title: Bloody Mary (2022 film)

text: "The Times of India" critic Neeshita Nyayapati called it a "passable thriller" writing, ""Bloody Mary" falls short of being a complex and grey film because it doesn't go the extra mile to tell a novel story." She appreciated the performances of Pethuraj and Ajay, in addition to the technical aspects of the film. Reviewing the film for "The Hindu", Sangeetha Devi called it "sketchy," criticising the narration and weak characterization.


title: Bloody Mary (2022 film)

text: Subbarao N of "NTV" criticized the story and screenplay while being appreciative of the performances, score and camerawork. A reviewer from "Pinkvilla" also felt the same, stating: "When 'Bloody Mary' doesn't frustrate you with its superficial performances, the sheer quantum of convenient plot points does."


title: Florida's 6th congressional district

text: Florida's 6th congressional district is a congressional district in the U.S. state of Florida. The district is located on the Eastern Florida Coast and stretches from the southern Jacksonville suburbs to New Smyrna Beach. It includes the city of Daytona Beach.


title: Florida's 6th congressional district

text: From 2003 to 2013 the district stretched from the St. Johns River and Jacksonville, sweeping through North Central Florida, encompassing portions of Gainesville and Ocala, and meandered down to the northern tip of the Greater Orlando area in Lake County. It included all of Bradford and Gilchrist counties and portions of Alachua, Clay, Duval, Lake, Levy, and Marion counties. Most of this district is now the 3rd District, while the current 6th covers most of the territory that was previously in the 7th district.


title: Florida's 6th congressional district

text: The district contains over 525,000 registered voters, of whom just over 39% are Democratic, while slightly more than 41% identify as Republican.


title: Ilse Hess

text: Ilse Hess (née Pröhl; 22 June 1900 – 7 September 1995) was the wife of Rudolf Hess. After World War II she became a well-known author.


title: Ilse Hess

text: Ilse Pröhl came from a nationalist conservative family. She was one of three daughters of the wealthy physician and doctor Friedrich Pröhl and his wife Elsa (née Meineke). Her father was killed in May 1917 during the First World War. Her mother then married artist Carl Horn, director of the Bremen Art Museum.


title: Ilse Hess

text: Ilse met Rudolf Hess in April 1920 in Munich. She was one of the first women to study at the University of Munich. In 1921, she joined the NSDAP (Nazi Party) for the first time, and rejoined in 1925 (member number 25,071) after the party had been banned. She felt drawn to Rudolf Hess from the beginning, but Hess was reluctant to enter into a relationship. Ilse introduced Hess to Adolf Hitler, who liked to travel in the circles of well-to-do ladies. Hitler finally gave the impetus to the marriage, which took place on 20 December 1927 in Munich. Hitler was also the godfather of her only child, Wolf Rüdiger Hess, who was born on 18 November 1937. After Rudolf Hess's flight to Scotland, Ilse left Munich with her son to live in Hindelang.


title: Ilse Hess

text: On 3 June 1947, Ilse Hess, like all the wives of the war criminals condemned or executed during the Nuremberg trials, was arrested and transferred to the internment camp in Augsburg-Göggingen. On 24 March 1948 she was released and settled down in the Allgäu, where she opened a pension in 1955.


title: Ilse Hess

text: Ilse Hess was a convinced National Socialist. Until her death, she remained loyal to Hitler and his views, and supported the Stille Hilfe after the war. Her 1952 book "England – Nürnberg – Spandau. Ein Schicksal in Briefen" was published by the far-right Druffel-Verlag. She maintained correspondence with, among others, Winifred Wagner, who also continued to admire Hitler.


title: Jebel Ali

text: Jebel Ali () is a port town south-west of Dubai. The Jebel Ali Port is located there. Al Maktoum International Airport has been constructed just outside the port area. Jebel Ali is connected to Dubai via the UAE Exchange (formerly Jebel Ali), Danube (formerly Jebel Ali Industrial) and Energy stations on the Dubai Metro. Among the infrastructure projects built to support the port and town is the world's largest desalination plant, the Jebel Ali Desalination Plant (Phase 2), providing an ample supply of freshwater.


title: Jebel Ali

text: Many Arab historians argue that it was named after Ali, who was the cousin and son-in-law of Muhammad while expanding the Islamic Caliphate had stood on one hill and looked towards the sea and perhaps known as well although no such recorded proof has been found. "Jebel" means mountain or hill in Arabic.


title: Jebel Ali

text: In 1968, Overseas AST started construction of Dubai's first Communications Station in the Jebal Ali area, because of the raised ground. It was commissioned in 1970, giving for the first time telecommunication links with the rest of the world. In the 1970s, Sheikh Rashid bin Saeed Al Maktoum planned to develop Jebel Ali into an industrial area with its own airport, port, and township.


title: Jebel Ali

text: On 23 September 1983, Gulf Air Flight 771 crashed in Jebel Ali killing all 112 people on board. The cause of the crash was a terrorist bomb that had been planted onboard the aircraft by the Abu Nidal Organization.


title: Jebel Ali

text: In 1985, the Jebel Ali Free Zone (JAFZA) was created: an industrial area surrounding the port. International companies that relocate there enjoy the special privileges of the free zone. These include exemption from corporate tax for 50 years, no personal income tax, no import or re-export duties, no restriction on currency, and easy labor supply and recruitment from authorized companies.


title: Jebel Ali

text: Jebel Ali Industrial Area (aka Jebel Ali Industrial) is one of Dubai's oldest industrial districts. It is located east of the Port of Jebel Ali, south of Jebel Ali Village, west of Dubai Investments Park, and north of the Jebel Ali Free Zone Extension. There have been a number of industrial fires in the area.


title: Jebel Ali

text: Jebel Ali has become the port most frequently visited by ships of the United States Navy outside the United States. Due to the depth of the harbour and size of the port facilities, a and several ships of the accompanying strike group can be accommodated pierside. Due to the frequency of these port visits, semi-permanent shore-leave facilities (referred to by US Navy service personnel as "The Sandbox") have been erected adjacent to the carrier berth.


title: Jebel Ali

text: The original Jebel Ali Village (JAV) was constructed in 1977 to provide accommodation to construction contractors' staff involved with the development of Jebel Ali. Jebel Ali Village was effectively a small British-style garden city. It was a project of Sir William Halcrow and Partners and acted as a prototype for further semi-autonomous residential areas in Dubai such as Emirates Hills and The Gardens. Construction for the redevelopment of Jebel Ali village began by 2008. More recently in 2021, it was announced that the previously isolated Jebel Ali Village was to be transformed to make way for luxury villas by Nakheel Properties.


title: Jebel Ali

text: Palm Jebel Ali is an artificially made archipelago in Jebel Ali which began construction in October 2002, was originally planned to be completed by mid-2008, and has been on hold since.


Tom Clancy's Ghost Recon Advanced Warfighter (GRAW) is a tactical shooter video game released for the Xbox 360, Xbox, PlayStation 2 and Microsoft Windows in 2006. As in previous "Ghost Recon" games, players command their team while neutralizing hostile forces and completing various mission objectives. These objectives can range from escorting friendly units across the map to rescuing hostages or taking out enemy artillery.


title: Tom Clancy's Ghost Recon Advanced Warfighter

text: The game places emphasis on using cover effectively in order to stay alive, together with sound strategic co-ordination to successfully complete mission objectives. A new feature is the "Integrated Warfighter System", a system based upon the Future Force Warrior program.


text: "Tom Clancy's Ghost Recon Advanced Warfighter 2", the game's direct sequel, was released a year later in 2007. The Xbox 360 version was released for Xbox One backwards compatibility in 2018.


text: Gameplay emphasis in "Tom Clancy's Ghost Recon Advanced Warfighter" is placed upon the player doing the bulk of the fighting in each mission, while relying on tactical combat rather than arcade run-and-gun shooting to overcome enemies. Combat in the game is more forgiving than in previous titles in the series, with players being able to survive several bullet hits before dying, instead of dying after only a few shots like in the original "Tom Clancy's Ghost Recon". However, players are unable to regain lost health in the middle of a mission, and a single headshot or a sustained burst of fire can still kill the player instantly.


text: "GRAW" features numerous firearms, many based on real weapons, including the Heckler & Koch G36, the Beretta M9, and the FN SCAR, in addition to prototypes such as the MR-C. The game also makes use of various other pieces of equipment, including tanks, helicopters, armored personnel carriers, and remote-controlled unmanned aerial vehicles, which can be used tactically to overcome obstacles, and scout out the environment, in order to complete objectives. "GRAW" features three versions of the game that each have slight minor variations in storyline/missions depending on the version; the main storyline, though, remains the same between the versions. While both the PlayStation 2 and Xbox versions are essentially the same title, the Xbox 360 and Windows versions each stand out on their own with their own unique features, including the ability to command a squad of soldiers indirectly.


text: One unique feature of the gameplay is the cross-com system, which is designed to allow indirect control over friendly units, assisting them on a mission (e.g. helicopters, UAV drones, etc.), by giving them commands on what to do, whether to locate enemies or deal with a particular threat encountered. This system can be used in conjunction with the game's Intel map - a tactical map showing the entire play area and location of objectives, enemies and friendlies/allies - to issue commands. Notably, this gameplay feature varies between the different versions. For the Xbox 360 version, the system highlights enemies spotted by both the player and both their squad and other allies, while the Microsoft Windows version only highlights enemies seen by the player's squad and not themselves.


