title: Gregory Harrison

text: Gregory Neale Harrison (born May 31, 1950) is an American actor. He is known for his roles as Chandler in the 1987 film "North Shore", as Dr. George Alonzo "Gonzo" Gates, the young surgeon assistant of Dr. Trapper John McIntyre (played by Pernell Roberts) on the CBS series "Trapper John, M.D." (1979–86), and as ruthless business tycoon Michael Sharpe in the CBS series "Falcon Crest" (1989–1990). Since 2015, he has played Joe O'Toole, father of Oliver, in the Hallmark Channel expansion films of "Signed, Sealed and Delivered".


title: Gregory Harrison

text: Harrison was born in Avalon, California, in 1950, the middle child of Ed Harrison, a ship's captain and poet, and Donna Lee Nagely, an aspiring dancer; they eventually divorced. He has an older sister, Kathleen (born 1948), and a younger brother, Christopher (born 1961). He served for two years in the United States Army during the Vietnam War era as a medic.


title: Gregory Harrison

text: He was the title character on the science fiction series "Logan's Run" (1977-78), after which he played Levi Zendt on the NBC miniseries "Centennial" (1978).


title: Gregory Harrison

text: He appeared on an episode of "M*A*S*H" before garnering fame in 1979 with the role of surgeon Dr. George "Gonzo" Gates on "Trapper John, M.D.", starring opposite Pernell Roberts. The medical drama was spun off from "M*A*S*H". Harrison remained as Gonzo on "Trapper John" until the middle of the show's seventh season, when he decided to leave for other ventures. The series continued for several more episodes without Harrison, but concluded its run at the end of the 1985–86 season.


title: Gregory Harrison

text: Harrison's later role as stripper John Phillips in the 1981 TV movie "For Ladies Only" made him a favorite with women and gay men in the 1980s. He spoofed that role in the 1986 miniseries "Fresno" where his character appeared shirtless at every opportunity. "Fresno" was also a spoof of the prime-time series "Falcon Crest", where Harrison became a regular three years later in the final season (1989–90). As "Falcon Crest" ended production in early 1990, the series' parent studio, Lorimar Television, cast Harrison in its upcoming CBS sitcom "The Family Man", from producers Thomas L. Miller and Robert L. Boyett. The series, which premiered that fall, featured Harrison as a single fireman father raising his four children with the help of his father-in-law, played by Al Molinaro. Despite a few attempts by CBS to help it build an audience, "The Family Man" was cancelled after one season.


title: Gregory Harrison

text: He later joined the cast of NBC's "Sisters" for the 1994–95 season, playing Daniel Albright. In 1996, he starred opposite Eric Roberts in "It's My Party", a film based on the true story of a man in the end stages of AIDS who planned a party to say goodbye to friends and family.


title: Gregory Harrison

text: He starred in the WB Network's "Safe Harbor" and "One Tree Hill" and has made guest appearances on other shows such as "Touched by an Angel", "Judging Amy", "Rizzoli & Isles",


title: Gregory Harrison

text: He played Billy Flynn in a touring production of "Chicago" and was the male lead in each film of the "Au Pair" trilogy. He also appeared as Benjamin Stone in the Broadway revival of Stephen Sondheim's "Follies" and starred in the original Broadway production of Kander & Ebb's musical "Steel Pier". In 2011, he had a guest-starring role as a doctor on ABC's "Body of Proof". More recently, he appeared on "General Hospital".


title: Gregory Harrison

text: Harrison has been married since 1980 to actress Randi Oakes. The couple have four children, three girls and a boy. He lived in Gold Beach, Oregon, for 15 years; as of 2007, he lives in Eugene, Oregon.


title: Gregory Harrison

text: Harrison became addicted to cocaine during the 1980s and was able to break the addiction with the help of the Betty Ford Center.


title: Spanish dollar

text: The Spanish dollar, also known as the piece of eight (, , , or ), is a silver coin of approximately diameter worth eight Spanish reales. It was minted in the Spanish Empire following a monetary reform in 1497 with content 25.563 g = 0.822 oz t fine silver. It was widely used as the first international currency because of its uniformity in standard and milling characteristics. Some countries countermarked the Spanish dollar so it could be used as their local currency.


title: Spanish dollar

text: Because the Spanish dollar was widely used in Europe, the Americas, and the Far East, it became the first world currency by the late 18th century.


title: Spanish dollar

text: The Spanish dollar was the coin upon which the original United States dollar was based (at 0.7735 oz t = 24.0566 g), and it remained legal tender in the United States until the Coinage Act of 1857. Many other currencies around the world, such as the Japanese yen and the Chinese yuan, were initially based on the Spanish dollar and other 8-real coins. Diverse theories link the origin of the "$" symbol to the columns and stripes that appear on one side of the Spanish dollar.


title: Spanish dollar

text: The term peso was used in Spanish to refer to this denomination, and it became the basis for many of the currencies in the former Spanish colonies, including the Argentine, Bolivian, Chilean, Colombian, Costa Rican, Cuban, Dominican, Ecuadorian, Guatemalan, Honduran, Mexican, Nicaraguan, Paraguayan, Philippine, Puerto Rican, Peruvian, Salvadoran, Uruguayan, and Venezuelan pesos. Of these, "peso" remains the name of the official currency in the Philippines, Mexico, Cuba, the Dominican Republic, Colombia, Chile, Argentina, and Uruguay.


title: Spanish dollar

text: In the 16th century, Count Hieronymus Schlick (Jeroným Šlik in Czech) of Bohemia began minting a silver coin known as a (from German , modern spelling , "valley", cognate with "dale" in English), named for "," the valley in the Ore Mountains where the silver was mined. was later shortened to or "thaler", a word that eventually found its way into many European languages including the Spanish and English as "dollar".


title: Spanish dollar

text: The weighed 451 Troy grains (29.2 g = 0.94 oz t) of silver. So successful were these coins that similar s were minted in Burgundy and France. This coin was then succeeded by the long-lived Reichsthaler of the Holy Roman Empire, used from the 16th to 19th centuries, of 25.984 g = 0.8355 oz t pure silver.


title: Spanish dollar

text: The Netherlands also introduced its own dollars in the 16th century: the Burgundian Cross Thaler ("Bourgondrische Kruisdaalder"), the German-inspired "Rijksdaalder", and the Dutch liondollar ("leeuwendaalder"). The latter coin was used for Dutch trade in the Middle East, in the Dutch East Indies and West Indies, and in the thirteen colonies of North America.


title: Spanish dollar

text: For the English North American colonists, however, the Spanish peso or "piece of eight" has always held first place, and this coin was also called the "dollar" as early as 1581. After the Declaration of American Independence the United States dollar was introduced in 1792 at par with this coin at 371.25 grains = 0.7735 troy ounces = 24.0566 g. Alexander Hamilton arrived at these numbers based on a treasury assay of the average fine silver content of a selection of worn Spanish dollars.


title: Spanish dollar

text: The term "cob" was used in Ireland and the British colonies to mean a piece of eight or a Spanish-American dollar, because Spanish gold and silver coins were irregularly shaped and crudely struck before the machine-milled dollar was introduced in 1732.


title: Spanish dollar

text: After the introduction of the Guldengroschen in Austria in 1486, the concept of a large silver coin with high purity (sometimes known as "specie" coinage) eventually spread throughout the rest of Europe. Monetary reform in Spain brought about the introduction of an 8-real (or 1-peso) coin in 1497, minted to the following standards-


title: Spanish dollar

text: This was supplemented in 1537 by the gold "escudo", minted at 68 to a mark of gold 0.917 fine (fineness reduced to 0.906 in 1742 and 0.875 in 1786). It was valued at 15–16 reales or approximately 2 dollars. The famed "Gold Doubloon" was worth 2 escudos or approximately 4 dollars.


title: Spanish dollar

text: From the 15th to the 19th centuries the coin was minted with several different designs at various mints in Spain and the New World, having gained wide acceptance beyond Spain's borders. Thanks to the vast silver deposits that were found mainly in Potosí in modern-day Bolivia and to a lesser extent in Mexico (for example, at Taxco and Zacatecas), and to silver from Spain's possessions throughout the Americas, mints in Mexico and Peru also began to strike the coin. The main New World mints for Spanish dollars were at Potosí, Lima, and Mexico City (with minor mints at Bogotá, Popayán, Guatemala City, and Santiago), and silver dollars from these mints could be distinguished from those minted in Spain by the Pillars of Hercules design on the reverse.


title: Spanish dollar

text: The dollar or peso was divided into 8 reales in Spanish Latin America until 19th century when the peso was divided instead into 100 centavos. However, monetary turbulence in Spain beginning under the reign of King Philip II resulted in the dollar being subdivided as follows in Spain only:


title: Spanish dollar

text: Spain's adoption of the peseta in 1869 and its joining the Latin Monetary Union meant the effective end of the last vestiges of the Spanish dollar in Spain itself. However, the 5-peseta coin (or "duro") was slightly smaller and lighter but was also of high purity (90%) silver.


title: Spanish dollar

text: In the 1990s, commemorative 2,000-peseta coins were minted, similar in size and weight to the dollar.


