did not challenge\n\nthe ALJ’s credibility finding in the district court, and therefore any challenge to\n\nthat finding is waived on appeal. ", "Edlund v. Massanari, 253 F.3d 1152, 1158 (9th\n\n\n 2\n\fCir. ", "2001). ", "The ALJ also resolved inconsistencies between conflicting medical\n\nevidence within the record, and gave clear and legitimate reasons for discounting\n\nor rejecting controverted opinions. ", "The ALJ is responsible for resolving\n\nconflicting or ambiguous testimony, Thomas v. Barnhart, 278 F.3d 947, 956–57\n\n(9th Cir. ", "2002), and here she fully summarized all medical evidence presented and\n\ngave specific reasons for rejecting portions provided by some of Hughes’s treating\n\nand examining physicians, see Tommasetti v. Astrue, 533 F.3d 1035, 1041 (9th Cir.", "\n\n2008).", "\n\n Second, the ALJ presented a proper hypothetical to the vocational expert\n\nbased on her assessment of Hughes’s functional limitations. ", "Though Hughes\n\nargues for a different reading of the record, we find that the ALJ’s interpretation of\n\nthe evidence was rational and should be upheld. ", "Id. at 1038 (“The court will\n\nuphold the ALJ’s conclusion when the evidence is susceptible to more than one\n\nrational interpretation.”). ", "The hypothetical posed to the vocational expert included\n\nall limitations in the RFC determination.", "\n\n Finally, the ALJ’s findings are supported by substantial evidence. ", "Reports\n\nby Dr. Bailey, Dr. Underwood, and Ms. Sjostrom indicated that Hughes suffered\n\nfrom depression, anxiety, and personality disorders, as well as substance abuse.", "\n\nThough these evaluations state that, without substance abuse, Hughes would\n\n\n 3\n\fexhibit only mild limitations, the ALJ found that his psychological impairments\n\nwould continue to be severe. ", "While Dr. Platter’s physical RFC assessment finds\n\nHughes able to perform medium work, after considering Hughes’s obesity and\n\ndiabetes, the ALJ found him able to perform only light work. ", "Both of these\n\nfindings are more favorable to Hughes than that of the reporting physicians. ", "Based\n\non Hughes’s RFC, education, work history, and age, along with the assistance of\n\nthe vocational expert, the ALJ did not err in finding that Hughes could perform\n\nwork existing in significant numbers in the national economy.", "\n\n AFFIRMED.", "\n\n\n\n\n 4\n\f" ]

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[ "Treadmill running starting 3 months after orchidectomy restores femoral bone mass in rats.", "\nThe present study was designed to provide data on the effects on femoral bone of endurance training starting only 3 months after orchidectomy in rats. ", "A total of 70 Wistar male rats were used at 8 weeks of age. ", "On day 0 of the experiment, 10 rats were killed by cervical dislocation to be used as first controls. ", "Among the 60 other animals, half was surgically castrated (CX) or sham operated (SH). ", "On day 90, 10 CX and 10 SH were killed and used as intermediary controls (ICX and ISH). ", "Among the other 20 CX and 20 SH, 10 within each group (CXE, SHE) were selected for treadmill running (60% maximal oxygen uptake, 1 h x day(-1), 5 days x week(-1) for 12 weeks). ", "The 20 other rats were used as sedentary controls (CXR, SHR) and killed (as runners) on day 180. ", "On day 90 femoral bone density (BMD) and mineral content (BMC) were lower in ICX than in ISH. ", "On day 180 total femoral BMD was lower in CXR than in CXE. ", "Simultaneously metaphyseal femoral BMD was lower in CXR than in CXE, SHR or SHE. ", "Furthermore, at that time, no significant difference concerning BMD and BMC was observed between SHR and CXE. ", "This would indicate that treadmill running starting only 3 months after orchidectomy is able to restore BMD and BMC to control values, mainly by inhibiting bone resorption (as shown by decreased urinary deoxypyridinoline excretion in CXE) without decreasing osteoblastic activity (evaluated by plasma osteocalcin concentration)." ]

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[ "package com.illposed.osc.utility;\n\nimport java.math.", "BigInteger;\nimport java.util.", "Date;\n\nimport com.illposed.osc.*;", "\n\n/**\n * Utility class to convert a byte array conforming to the OSC byte stream format\n * into Java objects.", "\n * <p>\n * Copyright (C) 2004-2006, C. Ramakrishnan / Illposed Software.", "\n * All rights reserved.", "\n * <p>\n * See license.txt (or license.rtf) for license information.", "\n *\n * @author Chandrasekhar Ramakrishnan\n * @version 1.0\n */\npublic class OSCByteArrayToJavaConverter {\n\n\tbyte[] bytes;\n\tint bytesLength;\n\tint streamPosition;\n\n\t/**\n\t * Create a helper object for converting from a byte array to an OSCPacket object.", "\n\t */\n\tpublic OSCByteArrayToJavaConverter() {\n\t\tsuper();\n\t}\n\n\t/**\n\t * Convert a byte array into an OSCPacket (either an OSCMessage or OSCBundle).", "\n\t * @return an OSCPacket\n\t */\n\tpublic OSCPacket convert(byte[] byteArray, int bytesLength) {\n\t\tbytes = byteArray;\n\t\tthis.bytesLength = bytesLength;\n\t\tstreamPosition = 0;\n\t\tif (isBundle())\n\t\t\treturn convertBundle();\n\t\telse\n\t\t\treturn convertMessage();\n\t}\n\n\t/**\n\t * Is my byte array a bundle?", "\n\t * @return true if it the byte array is a bundle, false o.w.", "\n\t */\n\tprivate boolean isBundle() {\n\t\t\t// only need the first 7 to check if it is a bundle\n\t\tString bytesAsString = new String(bytes, 0, 7);\n\t\treturn bytesAsString.startsWith(\"#bundle\");\n\t}\n\n\t/**\n\t * Convert the byte array a bundle. ", "Assumes that the byte array is a bundle.", "\n\t * @return a bundle containing the data specified in the byte stream\n\t */\n\tprivate OSCBundle convertBundle() {\n\t\t// skip the \"#bundle \" stuff\n\t\tstreamPosition = 8;\n\t\tDate timestamp = readTimeTag();\n\t\tOSCBundle bundle = new OSCBundle(timestamp);\n\t\tOSCByteArrayToJavaConverter myConverter = new OSCByteArrayToJavaConverter();\n\t\twhile (streamPosition < bytesLength) {\n\t\t\t// recursively read through the stream and convert packets you find\n\t\t\tint packetLength = ((Integer) readInteger()).intValue();\n\t\t\tbyte[] packetBytes = new byte[packetLength];\n\t\t\tfor (int i = 0; i < packetLength; i++)\n\t\t\t\tpacketBytes[i] = bytes[streamPosition++];\n\t\t\tOSCPacket packet = myConverter.convert(packetBytes, packetLength);\n\t\t\tbundle.addPacket(packet);\n\t\t}\n\t\treturn bundle;\n\t}\n\n\t/**\n\t * Convert the byte array a simple message. ", "Assumes that the byte array is a message.", "\n\t * @return a message containing the data specified in the byte stream\n\t */\n\tprivate OSCMessage convertMessage() {\n\t\tOSCMessage message = new OSCMessage();\n\t\tmessage.setAddress(readString());\n\t\tchar[] types = readTypes();\n\t\tif (null == types) {\n\t\t\t// we are done\n\t\t\treturn message;\n\t\t}\n\t\tmoveToFourByteBoundry();\n\t\tfor (int i = 0; i < types.length; ++i) {\n\t\t\tif ('[' == types[i]) {\n\t\t\t\t// we're looking at an array -- read it in\n\t\t\t\tmessage.addArgument(readArray(types, ++i));\n\t\t\t\t// then increment i to the end of the array\n\t\t\t\twhile (']' !", "= types[i])\n\t\t\t\t\ti++;\n\t\t\t} else\n\t\t\t\tmessage.addArgument(readArgument(types[i]));\n\t\t}\n\t\treturn message;\n\t}\n\n\t/**\n\t * Read a string from the byte stream.", "\n\t * @return the next string in the byte stream\n\t */\n\tprivate String readString() {\n\t\tint strLen = lengthOfCurrentString();\n\t\tchar[] stringChars = new char[strLen];\n\t\tfor (int i = 0; i < strLen; i++)\n\t\t\tstringChars[i] = (char) bytes[streamPosition++];\n\t\tmoveToFourByteBoundry();\n\t\treturn new String(stringChars);\n\t}\n\n\t/**\n\t * Read the types of the arguments from the byte stream.", "\n\t * @return a char array with the types of the arguments\n\t */\n\tprivate char[] readTypes() {\n\t\t// the next byte should be a \",\"\n\t\tif (bytes[streamPosition] !", "= 0x2C)\n\t\t\treturn null;\n\t\tstreamPosition++;\n\t\t// find out how long the list of types is\n\t\tint typesLen = lengthOfCurrentString();\n\t\tif (0 == typesLen) {\n\t\t\treturn null;\n\t\t}\n\t\t\n\t\t// read in the types\n\t\tchar[] typesChars = new char[typesLen];\n\t\tfor (int i = 0; i < typesLen; i++) {\n\t\t\ttypesChars[i] = (char) bytes[streamPosition++];\n\t\t}\t\t\t\t\n\t\treturn typesChars;\n\t}\n\n\t/**\n\t * Read an object of the type specified by the type char.", "\n\t * @param c type of argument to read\n\t * @return a Java representation of the argument\n\t */\n\tprivate Object readArgument(char c) {\n\t\tswitch (c) {\n\t\t\tcase 'i' :\n\t\t\t\treturn readInteger();\n\t\t\tcase 'h' :\n\t\t\t\treturn readBigInteger();\n\t\t\tcase 'f' :\n\t\t\t\treturn readFloat();\n\t\t\tcase 'd' :\n\t\t\t\treturn readDouble();\n\t\t\tcase 's' :\n\t\t\t\treturn readString();\n\t\t\tcase 'c' :\n\t\t\t\treturn readChar();\n\t\t\tcase 'T' :\n\t\t\t\treturn Boolean.", "TRUE;\n\t\t\tcase 'F' :\n\t\t\t\treturn Boolean.", "FALSE;\n\t\t}\n\n\t\treturn null;\n\t}\n\n\t/**\n\t * Read a char from the byte stream.", "\n\t * @return a Character\n\t */\n\tprivate Object readChar() {\n\t\treturn new Character((char) bytes[streamPosition++]);\n\t}\n\n\t/**\n\t * Read a double &mdash; this just read a float.", "\n\t * @return a Double\n\t */\n\tprivate Object readDouble() {\n\t\treturn readFloat();\n\t}\n\n\t/**\n\t * Read a float from the byte stream.", "\n\t * @return a Float\n\t */\n\tprivate Object readFloat() {\n\t\tbyte[] floatBytes = new byte[4];\n\t\tfloatBytes[0] = bytes[streamPosition++];\n\t\tfloatBytes[1] = bytes[streamPosition++];\n\t\tfloatBytes[2] = bytes[streamPosition++];\n\t\tfloatBytes[3] = bytes[streamPosition++];\n//\t\tint floatBits =\n//\t\t\t(floatBytes[0] << 24)\n//\t\t\t\t| (floatBytes[1] << 16)\n//\t\t\t\t| (floatBytes[2] << 8)\n//\t\t\t\t| (floatBytes[3]);\n\t\tBigInteger floatBits = new BigInteger(floatBytes);\n\t\treturn new Float(Float.intBitsToFloat(floatBits.intValue()));\n\t}\n\n\t/**\n\t * Read a Big Integer (64 bit int) from the byte stream.", "\n\t * @return a BigInteger\n\t */\n\tprivate Object readBigInteger() {\n\t\tbyte[] longintBytes = new byte[8];\n\t\tlongintBytes[0] = bytes[streamPosition++];\n\t\tlongintBytes[1] = bytes[streamPosition++];\n\t\tlongintBytes[2] = bytes[streamPosition++];\n\t\tlongintBytes[3] = bytes[streamPosition++];\n\t\tlongintBytes[4] = bytes[streamPosition++];\n\t\tlongintBytes[5] = bytes[streamPosition++];\n\t\tlongintBytes[6] = bytes[streamPosition++];\n\t\tlongintBytes[7] = bytes[streamPosition++];\n\t\treturn new BigInteger(longintBytes);\n\t}\n\n\t/**\n\t * Read an Integer (32 bit int) from the byte stream.", "\n\t * @return an Integer\n\t */\n\tprivate Object readInteger() {\n\t\tbyte[] intBytes = new byte[4];\n\t\tintBytes[0] = bytes[streamPosition++];\n\t\tintBytes[1] = bytes[streamPosition++];\n\t\tintBytes[2] = bytes[streamPosition++];\n\t\tintBytes[3] = bytes[streamPosition++];\n\t\tBigInteger intBits = new BigInteger(intBytes);\n\t\treturn new Integer(intBits.intValue());\n\t}\n\t\n\t/**\n\t * Read the time tag and convert it to a Java Date object. ", "A timestamp is a 64 bit number\n\t * representing the time in NTP format. ", "The first 32 bits are seconds since 1900, the\n\t * second 32 bits are fractions of a second.", "\n\t * @return a Date\n\t */\n\tprivate Date readTimeTag() {\n\t\tbyte[] secondBytes = new byte[8];\n\t\tbyte[] fractionBytes = new byte[8];\n\t\tfor (int i = 0; i < 4; i++) {\n\t\t\t// clear the higher order 4 bytes\n\t\t\tsecondBytes[i] = 0; fractionBytes[i] = 0;\n\t\t}\n\t\t\t// while reading in the seconds & fraction, check if\n\t\t\t// this timetag has immediate semantics\n\t\tboolean isImmediate = true;\t\t\n\t\tfor (int i = 4; i < 8; i++) {\n\t\t\tsecondBytes[i] = bytes[streamPosition++];\n\t\t\tif (secondBytes[i] > 0)\n\t\t\t\tisImmediate = false;\n\t\t}\n\t\tfor (int i = 4; i < 8; i++) {\n\t\t\tfractionBytes[i] = bytes[streamPosition++];\n\t\t\tif (i < 7) {\n\t\t\t\tif (fractionBytes[i] > 0)\n\t\t\t\t\tisImmediate = false;\n\t\t\t} else {\n\t\t\t\tif (fractionBytes[i] > 1)\n\t\t\t\t\tisImmediate = false;\n\t\t\t}\n\t\t}\n\t\t\n\t\tif (isImmediate) return OSCBundle.", "TIMESTAMP_IMMEDIATE;\n\n\t\tBigInteger secsSince1900 = new BigInteger(secondBytes);\t\t\n\t\tlong secsSince1970 = secsSince1900.longValue() - OSCBundle.", "SECONDS_FROM_1900_to_1970.longValue();\n\t\tif (secsSince1970 < 0) secsSince1970 = 0; // no point maintaining times in the distant past\n\t\tlong fraction = (new BigInteger(fractionBytes).longValue());\t\n\t\t\t// the next line was cribbed from jakarta commons-net's NTP TimeStamp code\n\t\tfraction = (fraction * 1000) / 0x100000000L;\n\t\t\t// I don't where, but I'm losing 1ms somewhere...\n\t\tfraction = (fraction > 0) ? ", "fraction + 1 : 0;\n\t\tlong millisecs = (secsSince1970 * 1000) + fraction;\n\t\treturn new Date(millisecs);\n\t}\n\n\t/**\n\t * Read an array from the byte stream.", "\n\t * @param types\n\t * @param i\n\t * @return an Array\n\t */\n\tprivate Object[] readArray(char[] types, int i) {\n\t\tint arrayLen = 0;\n\t\twhile (types[i + arrayLen] !", "= ']')\n\t\t\tarrayLen++;\n\t\tObject[] array = new Object[arrayLen];\n\t\tfor (int j = 0; j < arrayLen; j++) {\n\t\t\tarray[j] = readArgument(types[i + j]);\n\t\t}\n\t\treturn array;\n\t}\n\n\t/**\n\t * Get the length of the string currently in the byte stream.", "\n\t */\n\tprivate int lengthOfCurrentString() {\n\t\tint i = 0;\n\t\twhile (bytes[streamPosition + i] !", "= 0)\n\t\t\ti++;\n\t\treturn i;\n\t}\n\n\t/**\n\t * Move to the next byte with an index in the byte array divisable by four.", "\n\t */\t\n\tprivate void moveToFourByteBoundry() {\n\t\t// If i'm already at a 4 byte boundry, I need to move to the next one\n\t\tint mod = streamPosition % 4;\n\t\tstreamPosition += (4 - mod);\n\t}\n}\n" ]

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[ "Rawat's wife, Bahuguna's son, first lady DGP among 74 nominees\n\nDehradun: Political fate of Chief Minister Harish Rawat's wife Renuka, first woman DGP Kanchan Chaudhry Bhattacharya and former chief minister Vijay Bahuguna's son Saket and 71 others will be sealed when the five Lok Sabha seats in Uttarakhand will go to polls on Wednesday.", "\n\nThere are about 70 lakh voters in the five seats - Pauri, Tehri, Nainital, Almora (SC) and Haridwar where a straight contest between BJP and Congress appears to be on the cards.", "\n\nReflecting the personality-centric nature of the electioneering, BJP Prime Ministerial candidate Narendra Modi and Rawat of Congress are emerging as 'poster boys' in the respective party campaigns.", "\n\nIn Haridwar, Rawat's prestige is at stake with his wife Renuka battling it out with former chief minister and BJP nominee Ramesh Pokhariyal Nishank. ", "AAP nominee and the country's first woman DGP Bhattacharya is also in the fray from Haridwar, represented in Lok Sabha by Rawat himself in 2009.", "\n\nPopulation wise, Haridwar is the largest constituency in Uttarakhand with more than 16 lakh voters. ", "Both Congress chief Sonia Gandhi and Modi have held impressive rallies in the district.", "\n\nThe Chief Minister has taken the entire charge of campaigning for Congress in the state holding back-to-back rallies in different areas on a daily basis.", "\n\nHe is arguing that if poll verdict is not in favour of the party it may adversely affect the stability of his government and will not augur well for the development of the state.", "\n\nAnother interesting contest is in the offing at Tehri Lok Sabha seat where the daughter-in-law of erstwhile Tehri royal family Mala Rajya Laxmi Shah of BJP and Bahuguna's son Saket of Congress are battling it out against each other for the second time.", "\n\nSaket had lost to Shah in the 2012 bypoll after his father vacated it when he took over the reigns of power in the state.", "\n\nRawat, who is campaigning single-handedly for the party elsewhere, has left the onus entirely on his predecessor to canvass for his son.", "\n\nBahuguna, who resigned after his leadership skills came under scrutiny in the wake of 2013 natural calamity, is trying to cash in on the popular image his father Hemvati Nandan Bahuguna to give Saket an edge over Shah.", "\n\nShah, who entered the Lok Sabha from the seat in October 2012 making the most of public anger in the wake of the cap on LPG cylinders imposed by the Centre, hopes to return to the Lower House riding on 'Modi wave' this time.", "\n\nThe fight for Pauri seat is also being keenly watched with former BJP chief minister Bhuvan Chandra Khanduri and popular young Congress leader Harak Singh Rawat pitted against each other.", "\n\nThough Khanduri, who has won the seat four times in the past, is a formidable opponent, Harak Singh, who represents the Rudraprayag seat in the State Assembly, enjoys a connect with people.", "\n\nIn an apparent shot in the arm for Khanduri, sitting Pauri MP and former Congress heavyweight Satpal Maharaj switched over to BJP. ", "Satpal's following both as a political and spiritual leader is also likely to boost Khanduri's chances of winning the seat.", "\n\nThe chances of former BJP chief minister Bhagat Singh Koshiyari's, who is pitted against sitting Congress MP K C Singh Baba in Nainital Lok Sabha seat, appear good as he can expect support from eight assembly segments out of 14 in the constituency that are with the saffron party. ", "57-year-old Baba is the erstwhile king of Kumaon.", "\n\nIn Almora, sitting Congress MP Pradeep Tamta is in the fray against BJP's Ajay Tamta, who is an MLA. ", "While the BJP nominee hopes to make it to the Lok Sabha propelled by the Modi wave, the Congress MP is banking on the performances of both central and state governments for his victory.", "\n\nThere are 70.43 lakh voters in Uttarakhand where 10,094 polling booths and 7,471 polling stations have been set up to conduct the exercise." ]

