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https://en.wikipedia.org/wiki/1991%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 1991. Overview Liberia Petroleum Refining Company Oilers won the championship. References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1992%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 1992 season. Overview It was contested by 16 teams, and Liberia Petroleum Refining Company Oilers won the championship. Group stage Group A Group B Group C Group D Final References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1993%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 1993 season. Overview It was contested by 15 teams, and Mighty Barrolle won the championship. League standings References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1994%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 1994. Overview It was contested by 15 teams, and National Port Authority Anchors won the championship. References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1995%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 1995. Overview Mighty Barrolle won the championship. References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1996%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 1996. Overview Junior Professionals won the championship. References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1997%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 1997. Overview Liberia Petroleum Refining Company Oilers won the championship. References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1998%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 1998. Overview Invincible Eleven won the championship. References Liberia – List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/1999%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 1999. Overview It was contested by 9 teams, and Invincible Eleven won the championship. References Liberia – List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 2000–01 season. Overview It was contested by 18 teams, and Mighty Barrolle won the championship. Group stage Group A Group B Final Football competitions in Liberia Lea Lea
https://en.wikipedia.org/wiki/2002%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 2002 season. Overview It was contested by 16 teams, and Liberia Petroleum Refining Company Oilers won the championship. Group stage Group A Group B Final Top 4 clubs only Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2003%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 2003 season. Overview It was contested by 8 teams, and it was not finished. League standings References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2004%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 2004 season. Overview It was contested by 7 teams, and Mighty Barrolle won the championship. League standings References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2005%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 2005 season. Overview It was contested by 8 teams, and Liberia Petroleum Refining Company Oilers won the championship. League standings References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2006%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 2006 season. Overview It was contested by 9 teams, and Mighty Barrolle won the championship. League standings References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2007%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 2007. Overview It was contested by 12 teams, and Invincible Eleven won the championship. References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2008%20Liberian%20Premier%20League
Statistics of Liberian Premier League in season 2008. Overview It was contested by 16 teams, and Monrovia Black Star Football Club won the championship. References Liberia - List of final tables (RSSSF) Football competitions in Liberia Lea
https://en.wikipedia.org/wiki/2009%20Liberian%20Premier%20League
Statistics of Liberian Premier League for the 2009 season. League standings References Liberia - List of final tables (RSSSF) Football competitions in Liberia Liberian Premier League, 2009
https://en.wikipedia.org/wiki/1962%20Cameroonian%20Premier%20League
Statistics of the 1962 Cameroonian Premier League season. Overview Caïman Douala won the championship. References Cameroon - List of final tables (RSSSF) 1962 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1963%20Cameroonian%20Premier%20League
Statistics of the 1963 Cameroonian Premier League season. Overview Oryx Douala won the championship. References Cameroon - List of final tables (RSSSF) 1963 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1964%20Cameroonian%20Premier%20League
Statistics of the 1964 Cameroonian Premier League season. Overview Oryx Douala won the championship. References Cameroon - List of final tables (RSSSF) 1964 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1965%20Cameroonian%20Premier%20League
Statistics of the 1965 Cameroonian Premier League season. Overview Oryx Douala won the championship. References Cameroon - List of final tables (RSSSF) 1965 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1966%20Cameroonian%20Premier%20League
Statistics of the 1966 Cameroonian Premier League season. Overview Diamant Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1966 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1967%20Cameroonian%20Premier%20League
Statistics of the 1967 Cameroonian Premier League season. Overview Oryx Douala won the championship. References Cameroon - List of final tables (RSSSF) 1967 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1968%20Cameroonian%20Premier%20League
Statistics of the 1968 Cameroonian Premier League season. Overview Caïman Douala won the championship. References Cameroon 1968 - List of final tables (RSSSF) 1968 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1969%20Cameroonian%20Premier%20League
