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https://en.wikipedia.org/wiki/1912%E2%80%9313%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1912–13.
Overview
It was contested by 6 teams, and México FC won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1912-13
Mex
1912–13 in Mexican football |
https://en.wikipedia.org/wiki/1913%E2%80%9314%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1913–14.
Overview
It was contested by 5 teams, and Club España won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1913-14
Mex
1913–14 in Mexican football |
https://en.wikipedia.org/wiki/1914%E2%80%9315%20Primera%20Fuerza%20season | Statistics of the Primera Fuerza for the 1914–15 season.
Overview
It was contested by 6 teams, and Club España won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1914-15
Mex
1914–15 in Mexican football |
https://en.wikipedia.org/wiki/1915%E2%80%9316%20Primera%20Fuerza%20season | Statistics of the Primera Fuerza amateur football league in Mexico for the season 1915/1916.
Overview
It was contested by 6 teams, and Club España won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1915-16
Mex
1915–16 in Mexican football |
https://en.wikipedia.org/wiki/1916%E2%80%9317%20Primera%20Fuerza%20season | Statistics of the Primera Fuerza for the 1916–17 season.
Overview
It was contested by 6 teams, and Club España won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1916-17
Mex
1916–17 in Mexican football |
https://en.wikipedia.org/wiki/1917%E2%80%9318%20Primera%20Fuerza%20season | Statistics of the Primera Fuerza for the 1917–18 season.
Overview
It was contested by 6 teams, and Pachuca won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1917-18
Mex
1917–18 in Mexican football |
https://en.wikipedia.org/wiki/1918%E2%80%9319%20Primera%20Fuerza%20season | Statistics of the Primera Fuerza for the 1918–19 season.
Overview
It was contested by 7 teams, and Club España won the championship.
Iberia de Veracruz from "Liga Veracruzana" changes its name to España Veracruz and joined the league
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1918-19
Mex
1918–19 in Mexican football |
https://en.wikipedia.org/wiki/1919%E2%80%9320%20Primera%20Fuerza%20season | Statistics of the Primera Fuerza for the 1919–20 season.
Liga Nacional
Overview
It was contested by 9 teams, and Club España won the championship.
Centro Union switched their name to America.
League Standings
Top goalscorers
Players sorted first by goals scored, then by last name.
Liga Mexicana
Overview
Pachuca won the championship.
References
Mexico - List of final tables (RSSSF)
Primera Fuerza seasons
Mex
1919–20 in Mexican football |
https://en.wikipedia.org/wiki/1920%E2%80%9321%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1920–21.
Club España decided to form their own league called "Liga Nacional" playing alongside the official one (Liga Mexicana)
Liga Nacional (Unofficial)
Overview
It was contested by 5 teams, and Club España won the championship. Tournament played at a single leg.
Liga Mexicana (Official)
Overview
It was contested by 15 teams, and Germania won the championship.
References
Mexico - List of final tables (RSSSF)
Primera Fuerza seasons
Mex
1920–21 in Mexican football |
https://en.wikipedia.org/wiki/1921%E2%80%9322%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1921–22.
Overview
In 1921 the Campeonato del Centenario was played from August 7, 1921 - September 25, 1921. Club España became champion as Asturias walked off in the final. For 1922 no league was held, but 2 editions of the Copa Covadonga were played. Copa Covadonga II started in October 1922. Copa Covadonga was contested by 7 teams, and Club España won both championship. In Copa Covadonga II Club España beat Asturias after they had to abandon the game since 3 of their players were injured.
1921
Campeonato del Centenario
Matches
First round
Bye: Pachuca
Second round
Playoff First Round
Second round
Playoff
Bracket
Semifinals
Final
Torneo del Centenario
Once the "Torneo del Centenario" was finished, both leagues (Mexicana and Nacional), continue playing, apparently neither ended; however, there are reports that both conclude, if so, neither of the reported champions ( Asturias and RC España) were recognized. In the "Liga Mexicana" played only four teams (Deportivo Internacional and Morelos withdrew and joined Tranvías) and in the Liga Nacional joined Aurrera.
Liga Mexicana (official) participant teams:
Asturias
Germania FV
México FC
Tranvías
Liga Nacional (unofficial) participant teams:
RC España
América
Luz y Fuerza
L'Amicale Française
Reforma AC
Aurrerá
1922
Copa Covadonga 1922 I
Final
Copa Covadonga 1922 II
Final
References
Mexico - List of final tables (RSSSF)
Primera Fuerza seasons
Mex
1921–22 in Mexican football |
https://en.wikipedia.org/wiki/1922%E2%80%9323%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1922–23.
Overview
It was contested by 8 teams, and Asturias won the championship.
League standings
(All clubs from México City)
References
Mexico - List of final tables (RSSSF)
1922-23
Mex
1922–23 in Mexican football |
https://en.wikipedia.org/wiki/1923%E2%80%9324%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1923–24.
Overview
It was contested by 9 teams, and Club España won the championship.
League standings
(All clubs from México City)
Playoff
References
Mexico - List of final tables (RSSSF)
1923-24
Mex
1923–24 in Mexican football |
https://en.wikipedia.org/wiki/1924%E2%80%9325%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1924–25.
Overview
It was contested by 7 teams, and América won the championship.
League standings
(All clubs from México City)
Two points and a score of 1-0 awarded to teams who won by default vs. Club España, Aurrerá and Asturias which withdraw on February 29, 1925.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1924-25
Mex
1924–25 in Mexican football |
https://en.wikipedia.org/wiki/1925%E2%80%9326%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1925–26.
Overview
It was contested by 7 teams, and América won the championship.
League standings
This table includes four goals of a tie between América and Club España; this match was awarded to América because the players of Club España left the field during the game. The goals counted by determination of the league.
Championship Playoff
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1925-26
Mex
1925–26 in Mexican football |
https://en.wikipedia.org/wiki/1926%E2%80%9327%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1926–27.
Overview
It was contested by 7 teams, and América won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1926-27
Mex
1926–27 in Mexican football |
https://en.wikipedia.org/wiki/1927%E2%80%9328%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1927–28.
Overview
It was contested by 8 teams, and América won the championship.
League standings
Club España left the league and lost their last 4 matches by default vs. América, México FC, Atlante and Necaxa (0-1 each game)
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1927-28
Mex
1927–28 in Mexican football |
https://en.wikipedia.org/wiki/1928%E2%80%9329%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1928–29.
Overview
It was contested by 9 teams, and Marte won the championship.
League standings
Moves
Aurrerá retired.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1928-29
Mex
1928–29 in Mexican football |
https://en.wikipedia.org/wiki/1929%E2%80%9330%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1929–30.
Overview
It was contested by 8 teams, and Club España won the championship.
League standings
Moves
After this season México FC retired and Leonés joined.
The league was suspended for the 1930-31 season due to teams building new Parks to play in.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1929-30
1929–30 in Mexican football
1929–30 domestic association football leagues |
https://en.wikipedia.org/wiki/1931%E2%80%9332%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1931–32.
Overview
It was contested by 8 teams, and Atlante won the championship.
League standings
Championship playoff
The championship playoff was played between the top two teams.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1931-32
Mex
1931–32 in Mexican football |
https://en.wikipedia.org/wiki/1932%E2%80%9333%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1932–33.
Overview
It was contested by 10 teams, and Necaxa won the championship.
Group 1
Group 2
Relegation playoff
The relegation playoff was played between the last team of Group A and first team of Group B.
Moves
Before the start of the next season all the teams of Group B were dissolved (except México FC).
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
Primera Fuerza seasons
Mex
1932–33 in Mexican football |
https://en.wikipedia.org/wiki/1933%E2%80%9334%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1933–34.
Overview
It was contested by 6 teams, and Club España won the championship.
League standings
Playoff
Championship Playoffs Results
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1933-34
Mex
1933–34 in Mexican football |
https://en.wikipedia.org/wiki/1934%E2%80%9335%20Primera%20Fuerza%20season | Statistics of México Primera División Primera Fuerza in season 1934–35.
Overview
Necaxa won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
Primera Fuerza seasons
Mex
1934–35 in Mexican football |
https://en.wikipedia.org/wiki/1935%E2%80%9336%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1935–36.
