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https://en.wikipedia.org/wiki/Kosuke%20Ota%20%28footballer%2C%20born%201982%29 | is a Japanese former football player who plays as defender and midfielder.
Club statistics
Updated to the end of 2020 season.
References
External links
Profile at Zweigen Kanazawa
1982 births
Living people
Chuo University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Saitama SC players
Tokyo Musashino United FC players
FC Machida Zelvia players
Zweigen Kanazawa players
FC Imabari players
ReinMeer Aomori players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yoshinori%20Katsumata | Yoshinori Katsumata (勝又 慶典, born December 7, 1985) is a Japanese football player for Ococias Kyoto AC.
Club statistics
Updated to 23 February 2020.
1Includes J2/J3 Playoffs and 2019 Japanese Regional Promotion Series.
References
External links
Profile at Nagano Parceiro
1985 births
Living people
Toin University of Yokohama alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
FC Machida Zelvia players
Tochigi SC players
AC Nagano Parceiro players
Ococias Kyoto AC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Shohei%20Yanagizaki | is a Japanese football player for Kagoshima United FC.
Club statistics
Updated to 23 February 2016.
References
External links
Profile at Kagoshima United FC
1984 births
Living people
Komazawa University alumni
Association football people from Kagoshima Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
FC Machida Zelvia players
Kagoshima United FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yoshihiro%20Shoji | is a Japanese football player currently playing for FC Gifu.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at FC Gifu
Profile at Renofa Yamaguchi FC
1989 births
Living people
Senshu University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
FC Machida Zelvia players
Renofa Yamaguchi FC players
FC Gifu players
Vegalta Sendai players
Kyoto Sanga FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Takafumi%20Suzuki | Takafumi Suzuki (鈴木 崇文, born November 17, 1987) is a Japanese football player for Thespakusatsu Gunma.
Club statistics
Updated to 23 February 2019.
References
External links
Profile at Thespakusatsu Gunma
1987 births
Living people
Tokyo Gakugei University alumni
Association football people from Ibaraki Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
FC Machida Zelvia players
Fagiano Okayama players
Thespakusatsu Gunma players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kai%20Miki | Kai Miki (三鬼 海, born 19 April 1993) is a Japanese football player for FC Machida Zelvia.
Career statistics
Club
Updated to end of 2018 season.
1Includes Emperor's Cup.
2Includes J2/J3 Playoffs.
References
External links
Profile at Montedio Yamagata
Profile at Roasso Kumamoto
1993 births
Living people
Association football people from Mie Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
FC Machida Zelvia players
V-Varen Nagasaki players
Roasso Kumamoto players
Montedio Yamagata players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yuki%20Kitai | was a Japanese former football player belonged to SC Sagamihara. Now,he is a professional track cyclist.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Kataller Toyama
1990 births
Living people
Kindai University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
FC Machida Zelvia players
Matsumoto Yamaga FC players
Kataller Toyama players
SC Sagamihara players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Koji%20Suzuki%20%28footballer%29 | Koji Suzuki (鈴木 孝司, born 25 July 1989) is a Japanese professional footballer who plays as a forward for J1 League club Albirex Niigata.
Club statistics
Updated to 7 March 2019.
References
External links
Profile at Machida Zelvia
1989 births
Living people
Hosei University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
FC Machida Zelvia players
FC Ryukyu players
Cerezo Osaka players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yuki%20Nogami | is a Japanese professional footballer who plays as a centre back for club Nagoya Grampus.
Club statistics
Updated to 5 November 2022.
Honours
Club
Sanfrecce Hiroshima
J.League Cup: 2022
References
External links
Profile at Sanfrecce Hiroshima
1991 births
Living people
Toin University of Yokohama alumni
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Yokohama FC players
Sanfrecce Hiroshima players
Nagoya Grampus players
Men's association football defenders |
https://en.wikipedia.org/wiki/Raphael%20Macena | Raphael dos Santos Macena (born 25 February 1989) is a Brazilian professional footballer who plays as a forward for Mirassol.
Club statistics
References
External links
1989 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Greece
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in Greece
J2 League players
Campeonato Brasileiro Série B players
Campeonato Brasileiro Série C players
Esporte Clube Bahia players
Votoraty Futebol Clube players
União São João Esporte Clube players
Paulista Futebol Clube players
Shonan Bellmare players
Ceará Sporting Club players
Guarani FC players
Comercial Futebol Clube (Ribeirão Preto) players
Esporte Clube XV de Novembro (Piracicaba) players
Esporte Clube Juventude players
Rio Claro Futebol Clube players
Athens Kallithea F.C. players
Luverdense Esporte Clube players
União Recreativa dos Trabalhadores players
Veranópolis Esporte Clube Recreativo e Cultural players
Men's association football forwards |
https://en.wikipedia.org/wiki/Shuhei%20Otsuki | Shuhei Otsuki (大槻 周平, born May 26, 1989) is a Japanese football player for Renofa Yamaguchi FC.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Vissel Kobe
1989 births
Living people
Osaka Gakuin University alumni
Association football people from Kyoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
Shonan Bellmare players
Vissel Kobe players
Montedio Yamagata players
JEF United Chiba players
Renofa Yamaguchi FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Ryohei%20Yoshihama | is a Japanese football player.who plays as a Midfielder for Cambodian Premier League club, Boeung Ket
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Machida Zelvia
1992 births
Living people
Shoin University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Shonan Bellmare players
Fukushima United FC players
Thespakusatsu Gunma players
FC Machida Zelvia players
Renofa Yamaguchi FC players
FC Gifu players
Men's association football midfielders
Expatriate men's footballers in Cambodia
Japanese expatriate sportspeople in Cambodia |
https://en.wikipedia.org/wiki/Alex%20Martins%20%28footballer%2C%20born%201993%29 | Alex Martins Ferreira (born 8 July 1993) is a Brazilian professional footballer who plays as a forward for Liga 1 club Dewa United.
Club statistics
Updated to 28 October 2023.
References
External links
Profile at Fukushima United FC
Profile at Kagoshima United FC
1993 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J2 League players
J3 League players
China League One players
K League 2 players
Liga 1 (Indonesia) players
Shonan Bellmare players
Brasília Futebol Clube players
Esporte Clube Rio Verde players
Fukushima United FC players
Kagoshima United FC players
Tochigi SC players
Shanghai Shenxin F.C. players
Jeonnam Dragons players
Shanghai Jiading Huilong F.C. players
Bhayangkara Presisi Indonesia F.C. players
Dewa United F.C. players
Expatriate men's footballers in Japan
Expatriate men's footballers in China
Expatriate men's footballers in South Korea
Expatriate men's footballers in Indonesia
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in China
Brazilian expatriate sportspeople in South Korea
Brazilian expatriate sportspeople in Indonesia
Men's association football forwards |
https://en.wikipedia.org/wiki/Ryuji%20Ito%20%28footballer%29 | is a Japanese football player who plays for Tochigi SC in J2 League.
Club statistics
Updated to 23 February 2017.
References
External links
Profile at Fujieda MYFC
J. League (#2)
1990 births
Living people
Association football people from Tokyo
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Yokohama FC players
FC Ryukyu players
Matsumoto Yamaga FC players
Fujieda MYFC players
Vonds Ichihara players
Tochigi SC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Tetsuya%20Kijima | Tetsuya Kijima (, born August 20, 1983) is a Japanese football player for Kamatamare Sanuki.
Career
His elder brother Ryosuke is also a professional footballer.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Kamatamare Sanuki
1983 births
Living people
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Sagawa Shiga FC players
Blaublitz Akita players
FC Gifu players
Reilac Shiga FC players
Matsumoto Yamaga FC players
FC Machida Zelvia players
Kamatamare Sanuki players
Men's association football forwards
Association football people from Chiba (city) |
https://en.wikipedia.org/wiki/Atsuto%20Tatara | Atsuto Tatara (多々良 敦斗, born June 23, 1987) is a Japanese football player who currently plays for FC Maruyasu Okazaki.
Career
On 17 January 2019, Tatara joined FC Maruyasu Okazaki.
Club statistics
Updated to 23 February 2019.
References
External links
Profile at Roasso Kumamoto
1987 births
Living people
Shizuoka Sangyo University alumni
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Matsumoto Yamaga FC players
Vegalta Sendai players
JEF United Chiba players
Roasso Kumamoto players
FC Maruyasu Okazaki players
Men's association football defenders
Association football people from Shizuoka (city) |
https://en.wikipedia.org/wiki/Hiroki%20Higuchi | Hiroki Higuchi (樋口 寛規, born April 16, 1992) is a Japanese football player who plays as a forward for Fukushima United FC.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Fukushima United FC
1992 births
Living people
Association football people from Hyōgo Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Shimizu S-Pulse players
FC Gifu players
Shonan Bellmare players
SC Sagamihara players
Fukushima United FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Kim%20Dong-gwon | Kim Dong-Gwon () is a South Korean football player for Gimhae FC.
Club statistics
External links
1992 births
Living people
Men's association football defenders
South Korean men's footballers
South Korean expatriate men's footballers
K League 1 players
K League 2 players
Korea National League players
J2 League players
Pohang Steelers players
FC Gifu players
FC Osaka players
Chungju Hummel FC players
Ulsan Hyundai Mipo Dockyard FC players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan |
https://en.wikipedia.org/wiki/Hiroshi%20Sekita | Hiroshi Sekita (関田 寛士, born October 2, 1989) is a Japanese football former player.
Club statistics
Updated to 23 February 2017.
References
External links
Profile at Nagano Parceiro
1989 births
Living people
Toin University of Yokohama alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J2 League players
J3 League players
FC Gifu players
AC Nagano Parceiro players
Men's association football defenders
Sportspeople from Sagamihara |
https://en.wikipedia.org/wiki/Kohei%20Nakashima | is a Japanese football player. He plays for Verspah Oita.
Club statistics
References
External links
J. League (#27)
1989 births
Living people
Fukuyama University alumni
Association football people from Shimane Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
FC Gifu players
FC Machida Zelvia players
Vonds Ichihara players
Verspah Oita players
Men's association football forwards |
https://en.wikipedia.org/wiki/Ryuji%20Hirota | Ryuji Hirota (廣田 隆治, born 16 July 1993) is a Japanese professional footballer who plays as a winger for Thai League 2 club Chainat Hornbill.
Career statistics
Club
Updated to 23 February 2020.
1Includes JFL Relegation Playoffs.
References
External links
Profile at Gainare Tottori
Profile at Renofa Yamaguchi
1993 births
Living people
Association football people from Hyōgo Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
FC Gifu players
Gainare Tottori players
Renofa Yamaguchi FC players
Iwate Grulla Morioka players
Veertien Mie players
Ryuji Hirota
Men's association football midfielders |
https://en.wikipedia.org/wiki/Daiki%20Oizumi | is a former Japanese football player.
Club statistics
References
External links
J. League (#30)
1989 births
Living people
Osaka Gakuin University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
FC Gifu players
SP Kyoto FC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kim%20Jung-hyun%20%28footballer%2C%20born%201990%29 | Kim Jung-hyun (김정현; born January 3, 1990) is a South Korean football player.
Club statistics
References
External links
1990 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korean expatriate men's footballers
J2 League players
FC Gifu players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan |
https://en.wikipedia.org/wiki/Hwang%20Te-song | Hwang Te-Song (born December 20, 1989) is a South Korean football player.
Club statistics
References
External links
1989 births
Living people
Keio University alumni
Association football people from Gunma Prefecture
South Korean men's footballers
South Korean expatriate men's footballers
J2 League players
Kyoto Sanga FC players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Keisuke%20Kumazawa | is a Japanese football player. He plays for Maruyasu Okazaki.
Club statistics
References
External links
J. League (#24)
1989 births
Living people
Chukyo University alumni
Association football people from Aichi Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Gainare Tottori players
FC Maruyasu Okazaki players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Osamu%20Miura | is a Japanese football player. He plays for Arterivo Wakayama.
Club statistics
Updated to 18 November 2018.
References
External links
Profile at Nara Club
J. League (#27)
1989 births
Living people
Doshisha University alumni
Association football people from Hokkaido
Japanese men's footballers
J2 League players
Japan Football League players
Gainare Tottori players
Nara Club players
Arterivo Wakayama players
Men's association football defenders |
https://en.wikipedia.org/wiki/Partially%20ordered%20space | In mathematics, a partially ordered space (or pospace) is a topological space equipped with a closed partial order , i.e. a partial order whose graph is a closed subset of .
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
Equivalences
For a topological space equipped with a partial order , the following are equivalent:
is a partially ordered space.
For all with , there are open sets with and for all .
For all with , there are disjoint neighbourhoods of and of such that is an upper set and is a lower set.
The order topology is a special case of this definition, since a total order is also a partial order.
Properties
Every pospace is a Hausdorff space. If we take equality as the partial order, this definition becomes the definition of a Hausdorff space.
Since the graph is closed, if and are nets converging to x and y, respectively, such that for all , then .