Statistics of the 1969 Cameroonian Premier League season. Overview Union Douala won the championship. References Cameroon - List of final tables (RSSSF) 1969 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1970%20Cameroonian%20Premier%20League
Statistics of the 1970 Cameroonian Premier League season. Overview Canon Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1970 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1971%20Cameroonian%20Premier%20League
Statistics of the 1971 Cameroonian Premier League season. Overview Aigle Nkongsamba won the championship. References Cameroon - List of final tables (RSSSF) 1971 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1972%20Cameroonian%20Premier%20League
Statistics of the 1972 Cameroonian Premier League season. Overview Léopards Douala won the championship. References Cameroon - List of final tables (RSSSF) 1972 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1973%20Cameroonian%20Premier%20League
Statistics of the 1973 Cameroonian Premier League season. Overview Léopards Douala won the championship. References Cameroon - List of final tables (RSSSF) 1973 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1974%20Cameroonian%20Premier%20League
Statistics of the 1974 Cameroonian Premier League season. Overview Canon Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1974 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1975%20Cameroonian%20Premier%20League
Statistics of the 1975 Cameroonian Premier League season. Overview Caïman Douala won the championship. References Cameroon - List of final tables (RSSSF) 1975 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1976%20Cameroonian%20Premier%20League
Statistics of the 1976 Cameroonian Premier League season. Overview Union Douala won the championship. References Cameroon - List of final tables (RSSSF) 1976 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1977%20Cameroonian%20Premier%20League
Statistics of the 1977 Cameroonian Premier League season. Overview Canon Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1977 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1978%20Cameroonian%20Premier%20League
Statistics of the 1978 Cameroonian Premier League season. Overview Union Douala won the championship. References Cameroon - List of final tables (RSSSF) 1978 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1979%20Cameroonian%20Premier%20League
Statistics of the 1979 Cameroonian Premier League season. Overview Canon Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1979 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1980%20Cameroonian%20Premier%20League
Statistics of the 1980 Cameroonian Premier League season. Overview Canon Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1980 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1981%20Cameroonian%20Premier%20League
Statistics of the 1981 Cameroonian Premier League season. Overview Tonnerre Yaoundé won the championship. References Cameroon 1981 - List of final tables (RSSSF) 1981 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1982%20Cameroonian%20Premier%20League
Statistics of the 1982 Cameroonian Premier League season. Overview Canon Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1982 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1984%20Cameroonian%20Premier%20League
Statistics of the 1984 Cameroonian Premier League season. Overview Tonnerre Yaoundé won the championship. References Cameroon 1984 - List of final tables (RSSSF) 1984 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1985%20Cameroonian%20Premier%20League
Statistics of the 1985 Cameroonian Premier League season. Overview Canon Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) 1985 in Cameroonian football Cam Cam Elite One seasons
https://en.wikipedia.org/wiki/1987%20Cameroonian%20Premier%20League
Statistics of the 1987 Cameroonian Premier League season. Overview It was contested by 16 teams, and Tonnerre Yaoundé won the championship. References Cameroon 1987 - List of final tables (RSSSF) Cam Cam 1 Elite One seasons
https://en.wikipedia.org/wiki/1988%20Cameroonian%20Premier%20League
Statistics of the 1988 Cameroonian Premier League season. Overview Tonnerre Yaoundé won the championship. References Cameroon - List of final tables (RSSSF) Cam Cam 1 Elite One seasons
https://en.wikipedia.org/wiki/1992%20Cameroonian%20Premier%20League
Statistics of the 1992 Cameroonian Premier League season. Overview Racing Bafoussam won the championship. References Cameroon 1992 - List of final tables (RSSSF) Cam Cam 1 Elite One seasons
https://en.wikipedia.org/wiki/Algebraic%20Petri%20net
An algebraic Petri net (APN) is an evolution of the well known Petri net in which elements of user defined data types (called algebraic abstract data types (AADT)) replace black tokens. This formalism can be compared to coloured Petri nets (CPN) in many aspects. However, in the APN case, the semantics of the data types is given by an axiomatization enabling proofs and computations on it. Algebraic Petri nets were invented by Jacques Vautherin in 1985 in his PhD thesis and later improved by Wolfang Reisig. The formalism has two aspects : The control part which is handled by a Petri net. The data part which is handled by one or many AADTs. AADT can be themselves split in two parts: The signature (Sort and Ops in the example below) which gives the valid constants and operations of the term algebra. The axiomatization (Axioms in the example below) which gives the semantics of the operations described in the signature part. The following picture describes an algebraic Petri net model of the "dining philosophers problem". There are two AADT in this model, one for the forks algebra, one for the philosophers algebra. Please note that the philosophers AADT uses the fork AADT. Since all philosophers can take their left fork without taking their right fork, executing this model can result in a deadlock. The control part is composed of : Places contain multiset (bags) of tokens. Those tokens are