text: Both the PlayStation 2 and Xbox console versions of "Tom Clancy's Ghost Recon Advanced Warfighter" are played in first-person, but have notably fewer features than those found in the Xbox 360 or the Windows versions. The most notable differences to the other versions is that players only have a single AI-controlled teammate following them on missions (as opposed to a squad of 3 soldiers), there are more levels but of shorter lengths, and that player character Scott Mitchell never speaks throughout these versions.


text: Unlike all other versions of "GRAW", the Xbox 360 version gives players a choice between using either a third-person or first-person perspective during the campaign mode, and also features a unique covering system that allows the player to duck behind objects for protection from gun-fire while performing tactical maneuvers. In addition, some levels feature on-rail sections in which players must eliminate as many enemies as possible from their airborne transport via the use of a mounted gun, and if a mission provides them with a squad, orders given via the cross-com system apply to the entire squad, and not to individual squad members. Players cannot recover ammo from fallen enemies but can take their weapons if needed; otherwise they must re-arm at either rally points or ammo stations.


text: The Microsoft Windows version of "Tom Clancy's Ghost Recon Advanced Warfighter" is played from a first-person perspective. This version features larger levels than those of the console versions with enemy AI more likely to use ambush tactics, thus greater emphasis is placed upon using squad tactics and having more situational awareness in each mission. The cross-com system in this version is more flexible, capable of issuing more controlled commands to individual squad members, as well as plotting out attack routes and battle plans for other friendly units. In addition, players cannot pick up discarded weapons, unlike the other versions, but can take ammunition from the weapons of dead enemies or allies, if the player's weapon is of the same caliber.


text: Unlike other versions, players may select the loadouts for their selected team before the start of a mission, although this is done with no briefing given, with each member able to carry a primary weapon and a secondary weapon, while either carrying extra ammunition or a heat-seeking anti-tank weapon. Unlike other versions, every Ghost member has a maximum weight limit on how much they can carry, with them becoming fatigued from sprinting at a much faster rate the more equipment they are carrying. Weapons are able to be modified with various attachments, such as optical sights, grenade launchers, silencers, and vertical foregrips, and each modification has used on a weapon has its advantages and disadvantages. For example, attaching a grenade launcher reduces accuracy and adds weight, while adding a silencer reduces noise and muzzle flash of a weapon at the expense of accuracy and range.


text: The Microsoft Windows version can take advantage of the presence of a PhysX card, however the enhancements are only cosmetic and do not affect gameplay.


text: Online multiplayer on the Microsoft Windows edition has both versus and co-op mode. Versus mode features 32 players, double that of the Xbox 360 version. The online co-op mode only supports up to 4 players maximum on Windows, while the Xbox 360 version supports up to 16 co-op players.


text: On the Xbox 360 version, some new Xbox Live features are included: the co-operative campaign (which currently features 8 missions - 4 of which are available on the Xbox Live Marketplace) and multiplayer maps are all set in Nicaragua as a follow-up to the Mexico City missions of the single player campaign.


text: In the UK, a special Game of the Year edition was released in late 2006 containing a card enabling the player to obtain all marketplace content for free, including the Chapter 2 download.


text: The PlayStation 2 and Xbox versions have some similar online game modes and can be played with up to 8 players.


text: In 2013, seeking to unify their countries' security and to combat the smuggling of drugs and weapons, and the movement of terrorists between their borders, the Canadian prime minister and U.S. President Ballantine travel to Mexico City for a summit with Mexican President Ruiz-Peña, to sign a treaty known as the North American Joint Security Agreement (NAJSA). At the same time, a Ghost team, led by Captain Scott Mitchell, is sent into the city to intercept a deal taking place and recover Guardrail IX, a device capable of disrupting wireless communications, which had been taken from a spy plane that was shot down in Nicaragua. However, complications arise when a coup d'état takes place in Mexico City, forcing Ghost commander General Joshua Keating to abort the mission so that Mitchell's team can locate and extract both Ballantine and Ruiz-Peña, who both went into hiding shortly after Mexican rebels attacked the summit, killing the Canadian prime minister in the process.


text: Although the Ghosts rescue both VIPs, Ballantine is forced to hold at the city's airport until hostile anti-air emplacements are eliminated. Meanwhile, an attempt by rebels to kill Ruiz-Peña at the U.S. embassy is thwarted by the Ghosts. During this time, news reports confirm that the coup is being spearheaded by a rebel special forces group called the Aguila 7, led by General Ontiveros, and his son Colonel Carlos Ontiveros. Seeking to re-establish order, Ruiz-Peña allows U.S. forces to intervene, with Mitchell ordered by Major General Martin to help clear a path into the city, whereupon he neutralizes artillery and anti-air batteries, assaults Chapultepec Castle, and eventually secures some of the M1A2 Abrams tanks given to the Mexican government to help combat rebel armor. While securing the tanks however, Aguila 7 ambush Ballantine's position and capture him, forcing Mitchell to search the city's shanty district and extract him.


text: Although they rescue the U.S. president, Keating orders Mitchell and the Ghosts to recover the "Football", after the rebels steal it, but are thwarted by Guardrail IX, which Lieutenant Barnes reveals is being used to monitor and disrupt communications between U.S. forces. Mitchell quickly manages to find and destroy much of the device, before assisting allied forces pushing into Zocalo Plaza, and rescuing team pilot Martin when his Black Hawk is downed. During this time, Keating informs Ballantine that the rebels are still using elements of Guardrail IX to hack into NORAD in order to gain control over the United States' ICBMs, which in turn causes China and Russia to ready their own, buying time to gain international support for the coup. Mitchell is then ordered to assist in the final push on Palacio Nacional, destroying the last elements of Guardrail IX, and eventually helping to capture General Ontiveros before he can flee, but is unable to find the "Football" with him.


text: Mitchell quickly learns that Carlos has the "Football", after he manages to escape and hijacks Martin's Black Hawk, killing him in the process. As Mitchell and his team fight through the U.S. embassy, they overcome the last remnants of Aguila 7, successfully kill Carlos, and recover the "Football". As the team prepare for extraction, Barnes confirms suspicion of a traitor in the U.S. Army, and reveals through news footage that a retired U.S. Army general, who was in opposition to NAJSA, was responsible for giving Ontiveros access to Guardrail IX.


text: In May 2005, the Xbox 360 edition of "Advanced Warfighter" was shown to the press in trailer form at the E3 (Electronic Entertainment Expo) convention in Los Angeles.


text: The Xbox 360 version of "Ghost Recon: Advanced Warfighter" was released on March 9, 2006 to multiple rave reviews. GameRankings and Metacritic gave it a score of 90.47% and 90 out of 100. "Electronic Gaming Monthly" praised the game's stunning next-generation presentation and high entertainment value, giving it an average score of 9.5 out of 10. The game was noted for its deeply immersive atmosphere, smart A.I. (though flawed for the squad and VIP A.I.), wide variety of weapons, multiple action scenes and believable storyline. The Xbox 360's version was particularly noted for its realistic graphics. GameTrailers gave the game its highest review ever, a 9.9 out of 10, stating that GRAW is a landmark next-gen game that redefines shooters. TeamXbox gave the X360 version a score of 9.3 out of 10, saying that it has "some of the best modern combat scenarios you’ll have ever taken part in. Ubisoft is not only kicking the door in with "Advanced Warfighter", they are taking the whole damn building down along with it!"


text: Reviews for other platforms ranged from positive to mixed to negative. GameRankings and Metacritic gave the game a score of 80.07% and 80 out of 100 for the PC version; 64.57% and 66 out of 100 for the Xbox version; and 50.67% and 44 out of 100 for the PlayStation 2 version.


text: On October 5, 2006, "Ghost Recon: Advanced Warfighter" won two out of the eight nominated BAVGA awards including, "Best Technical Achievement" and "Best Game".


text: "Ghost Recon Advanced Warfighter" was a commercial success, selling more than 1.6 million copies by the end of March 2006. Its Xbox 360 release made up 800,000 of these sales, which made it the console's fastest-selling title by that date. The Xbox 360 version sold 360,000 units within a week of release, including 240,000 sales in North America. Europe bought 120,000 copies within three days of release. Sales of "Advanced Warfighter"s computer and Xbox 360 versions rose by another 560,000 copies between April 1 and June 30, 2006. By the end of September, the game's total sales had surpassed 2.4 million units.


text: "Advanced Warfighter" sold 1.2 million in the United States, 8,903 in Japan, and 200,000 in the United Kingdom.


title: Baalbek Stones

text: The Baalbek Stones are six massive Roman worked stone blocks in Baalbek (ancient Heliopolis), Lebanon, characterised by a megalithic gigantism unparallelled in antiquity.


title: Baalbek Stones

text: The smallest three are part of a podium wall in the Roman complex of the Temple of Jupiter Baal (Heliopolitan Zeus) on Tel Baalbek and are known as the "Trilithon". Each of these is estimated at about .