Overview
It was contested by 5 teams, and Club España won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1935-36
Mex
1935–36 in Mexican football |
https://en.wikipedia.org/wiki/1936%E2%80%9337%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1936–37.
Overview
It was contested by 5 teams, and Necaxa won the championship.
League standings
Moves
Marte joined for the 1937-38 season.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1936-37
Mex
1936–37 in Mexican football |
https://en.wikipedia.org/wiki/1937%E2%80%9338%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1937–38.
Overview
It was contested by 6 teams, and Necaxa won the championship.
League standings
Moves
After this season Euzkadi joined for the 1938-39 season.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1937-38
Mex
1937–38 in Mexican football |
https://en.wikipedia.org/wiki/1938%E2%80%9339%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1938–39:
Overview
It was contested by 7 teams, and Asturias won the championship.
League standings
Moves
After this season Euzkadi was broken up, with most of its players joining other teams in the league, except two who joined teams in Argentina.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
1938-39
Mex
1938–39 in Mexican football |
https://en.wikipedia.org/wiki/1939%E2%80%9340%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1939–40.
Overview
It was contested by 6 teams, and Club España won the championship.
League standings
Moves
After this season Moctezuma and Selección Jalisco joined.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1939-40
1939–40 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1940%E2%80%9341%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1940–41.
Overview
It was contested by 8 teams, and Atlante won the championship.
League standings
Table takes award of 3 matches against Asturias into account.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1940-41
1940–41 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1941%E2%80%9342%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1941–42.
Overview
It was contested by 8 teams, and Club España won the championship.
League standings
Championship playoff
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1941-42
1941–42 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1942%E2%80%9343%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1942–43.
Overview
It was contested by 8 teams, and Marte won the championship.
League standings
Results
Information
This is the last Liga Mexicana de Football Amateur tournament held in México, having Club Deportivo Marte as the last historic champion
After this season 10 teams began a professional football league system called Liga Mayor de Fútbol de México, today known as Liga MX.
The real great team, with 14 championships, in Liga Mexicana de Football Amateur era is Real Club España.
The championships won at the Liga Mexicana de Football Amateur are not accumulative for professional championships at the Liga MX
Moves
After this season Necaxa and Selección Jalisco retired, while Albinegros de Orizaba, Atlas, Guadalajara, and Veracruz joined.
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1942-43
Mex
1942–43 in Mexican football |
https://en.wikipedia.org/wiki/1944%E2%80%9345%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Primera División de México in season 1944–45.
Overview
It was contested by 13 teams, and Club España won the championship.
Teams
League standings
Results
Moves
After this season Monterrey, Tampico, and San Sebastián joined.
References
Mexico - List of final tables (RSSSF)
- Mexico 1944/45 (RSSSF)
1944-45
Mex
1944–45 in Mexican football |
https://en.wikipedia.org/wiki/1945%E2%80%9346%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1945–46 season.
Overview
It was contested by 16 teams, and Veracruz won the championship.
Teams
League standings
Results
Moves
After this season Monterrey folded.
References
Mexico - List of final tables (RSSSF)
- Mexico 1945/46 (RSSSF)
1945-46
Mex
1945–46 in Mexican football |
https://en.wikipedia.org/wiki/1946%E2%80%9347%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1946–47 season.
Overview
It was contested by 15 teams, and Atlante won the championship.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
- Mexico 1946/47 (RSSSF)
1946-47
Mex
1946–47 in Mexican football |
https://en.wikipedia.org/wiki/1947%E2%80%9348%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1947–48 season.
Overview
It was contested by 15 teams, and León won the championship.
Teams
League standings
Results
Playoff
References
Mexico - List of final tables (RSSSF)
Mexico 1947/48 (RSSSF)
1947-48
Mex
1947–48 in Mexican football |
https://en.wikipedia.org/wiki/1948%E2%80%9349%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1948–49 season.
Overview
It was contested by 15 teams, and León won the championship and became the first team to win consecutive championships.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1948-49
Mex
1948–49 in Mexican football |
https://en.wikipedia.org/wiki/1949%E2%80%9350%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1949–50 season.
Overview
It was contested by 14 teams, and Veracruz won the championship.
Teams
League standings
Results
Changes
After this season Asturias, Club España, and Moctezuma retired from the league due to differences with the Federation. Necaxa, an old member and who didn't accept professionalism in 1943, re-joined for next season.
References
Mexico - List of final tables (RSSSF)
1949-50
Mex
1949–50 in Mexican football |
https://en.wikipedia.org/wiki/1950%E2%80%9351%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1950–51 season.
Overview
The Segunda División starts in 1950, thus the "Liga Mayor" changed to "Primera Division", every year one team is promoted and one is relegated.
It was contested by 12 teams, and Atlas won the championship.
San Sebastián was relegated to Segunda Division.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1950-51
Mex
1950–51 in Mexican football |
https://en.wikipedia.org/wiki/1951%E2%80%9352%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1951–52 season.
Overview
Zacatepec was promoted to Primera División.
The season was contested by 12 teams, and León won the championship.
Veracruz was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1951-52
Mex
1951–52 in Mexican football |
https://en.wikipedia.org/wiki/1952%E2%80%9353%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1952–53 season.
Overview
La Piedad was promoted to Primera División.
The season was contested by 12 teams, and Tampico won the championship.
La Piedad was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1952-53
Mex
1952–53 in Mexican football |
https://en.wikipedia.org/wiki/1953%E2%80%9354%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1953–54 season.
Overview
Toluca was promoted to Primera División.
The season was contested by 12 teams, and Marte won the championship.
Atlas was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1953-54
Mex
1953–54 in Mexican football |
https://en.wikipedia.org/wiki/1954%E2%80%9355%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1954–55 season.
Overview
Irapuato was promoted to Primera División.
The season was contested by 12 teams, and Zacatepec won the championship.
Marte was relegated to Segunda División.
Teams
League standings
Results
Promotion Playoff
With the intention of increasing the number of teams in the First Division to 14 clubs, the Mexican Football Federation organized a promotional tournament between the two worst teams of the first level and the second, third and fourth teams of the Second Division. The playoff had five teams: Atlante, Marte, Cuautla, Querétaro and Zamora.
Standings
Results
References
Mexico - List of final tables (RSSSF)
1954-55
Mex
1954–55 in Mexican football |
https://en.wikipedia.org/wiki/1955%E2%80%9356%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1955–56 season.
Overview
Atlas (Segunda División 1954-55 Champion), Zamora and Cuautla (second and third place in a promotional tournament in Segunda División) were promoted to Primera División, to increase the number of teams to 14 clubs.
The season was contested by 14 teams, and León won the championship.
Zamora was relegated to Segunda División.
Teams
League standings
Results
Championship playoff
References
Mexico - List of final tables (RSSSF)
1955-56
Mex
1955–56 in Mexican football |
https://en.wikipedia.org/wiki/1956%E2%80%9357%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1956–57 season.
Overview
Monterrey was promoted to Primera División.
The season was contested by 13 teams, and Guadalajara won the championship.
Monterrey was relegated to Segunda División.
Due to economical problems Puebla advised the league that the club would not be playing next season and the club would return for the 1957-58 season.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1956-57
Mex
1956–57 in Mexican football |
https://en.wikipedia.org/wiki/1957%E2%80%9358%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1957–58 season.
Overview
Zamora (Segunda División Champion) were promoted to Primera División.
Atlético Morelia (second place in Segunda División) was also promoted to Primera División, to replace Puebla.
The season was contested by 14 teams, and Zacatepec won the championship.
Tampico was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1957-58
Mex
1957–58 in Mexican football |
https://en.wikipedia.org/wiki/1958%E2%80%9359%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1958–59 season.
Overview
Celaya was promoted to Primera División.
The season was contested by 14 teams, and Guadalajara won the championship.
Cuautla was relegated to Segunda División.
Guadalajara qualified to the 1959 Campeonato Centroamericano.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1958-59
Mex
1958–59 in Mexican football |
https://en.wikipedia.org/wiki/1959%E2%80%9360%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1959–60 season.
Overview
Tampico was promoted to Primera División.
It was contested by 14 teams, and Guadalajara won the championship and becomes second team to win consecutive championships.