elements of a term algebra built upon the signature of the AADT (in the example, terms that represent either a philosopher or a fork). Each place contains one and only one multiset of terms, the place is typed by its multiset. Arcs can be labeled with multisets of either closed or free terms. Again terms are built from the AADT signature. Transitions are events that can be fired whenever there are enough resources (namely enough tokens in the input places to satisfy all the input arcs) and the guard (firing conditions) of the transition holds. Then the produced tokens are put in the target places of the output arcs. Usually term rewriting is used for the operational semantics in order to check if conditions hold and to compute output terms. In the example below only transition goEat is firable at the beginning. One goEat has been fired, takeL and takeR are also enabled and thus can also be fired. Algebraic Petri nets are the basic formalism of more advanced ones such as CO-OPN. References Further reading External links An Introduction to the Algebraic Specification of Abstract Data Types Specification languages Petri nets
https://en.wikipedia.org/wiki/2009%E2%80%9310%20Scottish%20Football%20League
Statistics of the Scottish Football League in season 2009–10. Scottish First Division Playoffs Scottish Second Division Play-Offs Scottish Third Division See also 2009–10 in Scottish football References Scottish Football League seasons
https://en.wikipedia.org/wiki/Virgil%20Dr%C4%83ghia
Virgil Andrei Drăghia (born 31 July 1990) is a Romanian professional footballer who plays as a goalkeeper for Liga I club Rapid București. Career statistics Honours Rapid București Liga II: 2015–16 Liga III: 2018–19 References External links 1990 births Living people Footballers from Bucharest Romanian men's footballers Men's association football goalkeepers Romania men's under-21 international footballers Liga I players Liga II players FC Rapid București players CS Concordia Chiajna players ASC Daco-Getica București players FC Voluntari players
https://en.wikipedia.org/wiki/1973%E2%80%9374%20UAE%20Football%20League
Statistics of UAE Football League for the 1973–74 season. Overview The season was played as a trial season with two stages. First round being divided into three regional groups and final being played between the three winners. Al-Orouba won the championship after defeating Al Ahli and Oman. First round Group 1 Al Najah Al Nasr (Dubai) Al Orouba (winners) Al Shaab Shoala Al Ajmani Zamalek Group 2 Al Ahli (Dubai) (winners) Al Hilal (Ajman) Al Khaleej Al Nasr (Ajman) Al Shabab Group 3 Al Ahly (Fujairah) Al Hisn Al Rams Kalba Oman (winners) Ras Al Khaimah Final References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons 1973–74 in Emirati football Emir
https://en.wikipedia.org/wiki/1974%E2%80%9375%20UAE%20Football%20League
Statistics of UAE Football League for the 1974–75 season. Overview It was contested by 6 teams, and Al Ahli won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons 1 Emir
https://en.wikipedia.org/wiki/1975%E2%80%9376%20UAE%20Football%20League
Statistics of UAE Football League in season 1975–76. Overview Al-Ahli Football Club - Dubai won the championship. Ali Nawaz Baloch of Pakistan emerged as top scorer with 13 goals. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons 1 Emir
https://en.wikipedia.org/wiki/1976%E2%80%9377%20UAE%20Football%20League
Statistics of UAE Football League in season 1976/77. Overview Al Ain FC won the championship. References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons 1 Emir
https://en.wikipedia.org/wiki/1977%E2%80%9378%20UAE%20Football%20League
Statistics of UAE Football League in season 1977/78. Overview Al-Nasr Sports Club won the championship. References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons 1 Emir
https://en.wikipedia.org/wiki/1978%E2%80%9379%20UAE%20Football%20League
Statistics of UAE Football League in season 1978/79. Overview Al-Nasr Sports Club won the championship. References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons 1 Emir
https://en.wikipedia.org/wiki/1979%E2%80%9380%20UAE%20Football%20League
Statistics of UAE Football League in season 1979/80. Overview Al-Ahli Football Club - Dubai won the championship. References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons 1 Emir
https://en.wikipedia.org/wiki/1980%E2%80%9381%20UAE%20Football%20League
Statistics of UAE Football League in season 1980/81. Overview Al Ain FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1980–81 in Emirati football
https://en.wikipedia.org/wiki/1981%E2%80%9382%20UAE%20Football%20League
Statistics of UAE Football League in season 1981/82. Overview Al Wasl FC won the championship. They were captained by Danish international Peter Engelsen, 17. References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1981–82 in Emirati football
https://en.wikipedia.org/wiki/1982%E2%80%9383%20UAE%20Football%20League
Statistics of UAE Football League in season 1982/83. Overview Al Wasl FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1982–83 in Emirati football
https://en.wikipedia.org/wiki/1983%E2%80%9384%20UAE%20Football%20League
Statistics of UAE Football League for the 1983–84 season. Overview It was contested by 10 teams, and Al Ain FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1983–84 in Emirati football
https://en.wikipedia.org/wiki/1984%E2%80%9385%20UAE%20Football%20League
Statistics of UAE Football League in season 1984/85. Overview Al Wasl FC won the championship. References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1984–85 in Emirati football
https://en.wikipedia.org/wiki/1985%E2%80%9386%20UAE%20Football%20League
Statistics of UAE Football League in season 1985/86. Overview It was contested by 10 teams, and Al-Nasr Sports Club won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1985–86 in Emirati football
https://en.wikipedia.org/wiki/1986%E2%80%9387%20UAE%20Football%20League
