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text_id stringlengths 22 22 | page_url stringlengths 31 389 | page_title stringlengths 1 250 | section_title stringlengths 0 4.67k | context_page_description stringlengths 0 108k | context_section_description stringlengths 1 187k | media list | hierachy list | category list |
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projected-00310845-010 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Beverage products | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | Anheuser-Busch Companies is responsible for the production, importation and distribution of several AB InBev products, including three company-designated global brands, Budweiser, Stella Artois, and Beck's. Other multi-country brands distributed or produced by Anheuser-Busch Companies include Leffe and Hoegaarden, whil... | [] | [
"Operations and products",
"Beverage products"
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"Anheuser-Busch",
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"German-American culture in St. Louis",
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"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-011 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Advertising | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | Prior to its acquisition by InBev, the company was known in the United States for its advertising presence, including a sports marketing division which created advertising material for the Super Bowl and many other sporting events. Budweiser has sponsored horse racing events and motor sports including NASCAR, the "Miss... | [
"Clydesdale-busch-gardens.jpg",
"Malt-Nutrine, 1898.jpg"
] | [
"Advertising"
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"Anheuser-Busch",
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"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-013 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Spykes underaged alcohol marketing | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | In 2007, the company introduced a flavored 12% abv malt liquor under the name Spykes. It was sold in colorful, 2-ounce bottles. Available flavors included mango, lime, melon and chocolate. It was withdrawn in the same year after criticism from alcohol industry watchdog groups that it was being marketed to underage cust... | [] | [
"Controversies",
"Spykes underaged alcohol marketing"
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"Anheuser-Busch",
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"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-014 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Environmental record | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | In 2002, the Political Economy Research Institute ranked Anheuser–Busch 40th among the "Toxic 100", a list of U.S. corporations most responsible for air pollution. The study found that Anheuser–Busch released 1,002,786 kg (2,213,657 lbs) of toxic pollutants annually into the air. This is mainly because large amounts of... | [] | [
"Controversies",
"Environmental record"
] | [
"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-015 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Budweiser Bill | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | In 2003, after numerous deaths in football stadiums, Brazil passed a law outlawing alcohol sales in stadiums. FIFA demanded that Brazil allow alcohol sales at the 2014 FIFA World Cup because Budweiser, a major World Cup sponsor is the "Official Beer of the FIFA World Cup", a role it has played since 1986.
In response,... | [] | [
"Controversies",
"Budweiser Bill"
] | [
"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-016 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | "Up for Whatever" beer campaign controversy | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | In April 2015, Anheuser-Busch, in an effort to target new, younger consumers to buy its products, the company printed a slogan on Bud Light bottles that said "The perfect beer for removing 'no' from your vocabulary for the night. #UpForWhatever." The label triggered a backlash, and the company was immediately criticize... | [] | [
"Controversies",
"\"Up for Whatever\" beer campaign controversy"
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"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-017 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Liquor law violations in Seattle | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | The Washington State Liquor and Cannabis Board (WSLCB) issued a $150,000 violation to Anheuser-Busch in May 2016. Investigators determined that Anheuser-Busch entered into an illegal agreement of exclusivity with two concert venues in Seattle – the Showbox and the Showbox SoDo. | [] | [
"Controversies",
"Liquor law violations in Seattle"
] | [
"Anheuser-Busch",
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"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-018 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | 2017 Super Bowl advertising | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | In early February 2017, Anheuser-Busch's "Budweiser – Born the Hard Way" Super Bowl commercial was released online, prompting conservative political rebuke over its depiction of the immigration of founder Adolphus Busch from Germany to St. Louis. Although the ad had been conceived eight months before its release, some ... | [] | [
"Controversies",
"2017 Super Bowl advertising"
] | [
"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-019 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | 2019 Super Bowl advertising | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | On March 21, 2019, Anheuser-Busch was sued for false advertising by rival MillerCoors over a Bud Light commercial that aired during Super Bowl LIII. The commercial claimed MillerCoors' Miller Lite and Coors Light products contain corn syrup, but the lawsuit argues that corn syrup is only used during the brewing process... | [] | [
"Controversies",
"2019 Super Bowl advertising"
] | [
"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-020 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Ties with Washington University in St. Louis | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | Adolphus Busch joined Washington University's Board of Directors in 1895, back when the school was still located on its old downtown St. Louis campus. Adolphus would continue to serve on the board until his death in 1913, at which point his son, August Busch Sr. took over his seat. Though Adolphus Busch III and Gussie ... | [] | [
"Ties with Washington University in St. Louis"
] | [
"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-021 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | See also | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | Anheuser-Busch InBev
Anheuser-Busch Brewing Association Building
SeaWorld Entertainment
Manufacturers Railway (St. Louis)
Jacques Chirac – former President of France worked at the St Louis plant in the 1950s during his summer term at Harvard University. | [] | [
"See also"
] | [
"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310845-023 | https://en.wikipedia.org/wiki/Anheuser-Busch | Anheuser-Busch | Further reading | Anheuser-Busch Companies, LLC is an American brewing company headquartered in St. Louis, Missouri. Since 2008, it has been wholly owned by Anheuser-Busch InBev SA/NV (AB InBev), now the world's largest brewing company, which owns multiple global brands, notably Budweiser, Michelob, Stella Artois, and Beck’s.
The comp... | Dethroning the King – The Hostile Takeover of Anheuser-Busch, an American Icon. Julie MacIntosh (John Wiley & Sons Inc, 2011)
Bitter Brew – The Rise and Fall of Anheuser-Busch and America's Kings of Beer. William Knoedelseder (HarperCollins, 2012) | [] | [
"Further reading"
] | [
"Anheuser-Busch",
"Busch family",
"German-American culture in St. Louis",
"Amusement park companies",
"Beer brewing companies based in St. Louis",
"American companies established in 1852",
"American subsidiaries of foreign companies",
"Eclipse Award winners",
"AB InBev",
"1852 establishments in Mi... |
projected-00310847-000 | https://en.wikipedia.org/wiki/Hrunting | Hrunting | Introduction | Hrunting was a sword given to Beowulf by Unferth in the ancient Old English epic poem Beowulf. Beowulf used it in battle against Grendel's mother.