Zamora was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1959-60
Mex
1959–60 in Mexican football |
https://en.wikipedia.org/wiki/1960%E2%80%9361%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1960–61 season.
Overview
Monterrey was promoted to Primera División.
The season was contested by 14 teams, and Guadalajara won the championship and becomes first team to win three consecutive championships.
Celaya was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1960-61
Mex
1960–61 in Mexican football |
https://en.wikipedia.org/wiki/1961%E2%80%9362%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1961–62 season.
Overview
Nacional was promoted to Primera División.
The season was contested by 14 teams, and Guadalajara won the championship and becomes first team to win four consecutive championships.
Zacatepec was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1961-62
Mex
1961–62 in Mexican football |
https://en.wikipedia.org/wiki/1962%E2%80%9363%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Primera División de México in the 1962–63 season.
Overview
UNAM was promoted to Primera División.
The season was contested by 14 teams, and Oro won the championship.
Tampico was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1962-63
Mex
1962–63 in Mexican football |
https://en.wikipedia.org/wiki/1963%E2%80%9364%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1963–64 season.
Overview
Zacatepec was promoted to Primera División.
The season was contested by 14 teams, and Guadalajara won the championship.
No Relegation this season.
Promotional tournament was held to increase the number of teams for the 1964–65 season to 16. Nacional finished in first place, and was not to be relegated from the Primera Division. Veracruz finished in second place, and was promoted to Primera Division.
Teams
League standings
Results
Promotion tournament
In order to increase the number of team from 14 to 16 for the 1964-65 the league made a playoff. The top 2 teams would be in Primera Division. The playoff was composed of the lowest team from Primera division and the 2nd-5th teams in the standings from Segunda Division as Cruz Azul had earned automatic promotion. The playoff was played between January 16 and February 6 1964. All games were played in Estadio Olímpico Universitario, Mexico City.
Results
References
Mexico - List of final tables (RSSSF)
1963-64
Mex
1963–64 in Mexican football |
https://en.wikipedia.org/wiki/1964%E2%80%9365%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1964–65 season.
Overview
In order to increase the number of team from 14 to 16 for the 1964–65 season, the league made a Promotional Tournament. The top 2 teams would be in Primera Division. The playoff was composed of the lowest team from Primera division and the 2nd-5th teams in the standings from Segunda Division as Cruz Azul had earned automatic promotion. Nacional finished first securing another season in top flight. Veracruz finished in second place, and was promoted to Primera Division.
Cruz Azul (Jasso) was promoted to Primera División.
The season was contested by 16 teams, and Guadalajara won the championship.
Nacional was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1964-65
Mex
1964–65 in Mexican football |
https://en.wikipedia.org/wiki/1965%E2%80%9366%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1965–66 season.
Overview
Ciudad Madero was promoted to Primera División.
The season was contested by 16 teams, and America won the championship.
Zacatepec was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1965-66
Mex
1965–66 in Mexican football |
https://en.wikipedia.org/wiki/1966%E2%80%9367%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the México Primera División for the 1966–67 season.
Overview
Nuevo León was promoted to Primera División.
The season was contested by 16 teams, and Toluca won the championship.
Ciudad Madero was relegated to Segunda División.
Teams
League standings
Results
Notes
References
Mexico - List of final tables (RSSSF)
Mexico - Season 1966/67 (RSSSF)
1966-67
Mex
1966–67 in Mexican football |
https://en.wikipedia.org/wiki/1967%E2%80%9368%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1967–68 season.
Overview
Pachuca was promoted to Primera División.
The season was contested by 16 teams, and Toluca won the championship and becomes second team to win consecutive championships.
Morelia was relegated to Segunda División.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1967-68
Mex
1967–68 in Mexican football |
https://en.wikipedia.org/wiki/1968%E2%80%9369%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the México Primera División for the 1968–69 season.
Overview
Laguna was promoted to Primera División.
The season was contested by 16 teams, and Cruz Azul won the championship. The season was suspended from September 16 to November 8 for the celebration of the 1968 Summer Olympic Games in Mexico City.
Nuevo León was relegated to Segunda División
Teams
League standings
Results
Relegation Playoff
Aggregate tied. Replay will take place.
Aggregate tied. Replay will take place.
Oro won 4-3 on aggregate. Nuevo León is relegated to Segunda Division.
Notes
References
Mexico - List of final tables (RSSSF)
Mex
1968–69 in Mexican football
1968-69 |
https://en.wikipedia.org/wiki/1969%E2%80%9370%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1969–70 season.
Overview
Torreón was promoted to Primera División.
The season was contested by 16 teams, and Guadalajara won the championship.
No relegation this season, due to the increase of teams to 18 for the 1970-71 season.
Teams
League standings
Results
References
Mexico - List of final tables (RSSSF)
1969-70
Mex
1969–70 in Mexican football |
https://en.wikipedia.org/wiki/1970%E2%80%9371%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1970–71 season.
Overview
Zacatepec was promoted to Primera División (Segunda División 1969–70 Champion)
Puebla was also promoted to Primera División (winner of promotional tournament between Unión de Curtidores, Nacional and Naucalpan) to increase the number of teams to 18.
The season was contested by 18 teams, and America won the championship.
Atlas was relegated to Segunda División.
After this season Necaxa was sold and changed its name to Atlético Español.
Cruz Azul moves to Mexico City.
Teams
Group stage
Group 1
Group 2
Results
Relegation Playoffs
Aggregate score tied so a 3rd match will be played
Pachuca won on aggregate 5-3. Atlas was relegated to Segunda Division
Final
America won on aggregate 2-0
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1970–71 in Mexican football |
https://en.wikipedia.org/wiki/1971%E2%80%9372%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1971–72 season.
Overview
San Luis was promoted to Primera División.
This season was contested by 18 teams, and Cruz Azul won the championship.
Irapuato was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Results
Relegation Playoffs
Bracket
Semifinals
Atlético Español won 4-3 on aggregate
Veracruz won 3-1 on aggregate
Final
Irapuato relegated to Segunda Division
Championship Playoffs
Bracket
Semifinal
Cruz Azul won 2-1 on aggregate
Aggregate tied. 3rd match will be played
America won 5-4 on aggregate
Final
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1 |
https://en.wikipedia.org/wiki/1972%E2%80%9373%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1972–73 season.
Overview
Atlas was promoted to Primera División.
It was contested by 18 teams, and Cruz Azul won the championship.
Pachuca was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Results
Relegation playoff
Bracket
Semifinal
Aggregate tied. 3rd match will be played.
Laguna won 4-1 on aggregate.
Torreón won 3-1 on aggregate.
Relegation Final
Aggregate tied. A replay will be played
Zacatepec won 1-0 in aggregate. Pachuca was relegated to Segunda Division.
Championship playoff
Bracket
Semifinal
Cruz Azul won 4-2 on aggregate.
Aggregate tied. 3rd match will be played.
Aggregate 4-4. León won in penalties 6-5.
Final
Aggregate 1-1. 3rd match will be played.
Cruz Azul won 3-2 on aggregate.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1972–73 in Mexican football |
https://en.wikipedia.org/wiki/1973%E2%80%9374%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1973–74 season.
Overview
Ciudad Madero was promoted to Primera División.
This season was contested by 18 teams, and Cruz Azul won the championship.
San Luis was relegated to Segunda División.
UDG acquired Torreón franchise to reach the Primera Division for the 1974-75 season.
Segunda División semifinalists, UASLP-Pumas (re-branded to Atlético Potosino) and Unión de Curtidores were promoted to the Primera División to increase the number of teams to 20 for the 1974-75 season.
Teams
Group stage
Group 1
Group 2
Results
Relegation playoff
Laguna won 2-0 on aggregate. San Luis relegated to Segunda Division.
Championship playoff
Bracket
Semifinal
Cruz Azul won 7-2 on aggregate.
Atlético Español won 6-5 on aggregate.
Final
Cruz Azul won 4-2 on aggregate.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1973–74 in Mexican football |
https://en.wikipedia.org/wiki/1974%E2%80%9375%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Mexican Primera División for the 1974–75 season.
Overview
UANL was promoted to Primera División.