Statistics of UAE Football League in season 1986/87. Overview Sharjah FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1986–87 in Emirati football
https://en.wikipedia.org/wiki/1988%E2%80%9389%20UAE%20Football%20League
Statistics of UAE Football League for the 1988–89 season. Overview It was contested by 12 teams, and Sharjah FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1988–89 in Emirati football
https://en.wikipedia.org/wiki/1989%E2%80%9390%20UAE%20Football%20League
Statistics of the UAE Football League for the 1989–90 season. Overview It was contested by 14 teams, and Al-Shabab (United Arab Emirates) won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1989–90 in Emirati football
https://en.wikipedia.org/wiki/1991%E2%80%9392%20UAE%20Football%20League
Statistics of the UAE Football League for the 1991–92 season. Overview It was contested by 16 teams, and Al Wasl FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1991–92 in Emirati football
https://en.wikipedia.org/wiki/1992%E2%80%9393%20UAE%20Football%20League
Statistics of the UAE Football League for the 1992–93 season. Overview It was contested by 12 teams, and Al Ain FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1992–93 in Emirati football
https://en.wikipedia.org/wiki/1993%E2%80%9394%20UAE%20Football%20League
Statistics of UAE Football League for the 1993–94 season. Overview It was contested by 10 teams, and Sharjah FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1993–94 in Emirati football
https://en.wikipedia.org/wiki/1994%E2%80%9395%20UAE%20Football%20League
Statistics of UAE Football League for the 1994–95 season. Overview It was contested by 10 teams, and Al-Shabab (United Arab Emirates) won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1994–95 in Emirati football
https://en.wikipedia.org/wiki/1995%E2%80%9396%20UAE%20Football%20League
Statistics of UAE Football League for the 1995–96 season. Overview It was contested by 10 teams, and Sharjah won the championship. First stage Playoff References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1995–96 in Emirati football
https://en.wikipedia.org/wiki/1996%E2%80%9397%20UAE%20Football%20League
Statistics of the UAE Football League for the 1996–97 UAE Football League. Overview It was contested by 10 teams, and Al Wasl FC won the championship. First stage Playoff References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1996–97 in Emirati football
https://en.wikipedia.org/wiki/1997%E2%80%9398%20UAE%20Football%20League
Statistics of UAE Football League for the 1997–98 season. Overview It was contested by 10 teams, and Al Ain FC won the championship. First stage Playoff References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1997–98 in Emirati football
https://en.wikipedia.org/wiki/1998%E2%80%9399%20UAE%20Football%20League
Statistics of UAE Football League for the 1998–99 season. Overview It was contested by 12 teams, and Al-Wahda FC (Abu Dhabi) won the championship. League standings References United Arab Emirates – List of final tables (RSSSF) UAE Pro League seasons United 1998–99 in Emirati football
https://en.wikipedia.org/wiki/1999%E2%80%932000%20UAE%20Football%20League
Statistics of UAE Football League for the 1999–2000 season. Overview It was contested by 12 teams, and Al Ain FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1999–2000 in Emirati football
https://en.wikipedia.org/wiki/2000%E2%80%9301%20UAE%20Football%20League
Statistics of UAE Football League for the 2000–01 season. Overview It was contested by 12 teams, and Al-Wahda FC (Abu Dhabi) won the championship. League standings Goalscorers 22 goals Mohammed Salem Al-Enazi (Al-Wahda S.C.C.) 14 goals Farhad Majidi (Al Wasl SC) Baba Adamo (Al-Shabab) Rachid Benmahmoud (Al-Ahli) 13 goals Joël Tiéhi (Al Jazira Club) References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/2001%E2%80%9302%20UAE%20Football%20League
Statistics of UAE Football League for the 2001–02 season. Overview It was contested by 12 teams, and Al Ain FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/2002%E2%80%9303%20UAE%20Football%20League
Statistics of UAE Football League for the 2002–03 season. Overview It was contested by 12 teams, and Al Ain FC won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/2003%E2%80%9304%20UAE%20Football%20League
Statistics of UAE Football League for the 2003–04 season. Overview It was contested by 12 teams, and Al Ain FC won the championship. Group stage Group A Group B Playoff References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/2004%E2%80%9305%20UAE%20Football%20League
Statistics of UAE Football League for the 2004–05 season. Overview It was contested by 14 teams, and Al-Wahda FC (Abu Dhabi) won the championship. League standings References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/2005%E2%80%9306%20UAE%20Football%20League
Statistics of UAE Football League for the 2005–06 season. Overview It was contested by 12 teams, and Al-Ahli Football Club - Dubai won the championship. League standings Championship Playoff NB: Ahli played under protest following accusations of bribery against Wahda in their final match at Sharjah (where they were down 3-1 at half-time before winning 3-6; they allegedly bought 5 Sharjah players for 120,000 Euro). References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/2006%E2%80%9307%20UAE%20Football%20League
Statistics of UAE Football League for the 2006–07 season. Overview It was contested by 12 teams, and Al Wasl FC won the championship. League standings Top goalscorers Source: goalzz.com 19 goals Anderson Barbosa (Al Wasl) 17 goals Gregory du Frencie (Dubai Club) Ali Samereh (Al-Shaab) 15 goals Saeed Al Kass (Sharjah) 14 goals Nenad Jestrović (Al-Nasr) 13 goals Reza Enayati (Emirates Club) 12 goals Rasoul Khatibi (Sharja, Emirates Club) 11 goals Javad Kazemian (Al-Shaab) Faisal Khalil (Al-Ahli) 10 goals Prince Tagoe (Al-Shabab) References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/2007%E2%80%9308%20UAE%20Football%20League