Beowulf is described receiving the sword in lines 1455-1458:
And another item lent by Unferth
at that moment of need was of no small importance:
the brehon handed him a hi... | [] | [
"Introduction"
] | [
"Mythological swords",
"Beowulf"
] | |
projected-00310847-001 | https://en.wikipedia.org/wiki/Hrunting | Hrunting | Hrunting's significance | Hrunting was a sword given to Beowulf by Unferth in the ancient Old English epic poem Beowulf. Beowulf used it in battle against Grendel's mother.
Beowulf is described receiving the sword in lines 1455-1458:
And another item lent by Unferth
at that moment of need was of no small importance:
the brehon handed him a hi... | Swords have great significance in the war-centred Anglo-Saxon culture from which Beowulf arises. Therefore, emphasis is strongly placed on the exchange of weapons of war. Weapons such as swords circulated through Anglo-Saxon society as inheritance through family, birthed through the monsters, found under magic rocks,... | [] | [
"Hrunting's significance"
] | [
"Mythological swords",
"Beowulf"
] |
projected-00310847-002 | https://en.wikipedia.org/wiki/Hrunting | Hrunting | Symbolism of Hrunting | Hrunting was a sword given to Beowulf by Unferth in the ancient Old English epic poem Beowulf. Beowulf used it in battle against Grendel's mother.
Beowulf is described receiving the sword in lines 1455-1458:
And another item lent by Unferth
at that moment of need was of no small importance:
the brehon handed him a hi... | Unferth's very act of giving Hrunting to Beowulf and the sword's unexpected failure in the battle against Grendel's mother bear much symbolism in the poem. Given that Unferth shows his dislike for Beowulf early in the story, Unferth's choice to award Beowulf with Hrunting, which means “hunting,” can be interpreted as a... | [] | [
"Symbolism of Hrunting"
] | [
"Mythological swords",
"Beowulf"
] |
projected-00310847-003 | https://en.wikipedia.org/wiki/Hrunting | Hrunting | Hrunting's failure | Hrunting was a sword given to Beowulf by Unferth in the ancient Old English epic poem Beowulf. Beowulf used it in battle against Grendel's mother.
Beowulf is described receiving the sword in lines 1455-1458:
And another item lent by Unferth
at that moment of need was of no small importance:
the brehon handed him a hi... | The reason behind Hrunting's failing against Grendel's Mother has been a point of much scholarly debate. J.L. Rosier, in A Design for Treachery: The Unferth Intrigue, puts forth the contention that Unferth deliberately gave Beowulf a sword that he knew would fail, possibly for the purpose of preventing Beowulf from suc... | [] | [
"Hrunting's failure"
] | [
"Mythological swords",
"Beowulf"
] |
projected-00310847-004 | https://en.wikipedia.org/wiki/Hrunting | Hrunting | See also | Hrunting was a sword given to Beowulf by Unferth in the ancient Old English epic poem Beowulf. Beowulf used it in battle against Grendel's mother.
Beowulf is described receiving the sword in lines 1455-1458:
And another item lent by Unferth
at that moment of need was of no small importance:
the brehon handed him a hi... | Nægling, another sword used by Beowulf
Ulfberht | [] | [
"See also"
] | [
"Mythological swords",
"Beowulf"
] |
projected-00310847-005 | https://en.wikipedia.org/wiki/Hrunting | Hrunting | References | Hrunting was a sword given to Beowulf by Unferth in the ancient Old English epic poem Beowulf. Beowulf used it in battle against Grendel's mother.
Beowulf is described receiving the sword in lines 1455-1458:
And another item lent by Unferth
at that moment of need was of no small importance:
the brehon handed him a hi... | Category:Mythological swords
Category:Beowulf | [] | [
"References"
] | [
"Mythological swords",
"Beowulf"
] |
projected-00310848-000 | https://en.wikipedia.org/wiki/Beurre%20blanc | Beurre blanc | Introduction | Beurre blanc ("white butter" in French) is a warm emulsified butter sauce made with a reduction of vinegar and/or white wine (normally Muscadet) and shallots into which softened whole butter is whisked in off the heat to prevent separation. The small amount of emulsifiers naturally found in butter are used to form an ... | [] | [
"Introduction"
] | [
"Butter",
"French sauces"
] | |
projected-00310848-001 | https://en.wikipedia.org/wiki/Beurre%20blanc | Beurre blanc | Origin | Beurre blanc ("white butter" in French) is a warm emulsified butter sauce made with a reduction of vinegar and/or white wine (normally Muscadet) and shallots into which softened whole butter is whisked in off the heat to prevent separation. The small amount of emulsifiers naturally found in butter are used to form an ... | The chef Clémence Lefeuvre (née Clémence Prau) invented beurre blanc, apparently by accident, sometime around the beginning of the 20th century. She served this sauce at her restaurant "La Buvette de la Marine" in the hamlet of La Chebuette in the village of Saint-Julien-de-Concelles on the banks of the Loire River a f... | [] | [
"Origin"
] | [
"Butter",
"French sauces"
] |
projected-00310848-002 | https://en.wikipedia.org/wiki/Beurre%20blanc | Beurre blanc | Preparation | Beurre blanc ("white butter" in French) is a warm emulsified butter sauce made with a reduction of vinegar and/or white wine (normally Muscadet) and shallots into which softened whole butter is whisked in off the heat to prevent separation. The small amount of emulsifiers naturally found in butter are used to form an ... | A good beurre blanc is rich and buttery, with a neutral flavor that responds well to other seasonings and flavorings, thereby lending itself to the addition of herbs and spices. It should be light yet still liquid and thick enough to cling to food also known as nappe.