See also
References
External links
ordered space on Planetmath
Topological spaces |
https://en.wikipedia.org/wiki/Elongated%20pyramid | In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an pyramid to an prism. Along with the set of pyramids, these figures are topologically self-dual.
There are three elongated pyramids that are Johnson solids:
Elongated triangular pyramid (),
Elongated square pyramid (), and
Elongated pentagonal pyramid ().
Higher forms can be constructed with isosceles triangles.
Forms
See also
Gyroelongated bipyramid
Elongated bipyramid
Gyroelongated pyramid
Diminished trapezohedron
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Pyramids and bipyramids |
https://en.wikipedia.org/wiki/Gyroelongated%20pyramid | In geometry, the gyroelongated pyramids (also called augmented antiprisms) are an infinite set of polyhedra, constructed by adjoining an pyramid to an antiprism.
There are two gyroelongated pyramids that are Johnson solids made from regular triangles and square, and pentagons. A triangular and hexagonal form can be constructed with coplanar faces. Others can be constructed allowing for isosceles triangles.
Forms
See also
Gyroelongated bipyramid
Elongated bipyramid
Elongated pyramid
Diminished trapezohedron
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Pyramids and bipyramids |
https://en.wikipedia.org/wiki/Elongated%20cupola | In geometry, the elongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to an 2n-gonal prism.
There are three elongated cupolae that are Johnson solids made from regular triangles and square, and pentagons. Higher forms can be constructed with isosceles triangles. Adjoining a triangular prism to a cube also generates a polyhedron, but has adjacent parallel faces, so is not a Johnson solid. Higher forms can be constructed without regular faces.
Forms
See also
Gyroelongated bicupola
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Polyhedra |
https://en.wikipedia.org/wiki/Gyroelongated%20cupola | In geometry, the gyroelongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to an 2n-gonal antiprism.
There are three gyroelongated cupolae that are Johnson solids made from regular triangles and square, and pentagons. Higher forms can be constructed with isosceles triangles. Adjoining a triangular prism to a square antiprism also generates a polyhedron, but has adjacent parallel faces, so is not a Johnson solid. The hexagonal form can be constructed from regular polygons, but the cupola faces are all in the same plane. Topologically other forms can be constructed without regular faces.
Forms
See also
Elongated cupola
Elongated bicupola
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Polyhedra |
https://en.wikipedia.org/wiki/Gyroelongated%20bicupola | In geometry, the gyroelongated bicupolae are an infinite sets of polyhedra, constructed by adjoining two n-gonal cupolas to an n-gonal Antiprism. The triangular, square, and pentagonal gyroelongated bicupola are three of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form.
Adjoining two triangular prisms to a cube also generates a polyhedron, but has adjacent parallel faces, so is not a Johnson solid. The hexagonal form is also a polygon, but has coplanar faces. Higher forms can be constructed without regular faces.
See also
Elongated cupola
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Polyhedra |
https://en.wikipedia.org/wiki/Elongated%20bicupola | In geometry, the elongated bicupolae are two infinite sets of polyhedra, constructed by adjoining two n-gonal cupolas to an n-gonal prism. They have 2n triangles, 4n squares, and 2 n-gon. The ortho forms have the cupola aligned, while gyro forms are counter aligned.
See also
Bicupola
Elongated cupola
Gyroelongated bicupola
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Polyhedra |
https://en.wikipedia.org/wiki/Kazuya%20Okazaki | is a former Japanese football player.
Club statistics
Updated to 22 February 2016.
References
External links
J. League (#28)
Contract with Kazuya Okazaki.
Player Profile on Albirex Niigata (S) Official Website.
1991 births
Living people
Association football people from Hiroshima Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Fagiano Okayama players
Verspah Oita players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kenji%20Sekido | Kenji Sekido (関戸 健二, born January 7, 1990) is a Japanese football player.
Club statistics
Updated to 10 August 2022.
References
External links
1990 births
Living people
Ryutsu Keizai University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J2 League players
Fagiano Okayama players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Takumi%20Murakami | is a Japanese football player.
Club statistics
Updated to 23 February 2020.
References
External links
Profile at Roasso Kumamoto
1989 births
Living people
Ritsumeikan University alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J2 League players
J3 League players
Ehime FC players
Roasso Kumamoto players
Ococias Kyoto AC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Daisuke%20Ishizu | is a Japanese football player for FC Gifu.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Avispa Fukuoka
Profile at Vissel Kobe
1990 births
Living people
Fukuoka University alumni
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Avispa Fukuoka players
Vissel Kobe players
FC Gifu players
Men's association football forwards
Association football people from Fukuoka (city) |
https://en.wikipedia.org/wiki/Tokio%20Hatamoto | is a former Japanese football player.
Club statistics
Updated to 23 February 2019.
References
External links
Profile at Grulla Morioka
1992 births
Living people
Association football people from Kumamoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Avispa Fukuoka players
Zweigen Kanazawa players
Iwate Grulla Morioka players
Men's association football defenders |
https://en.wikipedia.org/wiki/Daiki%20Watari | Daiki Watari (渡 大生, born 25 June 1993) is a Japanese footballer who plays as a forward for club Tokushima Vortis.
Club statistics
.
References
External links
Profile at Sanfrecce Hiroshima
Profile at Tokushima Vortis
1993 births
Living people
Association football people from Hiroshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
Giravanz Kitakyushu players
Tokushima Vortis players
Sanfrecce Hiroshima players
Oita Trinita players
Avispa Fukuoka players
Men's association football forwards |
https://en.wikipedia.org/wiki/Miran%20Kabe | is a former Japanese football player.
Club statistics
References
External links
J. League (#27)
1992 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Ventforet Kofu players
Giravanz Kitakyushu players
Men's association football forwards |
https://en.wikipedia.org/wiki/Shuto%20Nakahara | is a Japanese football player currently playing for Kagoshima United FC.
Club statistics
Updated to end of 2018 season.
1Includes J1 Promotion Playoffs.
References
External links
Profile at Kagoshima United FC
Profile at Giravanz Kitakyushu
1990 births
Living people
University of Teacher Education Fukuoka alumni
Association football people from Kagoshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Giravanz Kitakyushu players
Avispa Fukuoka players
Kagoshima United FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Junki%20Goryo | Junki Goryo (五領 淳樹, born December 13, 1989) is a Japanese football player for Kagoshima United FC.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Kagoshima United FC
1989 births
Living people
Miyazaki Sangyo-keiei University alumni
Association football people from Kagoshima Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Roasso Kumamoto players
Kagoshima United FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yuki%20Yamazaki%20%28footballer%29 | is a former Japanese football player.
Club statistics
References
External links
J. League (#31)
1990 births
Living people
National Institute of Fitness and Sports in Kanoya alumni
Association football people from Tokyo
Japanese men's footballers
J2 League players
Japan Football League players
Roasso Kumamoto players
Renofa Yamaguchi FC players
Tochigi City FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kim%20Chang-hun | Kim Chang-Hun (born February 17, 1990) is a South Korean football player. He has represented South Korea at the U-22 level.
Club statistics
References
J. League (#19)
External links
1990 births
Living people
Men's association football defenders
South Korean men's footballers
South Korean expatriate men's footballers
J2 League players
Oita Trinita players
Ulsan Hyundai Mipo Dockyard FC players
Suwon FC players
Gimcheon Sangmu FC players
Korea National League players
K League 2 players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan |
https://en.wikipedia.org/wiki/Yusuke%20Goto | Yusuke Goto (後藤 優介, born April 23, 1993) is a Japanese football player for Shimizu S-Pulse.
Club statistics
Updated to 24 July 2022.
References
External links
Profile at Oita Trinita
1993 births
Living people
Association football people from Kagoshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Oita Trinita players
Verspah Oita players
J.League U-22 Selection players
Shimizu S-Pulse players
Men's association football forwards |
https://en.wikipedia.org/wiki/Minkowski%27s%20second%20theorem | In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.
Setting
Let be a closed convex centrally symmetric body of positive finite volume in -dimensional Euclidean space . The gauge or distance Minkowski functional attached to is defined by
Conversely, given a norm on we define to be
Let be a lattice in . The successive minima of or on are defined by setting the -th successive minimum to be the infimum of the numbers such that contains linearly-independent vectors of . We have .
Statement
The successive minima satisfy
Proof
A basis of linearly independent lattice vectors can be defined by .
The lower bound is proved by considering the convex polytope with vertices at , which has an interior enclosed by and a volume which is times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by along each basis vector to obtain -simplices with lattice point vectors).
To prove the upper bound, consider functions sending points in to the centroid of the subset of points in that can be written as for some real numbers . Then the coordinate transform has a Jacobian determinant . If and are in the interior of and (with ) then with , where the inclusion in (specifically the interior of ) is due to convexity and symmetry. But lattice points in the interior of are, by definition of , always expressible as a linear combination of , so any two distinct points of cannot be separated by a lattice vector. Therefore, must be enclosed in a primitive cell of the lattice (which has volume ), and consequently .
References
Hermann Minkowski |
https://en.wikipedia.org/wiki/Sara%20Errani%20career%20statistics | This is a list of the main career statistics of Italian professional tennis player Sara Errani.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup/Billie Jean King Cup and Olympic Games are included in win–loss records.
Singles
Current through the 2023 Wimbledon Championships.
Doubles
Current through the 2022 French Open.
Significant finals
Grand Slam finals
Singles: 1 (1 runner-ups)
Doubles: 8 (5 titles, 3 runner-ups)
WTA Premier Mandatory & Premier 5 finals
Singles: 1 (1 runner-up)
Doubles: 9 (5 titles, 4 runner-ups)
WTA career finals
Singles: 19 (9 titles, 10 runner-ups)
Doubles: 43 (28 titles, 15 runner-ups)
WTA Challenger finals
Singles: 5 (2 titles, 3 runner-ups)
Doubles: 1 (title)
National team competition finals
Fed Cup: 3 (3 titles)
ITF Circuit Finals
Singles: 8 (5 titles, 3 runner–ups)
Doubles: 11 (7 titles, 4 runner–ups)
Top 10 wins
Record against top 10 players
Errani's match record against certain players who have been ranked in the top 10. Players who are active are in boldface.
See also
Roberta Vinci career statistics
Notes
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Probability%20distribution%20fitting | Probability distribution fitting or simply distribution fitting is the fitting of a probability distribution to a series of data concerning the repeated measurement of a variable phenomenon.
The aim of distribution fitting is to predict the probability or to forecast the frequency of occurrence of the magnitude of the phenomenon in a certain interval.
There are many probability distributions (see list of probability distributions) of which some can be fitted more closely to the observed frequency of the data than others, depending on the characteristics of the phenomenon and of the distribution. The distribution giving a close fit is supposed to lead to good predictions.
In distribution fitting, therefore, one needs to select a distribution that suits the data well.
Selection of distribution
The selection of the appropriate distribution depends on the presence or absence of symmetry of the data set with respect to the central tendency.
Symmetrical distributions
When the data are symmetrically distributed around the mean while the frequency of occurrence of data farther away from the mean diminishes, one may for example select the normal distribution, the logistic distribution, or the Student's t-distribution. The first two are very similar, while the last, with one degree of freedom, has "heavier tails" meaning that the values farther away from the mean occur relatively more often (i.e. the kurtosis is higher). The Cauchy distribution is also symmetric.
Skew distributions to the right
When the larger values tend to be farther away from the mean than the smaller values, one has a skew distribution to the right (i.e. there is positive skewness), one may for example select the log-normal distribution (i.e. the log values of the data are normally distributed), the log-logistic distribution (i.e. the log values of the data follow a logistic distribution), the Gumbel distribution, the exponential distribution, the Pareto distribution, the Weibull distribution, the Burr distribution, or the Fréchet distribution. The last four distributions are bounded to the left.
Skew distributions to the left
When the smaller values tend to be farther away from the mean than the larger values, one has a skew distribution to the left (i.e. there is negative skewness), one may for example select the square-normal distribution (i.e. the normal distribution applied to the square of the data values), the inverted (mirrored) Gumbel distribution, the Dagum distribution (mirrored Burr distribution), or the Gompertz distribution, which is bounded to the left.
Techniques of fitting
The following techniques of distribution fitting exist:
Parametric methods, by which the parameters of the distribution are calculated from the data series. The parametric methods are:
Method of moments
Maximum spacing estimation
Method of L-moments
Maximum likelihood method
{| class="wikitable"
| bgcolor="white" | For example, the parameter (the expectation) can be estimated by the mea |
https://en.wikipedia.org/wiki/Computable%20topology | Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or computational topology, which studies the application of computation to topology.
Topology of lambda calculus
As shown by Alan Turing and Alonzo Church, the λ-calculus is strong enough to describe all mechanically computable functions (see Church–Turing thesis). Lambda-calculus is thus effectively a programming language, from which other languages can be built. For this reason when considering the topology of computation it is common to focus on the topology of λ-calculus. Note that this is not necessarily a complete description of the topology of computation, since functions which are equivalent in the Church-Turing sense may still have different topologies.