Statistics of UAE Football League for the 2007–08 season. Overview It was contested by 12 teams, and Al-Shabab (United Arab Emirates) won the championship. League standings Top goalscorers Source: goalzz.com 16 goals Faisal Khalil (Al-Ahli) Anderson Barbosa (Sharjah) 15 goals Mehrzad Madanchi (Al-Shaab) 14 goals André Dias (Al Wasl) Rasoul Khatibi (Emirates Club) 12 goals Reza Enayati (Emirates Club) 12 goals Antonin Koutouan (Al-Jazira) Ali Samereh (Al-Shaab) 11 goals Ousman Jallow (Al Ain) 10 goals Clederson (Al-Ahli) Renato Abreu (Al-Nasr) References United Arab Emirates - List of final tables (RSSSF) UAE Pro League seasons United 1
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Romania
In the NUTS (Nomenclature of Territorial Units for Statistics) codes of Romania (RO), the three levels are: NUTS codes RO1 Macroregion one (Macroregiunea Unu) RO11 Nord-Vest RO111 Bihor County RO112 Bistrița-Năsăud County RO113 Cluj County RO114 Maramureș County RO115 Satu Mare County RO116 Sălaj County RO12 Centru RO121 Alba County RO122 Brașov County RO123 Covasna County RO124 Harghita County RO125 Mureș County RO126 Sibiu County RO2 Macroregion two (Macroregiunea doi) RO21 Nord-Est RO211 Bacău County RO212 Botoșani County RO213 Iași County RO214 Neamț County RO215 Suceava County RO216 Vaslui County RO22 Sud-Est RO221 Brăila County RO222 Buzău County RO223 Constanța County RO224 Galați County RO225 Tulcea County RO226 Vrancea County RO3 Macroregion three (Macroregiunea trei) RO31 Sud-Muntenia RO311 Argeș County RO312 Călărași County RO313 Dâmbovița County RO314 Giurgiu County RO315 Ialomița County RO316 Prahova County RO317 Teleorman County RO32 București-Ilfov RO321 București RO322 Ilfov County RO4 Macroregion four (Macroregiunea patru) RO41 Sud-Vest Oltenia RO411 Dolj County RO412 Gorj County RO413 Mehedinți County RO414 Olt County RO415 Vâlcea County RO42 Vest RO421 Arad County RO422 Caraș-Severin County RO423 Hunedoara County RO424 Timiș County In the 2003 version, the codes were as follows: RO0 Romania RO01 Nord-Est RO011 Bacău County RO012 Botoșani County RO013 Iași County RO014 Neamț County RO015 Suceava County RO016 Vaslui County RO02 Sud-Est RO021 Brăila County RO022 Buzău County RO023 Constanța County RO024 Galați County RO025 Tulcea County RO026 Vrancea County RO03 Sud-Muntenia RO031 Argeș County RO032 Călărași County RO033 Dâmbovița County RO034 Giurgiu County RO035 Ialomița County RO036 Prahova County RO037 Teleorman County RO04 Sud-Vest Oltenia RO041 Dolj County RO042 Gorj County RO043 Mehedinți County RO044 Olt County RO045 Vâlcea County RO05 Vest RO051 Arad County RO052 Caraș-Severin County RO053 Hunedoara County RO054 Timiș County RO06 Nord-Vest RO061 Bihor County RO062 Bistrița-Năsăud County RO063 Cluj County RO064 Maramureș County RO065 Satu Mare County RO066 Sălaj County RO07 Centru RO071 Alba County RO072 Brașov County RO073 Covasna County RO074 Harghita County RO075 Mureș County RO076 Sibiu County RO08 București-Ilfov RO081 București RO082 Ilfov County Local administrative units Below the NUTS levels, the two LAU (Local Administrative Units) levels are: The LAU codes of Romania can be downloaded here: '' See also List of Romanian regions by Human Development Index Subdivisions of Romania ISO 3166-2 codes of Romania FIPS region codes of Romania Sources Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe Overview map of EU Countries - NUTS level 1 ROMÂNIA - NUTS level 2 ROMÂNIA - NUTS level 3 Correspondence between the NUTS levels and the national administrative units List of current NUTS codes Download current NUTS codes (ODS format) Cou
https://en.wikipedia.org/wiki/Sergei%20Berdnikov
Sergei Pavlovich Berdnikov (; born January 5, 1971, in Bratsk, USSR), is a Russian former ice hockey player. He played for Avangard Omsk from 1989–1994 and 1996-1998. Career statistics References External links 1971 births Avangard Omsk players Charlotte Checkers (1993–2010) players HC Lada Togliatti players Metallurg Novokuznetsk players HC MVD players Severstal Cherepovets players HC Sibir Novosibirsk players Living people Lokomotiv Yaroslavl players Providence Bruins players Russian ice hockey coaches Russian ice hockey left wingers Soviet ice hockey left wingers Portland Rage players Sacramento River Rats players People from Angarsk Sportspeople from Irkutsk Oblast
https://en.wikipedia.org/wiki/Perspectivity
In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. Graphics The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his De Pictura (1435). In English, Brook Taylor presented his Linear Perspective in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry". In a second book, New Principles of Linear Perspective (1719), Taylor wrote When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the Projection of the other Figure. The Lines producing that Projection, taken all together, are called the System of Rays. And when those Rays all pass thro’ one and same Point, they are called the Cone of Rays. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the Optic Cone Projective geometry In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil. Given two lines and in a projective plane and a point P of that plane on neither line, the bijective mapping between the points of the range of and the range of determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P). A special symbol has been used to show that points X and Y are related by a perspectivity; In this notation, to show that the center of perspectivity is P, write The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range. Projectivity The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a projectivity (projective transformation, projective collineation and homography are synonyms). There are several results concerning projectivities and perspectivities which hold in any pappian projective plane: Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities. Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities. Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity. Higher-dimensional perspectivities The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will also be called perspectivities. Let Sm and Tm be two di