Beurre blanc is prepared by reducing wine, vinegar... | [] | [
"Preparation"
] | [
"Butter",
"French sauces"
] |
projected-00310848-004 | https://en.wikipedia.org/wiki/Beurre%20blanc | Beurre blanc | Beurre rouge | Beurre blanc ("white butter" in French) is a warm emulsified butter sauce made with a reduction of vinegar and/or white wine (normally Muscadet) and shallots into which softened whole butter is whisked in off the heat to prevent separation. The small amount of emulsifiers naturally found in butter are used to form an ... | Beurre rouge (English: "red butter") is a variant of the beurre blanc sauce that is prepared by substituting a dry red wine for the white wine and red wine vinegar for the white wine vinegar. | [] | [
"Derivatives",
"Beurre rouge"
] | [
"Butter",
"French sauces"
] |
projected-00310848-005 | https://en.wikipedia.org/wiki/Beurre%20blanc | Beurre blanc | See also | Beurre blanc ("white butter" in French) is a warm emulsified butter sauce made with a reduction of vinegar and/or white wine (normally Muscadet) and shallots into which softened whole butter is whisked in off the heat to prevent separation. The small amount of emulsifiers naturally found in butter are used to form an ... | Beurre monté
Beurre noir
Beurre noisette | [] | [
"See also"
] | [
"Butter",
"French sauces"
] |
projected-00310848-006 | https://en.wikipedia.org/wiki/Beurre%20blanc | Beurre blanc | References | Beurre blanc ("white butter" in French) is a warm emulsified butter sauce made with a reduction of vinegar and/or white wine (normally Muscadet) and shallots into which softened whole butter is whisked in off the heat to prevent separation. The small amount of emulsifiers naturally found in butter are used to form an ... | Category:Butter
Category:French sauces | [] | [
"References"
] | [
"Butter",
"French sauces"
] |
projected-00310852-000 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Introduction | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | [] | [
"Introduction"
] | [
"1984 births",
"Living people",
"Footballers from Liverpool",
"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... | |
projected-00310852-002 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Liverpool | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | Wavertree born Welsh, joined Liverpool at the age of 10. He progressed through the ranks, becoming captain of the Reserve Team in the process. Welsh trained with the Liverpool first team squad at their Melwood training ground from the middle of the 2001–02 season.
In the 2002–03 season John Welsh played only one game ... | [] | [
"Career",
"Liverpool"
] | [
"1984 births",
"Living people",
"Footballers from Liverpool",
"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310852-003 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Hull City | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | After three starts and seven substitute appearances in all competitions for Liverpool, in August 2005, John went on a season-long loan to Championship side Hull City. The club were impressed, and in November 2005 agreed a deal which saw Liverpool take promising young winger Paul Anderson in exchange. The swap took plac... | [] | [
"Career",
"Hull City"
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"1984 births",
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"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310852-004 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Loans away from Hull | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | On 31 December 2007, Chester City announced they would be signing Welsh on loan when the transfer window reopened the following day. He made his Chester debut on 1 January 2008, in a 2–0 home defeat to Grimsby Town, and spent the rest of the month in Chester's starting line-up before returning to Hull.
He joined Carli... | [] | [
"Career",
"Loans away from Hull"
] | [
"1984 births",
"Living people",
"Footballers from Liverpool",
"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310852-005 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Tranmere Rovers | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | In July 2009, Welsh signed a deal with League One side Tranmere Rovers. He had a very consistent season for Tranmere, playing the holding midfield role successfully. Welsh also scored four goals in his first season at Prenton Park, including a volley in the 4–2 victory over Gillingham and a 25-yard strike in a 2–1 home... | [] | [
"Career",
"Tranmere Rovers"
] | [
"1984 births",
"Living people",
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"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310852-006 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Preston North End | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | In May 2012, it was announced that Welsh had signed for Preston North End on a two-year contract. He was voted Player of the Year and Players' Player of the Year of Preston North End for the season 2012/13. On 17 December 2013, Welsh extended his contract with Preston for a further twelve months, thus keeping him at th... | [] | [
"Career",
"Preston North End"
] | [
"1984 births",
"Living people",
"Footballers from Liverpool",
"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310852-007 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Grimsby Town | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | Welsh signed a two-year contract with League Two club Grimsby Town on 25 June 2018. He was firmly installed as club captain for the 2018–19 season. Welsh left the club by mutual consent on 20 March 2019. This may have been in part due to his participation in a fracas at a bus stop in which he broke a woman's leg. | [] | [
"Career",
"Grimsby Town"
] | [
"1984 births",
"Living people",
"Footballers from Liverpool",
"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310852-008 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Non-League | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | In October 2019 he joined Atherton Collieries on a short-term contract. It was announced in August 2020 that he had signed for Stafford Rangers, on step 10 of the English football pyramid.
On 8 October 2021, Welsh announced he was leaving Stafford and subsequently retired in order to focus on other work commitments. | [] | [
"Career",
"Non-League"
] | [
"1984 births",
"Living people",
"Footballers from Liverpool",
"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310852-010 | https://en.wikipedia.org/wiki/John%20Welsh%20%28English%20footballer%29 | John Welsh (English footballer) | Honours | John Joseph Welsh (born 10 January 1984) is an English former professional footballer who played as midfielder.