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[ "7 things to do to make your car last longer\n\nShocking news: Cars are expensive. ", "New or used, you’re dropping a minimum of four digits. ", "That kind of investment should be protected. ", "Luckily, there are simple things you can do to extend the life of your car. ", "Look, the difference between a ride that makes it 100,000 miles and one that pushes 200,000 could be quite a few years of driving, which means money in your pocket. ", "Here are a few easy things to do that can make your car last.", "\n\nWash it\n\nGuess what? ", "Washing and cleaning your car isn’t just about keeping it pretty. ", "A good scrubbing can help protect your car for the long haul. ", "Dirt build-up is detrimental to your car’s paint job, and that’s more important than you might think. ", "Deteriorating paint leads to the slow destruction of the body underneath. ", "Allowing the body to rust and corrode is like releasing termites in a house, as the damage can slowly spread to the most important elements. ", "While many will claim you have to wash your car every other week, we’d say a lot depends on a variety of factors. ", "Do you park outside? ", "How’s the weather where you live? ", "Do you have a long commute, during which pollen, bugs, and more are rocketed at your car? ", "Figure out if your car is more at risk and go from there. ", "For most, we’d say once a month is fine.", "\n\nYou need to change the oil … just not every 3,000 miles\n\nMost people fall in one of two categories when it comes to oil changes. ", "Either they get one every 3,000 miles like clockwork, or they push it until one of those “oh no” moments. ", "Both are foolish. ", "The latter isn’t good because you need oil changes. ", "The former is off the mark because it follows a mantra created by oil change companies trying to make a buck. ", "What you need to do is check your owner’s manual. ", "Yeah, you know—the thing in your glove box you haven’t looked at since the dealer showed you where it was. ", "It will tip you off to how often you should change your oil. ", "It’s probably in the every 5,000-mile department, but check it out. ", "If you push it too far, your oil will start to have far too much metal and dirt in it, and that can spell disaster for the most important components of your engine. ", "Your engine will become too hot and a host of problems are on the way if that happens.", "\n\nDon’t drive like you’re in the Daytona 500\n\nIf you treat your daily driver like a race car—taking turns at high speeds, braking hard, accelerating as fast as possible—you might have some fun, but it will come at the expense of your vehicle. ", "Those sharp turns are doing a number on your tires, which means balding and necessary rotation will come up quicker. ", "Braking hard leads to the deterioration of your brake pads. ", "It also puts added stress on a number of components, meaning they will have a shorter shelf life. ", "Strong acceleration isn’t the worst thing, but if you’re doing it all the time, and often while the engine is cold, you’re harming parts of the engine. ", "Oh, and that doesn’t even take into consideration the fact that you’re more likely to get in an accident, which is a quick way to keep your car from lasting forever.", "\n\nPay attention to your tires\n\nThere isn’t a ton of maintenance you’ll need to routinely be aware of, but if you think getting the oil changed is the only thing you should keep in mind, think again. ", "Your tires are the legs your car stands on, and when they’re not right, the structural integrity of the car is compromised. ", "If your tires aren’t properly inflated, not only are you at greater risk for a blowout, but you’re wearing your tires down unevenly, which means greater potential for an accident plus worse gas mileage. ", "Make sure you measure how inflated your tires are, as underinflated tires can often look like properly inflated tires. ", "But inflation isn’t the only thing to pay attention to—having your tires rotated as instructed in your owner’s manual is incredibly important as well. ", "Since tires naturally wear unevenly, if you don’t rotate your tires you’re cutting the life of those tires down and even putting your wheels and the rest of the car at greater risk every time you drive.", "\n\nPay attention to the little problems\n\nWe know how the story goes. ", "You hear a funny rattle when you’re driving your car. ", "You say to yourself, “Huh, that’s new.” ", "You say you’ll have it looked at. ", "Then, slowly, you get used to the rattle, accept it, and figure it’s no biggie if the car is still running fine. ", "One of the best things you can do for the life of your car is to nip problems in the bud. ", "Sure, that “check engine” light is probably nothing major, but get it checked. ", "All those new noises and unfamiliar things are warning signs and if you pay attention to them, your car will thank you with a longer life. ", "What’s a rattle today could be a snapped timing belt tomorrow.", "\n\nGetting your air filter changed is not a scam\n\nListen, we get it. ", "You go to one of those quickie oil change joints and feel like every additional service they try to sell you is just to fleece you out of some cash. ", "While it’s good to be skeptical, the air filter thing they’re talking about is important. ", "Dirt and debris can seriously harm your engine and chop years off its life. ", "The air filter blocks a lot of those particles from wreaking havoc. ", "However, if you don’t get the air filter changed, many will make it through. ", "It’s a cheap and easy way to keep your engine humming. ", "Check your manual but expect to change it every 15,000 – 20,000 miles or so.", "\n\nUse your parking brake\n\nWith the slow demise of the manual transmission, many have assumed the parking brake is an antiquated feature, a stick that’s the evolutionary equivalent of a human tailbone. ", "This isn’t the case. ", "When you put your transmission in park, a parking pawl in your engine locks the transmission. ", "That’s all fine and good but it’s an item that’s going to be stressed when you park on any sort of incline, and there are very few perfectly flat surfaces. ", "While rare, it can break and/or become dislodged. ", "A parking brake is a far more secure option, and it’s one that will keep your car from rolling away if the pawl fails or if your car is hit. ", "Also, if you don’t use the parking brake, the cables associated with it can age poorly and snap when you need them most. ", "Just make sure to disengage it when you start driving or you’ll end up with more problems to deal with.", "\n\nWas this article helpful?", "\n\nMike Newman, Cool Material\n\nManaging editor of Cool Material, a men’s lifestyle site dedicated to great gear. ", "What started as a small hobby blog has grown into a destination for insightful articles and original photography." ]

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[ "Foetz\n\nFoetz () is a small town located in the commune of Mondercange, in south-western Luxembourg. , ", "it has a population of 519 inhabitants.", "\n\nReferences\n\nCategory:Towns in Luxembourg\nCategory:Mondercange" ]

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[ "(Within our archive we collect a lot of software including: Speaking Notepad: I'll read your documents aloud with quality voices and record them into MP3 NHMTTS SAPI5: SAPI5-compliant Vietnamese text to speech engine Sounding Keyboard and Mouse: Lets you hear magic sound, music or speach when you press keys or click mouse ; these are very useful for tts engine)" ]

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[ "Sensitivity analysis of stochastic attractors and noise-induced transitions for population model with Allee effect.", "\nWe study a stochastically forced predator-prey model with Allee effect. ", "In the deterministic case, this model exhibits non-trivial stable equilibrium or limit cycle corresponding to the coexistence of both species. ", "Computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of the dispersion of random states in stochastic attractors. ", "Our method allows to construct confidence domains and estimate the threshold value of the intensity for noise generating a transition from the coexistence to the extinction." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "function res = ctranspose(obj,useconj);\n if nargin == 1\n useconj = 1;\n end\n\n if length(obj.dimensions) > 2\n error('Cannot transpose array');\n end\n \n % make new memory mapped data file\n % --------------------------------\n newFileName = mmo.getnewfilename;\n copyfile(obj.dataFile, newFileName);\n \n res = obj;\n res.dimensions = [ obj.dimensions(2) obj.dimensions(1) ];\n res.dataFile = newFileName;\n tmpMMO1 = memmapfile(obj.dataFile, 'writable', obj.writable, 'format', { 'single' obj.dimensions 'x' });\n tmpMMO2 = memmapfile(res.dataFile, 'writable', true, 'format', { 'single' res.dimensions 'x' });\n \n % copy the data\n % -------------\n if length(obj.dimensions) == 1 || obj.dimensions(1) > obj.dimensions(2)\n for index = 1:size(obj,2)\n s.type = '()';\n s.subs = { ':' index };\n if useconj, tmpMMO2.Data.x(index,:) = conj(subsref(tmpMMO1.Data.x,s));\n else tmpMMO2.Data.x(:,index) = subsref(tmpMMO1.Data.x,s);\n end\n end\n else\n for index = 1:size(obj,1)\n s.type = '()';\n s.subs = { index ':' };\n if useconj, tmpMMO2.Data.x(:,index) = conj(subsref(tmpMMO1.Data.x,s));\n else tmpMMO2.Data.x(:,index) = subsref(tmpMMO1.Data.x,s);\n end\n end\n end\n \n" ]

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[ "Q:\n\nHow can I import the System.", "Linq namespace to Boo?", "\n\nWhen I try to import the System.", "Linq namespace to Boo compiler, I get this error:\n\nBoo.", "Lang.", "Compiler.", "CompilerError: \nNamespace 'System.", "Linq' not found, maybe you forgot to add an assembly reference?", "\n\nI use \"Rhino.DSL.dll\" and my DSL engine code is here:\npublic class MyDslEngine : DslEngine\n{\n protected override void CustomizeCompiler(BooCompiler compiler, CompilerPipeline pipeline, string[] urls)\n {\n pipeline.", "Insert(1, new AnonymousBaseClassCompilerStep(typeof(DslBase), \"Prepare\",\n \"System.", "Linq\",\n \"Azarakhsh.", "Framework.", "Repository\" //it's my repository framework\n ));\n pipeline.", "Insert(2, new UseSymbolsStep());\n pipeline.", "Insert(3, new RunScriptCompilerStep());\n }\n}\n\nA:\n\nTry adding a reference to the System.", "Core assembly to your project. ", "Most of the classes in the System.", "Linq namespace are found in that assembly.", "\nIf that doesn't work, you might also try adding a reference to System.", "Data.", "Linq.", "\nAnd in the future, don't underestimate the usefulness of the error messages provided by the compiler. ", "Yes, sometimes they are cryptic and other times they are even misleading. ", "But they're certainly a good place to start when you're trying to figure out why something won't compile that you expected to work.", "\n\nA:\n\nWhy do you need System.", "Linq in your DSL? ", "Sytem.", "Linq must be \"hidden\" in your framework. ", "Besides using Linq in Boo, it's kinda of verbose (in my opinion) and your DSL should hide this verbose stuff...\nimport System.", "Linq.", "Enumerable from System.", "Core\nbar = List of string() \nbar.", "Add(\"foo\")\nbar.", "Add(\"baz\")\n\nbaz = bar.", "Where({x as string | x ==\"baz\"}).Single()\n\nAbout using the System.", "Linq, haven't tried but I found this link Boo Markmail, where the code above was copied...\n\n" ]

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[ "An electrifying iPSC disease model: long QT syndrome type 2 and heart cells in a dish.", "\nIn a recent Nature paper, Itzhaki et al. (", "2011) generate induced pluripotent stem cells (iPSCs) from patients with a potentially fatal inherited arrhythmia, long QT syndrome type 2. ", "Cardiomyocytes differentiated from the patient-derived iPSCs exhibit the hallmark cardiac electrical signatures of the disease, which can be reversed by pharmacological intervention." ]

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[ "Typically, it is difficult to determine how safe a route is that is taken while driving from point A to point B. Even more so in the event that a driver has no experience traveling the route." ]

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[ "Glycated albumin indicates peripheral diabetic neuropathy.", "\nWe investigated associations between serum levels of glycated albumin (GA) and glycated hemoglobin (HbA1c) and the presence of diabetic peripheral neuropathy (DPN) in patients with type 1 diabetes mellitus (T1DM). ", "Between September 2009 and April 2015, we evaluated 314 patients with T1DM in the Endocrinology Department of Shengjing Hospital of China Medical University. ", "We divided the patients into the DPN group (n = 72) and the non-DPN group (n = 242), on the basis of the presence of DPN. ", "The DPN group had significantly higher GA values than the non-DPN group. ", "After univariate logistic regression, we selected several factors for further analysis: HbA1c, GA, duration of T1DM, body mass index, smoking, hypertension, and the presence of diabetic complications, including nephropathy, retinopathy, and cardiovascular disease. ", "We performed a multivariate logistic regression analysis to examine the association between the presence of DPN and each of these variables. ", "We identified GA, HbA1c, hypertension, smoking, retinopathy, and cardiovascular disease as independent variables for indicating the presence of DPN. ", "Results of a receiver operating characteristic curve analysis revealed that the area under the curve of GA (0.771) was larger than that of HbA1c (0.629). ", "We defined the cutoff value of GA as 23.5 % (sensitivity 0.764, specificity 0.661) and the cutoff value of HbA1c as 8.45 % (sensitivity 0.667, specificity 0.595) for predicting DPN in patients with T1DM. ", "GA may be a better indicative marker of DPN in patients with T1DM than HbA1c." ]

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[ "Experimental evolution of an essential Bacillus gene in an E. coli host.", "\nThe acquisition of foreign genes by HGT potentially greatly speeds up adaptation by allowing faster evolution of beneficial traits. ", "The evolutionary integration of novel genes into host gene expression and physiology is critical for adaptation by HGT, but remains largely unknown. ", "We are exploring the evolutionary consequences of gene acquisition in populations of Escherichia coli in real time. ", "A plasmid bearing the genes necessary for sucrose catabolism was constructed and introduced into a single E. coli genotype. ", "Wild-type E. coli is generally incapable of utilizing sucrose, but E. coli transformants were able to grow on sucrose as a sole carbon and energy source, albeit poorly. ", "Twelve replicate populations were initiated and propagated in sucrose minimal media for 300 generations. ", "Over this time, we observed large fitness improvements in the selected environment. ", "These results demonstrate the potential for HGT to substantially increase microbial niche breadth." ]

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[ "The article by Melo et al.[@B1], published in the volume 26 of this journal showed that music therapy, one of the Integrative and Complementary Practices (PIC - *Práticas Integrativas e Complementares*) recently implemented in the Brazilian Unified Health System (SUS - *Sistema Único de Saúde*)[@B2], was significant in the reduction of anxiety in chronic patients. ", "Regarding the nursing care with music therapy, the amount of research is rather limited; however, the use of music therapy in Nursing, as well as other PIC, have been encouraged both nationally and internationally[@B1] ^,^ [@B3].", "\n\nDespite the local and global recommendation to offer PIC in national health systems[@B4], the topic is still quite controversial, with defenders and detractors. ", "In Brazil, since 1985, documents, resolutions and events marked the regulation process of those practices, and, in 2006, after an intense effort of different individual and institutional agents, the National Policy on Integrative and Complementary Practices (PNPIC - *Política Nacional de Práticas Integrativas e Complementares*)[@B5] was published. ", "Before its publication, a situational diagnosis was conducted on the PIC offered by SUS in Brazilian municipalities to identify the most common and viable practices. ", "Thus, in 2006, the Homeopathic Medical Rationality, the Traditional Chinese/Acupuncture Medical Rationality, the Anthroposophical Medical Rationality, Phytotherapy/medicinal plants and Thermalism became part of SUS.", "\n\nThroughout those 12 years, the institutionalization process of PNPIC was hampered by the lack of an official political national coordination in the Ministry of Health and by the lack of budget allocation to its deployment and implementation. ", "On March 12, 2018, the Minister of Health created the national coordination of PNPIC under the Primary Health Care Directorate of the Ministry of Health. ", "It is a historical achievement, although there is little to celebrate, as until now the national coordination was not regulated and there are many information gaps about, for example, the process of inclusion of fourteen Practices in March 2017[@B2] and the inclusion of other 10 Practices in March 2018.", "\n\nIt is known that data from the National Primary Care Access and Quality Improvement Program (PMAQ - *Programa Nacional de Melhoria do Acesso e de Qualidade da Atenção Básica*) are mentioned to justify the implantation of those 24 practices; however, there is no new policy in the current edition of PNPIC[@B5] ^)^ that details the recommended criteria in the *Estrategia de la OMS sobre medicina tradicional 2014-2023* [@B4] ^).^ The challenges of implementing PIC in health systems are not exclusively Brazilian, and a research performed in 39 European countries showed that 70% of them do not have specific regulation for PIC[@B6].", "\n\nThe commitment to the therapeutic plurality of SUS and its safe, effective, and quality offer, lead us to question the form and content of the implementation of new practices in SUS. ", "The alleged recent progress of PNPIC seems more like a threat to PIC and, in this context coated with challenges, it is observed that the political and scientific need to discuss the ingoing implementation and the relevance of PIC as a integrative care model in SUS.", "\n" ]

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[ "We’re used to making little adjustments all the time. ", "Most commonly, it’s because of ballpark environment. ", "A .350 wOBA in San Francisco is a hell of a lot more valuable than a .350 wOBA in Arizona. ", "Sometimes you’ll also see adjustments for era, which is relevant now given increasing strikeouts and decreasing runs. ", "There are raw stats, and there are adjusted stats, like, say, wRC+, but there’s one adjustment we seldom talk about even though it’s right there in front of our faces. ", "What about the opposition a player actually faces?", "\n\nIt’s like strength-of-schedule, on the player level. ", "No one debates the utility of strength-of-schedule measurements. ", "Now, in baseball, what’s convenient is that the samples get pretty big so we can generally get away with assuming that things even out. ", "Over broad windows, no one’s going to face exclusively awful opponents or awesome opponents. ", "But in certain cases, it’s worth digging in when we have a suspicion. ", "As such, I want to go into more detail on something I noted about Cole Hamels earlier.", "\n\nJust for the sake of easy comparison, let’s line Hamels up with Jon Lester, Max Scherzer, and James Shields. ", "Three of those are free agents, but Hamels is also highly likely to move, perhaps to someone who misses out on the FA market. ", "This is a table of WAR values, where I’ve taken regular WAR and RA9-WAR and averaged them 50/50. ", "Except for the Steamer line. ", "That’s just regular WAR, because that’s what’s projected.", "\n\nYear Hamels Lester Scherzer Shields 2012-2014 13.2 12.3 15.8 12.6 2014 4.9 6.0 5.5 3.8 2015, Steamer/200 2.7 3.5 3.9 3.0 2015 Age 31 31 30 33\n\nHamels, Lester, and Scherzer are of similar ages, with Shields a little older. ", "Over the last three years, Scherzer’s been the best, with the others more or less equal. ", "In just this past season, Lester was awesome, and then Scherzer was a little less awesome, and Hamels was a little less awesome, and Shields was still less awesome. ", "The projected order goes Scherzer — Lester — Shields — Hamels, but then those are just projections, right? ", "Hamels, for a long time, has been really good. ", "No one questions that he’s a good pitcher.", "\n\nThis is around where we usually stop. ", "But I wanted to investigate Hamels’ quality of opposition, compared to the other guys. ", "Hamels has lived in the National League East. ", "The others have all lived in the American League. ", "I went over the last three years, and matched each hitter seen to the same hitter’s single-season wRC+. ", "Then I weighted everything to come up with an average hitter wRC+ faced. ", "This is the best quality-of-opposition measure I can come up with in a day. ", "And the results, I’d say, are revealing:\n\nYear Hamels Lester Scherzer Shields 2012 90 97 100 102 2013 88 100 96 98 2014 89 98 100 103 Average 89 98 99 101 Hitter Comp Everth Cabrera Alex Avila Brian McCann Eric Hosmer\n\nYou’ve got single-season average wRC+, then you have the average of the last three seasons. ", "In the last line, you see a hitter who has posted the same wRC+ over the past three years. ", "On average, since 2012, Cole Hamels has faced a bunch of Everth Cabreras. ", "James Shields has faced a bunch of Eric Hosmers. ", "There’s little difference between the AL guys, but Hamels is far removed. ", "You have to assume this has worked to Hamels’ benefit.", "\n\nHow might one be able to adjust for this? ", "It’s easy, after all, to say that Hamels has faced weaker competition, but, what does that mean? ", "It’s time to try something. ", "Over the past three years, Hamels has allowed a .293 wOBA. ", "His batting opponents have averaged a .302 combined wOBA. ", "I decided to work through the odds ratio method backwards. ", "I wound up with a value of .304 — that is, based on the math, Hamels’ “true-talent wOBA” over the past three years would be .304. ", "Or, 11 points higher than what we observe, based on his results.", "\n\nOf course, that’s still a good mark. ", "Still a better mark than average. ", "But it does make a real difference. ", "There are a couple ways to figure out the impact. ", "For example, over the past three years, Hamels has averaged a 4.4 RA9-WAR per 200 innings. ", "Pitchers with wOBAs right around .304, however, have averaged a 3.2 RA9-WAR per 200 innings. ", "Alternatively, you can just calculate the difference between a .293 wOBA and a .304 wOBA over Hamels’ 2,601 plate appearances. ", "That comes out to about 21 runs. ", "Or, an average of almost a win a year.", "\n\nThat would all be a quality-of-opposition adjustment for Cole Hamels. ", "He’s faced relatively weak opponents. ", "If you retroactively have him face roughly average opponents, you’d expect him to be almost a win worse a year. ", "He’s still better than average, by a decent amount, but the gap shrinks. ", "To whatever extent the usual numbers favor Hamels over James Shields, Shields has faced far tougher opponents. ", "So interested teams don’t need to worry about adjusting him for that. ", "For Hamels, I think it’s necessary.", "\n\nThis obviously all gets complicated, and well removed from the field of play. ", "On the field of play, Hamels has the same fastball as ever, and a really fantastic changeup. ", "The quality of his stuff is independent of the hitters he faces. ", "It’s not because of the hitters that he’s been durable. ", "It’s not because of the hitters that he’s got playoff experience. ", "Hamels is good enough to pitch anywhere, and he’s recognized as a front-of-the-line kind of starter. ", "There’s a reason teams are interested in him.", "\n\nBut at the end of the day, expensive players get evaluated mostly by how they’ve done. ", "The numbers drive the contracts and the trades, and the numbers get adjusted by any half-smart organization. ", "Everybody understands ballpark effects. ", "There are also, sometimes, opponent effects. ", "Adjusting for the latter makes Cole Hamels look less valuable than his actual numbers, and that’s just one of the reasons why it’s proving so difficult for Ruben Amaro to find a deal to his liking. ", "The Red Sox don’t care so much about how Hamels might do in Philadelphia. ", "They care about how Hamels might do in Boston." ]

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[ "In advance of the Brexit vote, Republican and conservative leaders chose their words carefully. ", "It was easy to criticize President Obama for his unusual transatlantic campaigning in favor of Brexit; plenty of Republicans criticized him for saying Britain would head to \"the back of the queue\" in trade deals if it quit Europe.", "\n\nToday, while stock markets careened and media coverage has asked whether British voters just sparked a \"DIY recession,\" a few conservatives have embraced the vote. ", "Sen. Ted Cruz (R-Tex.), ", "one of the critics of the president's remarks, told supporters on Facebook that Americans needed to heed Brexit.", "\n\nAD\n\nAD\n\n\"The results of the ‪#‎Brexit‬ referendum should serve as a wake-up call for internationalist bureaucrats from Brussels to Washington, D.C. that some free nations still wish to preserve their national sovereignty,\" Cruz wrote. \"", "The British people have indicated that they will no longer outsource their future to the EU, and prefer to chart their own path forward. ", "The United States can learn from the referendum and attend to the issues of security, immigration and economic autonomy that drove this historic vote.\"", "\n\nSen. Jeff Sessions (R-Ala.), an occasional Cruz ally who has become the Senate's biggest booster of Donald Trump, had an even more supportive reaction.", "\n\n\"The people spoke from their hearts and with conviction,\" Sessions said in a statement. \"", "They considered deep and critical issues never discussed by the international elites. ", "Their strong vote arose not out of fear and pique but out of love for country and pride of place. ", "Their experience with a distant government in Brussels was given a long and fair chance to succeed. ", "In the end, however, they concluded that the costs outweighed the benefits.\"", "\n\nAD\n\nAD\n\nSessions compared the formation of the E.U. (\"a seemingly benign economic agreement\") to the Trans-Pacific Partnership, a trade deal opposed by protectionist Republicans and by the pro-labor left.", "\n\n\"Often, Britain makes changes that precede U.S. action,\" Sessions said. \"", "The Thatcher movement preceded the Reagan revolution. ", "Both were victories for the people over outdated and corrupt forces. ", "Both were achieved against powerful and determined establishment forces. ", "Both resulted in historic and positive periods in their nation’s history.\"" ]