title: Spanish dollar

text: Following independence in 1821, Mexican coinage of silver reales and gold escudos followed that of Spanish lines until decimalization and the introduction of the peso worth 8 reales or 100 centavos. It continued to be minted to Spanish standards throughout the 19th century, with the peso at 27.07 g of 0.9028 fine silver, and the escudo at 3.383 g of 0.875 fine gold. The Mexican peso or 8-real coin continued to be a popular international trading coin throughout the 19th century.


title: Spanish dollar

text: After 1918, the peso was reduced in size and fineness, with further reductions in the 1940s and 1950s. However, 2- (1921), 5- (1947) and 10- (1955) peso coins were minted during the same period with sizes and fineness similar to the old peso.


title: Spanish dollar

text: After the colony of New South Wales was founded in Australia in 1788, it ran into the problem of a lack of coinage, particularly since trading vessels took coins out of the colony in exchange for their cargo. In 1813, Governor Lachlan Macquarie made creative use of £10,000 in Spanish dollars sent by the British government. To make it difficult to take the coins out of the colony, and to double their number, the centres of the coins were punched out. The punched centre, known as the "dump", was valued at 15 pence, and the outer rim, known as the "holey dollar", was worth five shillings. This was indicated by overstamping the two new coins. The obverse of the holey dollar was stamped the words "New South Wales" and the date, 1813, and the reverse with the words "five shillings". The obverse of the dump was stamped with a crown, the words "New South Wales" and the date, 1813, and the reverse with the words "fifteen pence". The mutilated coins became the first official currency produced specifically for circulation in Australia. The expedient was relatively short lived. The British Parliament passed the Sterling Silver Money Act in 1825, which made British coins the only recognised form of currency and ended any legitimate use of the holey dollar and dump in the Australian colonies.


title: Spanish dollar

text: The Coinage Act of 1792 created the United States Mint and initially defined the United States dollar at par with the Spanish dollar due to its international reputation:


title: Spanish dollar

text: By far the leading specie coin circulating in America was the Spanish silver dollar, defined as consisting of 387 grains of pure silver. The dollar was divided into "pieces of eight," or "bits," each consisting of one-eighth of a dollar. Spanish dollars came into the North American colonies through lucrative trade with the West Indies. The Spanish silver dollar had been the world's outstanding coin since the early 16th century, and was spread partially by dint of the vast silver output of the Spanish colonies in Latin America. More important, however, was that the Spanish dollar, from the 16th to the 19th century, was relatively the most stable and least debased coin in the Western world.


title: Spanish dollar

text: The Coinage Act of 1792 specified that the U.S. dollar would contain 371.25 grains (24.057 g) pure or 416 grains (26.96 g) standard silver. This specification was based on the average weight of a random selection of worn Spanish dollars which Alexander Hamilton ordered to be weighed at the Treasury. Initially this dollar was comparable to the 371–373 grains found in circulating Spanish dollars and aided in its exportation overseas. The restoration of the old 0.9028 fineness in the Mexican peso after 1821, however, increased the latter's silver content to 24.44 g and reduced the export demand for U.S. dollars.


title: Spanish dollar

text: Before the American Revolution, owing to British mercantilist policies, there was a chronic shortage of British currency in Britain's colonies. Trade was often conducted with Spanish dollars that had been obtained through illicit trade with the West Indies. Spanish coinage was legal tender in the United States until the Coinage Act of 1857 discontinued the practice. The pricing of equities on U.S. stock exchanges in -dollar denominations persisted until the New York Stock Exchange converted first to pricing in sixteenths of a dollar on 24 June 1997, and then in 2001 to decimal pricing.


title: Spanish dollar

text: Long tied to the lore of piracy, "pieces of eight" were manufactured in the Spanish Americas and transported in bulk back to Spain, making them a very tempting target for seagoing pirates. In the Far East, it also arrived in the form of the Philippine peso in the Philippines as part of the Spanish East Indies of the Spanish colonial empire through the Manila galleons that transported Mexican silver peso to Manila in the Manila-Acapulco Galleon Trade, where it would be exchanged for Philippine and Chinese goods, since silver was the only foreign commodity China would accept. In Oriental trade, Spanish dollars were often stamped with Chinese characters known as "chop marks" which indicated that particular coin had been assayed by a well-known merchant and determined to be genuine. The specifications of the Spanish dollar became a standard for trade in the Far East, with later Western powers issuing trade dollars, and colonial currencies such as the Hong Kong dollar, to the same specifications.


title: Spanish dollar

text: The first Chinese yuan coins had the same specification as a Spanish dollar, leading to a continuing equivalence in some respects between the names "yuan" and "dollar" in the Chinese language. Other currencies also derived from the dollar include the Japanese yen, Korean won, Philippine peso, Malaysian ringgit, French Indochinese piastre, etc since it was widely traded across the Far East in the East Indies and the East Asia.


title: Spanish dollar

text: Contemporary names used for Spanish dollars in Qing Dynasty China include běnyáng (本洋), shuāngzhù (双柱), zhùyáng (柱洋), fóyáng (佛洋), fótóu (佛頭), fóyín (佛銀), and fótóuyín (佛頭銀). The "fó" element in those Chinese names referred to the King of Spain in those coins, as his face resembled that of images of the Buddha (佛 in chinese); and the "zhù" part of those names referred to the two pillars in the Spanish coat of arms.


title: Spanish dollar

text: In modern pop culture and fiction, pieces of eight are most often associated with the popular notion of pirates.


title: Money in the Bank (2021)

text: The 2021 Money in the Bank was the 12th annual Money in the Bank professional wrestling pay-per-view (PPV) and livestreaming event produced by WWE. It was held for wrestlers from the promotion's Raw and SmackDown brand divisions. The event took place on July 18, 2021, at Dickies Arena in Fort Worth, Texas, and was the first Money in the Bank to air on the livestreaming service Peacock. It was also the first Money in the Bank event to take place in the state of Texas.


title: Money in the Bank (2021)

text: During the height of the COVID-19 pandemic from March 2020 to July 2021, WWE had presented their shows behind closed doors, first from the WWE Performance Center, then moving in August 2020 to a bio-secure bubble called the WWE ThunderDome (excluding WrestleMania 37 in April 2021, which was the company's first major event to have live ticketed fans since the pandemic began). With declining cases and the availability of vaccines, WWE left the ThunderDome and returned to a live touring schedule in mid-July 2021, marking Money in the Bank as the first WWE PPV with a full crowd, and the first PPV held outside of the state of Florida since Elimination Chamber in March 2020.


title: Money in the Bank (2021)

text: Seven matches were contested at the event, including one on the Kickoff pre-show. In the main event, Roman Reigns defeated Edge to retain SmackDown's Universal Championship. Additionally, Big E and Nikki A.S.H. won their respective men's and women's Money in the Bank ladder matches. In other prominent matches, Bobby Lashley defeated Kofi Kingston to retain Raw's WWE Championship and Charlotte Flair defeated Rhea Ripley to win her record-tying fifth Raw Women's Championship and 13th overall singles championship in the women's division. The event also saw the surprise return of John Cena, appearing for the first time since WrestleMania 36 in April 2020.


title: Money in the Bank (2021)

text: This event is also notable for marking the beginning of the Usos' 5th reign as SmackDown Tag Team Champions, which currently stands as the longest reign in the championship's history, and the longest WWE Tag Team Championship reign in the history of WWE.


title: Money in the Bank (2021)

text: Money in the Bank is an annual gimmick pay-per-view (PPV) and WWE Network event produced by WWE since 2010, generally held between May and July. The concept of the show comes from WWE's established Money in the Bank ladder match, in which multiple wrestlers use ladders to retrieve a briefcase hanging above the ring. The briefcase contains a contract that guarantees the winner a match for a world championship of their choosing at any time within the next year. The 2021 event featured wrestlers from the Raw and SmackDown brands; male wrestlers competed for a contract to grant them a match for either Raw's WWE Championship or SmackDown's Universal Championship, while female wrestlers competed for a Raw Women's Championship or SmackDown Women's Championship match contract. While the previous year's event had featured a "Corporate Ladder" gimmick, which took place at WWE's Titan Towers headquarters and had only six competitors in each match due to the COVID-19 pandemic, the 2021 event reverted to the standard version of the titular ladder match as well as returning to having eight competitors for both matches, evenly divided between the two brands. It was the 12th event in the Money in the Bank chronology and the first to be held in July since the 2013 event. It was also the first Money in the Bank to air on Peacock after the American version of the WWE Network merged under Peacock in March.


title: Money in the Bank (2021)

text: The 2021 event was initially scheduled for May 16, however, it switched dates with WrestleMania Backlash and was to take place on June 20 until Hell in a Cell was announced for that date with Money in the Bank pushed back to July 18, which had originally been reported as the date for Extreme Rules. It was also reported that Money in the Bank would be held with live ticketed fans in attendance. Since August 2020, WWE had presented Raw and SmackDown's shows from a bio-secure bubble called the WWE ThunderDome due to the COVID-19 pandemic. With cases declining and vaccines available to most American citizens, the company announced that they would be leaving the ThunderDome and returning to live touring, starting with a 25-city tour beginning with the July 16, 2021, episode of "SmackDown" in Houston, Texas. Money in the Bank was in turn announced to take place at Dickies Arena in Fort Worth, Texas on July 18, thus it was the first WWE pay-per-view held outside of Florida since Elimination Chamber on March 8, 2020, and the first pay-per-view held following the end of the ThunderDome Era.