Welsh notably played in the Premier League for Liverpool, where he began his career before playing for Hull City, Chester City, Carlisle United, Bury, Tranmere Rovers and Grimsby Town. In 2019 he dropped dow... | Preston North End
Football League One play-offs: 2015 | [] | [
"Honours"
] | [
"1984 births",
"Living people",
"Footballers from Liverpool",
"England youth international footballers",
"England under-21 international footballers",
"Premier League players",
"English Football League players",
"English footballers",
"Association football midfielders",
"Liverpool F.C. players",
... |
projected-00310853-000 | https://en.wikipedia.org/wiki/Cementum | Cementum | Introduction | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | [] | [
"Introduction"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] | |
projected-00310853-001 | https://en.wikipedia.org/wiki/Cementum | Cementum | Structure | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | The cells of cementum are the entrapped cementoblasts, the cementocytes. Each cementocyte lies in its lacuna, similar to the pattern noted in bone. These lacunae also have canaliculi or canals. Unlike those in bone, however, these canals in cementum do not contain nerves, nor do they radiate outward. Instead, the canal... | [
"Human tooth diagram-en.svg"
] | [
"Structure"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-002 | https://en.wikipedia.org/wiki/Cementum | Cementum | Cementoenamel junction | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | The cementum joins the enamel to form the cementoenamel junction (CEJ), which is referred to as the cervical line.
Three possible types of transitional interfaces may be present at the CEJ. The traditional view was that certain interfaces dominated in certain oral cavities. The CEJ may exhibit all of these interfaces ... | [] | [
"Structure",
"Cementoenamel junction"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-003 | https://en.wikipedia.org/wiki/Cementum | Cementum | Dentinocemental junction | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | When the cementoid reaches the full thickness needed, the cementoid surrounding the cementocytes becomes mineralized, or matured, and is then considered cementum. Because of the apposition of cementum over the dentin, the dentinocemental junction (DCJ) is formed. This interface is not as defined, either clinically or h... | [] | [
"Structure",
"Dentinocemental junction"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-004 | https://en.wikipedia.org/wiki/Cementum | Cementum | Types | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | The different categories of cementum are based on the presence or absence of cementocytes, as well as whether the collagen fibres are extrinsic or intrinsic. It is thought that fibroblasts, and some cementoblasts, secrete extrinsic fibres, but only cementoblasts secrete intrinsic fibres. The extrinsic fibres within ace... | [] | [
"Structure",
"Types"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-005 | https://en.wikipedia.org/wiki/Cementum | Cementum | Composition | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | Cementum is slightly softer than dentin and consists of about 45% to 50% inorganic material (hydroxylapatite) by weight and 50% to 55% organic matter and water by weight. The organic portion is composed primarily of collagen and proteoglycans. Cementum is avascular, receiving its nutrition through its own imbedded cell... | [] | [
"Structure",
"Composition"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-006 | https://en.wikipedia.org/wiki/Cementum | Cementum | Development | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | Cementum is secreted by cells called cementoblasts within the root of the tooth and is thickest at the root apex. These cementoblasts develop from undifferentiated mesenchymal cells in the connective tissue of the dental follicle or sac.
Cementoblasts produces cementum in a rhythmic manner on intervals indicating perio... | [] | [
"Structure",
"Development"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-007 | https://en.wikipedia.org/wiki/Cementum | Cementum | Clinical significance | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | Some root resorption of the apical portion of the root may occur, however, if orthodontic pressures are excessive and movement is too fast. Some experts also agree on a third type of cementum, afibrillar cementum, which sometimes extends onto the enamel of the tooth.
The excessive buildup of cementum on the roots of ... | [] | [
"Clinical significance"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-008 | https://en.wikipedia.org/wiki/Cementum | Cementum | DNA studies | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | A 2010 archeological study has found that cementum has five times the amount of mitochondrial DNA compared to dentin, which is commonly sampled. Teeth are increasingly utilized as a source of nuclear DNA to aid identification of human remains. DNA extraction and the results of genetic analysis from the tissue are extre... | [] | [
"Clinical significance",
"DNA studies"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-009 | https://en.wikipedia.org/wiki/Cementum | Cementum | See also | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | Tooth development
Dentin
Periodontal ligament
Periodontium
Cementoma
Hypercementosis | [] | [
"See also"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310853-011 | https://en.wikipedia.org/wiki/Cementum | Cementum | Further reading | Cementum is a specialized calcified substance covering the root of a tooth. The cementum is the part of the periodontium that attaches the teeth to the alveolar bone by anchoring the periodontal ligament. | Category:Teeth
Category:Periodontology
Category:Human mouth anatomy | [] | [
"Further reading"
] | [
"Teeth",
"Periodontology",
"Human mouth anatomy"