https://en.wikipedia.org/wiki/Aleksandar%20Markoski
Aleksandar Markoski (Serbian Cyrillic: Александар Маркоски; born 17 September 1975) is a Serbian former football player. Career statistics References External links Living people 1975 births Footballers from Zrenjanin Serbian men's footballers FK Bežanija players FK Hajduk Kula players FK Banat Zrenjanin players FK Proleter Zrenjanin players Serbian SuperLiga players Men's association football midfielders
https://en.wikipedia.org/wiki/Elementary%20definition
In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic, and in particular without reference to set theory or using extensions such as plural quantification. Elementary definitions are of particular interest because they admit a complete proof apparatus while still being expressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms such as Zermelo–Fraenkel set theory (ZFC)). Saying that a definition is elementary is a weaker condition than saying it is algebraic. Related Elementary sentence Elementary theory References Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4. Mathematical logic
https://en.wikipedia.org/wiki/Algebraic%20definition
In mathematical logic, an algebraic definition is one that can be given using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Saying that a definition is algebraic is a stronger condition than saying it is elementary. Related Algebraic sentence Algebraic theory Algebraic expression Algebraic equation References Mathematical logic
https://en.wikipedia.org/wiki/Multiple-scale%20analysis
In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove (unwanted) secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions. Mathematics research from about the 1980s proposes that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see center manifold and slow manifold). Example: undamped Duffing equation Differential equation and energy conservation As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation: which is a second-order ordinary differential equation describing a nonlinear oscillator. A solution y(t) is sought for small values of the (positive) nonlinearity parameter 0 < ε ≪ 1. The undamped Duffing equation is known to be a Hamiltonian system: with q = y(t) and p = dy/dt. Consequently, the Hamiltonian H(p, q) is a conserved quantity, a constant, equal to H = ½ + ¼ ε for the given initial conditions. This implies that both y and dy/dt have to be bounded: Straightforward perturbation-series solution A regular perturbation-series approach to the problem proceeds by writing and substituting this into the undamped Duffing equation. Matching powers of gives the system of equations Solving these subject to the initial conditions yields Note that the last term between the square braces is secular: it grows without bound for large |t|. In particular, for this term is O(1) and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution. Method of multiple scales To construct a solution that is valid beyond , the method of multiple-scale analysis is used. Introduce the slow scale t1: and assume the solution y(t) is a perturbation-series solution dependent both on t and t1, treated as: So: using dt1/dt = ε. Similarly: Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become: Solution The zeroth-order problem has the general solution: with A(t1) a complex-valued amplitude to the zeroth-order solution Y0(t, t1) and i2 = −1. Now, in the first-order problem the forcing in the right hand side of the differential equation is where c.c. denotes the complex conjugate of the preceding terms. The occurrence of secular terms can be prevented by imposing on th
https://en.wikipedia.org/wiki/2009%20Urawa%20Red%20Diamonds%20season
2009 Urawa Red Diamonds season Competitions Domestic results J.League 1 Emperor's Cup J.League Cup Player statistics Other pages J.League official site Urawa Red Diamonds Urawa Red Diamonds seasons
https://en.wikipedia.org/wiki/Cover%20%28algebra%29
In abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup. This should not be confused with the concept of a cover in topology. When some object X is said to cover another object Y, the cover is given by some surjective and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In order to be interesting, the cover is usually endowed with additional properties, which are highly dependent on the context. Examples A classic result in semigroup theory due to D. B. McAlister states that every inverse semigroup has an E-unitary cover; besides being surjective, the homomorphism in this case is also idempotent separating, meaning that in its kernel an idempotent and non-idempotent never belong to the same equivalence class.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an F-inverse cover. McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover. Examples from other areas of algebra include the Frattini cover of a profinite group and the universal cover of a Lie group. Modules If F is some family of modules over some ring R, then an F-cover of a module M is a homomorphism X→M with the following properties: X is in the family F X→M is surjective Any surjective map from a module in the family F to M factors through X Any endomorphism of X commuting with the map to M is an automorphism. In general an F-cover of M need not exist, but if it does exist then it is unique up to (non-unique) isomorphism. Examples include: Projective covers (always exist over perfect rings) flat covers (always exist) torsion-free covers (always exist over integral domains) injective covers See also Embedding Notes References Abstract algebra
https://en.wikipedia.org/wiki/Canoe%20Creek%20Indian%20Reserve%20No.%201