title: Money in the Bank (2021)

text: The event comprised seven matches, including one on the Kickoff pre-show. The matches resulted from scripted storylines, where wrestlers portrayed heroes, villains, or less distinguishable characters in scripted events that built tension and culminated in a wrestling match or series of matches. Results were predetermined by WWE's writers on the Raw and SmackDown brands, while storylines were produced on WWE's weekly television shows, "Monday Night Raw" and "Friday Night SmackDown."


title: Money in the Bank (2021)

text: Qualifying matches for the men's Money in the Bank ladder match began on the June 21 episode of "Raw". Ricochet, John Morrison, and Riddle qualified by defeating AJ Styles, Randy Orton, and Drew McIntyre, respectively. Raw's final spot was determined in a triple threat match the following week, which was originally scheduled to be between Styles, Orton, and McIntyre; however, Orton was unable to compete for unknown reasons. A battle royal was held to fill Orton's spot in the triple threat match. Orton's RKBro tag team partner Riddle convinced WWE officials Adam Pearce and Sonya Deville to allow him to compete in the battle royal where if he won, he would represent Orton in the triple threat match. Although Riddle won the battle royal, McIntyre won the triple threat match to earn Raw's final spot in the Money in the Bank ladder match. SmackDown's first spot was filled by Big E, who qualified by defeating Intercontinental Champion Apollo Crews on the June 25 episode of "SmackDown". The following week, Kevin Owens qualified by defeating Sami Zayn in a Last Man Standing match, which was a rematch from Hell in a Cell that Zayn had won. SmackDown's last two spots were filled on the July 9 episode: Seth Rollins defeated Cesaro, which was another rematch from Hell in a Cell that Rollins had also won, and King Nakamura defeated Baron Corbin, previously King Corbin before he lost his King of the Ring crown to Shinsuke Nakamura, who took on the king moniker.


title: Money in the Bank (2021)

text: Qualifying matches for the women's Money in the Bank ladder match also began on the June 21 episode of "Raw". The brand's four participants were determined by two tag team matches in which the members of the winning teams qualified. Asuka and Naomi qualified by defeating the team of Eva Marie and Doudrop (formerly Piper Niven in NXT UK) and later, Alexa Bliss and Nikki Cross, the latter incorporating a superhero gimmick and a new nickname of Nikki A.S.H. (short for Almost a Super Hero), qualified by defeating the team of Nia Jax and Shayna Baszler. On the June 25 episode of "SmackDown", WWE official Sonya Deville named two-time Money in the Bank winner Carmella as the first of SmackDown's four participants. Liv Morgan, who had been feuding with Carmella and defeated her the previous week, took issue as Carmella did not compete in a qualifier match to earn her spot. Deville then had Morgan face Carmella to prove she deserved to be in the match and Morgan subsequently defeated Carmella; however, Deville did not add her to the ladder match. The following week, Deville named Zelina Vega, who made her return to WWE after her release in November 2020, as the second SmackDown entrant in the match. Morgan again confronted Deville about Vega not having to qualify. She also pointed out the fact that she proved herself by defeating Carmella and was not added, but would prove herself again in a match against Vega. Morgan subsequently defeated Vega, but was still not added. On the July 9 episode, Carmella was removed from the match as she was named as Bayley's replacement in the SmackDown Women's Championship match against Bianca Belair due to Bayley suffering an injury. Morgan once again confronted Deville about her giving Carmella an opportunity despite not earning one; although annoyed at Morgan's interruption, Deville named Morgan as Carmella's replacement in the Money in the Bank ladder match. WWE Women's Tag Team Champions Natalya and Tamina were announced as SmackDown's last two participants on July 12 and July 15, respectively, via Twitter.


title: Money in the Bank (2021)

text: On the June 21 episode of "Raw", as WWE Champion Bobby Lashley and his manager MVP were celebrating Lashley's win over Drew McIntyre at Hell in a Cell the previous night, they were interrupted by The New Day (Kofi Kingston and Xavier Woods). Kingston reminded Lashley that he had defeated him in a non-title match on the May 17 episode of "Raw" and that he had also never received a rematch for the title after he lost the championship to Brock Lesnar at SmackDown's 20th Anniversary in October 2019. Kingston then challenged Lashley for the title. Lashley accepted and the match was scheduled for Money in the Bank.


title: Money in the Bank (2021)

text: At Hell in a Cell, Charlotte Flair defeated Raw Women's Champion Rhea Ripley by disqualification after Ripley attacked Flair with the top cover of the announce table; however, since titles do not change hands via disqualification unless stipulated, Flair did not win the title. The following night on "Raw", Flair claimed she was "proud" of Ripley and told Ripley that she had never seen her as a "strategic champion". WWE official Sonya Deville then scheduled a rematch between the two for the title at Money in the Bank.


title: Money in the Bank (2021)

text: At WrestleMania 37 in April, Roman Reigns defeated Edge and Daniel Bryan in a triple threat match to retain the Universal Championship. On the June 25 episode of "SmackDown", Reigns and his special counsel Paul Heyman had a Universal Championship address. Heyman stated that Reigns had defeated all of SmackDown's top contenders and that there was no one left to challenge Reigns. Edge then made a surprise return after a two-month hiatus and attacked Reigns and also took out Jimmy Uso, who had come to Reigns' aid. On "Talking Smack" the next day, it was revealed that Edge had confronted WWE officials Adam Pearce and Sonya Deville and demanded a one-on-one match against Reigns for the title at Money in the Bank, which he was originally supposed to have at WrestleMania before Bryan worked his way into the match. Pearce then granted him the match. The following week, Edge explained that the loss at WrestleMania affected him mentally, which is why he had taken the past two months off.


title: Money in the Bank (2021)

text: At Hell in a Cell, Bianca Belair defeated Bayley in a Hell in a Cell match to retain the SmackDown Women's Championship. On the following episode of "SmackDown", both were involved in a mixed tag team match in which Bayley pinned Belair. The following week, Bayley claimed that Belair's Hell in a Cell victory was a fluke, citing her own victory over Belair in the mixed tag team match. Belair then came out and challenged Bayley to an "I quit" match with the title on the line at Money in the Bank and Bayley accepted. On July 9, however, WWE announced that Bayley suffered a legitimate injury while training that would sideline her for up to nine months; this was followed by an announcement that a replacement for Bayley would be revealed on that night's "SmackDown". Carmella was revealed as the replacement and the match was changed to a regular singles match that would instead occur on the July 16 episode of "SmackDown", where Belair retained.


title: Money in the Bank (2021)

text: During the Money in the Bank Kickoff pre-show, Rey Mysterio and Dominik Mysterio defended the SmackDown Tag Team Championship against The Usos (Jey and Jimmy). During the match, Jey caught Rey with a Superkick in mid-air and then performed the Uso Splash on Rey for a nearfall. In the end, as Rey attempted a 619 on Jey, Jimmy protected Jey with Jimmy receiving the 619 instead. Jey then performed a roll-up on Rey with an assist from Jimmy to win the title for the fifth time, and their seventh overall tag team championship in WWE.


title: Money in the Bank (2021)

text: The actual pay-per-view opened with the women's Money in the Bank ladder match, featuring Alexa Bliss, Asuka, Naomi, and Nikki A.S.H. from Raw and Liv Morgan, Zelina Vega, and WWE Women's Tag Team Champions, Natalya and Tamina, from SmackDown. During the match, Bliss used her supernatural powers to try and lower the contract into her hands only for Vega to stop her. Later atop a ladder, Bliss hypnotized Vega into climbing down the ladder and allowing Bliss an opportunity to unhook the briefcase, however, Natalya intercepted Bliss. The rest of the competitors eventually buried Bliss under a pile of ladders, taking Bliss out for the remainder of the match. With three ladders positioned under the briefcase, Asuka, Naomi, Morgan, Vega, Natalya, and Tamina all fought atop of the three ladders. Nikki, who had been down outside the ring, used the opportunity to quickly climb the middle ladder and reach over the other competitors to unhook the briefcase and win the match.


title: Money in the Bank (2021)

text: Next, AJ Styles and Omos defended the Raw Tag Team Championship against The Viking Raiders (Erik and Ivar). In the end, Erik and Ivar performed The Viking Experience on Styles, however, Omos stood on the ring apron and shoved Erik into Ivar to void the pin attempt. Omos then delivered a Two-Handed Chokeslam on Erik to retain the title.


title: Money in the Bank (2021)

text: After that, Bobby Lashley (accompanied by MVP) defended the WWE Championship against Kofi Kingston. At the start of the match, Kingston performed a Double Stomp on Lashley's chest for a nearfall. Following this, Lashley completely dominated Kingston for the remainder of the match. After Kingston refused to submit to the Hurt Lock, Lashley performed the Dominator three times on Kingston before applying the Hurt Lock again, forcing Kingston to submit and retain the title.


title: Money in the Bank (2021)

text: In the fourth match, Rhea Ripley defended the Raw Women's Championship against Charlotte Flair. After a well contested match, as Ripley went for the Riptide, Flair countered into a DDT. Flair performed the Natural Selection on Ripley off the second rope for a nearfall. Ripley performed a German Suplex on Flair who rolled out of the ring. Flair rammed Ripley's head in a turnbuckle and trapped her leg in between the steel steps and the corner post, kicking it several times. Flair then got Ripley back in the ring and applied the Figure Eight Leglock on her to win the title for a record-tying fifth time, marking her 13th overall singles championship in WWE and 14th title overall.