The topology of λ-calculus is the Scott topology, and when restricted to continuous functions the type free λ-calculus amounts to a topological space reliant on the tree topology. Both the Scott and Tree topologies exhibit continuity with respect to the binary operators of application ( f applied to a = fa ) and abstraction ((λx.t(x))a = t(a)) with a modular equivalence relation based on a congruency. The λ-algebra describing the algebraic structure of the lambda-calculus is found to be an extension of the combinatory algebra, with an element introduced to accommodate abstraction.
Type free λ-calculus treats functions as rules and does not differentiate functions and the objects which they are applied to, meaning λ-calculus is type free. A by-product of type free λ-calculus is an effective computability equivalent to general recursion and Turing machines. The set of λ-terms can be considered a functional topology in which a function space can be embedded, meaning λ mappings within the space X are such that λ:X → X. Introduced November 1969, Dana Scott's untyped set theoretic model constructed a proper topology for any λ-calculus model whose function space is limited to continuous functions. The result of a Scott continuous λ-calculus topology is a function space built upon a programming semantic allowing fixed point combinatorics, such as the Y combinator, and data types. By 1971, λ-calculus was equipped to define any sequential computation and could be easily adapted to parallel computations. The reducibility of all computations to λ-calculus allows these λ-topological properties to become adopted by all programming languages.
Computational algebra from λ-calculus algebra
Based on the operators within lambda calculus, application and abstraction, it is possible to develop an algebra whose group structure uses application and abstraction as binary operators. Application is defined as an operation between lambda terms producing a λ-term, e.g. the application of λ onto the lambda term a produces the lambda term λa. Abstraction incorporates undefined variables by denoting λx.t(x) as the function assigning the |
https://en.wikipedia.org/wiki/Da%20Ruan | Da Ruan (; September 10, 1960 – July 31, 2011) was a Chinese-Belgian mathematician, scientist, professor. He had a Ph.D. from Ghent University.
Bibliography
Fuzzy set theory and advanced mathematical applications (1995, Kluwer Academic Publishers)
References
1960 births
2011 deaths
Belgian mathematicians
Educators from Shanghai
Mathematicians from Shanghai
Ghent University alumni |
https://en.wikipedia.org/wiki/UEFA%20Euro%202012%20statistics | These are the statistics for the UEFA Euro 2012, which took place in Poland and Ukraine.
Goalscorers
Assists
Scoring
Sources: Opta Sports, UEFA
Attendance
Overall attendance: 1,440,896
Average attendance per match:
Highest attendance: 64,640 – Sweden (2–3) England
Lowest attendance: 31,840 – Denmark (2–3) Portugal
Wins and losses
Discipline
Sanctions against foul play at Euro 2012 were in the first instance the responsibility of the referee, but when if he deemed it necessary to give a caution, or dismiss a player, UEFA kept a record and may have enforced a suspension. UEFA's disciplinary committee had the ability to penalize players for offenses unpunished by the referee.
Overview
Red cards
A player receiving a red card was automatically suspended for the next match. A longer suspension was possible if the UEFA disciplinary committee had judged the offence as warranting it. In keeping with the FIFA Disciplinary Code (FDC) and UEFA Disciplinary Regulations (UDR), UEFA did not allow for appeals of red cards except in the case of mistaken identity. The FDC further stipulated that if a player was sent off during his team's final Euro 2012 match, the suspension would carry over to his team's next competitive international(s), which in this case would be the qualification matches for the 2014 FIFA World Cup.
Any player who was suspended due to a red card that was earned in Euro 2012 qualifying was required to serve the balance of any suspension unserved by the end of qualifying either in the Euro 2012 finals (for any player on a team that qualified, whether he was selected to the final squad or not) or in World Cup qualifying (for players on teams that did not qualify).
Yellow cards
Any player receiving a single yellow card during two of the three group stage matches and the quarter-final match was suspended for the following match. A single yellow card did not carry over to the semi-finals. This meant that no player could have been suspended for final unless he was sent off in semi-final or he was serving a longer suspension for an earlier incident. Suspensions due to yellow cards did carry over to the World Cup qualifiers. Yellow cards and any related suspensions earned in the Euro 2012 qualifiers were neither counted nor enforced in the final tournament.
In the event a player was sent off for two bookable offenses, only the red card was counted for disciplinary purposes. However, in the event a player received a direct red card after being booked in the same match, then both cards would have been counted. If the player was already facing a suspension for two tournament bookings when he was sent off, this would have resulted in separate suspensions that would have been served consecutively. The one match ban for the yellow cards would be served first unless the player's team was eliminated in the match in which he was sent off. If the player's team was eliminated in the match in which he was serving his ban for the yellow cards, th |
https://en.wikipedia.org/wiki/Puig%20subgroup | In mathematical finite group theory, the Puig subgroup, introduced by , is a characteristic subgroup of a p-group analogous to the Thompson subgroup.
Definition
If H is a subgroup of a group G, then LG(H) is the subgroup of G generated by the abelian subgroups normalized by H.
The subgroups Ln of G are defined recursively by
L0 is the trivial subgroup
Ln+1 = LG(Ln)
They have the property that
L0 ⊆ L2 ⊆ L4... ⊆ ...L5 ⊆ L3 ⊆ L1
The Puig subgroup L(G) is the intersection of the subgroups Ln for n odd, and the subgroup L*(G) is the union of the subgroups Ln for n even.
Properties
Puig proved that if G is a (solvable) group of odd order, p is a prime, and S is a Sylow p-subgroup of G, and the -core of G is trivial, then the center Z(L(S)) of the Puig subgroup of S is a normal subgroup of G.
References
Finite groups |
https://en.wikipedia.org/wiki/Richard%20Birkeland | Richard Birkeland (6 June 1879 – 10 April 1928) was a Norwegian mathematician, author and professor. He is known for his contributions to the theory of algebraic equations.
Biography
He was born at Farsund in Vest-Agder, Norway. He was the son of Theodor Birkeland (1834-1913) and Therese Karoline Overwien (1846-1883). He graduated from the Christiania Technical School in 1899. In 1906, he received a scholarship to study mathematics in Paris and Göttingen. He became a professor at the Norwegian Institute of Technology from 1910. He was rector of the Norwegian Institute of Technology and from 1923 professor at the University of Oslo.
He was a co-founder of the Norwegian Mathematical Society in 1918 and he was its vice chairman in the early years. He was for a time chairman of Trondheim Polytechnic Association. He was decorated Knight of the Order of St. Olav.
Selected works
Sur certaines singularités des équations différentielles (1909)
Lærebok i matematisk analyse : differential- og integralregning, differentialligninger tillæg (1917)
Personal life
He was a cousin of physics professor Kristian Birkeland (1867-1917). In 1909, he married Agnes Hoff (1883-1980). Their son Øivind (1910-2004) was a civil engineer.
References
1879 births
1928 deaths
People from Farsund
Norwegian mathematicians
Norwegian educators
Academic staff of the University of Oslo
Academic staff of the Norwegian Institute of Technology
Rectors of the Norwegian University of Science and Technology
Recipients of the St. Olav's Medal |
https://en.wikipedia.org/wiki/Friedrich%20Karl%20Schmidt | Friedrich Karl Schmidt (22 September 1901 – 25 January 1977) was a German mathematician, who made notable contributions to algebra and number theory.
Schmidt studied from 1920 to 1925 in Freiburg and Marburg. In 1925 he completed his doctorate at the Albert-Ludwigs-Universität Freiburg under the direction of Alfred Loewy. In 1927 he became a Privatdozent (lecturer) at the University of Erlangen, where he received his habilitation and in 1933 became a professor extraordinarius. In 1933/34 he was a Dozent at the University of Göttingen, where he worked with Helmut Hasse. Schmidt was then a professor ordinarius at the University of Jena from 1934 to 1945. During WW II, he was at the Deutsche Versuchsanstalt für Segelflug (German Research Station for Gliding) in Reichenhall. He was a professor from 1946 to 1952 at Westfälischen Wilhelms-Universität in Münster and from 1952 to 1966 at the University of Heidelberg, where he retired as professor emeritus.
In the mid-1930s Schmidt was on the editorial staff of .
Schmidt was elected in 1954 a member of the Heidelberger Akademie der Wissenschaften and was made in 1968 an honorary doctor of the Free University of Berlin.
Schmidt is known for his contributions to the theory of algebraic function fields and in particular for his definition of a zeta function for algebraic function fields and his proof of the generalized Riemann–Roch theorem for algebraic function fields (where the base field can be an arbitrary perfect field). He also made contributions to class field theory and valuation theory.
References
External links
1901 births
1977 deaths
Algebraists
20th-century German mathematicians
Number theorists
University of Freiburg alumni
Academic journal editors |
https://en.wikipedia.org/wiki/Treasure%20Valley%20Mathematics%20and%20Science%20Center | Treasure Valley Mathematics and Science Center, often referred to as Treasure Valley Math and Science Center or TVMSC, is a public magnet school in Boise, Idaho operated by the Boise School District that offers advanced secondary mathematics, science, technology, and research classes to students living in the Treasure Valley. After being accepted into the program, students attend TVMSC in its morning or afternoon sessions while they attend their normal public or non-public elementary, junior high, high schools, or are home-schooled, during the other part of the day. Since the school's founding in 2004, the program has been led by Dr. Holly Maclean, who, as the school principal, carefully leads the school's twelve teachers who have extensive experience in the subjects they teach.
Although the classes TVMSC offers are usually taught as advanced courses in grades seven through twelve, students at any age can apply for the school, as the program accepts students based on teacher recommendations that determine a student's desire to excel in math and science, report cards to determine academic capability, standardized test scores to determine knowledge, and fulfillment of prerequisite math classes, which can be taken as summer classes before entering the program. TVMSC was founded in 2004 with a donation of $1,000,000 from Micron Technology, whose headquarters are in Boise, as well as $300,000 worth of equipment from Hewlett-Packard, which has a large campus in Boise as well. Since cuts in funding caused by the 2008–2012 global recession inhibited the original plans for its own permanent location on the Boise State Campus, TVMSC holds its classes in the upper floor of the south wing at Riverglen Junior High School.
TVMSC has two very successful teams that compete in the National Science Bowl and the National Middle School Science Bowl. The middle school team placed 4th in the academic competition and 2nd in the electric car competition in 2014, 2013, and 2012, as well as placing 1st in the 2008 fuel cell car overall and 2nd fuel cell car race.
As of the 2018–19 school year, the TVMSC mascot is the Sloth, selected by a student vote. The following year, the students voted to name the mascot Slothy Joe.
References
Educational institutions established in 2004
Magnet schools in Idaho
High schools in Boise, Idaho
2004 establishments in Idaho |
https://en.wikipedia.org/wiki/Smooth%20algebra | In algebra, a commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose square is zero and a k-algebra map , there exists a k-algebra map such that u is v followed by the canonical map. If there exists at most one such lifting v, then A is said to be 0-unramified (or 0-neat). A is said to be 0-étale if it is 0-smooth and 0-unramified. The notion of 0-smoothness is also called formal smoothness.
A finitely generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k.
A separable algebraic field extension L of k is 0-étale over k. The formal power series ring is 0-smooth only when and (i.e., k has a finite p-basis.)
I-smooth
Let B be an A-algebra and suppose B is given the I-adic topology, I an ideal of B. We say B is I-smooth over A if it satisfies the lifting property: given an A-algebra C, an ideal N of C whose square is zero and an A-algebra map that is continuous when is given the discrete topology, there exists an A-algebra map such that u is v followed by the canonical map. As before, if there exists at most one such lift v, then B is said to be I-unramified over A (or I-neat). B is said to be I-étale if it is I-smooth and I-unramified. If I is the zero ideal and A is a field, these notions coincide with 0-smooth etc. as defined above.
A standard example is this: let A be a ring, and Then B is I-smooth over A.
Let A be a noetherian local k-algebra with maximal ideal . Then A is -smooth over if and only if is a regular ring for any finite extension field of .
See also
étale morphism
formally smooth morphism
Popescu's theorem
References
Algebra |
https://en.wikipedia.org/wiki/Jeon%20Hyeon-chul | Jeon Hyeon-chul (; born 3 July 1990) is a South Korean footballer who plays as a forward for Daegu FC.
Club career statistics
As of 17 July 2017
External links
1990 births
Living people
South Korean men's footballers
Men's association football forwards
Seongnam FC players
Jeonnam Dragons players
Busan IPark players
Daegu FC players
K League 1 players
K League 2 players
Ajou University alumni |
https://en.wikipedia.org/wiki/Fundamental%20theorem%20of%20ideal%20theory%20in%20number%20fields | In number theory, the fundamental theorem of ideal theory in number fields states that every nonzero proper ideal in the ring of integers of a number field admits unique factorization into a product of nonzero prime ideals. In other words, every ring of integers of a number field is a Dedekind domain.