] |
projected-00310854-000 | https://en.wikipedia.org/wiki/Follicle | Follicle | Introduction | Follicle may refer to:
Follicle (anatomy), a small spherical group of cells containing a cavity:
Dental follicle
Hair follicle
Lymph follicle
Ovarian follicle
Thyroid follicle
Follicle (fruit) | [] | [
"Introduction"
] | [] | |
projected-00310854-001 | https://en.wikipedia.org/wiki/Follicle | Follicle | See also | Follicle may refer to:
Follicle (anatomy), a small spherical group of cells containing a cavity:
Dental follicle
Hair follicle
Lymph follicle
Ovarian follicle
Thyroid follicle
Follicle (fruit) | Follicular lymphoma, a common form of blood cancer | [] | [
"See also"
] | [] |
projected-00310855-000 | https://en.wikipedia.org/wiki/Stardate | Stardate | Introduction | A stardate is a fictional system of time measurement developed for the television and film series Star Trek. In the series, use of this date system is commonly heard at the beginning of a voice-over log entry, such as "Captain's log, stardate 41153.7. Our destination is planet Deneb IV …". While the original method was... | [] | [
"Introduction"
] | [
"Star Trek terminology",
"Specific calendars"
] | |
projected-00310855-001 | https://en.wikipedia.org/wiki/Stardate | Stardate | Original stardate | A stardate is a fictional system of time measurement developed for the television and film series Star Trek. In the series, use of this date system is commonly heard at the beginning of a voice-over log entry, such as "Captain's log, stardate 41153.7. Our destination is planet Deneb IV …". While the original method was... | The original 1967 Star Trek Guide (April 17, 1967, p. 25) instructed writers for the original Star Trek TV series on how to select stardates for their scripts. Writers could pick any combination of four numbers plus a decimal point, and aim for consistency within a single script, but not necessarily between different s... | [] | [
"Original stardate"
] | [
"Star Trek terminology",
"Specific calendars"
] |
projected-00310855-002 | https://en.wikipedia.org/wiki/Stardate | Stardate | Revised stardate | A stardate is a fictional system of time measurement developed for the television and film series Star Trek. In the series, use of this date system is commonly heard at the beginning of a voice-over log entry, such as "Captain's log, stardate 41153.7. Our destination is planet Deneb IV …". While the original method was... | Subsequent Star Trek series followed a new numerical convention. Star Trek: The Next Generation revised the stardate system in the 1987 Star Trek: The Next Generation Writer's/Director's Guide, to five digits and one decimal place. According to the guide, the first digit "4" should represent the 24th century, with the ... | [] | [
"Revised stardate"
] | [
"Star Trek terminology",
"Specific calendars"
] |
projected-00310855-003 | https://en.wikipedia.org/wiki/Stardate | Stardate | Other stardates | A stardate is a fictional system of time measurement developed for the television and film series Star Trek. In the series, use of this date system is commonly heard at the beginning of a voice-over log entry, such as "Captain's log, stardate 41153.7. Our destination is planet Deneb IV …". While the original method was... | Additional Star Trek media have generated their own numbering systems. The 2009 MMORPG Star Trek Online takes place on stardate 86088.58, in the year 2409. Writer Roberto Orci revised the system for the 2009 film Star Trek so that the first four digits correspond to the year, while the remainder was intended to stand f... | [] | [
"Other stardates"
] | [
"Star Trek terminology",
"Specific calendars"
] |
projected-00310855-004 | https://en.wikipedia.org/wiki/Stardate | Stardate | See also | A stardate is a fictional system of time measurement developed for the television and film series Star Trek. In the series, use of this date system is commonly heard at the beginning of a voice-over log entry, such as "Captain's log, stardate 41153.7. Our destination is planet Deneb IV …". While the original method was... | Timeline of Star Trek
Ordinal date | [] | [
"See also"
] | [
"Star Trek terminology",
"Specific calendars"
] |
projected-00310857-000 | https://en.wikipedia.org/wiki/Nuggets%3A%20Original%20Artyfacts%20from%20the%20First%20Psychedelic%20Era%2C%201965%E2%80%931968 | Nuggets: Original Artyfacts from the First Psychedelic Era, 1965–1968 | Introduction | Nuggets: Original Artyfacts from the First Psychedelic Era is a compilation album of American psychedelic and garage rock singles that were released during the mid-to-late 1960s. It was created by Lenny Kaye, who was a writer and clerk at the Village Oldies record shop in New York. He would later become the lead guitar... | [] | [
"Introduction"
] | [
"Regional music compilation albums",
"Nuggets series albums",
"1972 compilation albums",
"Psychedelic rock compilation albums",
"Acid rock compilation albums",
"Garage rock compilation albums",
"Protopunk compilation albums",
"Elektra Records compilation albums",
"Sire Records compilation albums",
... | |
projected-00310857-001 | https://en.wikipedia.org/wiki/Nuggets%3A%20Original%20Artyfacts%20from%20the%20First%20Psychedelic%20Era%2C%201965%E2%80%931968 | Nuggets: Original Artyfacts from the First Psychedelic Era, 1965–1968 | Legacy | Nuggets: Original Artyfacts from the First Psychedelic Era is a compilation album of American psychedelic and garage rock singles that were released during the mid-to-late 1960s. It was created by Lenny Kaye, who was a writer and clerk at the Village Oldies record shop in New York. He would later become the lead guitar... | Jon Savage, in his history of the UK punk rock scene, England's Dreaming, cites Nuggets as a major influence on punk bands and includes it in his essential punk discography, alongside Iggy and the Stooges' Raw Power and The Velvet Underground's White Light/White Heat.