title: Money in the Bank (2021)

text: The penultimate match was the men's Money in the Bank ladder match, featuring Drew McIntyre, John Morrison, Ricochet, and Riddle from Raw and Big E, Kevin Owens, King Nakamura, and Seth Rollins from SmackDown. During the match, as McIntyre ascended the ladder, he was attacked by Jinder Mahal with a steel chair and was carried off by Mahal's henchmen Shanky and Veer—continuing a rivalry that had recently begun on "Raw". As Ricochet ascended the ladder, Riddle tipped the ladder over and Ricochet jumped onto the other competitors, who were outside the ring. Riddle and Owens performed the RKO and Stunner, respectively, on the remaining competitors. As Owens ascended the ladder, Rollins pulled Owens off, performing a Powerbomb on Owens through a ladder that was wedged between the apron and announce table. As Rollins ascended the ladder, Big E stopped him and performed the Big Ending on Rollins off the ladder. Big E then ascended the ladder and unhooked the briefcase to win the match.


title: Money in the Bank (2021)

text: In the main event, Roman Reigns (accompanied by Paul Heyman) defended the Universal Championship against Edge. During the match, Reigns and Edge brawled at ringside. As Reigns attempted the Spear, Edge moved out of the way and Reigns went through the barricade. Edge then performed a Spear on Reigns through the remainder of the barricade. Back in the ring, as Reigns performed the Superman Punch on Edge, Edge incapacitated the referee. Reigns obtained a broken piece of a chair and attempted to apply the Crossface on Edge, however, Edge reversed and applied the Crossface on Reigns. The Usos (Jey and Jimmy) then came out and attempted to interfere in the match, however this was thwarted by Rey and Dominik Mysterio, who attacked them before they reached the ring. Seth Rollins, frustrated over losing the Money in the Bank ladder match earlier as well as for Edge receiving the championship match before him, then came out and performed a Superkick on Edge. As Reigns attempted a Spear, Edge performed a Spear of his own for a nearfall. Rollins then returned to the ring, only for Edge to knock him off the apron. This brief distraction allowed Reigns to perform a Spear on Edge and retain the title. Following the match, as Reigns and Rollins had a stare down, Edge attacked Rollins and they brawled into the crowd. As Reigns obtained a microphone and proclaimed that the whole world needed to acknowledge him, John Cena made a surprise return—in his first appearance since WrestleMania 36 in April 2020—and entered the ring where Cena had a stare down with Reigns and performed his "You Can't See Me" taunt at Reigns to end the event.


title: Money in the Bank (2021)

text: Though the event itself was well-received, Peacock's streaming of the event in the United States was poorly received due to technical issues occurring throughout, particularly during the Men's Money in the Bank ladder match.


title: Money in the Bank (2021)

text: Following his dominant victory over Kofi Kingston, Bobby Lashley (accompanied by MVP) held an open challenge for the WWE Championship on the following night's episode of "Raw". The challenge was accepted by a returning Keith Lee, who had been out with health issues since February, but lost to Lashley. Following the match, Lashley and MVP were confronted by WWE Hall of Famer Goldberg, appearing for the first time since the Royal Rumble in January. Goldberg proclaimed that he would be Lashley's next challenger, which was scheduled for SummerSlam.


title: Money in the Bank (2021)

text: Also on the following "Raw", as Charlotte Flair boasted about winning the Raw Women's Championship for a fifth time, she was confronted by Rhea Ripley, who challenged her to a rematch that night which was granted by WWE officials Adam Pearce and Sonya Deville. During the match, Flair was disqualified after striking Ripley with the championship belt, thus Ripley won the match but not the title. Following the match, the two brawled at ringside where Ripley performed the Riptide on Flair. Nikki A.S.H. then ran down to the ring and cashed in her Money in the Bank contract and defeated Flair to win the Raw Women's Championship. The following week, Nikki was scheduled to defend the title against Flair and Ripley in a triple threat match at SummerSlam.


title: Money in the Bank (2021)

text: Also on "Raw", Riddle teamed with The Viking Raiders (Erik and Ivar) in a six-man tag team match against John Morrison and Raw Tag Team Champions AJ Styles and Omos in which The Viking Raiders performed The Viking Experience on Morrison to win the match. Styles and Omos were subsequently scheduled to defend the Raw Tag Team Championship in a rematch against The Viking Raiders on the following week's episode, where Styles and Omos again retained.


title: Money in the Bank (2021)

text: Jinder Mahal, along with Shanky and Veer, gloated about costing Drew McIntyre the Money in the Bank ladder match. McIntyre then appeared and attacked Mahal and Veer with a steel chair. As Shanky tried to calm McIntyre down, McIntyre attacked Shanky with multiple chair shots. After a couple more weeks of feuding, a match between McIntyre and Mahal was scheduled for SummerSlam with Shanky and Veer banned from ringside.


title: Money in the Bank (2021)

text: Instead of waiting until "SmackDown", John Cena first appeared on "Raw" to explain his appearance at Money in the Bank. He stated that he wanted to put Reigns in his place and officially challenged Reigns for the Universal Championship at SummerSlam. On "SmackDown", however, Reigns rejected Cena's challenge and instead accepted a challenge from Finn Bálor, who had returned to SmackDown the previous week. An altercation occurred during the contract signing the following week, which ultimately led to Cena signing the match contract, officially scheduling Reigns to defend the title against Cena at SummerSlam.


title: Money in the Bank (2021)

text: Also on the following episode of "SmackDown", Edge claimed that he would have won the Universal Championship had it not been for Seth Rollins' actions. He stated that their issues date back to December 2014 when Rollins had attempted to injure Edge's surgically repaired neck and definitively put an end to his career. Rollins then came out and said he despised older wrestlers like Edge and John Cena for coming back and taking opportunities from the current wrestlers who earned them. A brawl then ensued with Edge performing the Edgecution on Rollins. As Edge attempted a Spear, Rollins retreated. On the August 6 episode, Edge challenged Rollins to a match at SummerSlam that Rollins accepted.


title: Money in the Bank (2021)

text: Big E cut a promo on winning the Money in the Bank briefcase and called it the biggest victory of his career only for Intercontinental Champion Apollo Crews (accompanied by Commander Azeez) to interrupt and gloat about defeating Big E for the title back at WrestleMania 37. Dolph Ziggler and Robert Roode interrupted, followed by King Nakamura (accompanied by Rick Boogs) and Cesaro, all wanting to challenge Crews for the Intercontinental Championship. A brawl then ensued with Big E, Cesaro, and Nakamura taking out Crews, Azeez, Ziggler, and Roode. Big E then appeared on the September 13 episode of "Raw" and successfully cashed in his Money in the Bank contract on Bobby Lashley to win the WWE Championship, thus transferring to Raw in the process, and made Big E WWE's 33rd Triple Crown winner.


title: Money in the Bank (2021)

text: Also on "SmackDown", Jimmy Uso defeated Dominik Mysterio in a singles match, thanks to Jey assisting Jimmy during the pin, similar to the pin during Money in the Bank Kickoff that won The Usos the SmackDown Tag Team Championship from Dominik and Rey Mysterio.


title: Xeno (series)

text: Xeno is a Japanese science fantasy video game franchise created by Tetsuya Takahashi. The first entry was developed by SquareSoft, and subsequent entries have been developed by Monolith Soft, a company founded by Takahashi after he left Square in 1999. While the various games have no direct story connections, they have common thematic links and all sport the ""Xeno"" prefix, which Takahashi has variously described as a means of identifying his games and a symbolic representation of the series. All the games in the "Xeno" meta series take place within a science fiction setting with some fantasy elements, with its stories frequently featuring psychological, philosophical, and religious themes.


title: Xeno (series)

text: The first title, "Xenogears", was originally proposed as a storyline for "Final Fantasy VII", as well as a sequel to the 1995 RPG Chrono Trigger from the Chrono Series, but was allowed to be developed as its own project. After Square shifted its focus onto the "Final Fantasy" series, Takahashi and several other "Xenogears" staff founded Monolith Soft and began work on the "Xenosaga" games. Both "Xenogears" and "Xenosaga" were intended to be six-part series, but differing circumstances caused plans to be cut down. After the premature end of the "Xenosaga" series, Monolith Soft began developing "Xenoblade Chronicles", initially intended to be an original title. The games of the "Xeno" franchise have generally sold well and received positive press worldwide.


title: Xeno (series)

text: Xenogears was released for the PlayStation in 1998 in Japan and North America. Set in an alien world with warring rival human empires, the protagonist Fei Fong Wong is drawn into the battle against Deus, an ancient machine weapon worshiped as a god. The gameplay features turn-based combat on foot and inside large bipedal mechs called "Gears". Originally planned as part of a six-episode story, "Xenogears" represented the fifth episode in the saga. "Xenogears" is owned by Square Enix.


title: Xeno (series)

text: The Xenosaga series consists of a main trilogy of role-playing games for the PlayStation 2, as well as spin-off titles that form part of the main narrative. The games were released between 2002 and 2006. The "Xenosaga" trilogy boast similar gameplay to "Xenogears", although the balance of story and gameplay underwent drastic revisions for the second game. As with "Xenogears", "Xenosaga" was planned as a six-episode story, but was cut down to a trilogy due to various factors. "Xenosaga" is not a direct continuation or prequel to "Xenogears" despite similarities, instead being a spiritual successor. The "Xenosaga" series is owned by Bandai Namco Entertainment.