Segunda División semifinalists, UASLP-Pumas (who re-branded to Atlético Potosino) and Unión de Curtidores were promoted to the Primera División to increase the number of teams to 20.
UdeG acquired Torreón franchise.
This season was contested by 20 teams, and Toluca won the championship.
Ciudad Madero was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Results
Championship Group Stage Final
Championship Group Stage results
Round 1
Round 2
Round 3
Round 4
Round 5
Round 6
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1974–75 in Mexican football |
https://en.wikipedia.org/wiki/1975%E2%80%9376%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Mexican Primera División in season 1975–76.
Overview
Tecos was promoted to Primera División.
This season was contested by 20 teams, and América won the championship.
Atlante was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Relegation playoff
Atletico Potosino won 2-1 on aggregate. Atlante was relegated to Segunda División.
Championship Playoffs
Bracket
QuarterfinalAmérica won 1-0 on aggregate.Unión de Curtidores won 4-2 on aggregate.UdeG won 6-0 on aggregate.Monterrey won 7-2 on aggregate.Semi-finalsAmérica won 2-0 on aggregate.UdeG won 3-2 on aggregate.FinalAmérica won 4-0 on aggregate.''
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1975–76 in Mexican football |
https://en.wikipedia.org/wiki/1977%E2%80%9378%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1977–78 season.
Overview
Atlante was promoted to Primera División.
This season was contested by 20 teams, and UANL won the championship.
Atlas was relegated to Segunda División.
After this season Deportivo Neza bought the Laguna franchise.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Relegation playoff
Unión de Curtidores won 4-2 on aggregate. Atlas was relegated to Segunda División.
Championship playoff
Quarterfinal
Aggregate tied. Tampico Madero won 4-2 on penalty shootout.
UNAM won 3-2 on aggregate.
Cruz Azul won 3-2 on aggregate.
UANL won 3-2 on aggregate.
Semifinal
UNAM won 5-2 on aggregate.
UANL won 3-1 on aggregate.
Final
UANL won 3-1 on aggregate.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1977–78 in Mexican football |
https://en.wikipedia.org/wiki/1978%E2%80%9379%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1978–79 season.
Overview
Zacatepec was promoted to Primera División.
This season was contested by 20 teams, and Cruz Azul won the championship.
Veracruz was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Group Stage Playoff
Semifinal
Group A
Group A Results
Round 1América 1 - 0 Cruz AzulAtlético Potosino 0 - 2 Toluca
Round 2Cruz Azul 2 - 1 Atlético PotosinoToluca 1 - 0 América
Round 3Atlético Potosino 1 - 2 AméricaToluca 0 - 1 Cruz Azul
Round 4Cruz Azul 2 - 1 AméricaToluca 5 - 0 Atlético Potosino
Round 5Atlético Potosino 2 - 2 Cruz AzulAmérica 3 - 1 Toluca
Round 6América 0 - 1 Atlético PotosinoCruz Azul 1 - 0 Toluca
Group B
Group B Results
Group 2
Round 1U.N.A.M. 1 - 0 ZacatepecMonterrey 0 - 1 U.A.N.L.
Round 2U.N.A.M. 2 - 2 MonterreyU.A.N.L. 2 - 1 Zacatepec
Round 3Monterrey 2 - 1 ZacatepecU.N.A.M. 3 - 0 U.A.N.L.
Round 4Zacatepec 1 - 2 U.N.A.M.U.A.N.L. 1 - 1 Monterrey
Round 5Monterrey 1 - 1 U.N.A.M.Zacatepec 1 - 1 U.A.N.L.
Round 6U.A.N.L. 2 - 0 U.N.A.M.Zacatepec 2 - 2 Monterrey
Final
Cruz Azul won 2-0 on aggregate.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1978–79 in Mexican football |
https://en.wikipedia.org/wiki/1979%E2%80%9380%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1979–80 season.
Overview
Atlas was promoted to Primera División.
This season was contested by 20 teams, and Cruz Azul won the championship.
Jalisco was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Relegation playoff
Unión de Curtidores won 4-3 on aggregate. Jalisco was relegated to Segunda División.
Playoff
Semifinal
Group 1
Group A Results
Round 1América 1 - 2 U.A.N.L.U.N.A.M. 0 - 0 Zacatepec
Round 2América 2 - 0 ZacatepecU.A.N.L. 0 - 1 U.N.A.M.
Round 3U.N.A.M. 0 - 1 AméricaZacatepec 1 - 1 U.A.N.L.
Round 4Zacatepec 0 - 0 U.N.A.M.U.A.N.L. 0 - 0 América
Round 5Zacatepec 3 - 2 AméricaU.N.A.M. 2 - 2 U.A.N.L.
Round 6América 0 - 0 U.N.A.M.U.A.N.L. 4 - 1 Zacatepec
Group B
Group B Results
Round 1Tampico Madero 0 - 1 Cruz AzulNeza 1 – 0 Atlante
Round 2Cruz Azul 4 - 2 AtlanteNeza 1 - 1 Tampico Madero
Round 3Neza 0 - 2 Cruz AzulAtlante 5 - 2 Tampico
Round 4Cruz Azul 0 - 1 Tampico MaderoAtlante 1 – 1 Neza
Round 5Cruz Azul 1 - 0 NezaTampico Madero 1 - 1 Atlante
Round 6Tampico Madero 1 - 2 NezaAtlante 3 - 1 Cruz Azul
Final
Cruz Azul won 4-3 on aggregate.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1979–80 in Mexican football |
https://en.wikipedia.org/wiki/1980%E2%80%9381%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Primera División de México for the 1980–81 season.
Overview
It was contested by 20 teams, and UNAM won the championship.
Atletas Campesinos was promoted from Segunda División.
Unión de Curtidores was relegated to Segunda División.
Teams
Stadiums and locations
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Relegation play-offs
Atlas won 2-3 on aggregate. Unión de Curtidores was relegated to Segunda División.
Final stage
Semifinal
Group 1
Group 2
Final
UNAM won 4-2 on aggregate.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1980–81 in Mexican football |
https://en.wikipedia.org/wiki/1981%E2%80%9382%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Primera División de México for the 1981–82 season.
Overview
It was contested by 20 teams, and UANL won the championship.
Atlético Morelia was promoted from Segunda División.
After the season, the owner of Atlético Español decided to sell the team to the Mexican league. This gave way for Necaxa to come back to play in the 1982-83 season.
Tampico was relegated to Segunda División, however, the oil workers' syndicate acquired the Atletas Campesinos franchise and created a new team called Tampico Madero.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Relegation playoff
Atlas won 4-2 on aggregate. Tampico was relegated to Segunda División.
Playoff
U.A.N.L. won the championship.
Quarterfinal
Atlante won 5-3 on aggregate.
Deportivo Neza won 2-3 on aggregate.
América won 5-3 on aggregate.
UANL won 3-2 on aggregate.
Semi-finals
Atlante won 1-0 on aggregate.
UANL won 1-2 on aggregate.
Final
Aggregate tied. UANL won 1-3 on penalty shootout.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1981–82 in Mexican football |
https://en.wikipedia.org/wiki/1982%E2%80%9383%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1982–83 season.
Overview
It was contested by 20 teams, and Puebla won the championship.
C.F. Oaxtepec was promoted from Segunda División.
Zacatepec was relegated to Segunda División.
Teams
Moves
The owner of Atlético Español decided to sell the team to the Mexican league. This gave way for Necaxa to come back to play.
Tampico was relegated to Segunda División, however, the oil workers union bought the Atletas Campesinos franchise, they moved the team to Tampico and was renamed as Tampico Madero.
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Relegation playoff
Atlético Morelia won 5-4 on aggregate. Zacatepec was relegated to Segunda División
Playoff
Puebla won the championship.
Quarterfinal
América won 6-0 on aggregate.
Guadalajara won 1-6 on aggregate.
Puebla won 6-3 on aggregate.
UdeG won 2-4 on aggregate.
Semi-finals
Puebla won 4-3 on aggregate.
Guadalajara 2-4 on aggregate.
Final
Aggregate tied. Puebla won 7–6 on penalty shootout.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1982–83 in Mexican football |
https://en.wikipedia.org/wiki/1983%E2%80%9384%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1983–84 season.