Many other compilation albums took their cue from ... | [
"Nuggets, Volume 2.jpg",
"Nuggets, Volume 3.jpg",
"Nuggets, Volume 4.jpg"
] | [
"Legacy"
] | [
"Regional music compilation albums",
"Nuggets series albums",
"1972 compilation albums",
"Psychedelic rock compilation albums",
"Acid rock compilation albums",
"Garage rock compilation albums",
"Protopunk compilation albums",
"Elektra Records compilation albums",
"Sire Records compilation albums",
... |
projected-00310861-000 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | Introduction | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | [] | [
"Introduction"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... | |
projected-00310861-001 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | Early life | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | Paul Hill was born in Miami, Florida, on February 6, 1954, to Oscar Jennings Hill, an airline pilot, and his wife Louise. He was raised in Coral Gables, Florida. At the age of 17, Hill was charged with the assault of his father when his parents attempted to get him treatment for his drug problem. Hill said he experienc... | [] | [
"Early life"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... |
projected-00310861-002 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | Early career | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | Hill graduated from Reformed Theological Seminary, where he studied under Greg Bahnsen, a founder of the right-wing Christian Reconstructionist movement. He attended St. Paul Presbyterian Church, which espoused theonomy, a movement related to Reconstructionism. Following his ordination in 1984, Hill became a minister a... | [] | [
"Early career"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... |
projected-00310861-003 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | Crime, trial, and execution | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | On July 29, 1994, Hill approached the Ladies Center, an abortion clinic in Pensacola, Florida. When he spotted clinic doctor John Britton and his bodyguard, retired USAF Lieutenant Colonel James H. Barrett, outside the clinic, he fired on both of them at close range with a Mossberg Model 500A 12-gauge pump-action shotg... | [] | [
"Crime, trial, and execution"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... |
projected-00310861-004 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | Motives and aftermath | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | Prior to the murders, Hill sent two position papers to Reconstructionist author Gary North, which set out Hill's views of abortion and why he considered murder of abortionists to be warranted. The papers were followed by three additional letters to North in October 1994. North's responses, issued after the murders, com... | [] | [
"Motives and aftermath"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... |
projected-00310861-005 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | In media | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | While in police custody, Hill told the media "Now is the time to defend the unborn as to defend a slave that's about to be murdered." Hill's purported ties to the Army of God movement as well as his life and crimes are explored in the feature-length HBO Documentary film Soldiers in the Army of God (2000), directed by M... | [] | [
"In media"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... |
projected-00310861-006 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | See also | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | Capital punishment in Florida
Capital punishment in the United States
List of people executed in Florida
List of people executed in the United States in 2003
Donald Spitz
David Trosch
Eric Rudolph
John Britton (doctor)
Shelley Shannon
Anti-abortion violence | [] | [
"See also"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... |
projected-00310861-007 | https://en.wikipedia.org/wiki/Paul%20Jennings%20Hill | Paul Jennings Hill | References | Paul Jennings Hill (February 6, 1954 – September 3, 2003) was an American minister and anti-abortion terrorist who murdered physician John Britton and Britton's bodyguard, retired Air Force Lieutenant Colonel James Barrett, in 1994. Hill was sentenced to death by lethal injection and was executed on September 3, 2003. | Category:1954 births
Category:2003 deaths
Category:21st-century executions by Florida
Category:21st-century executions of American people
Category:1994 murders in the United States
Category:American assassins
Category:American people convicted of murder
Category:American anti-abortion activists
Category:Army of God (Un... | [] | [
"References"
] | [
"1954 births",
"2003 deaths",
"21st-century executions by Florida",
"21st-century executions of American people",
"1994 murders in the United States",
"American assassins",
"American people convicted of murder",
"American anti-abortion activists",
"Army of God (United States)",
"Belhaven Universit... |
projected-00310862-000 | https://en.wikipedia.org/wiki/Paul%20Hill | Paul Hill | Introduction | Paul Hill is the name of:
Paul Hill (musician) (1934–1999), American music director, founder of the Master Chorale of Washington (formerly the Paul Hill Chorale)
Paul Jennings Hill (1954–2003), the murderer of abortion provider John Britton
Paul Hill (Guildford Four) (born 1954), Irish man convicted and later clear... | [] | [
"Introduction"
] | [] | |
projected-00310864-000 | https://en.wikipedia.org/wiki/Intertwingularity | Intertwingularity | Introduction | Intertwingularity is a term coined by Ted Nelson to express the complexity of interrelations in human knowledge.
Nelson wrote in Computer Lib/Dream Machines : "EVERYTHING IS DEEPLY INTERTWINGLED. In an important sense there are no "subjects" at all; there is only all knowledge, since the cross-connections among the my... | [] | [
"Introduction"
] | [
"Ted Nelson",
"Knowledge"
] | |
projected-00310864-001 | https://en.wikipedia.org/wiki/Intertwingularity | Intertwingularity | Influence | Intertwingularity is a term coined by Ted Nelson to express the complexity of interrelations in human knowledge.
Nelson wrote in Computer Lib/Dream Machines : "EVERYTHING IS DEEPLY INTERTWINGLED. In an important sense there are no "subjects" at all; there is only all knowledge, since the cross-connections among the my... | Peter Morville, an influential figure in information architecture, discusses intertwingularity in some of his books. In Ambient Findability: What We Find Changes Who We Become (2005), Morville uses the concept of intertwingularity to describe the experience of using hypertext on the web and starting to use computers em... | [] | [
"Influence"
] | [
"Ted Nelson",
"Knowledge"
] |
projected-00310864-002 | https://en.wikipedia.org/wiki/Intertwingularity | Intertwingularity | See also | Intertwingularity is a term coined by Ted Nelson to express the complexity of interrelations in human knowledge.
Nelson wrote in Computer Lib/Dream Machines : "EVERYTHING IS DEEPLY INTERTWINGLED. In an important sense there are no "subjects" at all; there is only all knowledge, since the cross-connections among the my... | Connectedness
Directed graph
Multicategory
Multiclass classification, Multicriteria classification, Multi-label classification
Multigraph
Multiple inheritance
Polysemy | [] | [
"See also"
] | [
"Ted Nelson",
"Knowledge"
] |
projected-00310874-000 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Introduction | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | [
"NPS FortKentRRStation.jpg"
] | [
"Introduction"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
"Towns in Aroostook County, Maine",
"Towns in Maine"
] | |
projected-00310874-001 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | History | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | Fort Kent was erected in the summer of 1839 as an American border outpost during the undeclared Aroostook War. The blockhouse, the first structure built in what is present-day city of Fort Kent, was named after then-governor of Maine Edward Kent. The Saint John River was a log driving route from upstream forests to do... | [] | [
"History"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
"Towns in Aroostook County, Maine",
"Towns in Maine"
] |
projected-00310874-002 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | 2008 flooding | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | After receiving nearly of snow during the 2007–2008 winter season, the Saint John River began to flood on April 29. Nearly two days of nonstop rain also contributed to the flooding. As the river rose, it poured into the downtown area. More than 600 of Fort Kent's 4,233 residents were evacuated. The river's water leve... | [] | [
"History",
"2008 flooding"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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projected-00310874-003 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Relationship with French Canada | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | The town is economically and culturally linked to the Canadian towns of Clair and Saint-François-de-Madawaska, both in Madawaska County, New Brunswick, directly across the Saint John River. Most children on the Canadian side attend French-speaking school while their American counterparts are taught mainly in English.