title: Xeno (series)

text: Xenoblade Chronicles currently spans four games: the original game (2010), the spiritual successor "Xenoblade Chronicles X" (2015), "Xenoblade Chronicles 2" (2017) along with its expansion "" (2018) and "Xenoblade Chronicles 3" (2022). The games employ an action-based battle system, incorporating cooldown abilities and quick-time commands. "Xenoblade Chronicles X" also incorporates traversal using giant mechs known as Skells. While "Xenoblade Chronicles" adopts a story-driven design, "Xenoblade Chronicles X" uses a non-linear structure within an open world. No direct story connection exist between "Xenoblade Chronicles" and "Xenoblade Chronicles X". The Skells were also a deliberate callback to the Gears of "Xenogears". The "Xenoblade" series is owned by Nintendo and its subsidiary Monolith Soft.


title: Xeno (series)

text: The ""Xeno"" prefix, which means something strange or foreign to a group, was used in connection with the game's themes. The "Xeno" title has been repeated throughout the series: in an interview concerning "Xenoblade Chronicles", director Tetsuya Takahashi said that the prefix had become a symbol referencing the previous works of Monolith Soft. Later still, the "Xeno" title was described as a means of distinguishing Takahashi's work, phrased by the interviewer as a "director's signature". All of the games in the series have made use of a science fiction premise, although this has sometimes been placed in the background within settings more common to the fantasy genre.


title: Xeno (series)

text: According to Takahashi, each game in the series revolves around different themes. "Xenogears", "Xenosaga", and "Xenoblade" series make extensive use of various religious and philosophical themes. "Xenogears" incorporated concepts from the works of Friedrich Nietzsche, Sigmund Freud and Carl Jung. A particular reference is protagonist Fei, whose personality is split into multiple parts, one of which is a violent persona known as "Id". In addition, "Xenogears" uses motifs and references to abrahamic belief systems, along with other concepts such as reincarnation. During the development stage, main antagonist Deus' Japanese name was to have been "Yahweh", but the team were convinced by localization lead Richard Honeywood not to use it, and instead made the name a pun on a piece of Japanese slang.


title: Xeno (series)

text: The "Xenosaga" series made heavy use of biblical elements, particularly the New Testament. There are also extensive references to Gnosticism, Judaism and Jungian psychology. The games' principal writer said that the themes also paralleled many other world religions. The works of Nietzsche were also referenced in the "Xenosaga" games: the first "Xenosaga" drew its main theme from the "Will to power", a concept coined by Nietzsche to describe the driving motivation of humanity. The subtitles of each main "Xenosaga" also reference the ideas and works of Nietzsche.


title: Xeno (series)

text: The themes of "Xenoblade Chronicles" focused on the main characters overcoming a pre-determined fate and finding the meaning of human existence, along with what Takahashi described as "contrasting the realms of the micro and the macro". "Xenoblade Chronicles X" was a deliberate move away from this style of storytelling and the incorporation of philosophical themes: according to Takahashi, the concept was to create a solid gameplay foundation on which to base a future work which would feature a stronger story. Despite this, it covered similar thematic ground to "Xenoblade Chronicles".


title: Xeno (series)

text: "Xenogears", the first entry in what would become the "Xeno" series, was first proposed to Square by Tetsuya Takahashi and his wife, known under the pseudonym Soraya Saga, as a potential storyline for "Final Fantasy VII". While it was considered too mature for the "Final Fantasy" series, Takahashi was allowed to create an original work based on the premise. After initially attempting to create a sequel to "Chrono Trigger", Takahashi made the project entirely unique, beginning development approximately two years prior to its release. While a second "Xenogears" game was being planned, Square decided to focus on the "Final Fantasy" series, a decision that Takahashi did not agree with. Leaving Square, he established Monolith Soft in 1999 along with multiple other "Xenogears" staff. Monolith Soft was founded so that Takahashi could continue developing the concepts of "Xenogears", and with funding from Namco, the team began development on the first "Xenosaga".


title: Xeno (series)

text: The development cycle of the "Xenosaga" games was troubled: after the first game commercially underperformed, the development staff was changed, the proposed six-part series was cut down by half, and the second installment shifted focus from its story to its gameplay. After outcry, the third game shifted again in an attempt to balance story and gameplay while bringing the "Xenosaga" story to a close. After the failure of "Xenosaga", staff at Monolith Soft were in a state of low morale. During the development of "", Takahashi was struck with the idea of setting a game on the bodies of two frozen gods. After constructing a concept model for it with another member of staff, Monolith Soft began development. While originally intended to be an original title called "Monado: The Beginning of the World", Nintendo's then-CEO Satoru Iwata had the title changed to reflect Takahashi's previous games and hard work on the title, bringing the new game into the "Xeno" series. Working within deliberate restrictions and moving away from the previous cutscene-heavy style of "Xenogears", Monolith Soft worked hard to make the game a balance between gameplay and story. For "Xenoblade Chronicles X", the developers focused on the gameplay aspect, in particular creating an open world and online elements within a new set of self-imposed restrictions.


title: Xeno (series)

text: Near the release date of "Xenoblade Chronicles X," Takahashi stated that, if he were given the opportunity to make another game in the "Xenoblade" series, he would hope to create another traditional JRPG-style video game similar to "Xenoblade Chronicles". He stated that he hoped to continue the series by exploring many different settings instead of being confined to a specific genre. He also stated that he would like to use the core system set up in "Xenoblade Chronicles X" to further expand on the story and thematic elements for their next project. Two years later, "Xenoblade Chronicles 2" was released which returned to the story-driven JRPG style found in "Xenoblade Chronicles".


title: Xeno (series)

text: At the time of its development, Square said that "Xenogears" would likely not release in the west due to the religious content. The localization proved especially challenging, with the original translators either quitting the project or requesting transferral to other projects. This meant that Honeywood was under heavy pressure to render the game into English while both keeping it faithful to the original and stepping round some of the sensitive religious issues the title evoked. Although all three main "Xenosaga" were released in Japan and North America, the third game did not release in Europe, and the spin-offs remained exclusive to Japan. "Xenoblade Chronicles" was originally not announced for an overseas release, and despite being announced for release in Europe, its North American release was doubtful enough that a fan campaign dubbed Operation Rainfall began working to have the game, along with two other Wii role-playing games, released overseas. Later, during the run-up to the release of "Xenoblade Chronicles X", Takahashi stated that "Xenoblade Chronicles" was designed with an international audience in mind, and that he was pleased "Xenoblade Chronicles X" was receiving a western release in the same year as Japan.


title: Xeno (series)

text: The games have also undergone censorship. A sexually explicit scene featured in "Xenosaga" was toned down for its western release. The first and third "Xenosaga" games also received changes for their western release: in the first game, a scene between main antagonist Albedo and the character MOMO was toned down significantly; and the third game had all visible blood removed, which ended up making some scenes confusing. "Xenoblade Chronicles X" also received censorship in the form of the character Lin, who had her clothing made less sexually provocative.


title: Xeno (series)

text: An anime was produced by Namco and Monoliftsoft based on "Xenosaga", titled "", premiered on TV Asahi in Japan in January 2005. The story is based on "Episode I: Der Wille zur Macht". The series was licensed by ADV Films in June 2007 for $120,000.


title: Xeno (series)

text: Several characters from the "Xeno" series have been featured in the "Project X Zone" games, namely KOS-MOS and T-elos from the "Xenosaga" series and Fiora and Metal Face from the "Xenoblade" series.


title: Xeno (series)

text: Shulk, the main protagonist of "Xenoblade Chronicles", is a playable character in "Super Smash Bros. for Nintendo 3DS" and "Wii U", an entry in Nintendo's crossover fighting game series "Super Smash Bros.". He later returned in the sequel "Super Smash Bros. Ultimate". Additionally, Shulk is available as a Mystery Mushroom costume in "Super Mario Maker".


title: Xeno (series)

text: The outfit of "Xenoblade Chronicles 2"'s protagonist, Rex, is available as an extra costume for Link in "". It also appears as a wearable outfit for a Mii Fighter in "Super Smash Bros. Ultimate" as DLC. Pyra and Mythra from "Xenoblade Chronicles 2" were released as DLC characters for "Super Smash Bros. Ultimate" in March 2021.


title: Xeno (series)

text: The "Xeno" series has been very highly reviewed and praised by critics. "IGN" described "Xenogears" as the "hands-down best RPG" of the year, praising the storyline, gameplay, graphics, presentation, and soundtrack. The game placed 16th in "Famitsu"’s 2006 poll and 32nd in "IGN"’s poll of the greatest games of all time. "Famitsu" praised the world and character development of "Xenosaga Episode I", while "GamePro" stated that the story and characters "[rivaled] that of a good, hard, science-fiction novel or the best anime series available". "Xenosaga Episode II" was the least praised title of the series, with Eurogamer still praising its characters and plot, but finding the story uncomfortably long and complicated, requiring too much foreknowledge of the original game to be enjoyable for series newcomers. "GamePro" believed that "Episode 2" had changed game mechanics to make the title less approachable than the previous game, and "GameSpot" felt that the character growth system was simplified too much and made the game shorter as well. "Xenosaga Episode III" was met with divided feelings, as some such as Simon Parkin of "Eurogamer", felt that the sheer number of philosophical and religious elements in the story both stifled any relatable narrative and robbed the characters of any personality; despite this he felt the ending successfully tied up remaining narrative threads from previous games. "Famitsu" enjoyed the narrative's presentation, but noted the lack of a focused narrative due to the number of story threads needing addressing. "IGN" enjoyed the combat system and its blending of elements from past games, but was disappointed that no new elements had been added. After lower sales and still mixed reception, the "Xenosaga" development team's morale was low. To help revive their spirit, they started making a new game that eventually became "Xenoblade Chronicles".