Canoe Creek Indian Reserve No. 1, referred to by Statistics Canada as Canoe Creek 1, is an Indian reserve of the Canoe Creek Band/Dog Creek Indian Band of the Secwepemc people, located five miles northeast of the confluence of Canoe Creek with the Fraser River in British Columbia, Canada, in that province's Cariboo district. The reserve is 37.2 ha. in size. See also List of Indian Reserves in British Columbia Canoe Creek (disambiguation) References Indian reserves in British Columbia Secwepemc Geography of the Cariboo
https://en.wikipedia.org/wiki/Infrastructure%20%28number%20theory%29
In mathematics, an infrastructure is a group-like structure appearing in global fields. Historic development In 1972, D. Shanks first discovered the infrastructure of a real quadratic number field and applied his baby-step giant-step algorithm to compute the regulator of such a field in binary operations (for every ), where is the discriminant of the quadratic field; previous methods required binary operations. Ten years later, H. W. Lenstra published a mathematical framework describing the infrastructure of a real quadratic number field in terms of "circular groups". It was also described by R. Schoof and H. C. Williams, and later extended by H. C. Williams, G. W. Dueck and B. K. Schmid to certain cubic number fields of unit rank one and by J. Buchmann and H. C. Williams to all number fields of unit rank one. In his habilitation thesis, J. Buchmann presented a baby-step giant-step algorithm to compute the regulator of a number field of arbitrary unit rank. The first description of infrastructures in number fields of arbitrary unit rank was given by R. Schoof using Arakelov divisors in 2008. The infrastructure was also described for other global fields, namely for algebraic function fields over finite fields. This was done first by A. Stein and H. G. Zimmer in the case of real hyperelliptic function fields. It was extended to certain cubic function fields of unit rank one by R. Scheidler and A. Stein. In 1999, S. Paulus and H.-G. Rück related the infrastructure of a real quadratic function field to the divisor class group. This connection can be generalized to arbitrary function fields and, combining with R. Schoof's results, to all global fields. One-dimensional case Abstract definition A one-dimensional (abstract) infrastructure consists of a real number , a finite set together with an injective map . The map is often called the distance map. By interpreting as a circle of circumference and by identifying with , one can see a one-dimensional infrastructure as a circle with a finite set of points on it. Baby steps A baby step is a unary operation on a one-dimensional infrastructure . Visualizing the infrastructure as a circle, a baby step assigns each point of the next one. Formally, one can define this by assigning to the real number ; then, one can define . Giant steps and reduction maps Observing that is naturally an abelian group, one can consider the sum for . In general, this is not an element of . But instead, one can take an element of which lies nearby. To formalize this concept, assume that there is a map ; then, one can define to obtain a binary operation , called the giant step operation. Note that this operation is in general not associative. The main difficulty is how to choose the map . Assuming that one wants to have the condition , a range of possibilities remain. One possible choice is given as follows: for , define ; then one can define . This choice, seeming somewhat arbitrary, appears in a n
https://en.wikipedia.org/wiki/Differential%20algebraic%20geometry
Differential algebraic geometry is an area of differential algebra that adapts concepts and methods from algebraic geometry and applies them to systems of differential equations, especially algebraic differential equations. Another way of generalizing ideas from algebraic geometry is diffiety theory. References Differential algebraic geometry (three parts in one pdf), part of the Kolchin Seminar in Differential Algebra , Henri Gillet (2000), Differential algebra - A Scheme Theory Approach, Differential algebra and related topics: proceedings of the International Workshop, Newark Campus of Rutgers, The State University of New Jersey, 2-3 November 2000, Editors Li Guo, William F. Keigher, World Scientific, Differential algebra
https://en.wikipedia.org/wiki/Algebraic%20differential%20geometry
Algebraic differential geometry may refer to: Differential algebraic geometry Differential geometry of algebraic manifolds Manifolds equipped with a derivation
https://en.wikipedia.org/wiki/Prosolvable%20group
In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group. Examples Let p be a prime, and denote the field of p-adic numbers, as usual, by . Then the Galois group , where denotes the algebraic closure of , is prosolvable. This follows from the fact that, for any finite Galois extension of , the Galois group can be written as semidirect product , with cyclic of order for some , cyclic of order dividing , and of -power order. Therefore, is solvable. See also Galois theory References Mathematical structures Algebra Number theory Topology Properties of groups Topological groups
https://en.wikipedia.org/wiki/Daniela%20Hantuchov%C3%A1%20career%20statistics
This is a list of the main career statistics of professional Slovak tennis player Daniela Hantuchová. Performance timelines Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup and Olympic Games are included in win–loss records. Singles Doubles Mixed doubles Grand Slam finals Doubles: 3 (3 runner-ups) Mixed doubles: 5 (4 titles, 1 runner-up) By winning the 2005 US Open title, Hantuchová completed the mixed doubles Career Grand Slam. She became only the fifth female player in history to achieve this. Other significant finals WTA Premier Mandatory & 5 finals Singles: 3 (2 titles, 1 runner-up) Doubles: 7 (2 titles, 5 runner-ups) WTA career finals Singles: 16 (7 titles, 9 runner-ups) Doubles: 21 (9 titles, 12 runner-ups) ITF titles Singles: 3 titles Doubles: 1 title Fed Cup results Singles Doubles WTA Tour career earnings Hantuchová earned more than 10 million dollars during her career. Record against other players Record against top 10 players Hantuchová's record against players who have been ranked in the top 10. Active players are in boldface. No. 1 wins Top 10 wins Notes References Tennis career statistics