Overview
It was contested by 20 teams, and América won the championship.
Unión de Curtidores was promoted from Segunda División 1982-83 season, and was relegated from Primera División on this season.
Atlante F.C. was moved from Estadio Azteca to Estadio Azulgrana.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1983–84 in Mexican football |
https://en.wikipedia.org/wiki/1984%E2%80%9385%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Primera División de México for the 1984–85 season.
Overview
It was contested by 20 teams, and América won the championship.
On August 20, 1984, the Mexican Football Federation accepted the request of C.F. Oaxtepec. to move to Puebla, for this reason, the club changed its name to Ángeles de Puebla.
Zacatepec was promoted from Segunda División, also, this team was relegated on this season.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Relegation playoff
Zacatepec 1-2 ; 0-1 Necaxa
Playoff
Championship Playoff Game
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1984–85 in Mexican football |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Mexican Primera División for the 1985–86 season.
Prode-1985
Overview
It was contested by 20 teams, and América won the championship.
Irapuato F.C. was promoted from Segunda División.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Playoff
Mexico-1986
Overview
It was contested by 20 teams, and Monterrey won the championship.
Group stage
Group 1
Group 2
Playoff
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1985–86 in Mexican football |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of the Mexican Primera División for the 1986–87 season.
Overview
It was contested by 21 teams, and Guadalajara won the championship.
C.F. Cobras de Querétaro was promoted from Segunda División.
In the season were two teams relegated to Segunda División: Cobras and León.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
References
Mexico - List of final tables (RSSSF)
Mexico 1986/87
Liga MX seasons
Mex
1986–87 in Mexican football |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Mexican Primera División in season 1987–88.
Overview
It was contested by 20 teams, and América won the championship.
Correcaminos UAT was promoted from Segunda División.
Teams
Moves
After the season Correcaminos bought the Deportivo Neza franchise in order to remain in Primera División.
Ángeles was sold and transferred to Torreón and was made into Santos.
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1987–88 in Mexican football |
https://en.wikipedia.org/wiki/1988%E2%80%9389%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Primera División de México in season 1988–89.
Overview
It was contested by 20 teams, and América won the championship.
Cobras was promoted from Segunda División.
Atlético Potosino was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
Semifinal
Group 1
Group 2
Final
America won 5-4 on aggregate.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1988–89 in Mexican football |
https://en.wikipedia.org/wiki/1989%E2%80%9390%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Primera División de México in season 1989–90.
Overview
It was contested by 20 teams, and Puebla won the championship.
Potros Neza was promoted from Segunda División, however, the team sold its license to Veracruz.
Atlante F.C. was relocated from Mexico City to Querétaro. The same team was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
Final
Puebla won 6-4 on aggregate.
Moves
After this season Querétaro bought the Tampico Madero franchise in order to come into Primera Division for the 1990-91 season.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
Mex
1989–90 in Mexican football |
https://en.wikipedia.org/wiki/1990%E2%80%9391%20Mexican%20Primera%20Divisi%C3%B3n%20season | These are statistics of Primera División de México for the 1990–91 season.
Overview
It was contested by 20 teams, and Pumas de la UNAM won the championship.
León was promoted from Segunda División.
Querétaro bought the franchise of Tampico Madero, which allowed the team to return to Primera División.
Irapauto was relegated to Segunda División.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
Final
First leg
Second leg
3–3 on aggregate. UNAM won on away goals.
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
1990–91 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1991%E2%80%9392%20Mexican%20Primera%20Divisi%C3%B3n%20season | The following statistics encompass the 1991–92 Primera División de México season.
Overview
It was contested by 20 teams, and the León won the championship.
Atlante F.C. was promoted from Segunda División.
Starting in 1991–92, there was a change in the system of relegation to the Segunda División. The so-called "relegation table" was implemented, in which the points obtained in the last two seasons were divided between the games played in the current season to obtain a coefficient.
This is the relegation system used by Primera Div./Liga MX to this day. The difference is in that due to the change in the calendar which allowed for shorter tournaments, it encompasses the past five seasons (equivalent to about 2.5 years' worth of league games).
The club that was known at the time as Cobras de Ciudad Juárez was relegated to Segunda División, it was the first team relegated with the relegation table system.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
Repechaje round
América 2-0 ; 0-4 Cruz Azul
UAT 2-1 ; 1-4 Veracruz
Final
León won 2-0 on aggregate.
Relegation table
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
1991–92 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1992%E2%80%9393%20Mexican%20Primera%20Divisi%C3%B3n%20season | The following are statistics about Primera División de México for the 1992–93 season.
Overview
It was contested by 20 teams, and Atlante won the championship.
Pachuca, who was promoted from Segunda División the previous season, was relegated.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
Repechaje round
Veracruz 1-1 ; 0-1 Tecos
UANL 3-0 ; 1-4 UNAM
Final
Atlante won 4–0 on aggregate.
Relegation table
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
1992–93 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1993%E2%80%9394%20Mexican%20Primera%20Divisi%C3%B3n%20season | Statistics of Primera División de México for the 1993–94 season.
Overview
It was contested by 20 teams, and Tecos won the championship.
Toros Neza was promoted from Segunda División. The team started the season in Ciudad Nezahualcóyotl, however, the use of the Neza 86 Stadium was denied by the Mexican Football Federation, for this reason, the team was moved to Pachuca and changed its name to Toros Hidalgo.
Querétaro was relegated to Segunda División.
Teams
Moves
After this season U. de G. franchise was sold to a group of 15 Segunda Division clubs to form an intermediate division called Primera Division "A", between Primera Division and Segunda Division. The new owners disappeared the team.
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
If the two teams are tied after two legs, the higher-seeded team advances.
Repechaje round
Morelia won 5-4 on aggregate.
America won 4-2 on aggregate.
Quarterfinal
Tecos won 6-0 on aggregate.
América won 2-3 on aggregate.
Santos Laguna won 3-2 on aggregate.
Toluca won 4-3 on aggregate.
Semifinal
Aggregate tied 4-4. Tecos won the series because the team had a better position in general table
Santos Laguna won 1-2 on aggregate.
Final
Tecos won 2-1 on aggregate
Relegation table
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
1993–94 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1994%E2%80%9395%20Mexican%20Primera%20Divisi%C3%B3n%20season | The following are statistics of Primera División de México, the premier soccer division in Mexico, for the 1994–95 season.
Overview
It was contested by 19 teams, and Necaxa won the championship.
The U. de G. franchise briefly disappeared after being acquired by a few Primera Division A teams.
Tampico Madero was promoted from Segunda División. The team started the season playing in Tamaulipas, but on week 16, the team was moved to Querétaro and changed its name to TM Gallos Blancos, the owner cited leasing problems at his original venue.
TM Gallos Blancos and Correcaminos UAT were relegated to Primera División 'A', the new second-level league in the Mexican system. For the 1995–96 season, the number of participating teams was reduced to 18.
Toros Neza returned to Ciudad Nezahualcóyotl after one season playing in Pachuca.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Overall Table
Results
Playoff
Repechaje round
If the two teams are tied after two legs, the higher-seeded team advances.
Series tied 1–1 on aggregate. Puebla advanced as higher-seeded team.
Series tied 2–2 on aggregate. Tecos advanced as higher-seeded team.
Playoff round
Quarter-finals
Series tied 3–3 on aggregate. Guadalajara advanced to semi-finals by away goals rule.
Necaxa won 4–1 on aggregate.
America won 4–2 on aggregate.
Aggregate tied 1-1. Cruz Azul advanced to semi-finals as best seeded team.
Semi-finals
Aggregate tied 1-1. Necaxa advanced to final by away goals rule.
Cruz Azul won 2–3 on aggregate.
Final
Necaxa won 3–1 on aggregate.
Relegation table
References
Mexico – List of final tables (RSSSF)
Liga MX seasons
1994–95 in Mexican football
Mex |
https://en.wikipedia.org/wiki/1995%E2%80%9396%20Mexican%20Primera%20Divisi%C3%B3n%20season | The following are statistics about Primera División de México, the top professional soccer league in Mexico, for the 1995–96 season.
Overview
It was contested by 18 teams, and Necaxa won the championship.