title: Xeno (series)

text: "Xenoblade Chronicles" is thus far the highest praised series title by critics, earning perfect scores from Digital Spy, Joystiq, "GamePro" and RPGamer, and near-perfect scores from most other video gaming websites and magazines. The game received "universal acclaim", according to review aggregator Metacritic. It was also nominated for many awards, such as the 2011 Japan Game Awards where the title received the "Excellence" award. It was also nominated for many “Best RPG” and Best Wii/Wii U Game”. It was also nominated by "IGN" in the "Best Overall Role-playing Game" and "Best Overall Story" categories. "Xenoblade Chronicles X" also had its admirers, with Japanese gaming publication "Dengeki" giving the game a positive review, referring to it as a masterpiece; they praised the gameplay, detailed world, story direction, and music, and said it has a "very high degree of perfection". "Nintendo Life" praised the game's battle system, deep upgrade pathways, vast world size, and graphics, but criticized the occasional difficulty spike and fetch quest. "GameSpot" stated that, of "all the open-world games to come out this year, "Xenoblade Chronicles X" may be the most formidable" as a "truly enormous game, both in scale and scope," praising the landscapes, creature design, unlockables and quests, combat, and character progression and customization, but criticizing the inconsistent soundtrack, ambiguous systems, and disappointing story. "Xenoblade Chronicles 2" received wide acclaim. John Rairdin of "Nintendo World Report", who gave a 9.5 rating out of 10 to the title, considered the game "one of the finest JRPGs of the generation and perhaps of all time" and highly praising the music, "diverse world", "fresh and engaging combat", and "thrilling storyline", stating: "Washing over any minor issues is one of the most engaging stories I’ve ever played, a vastly improved and fun combat system, and an out-of-this-world soundtrack. It sets a precedent for JRPGs on the Switch that I doubt will be topped." "IGN Japan" gave a very positive review, stating that it "offers a timeless tale of adventure and an incredibly deep battle system." However, they criticized the fact that "its mechanics are not always well explained". The most recent title is an expansion called "" which was positively received, with some critics expressing enthusiasm for the story expansion. Other critics cited endearing, relatable characters, interesting story, and an improved battle system.


title: Xeno (series)

text: "Xenogears" shipped over one million copies worldwide by March 2003, with 910,000 being shipped in Japan and 280,000 overseas. The first "Xenosaga" was a commercial success in Japan, selling 450,000 units. It was also reported by Namco as one of their better-selling games overseas. The second "Xenosaga" sold over 256,000 copies in Japan by the end of 2004, and like its predecessor was considered commercially successful overseas. The third "Xenosaga" sold 343,000 units in all territories by the third quarter of 2006. Ultimately, the "Xenosaga" series was considered a commercial disappointment for Namco. "Xenoblade Chronicles" met with strong sales in Japan despite being released near to the end of its console's life cycle, and was commercially successful in the UK and North America. Despite low sales compared to other console titles, "Xenoblade Chronicles X" was also successful, with the majority of its sales coming from outside of Japan. "Xenoblade Chronicles 2" would become one of the highest selling games for the Nintendo Switch for 2017 and the highest selling Xeno series title of all time, with 1.53 million units sold by 2018.


title: Tegra

text: Tegra is a system on a chip (SoC) series developed by Nvidia for mobile devices such as smartphones, personal digital assistants, and mobile Internet devices. The Tegra integrates an ARM architecture central processing unit (CPU), graphics processing unit (GPU), northbridge, southbridge, and memory controller onto one package. Early Tegra SoCs are designed as efficient multimedia processors. The Tegra-line evolved to emphasize performance for gaming and machine learning applications without sacrificing power efficiency, before taking a drastic shift in direction towards platforms that provide vehicular automation with the applied "Drive" brand name on reference boards and its semiconductors; and with the "Jetson" brand name for boards adequate for AI applications within e.g. robots or drones, and for various smart high level automation purposes.


title: Tegra

text: The Tegra APX 2500 was announced on February 12, 2008. The Tegra 6xx product line was revealed on June 2, 2008, and the APX 2600 was announced in February 2009. The APX chips were designed for smartphones, while the Tegra 600 and 650 chips were intended for smartbooks and mobile Internet devices (MID).


title: Tegra

text: The first product to use the Tegra was Microsoft's Zune HD media player in September 2009, followed by the Samsung M1. Microsoft's Kin was the first cellular phone to use the Tegra; however, the phone did not have an app store, so the Tegra's power did not provide much advantage. In September 2008, Nvidia and Opera Software announced that they would produce a version of the Opera 9.5 browser optimized for the Tegra on Windows Mobile and Windows CE. At Mobile World Congress 2009, Nvidia introduced its port of Google's Android to the Tegra.


title: Tegra

text: On January 7, 2010, Nvidia officially announced and demonstrated its next generation Tegra system-on-a-chip, the Nvidia Tegra 250, at Consumer Electronics Show 2010. Nvidia primarily supports Android on Tegra 2, but booting other ARM-supporting operating systems is possible on devices where the bootloader is accessible. Tegra 2 support for the Ubuntu Linux distribution was also announced on the Nvidia developer forum.


title: Tegra

text: Nvidia announced the first quad-core SoC at the February 2011 Mobile World Congress event in Barcelona. Though the chip was codenamed Kal-El, it is now branded as Tegra 3. Early benchmark results show impressive gains over Tegra 2, and the chip was used in many of the tablets released in the second half of 2011.


title: Tegra

text: In January 2012, Nvidia announced that Audi had selected the Tegra 3 processor for its In-Vehicle Infotainment systems and digital instruments display. The processor will be integrated into Audi's entire line of vehicles worldwide, beginning in 2013. The process is ISO 26262-certified.


title: Tegra

text: In summer of 2012 Tesla Motors began shipping the Model S all electric, high performance sedan, which contains two NVIDIA Tegra 3D Visual Computing Modules (VCM). One VCM powers the 17-inch touchscreen infotainment system, and one drives the 12.3-inch all digital instrument cluster."


title: Tegra

text: In March 2015, Nvidia announced the Tegra X1, the first SoC to have a graphics performance of 1 teraflop. At the announcement event, Nvidia showed off Epic Games' Unreal Engine 4 "Elemental" demo, running on a Tegra X1.


title: Tegra

text: On October 20, 2016, Nvidia announced that the Nintendo Switch hybrid video game console will be powered by Tegra hardware. On March 15, 2017, TechInsights revealed the Nintendo Switch is powered by a custom Tegra X1 (model T210), with lower clockspeeds.


title: Tegra

text: The second generation Tegra SoC has a dual-core ARM Cortex-A9 CPU, an ultra low power (ULP) GeForce GPU, a 32-bit memory controller with either LPDDR2-600 or DDR2-667 memory, a 32KB/32KB L1 cache per core and a shared 1MB L2 cache. Tegra 2's Cortex A9 implementation does not include ARM's SIMD extension, NEON. There is a version of the Tegra 2 SoC supporting 3D displays; this SoC uses a higher clocked CPU and GPU.


title: Tegra

text: The Tegra 2 video decoder is largely unchanged from the original Tegra and has limited support for HD formats. The lack of support for high-profile H.264 is particularly troublesome when using online video streaming services.


title: Tegra

text: NVIDIA's Tegra 3 (codenamed "Kal-El") is functionally a SoC with a quad-core ARM Cortex-A9 MPCore CPU, but includes a fifth "companion" core in what Nvidia refers to as a "variable SMP architecture". While all cores are Cortex-A9s, the companion core is manufactured with a low-power silicon process. This core operates transparently to applications and is used to reduce power consumption when processing load is minimal. The main quad-core portion of the CPU powers off in these situations.


title: Tegra

text: The GPU in Tegra 3 is an evolution of the Tegra 2 GPU, with 4 additional pixel shader units and higher clock frequency. It can also output video up to 2560×1600 resolution and supports 1080p MPEG-4 AVC/h.264 40 Mbit/s High-Profile, VC1-AP, and simpler forms of MPEG-4 such as DivX and Xvid.


title: Tegra

text: The Tegra 4 (codenamed "Wayne") was announced on January 6, 2013 and is a SoC with a quad-core CPU, but includes a fifth low-power Cortex A15 companion core which is invisible to the OS and performs background tasks to save power. This power-saving configuration is referred to as "variable SMP architecture" and operates like the similar configuration in Tegra 3.