References
Keith Conrad, Ideal factorization
Algebraic numbers
Theorems in algebraic number theory
Factorization |
https://en.wikipedia.org/wiki/Sastry%20automorphism | In mathematics, a Sastry automorphism, is an automorphism of a field of characteristic 2 satisfying some rather complicated conditions related to the problem of embedding Ree groups of type 2F4 into Chevalley groups of type F4. They were introduced by , and named and classified by who showed that there are 22 families of Sastry automorphisms, together with 22 exceptional ones over some finite fields of orders up to 210.
References
Finite groups
Finite fields |
https://en.wikipedia.org/wiki/Allan%20M.%20Ramsay | Allan M. Ramsay is a Professor of Formal Linguistics in the Department of Computer Science at the University of Manchester.
Education
Ramsay's undergraduate degree was in Logic and Mathematics from the University of Sussex. After completing a Master of Science degree in Logic from the University of London, he returned to Sussex to complete a PhD in Artificial Intelligence. Prior to working at UMIST and the University of Manchester, he was Professor of Artificial Intelligence at University College Dublin.
Research
Ramsay's research focuses on Natural language processing, including morphology and syntax. He has published papers on the analysis of free word order languages, particularly morphology of the Arabic language, which poses a number of specific problems. Some of this research has been funded by the EPSRC.
References
Academics of the University of Manchester
People associated with the Department of Computer Science, University of Manchester
Linguists from England
Living people
Alumni of the University of London
Alumni of the University of Sussex
1953 births
Natural language processing researchers
Computational linguistics researchers |
https://en.wikipedia.org/wiki/Stefan%20Nemirovski | Stefan Yuryevich Nemirovski (; born 29 July 1973) is a Russian mathematician. He made notable contributions to topology and complex analysis, and was awarded an EMS Prize in 2000.
Nemirovski earned his Ph.D. from Moscow State University in 1998.
References
External links
EMS Prize Laudatio, Notices AMS
Website at the University of Bochum
1973 births
Living people
Russian mathematicians |
https://en.wikipedia.org/wiki/Rail%20Rozakov | Rail Rozakov (born March 29, 1981) is a Russian former professional ice hockey defenceman. He was drafted 106th overall in the 1999 NHL Entry Draft by the Calgary Flames.
Career statistics
External links
1981 births
Living people
Barys Nur-Sultan players
Calgary Flames draft picks
HC CSK VVS Samara players
HC CSKA Moscow players
HC Lada Togliatti players
Metallurg Novokuznetsk players
HC Sibir Novosibirsk players
Krylya Sovetov Moscow players
Lowell Lock Monsters players
Russian ice hockey defencemen
Severstal Cherepovets players
Traktor Chelyabinsk players
HC Vityaz players
Sportspeople from Murmansk |
https://en.wikipedia.org/wiki/Timor-Leste%20national%20football%20team%20records%20and%20statistics | The following table summarizes the all-time record for the Timor-Leste men's national football team. Timor-Leste has played matches against 16 current and former national teams, with the latest result, a loss, coming against Philippines on July 16, 2022.
Individual records
Player records
Players in bold are still active with Timor-Leste.
Most capped players
Top goalscorers
Most capped goalkeepers
note: 1. The above list pointed to a player who made his debut before they are even 18 years old.
See also
Timor-Leste national football team results
References
External links
Timor-Leste National football team results
Record
National association football team records and statistics |
https://en.wikipedia.org/wiki/Perpendicular%20bisector%20construction%20of%20a%20quadrilateral | In geometry, the perpendicular bisector construction of a quadrilateral is a construction which produces a new quadrilateral from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral. This construction arises naturally in an attempt to find a replacement for the circumcenter of a quadrilateral in the case that is non-cyclic.
Definition of the construction
Suppose that the vertices of the quadrilateral are given by . Let be the perpendicular bisectors of sides respectively. Then their intersections , with subscripts considered modulo 4, form the consequent quadrilateral . The construction is then iterated on to produce and so on.
An equivalent construction can be obtained by letting the vertices of be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of .
Properties
1. If is not cyclic, then is not degenerate.
2. Quadrilateral is never cyclic. Combining #1 and #2, is always nondegenrate.
3. Quadrilaterals and are homothetic, and in particular, similar. Quadrilaterals and are also homothetic.
3. The perpendicular bisector construction can be reversed via isogonal conjugation. That is, given , it is possible to construct .
4. Let be the angles of . For every , the ratio of areas of and is given by
5. If is convex then the sequence of quadrilaterals converges to the isoptic point of , which is also the isoptic point for every . Similarly, if is concave, then the sequence obtained by reversing the construction converges to the Isoptic Point of the 's.
6. If is tangential then is also tangential.
References
J. Langr, Problem E1050, Amer. Math. Monthly, 60 (1953) 551.
V. V. Prasolov, Plane Geometry Problems, vol. 1 (in Russian), 1991; Problem 6.31.
V. V. Prasolov, Problems in Plane and Solid Geometry, vol. 1 (translated by D. Leites), available at http://students.imsa.edu/~tliu/math/planegeo.eps.
D. Bennett, Dynamic geometry renews interest in an old problem, in Geometry Turned On, (ed. J. King), MAA Notes 41, 1997, pp. 25–28.
J. King, Quadrilaterals formed by perpendicular bisectors, in Geometry Turned On, (ed. J. King), MAA Notes 41, 1997, pp. 29–32.
G. C. Shephard, The perpendicular bisector construction, Geom. Dedicata, 56 (1995) 75–84.
A. Bogomolny, Quadrilaterals formed by perpendicular bisectors, Interactive Mathematics Miscellany and Puzzles, http://www.cut-the-knot.org/Curriculum/Geometry/PerpBisectQuadri.shtml.
B. Grünbaum, On quadrangles derived from quadrangles—Part 3, Geombinatorics 7(1998), 88–94.
O. Radko and E. Tsukerman, The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral, Forum Geometricorum 12: 161–189 (2012).
External links
Perpendicular-Bisectors of Circumscribed Quadrilateral Theorem at Dynamic Geometry Sketches, interactive dynamic geometry sketches.
Quadrilaterals |
https://en.wikipedia.org/wiki/2011%20Lao%20League | Statistics of Lao League in the 2011 season.
Clubs
Lao Army FC
Bank F.C.
Eastern Star Bilingual School FC
Ezra FC
Lao-American College FC
Lao Lane Xang FC
Pheuanphatthana FC
Lao Police Club (formerly Ministry of Public Security FC (MPS))
Vientiane F.C.
Yotha FC (formerly Ministry of Public Works and Transport FC)
Yotha FC were champions.
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/Bo%20Ericson%20%28ice%20hockey%29 | Bo Ragnar Ericson (born January 23, 1958, in Stockholm, Sweden) is an ice hockey player who played for the Swedish national team. He won a bronze medal at the 1984 Winter Olympics.
Career statistics
Regular season and playoffs
International
References
2. Bo Ericson's profile at ElitePropects.com
1958 births
AIK IF players
Colorado Rockies (NHL) draft picks
Ice hockey players at the 1984 Winter Olympics
Living people
Medalists at the 1984 Winter Olympics
Olympic bronze medalists for Sweden
Olympic ice hockey players for Sweden
Olympic medalists in ice hockey
Ice hockey people from Stockholm
Södertälje SK players |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20Athletic%20Bilbao%20season | The 2010–11 season was the 110th season in Athletic Bilbao's history and their 80th consecutive season in La Liga, the top division of Spanish football.
Squad statistics
Appearances and goals
|}
Competitions
La Liga
League table
Copa del Rey
External links
Athletic Bilbao
Athletic Bilbao seasons
2010 in the Basque Country (autonomous community)
2011 in the Basque Country (autonomous community) |
https://en.wikipedia.org/wiki/Joseph%20Madachy | Joseph Steven Madachy (March 16, 1927 – March 27, 2014) was a research chemist, technical editor and recreational mathematician. He was the lead editor of Journal of Recreational Mathematics for nearly 30 years and then served as editor emeritus. He was owner, publisher and editor of its predecessor, Recreational Mathematics Magazine, which appeared from 1961 to 1964.
Early life and education
Madachy was born in Star Junction, Pennsylvania, to Steven and Anne Madachy. He was raised in Cleveland, Ohio. He became interested in recreational mathematics after reading Eugene Northrop's 1944 book, Riddles in Mathematics. After service in World War II, he attended Western Reserve University on the G.I. Bill and earned a bachelor's and a master's in Chemistry.
Career
Madachy moved to Dayton, Ohio, and worked for Mound Laboratories. He made original contributions to the field of recreational mathematics. In 1960 he wrote to recreational mathematician Martin Gardner, asking whether Gardner knew of any publications devoted solely to recreational mathematics, as he was considering starting such a project. Gardner responded in the negative, including a box containing his correspondence and suggesting Madachy could use the addresses to promote the magazine. From February 1961 to 1964 Madachy published the bimonthly Recreational Mathematics Magazine.
In 1967, Greenwood Press asked him to start the journal again under the title Journal of Recreational Mathematics, which was published by Baywood Publishing starting in 1973. He authored several books on recreational mathematics, including Mathematics on Vacation (1966), Madachy's Mathematical Recreations and Mathematical Diversions. He served as the literary agent for Dmitri Borgmann's Language on Vacation. Longtime colleagues and co-authors include Martin Gardner, Harry L. Nelson, and Isaac Asimov, and Solomon Golomb (with pentominos).
He worked with polyominoes, pentominos, prime numbers, and amicable numbers. He worked developing mathematical concepts such as cryptarithmetic, used in cyber security applications. He made contributions to Fibonacci series and narcissistic numbers and devised puzzles using Fibonacci numbers. His recreational mathematics work included areas in chess, magic squares and calculator art.
Madachy retired from editing Journal of Recreational Mathematics in 2000.
In popular culture
Madachy is mentioned in the Jack Reacher novel series in the book Never Go Back, which uses perfect digit-to-digit invariant numbers in the plot: "Such numbers had been much discussed by a guy called Joseph Madachy, who once upon a time had been the owner, publisher, and editor of a magazine called Journal of Recreational Mathematics."
Personal life
Madachy and his wife, Juliana, lived in Dayton, Ohio and had six children.
References
1927 births
20th-century American mathematicians
21st-century American mathematicians
Recreational mathematicians
People from Pennsylvania
2014 deaths |
https://en.wikipedia.org/wiki/Weyl%E2%80%93von%20Neumann%20theorem | In mathematics, the Weyl–von Neumann theorem is a result in operator theory due to Hermann Weyl and John von Neumann. It states that, after the addition of a compact operator () or Hilbert–Schmidt operator () of arbitrarily small norm, a bounded self-adjoint operator or unitary operator on a Hilbert space is conjugate by a unitary operator to a diagonal operator. The results are subsumed in later generalizations for bounded normal operators due to David Berg (1971, compact perturbation) and Dan-Virgil Voiculescu (1979, Hilbert–Schmidt perturbation). The theorem and its generalizations were one of the starting points of operator K-homology, developed first by Lawrence G. Brown, Ronald Douglas and Peter Fillmore and, in greater generality, by Gennadi Kasparov.
In 1958 Kuroda showed that the Weyl–von Neumann theorem is also true if the Hilbert–Schmidt class is replaced by any Schatten class Sp with p ≠ 1. For S1, the trace-class operators, the situation is quite different. The Kato–Rosenblum theorem, proved in 1957 using scattering theory, states that if two bounded self-adjoint operators differ by a trace-class operator, then their absolutely continuous parts are unitarily equivalent. In particular if a self-adjoint operator has absolutely continuous spectrum, no perturbation of it by a trace-class operator can be unitarily equivalent to a diagonal operator.
References
Operator theory
Theorems in functional analysis
K-theory |
https://en.wikipedia.org/wiki/Wirtinger%20presentation | In mathematics, especially in group theory, a Wirtinger presentation is a finite presentation where the relations are of the form where is a word in the generators, Wilhelm Wirtinger observed that the complements of knots in 3-space have fundamental groups with presentations of this form.
Preliminaries and definition
A knot K is an embedding of the one-sphere S1 in three-dimensional space R3. (Alternatively, the ambient space can also be taken to be the three-sphere S3, which does not make a difference for the purposes of the Wirtinger presentation.) The open subspace which is the complement of the knot, is the knot complement. Its fundamental group is an invariant of the knot in the sense that equivalent knots have isomorphic knot groups. It is therefore interesting to understand this group in an accessible way.
A Wirtinger presentation is derived from a regular projection of an oriented knot. Such a projection can be pictured as a finite number of (oriented) arcs in the plane, separated by the crossings of the projection. The fundamental group is generated by loops winding around each arc. Each crossing gives rise to a certain relation among the generators corresponding to the arcs meeting at the crossing.
Wirtinger presentations of high-dimensional knots
More generally, co-dimension two knots in spheres are known to have Wirtinger presentations. Michel Kervaire proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied:
The abelianization of the group is the integers.