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[ "# Licensed to Elasticsearch B.V. under one or more contributor\n# license agreements. ", "See the NOTICE file distributed with\n# this work for additional information regarding copyright\n# ownership. ", "Elasticsearch B.V. licenses this file to you under\n# the Apache License, Version 2.0 (the \"License\"); you may\n# not use this file except in compliance with the License.", "\n# You may obtain a copy of the License at\n#\n# http://www.apache.org/licenses/LICENSE-2.0\n#\n# Unless required by applicable law or agreed to in writing,\n# software distributed under the License is distributed on an\n# \"AS IS\" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY\n# KIND, either express or implied. ", " See the License for the\n# specific language governing permissions and limitations\n# under the License.", "\n\n# frozen_string_literal: true\n\nmodule ElasticAPM\n # @api private\n module Spies\n # @api private\n class ElasticsearchSpy\n NAME_FORMAT = '%s %s'\n TYPE = 'db'\n SUBTYPE = 'elasticsearch'\n\n def self.sanitizer\n @sanitizer ||=\n begin\n config = ElasticAPM.agent.config\n ElasticAPM::Transport::Filters::HashSanitizer.new(\n key_patterns: config.custom_key_filters + config.sanitize_field_names\n )\n end\n end\n\n def install\n ::Elasticsearch::Transport::Client.class_eval do\n alias perform_request_without_apm perform_request\n\n def perform_request(method, path, *args, &block)\n unless ElasticAPM.current_transaction\n return perform_request_without_apm(method, path, *args, &block)\n end\n\n name = format(NAME_FORMAT, method, path)\n statement = []\n\n statement << { params: args&.[](0) }\n\n if ElasticAPM.agent.config.capture_elasticsearch_queries\n unless args[1].nil? ", "|| args[1].empty?", "\n statement << {\n body: ElasticAPM::Spies::ElasticsearchSpy.sanitizer.strip_from!(args[1])\n }\n end\n end\n\n context = Span::Context.new(\n db: { statement: statement.reduce({}, :merge).to_json },\n destination: {\n name: SUBTYPE,\n resource: SUBTYPE,\n type: TYPE\n }\n )\n\n ElasticAPM.with_span(\n name,\n TYPE,\n subtype: SUBTYPE,\n context: context\n ) { perform_request_without_apm(method, path, *args, &block) }\n end\n end\n end\n end\n\n register(\n 'Elasticsearch::Transport::Client',\n 'elasticsearch-transport',\n ElasticsearchSpy.new\n )\n end\nend\n" ]

{ "pile_set_name": "Github" }

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[ "Q:\n\nPython JSON schema validation for array of objects\n\nI am trying to validate a JSON file using the schema listed below, I can enter any additional fields, I don't understand, what I am doing wrong and why please?", "\nSample JSON Data\n{\n \"npcs\":\n [\n {\n \"id\": 0,\n \"name\": \"Pilot Alpha\",\n \"isNPC\": true,\n \"race\": \"1e\",\n \"testNotValid\": false\n },\n {\n \"id\": 1,\n \"name\": \"Pilot Beta\",\n \"isNPC\": true,\n \"race\": 1\n }\n ]\n}\n\nJSON Schema\nI have set \"required\" and \"additionalProperties\" so I thought the validation would fail....\nFileSchema = {\n \"definitions\":\n {\n \"NpcEntry\":\n {\n \"properties\":\n {\n \"id\": { \"type\": \"integer\" },\n \"name\": { \"type\" : \"string\" },\n \"isNPC\": { \"type\": \"boolean\" },\n \"race\": { \"type\" : \"integer\" }\n },\n \"required\": [ \"id\", \"name\", \"isNPC\", \"race\" ],\n \"additionalProperties\": False\n }\n },\n\n \"type\": \"object\",\n \"required\": [ \"npcs\" ],\n \"additionalProperties\": False,\n \"properties\": \n {\n \"npcs\":\n {\n \"type\": \"array\",\n \"npcs\": { \"$ref\": \"#/definitions/NpcEntry\" }\n }\n }\n}\n\nThe JSON file and schema are processed using the jsonschema package for Python, (I am using python 3.7 on a Mac).", "\nThe method I use to read and validate is below, I have removed a lot of the general validation to make the code as short and usable as possible:\nimport json\nimport jsonschema\n\ndef _ReadJsonfile(self, filename, schemaSystem, fileType):\n\n with open(filename) as fileHandle:\n fileContents = fileHandle.read()\n \n jsonData = json.loads(fileContents)\n\n try:\n jsonschema.validate(instance=jsonData, schema=schemaSystem)\n\n except jsonschema.exceptions.", "ValidationError as ex:\n print(f\"JSON schema validation failed for file '{filename}'\")\n return None\n\n return jsonData\n\nA:\n\nat: \"npcs\": { \"$ref\": \"#/definitions/NpcEntry\" }\nchange \"npcs\" to \"items\". ", "npcs is not a valid keyword so it is ignored. ", "The only validation that is happening is at the top level, verifying that the data is an object and that the one property is an array.", "\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "An out-of-print collection of essays about corporate America first published in 1969 has shot to the top of bestseller lists after Bill Gates revealed it was his favourite business book.", "\n\nJohn Brooks's Business Adventures is a collection of the late journalist's New Yorker articles from the 1960s, covering topics from the rise of Xerox and the $350m Edsel disaster to scandals at GE and Texas Gulf Sulphur. ", "Gates wrote in the Wall Street Journal this weekend that the long out-of-print title \"remains the best business book I've ever read\".", "\n\nHis praise galvanised other readers to get their hands on the book: Abebooks reported this weekend that on Friday and Saturday, Business Adventures by John Brooks was the top search term on its site. \"", "This morning (Sunday), we have no copies left on the site,\" said spokesman Richard Davies.", "\n\nGates's praise – and readers' subsequent interest – also led to the book being brought back into print. \"", "I was very pleased to learn Bill Gates was a fan of my father's book,\" the late author's son, Alex Brooks, told business news site Quartz . \"", "That's kind of great news so we called my father's agent, and asked them if we could get an ebook ready in a hurry. ", "They got in touch with the publisher Open Road, and Open Road seemed to think it was a good opportunity and jumped right in.\"", "\n\nOn Amazon.com, Open Road's new edition of the book is currently number five in the retailer's overall books bestseller list, ahead of Gillian Flynn's Gone Girl and Donna Tartt's The Goldfinch, and this summer's smash hit, Thomas Piketty's Capital. ", "The publisher will release a paperback edition in September, after putting out an ebook last week.", "\n\n\"It's certainly true that many of the particulars of business have changed. ", "But the fundamentals have not. ", "Brooks's deeper insights about business are just as relevant today as they were back then,\" wrote Gates of Business Adventures. \"", "Unlike a lot of today's business writers, Brooks didn't boil his work down into pat how-to lessons or simplistic explanations for success. (", "How many times have you read that some company is taking off because they give their employees free lunch?) ", "You won't find any listicles in his work. ", "Brooks wrote long articles that frame an issue, explore it in depth, introduce a few compelling characters and show how things went for them.\"", "\n\nGates highlights Brooks's coverage of Ford's Edsel fiasco, saying the journalist \"refutes the popular explanations for why Ford's flagship car was such a historic flop. ", "It wasn't because the car was overly poll-tested; it was because Ford's executives only pretended to be acting on what the polls said. '", "Although the Edsel was supposed to be advertised, and otherwise promoted, strictly on the basis of preferences expressed in polls, some old-fashioned snake-oil selling methods, intuitive rather than scientific, crept in.' ", "It certainly didn't help that the first Edsels 'were delivered with oil leaks, sticking hoods, trunks that wouldn't open and push buttons that … couldn't be budged with a hammer.'\"", "\n\nAlex Brooks told Quartz that \"I think he was one of the first to consider business journalism as a sort of topic for just general popular readership. ", "I think mostly before he came along business journalism was written for businessmen. ", "The idea of telling business stories as just kind of entertaining pieces of reading was a real innovation.\"", "\n\nGates, meanwhile, revealed that he first picked up the title after stock-market maven Warren Buffett lent it to him, not long after the pair met in 1991. \"", "Warren, if you're reading this, I still have your copy,\" writes Gates." ]

{ "pile_set_name": "Pile-CC" }

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[ "Comfort Woman\n\nA comfort woman was a woman forced into sexual slavery by the Imperial Japanese Army in occupied territories before and during World War II.", "\n\nComfort Woman may also refer to:\n\n Comfort Woman (album) by Me'shell Ndegeocello\n Comfort Woman (novel) by Nora Okja Keller" ]

{ "pile_set_name": "Wikipedia (en)" }

[ 0.0064516129032258064, 0.024 ]

[ "Transition metal free hydrolysis/cyclization strategy in a single pot: synthesis of fused furo N-heterocycles of pharmacological interest.", "\nA transition metal free tandem two-step strategy has been developed involving hydrolysis of 2-chloro-3-alkynyl quinoxalines/pyrazines followed by in situ cyclization of the corresponding 2-hydroxy-3-alkynyl intermediates in a single pot leading to fused furo N-heterocycles as potential inhibitors of sirtuins. ", "A representative compound showed promising pharmacological properties in vitro and in vivo." ]

{ "pile_set_name": "PubMed Abstracts" }

[ 0, 0, 0 ]

[ "Ash, Devon\n\nAsh is a village in Devon, England and is situated approximately south-west of Dartmouth.", "\n\nReferences\n\nExternal links\n\nCategory:Villages in Devon" ]

{ "pile_set_name": "Wikipedia (en)" }

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[ "皆様、こんにちは！むらちゃんです(*ﾟ∀ﾟ)！", "\n\n本日は書籍のご紹介です！", "\n\n大人気楽曲「リンちゃんなう！」を題材にした書籍の続編、「リンちゃんなう！SSs2」が本日発売致しました♪\n\n「リンちゃんに○○したい！」でおなじみの大人気楽曲「リンちゃんなう！」をベースにしたオリジナル短編小説集の第二巻です！", "\n\nリンちゃんが同級生や頼りになるお姉ちゃんなどなど、色んなリンちゃんになって登場！", "\n\nさまざまなリンちゃんの物語がぎゅっと詰まっています★\n\n執筆をsezuさん、監修をガルナ（オワタP）さん、イラストを田村ヒロさんが担当し、「リンちゃんなう！」楽曲制作メンバー勢ぞろいでお届けする一冊！", "\n\n小説「リンちゃんなう！SSs2」は本日10月2日（金）より、1,296円（税込）で発売！", "\n\nリンちゃん愛がいっぱい詰まった一冊なので、是非皆様お見逃しなく∠(｀・ω・´)\n\n楽曲も一緒に聞いて、リンちゃんへの愛を高めましょう！", "\n\n<a href=\"http://www.nicovideo.jp/watch/sm16539814\">【ニコニコ動画】【初音ミク・巡音ルカ】リンちゃんなう！【鏡音生誕祭2011】</a>\n\n(^ω^三^ω^)\n\nでは、またお会いしましょう！むらちゃんでした！" ]

{ "pile_set_name": "OpenWebText2" }

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[ "archiveclearly\n\nThe bottom line is, negatively. ", "Precisely what are GM foods? ", "Individuals foods whose DNA is modified using the idea to change on preferred traits or turn off undesired people that are injected with poisons to allegedly reduce pesticide use. ", "We have have have have been told by GM food producers..." ]

{ "pile_set_name": "Pile-CC" }

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[ "//----------------------------------------------------------------------------------------------\n//\n//\tQUIT MENU\n//\n//----------------------------------------------------------------------------------------------\n{\n\tmenuDef \n\t{\n\t\tname\t\t\t\t\t\"quitMenu\"\n\t\tvisible\t\t\t\t\t0\n\t\tfullScreen\t\t\t\t1\t\n\t\trect\t\t\t\t\t0 0 640 480\n\t\tfocusColor\t\t\t\t1 1 1 1\t\n\t\tdescX\t\t\t\t\t320\n\t\tdescY\t\t\t\t\t426\n\t\tdescScale\t\t\t\t1\n\t\tdescColor\t\t\t\t1 .682 0 .8\n\t\tdescAlignment\t\t\t\tITEM_ALIGN_CENTER\t\t\n\n\t\tonESC \n\t\t{ \n\t\t\tplay\t\t\t\t\"sound/interface/esc.wav\" \t\t\t\t\t\n\t\t\tclose\t\t\t\tquitMenu ; \n\t\t\topen\t\t\t\tmainMenu ;\n\t\t}\n\n\t\tonOpen \n\t\t{\n\t\t\tsetfocus\t\t\tquitgame_cancel\n\t\t}\n\n//----------------------------------------------------------------------------------------------\n//\tMENU BACKGROUND\n//----------------------------------------------------------------------------------------------\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\treally_background\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t156 154 320 240\n\t\t\tbackground\t\t\t\"gfx/menus/main_centerblue\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t1\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tbackground_text\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t0 0 160 480\n\t\t\tbackground\t\t\t\"gfx/menus/menu_side_text\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t1\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tbackground_text_b\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t480 0 160 480\n\t\t\tbackground\t\t\t\"gfx/menus/menu_side_text_right\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t1\n\t\t\tdecoration\n\t\t}\t\n\t\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tbackground\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t0 0 640 480\n\t\t\tbackground\t\t\t\"gfx/menus/main_background\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t1\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tstarwars\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t107 8 428 112\n\t\t\tbackground\t\t\t\"gfx/menus/jediacademy\"\t\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t1\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tleft_frame\n\t\t\tgroup\t\t\t\tlf_fr\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t0 50 320 160\n\t\t\tbackground\t\t\t\"gfx/menus/menu_boxes_left\"\t\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t1\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tright_frame\n\t\t\tgroup\t\t\t\trt_fr\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t320 50 320 160\n\t\t\tbackground\t\t\t\"gfx/menus/menu_boxes_right\"\t\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t1\n\t\t\tdecoration\n\t\t}\n\n//----------------------------------------------------------------------------------------------\n//\tTOP MAIN MENU BUTTONS\n//----------------------------------------------------------------------------------------------\n\t\t// Big button \"NEW\"\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tnewbutton_glow\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t7 126 130 24\n\t\t\tbackground\t\t\t\"gfx/menus/menu_buttonback\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t0\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tnewgamebutton\n\t\t\tgroup\t\t\t\tnbut\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_EMPTY\n\t\t\ttype\t\t\t\tITEM_TYPE_BUTTON\n\t\t\trect\t\t\t\t7 126 130 24\n\t\t\ttext\t\t\t\t@MENUS_NEW\n\t\t\tdescText\t\t\t@MENUS_START_A_NEW_GAME\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t1.1\n\t\t\ttextaligny\t\t\t0\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\n\t\t\ttextstyle\t\t\t1\n\t\t\ttextalignx\t\t\t65\n\t\t\tforecolor\t\t\t1 .682 0 1\n\t\t\tvisible\t\t\t1\n\n\t\t\tmouseEnter \n\t\t\t{ \n\t\t\t\tshow\t\t\tnewbutton_glow\n\t\t\t}\n\t\t\tmouseExit \n\t\t\t{ \n\t\t\t\thide\t\t\tnewbutton_glow \n\t\t\t}\t\n\t\t\taction \n\t\t\t{ \n\t\t\t\tplay\t\t\t\"sound/interface/button1.wav\" ; \n\t\t\t\tclose\t\t\tall ; \n\t\t\t\topen\t\t\tnewgameMenu \n\t\t\t} \t \n\t\t}\n\n\t\t// Big button \"LOAD\"\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tloadgamebutton_glow\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t170 126 130 24\n\t\t\tbackground\t\t\t\"gfx/menus/menu_buttonback\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t0\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tloadgamebutton\n\t\t\tgroup\t\t\t\tlbut\n\t\t\ttext\t\t\t\t@MENUS_LOAD\n\t\t\tdescText\t\t\t@MENUS_LOAD_A_SAVED_GAME\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_EMPTY\n\t\t\ttype\t\t\t\tITEM_TYPE_BUTTON\n\t\t\trect\t\t\t\t170 126 130 24\n\t\t\ttextaligny\t\t\t0\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t1.1\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\n\t\t\ttextstyle\t\t\t1\n\t\t\ttextalignx\t\t\t65\n\t\t\tforecolor\t\t\t1 .682 0 1\n\t\t\tvisible\t\t\t1\n\n\t\t\tmouseEnter \n\t\t\t{ \n\t\t\t\tshow\t\t\tloadgamebutton_glow \n\t\t\t}\n\t\t\tmouseExit \n\t\t\t{ \n\t\t\t\thide\t\t\tloadgamebutton_glow \n\t\t\t}\t \t \n\t\t\taction \n\t\t\t{ \n\t\t\t\tplay\t\t\t\"sound/interface/button1.wav\" ; \n\t\t\t\tclose\t\t\tall ; \n\t\t\t\topen\t\t\tloadgameMenu \n\t\t\t}\n\t\t}\n\n\t\t// Big button \"CONTROLS\"\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tcontrolsbutton_glow\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t340 126 130 24\n\t\t\tbackground\t\t\t\"gfx/menus/menu_buttonback\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t0\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef {\n\t\t\tname\t\t\t\tcontrolsbutton\n\t\t\tgroup\t\t\t\tcbut\n\t\t\ttext\t\t\t\t@MENUS_CONTROLS\n\t\t\tdescText\t\t\t@MENUS_CONFIGURE_GAME_CONTROLS\n\t\t\ttype\t\t\t\tITEM_TYPE_BUTTON\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_EMPTY\n\t\t\trect\t\t\t\t340 126 130 24\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t1.1\n\t\t\ttextaligny\t\t\t0\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\n\t\t\ttextstyle\t\t\t1\n\t\t\ttextalignx\t\t\t65\n\t\t\tbackcolor\t\t\t0 0 0 0\n\t\t\tforecolor\t\t\t1 .682 0 1\n\t\t\tvisible\t\t\t1\n\t\t\t \n\t\t\tmouseEnter \n\t\t\t{ \n\t\t\t\tshow\t\t\tcontrolsbutton_glow \n\t\t\t}\n\t\t\tmouseExit \n\t\t\t{ \n\t\t\t\thide\t\t\tcontrolsbutton_glow \n\t\t\t}\t \t \n\t\t\taction \n\t\t\t{\n\t\t\t\tplay\t\t\t\"sound/interface/button1.wav\" ; \n\t\t\t\tclose\t\t\tall ; \n\t\t\t\topen\t\t\tcontrolsMenu ; \n\t\t\t}\n\t\t}\n\n\t\t// Big button \"SETUP\"\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tsetupbutton_glow\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t502 126 130 24\n\t\t\tbackground\t\t\t\"gfx/menus/menu_buttonback\"\t\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t\t0\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tsetupbutton\n\t\t\tgroup\t\t\t\tsbut\n\t\t\ttext\t\t\t\t@MENUS_SETUP\n\t\t\tdescText\t\t\t@MENUS_CONFIGURE_GAME_SETTINGS\n\t\t\ttype\t\t\t\tITEM_TYPE_BUTTON\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_EMPTY\n\t\t\trect\t\t\t\t502 126 130 24\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t1.1\n\t\t\ttextaligny\t\t\t0\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\n\t\t\ttextstyle\t\t\t1\n\t\t\ttextalignx\t\t\t65\n\t\t\tbackcolor\t\t\t0 0 0 0\n\t\t\tforecolor\t\t\t1 .682 0 1\n\t\t\tvisible\t\t\t1\n\t\t\t \n\t\t\tmouseEnter \n\t\t\t{ \n\t\t\t\tshow\t\t\tsetupbutton_glow \n\t\t\t}\n\t\t\tmouseExit \n\t\t\t{ \n\t\t\t\thide\t\t\tsetupbutton_glow \n\t\t\t}\t \t \n\t\t\taction \n\t\t\t{\n\t\t\t\tplay\t\t\t\"sound/interface/button1.wav\" ; \n\t\t\t\tclose\t\t\tall ; \n\t\t\t\topen\t\t\tsetupMenu ; \n\t\t\t}\n\t\t}\n\n//----------------------------------------------------------------------------------------------\n// QUIT MENU specific stuff\n//----------------------------------------------------------------------------------------------\n\t\titemDef \n\t\t{\n\t \t\tname\t\t\t\tquit_title\n\t\t\tgroup\t\t\t\tnone\n \t\t\ttext\t\t\t\t@MENUS_LEAVING_JEDI_KNIGHT_2\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\tbackground\t\t\t\"gfx/menus/menu_blendbox\"\n \t\t\trect\t\t\t\t100 164 440 16\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t0.7\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\t\t\n\t\t\ttextalignx\t\t\t225\n\t\t\ttextaligny\t\t\t-1\n\t\t\tforecolor\t\t\t.549 .854 1 1\n\t \t\tvisible\t\t\t1\n \t\t\tdecoration \n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname \t\t\t\tconfirm\n\t\t\tgroup\t\t\t\tnone\n\t\t\ttext \t\t\t\t@MENUS_QUIT_JEDI_KNIGHT_II\n\t\t\trect \t\t\t\t0 250 640 20\n\t\t\tfont\t\t\t\t2\n\t\t\ttextstyle \t\t\t1\n\t\t\ttextscale \t\t\t1\n\t\t\ttextalign \t\t\tITEM_ALIGN_CENTER\n\t\t\ttextalignx \t\t\t320\n\t\t\ttextaligny\t \t\t0\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible \t\t\t1\n\t\t\tdecoration\n\t\t}\n\n//----------------------------------------------------------------------------------------------\n// CANCEL button\n//----------------------------------------------------------------------------------------------\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tquitgame_no_button\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t59 444 130 24\n\t\t\tbackground\t\t\t\"gfx/menus/menu_buttonback\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t0\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tquitgame_no\n\t\t\tgroup\t\t\t\tnone\n\t\t\ttext\t\t\t\t@MENUS_NO\n\t\t\tdescText\t\t\t@MENUS_DO_NOT_LEAVE_JEDI_KNIGHT\n\t\t\ttype\t\t\t\tITEM_TYPE_BUTTON\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_EMPTY\n\t\t\trect\t\t\t\t59 444 130 24\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t1\n\t\t\ttextalignx\t\t\t65\n\t\t\ttextaligny\t\t\t\t-1\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\n\t\t\ttextstyle\t\t\t1\n\t\t\tforecolor\t\t\t1 .682 0 1\n\t\t\tvisible\t\t1\n\t\t\taction \n\t\t\t{ \n\t\t\t\tplay\t\t\t\"sound/interface/esc.wav\" \t\t\t\t\t\n\t\t\t\tclose\t\t\tquitMenu\n\t\t\t\topen\t\t\tmainMenu\n\t\t\t}\n\t\t\tmouseEnter \n\t\t\t{ \n\t\t\t\tshow\t\t\tquitgame_no_button \n\t\t\t}\n\t\t\tmouseExit \n\t\t\t{ \n\t\t\t\thide\t\t\tquitgame_no_button \n\t\t\t}\t \t \n\t\t}\n\n//----------------------------------------------------------------------------------------------\n// CONFIRM button\n//----------------------------------------------------------------------------------------------\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tquitgame_yes_button\n\t\t\tgroup\t\t\t\tnone\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_SHADER\n\t\t\trect\t\t\t\t454 444 130 24\n\t\t\tbackground\t\t\t\"gfx/menus/menu_buttonback\"\n\t\t\tforecolor\t\t\t1 1 1 1\n\t\t\tvisible\t\t0\n\t\t\tdecoration\n\t\t}\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tquitgame_yes\n\t\t\tgroup\t\t\t\tnone\n\t\t\ttext\t\t\t\t@MENUS_YES\n\t\t\tdescText\t\t\t@MENUS_JEDI_KNIGHT_II\n\t\t\ttype\t\t\t\tITEM_TYPE_BUTTON\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_EMPTY\n\t\t\trect\t\t\t\t454 444 130 24\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t1\n\t\t\ttextalignx\t\t\t65\n\t\t\ttextaligny\t\t\t-1\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\n\t\t\ttextstyle\t\t\t1\n\t\t\tforecolor\t\t\t1 .682 0 1\n\t\t\tvisible\t\t1\n\t\t\tcvartest\t\t\t\"com_demo\"\n\t\t\thidecvar\t\t\n\t\t\t{\n\t\t\t\t\"1\"\n\t\t\t}\n\t\t \taction \n\t\t\t{ \n\t\t\t\tplay\t\t\t\"sound/interface/button1.wav\"\n\t\t\t\tuiScript\t\tQuit\t\t\t\n\t\t\t}\n\t\t\tmouseEnter \n\t\t\t{ \n\t\t\t\tshow\t\t\tquitgame_yes_button \n\t\t\t}\n\t\t\tmouseExit \n\t\t\t{ \n\t\t\t\thide\t\t\tquitgame_yes_button \n\t\t\t}\t \t \n\t\t}\n\n\n\t\titemDef \n\t\t{\n\t\t\tname\t\t\t\tquitgame_yes\n\t\t\tgroup\t\t\t\tnone\n\t\t\ttext\t\t\t\t@MENUS_YES\n\t\t\tdescText\t\t\t@MENUS_JEDI_KNIGHT_II\n\t\t\ttype\t\t\t\tITEM_TYPE_BUTTON\n\t\t\tstyle\t\t\t\tWINDOW_STYLE_EMPTY\n\t\t\trect\t\t\t\t454 444 130 24\n\t\t\tfont\t\t\t\t3\n\t\t\ttextscale\t\t\t1\n\t\t\ttextalignx\t\t\t65\n\t\t\ttextaligny\t\t\t-1\n\t\t\ttextalign\t\t\tITEM_ALIGN_CENTER\n\t\t\ttextstyle\t\t\t1\n\t\t\tforecolor\t\t\t1 .682 0 1\n\t\t\tvisible\t\t1\n\t\t\tcvartest\t\t\t\"com_demo\"\n\t\t\thidecvar\t\t\n\t\t\t{\n\t\t\t\t\"0\"\n\t\t\t}\n\t\t \taction \n\t\t\t{ \n\t\t\t\tplay\t\t\t\"sound/interface/button1.wav\"\n\t\t\t\tclose\t\t\tall\n\t\t\t\topen\t\t\tdemo_sellscreen1\n\t\t\t}\n\t\t\tmouseEnter \n\t\t\t{ \n\t\t\t\tshow\t\t\tquitgame_yes_button \n\t\t\t}\n\t\t\tmouseExit \n\t\t\t{ \n\t\t\t\thide\t\t\tquitgame_yes_button \n\t\t\t}\t \t \n\t\t}\n\n\t}\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n" ]