title: Tegra

text: The GeForce GPU in Tegra 4 is again an evolution of its predecessors. However, numerous feature additions and efficiency improvements were implemented. The number of processing resources was dramatically increased, and clock rate increased as well. In 3D tests, the Tegra 4 GPU is typically several times faster than that of Tegra 3. Additionally, the Tegra 4 video processor has full support for hardware decoding and encoding of WebM video (up to 1080p 60Mbit/s @ 60fps).


title: Tegra

text: Along with Tegra 4, Nvidia also introduced i500, an optional software modem based on Nvidia's acquisition of Icera, which can be reprogrammed to support new network standards. It supports category 3 (100Mbit/s) LTE but will later be updated to Category 4 (150Mbit/s).


title: Tegra

text: The Tegra 4i (codenamed "Grey") was announced on February 19, 2013. With hardware support for the same audio and video formats, but using Cortex-A9 cores instead of Cortex-A15, the Tegra 4i is a low-power variant of the Tegra 4 and is designed for phones and tablets. Unlike its Tegra 4 counterpart, the Tegra 4i also integrates the Icera i500 LTE/HSPA+ baseband processor onto the same die.


title: Tegra

text: Nvidia's Tegra K1 (codenamed "Logan") features ARM Cortex-A15 cores in a 4+1 configuration similar to Tegra 4, or Nvidia's 64-bit Project Denver dual-core processor as well as a Kepler graphics processing unit with support for Direct3D 12, OpenGL ES 3.1, CUDA 6.5, OpenGL 4.4/OpenGL 4.5, and Vulkan. Nvidia claims that it outperforms both the Xbox 360 and the PS3, whilst consuming significantly less power.


title: Tegra

text: In late April 2014, Nvidia shipped the "Jetson TK1" development board containing a Tegra K1 SoC and running Ubuntu Linux.


title: Tegra

text: In December 2015, the web page of wccftech.com published an article stating that Tesla is going to use a Tegra K1 based design derived from the template of the Nvidia Visual Computing Module (VCM) for driving the infotainment systems and providing visual driving aid in the respective vehicle models of that time. This news has, as of now, found no similar successor or other clear confirmation later on in any other place on such a combination of a multimedia with an auto pilot system for these vehicle models.


title: Tegra

text: Released in 2015, Nvidia's Tegra X1 (codenamed "Erista") features two cpu clusters, one with four ARM Cortex-A57 cores and the other with four ARM Cortex-A53 cores, as well as a Maxwell-based graphics processing unit.


title: Tegra

text: It supports Adaptive Scalable Texture Compression. Only one cluster of cores can be active at once, with the cluster switch being handled by software on the BPMP-L. Devices utilizing the Tegra X1 have only been seen to utilize the cluster with the more powerful ARM Cortex-A57 cores. The other cluster with four ARM Cortex-A53 cores cannot be accessed without first powering down the Cortex-A57 cores (both clusers must be in the CC6 off state). Nvidia has removed the ARM Cortex-A53 cores from later versions of technical documentation, implying that they have been removed from the die. The Tegra X1 was found to be vulnerable to a Fault Injection (FI) voltage glitching attack, which allowed for arbitrary code execution and homebrew software on the devices it was implemented in.


title: Tegra

text: A revision (codenamed "Mariko") with greater power efficiency, known officially as Tegra X1+ was released in 2019, fixing the Fusée Gelée exploit. It's also known as T214 and T210B01.


title: Tegra

text: CPU frequency may be clocked differently than the maximum validated by Nvidia at the OEM's discretion


title: Tegra

text: Nvidia's Tegra X2 (codenamed "Parker") features Nvidia's own custom general-purpose ARMv8-compatible core Denver 2 as well as code-named Pascal graphics processing core with GPGPU support. The chips are made using FinFET process technology using TSMC's 16 nm FinFET+ manufacturing process.


title: Tegra

text: The Xavier Tegra SoC, named after the comic book character Professor X, was announced on 28 September 2016, and by March 2019, it had been released. It contains 7 billion transistors and 8 custom ARMv8 cores, a Volta GPU with 512 CUDA cores, an open sourced TPU (Tensor Processing Unit) called DLA (Deep Learning Accelerator). It is able to encode and decode 8K Ultra HD (7680×4320). Users can configure operating modes at 10 W, 15 W, and 30 W TDP as needed and the die size is 350 mm. Nvidia confirmed the fabrication process to be 12 nm FinFET at CES 2018.


title: Tegra

text: On the Linux Kernel Mailing List, a Tegra194 based development board with type ID "P2972-0000" got reported: "The board consists of the P2888 compute module and the P2822 baseboard."


title: Tegra

text: Nvidia announced the next-gen SoC codename Orin on March 27, 2018 at GPU Technology Conference 2018. It contains 17 billion transistors and 12 Arm Hercules cores and is capable of 200 INT8 TOPs @ 65W.


title: Tegra

text: The Drive AGX Orin board system family was announced on December 18, 2019 at GTC China 2019. Nvidia has sent papers to the press documenting that the known (from Xavier series) clock and voltage scaling on the semiconductors and by pairing multiple such chips a wider range of application can be realized with the thus resulting board concepts. The vehicle company NIO got announced by Nvidia for receiving a 4 Orin chip based board design for use in their cars.


title: Tegra

text: Nvidia announced the latest member of the family, "Orin Nano" in September 2022 at the GPU Technology Conference 2022. The Orin product line now features SoC and SoM(System-On-Module) based on the core Orin design and scaled for different uses from 60W all the way down to 5W. While less is known about the exact SoC's that are being manufactured, Nvidia has publicly shared detailed technical specifications about the entire Jetson Orin SoM product line. These module specifications illustrate how Orin scales providing insight into future devices that contain an Orin derived SoC.


title: Tegra

text: Nvidia announced the next-gen SoC codename Atlan on April 12, 2021 at GPU Technology Conference 2021. Nvidia announced the cancellation of Atlan and their next SoC will be Thor.


title: Tegra

text: Nvidia announced the next-gen SoC codename Thor on September 20, 2022 at GPU Technology Conference 2022, replacing the cancelled Atlan.


title: Tegra

text: <nowiki>*</nowiki> VLIW-based Vec4: Pixel shaders + Vertex shaders. Since Kepler, Unified shaders are used.


title: Tegra

text: FreeBSD supports a number of different Tegra models and generations, ranging from Tegra K1, to Tegra 210.


title: Tegra

text: Nvidia distributes proprietary device drivers for Tegra through OEMs and as part of its "Linux for Tegra" (formerly "L4T") development kit. The newer and more powerful devices of the Tegra family are now supported by Nvidia's own Vibrante Linux distribution. Vibrante comes with a larger set of Linux tools plus several Nvidia provided libraries for acceleration in the area of data processing and especially image processing for driving safety and automated driving up to the level of deep learning and neuronal networks that make e.g. heavy use of the CUDA capable accelerator blocks, and via OpenCV can make use of the NEON vector extensions of the ARM cores.


title: Tegra

text: , due to different "business needs" from that of their GeForce line of graphics cards, Nvidia and one of their Embedded Partners, Avionic Design GmbH from Germany, are also working on submitting open-source drivers for Tegra upstream to the mainline Linux kernel.


title: Tegra

text: Nvidia co-founder & CEO laid out the Tegra processor roadmap using Ubuntu Unity in GPU Technology Conference 2013.


title: Tegra

text: By end of 2018 it is evident that Nvidia employees have contributed substantial code parts to make the T186 and T194 models run for HDMI display and audio with the upcoming official Linux kernel 4.21 in about Q1 2019. The affected software modules are the open source Nouveau and the closed source Nvidia graphics drivers along with the Nvidia proprietary CUDA interface.


title: Tegra

text: SoCs and platforms with comparable specifications (e.g. audio/video input, output and processing capability, connectivity, programmability, entertainment/embedded/automotive capabilities & certifications, power consumption) are:


title: Fireball Cinnamon Whisky

text: Fireball Cinnamon Whisky is a mixture of Canadian whisky, cinnamon flavoring and sweeteners that is produced by the Sazerac Company. Its foundation is Canadian whisky, and the taste otherwise resembles the candy with a similar name, Ferrara Candy Company's "Atomic Fireball" candy. It is bottled at 33% alcohol by volume (66 U.S. proof).


title: Fireball Cinnamon Whisky

text: According to the official Fireball website, the product was developed in Canada in the mid-1980s, and for a long time was little known outside of Canada.


title: Fireball Cinnamon Whisky

text: As of 2018, Fireball is among the top selling whisky-related brands in the United States. In addition to the United States and Canada, Fireball is now also available in other countries.


title: Fireball Cinnamon Whisky

text: Fireball was originally part of a line of flavored schnapps developed by Seagram in the mid-1980s. The manufacturer's story line is, in part, that it was the product of a Canadian bartender's efforts to warm up from an Arctic blast. The Sazerac Company purchased the brand rights and formula from Seagram in 1989.


title: Fireball Cinnamon Whisky

text: It was marketed as "Dr. McGillicuddy's Fireball Whisky". Ostensibly, the named doctor was Dr. Aloysius Percival McGillicuddy, allegedly more commonly referred to as "the shot doctor" who was born in the year 1808. Later, in 2007, the product was rebranded as "Fireball Cinnamon Whisky".