https://en.wikipedia.org/wiki/Georgy%20Gabulov
Georgy Borisovich Gabulov (, ; born 4 September 1988) is a Russian former footballer. He played as an attacking midfielder. Club career Career statistics Notes Personal life He is a younger brother of Vladimir Gabulov. Honours Lokomotiv 2007: Russian Cup Rostov 2013–14: Russian Cup External links 1988 births People from Mozdoksky District Living people Russian men's footballers Russia men's under-21 international footballers Russia men's B international footballers Men's association football midfielders FC Lokomotiv Moscow players FC Spartak Vladikavkaz players FC Anzhi Makhachkala players FC Rostov players PFC Krylia Sovetov Samara players FC SKA-Khabarovsk players FC Metalurgi Rustavi players Russian expatriate men's footballers Expatriate men's footballers in Georgia (country) Russian Premier League players Ossetian people Ossetian footballers FC Orenburg players Sportspeople from North Ossetia–Alania
https://en.wikipedia.org/wiki/Robert%20Morgan%20%28poet%29
Robert Morgan (born 1944) is an American poet, short story writer, and novelist. Life He studied at North Carolina State University as an engineering and mathematics major, transferred to the University of North Carolina at Chapel Hill as an English major, graduating in 1965, and completed an MFA degree at the University of North Carolina Greensboro in 1968. He has taught at Cornell University since 1971, and was appointed Professor of English in 1984. Awards Academy Award in Literature by the American Academy of Arts and Letters 2008 Thomas Wolfe Prize 2012 SIBA Book Award (nonfiction) for Lions of the West 2013 William "Singing Billy" Walker Award for Lifetime Achievement in Southern Letters Bibliography Poetry Collections Zirconia Poems. Northwood Narrows, New Hampshire: Lillabulero Press, 1969. Red Owl. New York: Norton, 1972. Trunk & Thicket. Fort Collins, Colorado: L’Epervier Press, 1978. Groundwork. Frankfort, Kentucky: Gnomon Press, 1979. Bronze Age. Emory, Virginia: Iron Mountain Press, 1981. At the Edge of the Orchard Country. Middletown, Connecticut: Wesleyan University Press, 1987. Green River: New and Selected Poems. Hanover, New Hampshire: University Press of New England, 1991. Wild Peavines: New Poems. Frankfort, Kentucky: Gnomon Press, 1996. List of poems "OPTION", The Atlantic, October 1997 "Wind From a Waterfall", The Atlantic, September 1999 "Girdling", The Atlantic, December 1997 "Holy Cussing", Southern Poetry Review, Vol. 43, No. 1, 2004 Short fiction Collections The Mountains Won’t Remember Us and Other Stories. Atlanta, Georgia: Peachtree Publishers, 1992. Stories Novels Non-fiction Good Measure: Essays, Interviews and Notes on Poetry. Baton Rouge: Louisiana State University Press, 1993. Boone: A Biography, Algonquin Books of Chapel Hill, 2007, Lions of the West - Heroes and Villains of the Westward Expansion, Shannon Ravenel Books, 2011, References External links "Author's website" "Nostalgia May Not Be the Right Word," an Interview with Robert Morgan, Southern Spaces, December 11, 2013. 1944 births Living people 20th-century American biographers 20th-century American essayists 20th-century American male writers 20th-century American novelists 20th-century American poets 20th-century American short story writers 21st-century American essayists 21st-century American male writers 21st-century American novelists 21st-century American poets 21st-century American short story writers American male essayists American male novelists American male poets American male short story writers The Atlantic (magazine) people Cornell University faculty North Carolina State University people Novelists from New York (state) University of North Carolina at Chapel Hill alumni University of North Carolina at Greensboro alumni American male biographers
https://en.wikipedia.org/wiki/Law%20of%20total%20covariance
In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then The nomenclature in this article's title parallels the phrase law of total variance. Some writers on probability call this the "conditional covariance formula" or use other names. Note: The conditional expected values E( X | Z ) and E( Y | Z ) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z ) is g(Z). Similar comments apply to the conditional covariance. Proof The law of total covariance can be proved using the law of total expectation: First, from a simple standard identity on covariances. Then we apply the law of total expectation by conditioning on the random variable Z: Now we rewrite the term inside the first expectation using the definition of covariance: Since expectation of a sum is the sum of expectations, we can regroup the terms: Finally, we recognize the final two terms as the covariance of the conditional expectations E[X | Z] and E[Y | Z]: See also Law of total variance, a special case corresponding to X = Y. Law of total cumulance, of this the law of total covariance is a special case. Notes and references Algebra of random variables Covariance and correlation Articles containing proofs Theory of probability distributions Theorems in statistics Statistical laws
https://en.wikipedia.org/wiki/1963%E2%80%9364%20Q-League
Statistics of Qatar Stars League for the 1963–64 season. Overview Al-Maref won the championship. References Qatar - List of final tables (RSSSF) Qatar Stars League seasons Qatar football
https://en.wikipedia.org/wiki/1964%E2%80%9365%20Qatar%20Stars%20League
Statistics of Qatar Stars League for the 1964–65 season. Overview Al-Maref won the championship. References Qatar - List of final tables (RSSSF) Qatar Stars League seasons Qatar football