The 2nd homology of the group is trivial.
The group is finitely presented.
The group is the normal closure of a single generator.
Conditions (3) and (4) are essentially the Wirtinger presentation condition, restated. Kervaire proved in dimensions 5 and larger that the above conditions are necessary and sufficient. Characterizing knot groups in dimension four is an open problem.
Examples
For the trefoil knot, a Wirtinger presentation can be shown to be
See also
Knot group
Further reading
, section 3D
Knot theory |
https://en.wikipedia.org/wiki/Poverty%20in%20Tanzania | Tanzania has a current population of 55.57 million people. Current statistics form the World Bank show that in 2011, 49.1% of Tanzanians lived below US$1.90 per day. This figure is an improvement over 2007's report indicating a poverty rate of 55.1%. Tanzania has seen annual GDP gains of 7% since 2010 and this economic growth is attributed to this positive trends for poverty alleviation in Tanzania. The 2019 World Bank report showed that in the last 10 years, poverty has reduced by 8 percent, from 34.4% in 2007 to 26.4% in 2018.
Recently there has been statistical reductions in the levels of extreme poverty, basic needs poverty, and food poverty. However, these reductions are occurring faster in urban areas as compared to rural areas.
Indicators of Poverty
GDP
Trends in GDP per capita also break along the same divisions, with Dar es Salaam's GDP per capita at TSh as compared to the Tanzania Mainland's of . High levels of economic growth in Tanzania has been sustained since 2001, yet the current high rates of poverty challenge whether pure economic growth can be realized in human development.
Food Poverty
The split between rural and urban poverty is most extreme in terms of food insecurity. As of 2012 only 1% of Tanzanian's in Dar es Salaam experience food poverty as compared to 11.3% of Tanzania's living in rural areas.
HDI
Utilizing the Human Development Index, urban areas Dar es Salaam and Arusha are classified as having Medium levels of HDI, while the remainder of Tanzania has Low HDI. HDI indicators also show the life expectancy is on the rise, as well as declines in infant mortality.
Rural poverty
Trends in poverty alleviation in Tanzania vary greatly between urban and rural areas in which about 70% of Tanzania's population dwells. Endowments play a large part in distributing economic growth unevenly, with urban households having better access to infrastructure, health services, and education. Migratory trends towards urbanization, which have risen from “5.6% in 1967 to 29.1% in 2012,” are only increasing the inequality. Another main factor of rural poverty in Tanzania is the lack of infrastructure to provide energy to a huge part of the population. Which means the electricity sector poses a significant liability to the government.
With most of Tanzania's population living in rural areas, there is a heavy dependency on rain-fed agriculture. 76% of Tanzanian's rely on agriculture or on access to natural resources for their livelihood. The reliance on agriculture leaves Tanzanian's especially susceptible to economic shocks due to climate change.
Child poverty
Slow economic growth is a contributory factor for child poverty in Tanzania. Based on 2012 estimates, more than a third of households "live below the basic needs poverty line" earning less than $1 a day, while 20% of the total population "live below the food poverty line". However, it is the rural communities of Mainland Tanzania and Zanzibar who are mostly affected. This dis |
https://en.wikipedia.org/wiki/Pseudo-reductive%20group | In mathematics, a pseudo-reductive group over a field k (sometimes called a k-reductive group) is a smooth connected affine algebraic group defined over k whose k-unipotent radical (i.e., largest smooth connected unipotent normal k-subgroup) is trivial. Over perfect fields these are the same as (connected) reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. A pseudo-reductive k-group need not be reductive (since the formation of the k-unipotent radical does not generally commute with non-separable
scalar extension on k, such as scalar extension to an algebraic closure of k). Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic (even over a perfect field of constants).
gives an exposition of Tits' results on pseudo-reductive groups, while builds on Tits' work to develop a general structure theory, including more advanced topics such as construction techniques, root systems and root groups and open cells, classification theorems, and applications to rational conjugacy theorems for smooth connected affine groups over arbitrary fields. The general theory (with applications) as of 2010 is summarized in , and later work in the second edition and in provides further refinements.
Examples of pseudo reductive groups that are not reductive
Suppose that k is a non-perfect field of characteristic 2, and a is an element of k that is not a square. Let G be the group of nonzero elements x + y in k[]. There is a morphism from G to the multiplicative group Gm taking x + y to its norm x2 – ay2, and the kernel is the subgroup of elements of norm 1. The underlying reduced scheme of the geometric kernel is isomorphic to the additive group Ga and is the unipotent radical of the geometric fiber of G, but this reduced subgroup scheme of the geometric fiber is not defined over k (i.e., it does not arise from a closed subscheme of G over the ground field k) and the k-unipotent radical of G is trivial. So G is a pseudo-reductive k-group but is not a reductive k-group. A similar construction works using a primitive nontrivial purely inseparable finite extension of any imperfect field in any positive characteristic, the only difference being that the formula for the norm map is a bit more complicated than in the preceding quadratic examples.
More generally, if K is a non-trivial purely inseparable finite extension of k and G is any non-trivial connected reductive K-group defined then the Weil restriction H=RK/k(G) is a smooth connected affine k-group for which there is a (surjective) homomorphism from HK onto G. The kernel of this K-homomorphism descends the unipotent radical of the geometric fiber of H and is not defined over k (i.e., does not arise from a closed subgroup scheme of H), so RK/k(G) is pseudo-reductive but not reductive. The previous example is the special case using th |
https://en.wikipedia.org/wiki/Equal%20parallelians%20point | In geometry, the equal parallelians point (also called congruent parallelians point) is a special point associated with a plane triangle. It is a triangle center and it is denoted by X(192) in Clark Kimberling's Encyclopedia of Triangle Centers. There is a reference to this point in one of Peter Yff's notebooks, written in 1961.
Definition
The equal parallelians point of triangle is a point in the plane of such that the three line segments through parallel to the sidelines of and having endpoints on these sidelines have equal lengths.
Trilinear coordinates
The trilinear coordinates of the equal parallelians point of triangle are
Construction for the equal parallelians point
Let be the anticomplementary triangle of triangle . Let the internal bisectors of the angles at the vertices of meet the opposite sidelines at respectively. Then the lines concur at the equal parallelians point of .
See also
Congruent isoscelizers point
References
Triangle centers |
https://en.wikipedia.org/wiki/UEFA%20Euro%202004%20statistics | These are the statistics for the Euro 2004 in Portugal.
Goalscorers
Penalty kicks
Not counting penalty shoot-outs, there were eight penalty kicks awarded during the tournament. England's David Beckham (in the match against France) was the only player who failed to convert his penalty.
Scored
Angelos Basinas in the first match against Portugal
Zinedine Zidane in a match against England
Milan Rapaić in a match against France
Zlatan Ibrahimović in a match against Bulgaria
Martin Petrov in a match against Italy
Henrik Larsson in a match against Denmark
Ruud van Nistelrooy in a match against Latvia
Missed
David Beckham in a match against France, saved by Fabien Barthez
Awards
UEFA Team of the Tournament
Golden Boot
Milan Baroš (5 goals)
UEFA Player of the Tournament
Theodoros Zagorakis
Scoring
Total number of goals scored: 77
Average goals per match: 2.48
Top scorer(s): 5 – Milan Baroš
Most goals scored by a team: 10 – ,
Fewest goals scored by a team: 1 – , ,
Most goals conceded by a team: 9 –
Fewest goals conceded by a team: 2 – ,
First goal of the tournament: Giorgos Karagounis vs.
Last goal of the tournament: Angelos Charisteas vs.
Fastest goal in a match: 68 seconds – Dmitri Kirichenko vs.
Latest goal in a match without extra time: 90+4 minutes – Antonio Cassano vs.
Latest goal in a match with extra time: 115 minutes – Frank Lampard vs.
Attendance
Overall attendance: 1,162,762
Average attendance per match: 37,508
Wins and losses
Most wins: 4 – Greece, Czech Republic, Portugal
Fewest wins: 0 – Bulgaria, Croatia, Germany, Latvia, Switzerland
Most losses: 3 – Bulgaria
Fewest losses: 0 – Italy
Discipline
Sanctions against foul play at UEFA Euro 2004 are in the first instance the responsibility of the referee, but when he deems it necessary to give a caution, or dismiss a player, UEFA keeps a record and may enforce a suspension. Referee decisions are generally seen as final. However, UEFA's disciplinary committee may additionally penalise players for offences unpunished by the referee.
Overview
Red cards
A player receiving a red card is automatically suspended for the next match. A longer suspension is possible if the UEFA disciplinary committee judges the offence as warranting it. In keeping with the FIFA Disciplinary Code (FDC) and UEFA Disciplinary Regulations (UDR), UEFA does not allow for appeals of red cards except in the case of mistaken identity. The FDC further stipulates that if a player is sent off during his team's final Euro 2004 match, the suspension carries over to his team's next competitive international(s). For Euro 2004 these were the qualification matches for the 2006 FIFA World Cup.
Any player who was suspended due to a red card that was earned in Euro 2004 qualifying was required to serve the balance of any suspension unserved by the end of qualifying either in the Euro 2004 finals (for any player on a team that qualified, whether he had been selected to the final squad or not) or in World |
https://en.wikipedia.org/wiki/Spherical%20nucleic%20acid | Spherical nucleic acids (SNAs) are nanostructures that consist of a densely packed, highly oriented arrangement of linear nucleic acids in a three-dimensional, spherical geometry. This novel three-dimensional architecture is responsible for many of the SNA's novel chemical, biological, and physical properties that make it useful in biomedicine and materials synthesis. SNAs were first introduced in 1996 by Chad Mirkin’s group at Northwestern University.
Structure and function
The SNA structure typically consists of two components: a nanoparticle core and a nucleic acid shell. The nucleic acid shell is made up of short, synthetic oligonucleotides terminated with a functional group that can be utilized to attach them to the nanoparticle core. The dense loading of nucleic acids on the particle surface results in a characteristic radial orientation around the nanoparticle core, which minimizes repulsion between the negatively charged oligonucleotides.
The first SNA consisted of a gold nanoparticle core with a dense shell of 3’ alkanethiol-terminated DNA strands. Repeated additions of salt counterions were used to reduce the electrostatic repulsion between DNA strands and enable more efficient DNA packing on the nanoparticle surface. Since then, silver, iron oxide, silica, and semiconductor materials have also been used as inorganic cores for SNAs. Other core materials with increased biocompatibility, such FDA-approved PLGA polymer nanoparticles, micelles, liposomes, and proteins have also been used to prepare SNAs. Single-stranded and double-stranded versions of these materials have been created using, for example, DNA, LNA, and RNA.
One- and two-dimensional forms of nucleic acids (e.g., single strands, linear duplexes, and plasmids) (Fig. 1) are important biological machinery for the storage and transmission of genetic information. The specificity of DNA interactions through Watson–Crick base pairing provides the foundation for these functions. Scientists and engineers have been synthesizing and, in certain cases, mass-producing nucleic acids for decades to understand and exploit this elegant chemical recognition motif. The recognition abilities of nucleic acids can be enhanced when arranged in a spherical geometry, which allows for polyvalent interactions to occur. This polyvalency, along with the high density and degree of orientation described above, helps explain why SNAs exhibit different properties than their lower-dimensional constituents (Fig. 2).
Over two decades of research has revealed that the properties of a SNA conjugate are a synergistic combination of those of the core and the shell. The core serves two purposes: 1) it imparts upon the conjugate novel physical and chemical properties (e.g., plasmonic, catalytic, magnetic, luminescent), and 2) it acts as a scaffold for the assembly and orientation of the nucleic acids. The nucleic acid shell imparts chemical and biological recognition abilities that include a greater binding stren |
https://en.wikipedia.org/wiki/Motion%20%28geometry%29 | In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. More generally, the term motion is a synonym for surjective isometry in metric geometry, including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.
Motions can be divided into direct and indirect motions.
Direct, proper or rigid motions are motions like translations and rotations that preserve the orientation of a chiral shape.
Indirect, or improper motions are motions like reflections, glide reflections and Improper rotations that invert the orientation of a chiral shape.
Some geometers define motion in such a way that only direct motions are motions.
In differential geometry
In differential geometry, a diffeomorphism is called a motion if it induces an isometry between the tangent space at a manifold point and the tangent space at the image of that point.
Group of motions
Given a geometry, the set of motions forms a group under composition of mappings. This group of motions is noted for its properties. For example, the Euclidean group is noted for the normal subgroup of translations. In the plane, a direct Euclidean motion is either a translation or a rotation, while in space every direct Euclidean motion may be expressed as a screw displacement according to Chasles' theorem. When the underlying space is a Riemannian manifold, the group of motions is a Lie group. Furthermore, the manifold has constant curvature if and only if, for every pair of points and every isometry, there is a motion taking one point to the other for which the motion induces the isometry.