title: Baalbek Stones

text: The remaining three are Roman monoliths, not part of a larger structure, conventionally known as the "Stone of the Pregnant Woman" (estimated at 1,000 t), the "Stone of the South" (est. 1,242 t), and the "Forgotten Stone" (est. 1,650 t). These are, in reverse order, the first, third, and tied fifth largest known stones ever quarried in human history. They are believed to have been intended for the nearby Jupiter Baal complex, possibly as an addition to the Trilithon; but, perhaps due to their size, they were never removed from their quarry. They have not been used since their extraction in ancient times.


title: Baalbek Stones

text: Numerous archaeological expeditions have gone to the site starting in the 19th century, primarily German and French groups, and research has continued into the 21st century.


title: Baalbek Stones

text: The Stone of the Pregnant Woman (), also called the First Monolith, still lies in the ancient quarry at a distance of 900 m from the Heliopolis temple complex. Although the smallest of the three monoliths, it is also the most famous due to its fine condition, the imposing angle at which it lies, and its never having been fully hidden by earth.


title: Baalbek Stones

text: In 1996, a geodetic team of the Austrian city of Linz conducted topographical measurements at the site which aimed at establishing the exact dimensions of the two monoliths and their possible use in the construction of the gigantic Jupiter temple. According to their calculations, the block weighs 1,000.12 t, thus practically confirming older estimations such as that of Jean-Pierre Adam.


title: Baalbek Stones

text: There are multiple stories behind the name. One says the monolith is named after a pregnant woman who tricked the people of Baalbek into believing that she knew how to move the giant stone if only they would feed her until she gave birth. Others say the name comes from the legends that pregnant jinn were assigned the task of cutting and moving the stone, while others say that the name reflects the belief that a woman who touches the stone experiences an increase in fertility.


title: Baalbek Stones

text: The Stone of the South (), also called the Second Monolith, was rediscovered in the same quarry in the 1990s. With its weight estimated at 1242 t, it surpasses even the dimension of the Stone of the Pregnant Woman. (There is some confusion over the naming, due to its location having been forgotten, and accordingly some sources identify "Stone of the South" as an alternate name of the Stone of the Pregnant Woman.)


title: Baalbek Stones

text: These are dimensions of the rectangular stone block, assuming that its shape is consistent in its still-buried parts:


title: Baalbek Stones

text: The Forgotten Stone, also called the Third Monolith, was discovered in the same quarry in 2014 by archaeologists from the German Archaeological Institute. Its weight is estimated at around 1,650 tonnes, making it the largest stone ever quarried.


title: Baalbek Stones

text: The Trilithon (), also called the Three Stones, is a group of three horizontally lying giant stones, which form part of the podium of the Temple of Jupiter Baal at Baalbek. The location of the megalithic structures is atop of a hill in the region, known as Tel Baalbek. Each one of these stones is 19 metres long, 4.2 metres high, and 3.6 metres thick, and weighs around 750–800 tonnes. The supporting stone layer beneath features a number of stones which weigh an estimated 350 tonnes and are 11 metres wide.


title: Baalbek Stones

text: Although they do not form a trilithon in the modern archaeological sense, they have been known as the Trilithon since at latest the early Byzantine period.


title: Cyndi Wang

text: Cyndi Wang Hsin-ling (; born 5 September 1982) is a Taiwanese singer and actress. Wang debuted her musical career as she released her debut album, "Begin..." (2003). Besides her musical career, Wang has starred in many TV dramas, including "The Car Is In Pursuit" (2000), "Westside Story" (2003), "Heaven's Wedding Gown" (2004), and "Smiling Pasta" (2006). She started her first tour in 2016, and second in 2019. Wang has been active in the Taiwanese music scene for more than eighteen years and at one point, she was one of the top female popstars in the Mandopop industry, regularly selling over a million copies of her every album across the Asia continent. She is widely known as one of the Mandopop Princess or The Taiwanese Sweetheart.


title: Cyndi Wang

text: Wang graduated from Okazaki Arts School and completed a drama course at Hwa Kang Arts School. Her father is "Waishengren" from Qingdao; while her mother is Hakka. Wang is a Buddhist, and has one younger brother. After their parents divorced in 1992, Wang and her brother were raised by their mother.


title: Cyndi Wang

text: Wang took part in a talent contest held by Avex Taiwan in 2003. After the competition, she was called again and given a chance to go to Japan. In order to pursue her dream, she decided to take on the challenge and go to Avex's music studio in Japan to learn singing and dancing for three months. She earned herself a graduation certificate with an impressive performance from Avex Trax. These achievements made Avex Taiwan impressed with her, and she was offered a contract by Avex Taiwan. From that time she has been involved in her entertainment career.


title: Jelly Babies

text: Jelly Babies are a type of soft sugar jelly sweets in the shape of plump babies, sold in a variety of colours. They were first manufactured in Lancashire, England, in the nineteenth century. Their popularity waned before being revived by Bassett's of Sheffield in Yorkshire, who began mass-producing Jelly Babies (initially sold as "Peace Babies") in 1918.


title: Jelly Babies

text: "Jelly Babies" are known at least since advertisements by Riches Confectionery Company of 22 Duke Street, London Bridge in 1885, along with a variety of other baby sweets, including "Tiny Totties" and "Sloper’s Babies". But the pricing of these, at one farthing each, suggests that they were very much larger than the modern Jelly Baby.


title: Jelly Babies

text: The sweets were invented in 1864 by an Austrian immigrant working at "Fryers of Lancashire", and were originally marketed as "Unclaimed Babies." By 1918 they were produced by Bassett's in Sheffield as "Peace Babies," to mark the end of World War I. Bassett's themselves have supported the "Peace Babies" name. Production was suspended during World War II due to wartime shortages. The product was relaunched as "Jelly Babies" in 1953.


title: Jelly Babies

text: A line of sweets called Jellyatrics was launched by Barnack Confectionery Ltd to commemorate the "Jelly Baby’s 80th Birthday" in March 1999. Jellyatrics celebrate "all that is great and good about the older generation".


title: Jelly Babies

text: The most noted modern manufacturer of Jelly Babies, Bassett's, now allocate individual name, shape, colour and flavour to different babies: Brilliant (red; strawberry), Bubbles (yellow; lemon), Baby Bonny (pink; raspberry), Boofuls (green; lime), Bigheart (purple; blackcurrant), and Bumper (orange).


title: Jelly Babies

text: The introduction of different shapes and names was an innovation, circa 1989, prior to which all colours of jelly baby were a uniform shape. Bassett's Jelly Babies changed in September 2007 to include only natural colours and ingredients. There are many brands of jelly babies, as well as supermarket own brands.


title: Jelly Babies

text: Jelly Babies manufactured in the United Kingdom tend to be dusted in starch, which is left over from the manufacturing process, where it is used to aid release from the mould. Jelly Babies manufactured in Australia generally lack this coating. Like most other gummy sweets, they contain gelatin.


title: Jelly Babies

text: Jelly Babies were referred to as "those kids’ candies" in an episode of "Supercar" in 1962, "Operation Superstork". In October 1963, as Beatlemania was breaking out, fans of The Beatles in the United Kingdom pelted the band with jelly babies (or, in the United States, the much harder jelly beans) after it was reported that George Harrison liked eating them.


title: Jelly Babies

text: In the television programme "Doctor Who", jelly babies were often mentioned in the classic series, as a confection The Doctor, an alien time traveller, favoured. They were first seen being consumed by the Second Doctor but they became most associated with Tom Baker’s Fourth Doctor, who had the predilection for offering them to strangers to defuse tense situations (and, in one episode, bluffing an opponent into believing them a weapon). The Fifth, Sixth, Seventh, Eighth, Eleventh and Twelfth Doctors, as well as the nemesis of the Doctor, The Master, also offered them up, in different episodes. In the series, they were often identified simply, by the fact the Doctor (and later the Master) usually carried them around, in a white paper bag.


title: Jelly Babies

text: In the series by Terry Pratchett, "Discworld", the country of Djelibeybi (a pun on "jelly baby", but putatively meaning "Child of the (River) Djel", and possibly derived from Djellaba), is the Discworld's analogue of Ancient Egypt. The main setting of "Pyramids", Djelibeybi is about two miles (3200 m) wide, along the 150-mile (240 km) length of the Djel.


title: Jelly Babies

text: In the episode "Ottery St. Mary" of the BBC Radio programme "Cabin Pressure" (season 3, episode 4), pilots Martin and Douglas perform a “pre-flight check” bit while preparing to drive Martin’s removals van to Devon. After cross-checking doors, seatbelts, and the piano the crew are delivering, Douglas continues the check, asking “Jelly Babies?”, followed by the sound of a bag crinkling as Martin replies, “Jelly Babies to manual”.


title: Jelly Babies

text: Australian singer, Alison Hams, released the "Jelly Baby Song" in May 2013 – its content alluding to the consumption of jelly babies by Type 1 Diabetics to overcome hypoglycaemic episodes – as a way to raise awareness for Type 1 Diabetes, for JDRF Australia (Juvenile Diabetes Research Foundation) who sell specially packaged jelly babies, as the focus of their annual campaign "Jelly Baby Month".


title: Jelly Babies

text: A popular school chemistry experiment, is to put them in a strong oxidising agent, and see the resulting spectacular reaction. The experiment is commonly referred to as "screaming jelly babies".


title: Jelly Babies

text: A poll of 4,000 adults in Britain voted jelly babies their sixth favourite sweet in August 2009. Jelly Babies are the favourite snack of the British children's sitcom character Basil Brush, a puppet fox.