Celaya was promoted from Primera División 'A'.
UANL was relegated to Primera División 'A'.
Teams
Group stage
Group 1
Group 2
Group 3
Group 4
Results
Playoff
Repechage
UANL won 4-5 on aggregate.
América won 0-2 on aggregate.
Playoff Series
Quarter-finals
Series tied 3-3 on aggregate. Veracruz advanced to semi-finals by away goals rule.
Series tied 2-2 on aggregate. Celaya advanced to semi-finals by away goals rule.
Necaxa won 2-1 on aggregate.
América won 3-2 on aggregate.
Semi-finals
Celaya won 6-1 on aggregate.
Necaxa won 3-1 on aggregate.
Final
1-1 on aggregate. Necaxa won on away goals.
Relegation table
References
Mexico - List of final tables (RSSSF)
Liga MX seasons
1995–96 in Mexican football
Mex |
https://en.wikipedia.org/wiki/Covering%20groups%20of%20the%20alternating%20and%20symmetric%20groups | In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classified in : for , the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold.
For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4.
Definition and classification
A group homomorphism from D to G is said to be a Schur cover of the finite group G if:
the kernel is contained both in the center and the commutator subgroup of D, and
amongst all such homomorphisms, this D has maximal size.
The Schur multiplier of G is the kernel of any Schur cover and has many interpretations. When the homomorphism is understood, the group D is often called the Schur cover or Darstellungsgruppe.
The Schur covers of the symmetric and alternating groups were classified in . The symmetric group of degree n ≥ 4 has
Schur covers of order 2⋅n! There are two isomorphism classes if n ≠ 6 and one isomorphism class if n = 6.
The alternating group of degree n has one isomorphism class of Schur cover, which has order n! except when n is 6 or 7, in which case the Schur cover has order 3⋅n!.
Finite presentations
Schur covers can be described by means of generators and relations. The symmetric group Sn has a presentation on n−1 generators ti for i = 1, 2, ..., n−1 and relations
titi = 1, for 1 ≤ i ≤ n−1
ti+1titi+1 = titi+1ti, for 1 ≤ i ≤ n−2
tjti = titj, for 1 ≤ i < i+2 ≤ j ≤ n−1.
These relations can be used to describe two non-isomorphic covers of the symmetric group. One covering group has generators z, t1, ..., tn−1 and relations:
zz = 1
titi = z, for 1 ≤ i ≤ n−1
ti+1titi+1 = titi+1ti, for 1 ≤ i ≤ n−2
tjti = titjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The same group can be given the following presentation using the generators z and si given by ti or tiz according as i is odd or even:
zz = 1
sisi = z, for 1 ≤ i ≤ n−1
si+1sisi+1 = sisi+1siz, for 1 ≤ i ≤ n−2
sjsi = sisjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The other covering group has generators z, t1, ..., tn−1 and relations:
zz = 1, zti = tiz, for 1 ≤ i ≤ n−1
titi = 1, for 1 ≤ i ≤ n−1
ti+1titi+1 = titi+1tiz, for 1 ≤ i ≤ n−2
tjti = titjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The same group can be given the following presentation using the generators z and si given by ti or tiz according as i is odd or even:
zz = 1, zsi = siz, for 1 ≤ i ≤ n−1
sisi = 1, for 1 ≤ i ≤ n−1
si+1sisi+1 = sisi+1si, for 1 ≤ i ≤ n−2
sjsi = sisjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
Sometimes all of the relations of the symmetric group are expressed as where mij are non-negative integers, namely mii = 1, mi,i+1 = 3, and mij = 2, for 1 ≤ i < i+2 ≤ j ≤ n−1. The presentation of becomes particularly simple in this form: and zz = 1. T |
https://en.wikipedia.org/wiki/2010%20Brisbane%20Lions%20season | This covers all the results and statistics for the Brisbane Lions in the 2010 AFL season.
2009 off-season
The Lions lost in their semi-final clash against the Western Bulldogs at the Melbourne Cricket Ground by 51 points. It was the last game for veteran Tim Notting, who announced his retirement almost a week from the match. Michael Voss was desperate to make some changes to the playing list. The Lions secured three players on the first day of the 2009 AFL Trade Week. 's Amon Buchanan and West Coast's Brent Staker were traded in a complex three-team deal that also involved Brisbane's Bradd Dalziell who went to West Coast and West Coast's Mark Seaby who went to . In a separate trade on the same day, 's Andrew Raines joined the Lions. The Lions continued to be active in the trade week and on the fourth day they secured 's Xavier Clarke. On the final day of Trade Week, Fevola was traded to the Lions for Lachlan Henderson. Daniel Bradshaw and Michael Rischitelli originally were put up trade for Fevola but the deal fell through after Rischitelli declined. Bradshaw was incensed at being put up for trade and eventually quit the Lions.
With one of the best team rosters ahead of the 2010 season, Brisbane were expected to once again make the finals. Four straight wins saw them on top of the ladder after round four, but from there the Lions went into freefall winning only three more games for the season.
Squad for 2010
Statistics are correct as of start of 2010 season.
Player changes for 2010
In
Out
Ladder
Results
NAB Cup
First Round
NAB Challenge
Week 2
Week 3
Week 4
Premiership season
Round 1
Round 2
Round 3
Round 4
Round 5
Round 6
Round 7
Round 8
Round 9
Round 10
Round 11
Round 12
Round 13
Round 14
Round 15
Round 16
Round 17
Round 18
Round 19
Round 20
Round 21
Round 22
Exhibition Match
Milestones
References
Notes
External links
Official Brisbane Lions website
Official AFL Website
Brisbane Lions
2010
Brisbane Lions |
https://en.wikipedia.org/wiki/Balanced%20polygamma%20function | In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.
It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
Definition
The generalized polygamma function is defined as follows:
or alternatively,
where is the polygamma function and , is the Hurwitz zeta function.
The function is balanced, in that it satisfies the conditions
.
Relations
Several special functions can be expressed in terms of generalized polygamma function.
where are the Bernoulli polynomials
where is the -function and is the Glaisher constant.
Special values
The balanced polygamma function can be expressed in a closed form at certain points (where is the Glaisher constant and is the Catalan constant):
References
Gamma and related functions |
https://en.wikipedia.org/wiki/Liberman%27s%20lemma | Liberman's lemma is a theorem used in studying intrinsic geometry of convex surface.
It is named after Joseph Liberman.
Formulation
If is a unit-speed minimizing geodesic on the surface of a convex body K in Euclidean space then for any point p ∈ K, the function
is concave.
References
Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313.
Differential geometry of surfaces
Lemmas |
https://en.wikipedia.org/wiki/Overcompleteness | Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R. J. Duffin and A. C. Schaeffer in 1952.
Formally, a subset of the vectors of a Banach space , sometimes called a "system", is complete if every element in can be approximated arbitrarily well in norm by finite linear combinations of elements in . A complete system is additionally overcomplete if there exists a which can be removed from the system while maintaining completeness (i.e., ). In this sense, the system contains more vectors than necessary to be complete, hence "overcomplete".
In research areas such as signal processing and function approximation, overcompleteness can help researchers to achieve a more stable, more robust, or more compact decomposition than using a basis.
Relation between overcompleteness and frames
Overcompleteness is usually discussed as a property of overcomplete frames. The theory of frame originates in a paper by Duffin and Schaeffer on non-harmonic Fourier series. The frame is defined to be a set of non-zero vectors such that for an arbitrary ,
where denotes the inner product, and are positive constants called bounds of the frame. When and can be chosen such that , the frame is called a tight frame.
It can be seen that .
An example of frame can be given as follows.
Let each of and be an orthonormal basis of , then
is a frame of with bounds .
Let be the frame operator,
A frame that is not a Riesz basis, in which case it consists of a set of functions more than a basis, is said to be overcomplete. In this case, given , it can have different decompositions based on the frame. The frame given in the example above is an overcomplete frame.
When frames are used for function estimation, one may want to compare the performance of different frames. The parsimony of the approximating functions by different frames may be considered as one way to compare their performances.