The idea of a group of motions for special relativity has been advanced as Lorentzian motions. For example, fundamental ideas were laid out for a plane characterized by the quadratic form in American Mathematical Monthly.
The motions of Minkowski space were described by Sergei Novikov in 2006:
The physical principle of constant velocity of light is expressed by the requirement that the change from one inertial frame to another is determined by a motion of Minkowski space, i.e. by a transformation
preserving space-time intervals. This means that
for each pair of points x and y in R1,3.
History
An early appreciation of the role of motion in geometry was given by Alhazen (965 to 1039). His work "Space and its Nature" uses comparisons of the dimensions of a mobile body to quantify the vacuum of imaginary space. He was criticised by Omar Khayyam who pointed that Aristotle had condemned the use of motion in geometry.
In the 19th century Felix Klein became a proponent of group theory as a means to classify geometries according to their "groups of motions". He proposed using symmetry groups in his Erlangen program, a suggestion that was widely adopted. He noted that every Euclidean congruence is an affi |
https://en.wikipedia.org/wiki/Van%20Amringe%20Mathematical%20Prize | The Department of Mathematics at Columbia University has presented a Professor Van Amringe Mathematical Prize each year (since 1910). The prize was established in 1910 by George G. Dewitt, Class of 1867. It was named after John Howard Van Amringe, who taught mathematics at Columbia (holding a professorship from 1865 to 1910), was the first Dean of Columbia College, and was the first president of the American Mathematical Society (between 1888 and 1890).
For many years, the prize was awarded to the freshman or sophomore mathematics student at Columbia College deemed most proficient in the mathematical subjects designated during the year of the award. More recently (since 2003), the prize has been awarded to three Columbia College students majoring in math (a freshman, a sophomore, and a junior) who are deemed proficient in their class in the mathematical subjects designated during the year of the award.
Recipients
External links
Columbia College Prizes
Columbia College Prizes and Fellowships
Past Prize Exams
Notes
Mathematics awards
Student awards
Awards established in 1910
Awards and prizes of Columbia University |
https://en.wikipedia.org/wiki/Eduard%20Zehnder | Eduard J. Zehnder is a Swiss mathematician, considered one of the founders of symplectic topology.
Biography
Zehnder studied mathematics and physics at ETH Zurich from 1960 to 1965, where he also did his Ph.D. in theoretical physics, defending his thesis on the three-body problem in 1971 under the direction of Res Jost. He was a visiting professor at Courant Institute of Mathematical Sciences (invited by Jürgen Moser), visiting member of Institute for Advanced Study in Princeton from 1972 to 1974. He passed his habilitation in mathematics in 1974 at the University of Erlangen-Nuremberg. He had appointments at the University of Bochum from 1976 to 1986; at the University of Aix-la-Chapelle during the academic year 1987–88, where he was director of the Mathematical Institute. From 1988, he had a chair at ETH Zurich, where he became emeritus in 2006. He was plenary speaker at the International Congress of Mathematicians (ICM) in 1986 at the University of California, Berkeley. In 2012 he became a fellow of the American Mathematical Society.
He has made fundamental contributions to the field of dynamical systems. In particular, in one of his groundbreaking works with Charles C. Conley, he established the celebrated Arnold conjecture for fixed points of Hamiltonian diffeomorphisms, and paved the way for the development of the new field of symplectic topology.
He directed the thesis of several mathematicians. His first student was Andreas Floer, who defended his thesis in 1984.
Major publications
Textbooks.
Jürgen Moser and Eduard J. Zehnder. Notes on dynamical systems. Courant Lecture Notes in Mathematics, 12. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2005. viii+256 pp.
Eduard Zehnder. Lectures on dynamical systems. Hamiltonian vector fields and symplectic capacities. EMS Textbooks in Mathematics. European Mathematical Society, Zürich, 2010. x+353 pp.
Helmut Hofer and Eduard Zehnder. Symplectic invariants and Hamiltonian dynamics. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2011. xiv+341 pp.
Research articles.
E. Zehnder. Generalized implicit function theorems with applications to some small divisor problems. I. Comm. Pure Appl. Math. 28 (1975), 91–140.
H. Amann and E. Zehnder. Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 4, 539–603.
C.C. Conley and E. Zehnder. The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnolʹd. Invent. Math. 73 (1983), no. 1, 33–49.
Charles Conley and Eduard Zehnder. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. 37 (1984), no. 2, 207–253.
Dietmar Salamon and Eduard Zehnder. Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Comm. Pure Appl. Math. 45 (1992), no. 10, 1303–1 |
https://en.wikipedia.org/wiki/Ferran%20Sunyer%20i%20Balaguer | Ferran Sunyer i Balaguer (born 1912, Figueres - 27 December 1967, Barcelona) was a Spanish mathematician. He is the namesake of the Ferran Sunyer i Balaguer Prize in mathematics. Ferran Sunyer was born with almost complete physical disabilities and never went to school because his doctor advised that Ferran should not be submitted to such stress. Ferran was home schooled by his mother and developed great interest in mathematics.
It wasn't until 9 December 1967, 18 days prior to his death, that his confirmation as a scientific member was made public by the Divisió de Ciencias Matemá, Médicas y de Naturaleza of the Council.
References
External links
Ferran Sunyer i Balaguer Foundation
Scientists from Catalonia
Spanish scientists
1912 births
1967 deaths
20th-century Spanish mathematicians |
https://en.wikipedia.org/wiki/Gilbert%20tessellation | In applied mathematics, a Gilbert tessellation or random crack network is a mathematical model for the formation of mudcracks, needle-like crystals, and similar structures. It is named after Edgar Gilbert, who studied this model in 1967.
In Gilbert's model, cracks begin to form at a set of points randomly spread throughout the plane according to a Poisson distribution. Then, each crack spreads in two opposite directions along a line through the initiation point, with the slope of the line chosen uniformly at random. The cracks continue spreading at uniform speed until they reach another crack, at which point they stop, forming a T-junction. The result is a tessellation of the plane by irregular convex polygons.
A variant of the model that has also been studied restricts the orientations of the cracks to be axis-parallel, resulting in a random tessellation of the plane by rectangles.
write that, in comparison to alternative models in which cracks may cross each other or in which cracks are formed one at a time rather than simultaneously, "most mudcrack patterns in nature topologically resemble" the Gilbert model.
References
Tessellation |
https://en.wikipedia.org/wiki/MathStar | MathStar, Inc., was an American, fabless semiconductor company based in Oregon. Founded in Minnesota in 1999, the company moved to the Portland metropolitan area where it remained until it completed a reverse merger with Sajan, Inc. in 2010. MathStar never made a profit after raising $137 million over the lifetime of the company, including via several stock offerings while the company was publicly traded on the NASDAQ market. The company's only product was a field programmable object array (FPOA) chip.
History
Bob Johnson and Douglas Pihl started discussing the formation of a company in 1999 to design a new type of digital signal processors (DSP) microprocessor chip, and founded MathStar the next year and began raising funds. The two founded the company in Minneapolis, Minnesota, and raised $18 million for the venture by September 2000 when they had grown to approximately 15 employees. MathStar's new processor was to be based on using a series of algorithms developed by Johnson that were imprinted directly into the processor.
In 2002, the company raised another $15.3 million in capital followed by $6 million in 2003. At one point in 2003 the company planned to merge with Digital MediaCom as MathStar still worked to finish developing its chip. MathStar first started producing its processor in 2003, but technical problems led to additional design changes with hopes to restart production in April 2004 after raising an additional $10 million.
In May 2005, the company announced plans for an initial public offering (IPO) in hopes of securing $28 million for the then Minnetonka-based company. The company then priced the offering at $6 per share in October of that year with the plan of selling 4 million shares on the Nasdaq market under the ticker symbol MATH. MathStar hoped to raise $21 million at that point to pay down debt and fund research. The company then held the IPO in October 2005 and raised $24 million.
MathStar opened an office in Oregon in May 2005 and announced in December that year they would move company headquarters to Hillsboro, Oregon, to have better access to microprocessor talent in the area's Silicon Forest. The company already had 22 employees there at the time, but planned to keep an office in Minnesota as well. At that time the company's market capitalization valued the company at $93 million.
MathStar officially relocated to Hillsboro in March 2006 from Plymouth, Minnesota. At that point MathStar had 35 employees in Hillsboro with plans to hire 15 more. In early 2006 the company's auditors raised concerns over MathStar's ability to continue as a going concern, with the company announcing they would raise more funds to address the issue. MathStar raised an additional $12.6 million by selling stock and warrants in September 2006, and used part of the proceeds to increase staffing to 56 people. At that time, the company also finished its first run of production of its chips using Taiwan Semiconductor Manufacturing Company as th |
https://en.wikipedia.org/wiki/Glennie%27s%20identity | In mathematics, Glennie's identity is an identity used by Charles M. Glennie to establish some s-identities that are valid in special Jordan algebras but not in all Jordan algebras. A Jordan s-identity ("s" for special) is a Jordan polynomial which vanishes in all special Jordan algebras but not in all Jordan algebras. What is now known as Glennie's identity first appeared in his 1963 Yale PhD thesis with Nathan Jacobson as thesis advisor.
Formal definition
Let • denote the product in a special Jordan algebra . For all X, Y, Z in A, define the Jordan triple product
{X,Y,Z} = X•(Y•Z) − Y•(Z•X) + Z•(X•Y) then Glennie's identity G8 holds in the form:
2{ {Z,{X,Y,X},Z}, Y, Z•X} − {Z, {X, {Y, X•Z, Y}, X}, Z} = 2{ X•Z, Y, {X, {Z,Y,Z}, X} } − {X, {Z, {Y,X•Z,Y}, Z}, X}.
References
Non-associative algebras |
https://en.wikipedia.org/wiki/UEFA%20European%20Championship%20records%20and%20statistics | This is a list of records and statistics of the UEFA European Championship.
Ranking of teams by number of appearances
Debut of national teams
Each final tournament has had at least one team appearing for the first time. A total of 35 UEFA members have reached the finals.
Overall team records
The system used in the European Championship up to 1992 was 2 points for a win, and 3 points for a win from 1996 onwards. In this ranking 3 points are awarded for a win, 1 for a draw and 0 for a loss. As per statistical convention in football, matches decided in extra time are counted as wins and losses, while matches decided by penalty shoot-outs are counted as draws. Teams are ranked by total points, then by goal difference, then by goals scored.
Notes
Former countries
Medal table
The Third place playoff has been removed since 1984, meaning the losing semi-finalists are both counted under bronze since then.
Comprehensive team results by tournament
Legend
– Champions
– Runners-up
– Third place
– Fourth place
– Semi-finals
– Quarter-finals
R16 – Round of 16
GS – Group stage
Q – Qualified for upcoming tournament
– Did not qualify
– Disqualified
– Did not enter / Withdrew / Banned
– Hosts
For each tournament, the number of teams in each finals tournament (in brackets) are shown.
Notes
Hosts
From 1960 to 1976 the host was decided between one of the four semi-finalists. Since 1980 the hosts have automatically qualified, except in 2020 when every country had to qualify through qualification. Germany will host the next finals in 2024.
Notes
Results of defending finalists
Active consecutive participations
This is a list of active consecutive participations of national teams in the UEFA European Championships.
Teams not yet qualified for UEFA Euro 2024.
Notes
Droughts
This is a list of droughts associated with the participation of national teams in the UEFA European Championships.
Longest active UEFA European Championship droughts
Does not include teams that have not yet made their first appearance or teams that no longer exist.
Teams not yet qualified for UEFA Euro 2024.
Longest UEFA European Championship droughts overall
Only includes droughts begun after a team's first appearance and until the team ceased to exist.
Notes
Countries that have never qualified
The following teams which are current UEFA members have never qualified for the European Championship. is the only one of these teams which appeared in the FIFA World Cup, although qualified for the 1970 tournament when it was part of AFC.
Legend
– Did not qualify
– Did not enter / Withdrew / Banned
– Co-host of the final tournament
For each tournament, the number of teams in each finals tournament (in brackets) are shown.
Notes
Former countries
East Germany played in eight qualification competitions before the reunification of Germany in 1990.
Notes
General statistics by tournament
Note: Matthias Sammer was the first player to officially win the MVP of the tournam |
https://en.wikipedia.org/wiki/Electronic%20Journal%20of%20Probability | The Electronic Journal of Probability is a peer-reviewed open access scientific journal published by the Institute of Mathematical Statistics and the Bernoulli Society. It covers all aspects of probability theory and the current editor-in-chief is Bénédicte Haas (Université Sorbonne Paris Nord). Electronic Communications in Probability is a sister journal that publishes short papers. The two journals share the same editorial board, but have different editors-in-chiefs, each chosen for a three-year period. According to the Journal Citation Reports, the Electronic Journal of Probability has a 2016 impact factor of 0.904.