{ "pile_set_name": "Github" }

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[ "Q:\n\nEmpty object prototype is undefined\n\n(I'm new to JavaScript) If all objects inherit their properties from a prototype, and if the default object is Object, why does the following script return undefined in both cases (I was expecting 'Object')?", "\nobj1 = {}; //empty object\nobj2 = new Object();\nconsole.log(obj1.prototype);\nconsole.log(obj2.prototype);\n\nPardon me if it's a silly question!", "\n\nA:\n\n.prototype is not a property of a live object and thus it doesn't exist so it reports undefined. ", " The .prototype property is on the constructor which in this case is Object.prototype. ", "For a given object in a modern browser, you can get the active prototype with this:\nvar obj1 = {}; \nvar p = Object.getPrototypeOf(obj1);\n\nA non-standard and now deprecated way to get the prototype is:\nvar obj1 = {}; \nvar p = obj1.__proto__;\n\n" ]

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[ "Graduate Employment 2015 Statistics\n\nFor most college graduates, life after college is inevitably connected with making the dream career. ", "However, many fail to have the desired career and the graduate employment statistics evidence that fact. ", "Therefore, a graduate employment survey is conducted on a regular basis to summarize the problems. ", "The following graduate employment infographic provides statistics about graduate employment 2015. ", "Particularly, the graduate employment statistics is divided into industries and attitudes to employment.", "\n\nMoreover, the following infographic shows the top 10 college degrees by early career salary potential. ", "Among them are nuclear engineering, actuarial mathematics and aerospace engineering. ", "Thanks to the graduate employment surveys so far a comparison between expectations and reality is given as well. ", "Besides, this graduate employment infographic provides the attracting top college graduates. ", "According to the statistics, only 15% want to work for large companies opposed to 64% who believe that they need to pursue further education to get the job they want." ]

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[ "A date with yourself\n\n2019-03-07\n\nAt our STENDERS office, laboratory, shops and factory – we spend every moment to make your day a bit more beautiful. ", "We believe that every day can be Valentine's Day and every month – full of love. ", "We also believe that love should be shared not only with others, but also with yourself.", "\n\nAnd what a better partner for a romantic, thought-out, love-infused date than yourself? ", "Where to start? ", "What to do on this date? ", "Here are couple of ideas:\n\nGet party-ready. ", "Maybe for you a whole day spent in pyjamas is a dream come true (if so, go with this option), but nothing creates a festive feeling better than getting ready. ", "A pair of ballroom shoes, a glittering dress or jacket and your favourite perfume. ", "A wonderful start of a great day!", "\n\nSurprise yourself with a gift. ", "A small piece of jewellery or a frying pan… whatever your choice, treat yourself with a gift you have been longing for. ", "Let it be a reminder that you should go on dates as often as you visit a nail salon or a hairdresser.", "\n\nA handwritten greeting card. ", "Instead of writing a card to a birthday boy/girl or a loved one, write a greeting card or letter to yourself. ", "Say thank you for your accomplishments, for all that you have been through and promise that you will continue to love yourself today, tomorrow and always.", "\n\nMake your day with flowers. ", "Of all the winter months February makes you long for the sun and spring the most. ", "Tulips, daffodils, hyacinths ... flowers will create the spring mood and will give off a fragrance that is hard to resist.", "\n\nMovie/book marathon. ", "There is definitely a movie, a TV series or a book that you have long been craving for but have not managed to try out yet. ", "A comedy, horror movie or animation, autobiography, novel or collection of poems – reach for what your heart desires the most on your date with yourself.", "\n\nDinner. ", "If cooking is one of your hobbies, make a gastronomic three-course dinner for yourself. ", "Plan a menu, get the necessary ingredients, experiment, and try new flavours, aromas and textures. ", "This will be an unforgettable, particularly inspiring experience that will give you a lot of new knowledge." ]

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[ "The woman who as a teen encouraged her boyfriend’s suicide will serve her full sentence after the U.S. Supreme Court declined to hear her appeal.", "\n\nMichelle Carter’s appeal appeared on a list on Monday titled “Certiorari Denied,” which means the high court won’t review those cases.", "\n\nCarter, now 23, was convicted of involuntary manslaughter after Conrad Roy III took his life in a Kmart parking lot in Fairhaven, Massachusetts. ", "Sentenced as an adult, she was given 15 months in prison.", "\n\nRoy was 18 and Carter was 17 on July 12, 2014, the evening when he called her and talked of killing himself. ", "At one point, he appeared to have second thoughts and she commanded him to “get back in” his carbon monoxide-filled pickup truck.", "\n\nThe phone call wasn’t recorded, and the judge relied on a text Carter sent her friend in which she admitted she told Roy to get back in the truck. ", "In text messages sent in the days leading up to Roy’s death, Carter also encouraged him to follow through with his suicide plan, Massachusetts courts found.", "\n\nCarter’s lawyer Joseph P. Cataldo called the Supreme Court’s decision “unfortunate” in an email to the Boston Herald on Monday and left the door open to other possible action in the case.", "\n\nCataldo had argued Carter’s conviction violated the First Amendment guarantee of free speech because it was based solely on words that she texted or spoke. ", "An appeal on the same grounds was previously rejected by the Massachusetts Supreme Judicial Court.", "\n\nCarter began serving her sentence in February 2019 at the Bristol County House of Corrections. ", "Her request for early parole was denied in September." ]

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[ "Q:\n\nHow to get validation error value(=inputed value, but not in model) angularjs way\n\nI’m trying to get user.name.length value at error message.", "\nIf jQuery is used, it is realizable, but isn't there any method unique to Angularjs?", "\n<form novalidate name=\"myForm\" ng-submit=\"addUser()\">\n<p>Name:\n<input type=\"text\" name=\"name\" ng-model=\"user.name\" required ng-minlength=\"5\" ng-maxlength=\"8\">\n<span ng-show=\"myForm.name.$error.minlength\">{{user.name.length}} too short!</span>\n\nA:\n\nCan you try with {{myForm.name.$viewValue.length}}, like below:\n<input type=\"text\" name=\"name\" ng-model=\"user.name\" ng-minlength=\"5\" ng-maxlength=\"8\" />\n <span ng-show=\"myForm.name.$error.minlength\">{{myForm.name.$viewValue.length}} too short!</span>\n <span ng-show=\"myForm.name.$error.maxlength\">{{myForm.name.$viewValue.length}} too long!</span>\n\n" ]

{ "pile_set_name": "StackExchange" }

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[ "Q:\n\nMultiply digits of a number using recursion\n\nI am doing the following exercise:\n\nGiven a four digit number such as 3183, compare each digit with the last and if greater or equal multiply it with the following\n\nExample: for the number 3183 it would be n = 3*8*3 = 72.", "\nMy code:\n#include <stdio.h>\n\nint f ( int n )\n{\n if ( n < 10 ) \n return n ;\n return (((n/10) % 10) >= (n%10) ? ((", "n/10)10) : 1) * f((n/100 )* 10 + n % 10 ) ;\n}\n\nint main() \n{\n printf( \"%d\", f( 3183 );\n\n return(0);\n}\n\nIs there any way to shorten it or make it better?", "\n\nA:\n\nLeave another approach more compacted than the original:\n#include <stdio.h>\n\nint f (int n, int u)\n{\n if (u > n) return(1);\n return (n % 10 >= u ? ", "n % 10 : 1) * f(n/10, u);\n}\n\nint main (void)\n{\n int n = 3284;\n printf (\"%d\", f (n , n%10));\n\n return(0);\n}\n\nA:\n\nEDIT I mis-read this yesterday. ", "No point in effectively re-creating @Red Alert's answer now, but I can't delete it either since't it's accepted so here goes.", "\nI assume we can create our own \"inner\" function to maintain state. ", "I also assume digits are to be processed from the right, the original example isn't clear.", "\nstatic int g(int n, int ack, int last)\n{\n const int here = n % 10;\n const bool mult = here >= last;\n\n if(n < 10)\n return mult ? ", "here * ack : here;\n return g(n / 10, mult ? ", "here * ack : ack, here);\n}\n\nint f(int n)\n{\n return g(n, 1, 0);\n}\n\n" ]

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[ "Constructed Universe\n\nI saw this interesting piece “Constructed Universe” (1983) by Daniel Faust from the Metropolitan Museum of Art via Twitter and it intrigued me enough to share it here, although some of you might think it’s just a load of balls." ]

{ "pile_set_name": "Pile-CC" }

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[ "Q:\n\nCSS transition is not working in Safari\n\nWhen you hover over image1div, it scales to 0.95 and fades to 80% opacity. ", "It works in Chrome and Firefox but not Safari. ", "It fades and scales instantly in Safari rather than smoothly in 0.5s.", "\n.image1div {\n width: 350px;\n height: 350px;\n margin-top: 0px;\n float: right;\n background-color: #5a89ad;\n background-size: cover;\n filter:alpha(opacity=100);\n -webkit-transform: scale(1,1);\n -ms-transform: scale(1,1);\n transform: scale(1,1);\n -webkit-transition: opacity 0.5s ease, transform 0.5s ease;\n transition: opacity 0.5s ease, transform 0.5s ease; \n}\n\n.image1div:not(.no-hover):hover {\n -webkit-transform: scale(0.95,0.95);\n -ms-transform: scale(0.95,0.95);\n transform: scale(0.95,0.95);\n opacity:0.8;\n filter:alpha(opacity=80);\n}\n\nA:\n\nI think it has to do with the filter property.", "\nTransition is supported by safari: http://caniuse.com/#feat=css-transitions\nAlso the filter property, but you need to add a prefix: http://caniuse.com/#feat=css-filters \nLet me know if it helps, if not, provide more details and we will find a workaround.", "\n-- EDIT\nInstead of transition: opacity, transform. ", "Use all, or check out how you can add multiple properties CSS transition shorthand with multiple properties?", "\n\n" ]

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[ "1. ", "Field of the Invention\nThe present invention relates to semiconductor devices, and more particularly, to improvement of the operating characteristics of FETs, IGBTs and MCTs.", "\n2. ", "Background of the Invention\nIn an FET in which an N+ source region is spaced from an N type drift/drain region by a P type body region with an insulated gate electrode disposed over the P type body region and extending from the source region to the drift/drain region, a parasitic NPN bipolar transistor is disposed in parallel with the desired field effect transistor. ", "A first main electrode is disposed in ohmic contact with the source and body regions and a second main electrode is disposed in ohmic contact with the drain region. ", "In this parasitic bipolar transistor, the N type source region functions as an emitter, the P type body region functions as the base region and the N type drain region functions as the collector and the emitter/base function is shorted. ", "During FET turn-off of an inductive load, this parasitic NPN transistor can become conductive and have a detrimental effect on the operation of the field effect transistor or result in its destruction. ", "In circuits in which the body/drift region PN junction (an inherent diode) becomes forward biased, there is a substantial hole current in the P type body region extension beneath the source region. ", "This current, in combination with the relative high resistance of that portion of the body region, can produce a sufficient voltage drop to forward bias the source/body junction far from the first main electrode contact which shorts that junction. ", "When that portion of the source/body PN junction injects carriers, the NPN parasitic transistor becomes conductive, which can destroy the device. ", "In power FETs, a large gain in this parasitic NPN transistor has an adverse effect on the ruggedness of the FET because the larger the gain of this NPN transistor, the smaller the current required to turn it on and the smaller the safe operating area of the FET is. ", "The safe operating area is the combination of current and voltage which the FET can withstand during the process of turning off without destruction of the device. ", "At a given voltage, increasing the current through the FET to more than the maximum SOA value for that voltage leads to breakdown of the device. ", "Similarly, increasing the voltage at a fixed current level to more than the SOA maximum voltage for that current also leads to breakdown of the device which prevents the device from turning off. ", "Such breakdown can also result in destruction of the device. ", "For these reasons, it is considered desirable to minimize the gain of the parasitic NPN transistor in power FETs. ", "The gain of this NPN transistor depends on the doping level of the P type body region. ", "The higher the doping level of the P type body region, the lower is the gain of this NPN transistor. ", "Consequently, in order to minimize the adverse effect of this parasitic NPN transistor, it is desirable to dope the body region as heavily P type as possible. ", "Unfortunately, heavy doping of the body region has an adverse effect on the operation of the desired field effect transistor. ", "That is, increasing the doping level of the P type body region increases the threshold voltage of the N type channel extending through the P type body region from the source to drain region, i.e. the gate voltage necessary to render that channel conductive to electrons.", "\nIt is known in the prior art to heavily dope the P type body region and to counter dope just the channel portion of the body region by implanting N type dopant material into the surface of the body region to reduce its net P type doping level to adjust the threshold voltage of the field effect transistor independent of the doping level of the P type body region.", "\nThe presence of this parasitic NPN bipolar transistor does not have any significant effect on the ON-state operation of the field effect transistor. ", "It is only during turn-off of the field effect transistor in the presence of an inductive load or during turn-off of the inherent diode that this parasitic NPN bipolar transistor has an effect on the operating characteristics of the device. ", "Thus, this parasitic bipolar transistor only affects the operating characteristics of a field effect transistor if that field effect transistor is a power device which will be operated near the limits of its safe operating area.", "\nAn insulated gate bipolar transistor (IGBT) is similar in structure to an FET, except that a P+ collector region is disposed between the N type drift region and second main electrode. ", "Consequently, an IGBT contains a similar parasitic NPN bipolar transistor. ", "However, this parasitic NPN transistor does not affect the SOA of the device because it is the lower or PNP transistor which limits device SOA. ", "Rather, in the IGBT, this parasitic transistor affects the main current level at which the IGBT latches in the ON-state. ", "The larger the gain of this parasitic NPN transistor, the lower the latching current of the IGBT becomes. ", "As is well known, the IGBT is intended to be a non-latching device and, therefore, such decreased latching current is undesirable. ", "Consequently, it is desirable to minimize the gain of the parasitic NPN transistor in an IGBT in order to maximize the latching current of the IGBT. ", "The same techniques may be used to minimize the gain of the parasitic NPN transistor in an IGBT as are used in FETs. ", "Thus, the only reason for reducing the gain of the parasitic NPN transistor in an IGBT is to increase latching current.", "\nAn MOS controlled thyristor (MCT) incorporates a field effect structure designed to carry current from the N type base region directly to the first power electrode to bypass the P type emitter/N type base region junction of the thyristor and thereby interrupt the regenerative action in the thyristor to turn it off. ", "Introduction of this field effect structure creates what is in effect a field effect transistor of the N+ source region of the turn-off structure, the P type emitter region of the thyristor and the N type base region of the thyristor. ", "This structure includes a parasitic NPN bipolar transistor. ", "By analogy to the IGBT, decreasing the gain of that parasitic NPN bipolar transistor is expected to increase the latching current of the MCT, which is undesirable since the MCT is intended to latch on in the ON-state. ", "Consequently, the prior art provides no reason to attempt to reduce the gain of this parasitic NPN bipolar transistor.", "\nConsequently, the prior art provides no teaching or suggestion for heavily doping the portion of the P type emitter region of an MCT in the vicinity of the turn-off channel of the MOS structure in combination with counter doping of that channel. ", "The prior art incentives for such structures in FETs or IGBTs are not applicable to MCTs because the problems that structure ameliorates in FETs and IGBT do not exist in the MCTs.", "\nIt has become desirable to operate circuits including MCTs at as high a frequency as possible, in such systems as dc-to-dc power converters. ", "Consequently, new MCT structures which facilitate operation at higher frequencies are desirable.", "\nWith the advent of trench gate FETs and IGBTs, control of the gain of the parasitic NPN bipolar transistor has been one of the considerations in the trade-off decision as to whether to use a trench structure or a planar structure because ion implantation counter doping of the channel portion of the body or base region of a trench gate device is not possible because the trench walls are normally perpendicular to the major surface of the wafer whereby ion implantation ions are traveling parallel to the trench walls and thus do not implant in the body or base region. ", "Since counter doping is not possible, a designer, in designing a trench gate FET or IGBT, must choose between a high threshold voltage with a low gain parasitic NPN transistor and a low threshold voltage with a high gain parasitic NPN transistor. ", "Consequently, an improved structure for minimizing the gain of the parasitic NPN transistor in FETs and IGBTs, especially trench gate versions is needed, as is a method of providing such a structure.", "\nWhile it is known that indium and aluminum are P type dopants, it is standard practice in the semiconductor industry to avoid using indium and aluminum as dopants in silicon because their small segregation coefficients lead to unpredictability of a process to obtain a desired structure. ", "Use of boron as a P type dopant produces device structures and characteristics which are quite predictable. ", "The reasons behind the essentially exclusive use of boron as a P type dopant are discussed in xe2x80x9cThe Theory and Practice of Microelectronicsxe2x80x9d by Sorab K. Ghandi published by John Wiley and Sons, especially in the xe2x80x9cChoice of p-type Impurityxe2x80x9d section. ", "While there is much literature on the characteristics of boron and phosphorous as dopants in silicon, there is much less literature dealing with the use of indium and aluminum to create P type doped regions. ", "An article entitled xe2x80x9cDiffusion of Indium in Silicon Inert and Oxidizing Ambientsxe2x80x9d, by D.A. Antoniadis et al., ", "which appeared in the Journal of Applied Physics, Volume 53, No. ", "12, December 1982, pages 9214-9216 presents data on the diffusion characteristics of indium and concludes that indium has a segregation coefficient of 0.1. ", "Both of these references are incorporated herein by reference.", "\nThose skilled in the semiconductor art rely on established, controllable processes for the fabrication of devices because of their established, relatively high yields, rather than experimenting with processes which are known to be difficult to predict or control, unless there is a clear incentive for the use of the xe2x80x9cunpredictablexe2x80x9d process. ", "There has been no such incentive for the use of indium or aluminum as dopants in silicon in the fabrication of field effect devices because of the availability and established techniques for controlling ion implantation of the counter dopant material in the channel portion of the body region to control the threshold voltage of the insulated gate structure in the presence of heavy body region dopant concentrations.", "\nAccordingly, a primary object of the present invention is to provide an improved structure for controlling the threshold voltage of FETs and IGBTs having trench gates.", "\nAnother object of the present invention is to provide an MCT structure having a faster turn-off than prior art MCT structures.", "\nAnother object is to provide an improved method of fabricating FETs, IGBTs and MCTs.", "\nIn accordance with the above objects and others which will become apparent from the specification as a whole, including the drawings, an FET is provided with a heavily doped P type body region and a more lightly doped channel region by doping the body region with one or more of indium, aluminum and gallium or a combination of dopants including boron and one or more of indium, aluminum and gallium. ", "Indium and aluminum are depleted in the body region adjacent the gate oxide because indium and aluminum both have small silicon/silicon dioxide segregation coefficients. ", "The concentration of gallium is substantially reduced because of its higher diffusion rate in the oxide. ", "The resulting structure is functionally equivalent to counter doping of the channel where only boron has been used to dope the base region. ", "However, unlike counter doping, this technique is equally effective for trench gate structures and planar structures. ", "Consequently, it is a substantial advance over the use of ion implantation counter doping since it provides independent control over body doping level and threshold voltage in trench gate devices.", "\nUse of a combination of boron and one or more of aluminum, indium and gallium as the base dopant in an IGBT is similar in providing the same benefits provided by counter doping the channel portion of the base region.", "\nWe have found that in an MCT, providing a heavily doped P type emitter region along with a low threshold voltage channel in the MOS turn-off structure in that emitter region provides a substantial improvement in the MCT\"\"s operating characteristics, in particular, by providing a much more rapid turn-off of the device upon application of a turn-off voltage to the insulated gate. ", "This newly desirable structure may be provided either by doping the P type emitter region with a combination of boron and one or more of indium, aluminum and gallium or by counter doping the channel portion of that P type emitter region of a surface gate device to provide a low threshold voltage along with a heavy doping of the emitter region everywhere except in the channel itself." ]