title: Jelly Babies

text: In the 2018 film "Johnny English Strikes Again", the titular character (played by Rowan Atkinson) carries a box of jelly babies with him, but they are actually disguised explosives, as in said context, "jelly" is actually short for gelignite, and they blow up whoever eats them.


title: Cheddi Jagan International Airport

text: Cheddi Jagan International Airport , formerly Timehri International Airport, is the primary airport of Guyana. The airport is located on the right bank of the Demerara River in the city of Timehri, south of Guyana's capital, Georgetown. It is the larger of the two international airports serving Georgetown with the other airport being the Eugene F. Correira International Airport.


title: Cheddi Jagan International Airport

text: The United States obtained rights to locate military facilities in British Guiana as part of the Destroyers for Bases Agreement with the United Kingdom in 1941. On 14 June 1941, the first United States Army forces arrived to survey land for a bomber airfield near Georgetown.


title: Cheddi Jagan International Airport

text: Atkinson Field was built 28 miles (45 km) from Georgetown on of land formerly known as Hyde Park, on the Demerara River. The forest was cleared and hills were levelled and a long concrete runway was constructed. On 20 June 1941, the airfield officially opened with the activation of a weather station. The station was named after Lieutenant Colonel Bert M. Atkinson, a United States Army Air Service World War I aviator. Colonel Atkinson was the commander of the 1st Pursuit Wing on the western front in 1918. Colonel Atkinson retired from the Army in 1922 after a distinguished career and died on 27 April 1937.


title: Cheddi Jagan International Airport

text: The mission of the station was the defense of the colony against German U-boats. The airfield was also a major staging point for American aircraft crossing the Atlantic Ocean heading to the European Theatre on the South Atlantic transport route. Aircraft supplied to the British forces by the United States were flown to Atkinson where they were turned over and ferried to North Africa. With the discovery of bauxite deposits in northeast Brazil in 1943, the mission of the airfield was expanded to protect the coastline of northeast South America and prevent any submarine landings by Axis forces on the continent.


title: Cheddi Jagan International Airport

text: United States Army Air Forces 430th Bombardment Squadron 9th Bombardment Group was assigned to Atkinson field from 4 November 1941 to 31 October 1942 flying anti-submarine sorties in Douglas B-18 bombers. The 430th was replaced by a detachment of the 35th Bombardment Squadron (25th Bombardment Group) from 1 November 1942 to 7 October 1943. After the detection of U-boat activity was taken over by the United States Navy, the 91st Reconnaissance Squadron (344th Reconnaissance Group) was assigned to the airfield during 1944 and 1945 flying the F-10 photo-recon version of the B-25 Mitchell bomber on various mapping missions.


title: Cheddi Jagan International Airport

text: At the end of the war, Atkinson Field was reduced in scope to a skeleton staff. The facility was opened for all air travel, including commercial air flights on 1 October 1946. The same year, British West Indian Airways (BWIA, now Caribbean Airlines) was operating nonstop passenger service to Port of Spain, Trinidad. The military airfield was designated Atkinson Air Force Base on 26 March 1948 by Department of the Air Force General Order Number 10. The base was ordered closed on 31 July 1949 due to budgetary cutbacks. The final military cadre was 3 officers and 25 enlisted men upon closure, and the base was officially turned over to British authorities on 1 August 1949.


title: Cheddi Jagan International Airport

text: A modern terminal building was built and opened on 15 March 1952. When the new building was ravaged by fire on 5 August 1959 the old terminal building was renovated and used again until the destroyed building was replaced.


title: Cheddi Jagan International Airport

text: The airport had jet service in 1961 when Pan American World Airways (Pan Am) was operating Boeing 707 flights with a weekly round trip routing of Paramaribo - Georgetown - Port of Spain - Barbados - New York City. At this same time, Pan Am was also operating weekly Douglas DC-6 propliner service on a routing of Belém - Cayenne - Paramaribo - Georgetown - Port of Spain - Caracas - Curacao - Port Au Prince - Miami. Also in 1961, British West Indian Airways (BWIA) was operating all flights from the airport with Vickers Viscount turboprop aircraft with nonstop service to Port of Spain with direct one stop service to Barbados. BWIA then introduced jet service with Boeing 727-100 aircraft during the mid 1960s and by 1966 was operating all of its passenger service into the airport with the 727 on nonstop flights to Port of Spain and also to Paramaribo in addition to all-cargo flights nonstop to Port of Spain and direct to Miami operated with Douglas DC-4 prop aircraft.


title: Cheddi Jagan International Airport

text: The lease of the facility by the United States was formally terminated on 26 May 1966 (Guyana's Independence Day). Because the lease was terminated 74 years before its end date, a new agreement was reached giving certain specified rights to the United States in relation to the air base for the next 17 years.


title: Cheddi Jagan International Airport

text: In 1965 and 1968 additions were made to the airport facilities. Also in 1968, British West Indian Airways (BWIA) was operating direct, no change of plane Boeing 727-100 jet service to New York City via JFK Airport via intermediate stops in Port of Spain and Barbados. On 1 May 1969, the Atkinson Aerodrome was renamed the Timehri International Airport – "Timehri" is a Carib word for rock motifs located deep in the Guyana hinterland that pre-date the arrival of Europeans in the New World. The airport featured murals employing Amerindian motifs by Guyanese painter Aubrey Williams. In 1971, BWIA was operating daily nonstop service to Port of Spain from the airport with Boeing 707 jetliners. By 1973, ALM Antillean Airlines was operating Douglas DC-9 jet service on a routing of Paramaribo - Georgetown - Port of Spain - Curacao - Kingston - Miami three times a week according to the Official Airline Guide (OAG). Air France was serving the airport in 1977 with round trip service flown twice weekly on a Pointe-a-Pitre - Fort-de-France - Port of Spain - Georgetown - Paramaribo - Cayenne routing operated with a Boeing 737-200. Pan American World Airways (Pan Am) was serving the airport in 1980 with daily one stop Boeing 707 jet service to New York JFK Airport via an intermediate stop in Port of Spain. In 1981, Guy-America Airways was operating nonstop Boeing 707 jet service between New York JFK Airport and Guyana three days a week.


title: Cheddi Jagan International Airport

text: In 1983, according to the Official Airline Guide (OAG), four airlines were operating scheduled passenger air service at the airport: ALM Antillean Airlines, Arrow Air, British West Indies Airways (BWIA, formerly British West Indian Airways) and Guyana Airways. At this time, ALM was operating McDonnell Douglas MD-80 jet service nonstop to Port of Spain, Trinidad, with one stop direct MD-80 flights to Curacao, U.S.-based Arrow Air was flying nonstop Boeing 707 jet service to JFK Airport in New York City, BWIA was operating nonstop McDonnell Douglas DC-9-50 jet flights to Port of Spain and locally based Guyana Airways was flying Boeing 707 nonstop service to Barbados, Miami, New York JFK Airport, Paramaribo, Suriname and Port of Spain. The OAG also lists local and regional flights operated by Guyana Airways with Hawker Siddeley 748 and de Havilland Canada DHC-6 Twin Otter turboprop aircraft at this time. Guyana Airways had earlier operated Boeing 737-200 jet service from the airport in 1981 with nonstop flights to Barbados, Miami, Paramaribo and Port of Spain. By 1989, two airlines were operating nonstop flights between the airport and New York JFK: Guyana Airways with three Boeing 707 flights a week and Tropical Airways flying stretched Super Douglas DC-8 jets twice a week. In 1993, Guyana Airways was operating nonstop Boeing 757-200 flights from GEO to New York JFK Airport three days a week while BWIA and Leisure Air were operating direct one stop flights from New York JFK Airport, BWIA with McDonnell Douglas MD-80 jets five days a week via an intermediate stop in Antigua and Leisure Air with Airbus A320 jets once a week via an intermediate stop in St. Lucia. In 1995 Barbados-based Carib Express was operating British Aerospace BAe 146-100 jet service to destinations in the Caribbean.


title: Cheddi Jagan International Airport

text: In March 1997, following the death of President Dr. Cheddi Jagan, then-President Samuel Hinds decided to rename the airport the Cheddi Jagan International Airport. The proposition to rename the airport was proposed in the Parliament (National Assembly) by the Minister of Amerindian Affairs, Vibert De Souza, who noted that it would be a fitting tribute to a man who had spent his life committed to the betterment of Amerindian people and fighting for the freedom and unity of all Guyanese. A plaque bearing the new name was unveiled on 21 May 1997 by the Prime Minister, Janet Jagan.


title: Cheddi Jagan International Airport

text: The Cheddi Jagan International Airport underwent a 150 million dollar modernization and expansion. The runway was extended to about 10,500 feet and the expansion provided a new arrivals terminal building with eight boarding bridges, elevators and CCTV. This was completed in December 2018. However, the check-in facility opened at the same time.


title: Cheddi Jagan International Airport

text: The airport sits at an elevation of above mean sea level. It has two asphalt paved runways: 06/24 measuring and 11/29 measuring . The terminal has six ground level gates and four boarding bridges.


title: Algebraic variety

text: Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.


title: Algebraic variety

text: Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.


title: Algebraic variety

text: The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the "Nullstellensatz" and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a defining feature of algebraic geometry.


title: Algebraic variety

text: Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces.