Recent editors-in-chief
Bénédicte Hass (2021-2023)
Andreas Kyprianou (2018-2020)
Brian Rider (2015-2017)
Michel Ledoux (2012-2014)
Bálint Tóth (2009-2011)
Andreas Greven (2005-2008)
J Theodore Cox (2002-2004)
Richard Bass (1999-2002)
Krzysztof Burdzy and Gregory Lawler (1995-1999)
References
External links
Probability journals
English-language journals
Academic journals established in 1995
Institute of Mathematical Statistics academic journals
Creative Commons Attribution-licensed journals |
https://en.wikipedia.org/wiki/Electronic%20Communications%20in%20Probability | The Electronic Communications in Probability is a peer-reviewed open access scientific journal published by the Institute of Mathematical Statistics and the Bernoulli Society. The editor-in-chief is Siva Athreya (Indian Statistical Institute). It contains short articles covering probability theory, whereas its sister journal, the Electronic Journal of Probability, publishes full-length papers and shares the same editorial board, but with a different editor-in-chief.
External links
Probability journals
English-language journals
Academic journals established in 1995
Institute of Mathematical Statistics academic journals
Creative Commons Attribution-licensed journals |
https://en.wikipedia.org/wiki/Octant%20%28solid%20geometry%29 | An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. It is similar to the two-dimensional quadrant and the one-dimensional ray.
The generalization of an octant is called orthant.
Naming and numbering
A convention for naming an octant is to give its list of signs, e.g. (+,−,−) or (−,+,−). Octant (+,+,+) is sometimes referred to as the first octant, although similar ordinal name descriptors are not defined for the other seven octants. The advantages of using the (±,±,±) notation are its unambiguousness, and extensibility for higher dimensions.
The following table shows the sign tuples together with likely ways to enumerate them.
A binary enumeration with − as 1 can be easily generalized across dimensions. A binary enumeration with + as 1 defines the same order as balanced ternary.
The Roman enumeration of the quadrants is in Gray code order, so the corresponding Gray code is also shown for the octants.
Verbal descriptions are ambiguous, because they depend on the representation of the coordinate system.
In the two depicted representations of a right-hand coordinate system, the first octant could be called right-back-top or right-top-front respectively.
See also
Orthant
Octant (plane geometry)
Octree
References
Euclidean solid geometry |
https://en.wikipedia.org/wiki/Morley%20centers | In plane geometry, the Morley centers are two special points associated with a triangle. Both of them are triangle centers. One of them called first Morley center (or simply, the Morley center ) is designated as X(356) in Clark Kimberling's Encyclopedia of Triangle Centers, while the other point called second Morley center (or the 1st Morley–Taylor–Marr Center) is designated as X(357). The two points are also related to Morley's trisector theorem which was discovered by Frank Morley in around 1899.
Definitions
Let be the triangle formed by the intersections of the adjacent angle trisectors of triangle . is called the Morley triangle of . Morley's trisector theorem states that the Morley triangle of any triangle is always an equilateral triangle.
First Morley center
Let be the Morley triangle of . The centroid of is called the first Morley center of .
Second Morley center
Let be the Morley triangle of . Then, the lines are concurrent. The point of concurrence is called the second Morley center of triangle .
Trilinear coordinates
First Morley center
The trilinear coordinates of the first Morley center of triangle are
Second Morley center
The trilinear coordinates of the second Morley center are
References
Triangle centers |
https://en.wikipedia.org/wiki/Tychonoff%20cube | In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tychonoff cube is named after Andrey Tychonoff, who first considered the arbitrary product of topological spaces and who proved in the 1930s that the Tychonoff cube is compact. Tychonoff later generalized this to the product of collections of arbitrary compact spaces. This result is now known as Tychonoff's theorem and is considered one of the most important results in general topology.
Definition
Let denote the unit interval . Given a cardinal number , we define a Tychonoff cube of weight as the space with the product topology, i.e. the product where is the cardinality of and, for all , .
The Hilbert cube, , is a special case of a Tychonoff cube.
Properties
The axiom of choice is assumed throughout.
The Tychonoff cube is compact.
Given a cardinal number , the space is embeddable in .
The Tychonoff cube is a universal space for every compact space of weight .
The Tychonoff cube is a universal space for every Tychonoff space of weight .
The character of is .
See also
Tychonoff plank – the topological product of the two ordinal spaces and , where is the first infinite ordinal and the first uncountable ordinal
Long line (topology) – a generalization of the real line from a countable number of line segments [0, 1) laid end-to-end to an uncountable number of such segments.
References
Ryszard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, .
Notes
General topology |
https://en.wikipedia.org/wiki/Angelique%20Kerber%20career%20statistics | This is a list of the main career statistics of German professional tennis player, Angelique Kerber. To date, Kerber has won 14 career singles titles, including three Grand Slam singles titles at the 2016 Australian Open, 2016 US Open and 2018 Wimbledon Championships. She has also won titles on each playing surface (namely, hard, clay and grass). She was also the runner-up at the 2016 Wimbledon Championships and won Silver at the 2016 Rio Olympics. Kerber became the world No. 1 for the first time in her career on 12 September 2016.
Career achievements
In 2010, Kerber made her first WTA final, at the Copa Colsanitas where she finished runner-up to Mariana Duque-Mariño. She also recorded the first top-50 finish of her career that year, at world No. 47. The following year, her breakthrough occurred at the US Open where, as the world No. 92, she soared to her first Grand Slam semifinal where she fell in three sets to the ninth seed and eventual champion Samantha Stosur. After the tournament, she rose into the world's top 40, and eventually finished the year ranked No. 32.
In February 2012, Kerber scored her maiden career title, at the Open GDF Suez before reaching her first Premier Mandatory semifinal in Indian Wells where she lost to eventual champion Victoria Azarenka. Her second career title came shortly later, in April, at the Danish Open where she beat former world No. 1, Caroline Wozniacki. She then achieved her career-best result at the French Open by making the quarterfinals, before posting a runner-up result in Eastbourne and a semifinal showing at Wimbledon where she lost to Agnieszka Radwańska. A few weeks later, Kerber progressed to the quarterfinals of the London Olympics, falling to Azarenka once more. before upsetting Serena Williams en route to her maiden Premier 5 final in Cincinnati. Kerber cracked the world's top 5 before the WTA Championships that year, and subsequently finished the year ranked world No. 5.
Between 2013 and 2014, won one title at the Linz Open in 2013, while making Premier 5 finals in Tokyo (2013) and Doha (2014), and advancing to her second quarterfinal at Wimbledon in 2014, where she lost to eventual runner-up Eugenie Bouchard. 2015 saw Kerber reverse her previous season's 0–4 record in singles finals by winning her first four finals in succession, which includes her maiden titles on clay and grass courts, at the Family Circle Cup and Aegon Classic, respectively. She also won her first title on home soil in 2015, doing so in Stuttgart where she beat Wozniacki in the final. Her other finals in 2015 came in Stanford and Hong Kong, the former being her fourth and final title win of the year.
In 2016, Kerber posted a runner-up finish in Brisbane International, and then lifting her maiden Grand Slm title at the Australian Open, where she overcame Serena Williams in the final in three sets. Kerber ascended to a new career-high ranking of world No. 2 as a result. In April, she defended a title for the first time b |
https://en.wikipedia.org/wiki/ASOR | Asor or ASOR may refer to:
Asor, musical instrument "of ten strings" mentioned in the Bible
Maor Asor, Israeli footballer
American Society of Overseas Research
Applied Statistics and Operations Research
Australian Society for Operations Research |
https://en.wikipedia.org/wiki/Allen%20Knutson | Allen Ivar Knutson is an American mathematician who is a professor of mathematics at Cornell University.
Education
Knutson completed his undergraduate studies at the California Institute of Technology and received a Ph.D. from the Massachusetts Institute of Technology in 1996 under the joint advisorship of Victor Guillemin and Lisa Jeffrey.
Career
He was on the faculty at the University of California, Berkeley before moving to the University of California, San Diego in 2005 and then to Cornell University in 2009. In 2005, he and Terence Tao won the Levi L. Conant Prize of the American Mathematical Society for their paper "Honeycombs and Sums of Hermitian Matrices".
Knutson is also known for his studies of the mathematics of juggling. For five years beginning in 1990, he and fellow Caltech student David Morton held a world record for passing 12 balls.
References
Living people
20th-century American mathematicians
21st-century American mathematicians
Jugglers
California Institute of Technology alumni
Massachusetts Institute of Technology alumni
University of California, Berkeley faculty
University of California, San Diego faculty
Cornell University faculty
Scientists from California
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Cheboksary%20Physics%20and%20Mathematics%20School | Cheboksary Physics and Mathematics School — special boarding school in the Chuvash State University.
Location: 24, str. Urukova, Cheboksary, Chuvash Republic, RSFSR.
Now closed.
Overview
For the school selected the best trained in physics and math students of the Chuvash Republic, graduated seven classes. In school education took place in 8-10 grades.
History
The school was opened in 1968 near the Chuvash University.
In 1981 the school closed. The school was closed due to a change in the school's management, squabbles in the Ministry of education of Chuvashia and among teachers of the city of Cheboksary, as well as due to the dropping number of children in villages and towns, and a General drop in the birth rate in rural areas.
Then, in place of boarding school works Teacher Training College, now it is a Children's art school.
Reunion
Meeting of graduates of the school is held annually the first Saturday in August.
See also
Alikovo middle school
Chuvash State Academic Song and Dance Ensemble
Chuvash State Symphony Capella
References
External links
Чебоксарская физмат школа-интернат №2 с углубленным изучением физики и математики
Middle schools
Schools in Russia
Cheboksary
Boarding schools in Russia |
https://en.wikipedia.org/wiki/Zinbiel%20algebra | In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:
Zinbiel algebras were introduced by . The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.
In any Zinbiel algebra, the symmetrised product
is associative.
A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over V is the tensor algebra with product
where the sum is over all shuffles.
References
Lie algebras
Non-associative algebras
Algebra of random variables |
https://en.wikipedia.org/wiki/UEFA%20Euro%202000%20statistics | These are the statistics for UEFA Euro 2000, held in Belgium and Netherlands.
Goalscorers
Assists
Best goalkeepers
Awards
UEFA Best XI of the Tournament
Golden Boot
Savo Milošević (5 goals)
Patrick Kluivert (5 goals)
UEFA Player of the Tournament
Zinedine Zidane
Man of the Match
Scoring
Overview
Average goals per match: 2.74
Top scorer(s): 5 - Patrick Kluivert, Savo Milošević
Most goals scored by a team: 13 – ,
Fewest goals scored by a team: 0 –
Most goals conceded by a team: 13 –
Fewest goals conceded by a team: 1 –
First goal of the tournament: Bart Goor vs.
Last goal of the tournament: David Trezeguet vs.
Fastest goal in a match: 3 minutes – Paul Scholes vs.
Latest goal in a match without extra time: 90+6 minutes – Alfonso vs.
Latest goal in a match with extra time: 117 minutes – Zinedine Zidane vs.
Wins and losses
Most wins: 5 - France, Italy
Fewest wins: 0 - Denmark, Germany, Slovenia, Sweden
Most losses: 3 - Denmark
Fewest losses: - 0 - Netherlands
Discipline
Sanctions against foul play at UEFA Euro 2000 are in the first instance the responsibility of the referee, but when he deems it necessary to give a caution, or dismiss a player, UEFA keeps a record and may enforce a suspension. Referee decisions are generally seen as final. However, UEFA's disciplinary committee may additionally penalise players for offences unpunished by the referee.
Overview
Red cards
A player receiving a red card is automatically suspended for the next match. A longer suspension is possible if the UEFA disciplinary committee judges the offence as warranting it. In keeping with the FIFA Disciplinary Code (FDC) and UEFA Disciplinary Regulations (UDR), UEFA does not allow for appeals of red cards except in the case of mistaken identity. The FDC further stipulates that if a player is sent off during his team's final Euro 2008 match, the suspension carries over to his team's next competitive international(s). For Euro 2000 these were the qualification matches for the 2002 FIFA World Cup.
Any player who was suspended due to a red card that was earned in Euro 2000 qualifying was required to serve the balance of any suspension unserved by the end of qualifying either in the Euro 2000 finals (for any player on a team that qualified, whether he had been selected to the final squad or not) or in World Cup qualifying (for players on teams that did not qualify).
Yellow cards
Any player receiving a single yellow card during two of the three group stage matches plus the quarter-final match was suspended for the next match. A single yellow card does not carry over to the semi-finals. This means that no player will be suspended for final unless he gets sent off in semi-final or he is serving a longer suspension for an earlier incident. Suspensions due to yellow cards will not carry over to the World Cup qualifiers. Yellow cards and any related suspensions earned in the Euro 2004 qualifiers are neither counted nor enforced in the final tournament.
In |
https://en.wikipedia.org/wiki/2E6%20%28mathematics%29 | {{DISPLAYTITLE:2E6 (mathematics)}}
In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E6, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L) (thinking of the group as a subgroup of E6(L) fixed by an outer involution).
Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by and .
Over finite fields
The group 2E6(q2) has order
q36
(q12 − 1)
(q9 + 1)
(q8 − 1)
(q6 − 1)
(q5 + 1)
(q2 − 1)
/(3,q + 1).
This is similar to the order q36
(q12 − 1)
(q9 − 1)
(q8 − 1)
(q6 − 1)
(q5 − 1)
(q2 − 1)
/(3,q − 1)
of E6(q).
Its Schur multiplier has order (3, q + 1) except for q=2, i. e. 2E6(22), when it has order 12 and is a product of cyclic groups of orders 2,2,3. One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group,
and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.
The outer automorphism group has order (3, q + 1) · f where q2 = pf.
Over the real numbers
Over the real numbers, 2E6 is the quasisplit form of E6, and is one of the five real forms of E6 classified by Élie Cartan. Its maximal compact subgroup is of type F4.
Remarks
References
Robert Wilson: Atlas of Finite Group Representations: Sporadic groups
Finite groups
Lie groups |
https://en.wikipedia.org/wiki/UEFA%20Euro%201992%20statistics | These are the statistics for the Euro 1992 in Sweden.
Goalscorers
Awards
UEFA Team of the Tournament
Scoring
Total number of goals scored: 32
Average goals per match: 2.13
Top scorer: Dennis Bergkamp, Tomas Brolin, Henrik Larsen, Karl-Heinz Riedle (3 goals)
Most goals scored by a team: 7 – Germany
Fewest goals scored by a team: 1 – CIS, England
Most goals conceded by a team: 8 – Germany
Fewest goals conceded by a team: 2 – England
First goal of the tournament: Jan Eriksson against France
Last goal of the tournament: Kim Vilfort against Germany
Fastest goal in a match: 3 minutes and 4 seconds: Frank Rijkaard (for the Netherlands against Germany)
No late goals were scored during a match with extra time.
Latest goal in a match without extra time: 90 minutes: Thomas Häßler (for Germany against the CIS)
No hat-tricks were scored during the tournament.
Most goals scored by one player in a match: 2 – Karl-Heinz Riedle against Sweden, Henrik Larsen against Netherlands
No own goals were scored during the tournament.
Attendance
Overall attendance: 430,111
Average attendance per match: 28,674
Highest attendance: 37,800 – Denmark vs Germany (Final)
Lowest attendance: 14,660 – Scotland vs CIS (Group 2)
Wins, draws and losses
Most wins: 3 – Denmark
Fewest wins: 0 – CIS, England, France
Most losses: 2 – Germany, Scotland
Fewest losses: 1 – Denmark, Netherlands, Sweden
Most draws: 2 - CIS, England, France
Fewest draws: 0 - Scotland
Discipline
Sanctions against foul play at UEFA Euro 1992 are in the first instance the responsibility of the referee, but when he deems it necessary to give a caution, or dismiss a player, UEFA keeps a record and may enforce a suspension. Referee decisions are generally seen as final. However, UEFA's disciplinary committee may additionally penalise players for offences unpunished by the referee.
Overview
Red cards
A player receiving a red card is automatically suspended for the next match. A longer suspension is possible if the UEFA disciplinary committee judges the offence as warranting it. In keeping with the FIFA Disciplinary Code (FDC) and UEFA Disciplinary Regulations (UDR), UEFA does not allow for appeals of red cards except in the case of mistaken identity. The FDC further stipulates that if a player is sent off during his team's final Euro 1996 match, the suspension carries over to his team's next competitive international(s). For Euro 1992 these were the qualification matches for the 1994 FIFA World Cup.
Any player who was suspended due to a red card that was earned in Euro 1992 qualifying was required to serve the balance of any suspension unserved by the end of qualifying either in the Euro 1992 finals (for any player on a team that qualified, whether he had been selected to the final squad or not) or in World Cup qualifying (for players on teams that did not qualify).
Yellow cards
Any player receiving a single yellow card during two of the three group stage matches plus the quarter-final match was suspended for |
https://en.wikipedia.org/wiki/Skills-based%20hiring | Skills-based hiring refers to the practice of employers setting specific skill or competency requirements or targets. Skills and competencies may be cognitive (such as mathematics or reading) or other professional skills, often commonly called "soft" skills (such as "drive for results" or customer service).
Purpose
The intent of skills-based hiring is for applicants to demonstrate, independent of an academic degree the skills required to be successful on the job. It is also a mechanism by which employers may clearly and publicly advertise the expectations for the job – for example indicating they are looking for a particular set of skills at an appropriately communicated level of proficiency. The result of matching the specific skill requirements of a particular job to with the skills an individual has is both more efficient for the employer to identify qualified candidates, as well as provides an alternative, more precise method for candidates to communicate their knowledge, skills, abilities and behaviors to the employer.
Process
In skills-based hiring, the applicant is tested by a third party and presents the scores to the employer as part of the application process. In this sense, skills-based hiring is similar to the U.S. practice of individuals taking third party (e.g., SAT or ACT) tests, and then using those scores as part of a college application. Skills-based hiring is distinct from pre-employment testing, in that it is not the employer who issues the test or controls who sees the scores.
The specific skills needed for a job, and their corresponding levels, are established by the employer through job profiling. Thus, skills-based hiring requires not only that suitable tests be available for applicants, but also that employers have a legally compliant process for defining the levels and suite of skills required for each distinct job title for which they wish to hire.
Advantages
Advocates of skills-based hiring claim it has the following beneficial effects for employers:
Turnover: 25-70% reductions in turnover, often to levels of 4% or less, due to a more exact match of applicant to position.
Training: 25-75% reductions in employee training time, training cost, and/or time-to-full-productivity
Hiring: 70% Reductions in cost-to-hire; 50%-70% reductions in time-to-hire
Productivity: “Significant,” though usually unspecified, increases in total employee productivity
Universality: The same skills-qualification methodology can be used for all jobs within the same company, from entry-level through upper management. This is because skills tests are designed to assess across a far larger range of ability than typical academic, placement, or certification exams.
Ability to locate applicants (by skill scores) for “hard-to-fill” jobs requiring unique skill combinations or jobs for which there is no formal degree program
Shifting of the testing burden from the employer (typical in pre-employment testing) to the applicant (typical |
https://en.wikipedia.org/wiki/3D4 | {{DISPLAYTITLE:3D4}}In mathematics, the Steinberg triality groups of type 3D4 form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D4, depending on a cubic Galois extension of fields K ⊂ L, and using the triality automorphism of the Dynkin diagram D4. Unfortunately the notation for the group is not standardized, as some authors write it as 3D4(K) (thinking of 3D4 as an algebraic group taking values in K) and some as 3D4(L) (thinking of the group as a subgroup of D4(L) fixed by an outer automorphism of order 3). The group 3D4 is very similar to an orthogonal or spin group in dimension 8.
Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced by . They were independently discovered by Jacques Tits in and .
Construction
The simply connected split algebraic group of type D4 has a triality automorphism σ of order 3 coming from an order 3 automorphism of its Dynkin diagram. If L is a field with an automorphism τ of order 3, then this induced an order 3 automorphism τ of the group D4(L). The group 3D4(L) is the subgroup of D4(L) of points fixed by στ. It has three 8-dimensional representations over the field L, permuted by the outer automorphism τ of order 3.
Over finite fields
The group 3D4(q3) has order
q12
(q8 + q4 + 1)
(q6 − 1)
(q2 − 1).
For comparison, the split spin group D4(q) in dimension 8 has order
q12
(q8 − 2q4 + 1)
(q6 − 1)
(q2 − 1)
and the quasisplit spin group 2D4(q2) in dimension 8 has order
q12
(q8 − 1)
(q6 − 1)
(q2 − 1).
The group 3D4(q3) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclic of order f where q3 = pf and p is prime.
This group is also sometimes called 3D4(q), D42(q3), or a twisted Chevalley group.
3D4(23)
The smallest member of this family of groups has several exceptional properties not shared by other members of the family. It has order 211341312 = 212⋅34⋅72⋅13 and outer automorphism group of order 3.
The automorphism group of 3D4(23) is a maximal subgroup of the Thompson sporadic group, and is also a subgroup of the compact Lie group of type F4 of dimension 52. In particular it acts on the 26-dimensional representation of F4. In this representation it fixes a 26-dimensional lattice that is the unique 26-dimensional even lattice of determinant 3 with no norm 2 vectors, studied by . The dual of this lattice has 819 pairs of vectors of norm 8/3, on which 3D4(23) acts as a rank 4 permutation group.
The group 3D4(23) has 9 classes of maximal subgroups, of structure
21+8:L2(8) fixing a point of the rank 4 permutation representation on 819 points.
[211]:(7 × S3)
U3(3):2
S3 × L2(8)
(7 × L2(7)):2
31+2.2S4
72:2A4
32:2A4
13:4
See also
List of finite simple groups
2E6
References
External links
3D4(23) at the atlas of finite groups
3D4(33) at the atlas of finite groups
Finite groups
Lie groups |
https://en.wikipedia.org/wiki/Intrinsic%20flat%20distance | In mathematics, the intrinsic flat distance is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral currents lying in Euclidean space.
Overview
The Sormani–Wenger intrinsic flat (SWIF) distance is a distance between compact oriented Riemannian manifolds of the same dimension. More generally it defines the distance between two integral current spaces, (X,d,T), of the same dimension (see below). This class of spaces and this distance were first announced by mathematicians Sormani and Wenger at the Geometry Festival in 2009 and the detailed development of these notions appeared in the Journal of Differential Geometry in 2011.
The SWIF distance is an intrinsic notion based upon the (extrinsic) flat distance between submanifolds and integral currents in Euclidean space developed by Federer and Fleming. The definition imitates Gromov's definition of the Gromov–Hausdorff distance in that it involves taking an infimum over all distance-preserving maps of the given spaces into all possible ambient spaces Z. Once in a common space Z, the flat distance between the images is taken by viewing the images of the spaces as integral currents in the sense of Ambrosio–Kirchheim.
The rough idea in both intrinsic and extrinsic settings is to view the spaces as the boundary of a third space or region and to find the smallest weighted volume of this third space. In this way, spheres with many splines that contain increasingly small amounts of volume converge "SWIF-ly" to spheres.
Riemannian setting
Given two compact oriented Riemannian manifolds, Mi, possibly with boundary:
dSWIF(M1, M2) = 0
iff there is an orientation preserving isometry from M1 to M2. If Mi converge in the Gromov–Hausdorff sense to a metric space Y then a subsequence of the Mi converge SWIF-ly to an integral current space contained in Y but not necessarily equal to Y. For example, the GH limit of a sequence of spheres with a long thin neck pinch is a pair of spheres with a line segment running between them while the SWIF limit is just the pair of spheres. The GH limit of a sequence of thinner and thinner tori is a circle but the flat limit is the 0 space. In the setting with nonnegative Ricci curvature and a uniform lower bound on volume, the GH and SWIF limits agree. If a sequence of manifolds converge in the Lipschitz sense to a limit Lipschitz manifold then the SWIF limit exists and has the same limit.
Wenger's compactness theorem states that if a sequence of compact Riemannian manifolds, Mj, has a uniform upper bound on diameter, volume and boundary volume, then a subsequence converges SWIF-ly to an integral current space.
Integral current spaces
An m dimensional integral current space (X,d,T) is a metric space (X,d) with an m-dimensional integral current structure T. More precisely, using notions of Ambrosio–Kirchheim, T is an m-dimensional integral current on the me |
https://en.wikipedia.org/wiki/Flat%20convergence | In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find solutions to Plateau's problem. In 2001 the notion of an integral current was extended to arbitrary metric spaces by Ambrosio and Kirchheim.
Integral currents
A k-dimensional current T is a linear functional on the space of smooth, compactly supported k-forms. For example, given a Lipschitz map from a manifold into Euclidean space, , one has an integral current T(ω) defined by integrating the pullback of the differential k-form, ω, over N. Currents have a notion of boundary
(which is the usual boundary when N is a manifold with boundary) and a notion of mass, M(T), (which is the volume of the image of N). An integer rectifiable current is defined as a countable sum of currents formed in this respect. An integral current is an integer rectifiable current whose boundary has finite mass. It is a deep theorem of Federer-Fleming that the boundary is then also an integral current.
Flat norm and flat distance
The flat norm |T| of a k-dimensional integral current T is the infimum of M(A) + M(B), where the infimum is taken over all integral currents A and B such that .
The flat distance between two integral currents is then dF(T,S) = |T − S|.
Compactness theorem
Federer-Fleming proved that if one has a sequence of integral currents whose supports lie in a compact set K with a uniform upper bound on , then a subsequence converges in the flat sense to an integral current.
This theorem was applied to study sequences of submanifolds of fixed boundary whose volume approached the infimum over all volumes of submanifolds with the given boundary. It produced a candidate weak solution to Plateau's problem.
References
Metric geometry
Riemannian geometry
Convergence (mathematics) |
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