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[ "Q:\n\nTrain Cascade Object Detector bounding box out of bound?", "\n\nI am trying to train a cascade object detector using the built-in function in Matlab (vision toolbox). ", "However, the following message came up after running the command.", "\n*\nError using trainCascadeObjectDetector (line 245)\nError reading instance 1 from image 2, bounding box possibly out of image bounds.", "\n\n*\nI don't understand why the bounding box can be out of bounds. ", "All the parameters for my positive images are set up correctly (starting point x,y, width, and height. ", "I used createMask(h) to create a mask and find the minimum coordinates for x and y to be starting point and max-min for each dimension to be the width and height), and the negative images (as far as I know) are just images without any setup needed.", "\nAnyone ever ran into the same problem? ", "How did you solve it?", "\nEDIT:\nHere's the code. ", "I don't have the toolbox for training the \"data\" struct, so I wrote one myself\npositive_samples=struct;\nlist=dir('my_folder_name_which_I_took_out');\n\nL=length(list)-3; %Set L to be the length of the image list.", "\nfor i=1:length(list)\n positive_samples(i).imageFilename=list(i).name;\nend\npositive_samples(:,1)=[]; %first 3 lines do not contain file names\npositive_samples(:,1)=[];\npositive_samples(:,1)=[];\n\nfor j=1:1\n imshow(positive_samples(j).imageFilename);\n title(positive_samples(j).imageFilename);\n h=imrect;\n h1=createMask(h);\n I=imread(positive_samples(j).imageFilename);\n [le, wi, hi]=size(I);\n tempmat=[];\n count=1;\n for l=1:le\n for m=1:wi\n if h1(l,m)==1\n tempmat(count,1)=l;\n tempmat(count,2)=m;\n count=count+1;\n end\n end\n end\n positive_samples(j).objectBoundingBoxes(1,1)=min(tempmat(:,1));\n positive_samples(j).objectBoundingBoxes(1,2)=min(tempmat(:,2));\n positive_samples(j).objectBoundingBoxes(1,3)=max(tempmat(:,2))-min(tempmat(:,2));\n positive_samples(j).objectBoundingBoxes(1,4)=max(tempmat(:,1))-min(tempmat(:,1));\n imtool close all\nend\ntrainCascadeObjectDetector('animalfinder.xml', positive_samples, 'my_neative_folder_name', 'FalseAlarmRate', 0.2, 'NumCascadeStages', 3);\n\nsorry if it's messy......\n\nA:\n\nI did not run the code, because I don't own the toolbox, but the following lines are very \"suspicious\":\npositive_samples(j).objectBoundingBoxes(1,1)=min(tempmat(:,1));\npositive_samples(j).objectBoundingBoxes(1,2)=min(tempmat(:,2));\npositive_samples(j).objectBoundingBoxes(1,3)=max(tempmat(:,2))-min(tempmat(:,2));\npositive_samples(j).objectBoundingBoxes(1,4)=max(tempmat(:,1))-min(tempmat(:,1));\n\nI would expect:\npositive_samples(j).objectBoundingBoxes(1,1)=min(tempmat(:,2));\npositive_samples(j).objectBoundingBoxes(1,2)=min(tempmat(:,1));\npositive_samples(j).objectBoundingBoxes(1,3)=max(tempmat(:,2))-min(tempmat(:,2));\npositive_samples(j).objectBoundingBoxes(1,4)=max(tempmat(:,1))-min(tempmat(:,1));\n\nSome suggestions to shorten your code, they are not related to the problem:\nYou can shorten line 4 to 9 to a single line, avoiding the loop: [positive_samples(1:L).im]=list(4:end).name\nAnd this loop can be replaced as well:\ntempmat=[];\ncount=1;\nfor l=1:le\n for m=1:wi\n if h1(l,m)==1\n tempmat(count,1)=l;\n tempmat(count,2)=m;\n count=count+1;\n end\n end\nend\n\nshorter and faster code:\n[y,x]=find(h1);\ntempmat=[y x];\n\n" ]

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[ "Q:\n\nBasic self-signed certificate questions\n\nA few basic questions around self-signed certificates\nQuestion 1 \nDoes a self-signed certificate has to have a CA? ", "I realize that a \"real\" certificate must as it otherwise can be chain validated but what about self-signed?", "\nQuestion 2\nIf a self-signed hasn't got a CA - should it still go in the trusted CA store to not cause chain validation errors when using it for test?", "\nQuestion 3\nIs there a way to see if a self-signed cert is signed by a CA - if it has a CA does that effect the question above? ", "Do I then have to have the CA to trust it or can I still just put the actual cert in the CA store to create a secure SSL channel for example?", "\n\nA:\n\nA certificate is signed by the CA which issues it. ", "A self-signed certificate, by definition, is not issued by a CA (or is its own CA, if you want to view it like this).", "\nA certificate may have CA power, i.e. be trusted to sign other certificates, or not, depending on whether it contains a Basic Constraints extension with the cA flag set to TRUE (cf the standard).", "\nWhat happens is the following: when some application wants to validate a certificate (e.g. a Web browser who wants to do SSL with a server, and just obtained the server's certificate), it will want to build a certificate chain which begins with a \"trust anchor\" (one of the certificates in its \"trusted root\" certificate store) and ends with the certificate to validate (called \"EE\" as \"end-entity\"). ", "Exact rules for a chain to be valid are intricate and full of details; for the purposes of this answer, let's limit ourselves to these necessary conditions:\n\nThe chain must start with a trust anchor.", "\nEach certificate is signed by the previous certificate in the chain (i.e. the signature on each certificate is to be verified relatively to the public key as is stored in the previous certificate).", "\nEach certificate except the end-entity has a Basic Constraints extension with the cA flag set to TRUE.", "\n\nSo a self-signed but not CA certificate, when used as a trust anchor, will be accepted as valid as an end-entity certificate (i.e. in a chain reduced to that certificate exactly) but not otherwise. ", "This is the normal case. ", "When, as a browser user, you want to accept a given self-signed certificate as valid, you actually tell your browser that the self-signed certificate should become a trust anchor -- but you certainly do not want to trust that certificate for issuing other certificates with other names ! ", "You want to trust it only for authenticating a specific site.", "\nAs usual, details may vary -- not all browsers react in the exact same ways. ", "But the core concepts remain:\n\nA self-signed certificate lives outside of the CA world: it is not issued by a CA.", "\nA client (browser) uses trust anchors as the basis for what it trusts.", "\nA self-signed certificate for a Web server, usually, should be trusted (if at all) only for that server, i.e. added as a trust anchor, but not tagged as \"good for issuing certificates\".", "\n\nWhen a chain is reduced to a single certificate, i.e. the end-entity is also the trust anchor, then this is known as direct trust: a specific certificate is trusted by being already known, exactly, instead of being trusted by virtue of being issued by a trusted CA.", "\n\nA:\n\nNo. ", " A self-signed certificate is not signed by a CA - it is signed by itself. ", " However, all CAs publish a self-signed certificate that is used to sign other certificates. ", " A \"trusted CA\" is one that is included in your operating system or browser, or one that you explicitly choose to trust.", "\nDepends on the self-signed cert. ", " If you are using it as a CA and to sign other certificates in your organization, then I might add it to my Trusted CA store. ", " If it's used only to authenticate a particular server, and is not used as a CA, I would include it in a different store, such as \"trusted publishers\". ", " Even then, I would be careful to make sure the certificate is being stored and handled carefully - anyone who gets it can do whatever they want to your machines that trust it. ", " I would never place a random certificate in my trusted CA store.", "\nA self-signed certificate is found at the end a chain of trust - it says \"I am my own parent.\" ", " Validating the chain of certificates that signs an ordinary certificate means validating the signatures of each signer, all the way back until you find a self-signed certificate. ", " At that point, the self-signed certificate needs to be in your trusted CA store.", "\n\n" ]

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[ "Gooding railway station\n\nGooding was a railway station on the Walhalla narrow gauge line in Gippsland, Victoria, Australia. ", "The station was opened in 1910 and was the lowest altitude station on the line. ", "The station consisted of a platform area and a name board.", "\n\nThe station was named after the Gooding family, who owned a farm in the area. ", " Prior to 1910, the Goodings' farm had been a stopping place for coaches traveling towards Walhalla, this coach traffic ceasing after the railway opened. ", "With the construction of the railway, the Railways Commission of Victoria allotted the area a siding at a point where the line crossed the Walhalla Road, which was named Gooding Station. ", " Passengers who wished to board the train could do so by signalling the train driver to stop and take them aboard. ", "Similarly, passengers wishing to disembark could also do so by arrangement with the train's crew.", "\n\nReferences\n\nCategory:Disused railway stations in Victoria (Australia)" ]

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[ "Aegle marmelos (L.) Correa (Bael) and its phytochemicals in the treatment and prevention of cancer.", "\nAegle marmelos, commonly known as Bael and belonging to the family Rutaceae is an important medicinal plant in the traditional Indian system of medicine, the Ayurveda. ", "The extract prepared by boiling the bark, leaves or roots in water is useful as laxative, febrifuge, and expectorant. ", "The extract is also useful in ophthalmia, deafness, inflammations, catarrh, diabetes, and asthmatic complaints. ", "The fruits are used in treating diarrhea, dysentery, stomach ache, and cardiac ailments. ", "Scientific studies have validated many of Bael's ethnomedicinal properties and its potential antimicrobial effects, hypoglycemic, astringent, antidiarrheal, antidysenteric, demulcent, analgesic, anti-inflammatory, antipyretic, wound-healing, insecticidal, and gastroprotective properties. ", "In addition, studies have also shown that Bael and some of the Bael phytochemicals possess antineoplastic, radioprotective, chemoprotective, and chemopreventive effects, properties efficacious in the treatment and prevention of cancer. ", "For the first time, the current review summarizes the results related to these properties and emphasizes aspects that require further investigation for Bael's safe and effective use in the near future." ]

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[ "Christopher Booker, the former Telegraph journalist and first editor of Private Eye, has died aged 81.", "\n\nMr Booker was a columnist for The Sunday Telegraph from its launch in 1961 to March this year when he retired.", "\n\nAllister Heath, Editor of The Sunday Telegraph, said: \"Mr Booker was one of the great social critics of his generation, and one of the leading campaigning journalists of the post-war era.", "\n\n\"We are honoured to have published him and, on behalf of everyone at The Telegraph, we send our condolences to his family and to all those who knew him.\"", "\n\nHe penned his final article for this paper on March 31, citing health reasons as his reason for leaving.", "\n\n\"One of my guiding principles as a journalist had long been to bring to light what I considered to be some shockingly important story which was not being properly covered elsewhere,\" he wrote.", "\n\n\"And one of the reasons I have been very grateful to The Sunday Telegraph is the unusual freedom it gave me to pursue a long succession of such campaigning stories, if necessary for years on end.\"", "\n\nMr Booker has written about literature, butterflies, religion, architecture, climate change and the European Union among other subjects." ]

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[ "FLORENCE - Charles G. Elnicki, 93, died peacefully on Friday, January 18, 2013, at Mountain View Center in Rutland, after a long illness.", "\n\nHe was born April 1, 1919, in Brackenridge, Pa., the son of Michael and Mary (Paszuhance) Elnicki.", "\n\nCharlie was a WWII U.S. Army veteran serving in Africa, Casablanca, Tunisia, Salano, Naples and Anzio. ", "He earned European, African and Middle Eastern Theatre Ribbons with five Battle Stars, and received a Purple Heart for injuries he received.", "\n\nAfter an honorable discharge, he was employed for 25 years as a field inspector for H.P. Hood.", "\n\nCharlie loved gardening, ice fishing, woodworking and camping, and one of his most memorable camping trips was from coast to coast with his wife, Aline, in their slide-on camper, but above all, he loved spending time with family and friends.", "\n\nHe is survived by his wife of 63 years, Aline of Florence; a son, Michael C. Elnicki and his wife Colleen of Pittsford; a daughter, Donna Whitman and her husband Gary of Ripton; a granddaughter, Jill Merrill and her husband Michael of Pittsford, and a grandson, Richard Elnicki and his wife Nicole of Rutland; two sisters, Pauline McMahon of Rutland and Mildred Higgins of N.H.; and many nieces and nephews.", "\n\nHe was predeceased by a sister, Anna, and three brothers, Mike, Jim and Frank.", "\n\nFuneral services will be held on Wednesday, January 23, 2013, at 2:00 p.m. at the Pittsford Congregational Church, immediately followed by a reception downstairs in the church fellowship hall. ", "Burial will be held at a later date in Evergreen Cemetery in Pittsford.", "\n\nFamily and friends may call on Wednesday from 12:00 noon until 1:45 p.m. at the Barnard Funeral Home in Pittsford.", "\n\nThe family suggests, for those who wish, memorial contributions may be made to Rutland County Humane Society, 765 Stevens Road, Pittsford, VT 05763; or Pittsford First Response Squad, PO Box 433, Pittsford, VT 05763." ]

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[ "// Copyright (c) 2016 The Decred developers\n// Copyright (c) 2017 The btcsuite developers\n// Use of this source code is governed by an ISC\n// license that can be found in the LICENSE file.", "\n\npackage wallet\n\nimport (\n\t\"github.com/btcsuite/btcd/chaincfg/chainhash\"\n\t\"github.com/btcsuite/btcwallet/walletdb\"\n\t\"github.com/btcsuite/btcwallet/wtxmgr\"\n)\n\ntype unstableAPI struct {\n\tw *Wallet\n}\n\n// UnstableAPI exposes additional unstable public APIs for a Wallet. ", " These APIs\n// may be changed or removed at any time. ", " Currently this type exists to ease\n// the transation (particularly for the legacy JSON-RPC server) from using\n// exported manager packages to a unified wallet package that exposes all\n// functionality by itself. ", " New code should not be written using this API.", "\nfunc UnstableAPI(w *Wallet) unstableAPI { return unstableAPI{w} }\n\n// TxDetails calls wtxmgr.", "Store.", "TxDetails under a single database view transaction.", "\nfunc (u unstableAPI) TxDetails(txHash *chainhash.", "Hash) (*wtxmgr.", "TxDetails, error) {\n\tvar details *wtxmgr.", "TxDetails\n\terr := walletdb.", "View(u.w.db, func(dbtx walletdb.", "ReadTx) error {\n\t\ttxmgrNs := dbtx.", "ReadBucket(wtxmgrNamespaceKey)\n\t\tvar err error\n\t\tdetails, err = u.w.", "TxStore.", "TxDetails(txmgrNs, txHash)\n\t\treturn err\n\t})\n\treturn details, err\n}\n\n// RangeTransactions calls wtxmgr.", "Store.", "RangeTransactions under a single\n// database view tranasction.", "\nfunc (u unstableAPI) RangeTransactions(begin, end int32, f func([]wtxmgr.", "TxDetails) (bool, error)) error {\n\treturn walletdb.", "View(u.w.db, func(dbtx walletdb.", "ReadTx) error {\n\t\ttxmgrNs := dbtx.", "ReadBucket(wtxmgrNamespaceKey)\n\t\treturn u.w.", "TxStore.", "RangeTransactions(txmgrNs, begin, end, f)\n\t})\n}\n" ]

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[ "On March 10, Charles England team one took home the first place price in the second annual Geography Bee competition for Lexington City Schools.", "\n\nThe event was held at Pickett Elementary School and consisted of two teams from each of the elementary schools in the district.", "\n\nPickett Team one and Southwest Team one both tied for second place.", "\n\nThe Geo Bee consisted of four rounds of competition, and each round was composed of a labeling and questioning component. ", "Round one consisted of labeling the 50 states, round two consisted of labeling 25 state capitals, round three included labeling the seven continents and major oceans, and in round four, students had to label 10 countries found throughout the world.", "\n\nThe schools' teams were as followed:\n\nCharles England team one: Natalie Boger, Brooks Calderone, Jayden Moore, Citlaly Razura, Luke Staten, and Addie Walser\n\nCharles England team two: Tyzick Carranco, Josie Gibson, Caalib Grimmett, Juliana Mamaril, Ian Reedy, and Micah Thompson\n\nPickett Team one: Wes Rummage, Lucas Burelson, Charlotte Pyle, Victoria Ross, Maria Cuanas Pedro, and Mailee Kearn\n\nPickett Team two: Colin Weeks, Robert Ross, Nirena Neang, Abigail Lopez, Marie Macauley, and John Jahn\n\nSouthwest Team one: Addison Adinarain, Victoria Freeman, Adyson Griffin, Amaya Hairston, Bradley Johnson, and Jacob Story\n\nSouthwest Team two: Katie Hodges, Serenity Jordan, Jasmine Martinez Carranza, Benny Perez, Niyana Saunders, and Zahari Smith" ]

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[ "Operation Buffalo\n\nOperation Buffalo may refer to:\n\n The breakout from the Operation Shingle Anzio beachhead by the U.S. VI Corps commencing on 23 May 1944\n Operation Buffalo (1956), four open-air nuclear tests in South Australia in late 1956\n Operation Buffalo (1967), a Vietnam War operation in July 1967 by the U.S. 9th Marine Regiment\n Operation Büffel (Buffalo), the withdrawal of the German 9th Army under General Walter Model in March 1943" ]

{ "pile_set_name": "Wikipedia (en)" }

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[ "Anti-neutrophil cytoplasmic antibodies in children with ulcerative colitis.", "\nTests that positively identify individuals with ulcerative colitis, distinguishing them from patients with Crohn disease or other causes of colitis, have not been reliable. ", "Genetic predisposition to inflammatory bowel diseases and genetic influence on immune regulation resulted in the clinical evaluation of potential serologic markers. ", "In adults the presence of anti-neutrophil cytoplasmic antibody (ANCA) in serum identifies patients with ulcerative colitis. ", "In this study we demonstrated that high levels of ANCA are present in 83% of children and adolescents with ulcerative colitis. ", "Furthermore, the majority of patients with ulcerative colitis had a perinuclear pattern of these antibodies by indirect immunofluorescence. ", "The combination of a positive ANCA and perinuclear indirect immunofluorescence pattern was 97% specific for ulcerative colitis. ", "We conclude that determination of ANCA is a sensitive and specific clinical test for identification of children and adolescents with ulcerative colitis." ]

{ "pile_set_name": "PubMed Abstracts" }

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[ "Q:\n\nEliminate The Need For Casting\n\nI am busy writing an API which modifies a graph. ", "To control what I am exposing to the users of the API I am using interfaces.", "\nThe interfaces exposing the methods which will be used are as follows. ", "Essentially this is what the user will see:\npublic interface GraphEditor{\n Edge addEdge(AsbtractNode f, AbstractNode t)\n}\npublic interface AbstractNode{\n}\npublic interface ExampleNode1 extends AbstractNode{\n}\npublic interface ExampleNode2 extends AbstractNode{\n}\n\nThe implementation structure is:\npublic class GraphEditorImpl implements GraphEditor{\n public Edge addEdge(AsbtractNode f, AbstractNode t){\n AbstractNodeImpl from = (AbstractNodeImpl) f;\n AbstractNodeImpl t = (AbstractNodeImpl) f;\n from.getVertex().addEdge(t.getVertex);\n }\n}\npublic class AbstractNodeImpl implements AbstractNode{\n Vertex vertex; //Must remain hidden from users\n public Vertex getVertex(){\n return vertex;\n }\n}\n\nInside graph editor I have a method which allows the user to add and edge between two nodes addEdge. ", "This method has to cast AbstractNode to AbstractNodeImpl in order to access the getVertex() method which is needed to get the 'Vertex'.", "\nI cannot expose getVertex() within the AbstractNode interface as I cannot have users working with the Vertex directly. ", "Is there any way to achieve this sort of functionality without having to cast from the interface into the implementation? ", " \n\nA:\n\nYour current model will not work. ", "You can make it so that the AbstractNode interface only contains information on on the node, and the GraphEditor interface contains all the methods need to edit the graph:\ninterface AbstractNode {\n // identify this node\n // this class only contains the ID of this node\n}\n\ninterface GraphEditor<T extends AbstractNode> {\n // this class stores the graph\n void addEdge(T a, T b);\n}\n\n" ]