title: Algebraic variety

text: In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type.


title: Algebraic variety

text: An "affine variety" over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s.


title: Algebraic variety

text: For an algebraically closed field and a natural number , let be an affine -space over , identified to formula_1 through the choice of an affine coordinate system. The polynomials in the ring can be viewed as "K"-valued functions on by evaluating at the points in , i.e. by choosing values in "K" for each "x". For each set "S" of polynomials in , define the zero-locus "Z"("S") to be the set of points in on which the functions in "S" simultaneously vanish, that is to say


title: Algebraic variety

text: A subset "V" of is called an affine algebraic set if "V" = "Z"("S") for some "S". A nonempty affine algebraic set "V" is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is also called an affine variety. (Many authors use the phrase "affine variety" to refer to any affine algebraic set, irreducible or not.)


title: Algebraic variety

text: Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets. This topology is called the Zariski topology.


title: Algebraic variety

text: Given a subset "V" of , we define "I"("V") to be the ideal of all polynomial functions vanishing on "V":


title: Algebraic variety

text: For any affine algebraic set "V", the coordinate ring or structure ring of "V" is the quotient of the polynomial ring by this ideal.


title: Algebraic variety

text: Let be an algebraically closed field and let be the projective "n"-space over . Let in be a homogeneous polynomial of degree "d". It is not well-defined to evaluate on points in in homogeneous coordinates. However, because is homogeneous, meaning that , it "does" make sense to ask whether vanishes at a point . For each set "S" of homogeneous polynomials, define the zero-locus of "S" to be the set of points in on which the functions in "S" vanish:


title: Algebraic variety

text: A subset "V" of is called a projective algebraic set if "V" = "Z"("S") for some "S". An irreducible projective algebraic set is called a projective variety.


title: Algebraic variety

text: Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.


title: Algebraic variety

text: Given a subset "V" of , let "I"("V") be the ideal generated by all homogeneous polynomials vanishing on "V". For any projective algebraic set "V", the coordinate ring of "V" is the quotient of the polynomial ring by this ideal.


title: Algebraic variety

text: A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice also that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine varieties, such a quasi-projective variety is usually not called a variety but a constructible set.


title: Algebraic variety

text: In classical algebraic geometry, all varieties were by definition quasi-projective varieties, meaning that they were open subvarieties of closed subvarieties of projective space. For example, in Chapter 1 of Hartshorne a "variety" over an algebraically closed field is defined to be a quasi-projective variety, but from Chapter 2 onwards, the term variety (also called an abstract variety) refers to a more general object, which locally is a quasi-projective variety, but when viewed as a whole is not necessarily quasi-projective; i.e. it might not have an embedding into projective space. So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the regular functions on the variety. The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space. For example, under this definition, the product is not a variety until it is embedded into the projective space; this is usually done by the Segre embedding. However, any variety that admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding. Consequently, many notions that should be intrinsic, such as the concept of a regular function, are not obviously so.


title: Algebraic variety

text: The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by André Weil. In his "Foundations of Algebraic Geometry", Weil defined an abstract algebraic variety using valuations. Claude Chevalley made a definition of a scheme, which served a similar purpose, but was more general. However, Alexander Grothendieck's definition of a scheme is more general still and has received the most widespread acceptance. In Grothendieck's language, an abstract algebraic variety is usually defined to be an integral, separated scheme of finite type over an algebraically closed field, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed. Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.


title: Algebraic variety

text: One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata. Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non-projective. Since then other examples have been found; for example, it is straightforward to construct a toric variety that is not quasi-projective but complete.


title: Algebraic variety

text: A subvariety is a subset of a variety that is itself a variety (with respect to the structure induced from the ambient variety). For example, every open subset of a variety is a variety. See also closed immersion.


title: Algebraic variety

text: Hilbert's Nullstellensatz says that closed subvarieties of an affine or projective variety are in one-to-one correspondence with the prime ideals or homogeneous prime ideals of the coordinate ring of the variety.


title: Algebraic variety

text: Let , and A be the two-dimensional affine space over C. Polynomials in the ring C["x", "y"] can be viewed as complex valued functions on A by evaluating at the points in A. Let subset "S" of C["x", "y"] contain a single element :


title: Algebraic variety

text: The zero-locus of is the set of points in A on which this function vanishes: it is the set of all pairs of complex numbers ("x", "y") such that "y" = 1 − "x". This is called a line in the affine plane. (In the classical topology coming from the topology on the complex numbers, a complex line is a real manifold of dimension two.) This is the set :


title: Algebraic variety

text: Thus the subset of A is an algebraic set. The set "V" is not empty. It is irreducible, as it cannot be written as the union of two proper algebraic subsets. Thus it is an affine algebraic variety.


title: Algebraic variety

text: Let , and A be the two-dimensional affine space over C. Polynomials in the ring C["x", "y"] can be viewed as complex valued functions on A by evaluating at the points in A. Let subset "S" of C["x", "y"] contain a single element "g"("x", "y"):


title: Algebraic variety

text: The zero-locus of "g"("x", "y") is the set of points in A on which this function vanishes, that is the set of points ("x","y") such that "x" + "y" = 1. As "g"("x", "y") is an absolutely irreducible polynomial, this is an algebraic variety. The set of its real points (that is the points for which "x" and "y" are real numbers), is known as the unit circle; this name is also often given to the whole variety.


title: Algebraic variety

text: The following example is neither a hypersurface, nor a linear space, nor a single point. Let A be the three-dimensional affine space over C. The set of points ("x", "x", "x") for "x" in C is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane. It is the twisted cubic shown in the above figure. It may be defined by the equations


title: Algebraic variety

text: The irreducibility of this algebraic set needs a proof. One approach in this case is to check that the projection ("x", "y", "z") → ("x", "y") is injective on the set of the solutions and that its image is an irreducible plane curve.


title: Algebraic variety

text: For more difficult examples, a similar proof may always be given, but may imply a difficult computation: first a Gröbner basis computation to compute the dimension, followed by a random linear change of variables (not always needed); then a Gröbner basis computation for another monomial ordering to compute the projection and to prove that it is generically injective and that its image is a hypersurface, and finally a polynomial factorization to prove the irreducibility of the image.


title: Algebraic variety

text: The set of "n"-by-"n" matrices over the base field "k" can be identified with the affine "n"-space formula_9 with coordinates formula_10 such that formula_11 is the ("i", "j")-th entry of the matrix formula_12. The determinant formula_13 is then a polynomial in formula_10 and thus defines the hypersurface formula_15 in formula_9. The complement of formula_17 is then an open subset of formula_9 that consists of all the invertible "n"-by-"n" matrices, the general linear group formula_19. It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, consider formula_20 where the affine line is given coordinate "t". Then formula_19 amounts to the zero-locus in formula_20 of the polynomial in formula_23:


title: Algebraic variety

text: i.e., the set of matrices "A" such that formula_25 has a solution. This is best seen algebraically: the coordinate ring of formula_19 is the localization formula_27, which can be identified with formula_28.


title: Algebraic variety

text: The multiplicative group k of the base field "k" is the same as formula_29 and thus is an affine variety. A finite product of it formula_30 is an algebraic torus, which is again an affine variety.


title: Algebraic variety

text: A general linear group is an example of a linear algebraic group, an affine variety that has a structure of a group in such a way the group operations are morphism of varieties.


title: Algebraic variety

text: A projective variety is a closed subvariety of a projective space. That is, it is the zero locus of a set of homogeneous polynomials that generate a prime ideal.


title: Algebraic variety

text: A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The projective line P is an example of a projective curve; it can be viewed as the curve in the projective plane } defined by . For another example, first consider the affine cubic curve


title: Algebraic variety

text: in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation:


title: Algebraic variety

text: which defines a curve in P called an elliptic curve. The curve has genus one (genus formula); in particular, it is not isomorphic to the projective line P, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of moduli of algebraic curves).


title: Algebraic variety

text: Let "V" be a finite-dimensional vector space. The Grassmannian variety "G"("V") is the set of all "n"-dimensional subspaces of "V". It is a projective variety: it is embedded into a projective space via the Plücker embedding:


title: Algebraic variety

text: where "b" are any set of linearly independent vectors in "V", formula_34 is the "n"-th exterior power of "V", and the bracket ["w"] means the line spanned by the nonzero vector "w".


title: Algebraic variety

text: The Grassmannian variety comes with a natural vector bundle (or locally free sheaf in other terminology) called the tautological bundle, which is important in the study of characteristic classes such as Chern classes.


title: Algebraic variety

text: Let "C" be a smooth complete curve and formula_35 the Picard group of it; i.e., the group of isomorphism classes of line bundles on "C". Since "C" is smooth, formula_35 can be identified as the divisor class group of "C" and thus there is the degree homomorphism formula_37. The Jacobian variety formula_38 of "C" is the kernel of this degree map; i.e., the group of the divisor classes on "C" of degree zero. A Jacobian variety is an example of an abelian variety, a complete variety with a compatible abelian group structure on it (the name "abelian" is however not because it is an abelian group). An abelian variety turns out to be projective (theta functions in the algebraic setting gives an embedding); thus, formula_38 is a projective variety. The tangent space to formula_38 at the identity element is naturally isomorphic to formula_41 hence, the dimension of formula_38 is the genus of formula_43.