... | [] | [
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"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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"Towns in Maine"
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projected-00310874-004 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Geography | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | According to the United States Census Bureau, the town has a total area of , of which is land and is water. | [] | [
"Geography"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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"Towns in Maine"
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projected-00310874-005 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Climate | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | Fort Kent has a humid continental climate (Koppen: Dfb). There are four distinct seasons, with winter being the longest, typically beginning during November or late October and lasting well into April many years. Summers are very pleasant, with warm days and cool nights. Fort Kent receives a large amount of snowfall ev... | [] | [
"Geography",
"Climate"
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"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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"Towns in Maine"
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projected-00310874-007 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | 2010 census | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | As of the census of 2010, there were 4,097 people, 1,747 households, and 1,062 families living in the town. The population density was . There were 1,922 housing units at an average density of . The racial makeup of the town was 94.9% White, 0.8% African American, 0.8% Native American, 0.7% Asian, 0.1% from other races... | [] | [
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"Fort Kent, Maine",
"French-Canadian culture in Maine",
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projected-00310874-008 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | 2000 census | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | As of the census of 2000, there were 4,233 people, 1,735 households, and 1,106 families living in the town. The population density was 78.2 people per square mile (30.2/km2). There were 1,824 housing units at an average density of 33.7 per square mile (13.0/km2). The racial makeup of the town was 96.95% White, 0.38%... | [] | [
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"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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"Towns in Maine"
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projected-00310874-010 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | International Muskie Derby | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | Fort Kent is home to the International Muskie Fishing Derby. The Derby was started in 2003 to provide a platform showcase the new fishery and continues today. It has brought attention to the new fishery and helped bring attention to national TV shows such as ESPN 2's Beat Charlie Moore. This show and other future shows... | [] | [
"Arts and culture",
"International Muskie Derby"
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"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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"Towns in Maine"
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projected-00310874-011 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Ploye Festival | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | Along with the Muskie Derby is the Fort Kent Ploye Festival, one of the featured events is the making of the world's largest ploye. This event attracts hundreds of spectators, making it the highlight of the Ploye Festival. The massive ploye, which thus far, measures 12 feet in diameter, can only be accomplished with th... | [] | [
"Arts and culture",
"Ploye Festival"
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"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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"Towns in Maine"
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projected-00310874-012 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | World Cup IBU | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | The Fort Kent Outdoor Center (previously known as 10th Mountain) has hosted a multitude of World Cup biathlon events, including the largest events in 2004 and 2011, where racers from around the globe took part in various biathlon events in Fort Kent. During the 2011 World Cup Biathlon, Chicago Bulls superstar Scotty Pi... | [] | [
"Arts and culture",
"World Cup IBU"
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"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
"Towns in Aroostook County, Maine",
"Towns in Maine"
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projected-00310874-013 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | World Acadian Congress | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | Northern Maine and parts of New Brunswick and Quebec participated in the World Acadian Congress to celebrate the rich Acadians history of the area in 2014. Multiple family reunions and events took place during the time, including concerts, parades, and festivals. | [] | [
"Arts and culture",
"World Acadian Congress"
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"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
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"Towns in Maine"
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projected-00310874-014 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Can Am Crown International Sled Dog Race | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | The Can-Am Crown International Sled Dog Race was first held on Tuesday, February 16, 1993. Nine teams from Maine, Ontario, Quebec, and New Brunswick competed in that race. In 1994 sixteen teams competed in the 250-mile race and eleven in the newly introduced 60-mile race. In 1997 a 30-mile race was added to the itinera... | [] | [
"Arts and culture",
"Can Am Crown International Sled Dog Race"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
"Towns in Aroostook County, Maine",
"Towns in Maine"
] |
projected-00310874-015 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Education | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | Fort Kent is served by Maine School Administrative District 27. Schools in the town include Fort Kent Elementary School, Valley Rivers Middle School and Fort Kent Community High School.
The town is home to the campus of the University of Maine at Fort Kent. | [] | [
"Education"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
"Towns in Aroostook County, Maine",
"Towns in Maine"
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projected-00310874-017 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Transportation | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | Fort Kent signed a lease agreement with the Fish River Flying Club on July 8, 2011 to repair, maintain, and operate the local municipal airport, which had been closed since the 1980s. | [] | [
"Infrastructure",
"Transportation"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
"Towns in Aroostook County, Maine",
"Towns in Maine"
] |
projected-00310874-018 | https://en.wikipedia.org/wiki/Fort%20Kent%2C%20Maine | Fort Kent, Maine | Notable people | Fort Kent is a town in Aroostook County, Maine, United States, situated at the confluence of the Fish River and the Saint John River, on the border with New Brunswick, Canada. The population was 4,067 in the 2020 census. Fort Kent is home to an Olympic biathlete training center, an annual CAN-AM dogsled race, and the F... | People born in Fort Kent:
Benjamin Collings (born 1976), politician
Mike Daisey (born 1976), monologist
Clair Goodblood (1929-1951), recipient of the Medal of Honor
Helen Hamlin (1917-2004), author
Troy Dale Jackson (born 1968), politician
Ellis Paul (born 1965), singer-songwriter
Austin Theriault (born 1994), ... | [] | [
"Notable people"
] | [
"Fort Kent, Maine",
"French-Canadian culture in Maine",
"Populated places on the Saint John River (Bay of Fundy)",
"Towns in Aroostook County, Maine",
"Towns in Maine"
] |
projected-00310883-000 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Introduction | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | [] | [
"Introduction"
] | [
"Lattice theory"
] | |
projected-00310883-001 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Definition | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient.... | [] | [
"Definition"
] | [
"Lattice theory"
] |
projected-00310883-002 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Morphisms | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | A morphism of distributive lattices is just a lattice homomorphism as given in the article on lattices, i.e. a function that is compatible with the two lattice operations. Because such a morphism of lattices preserves the lattice structure, it will consequently also preserve the distributivity (and thus be a morphism o... | [] | [
"Morphisms"
] | [
"Lattice theory"
] |
projected-00310883-003 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Examples | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | Distributive lattices are ubiquitous but also rather specific structures. As already mentioned the main example for distributive lattices are lattices of sets, where join and meet are given by the usual set-theoretic operations. Further examples include:
The Lindenbaum algebra of most logics that support conjunction ... | [
"Young's lattice.svg"
] | [
"Examples"
] | [
"Lattice theory"
] |
projected-00310883-004 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Characteristic properties | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | Various equivalent formulations to the above definition exist. For example, L is distributive if and only if the following holds for all elements x, y, z in L:
(xy)(yz)(zx) = (xy)(yz)(zx).