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[ "You Deserve Someone Who Will Love Every Piece of You\n\nYou deserve someone who will look at you and know that you are good enough. ", "Someone who will not let you struggle on whether or not you’re worth it, because you are. ", "How people see you or label you doesn’t add, change or deduct who you are.", "\n\nYou deserve someone who will never require you to change in order to fit other people’s standards. ", "Because they understand that our eyes see things, places and people out of our own perspectives and lenses. ", "That in actuality, we don’t see all these by what they are, we see it by what we really are—a reflection of our heart and soul.", "\n\nYou deserve someone whom you don’t have to verify yourself to. ", "Someone who will never see you as too much of this, too much of that, someone who will embrace all your edges and be sure that if not for your edges, you won’t be YOU.", "\n\nYou deserve someone who will never make you feel like you have to be their monumental idea of who you should be. ", "Someone who will understand that the way you talk, the way you dress, the way you carry yourself is what makes you unique, and unique is beautiful. ", "Someone who will look at you and will love your bad parts as much as they love your good parts.", "\n\nYou deserve someone who will never overthink your flaws. ", "Someone who will realize how difficult it has been for you to love and embrace who you are, thus never allowing conclusively insignificant people ruin it in second tops.", "\n\n[video-ad]\n\nYou deserve someone who will exist to see the beauty in you. ", "Someone who will see you with your morning face, smell your morning breath and hear you fail a note as you belt out your favorite song. ", "Someone who will see you lick your chocolate coned ice-cream in the most unattractive way, see you fart, see you eat a hotdog you’ve already dropped—and will still find you beautifully attractive.", "\n\nYou deserve someone who will stay with you during the stormy days and assures you that there will be a rainbow somewhere along the way. ", "Someone who will inspire you, encourage you, push you beyond your own limits. ", "Someone who will trust your talents and abilities even when you can’t trust your own. ", "Someone who will love you at your most critical moments, even those moments when you can’t even love yourself.", "\n\nYou deserve someone who will see you with your bad days and also with your good days. ", "Someone who will see you in your most despairing moments and also in your joyous ones. ", "Someone who will see you from flaw to flaw and embrace every single thing about it. ", "Someone who will see you at your ugliest and bitchiest moments, someone who will see you at your worst and will STILL love you for it.", "\n\nYou deserve someone who will see an intelligent human being even when you don’t make any sense. ", "Someone who will see a genuinely happy person even when you’re being too foolish. ", "Someone who will see the smiles, the misadventures and the facades, but more than that, will see your real substance.", "\n\nYou deserve someone who will see beyond the outer, who will see a smart, beautiful person who fiercely fights to survive life. ", "Someone who will regard the intensity of bravery she’s amounted to. ", "Someone who will recognize her strong pursuit to overcome life’s hardest moments.", "\n\nYou deserve someone who will never give you shit talk about who you are nor lectures you to change. ", "You deserve someone who will be there for you. ", "Patiently loving you for every good and every bad part of yourself. ", "Constantly understanding that this is who you are, and as long as you don’t overstep your boundaries and you respect the people around you, you’re good. ", "They’re good.", "\n\nYou deserve someone who will see you and know that you are worth so much more than meets the eye." ]

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[ "Q:\n\nWorklight 6.1 Encrypted Cache error\n\nI am writing a code to store some information to worklight Encrypted cache.", "\nI am trying to encrypt a value which is primary key in my local DB which looks like 50005 it is a number and I am passing it to write method of encrypted cache\nI am running project in web preview environment.", "\nThe error is Invalid argument value '50005', expected null or 'string'.", "\nFollowing is the code snippet \nfunction setUserId(userId){\n WL.EncryptedCache.write(\"USER_ID\",userId, onCompleteHandler, onErrorHandler);\n }\n\nfunction onCompleteHandler(status){\n console.log(\"Global cache write success.\");", "\n }\n\n function onErrorHandler(status){\n console.log(\"Global cache open error.", "\"+status);\n switch(status){\n case WL.EncryptedCache.", "ERROR_KEY_CREATION_IN_PROGRESS:\n console.log(\"ERROR: KEY CREATION IN PROGRESS\");\n break;\n case WL.EncryptedCache.", "ERROR_LOCAL_STORAGE_NOT_SUPPORTED:\n console.log(\"ERROR: LOCAL STORAGE NOT SUPPORTED\");\n break;\n case WL.EncryptedCache.", "ERROR_NO_EOC:\n console.log(\"ERROR: NO EOC\");\n break;\n case WL.EncryptedCache.", "ERROR_COULD_NOT_GENERATE_KEY:\n console.log(\"ERROR: COULD NOT GENERATE KEY\");\n break;\n case WL.EncryptedCache.", "ERROR_CREDENTIALS_MISMATCH:\n console.log(\"ERROR: CREDENTIALS MISMATCH\");\n break;\n default:\n console.log(\"AN ERROR HAS OCCURED. ", "STATUS :: \" + status);\n }\n }\n\nA:\n\nAlways look at the API documentation before using an API call. ", "Here's the documentation for write.", "\nIt says: \n\nParameters:\nvalue - Mandatory. ", "String. ", "The data to encrypt. ", "When set to null, the key is removed.", "\n\nChange:\nWL.EncryptedCache.write(\"USER_ID\",userId, onCompleteHandler, onErrorHandler);\n\nto:\nWL.EncryptedCache.write(\"USER_ID\",userId.toString(), onCompleteHandler, onErrorHandler);\n\nYou can only store strings using that API. ", "If you want to store objects, you must use JSON.stringify (object to string) and JSON.parse (string to object). ", "If you want to go from string to int, you can use the parseInt function like this: parseInt(userId).", "\nAlternatively, you could use the JSONStore API instead. ", "Note it's only supported on Android and iOS (in Worklight v6.2 it's supported on WP8 and W8 too). ", "The code would look something like this:\nvar collections = {\n users : {\n searchFields : {'userid' : 'integer'}\n }\n};\n\nvar options = {\n password: '123'\n};\n\nWL.JSONStore.init(collections, options)\n.then(function () {\n return WL.JSONStore.get('users').add({userid: 50005});\n})\n.then(function () {\n return WL.JSONStore.get('users').findAll();//or .find({userid: 50005})\n})\n.then(function (results) {\n WL.Logger.debug(results);\n})\n.fail(function () {\n //handle failure in any of the API calls above\n});\n\nThere's documentation here for JSONStore.", "\n\n" ]

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[ "Surgical staplers have found wide-spread acceptance in surgical procedures since such devices drastically reduce the amount of time required by surgeons to suture body tissue.", "\nTypically, such surgical staplers have a pair of upper and lower jaw members to clamp body tissue therebetween. ", "The lower jaw member typically carries a staple cartridge which contains a plurality of staple drive members having cam contact surfaces in association with one or more laterally spaced rows of surgical staples. ", "The upper jaw member has an anvil portion to close the surgical staples which have passed through body tissue. ", "The stapler is further provided with at least one pusher bar longitudinally moveable relative to the jaw members and having a cam surface for engaging the cam contact surfaces of the staple drive members so that the longitudinal movement of the pusher bar sequentially drives the staple drive members, through a camming action, to fire surgical staples from the cartridge.", "\nFor some uses, double-row stapling is preferable to single-row stapling in providing secure hemostasis. ", "Accordingly, some currently available surgical staplers include double staple drive members respectively formed of a single piece construction having two staple driving surfaces for firing two rows of surgical staples. ", "These double staple drive members are longitudinally arranged in a row within the cartridge to be driven with a single stroke of the pusher bar. ", "Typical arrangements include double staple drive members each having a longitudinally-staggered pair of staple driving surfaces which bridge the pusher bar path.", "\nU.S. Pat. ", "No. ", "4,978,049, to Green, discloses a modification which includes triple staple drive members respectively formed of a single piece construction having three laterally spaced staple driving surfaces capable of firing three rows of surgical staples. ", "However, this arrangement has distinct disadvantages.", "\nAs the number of the staple driving surfaces for firing individual staples increases, greater resistance is created to the longitudinal movement of pusher bar. ", "This requires a surgeon to apply a greater force in operating the surgical stapler. ", "Furthermore, there can be a tendency for the staple drive members to bind against the walls of cartridge slots and jam due to an unbalanced force distribution applied to staple driving surfaces.", "\nTherefore, it is highly desirable to design and arrange the pattern of individual staple drive members and driving surfaces to facilitate application of the driving force by the surgeon throughout the pusher bar stroke to provide a surgical stapler which allows a smooth stapling operation." ]

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[ "[Management of massive hemoptysis].", "\nMassive hemoptyses are serious clinical conditions that can quickly jeopardize the vital prognosis. ", "The major risk is asphyxiation, due to the bleeding into the tracheobronchial tree. ", "The clinician should provide in parallel support for diagnosis and treatment, locating the bleeding but also finding its cause. ", "Such patients should be cared for by a multidisciplinary team, having quick access to an important technical support. ", "The association fiberoptic bronchoscopy-chest CT scan seems to be the most effective to locate and identify the cause of the bleeding. ", "The development of bronchial artery embolization has revolutionized the management of these patients, replacing surgery in many of its indications. ", "The latter still keeps a place in the management of these patients. ", "Indeed, it is the main etiological treatment, preventing the vast majority of recidivism. ", "It is absolutely indicated in the treatment of bleeding from the pulmonary vessels, and in case of failure of other techniques. ", "It should be performed whenever possible away from the episode of hemoptysis, in order to minimize the operative risk." ]