title: Algebraic variety

text: where formula_48 is the product of "n" copies of "C". For formula_49 (i.e., "C" is an elliptic curve), the above morphism for formula_50 turns out to be an isomorphism; in particular, an elliptic curve is an abelian variety.


title: Algebraic variety

text: Given an integer formula_51, the set of isomorphism classes of smooth complete curves of genus formula_52 is called the moduli of curves of genus formula_52 and is denoted as formula_54. There are few ways to show this moduli has a structure of a possibly reducible algebraic variety; for example, one way is to use geometric invariant theory which ensures a set of isomorphism classes has a (reducible) quasi-projective variety structure. Moduli such as the moduli of curves of fixed genus is typically not a projective variety; roughly the reason is that a degeneration (limit) of a smooth curve tends to be non-smooth or reducible. This leads to the notion of a stable curve of genus formula_55, a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curves formula_56, the set of isomorphism classes of stable curves of genus formula_55, is then a projective variety which contains formula_54 as an open subset. Since formula_56 is obtained by adding boundary points to formula_54, formula_56 is colloquially said to be a compactification of formula_54. Historically a paper of Mumford and Deligne introduced the notion of a stable curve to show formula_54 is irreducible when formula_55.


title: Algebraic variety

text: The moduli of curves exemplifies a typical situation: a moduli of nice objects tend not to be projective but only quasi-projective. Another case is a moduli of vector bundles on a curve. Here, there are the notions of stable and semistable vector bundles on a smooth complete curve formula_43. The moduli of semistable vector bundles of a given rank formula_66 and a given degree formula_67 (degree of the determinant of the bundle) is then a projective variety denoted as formula_68, which contains the set formula_69 of isomorphism classes of stable vector bundles of rank formula_66 and degree formula_67 as an open subset. Since a line bundle is stable, such a moduli is a generalization of the Jacobian variety of formula_43.


title: Algebraic variety

text: In general, in contrast to the case of moduli of curves, a compactification of a moduli need not be unique and, in some cases, different non-equivalent compactifications are constructed using different methods and by different authors. An example over formula_73 is the problem of compactifying formula_74, the quotient of a bounded symmetric domain formula_75 by an action of an arithmetic discrete group formula_76. A basic example of formula_74 is when formula_78, Siegel's upper half-space and formula_76 commensurable with formula_80; in that case, formula_74 has an interpretation as the moduli formula_82 of principally polarized complex abelian varieties of dimension formula_52 (a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactify formula_74, a toroidal compactification of it. But there are other ways to compactify formula_74; for example, there is the minimal compactification of formula_74 due to Baily and Borel: it is the projective variety associated to the graded ring formed by modular forms (in the Siegel case, Siegel modular forms). The non-uniqueness of compactifications is due to the lack of moduli interpretations of those compactifications; i.e., they do not represent (in the category-theory sense) any natural moduli problem or, in the precise language, there is no natural moduli stack that would be an analog of moduli stack of stable curves.


title: Algebraic variety

text: An algebraic variety can be neither affine nor projective. To give an example, let and the projection. It is an algebraic variety since it is a product of varieties. It is not affine since P is a closed subvariety of "X" (as the zero locus of "p"), but an affine variety cannot contain a projective variety of positive dimension as a closed subvariety. It is not projective either, since there is a nonconstant regular function on "X"; namely, "p".


title: Algebraic variety

text: Consider the affine line formula_87 over formula_73. The complement of the circle formula_89 in formula_90 is not an algebraic variety (not even algebraic set). Note that formula_91 is not a polynomial in formula_92 (although a polynomial in real variables formula_93.) On the other hand, the complement of the origin in formula_90 is an algebraic (affine) variety, since the origin is the zero-locus of formula_92. This may be explained as follows: the affine line has dimension one and so any subvariety of it other than itself must have strictly less dimension; namely, zero.


title: Algebraic variety

text: For similar reasons, a unitary group (over the complex numbers) is not an algebraic variety, while the special linear group formula_96 is a closed subvariety of formula_97, the zero-locus of formula_98. (Over a different base field, a unitary group can however be given a structure of a variety.)


title: Algebraic variety

text: Let be algebraic varieties. We say and are isomorphic, and write , if there are regular maps and such that the compositions and are the identity maps on and respectively.


title: Algebraic variety

text: The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required. The modern notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An "abstract algebraic variety" is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a spectrum of a ring. Basically, a variety over is a scheme whose structure sheaf is a sheaf of -algebras with the property that the rings "R" that occur above are all integral domains and are all finitely generated -algebras, that is to say, they are quotients of polynomial algebras by prime ideals.


title: Algebraic variety

text: This definition works over any field . It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space. This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. Such objects are usually not considered varieties, and are eliminated by requiring the schemes underlying a variety to be "separated". (Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.)


title: Algebraic variety

text: Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety only require that the affine charts have trivial nilradical.


title: Algebraic variety

text: A complete variety is a variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa.


title: Algebraic variety

text: These varieties have been called "varieties in the sense of Serre", since Serre's foundational paper FAC


title: Algebraic variety

text: on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way.


title: Algebraic variety

text: One way that leads to generalizations is to allow reducible algebraic sets (and fields that aren't algebraically closed), so the rings "R" may not be integral domains. A more significant modification is to allow nilpotents in the sheaf of rings, that is, rings which are not reduced. This is one of several generalizations of classical algebraic geometry that are built into Grothendieck's theory of schemes.


title: Algebraic variety

text: Allowing nilpotent elements in rings is related to keeping track of "multiplicities" in algebraic geometry. For example, the closed subscheme of the affine line defined by "x" = 0 is different from the subscheme defined by "x" = 0 (the origin). More generally, the fiber of a morphism of schemes "X" → "Y" at a point of "Y" may be non-reduced, even if "X" and "Y" are reduced. Geometrically, this says that fibers of good mappings may have nontrivial "infinitesimal" structure.


title: Algebraic variety

text: An algebraic manifold is an algebraic variety that is also an "m"-dimensional manifold, and hence every sufficiently small local patch is isomorphic to "k". Equivalently, the variety is smooth (free from singular points). When is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Projective algebraic manifolds are an equivalent definition for projective varieties. The Riemann sphere is one example.


title: Swing Shift (film)

text: Swing Shift is a 1984 American romantic-drama directed by Jonathan Demme and produced by and starring Goldie Hawn with Kurt Russell. It also features Christine Lahti, Fred Ward, Ed Harris, and Holly Hunter, in one of her earlier movie roles. The film was a box office bomb, grossing just $6.6 million against its $15 million budget. Christine Lahti earned a nomination for the Academy Award for Best Supporting Actress.


title: Swing Shift (film)

text: During World War II, Kay Walsh signs up to work in an armaments factory in California while her husband Jack, a U.S. Naval seaman, is overseas in naval service. Lonely and vulnerable, Kay falls for the charms of another man, a musician named Lucky, and befriends her experienced, skeptical neighbor Hazel, a former singer. The three of them enjoy their time together until Kay's husband comes home and realizes what has occurred.


title: Swing Shift (film)

text: "Swing Shift" has become a case study in star/producer/director conflict. Over Demme's objections, an additional 30 minutes' worth of new scenes were written and shot to emphasize Hawn's onscreen love triangle, pushing back the intended release date from Christmas 1983 and inflating the budget. There were reports that the reshoots were ordered because Christine Lahti was upstaging Hawn, but Hawn maintained that she and her producing partner Anthea Sylbert were "just trying to get the movie to work." Demme had the phrase "A Jonathan Demme Film" removed from the credits and promotional materials, and screenwriter Nancy Dowd requested to be credited under the pseudonym "Rob Morton."


title: Swing Shift (film)

text: Roger Ebert of the "Chicago Sun-Times" gave the film three stars out of four and wrote "There's no suspense and no big emotional payoff, but the movie is always absorbing." Gene Siskel of the "Chicago Tribune" awarded two-and-a-half stars out of four and wrote "Although the World War II drama does manage to work up considerable emotion for a few big moments, we also find the script wanting at as many moments. One more run through the typewriter would have helped." Vincent Canby of "The New York Times" wrote "Despite what seem to have been certain differences of opinion in the course of the production, 'Swing Shift' plays very smoothly. No one need be ashamed." Gary Arnold of "The Washington Post" wrote "Its elaborate and meticulously re-created period settings and moods prove far more interesting and diverting than the undernourished characterizations and love stories that flutter and sputter across the foregrounds." "Variety" found that the characters "were not people worth fighting a war for" and remarked that while "[g]reat drama, to be sure, does not depend on likeable characters...the writing and acting are too flat for the challenge." Sheila Benson of the "Los Angeles Times" stated "Weak, flat, mystifyingly inconsistent, the present version is understandably disownable." Pauline Kael of "The New Yorker" wrote "There are no high spots, no exciting moments. The picture just goes popping from one recessive, undeveloped scene to the next." Steve Jenkins of "The Monthly Film Bulletin" wrote "Given the pseudonymous script credit, covering the contributions of three writers, and the serious disputes between Jonathan Demme and Goldie Hawn during production, it is perhaps not surprising that "Swing Shift" should emerge as a disappointingly bland, muddled and inconclusive affair."


title: Swing Shift (film)

text: Steve Vineberg of "Sight & Sound" called Jonathan Demme's original cut "extraordinary – one of the best movies made by an American in the 80s." He described the story of the film as "a Hollywood tragedy. It echoes what RKO did to Orson Welles' "The Magnificent Ambersons.""