Similarly, L is distributive if and only if
xz = yz and xz = yz always imply x=y.
The simplest non-distributive lattices are ... | [
"Non-dstrbtive lattices-warning.png"
] | [
"Characteristic properties"
] | [
"Lattice theory"
] |
projected-00310883-005 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Representation theory | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive if and only if it is isomorphic to a lattice of sets (closed under set union and intersection). (The latter structure is sometimes called a ring of sets in this context.) That set union and inters... | [] | [
"Representation theory"
] | [
"Lattice theory"
] |
projected-00310883-006 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Free distributive lattices | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | The free distributive lattice over a set of generators G can be constructed much more easily than a general free lattice. The first observation is that, using the laws of distributivity, every term formed by the binary operations and on a set of generators can be transformed into the following equivalent normal form:... | [
"Monotone Boolean functions.svg"
] | [
"Free distributive lattices"
] | [
"Lattice theory"
] |
projected-00310883-007 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | See also | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | Completely distributive lattice — a lattice in which infinite joins distribute over infinite meets
Duality theory for distributive lattices
Spectral space | [] | [
"See also"
] | [
"Lattice theory"
] |
projected-00310883-009 | https://en.wikipedia.org/wiki/Distributive%20lattice | Distributive lattice | Further reading | In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery ... | Category:Lattice theory | [] | [
"Further reading"
] | [
"Lattice theory"
] |
projected-00310885-000 | https://en.wikipedia.org/wiki/Dual%20%28category%20theory%29 | Dual (category theory) | Introduction | In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... | [] | [
"Introduction"
] | [
"Category theory",
"Duality theories"
] | |
projected-00310885-001 | https://en.wikipedia.org/wiki/Dual%20%28category%20theory%29 | Dual (category theory) | Formal definition | In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... | We define the elementary language of category theory as the two-sorted first order language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms.
Let σ be any statement in this language. We form the dual... | [] | [
"Formal definition"
] | [
"Category theory",
"Duality theories"
] |
projected-00310885-002 | https://en.wikipedia.org/wiki/Dual%20%28category%20theory%29 | Dual (category theory) | Examples | In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... | A morphism is a monomorphism if implies . Performing the dual operation, we get the statement that implies For a morphism , this is precisely what it means for f to be an epimorphism. In short, the property of being a monomorphism is dual to the property of being an epimorphism.
Applying duality, this means that a... | [] | [
"Examples"
] | [
"Category theory",
"Duality theories"
] |
projected-00310885-003 | https://en.wikipedia.org/wiki/Dual%20%28category%20theory%29 | Dual (category theory) | See also | In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... | Dual object
Duality (mathematics)
Opposite category
Adjoint functor | [] | [
"See also"
] | [
"Category theory",
"Duality theories"
] |
projected-00310885-004 | https://en.wikipedia.org/wiki/Dual%20%28category%20theory%29 | Dual (category theory) | References | In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... | Category:Category theory
Category theory | [] | [
"References"
] | [
"Category theory",
"Duality theories"
] |
projected-00310886-000 | https://en.wikipedia.org/wiki/Coalgebra | Coalgebra | Introduction | In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.
Every c... | [] | [
"Introduction"
] | [
"Coalgebras"
] | |
projected-00310886-001 | https://en.wikipedia.org/wiki/Coalgebra | Coalgebra | Informal discussion | In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.
Every c... | One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angular momentum and spin. For this purpose, one uses the Cleb... | [] | [
"Informal discussion"
] | [
"Coalgebras"
] |
projected-00310886-002 | https://en.wikipedia.org/wiki/Coalgebra | Coalgebra | Formal definition | In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.
Every c... | Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that
.
(Here ⊗ refers to the tensor product over K and id is the identity function.)
Equivalently, the following two diagrams commute:
In the first diagram, C ⊗ (C ⊗ C) is identified with (C... | [
"Defining diagrams of coalgebra.png"
] | [
"Formal definition"
] | [
"Coalgebras"
] |
projected-00310886-003 | https://en.wikipedia.org/wiki/Coalgebra | Coalgebra | Examples | In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.
Every c... | Take an arbitrary set S and form the K-vector space C = K(S) with basis S, as follows. The elements of this vector space C are those functions from S to K that map all but finitely many elements of S to zero; identify the element s of S with the function that maps s to 1 and all other elements of S to 0. Define
Δ(s) =... | [] | [
"Examples"
] | [
"Coalgebras"
] |
projected-00310886-004 | https://en.wikipedia.org/wiki/Coalgebra | Coalgebra | Finite dimensions | In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.
Every c... | In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finite-dimensional (unital associative) algebra is a coalgebra, while the dual of a finite-dimensional coalgebra is a (unital associative) algebra. In general, the dual of an algebra may not be a coalgebra.
The key point is that... | [] | [
"Finite dimensions"
] | [
"Coalgebras"
] |