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[ "---\nabstract: 'This paper presents a novel approach for learning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models from examples that compiles this inductive learning task into a classical planning task. ", "Interestingly, the compilation approach is flexible to different amounts of available input knowledge; the learning examples can range from a set of plans (with their corresponding initial and final states) to just a pair of initial and final states (no intermediate action or state is given). ", "Moreover, the compilation accepts partially specified action models and it can be used to validate whether the observation of a plan execution follows a given [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model, even if this model is not fully specified.'", "\nauthor:\n- '\\#39'\n- Diego Aineto\n- Sergio Jiménez\n- |\n Eva Onaindia\\\n [Departamento de Sistemas Informáticos y Computación]{}\\\n [Universitat Politècnica de València.]{}\\\n [Camino de Vera s/n. 46022 Valencia, Spain]{}\\\n [{dieaigar,serjice,onaindia}@dsic.upv.es]{}\nbibliography:\n- 'planlearnbibliography.bib'\ntitle: 'Learning [<span style=\"font-variant:small-caps;\">Strips</span>]{} Action Models with Classical Planning'\n---\n\nIntroduction\n============\n\nBesides [*plan synthesis*]{} [@ghallab2004automated], planning action models are also useful for [*plan/goal recognition*]{} [@ramirez2012plan]. ", "In both planning tasks, an automated planner is required to reason about action models that correctly and completely capture the possible world transitions [@geffner:book:2013]. ", "Unfortunately, building planning action models is complex, even for planning experts, and this knowledge acquisition task is a bottleneck that limits the potential of AI planning [@kambhampati:modellite:AAAI2007].", "\n\nOn the other hand, Machine Learning (ML) has shown to be able to compute a wide range of different kinds of models from examples [@michalski2013machine]. ", "The application of inductive ML to learning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models, the vanilla action model for planning [@fikes1971strips], is not straightforward though:\n\n- The [*input*]{} to ML algorithms (the learning/training data) is usually a finite set of vectors that represent the value of some fixed object features. ", "The input for learning planning action models is, however, observations of plan executions (where each plan possibly has a different length).", "\n\n- The [*output*]{} of ML algorithms is usually a scalar value (an integer, in the case of [*classification*]{} tasks, or a real value, in the case of [*regression*]{} tasks). ", "When learning action models the output is, for each action, the preconditions, negative and positive effects that define the possible state transitions.", "\n\nLearning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models is a well-studied problem with sophisticated algorithms such as [ARMS]{} [@yang2007learning], [SLAF]{} [@amir:alearning:JAIR08] or [LOCM]{} [@cresswell2013acquiring], which do not require full knowledge of the intermediate states traversed by the example plans. ", "Motivated by recent advances on the synthesis of different kinds of generative models with classical planning [@bonet2009automatic; @segovia2016hierarchical; @segovia2017generating], this paper introduces an innovative planning compilation approach for learning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models. ", "The compilation approach is appealing by itself because it opens up the door to the bootstrapping of planning action models, but also because:\n\n1. ", " It is flexible to various amounts of input knowledge. ", "Learning examples range from a set of plans (with their corresponding initial and final states) to just a pair of initial and final states where no intermediate state or action is observed.", "\n\n2. ", " It accepts previous knowledge about the structure of the actions in the form of partially specified action models. ", "In the extreme, the compilation can validate whether an observed plan execution is valid for a given [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model, even if this model is not fully specified.", "\n\nThe second section of the paper formalizes the classical planning model, its extension to [*conditional effects*]{} (a requirement of the proposed compilation) and the [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model (the output of the addressed learning task). ", "The third section formalizes the task of learning action models with different amounts of available input knowledge. ", "The fourth and fifth sections describe our compilation approach to tackle the formalized learning tasks. ", "Finally, the last sections show the experimental evaluation, discuss the strengths and weaknesses of the compilation approach and propose several opportunities for future research.", "\n\nBackground\n==========\n\nThis section defines the planning model and the output of the learning tasks addressed in the paper.", "\n\nClassical planning with conditional effects\n-------------------------------------------\n\nOur approach to learning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models is compiling this learning task into a classical planning task with conditional effects. ", "Conditional effects allow us to compactly define actions whose effects depend on the current state. ", "Supporting conditional effects is now a requirement of the IPC [@vallati:IPC:AIM2015] and many classical planners cope with conditional effects without compiling them away.", "\n\nWe use $F$ to denote the set of [*fluents*]{} (propositional variables) describing a state. ", "A [*literal*]{} $l$ is a valuation of a fluent $f\\in F$; i.e. either $l=f$ or $l=\\neg f$. A set of literals $L$ represents a partial assignment of values to fluents (without loss of generality, we will assume that $L$ does not contain conflicting values). ", "We use $\\mathcal{L}(F)$ to denote the set of all literal sets on $F$; i.e. all partial assignments of values to fluents.", "\n\nA [*state*]{} $s$ is a full assignment of values to fluents; $|s|=|F|$, so the size of the state space is $2^{|F|}$. Explicitly including negative literals $\\neg f$ in states simplifies subsequent definitions but often we will abuse of notation by defining a state $s$ only in terms of the fluents that are true in $s$, as it is common in [<span style=\"font-variant:small-caps;\">Strips</span>]{} planning.", "\n\nA [*classical planning frame*]{} is a tuple $\\Phi={{\\langle F,A \\rangle}}$, where $F$ is a set of fluents and $A$ is a set of actions. ", "An action $a\\in A$ is defined with [*preconditions*]{}, ${\\mathsf{pre}}(a)\\subseteq\\mathcal{L}(F)$, [*positive effects*]{}, ${\\mathsf{eff}}^+(a)\\subseteq\\mathcal{L}(F)$, and [*negative effects*]{} ${\\mathsf{eff}}^-(a)\\subseteq\\mathcal{L}(F)$. We say that an action $a\\in A$ is [*applicable*]{} in a state $s$ iff ${\\mathsf{pre}}(a)\\subseteq s$. The result of applying $a$ in $s$ is the [*successor state*]{} denoted by $\\theta(s,a)=\\{s\\setminus{\\mathsf{eff}}^-(a))\\cup{\\mathsf{eff}}^+(a)\\}$.\n\nAn action $a\\in A$ with conditional effects is defined as a set of [*preconditions*]{} ${\\mathsf{pre}}(a)$ and a set of [*conditional effects*]{} ${\\mathsf{cond}}(a)$. Each conditional effect $C\\rhd E\\in{\\mathsf{cond}}(a)$ is composed of two sets of literals: $C \\subseteq \\mathcal{L}(F)$, the [*condition*]{}, and $E \\subseteq \\mathcal{L}(F)$, the [*effect*]{}. ", "An action $a\\in A$ is [*applicable*]{} in a state $s$ iff ${\\mathsf{pre}}(a)\\subseteq s$, and the [*triggered effects*]{} resulting from the action application are the effects whose conditions hold in $s$: $$triggered(s,a)=\\bigcup_{C\\rhd E\\in{\\mathsf{cond}}(a),C\\subseteq s} E,$$\n\nThe result of applying action $a$ in state $s$ is the [*successor*]{} state $\\theta(s,a)=\\{s\\setminus{\\mathsf{eff}}_c^-(s,a))\\cup{\\mathsf{eff}}_c^+(s,a)\\}$ where ${\\mathsf{eff}}_c^-(s,a)\\subseteq triggered(s,a)$ and ${\\mathsf{eff}}_c^+(s,a)\\subseteq triggered(s,a)$ are, respectively, the triggered [*negative*]{} and [*positive*]{} effects.", "\n\nA [*classical planning problem*]{} is a tuple $P={{\\langle F,A,I,G \\rangle}}$, where $I$ is an initial state and $G\\subseteq\\mathcal{L}(F)$ is a goal condition. ", "A [*plan*]{} for $P$ is an action sequence $\\pi={{\\langle a_1, \\ldots, a_n \\rangle}}$ that induces the [*state trajectory*]{} ${{\\langle s_0, s_1, \\ldots, s_n \\rangle}}$ such that $s_0=I$ and $a_i$ ([$1\\leq i\\leq n$]{}) is applicable in $s_{i-1}$ and generates the successor state $s_i=\\theta(s_{i-1},a_i)$. The [*plan length*]{} is denoted with $|\\pi|=n$ . ", "A plan $\\pi$ [*solves*]{} $P$ iff $G\\subseteq s_n$; i.e. if the goal condition is satisfied in the last state resulting from the application of the plan $\\pi$ in the initial state $I$.\n\n[<span style=\"font-variant:small-caps;\">Strips</span>]{} action schemas and [*variable name*]{} objects\n-------------------------------------------------------------------------------------------------------\n\nOur approach is aimed at learning PDDL action schemas that follow the [<span style=\"font-variant:small-caps;\">Strips</span>]{} requirement [@mcdermott1998pddl; @fox2003pddl2]. ", "Figure \\[fig:stack\\] shows the [*stack*]{} schema of a four-operator [*blocksworld*]{} domain [@slaney2001blocks] encoded in PDDL.", "\n\n (:action stack\n :parameters (?", "v1 ?", "v2 - object)\n :precondition (and (holding ?", "v1) (clear ?", "v2))\n :effect (and (not (holding ?", "v1))\n (not (clear ?", "v2))\n (handempty) (clear ?", "v1)\n (on ?", "v1 ?", "v2)))\n\nTo formalize the output of the learning task, we assume that fluents $F$ are instantiated from a set of [*predicates*]{} $\\Psi$, as in PDDL. ", "Each predicate $p\\in\\Psi$ has an argument list of arity $ar(p)$. Given a set of [*objects*]{} $\\Omega$, the set of fluents $F$ is induced by assigning objects in $\\Omega$ to the arguments of the predicates in $\\Psi$; i.e. $F=\\{p(\\omega):p\\in\\Psi,\\omega\\in\\Omega^{ar(p)}\\}$, where $\\Omega^k$ is the $k$-th Cartesian power of $\\Omega$.\n\nLet $\\Omega_v=\\{v_i\\}_{i=1}^{\\operatorname*{max}_{a\\in A} ar(a)}$ be a new set of objects denoted as [*variable names*]{} ($\\Omega\\cap\\Omega_v=\\emptyset$). ", "$\\Omega_v$ is bound to the maximum arity of an action in a given planning frame. ", "For instance, in a three-block *blocksworld*, $\\Omega=\\{block_1, block_2, block_3\\}$ while $\\Omega_v=\\{v_1, v_2\\}$ because the operators with the maximum arity, [stack]{} and [unstack]{}, have two parameters each.", "\n\nLet $F_v$ be a new set of fluents, $F\\cap F_v=\\emptyset$, that results from instantiating the predicates in $\\Psi$ using exclusively objects of $\\Omega_v$. $F_v$ defines the elements of the preconditions and effects of an action schema. ", "For instance, in the *blocksworld* domain, $F_v$=[{handempty, holding($v_1$), holding($v_2$), clear($v_1$), clear($v_2$), ontable($v_1$), ontable($v_2$), on($v_1,v_1$), on($v_1,v_2$), on($v_2,v_1$), on($v_2,v_2$)}]{}.", "\n\nFinally, we assume that an action $a\\in A$ is instantiated from a [<span style=\"font-variant:small-caps;\">Strips</span>]{} operator schema $\\xi={{\\langle head(\\xi),pre(\\xi),add(\\xi),del(\\xi) \\rangle}}$ where:\n\n- $head(\\xi)={{\\langle name(\\xi),pars(\\xi) \\rangle}}$ is the operator [*header*]{} defined by its name and the corresponding [*variable names*]{}, $pars(\\xi)=\\{v_i\\}_{i=1}^{ar(\\xi)}$. For instance, the headers of a four-operator *blocksworld* domain are: [pickup($v_1$), putdown($v_1$), stack($v_1,v_2$)]{} and [unstack($v_1,v_2$)]{}.", "\n\n- $pre(\\xi)\\subseteq F_v$ is the set of preconditions, $del(\\xi)\\subseteq F_v$ the negative effects and $add(\\xi)\\subseteq F_v$ the positive effects such that $del(\\xi)\\subseteq pre(\\xi)$, $del(\\xi)\\cap add(\\xi)=\\emptyset$ and $pre(\\xi)\\cap add(\\xi)=\\emptyset$.\n\nLearning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models\n===============================================================================\n\nLearning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models from fully available input knowledge, i.e. from plans where the [*pre-*]{} and [*post-states*]{} of every action in the plans are known, is straightforward. ", "When intermediate states are available, operator schemas are derived lifting the literals that change between the pre and post-state of each action execution. ", "Preconditions of an action are derived lifting the minimal set of literals that appears in all the pre-states of the corresponding action [@jimenez2012review].", "\n\nThis section formalizes more challenging learning tasks, where less input knowledge is available:\n\n### Learning from (initial, final) state pairs.", "\n\nThis learning task amounts to observing agents acting in the world but watching only the result of their plans execution. ", "No intermediate information about the actions in the plans is given. ", "This learning task is formalized as $\\Lambda={{\\langle \\Psi,\\Sigma \\rangle}}$:\n\n- $\\Psi$ is the set of predicates that define the abstract state space of a given planning domain.", "\n\n- $\\Sigma=\\{\\sigma_1,\\ldots,\\sigma_{\\tau}\\}$ is a set of $(initial, final)$ state pairs called [*labels*]{}. ", "Each label $\\sigma_t=(s_0^t,s_{n}^t)$, [$1\\leq t\\leq \\tau$]{}, comprises the [*final*]{} state $s_{n}^t$ resulting from executing an unknown plan $\\pi_t={{\\langle a_1^t, \\ldots, a_n^t \\rangle}}$ in the [*initial*]{} state $s_0^t$.\n\n### Learning from labeled plans.", "\n\nWe augment the input knowledge with the actions executed by the observed agent and define the learning task $\\Lambda'={{\\langle \\Psi,\\Sigma,\\Pi \\rangle}}$:\n\n- $\\Pi=\\{\\pi_1,\\ldots,\\pi_{\\tau}\\}$ is a given set of example plans where $\\pi_t={{\\langle a_1^t, \\ldots, a_n^t \\rangle}}$, [$1\\leq t\\leq \\tau$]{}, is an action sequence that induces the corresponding state sequence ${{\\langle s_0^t, s_1^t, \\ldots, s_n^t \\rangle}}$ such that $a_i^t$, [$1\\leq i\\leq n$]{}, is applicable in $s_{i-1}^t$ and generates $s_i^t=\\theta(s_{i-1}^t,a_i^t)$.\n\nFigure \\[fig:lexample\\] shows an example of a learning task $\\Lambda'$ of the [*blocksworld*]{} domain. ", "This task has a single learning example, $\\Pi=\\{\\pi_1\\}$ and $\\Sigma=\\{\\sigma_1\\}$, that corresponds to observing the execution of an eight-action plan $(|\\pi_1|=8)$ for inverting a four-block tower.", "\n\n### Learning from partially specified action models.", "\n\nIn case that partially specified operator schemas are available, we can incorporate this information within the learning task. ", "The new leaning task is defined as $\\Lambda''={{\\langle \\Psi,\\Sigma,\\Pi,\\Xi_0 \\rangle}}$:\n\n- $\\Xi_0$ is a partially specified action model in which some preconditions and effects are known a priori.", "\n\nA solution to $\\Lambda$ is a set of operator schemas $\\Xi$ that is compliant with the predicates in $\\Psi$ and the set of initial and final states $\\Sigma$. In a $\\Lambda$ learning scenario, a solution must not only determine a possible [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model but also the plans $\\pi_t$, [$1\\leq t\\leq \\tau$]{}, that explain the given labels $\\Sigma$ using the learned model. ", "A solution to $\\Lambda'$ is a set of [<span style=\"font-variant:small-caps;\">Strips</span>]{} operator schemas $\\Xi$ (one schema $\\xi={{\\langle head(\\xi),pre(\\xi),add(\\xi),del(\\xi) \\rangle}}$ for each action that has a different name in the example plans $\\Pi$) that is compliant with the predicates $\\Psi$, the example plans $\\Pi$, and their corresponding labels $\\Sigma$. Finally, a solution to $\\Lambda''$ is a set of [<span style=\"font-variant:small-caps;\">Strips</span>]{} operator schemas $\\Xi$ that is also compliant with the provided partially specified action model $\\Xi_0$.\n\n[;;; Predicates in $\\Psi$]{}\n\n (handempty) (holding ?", "o - object)\n (clear ?", "o - object) (ontable ?", "o - object)\n (on ?", "o1 - object ?", "o2 - object)\n\n[.25]{} [;;; Plan $\\pi_1$]{}\n\n 0: (unstack A B)\n 1: (putdown A)\n 2: (unstack B C)\n 3: (stack B A)\n 4: (unstack C D)\n 5: (stack C B)\n 6: (pickup D)\n 7: (stack D C)\n\n[.6]{} [;;; Label $\\sigma_1=(s_0^1,s_{n}^1)$]{}\n\n``` {mathescape=\"\"}\n```\n\n(initD) \\[draw\\][D]{}; (initC) \\[draw, above=of initD.north\\][C]{}; (initB) \\[draw, above=of initC.north\\][B]{}; (initA) \\[draw, above=of initB.north\\][A]{}; (-1,-0.25) – (2.5,-0.25);\n\n(goalA) \\[draw, right=10mm of initD\\][A]{}; (goalB) \\[draw, right=10mm of initC\\][B]{}; (goalC) \\[draw, right=10mm of initB\\][C]{}; (goalD) \\[draw, right=10mm of initA\\][D]{};\n\nLearning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models with planning\n=============================================================================================\n\nIn our approach, a learning task $\\Lambda$, $\\Lambda'$ or $\\Lambda''$ is solved by compiling it into a classical planning task with conditional effects. ", "The intuition behind the compilation is that a solution to the resulting classical planning task is a sequence of actions that:\n\n1. ", " [*Programs the [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model $\\Xi$*]{}. ", "A solution plan has a [*prefix*]{} that, for each $\\xi\\in\\Xi$, determines the fluents from $F_v$ that belong to $pre(\\xi)$, $del(\\xi)$ and $add(\\xi)$.\n\n2. ", " [*Validates the programmed [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model $\\Xi$ in the given input knowledge*]{} (the labels $\\Sigma$ and $\\Pi$, and/or $\\Xi_0$ if available). ", "For every label $\\sigma_t\\in \\Sigma$, a solution plan has a postfix that produces a final state $s_{n}^t$ using the programmed action model $\\Xi$ in the corresponding initial state $s_0^t$. This process is the validation of the programmed action model $\\Xi$ with the set of learning examples [$1\\leq t\\leq \\tau$]{}.", "\n\nTo formalize our compilation we first define a set of classical planning instances $P_t={{\\langle F,\\emptyset,I_t,G_t \\rangle}}$ that belong to the same planning frame (i.e. same fluents and actions but different initial states and goals). ", "Fluents $F$ are built instantiating the predicates in $\\Psi$ with the objects of the input labels $\\Sigma$. Formally, $\\Omega=\\bigcup_{\\small 1\\leq t\\leq \\tau} obj(s_0^t)$, where $obj$ is a function that returns the objects that appear in a fully specified state. ", "The set of actions, $A=\\emptyset$, is empty because the action model is initially unknown. ", "Finally, the initial state $I_t$ is given by the state $s_0^t\\in \\sigma_t$, and the goals $G_t$ are defined by the state $s_n^t\\in \\sigma_t$.\n\nWe can now formalize the compilation approach. ", "We start with $\\Lambda$ as it requires the least input knowledge. ", "Given a learning task $\\Lambda={{\\langle \\Psi,\\Sigma \\rangle}}$, the compilation outputs a classical planning task $P_{\\Lambda}={{\\langle F_{\\Lambda},A_{\\Lambda},I_{\\Lambda},G_{\\Lambda} \\rangle}}$:\n\n- $F_{\\Lambda}$ extends $F$ with:\n\n - Fluents $pre_f(\\xi)$, $del_f(\\xi)$ and $add_f(\\xi)$, for every $f\\in F_v$ and $\\xi \\in \\Xi$ that represent the programmed action model. ", "If a fluent of $pre_f(\\xi)/del_f(\\xi)/add_f(\\xi)$ holds, it means that $f$ is a precondition/negative effect/positive effect of the operator schema $\\xi\\in \\Xi$. For instance, the preconditions of the $stack$ schema (Figure \\[fig:stack\\]) are represented by fluents [pre\\_holding\\_stack\\_$v_1$]{} and [pre\\_clear\\_stack\\_$v_2$]{}.", "\n\n - A fluent $mode_{prog}$ to indicate whether the operator schemas are being programmed or validated (when already programmed)\n\n - Fluents $\\{test_t\\}_{1\\leq t\\leq \\tau}$ which represent the examples where the action model will be validated.", "\n\n- $I_{\\Lambda}$ contains the fluents from $F$ that encode $s_0^1$ (the initial state of the first label), the fluents in every $pre_f(\\xi)\\in F_{\\Lambda}$ and the fluent $mode_{prog}$ set to true. ", "Our compilation assumes that any operator schema is initially programmed with every possible precondition (the most specific learning hypothesis), no negative effect and no positive effect.", "\n\n- $G_{\\Lambda}=\\bigcup_{1\\leq t\\leq \\tau}\\{test_t\\}$ indicates that the programmed action model is validated in all the learning examples.", "\n\n- $A_{\\Lambda}$ contains actions of three kinds:\n\n 1. ", " Actions for [*programming*]{} an operator schema $\\xi\\in\\Xi$:\n\n - Actions for [**removing**]{} a [*precondition*]{} $f\\in F_v$ from $\\xi$.\n\n $$\\begin{aligned}\n \\hspace*{7pt}{\\mathsf{pre}}(\\mathsf{programPre_{f,\\xi}})=&\\{\\neg del_{f}(\\xi),\\neg add_{f}(\\xi),\\\\\n & mode_{prog}, pre_{f}(\\xi)\\},\\\\\n {\\mathsf{cond}}(\\mathsf{programPre_{f,\\xi}})=&\\{\\emptyset\\}\\rhd\\{\\neg pre_{f}(\\xi)\\}.\\end{aligned}$$\n\n - Actions for [**adding**]{} a [*negative*]{} or [*positive*]{} effect $f\\in F_v$ to $\\xi$.\n\n $$\\begin{aligned}\n \\hspace*{7pt}{\\mathsf{pre}}(\\mathsf{programEff_{f,\\xi}})=&\\{\\neg del_{f}(\\xi),\\neg add_{f}(\\xi),\\\\\n & mode_{prog}\\},\\\\\n {\\mathsf{cond}}(\\mathsf{programEff_{f,\\xi}})=&\\{pre_{f}(\\xi)\\}\\rhd\\{del_{f}(\\xi)\\},\\\\\n &\\{\\neg pre_{f}(\\xi)\\}\\rhd\\{add_{f}(\\xi)\\}.\\end{aligned}$$\n\n 2. ", " Actions for [*applying*]{} an already programmed operator schema $\\xi\\in\\Xi$ bound to the objects $\\omega\\subseteq\\Omega^{ar(\\xi)}$. We assume operators headers are known so the binding of $\\xi$ is done implicitly by order of appearance of the action parameters, i.e. variables $pars(\\xi)$ are bound to the objects in $\\omega$ that appear in the same position. ", "Figure \\[fig:compilation\\] shows the PDDL encoding of the action for applying a programmed operator $stack$.\n\n $$\\begin{aligned}\n \\hspace*{7pt}{\\mathsf{pre}}(\\mathsf{apply_{\\xi,\\omega}})=&\\{pre_{f}(\\xi)\\implies p(\\omega)\\}_{\\forall p\\in\\Psi,f=p(pars(\\xi))},\\\\\n {\\mathsf{cond}}(\\mathsf{apply_{\\xi,\\omega}})=&\\{del_{f}(\\xi)\\}\\rhd\\{\\neg p(\\omega)\\}_{\\forall p\\in\\Psi,f=p(pars(\\xi))},\\\\\n &\\{add_{f}(\\xi)\\}\\rhd\\{p(\\omega)\\}_{\\forall p\\in\\Psi,f=p(pars(\\xi))},\\\\\n &\\{mode_{prog}\\}\\rhd\\{\\neg mode_{prog}\\}.\\end{aligned}$$\n\n 3. ", " Actions for [*validating*]{} the learning example [$1\\leq t\\leq \\tau$]{}.", "\n\n $$\\begin{aligned}\n \\hspace*{7pt}{\\mathsf{pre}}(\\mathsf{validate_{t}})=&G_t\\cup\\{test_j\\}_{1\\leq j<t}\\\\\n &\\cup\\{\\neg test_j\\}_{t\\leq j\\leq \\tau}\\cup \\{\\neg mode_{prog}\\},\\\\\n {\\mathsf{cond}}(\\mathsf{validate_{t}})=&\\{\\emptyset\\}\\rhd\\{test_t\\} \\cup \\{\\neg f\\}_{\\forall f\\in G_t, f \\notin I_{t+1}}\\\\\n &\\cup \\{f\\}_{\\forall f\\in I_{t+1}, f \\notin G_t}.\\end{aligned}$$\n\nA classical plan $\\pi$ that solves $P_{\\Lambda}$ induces an action model $\\Xi$ that solves the learning task $\\Lambda$.\n\nOnce operator schemas $\\Xi$ are programmed, they can only be applied and validated according to the $mode_{prog}$ fluent. ", "To solve $P_{\\Lambda}$, goals $\\{test_t\\}$, [$1\\leq t\\leq \\tau$]{} can only be achieved by executing an applicable sequence of programmed operator schemas that reaches the final state $s_n^t$, defined in $\\sigma_t$, starting from $s_0^t$. If this is achieved for all the input examples [$1\\leq t\\leq \\tau$]{}, it means that the programmed action model $\\Xi$ is compliant with the provided input knowledge and hence it is a solution to $\\Lambda$.\n\nThe compilation is [*complete*]{} in the sense that it does not discard any possible [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model. ", "The size of the classical planning task $P_{\\Lambda}$ depends on:\n\n- The arity of the actions headers in $\\Xi$ and the predicates $\\Psi$ of the learning task. ", "The larger the arity, the larger the $F_v$ set, which in turn defines the size of the fluent sets $pre_f(\\xi)/del_f(\\xi)/add_f(\\xi)$ and the corresponding set of [*programming*]{} actions.", "\n\n- The number of learning examples. ", "The larger this number, the more $test_t$ fluents and $\\mathsf{validate_{t}}$ actions in $P_{\\Lambda}$.\n\n<!-- --", ">\n\n (:action apply_stack\n :parameters (?", "o1 - object ?", "o2 - object)\n :precondition\n (and (or (not (pre_on_stack_v1_v1)) (on ?", "o1 ?", "o1))\n (or (not (pre_on_stack_v1_v2)) (on ?", "o1 ?", "o2))\n (or (not (pre_on_stack_v2_v1)) (on ?", "o2 ?", "o1))\n (or (not (pre_on_stack_v2_v2)) (on ?", "o2 ?", "o2))\n (or (not (pre_ontable_stack_v1)) (ontable ?", "o1))\n (or (not (pre_ontable_stack_v2)) (ontable ?", "o2))\n (or (not (pre_clear_stack_v1)) (clear ?", "o1))\n (or (not (pre_clear_stack_v2)) (clear ?", "o2))\n (or (not (pre_holding_stack_v1)) (holding ?", "o1))\n (or (not (pre_holding_stack_v2)) (holding ?", "o2))\n (or (not (pre_handempty_stack)) (handempty)))\n :effect\n (and (when (del_on_stack_v1_v1) (not (on ?", "o1 ?", "o1)))\n (when (del_on_stack_v1_v2) (not (on ?", "o1 ?", "o2)))\n (when (del_on_stack_v2_v1) (not (on ?", "o2 ?", "o1)))\n (when (del_on_stack_v2_v2) (not (on ?", "o2 ?", "o2)))\n (when (del_ontable_stack_v1) (not (ontable ?", "o1)))\n (when (del_ontable_stack_v2) (not (ontable ?", "o2)))\n (when (del_clear_stack_v1) (not (clear ?", "o1)))\n (when (del_clear_stack_v2) (not (clear ?", "o2)))\n (when (del_holding_stack_v1) (not (holding ?", "o1)))\n (when (del_holding_stack_v2) (not (holding ?", "o2)))\n (when (del_handempty_stack) (not (handempty)))\n (when (add_on_stack_v1_v1) (on ?", "o1 ?", "o1))\n (when (add_on_stack_v1_v2) (on ?", "o1 ?", "o2))\n (when (add_on_stack_v2_v1) (on ?", "o2 ?", "o1))\n (when (add_on_stack_v2_v2) (on ?", "o2 ?", "o2))\n (when (add_ontable_stack_v1) (ontable ?", "o1))\n (when (add_ontable_stack_v2) (ontable ?", "o2))\n (when (add_clear_stack_v1) (clear ?", "o1))\n (when (add_clear_stack_v2) (clear ?", "o2))\n (when (add_holding_stack_v1) (holding ?", "o1))\n (when (add_holding_stack_v2) (holding ?", "o2))\n (when (add_handempty_stack) (handempty))\n (when (modeProg) (not (modeProg)))))\n\nConstraining the learning hypothesis space with additional input knowledge {#sec:Constraining}\n==========================================================================\n\nIn this section, we show that further input knowledge can be used to constrain the space of possible action models and to make the learning task more practicable.", "\n\nLabeled plans\n-------------\n\nWe extend the compilation to consider labeled plans. ", "Given a learning task $\\Lambda'={{\\langle \\Psi,\\Sigma,\\Pi \\rangle}}$, the compilation outputs a classical planning task $P_{\\Lambda'}={{\\langle F_{\\Lambda'},A_{\\Lambda'},I_{\\Lambda'},G_{\\Lambda'} \\rangle}}$:\n\n- $F_{\\Lambda'}$ extends $F_{\\Lambda}$ with $F_{\\Pi}=\\{plan(name(\\xi),\\Omega^{ar(\\xi)},j)\\}$, the fluents to code the steps of the plans in $\\Pi$, where $F_{\\pi_t}\\subseteq F_{\\Pi}$ encodes $\\pi_t\\in \\Pi$. Fluents $at_j$ and $next_{j,j+1}$, [$1\\leq j< n$]{}, are also added to represent the current plan step and to iterate through the steps of a plan.", "\n\n- $I_{\\Lambda'}$ extends $I_{\\Lambda}$ with fluents $F_{\\pi_1}$ plus fluents $at_1$ and $\\{next_{j,j+1}\\}$, [$1\\leq j<n$]{}, to indicate the plan step where the action model is validated. ", "As in the original compilation, $G_{\\Lambda'}=G_{\\Lambda}=\\bigcup_{1\\leq t\\leq \\tau}\\{test_t\\}$.\n\n- With respect to $A_{\\Lambda'}$.\n\n 1. ", " The actions for [*programming*]{} the preconditions/effects of a given operator schema $\\xi\\in\\Xi$ are the same.", "\n\n 2. ", " The actions for [*applying*]{} an already programmed operator have an extra precondition $f\\in F_{\\Pi}$ that encodes the current plan step, and extra conditional effects $\\{at_{j}\\}\\rhd\\{\\neg at_{j},at_{j+1}\\}_{\\forall j\\in [1,n]}$ for advancing to the next plan step. ", "With this mechanism we ensure that these actions are applied in the same order as in the example plans.", "\n\n 3. ", " The actions for [*validating*]{} the current learning example have an extra precondition, $at_{|\\pi_t|}$, to indicate that the current plan $\\pi_t$ is fully executed and extra conditional effects to remove plan $\\pi_{t}$ and initiate the next plan $\\pi_{t+1}$:\n\n $$\\begin{aligned}\n &\\{\\emptyset\\}\\rhd\\{\\neg at_{|\\pi_t|},at_1\\} \\cup \\{\\neg f\\}_{f\\in F_{\\pi_t}} \\cup \\{f\\}_{f\\in F_{\\pi_t+1}}.\\end{aligned}$$\n\nPartially specified action models\n---------------------------------\n\nThe known preconditions and effects of a partially specified action model are encoded as fluents $pre_f(\\xi)$, $del_f(\\xi)$ and $add_f(\\xi)$ set to true in the initial state $I_{\\Lambda'}$. The programming actions, $\\mathsf{programPre_{f,\\xi}}$ and $\\mathsf{programEff_{f,\\xi}}$, become now unnecessary and they are removed from $A_{\\Lambda'}$, thus making the planning task $P_{\\Lambda'}$ be easier to solve.", "\n\nTo illustrate this, the plan of Figure \\[fig:plan\\] is a solution to a learning task $\\Lambda''={{\\langle \\Psi,\\Sigma,\\Pi,\\Xi_0 \\rangle}}$ for acquiring the [*blocksworld*]{} action model where operator schemas for [pickup]{}, [putdown]{} and [unstack]{} are specified in $\\Xi_0$. This plan programs and validates the operator schema [stack]{} from [*blocksworld*]{} using the plan $\\pi_1$ and label $\\sigma_1$ shown in Figure \\[fig:lexample\\]. ", "Plan steps $[0,8]$ program the preconditions of the [stack]{} operator, steps $[9,13]$ program the operator effects and steps $[14,22]$ validate the programmed operators following the plan $\\pi_1$ shown in Figure \\[fig:lexample\\].", "\n\nIn the extreme, when a fully specified [<span style=\"font-variant:small-caps;\">Strips</span>]{} action model is given in $\\Xi_0$, the compilation validates whether an observed plan follows the given model. ", "In this case, if a solution plan is found for $P_{\\Lambda'}$, it means that the given action model is [*valid*]{} for the provided examples. ", "If $P_{\\Lambda'}$ is unsolvable then it means that the action model is invalid because it is not compliant with all the given examples. ", "Tools for plan validation like VAL [@howey2004val] could also be used at this point.", "\n\n[ : (program\\_pre\\_clear\\_stack\\_v1)\\\n01 : (program\\_pre\\_handempty\\_stack)\\\n02 : (program\\_pre\\_holding\\_stack\\_v2)\\\n03 : (program\\_pre\\_on\\_stack\\_v1\\_v1)\\\n04 : (program\\_pre\\_on\\_stack\\_v1\\_v2)\\\n05 : (program\\_pre\\_on\\_stack\\_v2\\_v1)\\\n06 : (program\\_pre\\_on\\_stack\\_v2\\_v2)\\\n07 : (program\\_pre\\_ontable\\_stack\\_v1)\\\n08 : (program\\_pre\\_ontable\\_stack\\_v2)\\\n[**09**]{} : (program\\_eff\\_clear\\_stack\\_v1)\\\n10 : (program\\_eff\\_clear\\_stack\\_v2)\\\n11 : (program\\_eff\\_handempty\\_stack)\\\n12 : (program\\_eff\\_holding\\_stack\\_v1)\\\n13 : (program\\_eff\\_on\\_stack\\_v1\\_v2)\\\n[**14**]{} : (apply\\_unstack a b i1 i2)\\\n15 : (apply\\_putdown a i2 i3)\\\n16 : (apply\\_unstack b c i3 i4)\\\n17 : (apply\\_stack b a i4 i5)\\\n18 : (apply\\_unstack c d i5 i6)\\\n19 : (apply\\_stack c b i6 i7)\\\n20 : (apply\\_pickup d i7 i8)\\\n21 : (apply\\_stack d c i8 i9)\\\n[**22**]{} : (validate\\_1) ]{}\n\nStatic predicates\n-----------------\n\nA [*static predicate*]{} $p \\in \\Psi$ is a predicate that does not appear in the effects of any action [@fox:TIM:JAIR1998]. ", "Therefore, one can get rid of the mechanism for programming these predicates in the effects of any action schema while keeping the compilation complete. ", "Given a static predicate $p$:\n\n- Fluents $del_f(\\xi)$ and $add_f(\\xi)$, such that $f\\in F_v$ is an instantiation of the static predicate $p$ in the set of [*variable names*]{} $\\Omega_v$, can be discarded for every $\\xi\\in\\Xi$.\n\n- Actions $\\mathsf{programEff_{f,\\xi}}$ (s.t. ", "$f\\in F_v$ is an instantiation of $p$ in $\\Omega_v$) can also be discarded for every $\\xi\\in\\Xi$.\n\nStatic predicates can also constrain the space of possible preconditions by looking at the given set of labels $\\Sigma$. One can assume that if a precondition $f\\in F_v$ (s.t. ", "$f\\in F_v$ is an instantiation of a static predicate in $\\Omega_v$) is not compliant with the labels in $\\Sigma$ then fluents $pre_f(\\xi)$ and actions $\\mathsf{programPre_{f,\\xi}}$ can be discarded for every $\\xi\\in\\Xi$. For instance, in the [*zenotravel*]{} domain, $pre\\_next\\_board\\_v1\\_v1$, $pre\\_next\\_debark\\_v1\\_v1$, $pre\\_next\\_fly\\_v1\\_v1$, $pre\\_next\\_zoom\\_v1\\_v1$, $pre\\_next\\_refuel\\_v1\\_v1$ can be discarded (and their corresponding programming actions) because a precondition [(next ?", "v1 ?", "v1 - flevel)]{} will never hold in any state of $\\Sigma$.\n\nOn the other hand, fluents $pre_f(\\xi)$ and actions $\\mathsf{programPre_{f,\\xi}}$ are discardable for every $\\xi\\in\\Xi$ if a precondition $f\\in F_v$ (s.t. ", "$f\\in F_v$ is an instantiation of a static predicate in $\\Omega_v$) is not possible according to $\\Pi$. Back to the [*zenotravel*]{} domain, if an example plan $\\pi_t\\in \\Pi$ contains the action [(fly plane1 city2 city0 fl3 fl2)]{} and the corresponding label $\\sigma_t\\in\\Sigma$ contains the static literal [(next fl2 fl3)]{} but does not contain [(next fl2 fl2)]{}, [(next fl3 fl3)]{} or [(next fl3 fl2)]{}, the only possible precondition that would include the static predicate is $pre\\_next\\_fly\\_v5\\_v4$.\n\nEvaluation\n==========\n\nThis section evaluates the performance of our approach for learning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models with different amounts of available input knowledge.", "\n\n### Setup.", "\n\nThe domains used in the evaluation are IPC domains that satisfy the [<span style=\"font-variant:small-caps;\">Strips</span>]{} requirement [@fox2003pddl2], taken from the [planning.domains]{} repository [@muise2016planning]. ", "We only used 5 learning examples for each domain and we fixed the examples for all the experiments so that we can evaluate the impact of the input knowledge in the quality of the learned models. ", "All experiments are run on an Intel Core i5 3.10 GHz x 4 with 8 GB of RAM.", "\n\nThe classical planner we used to solve the instances that result from our compilations is [Madagascar]{} [@rintanen2014madagascar]. ", "We used [Madagascar]{} due to its ability to deal with planning instances populated with dead-ends. ", "In addition, SAT-based planners can apply the actions for programming preconditions in a single planning step (in parallel) because these actions do not interact. ", "Actions for programming action effects can also be applied in a single planning step reducing significantly the planning horizon.", "\n\n### Metrics.", "\n\nThe quality of the learned models is measured with the [*precision*]{} and [*recall*]{} metrics. ", "These two metrics are frequently used in [*pattern recognition*]{}, [*information retrieval*]{} and [*binary classification*]{} and are more informative that simply counting the number of errors in the learned model or computing the [*symmetric difference*]{} between the learned and the reference model [@davis2006relationship].", "\n\nIntuitively, precision gives a notion of [*soundness*]{} while recall gives a notion of the [*completeness*]{} of the learned models. ", "Formally, $Precision=\\frac{tp}{tp+fp}$, where $tp$ is the number of true positives (predicates that correctly appear in the action model) and $fp$ is the number of false positives (predicates appear in the learned action model that should not appear). ", "Recall is formally defined as $Recall=\\frac{tp}{tp+fn}$ where $fn$ is the number of false negatives (predicates that should appear in the learned action model but are missing).", "\n\nGiven the syntax-based nature of these metrics, it may happen that they report low scores for learned models that are actually good but correspond to [*reformulations*]{} of the actual model; i.e. a learned model semantically equivalent but syntactically different to the reference model. ", "This mainly occurs when the learning task is under-constrained.", "\n\nLearning from labeled plans\n---------------------------\n\nWe start evaluating our approach with tasks $\\Lambda'={{\\langle \\Psi,\\Sigma,\\Pi \\rangle}}$, where [*labeled plans*]{} are available. ", "We then repeat the evaluation but exploiting potential *static predicates* computed from $\\Sigma$, which are the predicates that appear unaltered in the initial and final states in every $\\sigma_t\\in\\Sigma$. Static predicates are used to constrain the space of possible action models as explained in the previous section.", "\n\nTable \\[tab:results\\_plans\\] shows the obtained results. ", "Precision ([**P**]{}) and recall ([**R**]{}) are computed separately for the preconditions ([**Pre**]{}), positive effects ([**Add**]{}) and negative Effects ([**Del**]{}), while the last two columns of each setting and the last row report averages values. ", "We can observe that identifying static predicates leads to models with better precondition [*recall*]{}. ", "This fact evidences that many of the missing preconditions corresponded to static predicates because there is no incentive to learn them as they always hold [@gregory2015domain].", "\n\nTable \\[tab:time\\_plans\\] reports the total planning time, the preprocessing time (in seconds) invested by [Madagascar]{} to solve the planning instances that result from our compilation as well as the number of actions of the solution plans. ", "All the learning tasks are solved in a few seconds. ", "Interestingly, one can identify the domains with static predicates by just looking at the reported plan length. ", "In these domains some of the preconditions that correspond to static predicates are directly derived from the learning examples and therefore fewer programming actions are required. ", "When static predicates are identified, the resulting compilation is also much more compact and produces smaller planning/instantiation times.", "\n\n ------------ ------- ------------ -------- ------- ------------ --------\n \n Total Preprocess Length Total Preprocess Length\n Blocks 0.04 0.00 72 0.03 0.00 72\n Driverlog 0.14 0.09 83 0.06 0.03 59\n Ferry 0.06 0.03 55 0.06 0.03 55\n Floortile 2.42 1.64 168 0.67 0.57 77\n Grid 4.82 4.75 88 3.39 3.35 72\n Gripper 0.03 0.01 43 0.01 0.00 43\n Hanoi 0.12 0.06 48 0.09 0.06 39\n Miconic 0.06 0.03 57 0.04 0.00 41\n Satellite 0.20 0.14 67 0.18 0.12 60\n Transport 0.59 0.53 61 0.39 0.35 48\n Visitall 0.21 0.15 40 0.17 0.15 36\n Zenotravel 2.07 2.04 71 1.01 1.00 55\n ------------ ------- ------------ -------- ------- ------------ --------\n\n : Total planning time, preprocessing time and plan length for learning tasks from labeled plans without/with static predicates.[]{data-label=\"tab:time_plans\"}\n\nLearning from partially specified action models\n-----------------------------------------------\n\nWe evaluate now the ability of our approach to support partially specified action models; that is, addressing learning tasks of the kind $\\Lambda''={{\\langle \\Psi,\\Sigma,\\Pi,\\Xi_0 \\rangle}}$. In this experiment, the model of half of the actions is given in $\\Xi_0$ as an extra input of the learning task.", "\n\nTables \\[tab:results\\_plans\\_partial\\] and \\[tab:time\\_plans\\_partial\\] summarize the obtained results, which include the identification of static predicates. ", "We only report the [*precision*]{} and [*recall*]{} of the [*unknown*]{} actions since the values of the metrics of the [*known*]{} action models is 1.0. ", "In this experiment, a low value of [*precision*]{} or [*recall*]{} has a greater impact than in the corresponding $\\Lambda'$ tasks because the evaluation is done only over half of the actions. ", "This occurs, for instance, in the precondition *recall* of domains such as [*Floortile*]{}, [*Gripper*]{} or [*Satellite*]{}.", "\n\nRemarkably, the overall *precision* is now $0.98$, which means that the contents of the learned models is highly reliable. ", "The value of *recall*, 0.87, is an indication that the learned models still miss some information (preconditions are again the component more difficult to be fully learned). ", "Overall, the results confirm the previous trend: the more input knowledge of the task, the better the models and the less planning time. ", "Additionally, the solution plans required for this task are smaller because it is only necessary to program half of the actions (the other half are included in the input knowledge $\\Xi_0$). [*", "Visitall*]{} and [*Hanoi*]{} are excluded from this evaluation because they only contain one action schema.", "\n\nLearning from (initial,final) state pairs\n-----------------------------------------\n\nFinally, we evaluate our approach when input plans are not available and thereby the planner must not only compute the action models but also the plans that satisfy the input labels. ", "Table \\[tab:results\\_states\\] and  \\[tab:time\\_states\\] summarize the results obtained for the task $\\Lambda={{\\langle \\Psi,\\Sigma,\\Xi_0 \\rangle}}$ using static predicates. ", "Values for the [*Zenotravel*]{} and [*Grid*]{} domains are not reported because [Madagascar]{} was not able to solve the corresponding planning tasks within a 1000 sec. ", "time bound. ", "The values of *precision* and *recall* are significantly lower than in Table  \\[tab:results\\_plans\\]. ", "Given that the learning hypothesis space is now fairly under-constrained, actions can be reformulated and still be compliant with the inputs (e.g. the [*blocksworld*]{} operator [stack]{} can be [*learned*]{} with the preconditions and effects of the [unstack]{} operator and vice versa). ", "We tried to minimize this effect with the additional input knowledge (static predicates and partially specified action models) and yet the results are below the scores obtained when learning from labeled plans.", "\n\nRelated work\n============\n\nAction model learning has also been studied in domains where there is partial or missing state observability. [", "ARMS]{} works when no partial intermediate state is given. ", "It defines a set of weighted constraints that must hold for the plans to be correct, and solves the weighted propositional satisfiability problem with a MAX-SAT solver [@yang2007learning]. ", "In order to efficiently solve the large MAX-SAT representations, [ARMS]{} implements a hill-climbing method that models the actions approximately. ", "In contrast to our model comparison validation which aims at covering all the training examples, ARMS maximizes the number of covered examples from a testing set.", "\n\n[SLAF]{} also deals with partial observability [@amir:alearning:JAIR08]. ", "Given a formula representing the initial belief state, a sequence of executed actions and the corresponding partially observed states, it builds a complete explanation of observations by models of actions through a CNF formula. ", "The learning algorithm updates the formula of the belief state with every action and observation in the sequence and thus the final returned formula includes all consistent models. ", "SLAF assesses the quality of the learned models with respect to the actual generative model.", "\n\n[LOCM]{} only requires the example plans as input without need for providing information about predicates or states [@cresswell2013acquiring; @cresswell2011generalised]. ", "The lack of available information is overcome by exploiting assumptions about the kind of domain model it has to generate. ", "Particularly, it assumes a domain consists of a collection of objects (sorts) whose defined set of states can be captured by a parameterized Finite State Machine. [", "LOP]{} ([LOCM]{} with Optimized Plans  [@gregory2015domain]) incorporates static predicates and applies a post-processing step after the [LOCM]{} analysis that requires a set of optimal plans to be used in the learning phase. ", "This is done to mitigate the limitation of [LOCM]{} of inducing similar models for domains with similar structures. [", "LOP]{} compares the learned models with the corresponding reference model.", "\n\nCompiling an action model learning task into classical planning is a general and flexible approach that allows to accommodate various amounts and kinds of input knowledge and opens up a path for addressing further learning and validation tasks. ", "For instance, the example plans in $\\Pi$ can be replaced or complemented by a set $\\mathcal{O}$ of sequences of observations (i.e., fully or partial state observations with noisy or missing fluents [@SohrabiRU16]), and learning tasks of the kind $\\Lambda={{\\langle \\Psi,\\Sigma,\\mathcal{O},\\Xi_0 \\rangle}}$ would also be attainable. ", "Furthermore, our approach seems extensible to learning other types of generative models (e.g. hierarchical models like HTN or behaviour trees) that can be more appealing than [<span style=\"font-variant:small-caps;\">Strips</span>]{} models since they require less search effort to compute a a planning solution.", "\n\n Total time Preprocess Plan length\n ------------ ------------ ------------ -------------\n Blocks 0.07 0.01 54\n Driverlog 0.03 0.01 40\n Ferry 0.06 0.03 45\n Floortile 0.43 0.42 55\n Grid 3.12 3.07 53\n Gripper 0.03 0.01 35\n Miconic 0.03 0.01 34\n Satellite 0.14 0.14 47\n Transport 0.23 0.21 37\n Zenotravel 0.90 0.89 40\n\n : Time and plan length learning for learning tasks with partially specified action models.[]{data-label=\"tab:time_plans_partial\"}\n\n Total time Preprocess Plan length\n ------------ ------------ ------------ -------------\n Blocks 2.14 0.00 58\n Driverlog 0.09 0.00 88\n Ferry 0.17 0.01 65\n Floortile 6.42 0.15 126\n Grid - - -\n Gripper 0.03 0.00 47\n Miconic 0.04 0.00 68\n Satellite 4.34 0.10 126\n Transport 2.57 0.21 47\n Zenotravel - - -\n\n : Time and plan length when learning from (initial,final) state pairs.[]{data-label=\"tab:time_states\"}\n\nConclusions\n===========\n\nWe presented a novel approach for learning [<span style=\"font-variant:small-caps;\">Strips</span>]{} action models from examples using classical planning. ", "The approach is flexible to various amounts of input knowledge and accepts partially specified action models. ", "We also introduced the [*precision*]{} and [*recall*]{} metrics, widely used in ML, for evaluating the learned action models. ", "These two metrics measure the soundness and completeness of the learned models and facilitate the identification of model flaws.", "\n\nTo the best of our knowledge, this is the first work on learning action models that is exhaustively evaluated over a wide range of domains and uses exclusively an [*off-the-shelf*]{} classical planner. ", "The work in [@SternJ17] proposes a planning compilation for learning action models from plan traces following the [*finite domain*]{} representation for the state variables. ", "This is a theoretical study on the boundaries of the learned models and no experimental results are reported.", "\n\nWhen example plans are available, we can compute accurate action models from small sets of learning examples (five examples per domain) in little computation time (less than a second). ", "When action plans are not available, our approach still produces action models that are compliant with the input information. ", "In this case, since learning is not constrained by actions, operators can be reformulated changing their semantics, in which case the comparison with a reference model turns out to be tricky.", "\n\nGenerating [*informative*]{} examples for learning planning action models is still an open issue. ", "Planning actions include preconditions that are only satisfied by specific sequences of actions which have low probability of being chosen by chance [@fern2004learning]. ", "The success of recent algorithms for exploring planning tasks [@FrancesRLG17] motivates the development of novel techniques that enable to autonomously collect informative learning examples. ", "The combination of such exploration techniques with our learning approach is an appealing research direction that opens up the door to the bootstrapping of planning action models.", "\n\nIn many applications, the actual actions executed by the observed agent are not available but, instead, the resulting states can be observed. ", "We plan to extend our approach for learning from state observations as it broadens the range of application to external observers and facilitates the representation of imperfect observability, as shown in plan recognition [@SohrabiRU16], as well as learning from unstructured data, like state images [@AsaiF18].", "\n\nAcknowledgments {#acknowledgments .unnumbered}\n---------------\n\nThis work is supported by the Spanish MINECO project TIN2017-88476-C2-1-R. Diego Aineto is partially supported by the [*FPU16/03184*]{} and Sergio Jiménez by the [*RYC15/18009*]{}, both programs funded by the Spanish government.", "\n" ]