title: Plane (tool)

text: A hand plane is a tool for shaping wood using muscle power to force the cutting blade over the wood surface. Some rotary power planers are motorized power tools used for the same types of larger tasks, but are unsuitable for fine-scale planing, where a miniature hand plane is used.


title: Plane (tool)

text: Generally, all planes are used to flatten, reduce the thickness of, and impart a smooth surface to a rough piece of lumber or timber. Planing is also used to produce horizontal, vertical, or inclined flat surfaces on workpieces usually too large for shaping, where the integrity of the whole requires the same smooth surface. Special types of planes are designed to cut joints or decorative mouldings.


title: Plane (tool)

text: Hand planes are generally the combination of a cutting edge, such as a sharpened metal plate, attached to a firm body, that when moved over a wood surface, take up relatively uniform shavings, by nature of the body riding on the 'high spots' in the wood, and also by providing a relatively constant angle to the cutting edge, render the planed surface very smooth. A cutter that extends below the bottom surface, or sole, of the plane slices off shavings of wood. A large, flat sole on a plane guides the cutter to remove only the highest parts of an imperfect surface, until, after several passes, the surface is flat and smooth. When used for flattening, bench planes with longer soles are preferred for boards with longer longitudinal dimensions. A longer sole registers against a greater portion of the board's face or edge surface which leads to a more consistently flat surface or straighter edge. Conversely, using a smaller plane allows for more localized low or high spots to remain.


title: Plane (tool)

text: Though most planes are pushed across a piece of wood, holding it with one or both hands, Japanese planes are pulled toward the body, not pushed away.


title: Plane (tool)

text: Woodworking machinery that perform a similar function as hand planes include the jointer and the thickness planer, also called a thicknesser; the job these specialty power tools can still be done by hand planers and skilled manual labor as it was for many centuries. When rough lumber is reduced to dimensional lumber, a large electric motor or internal combustion engine will drive a thickness planer that removes a certain percentage of excess wood to create a uniform, smooth surface on all four sides of the board and in specialty woods, may also plane the cut edges.


title: Plane (tool)

text: Hand planes are ancient, originating thousands of years ago. Early planes were made from wood with a rectangular slot or mortise cut across the center of the body. The cutting blade or iron was held in place with a wooden wedge. The wedge was tapped into the mortise and adjusted with a small mallet, a piece of scrap wood or with the heel of the user's hand. Planes of this type have been found in excavations of old sites as well as drawings of woodworking from medieval Europe and Asia. The earliest known examples of the woodworking plane have been found in Pompeii although other Roman examples have been unearthed in Britain and Germany. The Roman planes resemble modern planes in essential function, most having iron wrapping a wooden core top, bottom, front and rear, and an iron blade secured with a wedge. One example found in Cologne has a body made entirely of bronze without a wooden core. A Roman plane iron used for cutting moldings was found in Newstead, England. Histories prior to these examples are not clear although furniture pieces and other woodwork found in Egyptian tombs show surfaces carefully smoothed with some manner of cutting edge or scraping tool. There are suggestions that the earliest planes were simply wooden blocks fastened to the soles of adzes to effect greater control of the cutting action.


title: Plane (tool)

text: In the mid-1860s, Leonard Bailey began producing a line of cast iron-bodied hand planes, the patents for which were later purchased by Stanley Rule & Level, now Stanley Works. The original Bailey designs were further evolved and added to by Justus Traut and others at Stanley Rule & Level. The Bailey and Bedrock designs became the basis for most modern metal hand plane designs manufactured today. The Bailey design is still manufactured by Stanley Works.


title: Plane (tool)

text: In 1918 an air-powered handheld planing tool was developed to reduce shipbuilding labor during World War I. The air-driven cutter spun at 8000 to 15000 rpm and allowed one man to do the planing work of up to fifteen men who used manual tools.


title: Plane (tool)

text: Modern hand planes are made from wood, ductile iron or bronze which produces a tool that is heavier and will not rust.


title: Plane (tool)

text: Most planes fall within the categories (by size) of block plane, smoothing plane, and jointing plane. Specialty planes include the shoulder plane, router plane, bullnose plane, and chisel plane, among others.


title: Plane (tool)

text: Electrically powered hand planers (loosely referred to as power planes) have joined the hand-held plane family.


title: Plane (tool)

text: "Bench planes" are characterized by having their cutting bevel facing down and attached to a chipbreaker. Most metal bench planes, and some larger wooden ones, are designed with a rear handle known as a tote. "Block planes" are characterized by the absence of a chipbreaker and the cutting iron bedded with the bevel up. The block plane is a smaller tool that can be held with one hand which excels at working against the grain on a cut end of a board. It is also good for general purpose work such as taking down a knot in the wood, smoothing small pieces, and chamfering edges.


title: Plane (tool)

text: Different types of bench planes are designed to perform different tasks, with the name and size of the plane being defined by the use. Bailey iron bench planes were designated by number respective to the length of the plane. This has carried over through the type, regardless of manufacturer. A No. 1 plane is but little more than five inches long. A typical smoothing plane (approx. nine inches) is usually a No. 4, jack planes at about fourteen inches are No. 5, an eighteen-inch fore plane will be a No. 6, and the jointer planes at twenty-two to twenty-four inches in length are No. 7 or 8 respectively. A designation, such as No. 4½ indicates a plane of No. 4 length but slightly wider. A designation, such as 5-1/2 indicates the length of a No. 5 but slightly wider (actually, the width of a No. 6 or a No. 7), while a designation, such as 5-1/4 indicates the length of a No. 5 but slightly narrower (actually, the width of a No. 3). "Bedrock" versions of the above are simply 600 added to the base number (although no "601" was ever produced, such plane is indeed available from specialist dealers; 602 through 608, including all the fractionals, were made).


title: Plane (tool)

text: Planing wood along its side grain should result in thin shavings rising above the surface of the wood as the edge of the plane iron is pushed forward, leaving a smooth surface, but sometimes splintering occurs. This is largely a matter of cutting "with the grain" or "against the grain" respectively, referring to the side grain of the piece of wood being worked.


title: Plane (tool)

text: The grain direction can be determined by looking at the edge or side of the work piece. Wood fibers can be seen running out to the surface that is being planed. When the fibers meet the work surface it looks like the point of an arrow that indicates the direction. With some very figured and difficult woods, the grain runs in many directions and therefore working against the grain is inevitable. In this case, a very sharp and finely-set blade is required.


title: Plane (tool)

text: When planing "against the grain", the wood fibers are lifted by the plane iron, resulting in a jagged finish, called tearout.


title: Plane (tool)

text: Planing the end grain of the board involves different techniques and frequently different planes designed for working end grain. Block planes and other bevel-up planes are often effective in planing the difficult nature of end grain. These planes are usually designed to use an iron bedded at a "low angle," typically about 12 degrees.


title: Propagator

text: In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called "(causal) Green's functions" (called ""causal"" to distinguish it from the elliptic Laplacian Green's function).


title: Propagator

text: In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t).


title: Propagator

text: Consider a system with Hamiltonian . The Green's function (fundamental solution) for the Schrödinger equation is a function


title: Propagator

text: where denotes the Hamiltonian written in terms of the coordinates, denotes the Dirac delta-function, is the Heaviside step function and is the kernel of the above Schrödinger differential operator in the big parentheses. The term propagator is sometimes used in this context to refer to , and sometimes to . This article will use the term to refer to (see Duhamel's principle).


title: Propagator

text: where is the unitary time-evolution operator for the system taking states at time to states at time . Note the initial condition enforced by formula_4.


title: Propagator

text: where the boundary conditions of the path integral include . Here denotes the Lagrangian of the system. The paths that are summed over move only forwards in time and are integrated with the differential formula_6 following the path in time.


title: Propagator

text: In non-relativistic quantum mechanics, the propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is specified by the equation


title: Propagator

text: If only depends on the difference , this is a convolution of the initial wave function and the propagator.


title: Propagator

text: For a time-translationally invariant system, the propagator only depends on the time difference , so it may be rewritten as


title: Propagator

text: The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,


title: Propagator

text: In relativistic quantum mechanics and quantum field theory the propagators are Lorentz-invariant. They give the amplitude for a particle to travel between two spacetime points.


title: Propagator

text: In quantum field theory, the theory of a free (or non-interacting) scalar field is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin-zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.


title: Propagator

text: The position space propagators are Green's functions for the Klein–Gordon equation. This means that they are functions satisfying


title: Propagator

text: We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform of the equation for the propagator, obtaining


title: Propagator

text: This equation can be inverted in the sense of distributions, noting that the equation has the solution (see Sokhotski–Plemelj theorem)


title: Propagator

text: with implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.


title: Propagator

text: The different choices for how to deform the integration contour in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the formula_20 integral.


title: Propagator

text: A contour going clockwise over both poles gives the causal retarded propagator. This is zero if is spacelike or if (i.e. if is to the future of ).


title: Propagator

text: is the proper time from to and formula_25 is a Bessel function of the first kind. The expression formula_26 means causally precedes which, for Minkowski spacetime, means


title: Propagator

text: This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator,


title: Propagator

text: A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if is spacelike or if (i.e. if is to the past of ).