title: Fireball Cinnamon Whisky

text: An April 2014 article in "Bloomberg Business Week" said "It's also one of the most successful liquor brands in decades. In 2011, Fireball accounted for a mere $1.9 million in sales in U.S. gas stations, convenience stores, and supermarkets, according to IRI, a Chicago-based market research firm. In 2013, sales leapt to $61 million, passing Jameson Irish whiskey and Patrón tequila." In 2012 and 2013, the product had a surge in popularity, which the company achieved by using social media, cultivating bartenders, word of mouth, and a relatively small advertising budget. It is said that the sharp increase in sales early in its resurgence (late 2011/early 2012) can be attributed to a grass roots effort by Beer Can Alley, a Des Moines, Iowa Country bar. Several national country music acts would perform at the establishment during this time and inspired multiple references in many popular songs In 2013, it became one of the top ten most popular liquors, displacing Jose Cuervo tequila. In 2016, "Bloomberg" reported that with estimated sales of at least $150 million in 2015, the brand had overtaken Jägermeister in popularity to become the top-selling liqueur in the United States.


title: Fireball Cinnamon Whisky

text: The brand's label was designed by Ross Sutherland, who's also designed labels for Black Magic rum, Wheatley Vodka, and some other brands.


title: Fireball Cinnamon Whisky

text: Fireball is usually consumed as a "straight shot" or on the rocks. The Sazerac website says "the cinnamon flavor is often used for shooters but can add character to a mixed drink."


title: Fireball Cinnamon Whisky

text: There are, however, various cocktails that use Fireball as an ingredient, and the company has encouraged a variety of such cocktails through its official website and social media channels. They include "Cinna-Rita", "Fire Nut Ball", "Hot Cherry Fizz", "Red Apple Spice", and "Ring-of-Fire". Another is a blend of Irish coffee called "Fire Starter Coffee". An "Angry Balls" mixer is a combination of Angry Orchard cider and Fireball.


title: Fireball Cinnamon Whisky

text: In 2014, Finland and Sweden reported that Fireball contained amounts of propylene glycol that surpassed the EU limitations of 1g/kg.


title: Fireball Cinnamon Whisky

text: There are a variety of cinnamon-flavored liquor products available and they are a popular form of flavored liquor. They include cinnamon vodka, tequila, and bourbon.


title: Fireball Cinnamon Whisky

text: In early 2012, the Sazerac company sued Hood River Distillers over the allegedly confusing trade name and marketing of its product, "SinFire Cinnamon Whisky". The case was settled in 2013 and SinFire Cinnamon Whisky continues to exist.


title: Fireball Cinnamon Whisky

text: In 2015, the Sazerac company filed a lawsuit complaining that the Jack Daniel's division of Brown-Forman had infringed its Fireball trademark while marketing Tennessee Fire, a cinnamon flavored Tennessee whiskey. The lawsuit was dropped later that year.


title: Lie group

text: In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.


title: Lie group

text: Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group formula_1). Lie groups are widely used in many parts of modern mathematics and physics.


title: Lie group

text: Lie groups were first found by studying matrix subgroups formula_2 contained in formula_3 or formula_4, the groups of formula_5 invertible matrices over formula_6 or formula_7. These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.


title: Lie group

text: According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1), Sophus Lie himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation ("ibid"). Some of Lie's early ideas were developed in close collaboration with Felix Klein. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years ("ibid", p. 2). Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe ("ibid", p. 76). In 1884 a young German mathematician, Friedrich Engel, came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume "Theorie der Transformationsgruppen", published in 1888, 1890, and 1893. The term "groupes de Lie" first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse.


title: Lie group

text: Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany (Hawkins, p. 43). Lie's "idée fixe" was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of "continuous groups", to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.


title: Lie group

text: Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.


title: Lie group

text: Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled "Die Zusammensetzung der stetigen endlichen Transformationsgruppen" ("The composition of continuous finite transformation groups") (Hawkins, p. 100). The work of Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights.


title: Lie group

text: In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.


title: Lie group

text: Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's "infinitesimal groups" (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups. The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.


title: Lie group

text: Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the "global" object, the group, with its "local" or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra.


title: Lie group

text: Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where "G" is a Lie group of "local" symmetries of a manifold.


title: Lie group

text: Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the representations of the Lie group (or of its Lie algebra) are especially important. Representation theory is used extensively in particle physics. Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group.


title: Lie group

text: On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory.


title: Lie group

text: In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; "p"-adic Lie groups play an important role, via their connections with Galois representations in number theory.


title: Lie group

text: A real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication


title: Lie group

text: means that "μ" is a smooth mapping of the product manifold into "G". The two requirements can be combined to the single requirement that the mapping


title: Lie group

text: We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by


title: Lie group

text: with formula_17 a "fixed" irrational number, is a subgroup of the torus formula_18 that is not a Lie group when given the subspace topology. If we take any small neighborhood formula_19 of a point formula_20 in formula_21, for example, the portion of formula_21 in formula_19 is disconnected. The group formula_21 winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of formula_18.


title: Lie group

text: The group formula_21 can, however, be given a different topology, in which the distance between two points formula_27 is defined as the length of the shortest path "in the group " formula_21 joining formula_29 to formula_30. In this topology, formula_21 is identified homeomorphically with the real line by identifying each element with the number formula_32 in the definition of formula_21. With this topology, formula_21 is just the group of real numbers under addition and is therefore a Lie group.


title: Lie group

text: The group formula_21 is an example of a "Lie subgroup" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.


title: Lie group

text: Let formula_36 denote the group of formula_5 invertible matrices with entries in formula_7. Any closed subgroup of formula_36 is a Lie group; Lie groups of this sort are called matrix Lie groups. Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall, Rossmann, and Stillwell.


title: Lie group

text: Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.


title: Lie group

text: A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: formula_59), and holomorphic maps. Similarly, using an alternate metric completion of formula_60, one can define a "p"-adic Lie group over the "p"-adic numbers, a topological group which is also an analytic "p"-adic manifold, such that the group operations are analytic. In particular, each point has a "p"-adic neighborhood.


title: Lie group

text: Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, Montgomery and Zippin showed that if "G" is a topological manifold with continuous group operations, then there exists exactly one analytic structure on "G" which turns it into a Lie group (see also Hilbert–Smith conjecture). If the underlying manifold is allowed to be infinite-dimensional (for example, a Hilbert manifold), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups.


title: Lie group

text: The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups. This categorical point of view leads also to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement.


title: Lie group

text: A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup "G" of the general linear group formula_36 such that


title: Lie group

text: (For example, a closed subgroup of formula_36; that is, a matrix Lie group satisfies the above conditions.)


title: Lie group

text: Then a "Lie group" is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:


title: Lie group

text: The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, "the topology of a Lie group" together with the group law determines the geometry of the group.


title: Lie group

text: Lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.


title: Lie group

text: The only connected Lie groups with dimension one are the real line formula_6 (with the group operation being addition) and the circle group formula_70 of complex numbers with absolute value one (with the group operation being multiplication). The formula_70 group is often denoted as formula_72, the group of formula_73 unitary matrices.


title: Lie group

text: In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are formula_74 (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples".


title: Lie group

text: Some examples of groups that are "not" Lie groups (except in the trivial sense that any group having at most countably many elements can be viewed as a 0-dimensional Lie group, with the discrete topology), are:


title: Lie group

text: To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples:


title: Lie group

text: The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of the representation we use. To get around these problems we give


title: Lie group

text: This Lie algebra formula_93 is finite-dimensional and it has the same dimension as the manifold "G". The Lie algebra of "G" determines "G" up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element.


title: Lie group

text: Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily.


title: Lie group

text: For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.


title: Lie group

text: We could also define a Lie algebra structure on "T" using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on "G" can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space "T".


title: Lie group

text: on "G" × "G" sends ("e", "e") to "e", so its derivative yields a bilinear operation on "TG". This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.


title: Lie group

text: If "G" and "H" are Lie groups, then a Lie group homomorphism "f" : "G" → "H" is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic.


title: Lie group

text: The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let formula_95 be a Lie group homomorphism and let formula_96 be its derivative at the identity. If we identify the Lie algebras of "G" and "H" with their tangent spaces at the identity elements, then formula_96 is a map between the corresponding Lie algebras:


title: Lie group

text: which turns out to be a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.


title: Lie group

text: Two Lie groups are called "isomorphic" if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group formula_2 to a Lie group formula_21 is an isomorphism of Lie groups if and only if it is bijective.


title: Lie group

text: Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras.


title: Lie group

text: The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.


title: Lie group

text: On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the "global" structure of a Lie group is not determined by its Lie algebra; for example, if "Z" is any discrete subgroup of the center of "G" then "G" and "G"/"Z" have the same Lie algebra (see the table of Lie groups for examples). An example of importance in physics are the groups SU(2) and SO(3). These two groups have isomorphic Lie algebras, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.


title: Lie group

text: On the other hand, if we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.


title: Lie group

text: A Lie group formula_2 is said to be simply connected if every loop in formula_2 can be shrunk continuously to a point in formula_2. This notion is important because of the following result that has simple connectedness as a hypothesis:


title: Lie group

text: Lie's third theorem says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a "unique" simply connected Lie group.


title: Lie group

text: An example of a simply connected group is the special unitary group SU(2), which as a manifold is the 3-sphere. The rotation group SO(3), on the other hand, is not simply connected. (See Topology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group SU(n), the spin group (double cover of rotation group) Spin(n) for formula_114, and the compact symplectic group Sp(n).