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[ "Q:\n\nПроблема с сериализацией на с#\n\nЗдравствуйте.", "\nЧто должно быть реализовано мной:\nС клавиатуры вводятся данные, которые потом должны быть сохранены в XML файл.", "\nМоя реализация:\n XmlSerializer serialiser = new XmlSerializer(typeof(Data_Holder));\n TextWriter Filestream = new StreamWriter(@\"C:\\Users\\Roman-PC\\Documents\\Visual Studio 2017\\Projects\\Lab5Prog\\Lab5Prog\\bin\\Debug\\test2.xml\");\n for (int i = 0; i < Number; ++i)\n {\n //Данные вводятся с клавиатуры (вырезал, ибо не важно)\n MyClassVariable[i] = new Data_Holder(Date2, i, Name2, Surname2, Otchestvo2, Dolznost2);\n serialiser.", "Serialize(Filestream, MyClassVariable[i]);\n }\n Filestream.", "Close();\n\n public class Data_Holder\n{\n public int Date { get; set; }\n public int ID { get; set; }\n public string Name { get; set; }\n public string Surname { get; set; }\n public string Otchestvo { get; set; }\n public string Dolznost { get; set; }\n Data_Holder() { }\n public Data_Holder(int date, int id, string name, string surname, string otchestvo, string dolznost)\n {\n ID = id;\n Date = date;\n Name = name;\n Surname = surname;\n Otchestvo = otchestvo;\n Dolznost = dolznost;\n }\n}\n\nВ чем заключается проблема:\nПосле работы программы я получаю такой XML файл:\n<?", "xml version=\"1.0\" encoding=\"utf-8\"?", ">\n<Data_Holder xmlns:xsi=\"http://www.w3.org/2001/XMLSchema-instance\" xmlns:xsd=\"http://www.w3.org/2001/XMLSchema\">\n <Date>1</Date>\n <ID>0</ID>\n<!--- ", "Прочие данные с ID 0 -->\n</Data_Holder><?xml version=\"1.0\" encoding=\"utf-8\"?", ">\n<Data_Holder xmlns:xsi=\"http://www.w3.org/2001/XMLSchema-instance\" xmlns:xsd=\"http://www.w3.org/2001/XMLSchema\">\n <Date>6</Date>\n <ID>1</ID>\n<!--- ", "Прочие данные с ID 1 -->\n</Data_Holder>\n\nЧто я хочу получить:\n<?", "xml version=\"1.0\" encoding=\"utf-8\"?", ">\n<Data_Holder xmlns:xsi=\"http://www.w3.org/2001/XMLSchema-instance\" \nxmlns:xsd=\"http://www.w3.org/2001/XMLSchema\">\n<Date>1</Date>\n<ID>0</ID>\n <!--- ", "Прочие данные с ID 0 -->\n <Date>6</Date>\n <ID>1</ID>\n <!--- ", "Прочие данные с ID 1 -->\n </Data_Holder>\n\nВопрос:\nКак это пофиксить? ", "Спасибо.", "\nP.S. Если добавить Filestream.", "Close(); в тело цикла, то будет сохраняться лишь данные, которые введены последними.", "\n\nA:\n\nПо идее, нужно сохранять всю коллекцию целиком. ", "Для этого создайте сериализатор нужного типа:\nXmlSerializer serialiser = new XmlSerializer(MyClassVariable.", "GetType());\n\nИ вынесите сохранение из цикла.", "\n// Ввод данных\nfor (int i = 0; i < Number; ++i)\n{\n MyClassVariable[i] = new Data_Holder(...);\n}\n\n// Сохранение всей коллекции целиком\nserialiser.", "Serialize(Filestream, MyClassVariable);\n\n" ]

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[ "Monday, November 13, 2006\n\nThat's right folks, my good uni friend Beth has decided to head overseas in search of adventure, a bloody great big frost, and a hot Canadian to teach her how to snowboard (or something).", "\n\nSo we (Beth, Posty, Millgate, Ezre, Kelly, Karla, Bonnie, Digi, Crump, Deako and a heap of Beth's other mates and family) went to the races to farewell the filly, enjoying a glorious Saturday in the sun. ", "We also went to Bondi the next day, but I was too tired from the previous night's festivities to lift a camera.", "\n\nI agree with Tim, HOT HOT HOT!!!Wow looks like a truely amazing day. ", "PS love the mo!! ", "although I don't really look forward to the rash I could get. ", "Just ask your mum steve, she can give u a lesson on hygiene about not shaving and the consequences. ", "MWAH" ]

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