title: Propagator

text: This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field.


title: Propagator

text: A contour going under the left pole and over the right pole gives the Feynman propagator, introduced by Richard Feynman in 1948.


title: Propagator

text: where and are two points in Minkowski spacetime, and the dot in the exponent is a four-vector inner product. is a and is a .


title: Propagator

text: This expression can be derived directly from the field theory as the vacuum expectation value of the "time-ordered product" of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same,


title: Propagator

text: This expression is Lorentz invariant, as long as the field operators commute with one another when the points and are separated by a spacelike interval.


title: Propagator

text: The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, and then to show that the functions providing the causal time ordering may be obtained by a contour integral along the energy axis, if the integrand is as above (hence the infinitesimal imaginary part), to move the pole off the real line.


title: Propagator

text: The Fourier transform of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators.


title: Propagator

text: They are often written with an explicit term although this is understood to be a reminder about which integration contour is appropriate (see above). This term is included to incorporate boundary conditions and causality (see below).


title: Propagator

text: For purposes of Feynman diagram calculations, it is usually convenient to write these with an additional overall factor of (conventions vary).


title: Propagator

text: The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is "nonzero" outside of the light cone, though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle travelling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages?


title: Propagator

text: The answer is no: while in classical mechanics the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is commutators that determine which operators can affect one another.


title: Propagator

text: So what "does" the spacelike part of the propagator represent? In QFT the vacuum is an active participant, and particle numbers and field values are related by an uncertainty principle; field values are uncertain even for particle number "zero". There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlation. Indeed, the propagator is often called a "two-point correlation function" for the free field.


title: Propagator

text: Since, by the postulates of quantum field theory, all observable operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables.


title: Propagator

text: Regarding virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle-antiparticle pair that eventually disappears into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone. However, no signaling back in time is allowed.


title: Propagator

text: This is the usual definition but normalised by a factor of formula_40. Then the rule is that one only takes the limit formula_41 at the end of a calculation.


title: Propagator

text: Hence this means that a single photon will always stay on the light cone. It is also shown that the total probability for a photon at any time must be normalised by the reciprocal of the following factor:


title: Propagator

text: We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.


title: Propagator

text: The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every "internal line", that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's Lagrangian for every internal vertex where lines meet. These prescriptions are known as "Feynman rules".


title: Propagator

text: Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be off shell. In fact, since the propagator is obtained by inverting the wave equation, in general, it will have singularities on shell.


title: Propagator

text: The energy carried by the particle in the propagator can even be "negative". This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticle is going the "other" way, and therefore carrying an opposing flow of positive energy. The propagator encompasses both possibilities. It does mean that one has to be careful about minus signs for the case of fermions, whose propagators are not even functions in the energy and momentum (see below).


title: Propagator

text: Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed "loop", the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of renormalization.


title: Propagator

text: If the particle possesses spin then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The differential equation satisfied by the propagator for a spin particle is given by


title: Propagator

text: where is the unit matrix in four dimensions, and employing the Feynman slash notation. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation,


title: Propagator

text: where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus


title: Propagator

text: the momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics is found to have form


title: Propagator

text: The downstairs is a prescription for how to handle the poles in the complex -plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written


title: Propagator

text: for short. It should be remembered that this expression is just shorthand notation for . "One over matrix" is otherwise nonsensical. In position space one has


title: Propagator

text: The propagator for a gauge boson in a gauge theory depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg, the propagator for a photon is


title: Propagator

text: The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter , up to overall sign and the factor of formula_59, reads


title: Propagator

text: With these general forms one obtains the propagators in unitary gauge for , the propagator in Feynman or 't Hooft gauge for and in Landau or Lorenz gauge for . There are also other notations where the gauge parameter is the inverse of , usually denoted (see gauges). The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.


title: Propagator

text: where formula_67 is the number of spacetime dimensions, formula_68 is the transverse and traceless spin-2 projection operator and formula_69 is a spin-0 scalar multiplet.


title: Propagator

text: where formula_71 is the Hubble constant. Note that upon taking the limit formula_72 and formula_73, the AdS propagator reduces to the Minkowski propagator.


title: Propagator

text: The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. We follow the notation in Bjorken and Drell. See also Bogolyubov and Shirkov (Appendix A). These functions are most simply defined in terms of the vacuum expectation value of products of field operators.


title: Propagator

text: We can define the positive and negative frequency parts of formula_74, sometimes called cut propagators, in a relativistically invariant way.


title: Propagator

text: The retarded, advanced and Feynman propagators defined above are all Green's functions for the Klein–Gordon equation.


title: Black Emanuelle

text: Black Emanuelle (Italian: "Emanuelle nera") is a softcore sexploitation film from 1975 directed by Bitto Albertini. This Africa set film was shot mostly in Kenya. The music was composed by Nico Fidenco.


title: Black Emanuelle

text: "Black Emanuelle" was followed by a number of sequels, all revolving around the erotic adventures of Mae Jordan (played by Laura Gemser), a globe-trotting, hedonistic investigative journalist and photographer known to her readers as "Emanuelle". Her character has been described as "a strong and independent woman, sexually proactive, at the centre of wealthy young and old white men of power, and involved in any sort of depraved set and situation."


title: Black Emanuelle

text: Journalist and photographer Mae Jordan (Laura Gemser) publishes her work under the name Emanuelle. She accepts an assignment from a diplomatic couple in Nairobi, and starts a sexual relationship with both. Together they teach her the ways of the country and love.


title: Black Emanuelle

text: According to the account in his memoir, Albertini came across a picture of Gemser at a travel agency while he was shooting in Kenya. Struck by her beauty, he located her agent in Gand (Belgium). Upon travelling there and telling him that he wanted her to star in a film, the agent tried to dissuade him, pointing out that she could hardly pose for a photograph properly, let alone act. Albertini, however, insisted. In the meantime, the producer had called together Angelo Infanti, Karin Schubert and Gabriele Tinti, and they started shooting in Kenya. After a few days, Gemser loosened up and began to act.


title: Black Emanuelle

text: Albertini's account omits that Gemser had already appeared in two films: "Amore libero - Free Love" and "".


title: Black Emanuelle

text: "Black Emanuelle" was made to cash in on the success of the French film "Emmanuelle" with Sylvia Kristel, which was released the year before.


title: Black Emanuelle

text: Albertini claims it was he who omitted one "m" in the name of the lead character, in contrast to the French Emmanuelle character.


title: Black Emanuelle

text: Director Albertini remembers that the film was a huge success. He states that this was also due to the "beautiful costumes" () by Adriana Spadaro and to the soundtrack by Nico Fidenco, who Albertini had invited for the job. As Albertini remembers, the soundtrack record "Black Emanuelle" stayed in the hit parades for a long time.


title: Black Emanuelle

text: By 1976 came two "Black Emanuelle" follow-ups, one in title ("Black Emanuelle 2" by Albertini) and one in plot ("Emanuelle in Bangkok", a.k.a. "Black Emanuelle 2", by Joe D'Amato), also triggering four quasi-sequels from 1977 to 1978 by D'Amato (under the title "Emanuelle") and two "Emanuelle" women in prison films by Bruno Mattei in 1982 and 1983.


title: Black Emanuelle

text: "Black Emanuelle 2" differs greatly in plot than the first film, featuring Israeli actress Shulamith Lasri as Emanuelle Richmond, a supermodel going through a state of amnesia and locked in a mental institution in New York. The lead actor, as in the first film, is Angelo Infanti.


title: Black Emanuelle

text: Albertini's later movie, "Il Mondo dei sensi di Emy Wong" (1977, starring Chai Lee) was released as "Emanuelle Gialla" and "Yellow Emanuelle" in some markets.


title: Black Emanuelle

text: "Emanuelle in Bangkok" that stars Laura Gemser as the journalist lead character 'Emanuelle' of the first "Black Emanuelle" film has the original Italian title of "Emanuelle nera - Orient Reportage" and is considered a genuine sequel directed by Joe D'Amato. The later D'Amato sequels that all have the same lead character but do not use the word "nera" (black) in their titles are noted to feature scenes of extreme violence and depravity (one controversial scene in "Emanuelle in America" shows a naked woman masturbating a horse).


title: Black Emanuelle

text: Kristopher Spencer calls Fidenco's scores to the Black Emanuelle films "by turns sultry and serious, fun and funky", and describes the instrumentation and sound typical of these soundtracks:


title: Black Emanuelle

text: Four years after the release of the last Emanuelle film by D'Amato, the journalist character 'Emanuelle' played by Laura Gemser was revived by Bruno Mattei in two women in prison films: "Violence in a Women's Prison" where the real name of the character is given as Laura Kendall and "Emanuelle Escapes from Hell" ("Emanuelle fuga dall'inferno" a.k.a. "Blade Violent - I violenti"). "Emanuelle Escapes from Hell" was directed by Mattei and Claudio Fragasso under the collective pseudonym Gilbert Roussel.


title: Black Emanuelle

text: Uncut versions of several Emanuelle films contain scenes depicting actual penetration. Also "Black Emanuelle" and "Emanuelle Around the World" contain scenes where the Emanuelle character is seen having explicit sex. These scenes were created with hardcore inserts, using a body double. Laura Gemser never performed explicit sexual acts on film, nor was she informed that a body double would be used.