Given a tolerance and a frame in , for any function , define the set of all approximating functions that satisfy
Then let
indicates the parsimony of utilizing frame to approximate . Different may have different based on the hardness to be approximated with elements in the frame. The worst case to estimate a function in is defined as
For another frame , if , then frame is better than frame at level . And if there exists a that for each , we have , then is better than broadly.
Overcomplete frames are usually constructed in three ways.
Combine a set of bases, such as wavelet basis and Fourier basis, to obtain an overcomplete frame.
Enlarge the range of parameters in some frame, such as in Gabor frame and wavelet frame, to have an overcomplete frame.
Add some other functions to an existing complete basis to achieve an overcomplete frame.
An example of an overcomplete frame is shown below. The collecte |
https://en.wikipedia.org/wiki/L%C3%A9vy%20hierarchy | In set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy of formulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language of set theory. This is analogous to the arithmetical hierarchy, which provides a similar classification for sentences of the language of arithmetic.
Definitions
In the language of set theory, atomic formulas are of the form x = y or x ∈ y, standing for equality and set membership predicates, respectively.
The first level of the Lévy hierarchy is defined as containing only formulas with no unbounded quantifiers and is denoted by . The next levels are given by finding a formula in prenex normal form which is provably equivalent over ZFC, and counting the number of changes of quantifiers:p. 184
A formula is called:
if is equivalent to in ZFC, where is
if is equivalent to in ZFC, where is
If a formula has both a form and a form, it is called .
As a formula might have several different equivalent formulas in prenex normal form, it might belong to several different levels of the hierarchy. In this case, the lowest possible level is the level of the formula.
Lévy's original notation was (resp. ) due to the provable logical equivalence, strictly speaking the above levels should be referred to as (resp. ) to specify the theory in which the equivalence is carried out, however it is usually clear from context.pp. 441–442 Pohlers has defined in particular semantically, in which a formula is " in a structure ".
The Lévy hierarchy is sometimes defined for other theories S. In this case and by themselves refer only to formulas that start with a sequence of quantifiers with at most i−1 alternations, and and refer to formulas equivalent to and formulas in the language of the theory S. So strictly speaking the levels and of the Lévy hierarchy for ZFC defined above should be denoted by and .
Examples
Σ0=Π0=Δ0 formulas and concepts
x = {y, z}p. 14
x ⊆ y
x is a transitive set
x is an ordinal, x is a limit ordinal, x is a successor ordinal
x is a finite ordinal
The first countable ordinal ω
x is an ordered pair. The first entry of the ordered pair x is a. The second entry of the ordered pair x is b p. 14
f is a function. x is the domain/range of the function f. y is the value of f on x p. 14
The Cartesian product of two sets.
x is the union of y
x is a member of the αth level of Godel's L
R is a relation with domain/range/field a p. 14
Δ1-formulas and concepts
x is a well-founded relation on y
x is finite.
Ordinal addition and multiplication and exponentiation
The rank (with respect to Gödel's constructible universe) of a set p. 61
The transitive closure of a set.
Σ1-formulas and concepts
x is countable.
|X|≤|Y|, |X|=|Y|.
x is constructible.
g is the restriction of the function f to a p. 23
g is the image of f on a p. 23
b is the successor ordinal of a p. 23
rank(x) p. 29
The |
https://en.wikipedia.org/wiki/Australian%20cricket%20team%20in%202009%E2%80%9310 | This article contains information, results and statistics regarding the Australian national cricket team in the 2009 and 2009-10 cricket seasons. Statisticians class the 2009 season and 2009–10 season as those matches played on tours that started between May 2009 and April 2010.
Player contracts
The 2009-10 list was announced on 14 May 2009. Note that uncontracted players are still available to be selected to play for the national cricket team.
Match summary
P = Matches Played/Total Matches, W = Won, L = Lost, D = Drawn, T = Tied, NR = No Result
N.B. Result from All-Star match not included as it was only an exhibition match.
2009 season
Pakistan v Australia in UAE
This series was moved from Pakistan to UAE for security reasons.
ICC World Twenty20
Tour of England
Tour of Scotland
2009–10 Season
2009 ICC Champions Trophy
The 2009 ICC Champions Trophy was scheduled to take place in the 2008–09 season in Pakistan, but because of an unstable security situation, it was rescheduled for the 2009–10 season. The hosting rights were also moved from Pakistan to South Africa. Sri Lanka was considered as a potential host, but was discarded due to worries related to the weather during that time of the year in Sri Lanka.
Tour of India
The Australian Cricket Team toured India from 25 October to 11 November 2009. The tour consists of 7 ODIs which Australia won 4–2.
Johnnie Walker All*Star Twenty20 match
The Australian Cricket team played an exhibition Twenty20 match against an all star cricket team featuring up and coming first class cricketers and Australian cricket legends to open the 2009/10 summer of international cricket in Australia. The Australian Cricketers' Association (ACA) XI all star team consisted of legends Adam Gilchrist, Matthew Hayden, Glenn McGrath & Shane Warne who is the captain. They were coached by former Australian fast bowler Dennis Lillee.
West Indies in Australia
The tour began with a practice match for the West Indians against Queensland starting 18 November. The first test began on 26 November and the tour will conclude on 23 February with a T20I. In all the tour included one First Class match, one List A match, three Tests, five ODIs and two T20Is.
Pakistan in Australia
The tour started on 19 December with a three-day first class match against Tasmania. This was followed on 26 December with the Boxing Day Test match at the MCG. It concluded on 5 February with a T20I at the MCG. In all the tour included three tests, five ODIs and a T20I.
Tour of New Zealand
The Australia Cricket Team toured New Zealand from 26 February to 31 March 2010. The tour consisted of 2 Twenty20s, 5 One Day Internationals and 2 Tests.
Important events
See also
Australia national cricket team
References
Australia in international cricket
International cricket competitions in 2009–10
National team |
https://en.wikipedia.org/wiki/Free%20independence | In the mathematical theory of free probability, the notion of free independence was introduced by Dan Voiculescu. The definition of free independence is parallel to the classical definition of independence, except that the role of Cartesian products of measure spaces (corresponding to tensor products of their function algebras) is played by the notion of a free product of (non-commutative) probability spaces.
In the context of Voiculescu's free probability theory, many classical-probability theorems or phenomena have free probability analogs: the same theorem or phenomenon holds (perhaps with slight modifications) if the classical notion of independence is replaced by free independence. Examples of this include: the free central limit theorem; notions of free convolution; existence of free stochastic calculus and so on.
Let be a non-commutative probability space, i.e. a unital algebra over equipped with a unital linear functional . As an example, one could take, for a probability measure ,
Another example may be , the algebra of matrices with the functional given by the normalized trace . Even more generally, could be a von Neumann algebra and a state on . A final example is the group algebra of a (discrete) group with the functional given by the group trace .
Let be a family of unital subalgebras of .
Definition. The family is called freely independent if
whenever , and .
If , is a family of elements of (these can be thought of as random variables in ), they are called
freely independent if the algebras generated by and are freely independent.
Examples of free independence
Let be the free product of groups , let be the group algebra, be the group trace, and set . Then are freely independent.
Let be unitary random matrices, taken independently at random from the unitary group (with respect to the Haar measure). Then become asymptotically freely independent as . (Asymptotic freeness means that the definition of freeness holds in the limit as ).
More generally, independent random matrices tend to be asymptotically freely independent, under certain conditions.
References
Sources
James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
Functional analysis
Free probability theory
Free algebraic structures |
https://en.wikipedia.org/wiki/Joint%20spectral%20radius | In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research.
General description
The joint spectral radius of a set of matrices is the maximal asymptotic growth rate of products of matrices taken in that set. For a finite (or more generally compact) set of matrices the joint spectral radius is defined as follows:
It can be proved that the limit exists and that the quantity actually does not depend on the chosen matrix norm (this is true for any norm but particularly easy to see if the norm is sub-multiplicative). The joint spectral radius was introduced in 1960 by Gian-Carlo Rota and Gilbert Strang, two mathematicians from MIT, but started attracting attention with the work of Ingrid Daubechies and Jeffrey Lagarias. They showed that the joint spectral radius can be used to describe smoothness properties of certain wavelet functions. A wide number of applications have been proposed since then. It is known that the joint spectral radius quantity is NP-hard to compute or to approximate, even when the set consists of only two matrices with all nonzero entries of the two
matrices which are constrained to be equal. Moreover, the question "" is an undecidable problem. Nevertheless, in recent years much progress has been done on its understanding, and it appears that in practice the joint spectral radius can often be computed to satisfactory precision, and that it moreover can bring interesting insight in engineering and mathematical problems.
Computation
Approximation algorithms
In spite of the negative theoretical results on the joint spectral radius computability, methods have been proposed that perform well in practice. Algorithms are even known, which can reach an arbitrary accuracy in an a priori computable amount of time. These algorithms can be seen as trying to approximate the unit ball of a particular vector norm, called the extremal norm. One generally distinguishes between two families of such algorithms: the first family, called polytope norm methods, construct the extremal norm by computing long trajectories of points. An advantage of these methods is that in the favorable cases it can find the exact value of the joint spectral radius and provide a certificate that this is the exact value.
The second family of methods approximate the extremal norm with modern optimization techniques, such as ellipsoid norm approximation, semidefinite programming, Sum Of Squares, and conic programming. The advantage of these methods is that they are easy to implement, and in practice, they provide in general the best bounds on the joint spectral radius.
The finiteness conjecture
Related to the computability of the joint spectral radius is the following conjecture:
"For any finite set of matrices there is a product of matrices |
https://en.wikipedia.org/wiki/Leconte%20Prize | The Leconte Prize (French: ) is a prize created in 1886 by the French Academy of Sciences to recognize important discoveries in mathematics, physics, chemistry, natural history or medicine. In recent years the prize has been awarded in the specific categories of mathematics, physics, and biology. Scientists and mathematicians of all nationalities are eligible for the award. The value of the award in the late 19th and early 20th century was F50,000 (at the time equivalent to £2,000, or US$10,000), about five times as much as the annual salary of the average professor in France. The award was F22,000 in 1984, F20,000 in 2001, €3,000 in 2008, €2,500 in 2010, €2,000 in 2014, and €1,500 in 2019.
The Leconte Prize was established with a donation from a businessman, Victor Eugene Leconte, to the academy. The donation specified that a F50,000 prize would be awarded every three years for outstanding past work, and that up to 1/8th of the interest earned by the fund each year could be awarded as encouragements, i.e., support for ongoing and future research. The academy did not award any large (F50,000) prizes between 1905 and 1916, but did award a total of F30,000 in encouragements during that period.
Recipients
See also
List of biology awards
List of mathematics awards
List of physics awards
References
Physics awards
Mathematics awards
Biology awards
Awards of the French Academy of Sciences
1886 establishments in France
Awards established in 1886 |
https://en.wikipedia.org/wiki/Schoenflies%20problem | In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
Original formulation
The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane.
An alternative statement is that if is a simple closed curve, then there is a homeomorphism such that is the unit circle in the plane. Elementary proofs can be found in , , and . The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case of a continuous curve is then deduced by approximating by polygons. The theorem is also an immediate consequence of Carathéodory's extension theorem for conformal mappings, as discussed in .
If the curve is smooth then the homeomorphism can be chosen to be a diffeomorphism. Proofs in this case rely on techniques from differential topology. Although direct proofs are possible (starting for example from the polygonal case), existence of the diffeomorphism can also be deduced by using the smooth Riemann mapping theorem for the interior and exterior of the curve in combination with the Alexander trick for diffeomorphisms of the circle and a result on smooth isotopy from differential topology.
Such a theorem is valid only in two dimensions. In three dimensions there are counterexamples such as Alexander's horned sphere. Although they separate space into two regions, those regions are so twisted and knotted that they are not homeomorphic to the inside and outside of a normal sphere.
Proofs of the Jordan–Schoenflies theorem
For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way. Indeed, the curve has a tubular neighbourhood, defined in the smooth case by the field of unit normal vectors to the curve or in the polygonal case by points at a distance of less than ε from the curve.
In a neighbourhood of a differentiable point on the curve, there is a coordinate change in which the curve becomes the diameter of an open disk. Taking a point not on the curve, a straight line aimed at the curve starting at the point will eventually meet the tubular neighborhood; the path can be continued next to the curve until it meets the disk. It will meet it on one side or the other. This proves that the complement of the curve has at most two connected components. On the other hand, using the Cauchy integral formula for the winding number, it can be seen that the winding number is constant on connected components of the complement of the curve, is zero near infinity and increases by 1 when crossing the curve. He |
https://en.wikipedia.org/wiki/Orthogonal%20diagonalization | In linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates.
The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on Rn by means of an orthogonal change of coordinates X = PY.
Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial
Step 2: find the eigenvalues of A which are the roots of .
Step 3: for each eigenvalue of A from step 2, find an orthogonal basis of its eigenspace.
Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of Rn.
Step 5: let P be the matrix whose columns are the normalized eigenvectors in step 4.
Then X=PY is the required orthogonal change of coordinates, and the diagonal entries of will be the eigenvalues which correspond to the columns of P.
References
Maxime Bôcher (with E.P.R. DuVal)(1907) Introduction to Higher Algebra, § 45 Reduction of a quadratic form to a sum of squares via HathiTrust
Linear algebra |
https://en.wikipedia.org/wiki/Sled%20Lake%2C%20Saskatchewan | Sled Lake is a northern settlement in Northern Saskatchewan. The community is accessed from Highway 924.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Sled Lake had a population of 24 living in 12 of its 27 total private dwellings, a change of from its 2016 population of 10. With a land area of , it had a population density of in 2021.
See also
List of communities in Northern Saskatchewan
List of communities in Saskatchewan
References
Designated places in Saskatchewan
Division No. 18, Unorganized, Saskatchewan
Northern settlements in Saskatchewan |
https://en.wikipedia.org/wiki/Salem%20Abdullah%20%28footballer%2C%20born%201986%29 | Salem Abdullah Omar Salem Ba Abdullah (; born 5 July 1986) is an Emarati footballer who plays as a midfielder.
External links
Salem Statistics At Goalzz.com
http://www.alainfc.net/en/index.php?p=playerinfo&pid=343
Emirati men's footballers
Al Ain FC players
Al Wasl F.C. players
Al Jazira Club players
Living people
1986 births
UAE Pro League players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Fawzi%20Fayez | Fawzi Fayez Subait Khalifa Alalawi (; born 14 July 1987) is an Emirati footballer who plays as a right back.
He played 3 times at 2010 AFC Champions League.
External links
Fawzi Statistics At Goalzz.com
1987 births
Living people
Emirati men's footballers
United Arab Emirates men's international footballers
Al Ain FC players
Sharjah FC players
Khor Fakkan Club players
Al Urooba Club players
Al Jazirah Al Hamra Club players
UAE Pro League players
UAE First Division League players
Men's association football fullbacks |
https://en.wikipedia.org/wiki/Sultan%20Musabah | Sultan Musabah is an Emirati footballer.
External links
Sultan Profile At Al Ain Fc.net Official Site
Sultan Statistics At Goalzz.com
Emirati men's footballers
Al Ain FC players
Baniyas Club players
Living people
UAE Pro League players
1988 births
Men's association football goalkeepers
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Ahmed%20Moadhed | Ahmed Ali Moadhed Al Samahi (born 13 December 1980) is an Emarati footballer .
Moadhed play all six matches at 2010 AFC Champions League.
External links
Ahmad Statistics At Goalzz.com
http://www.alainfc.net/en/index.php?p=playerinfo&pid=336
Emirati men's footballers
Al Ain FC players
Shabab Al Ahli Club players
Fujairah FC players
Masafi Club players
1980 births
Living people
UAE First Division League players
UAE Pro League players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Rami%20Yaslam | Rami Yaslam Al-Jaberi (born 11 June 1981) is an Emirati footballer.
External links
Rami Statistics At Goalzz.com
Rami Yaslam Player Information, Al Ain Sports & Cultural Club
1981 births
Living people
Emirati men's footballers
Al Ain FC players
Dubai CSC players
2004 AFC Asian Cup players
UAE Pro League players
Men's association football midfielders
United Arab Emirates men's international footballers |
https://en.wikipedia.org/wiki/Nasser%20Khamis | Nasser Khamis Abdullah Hassan (born 20 December 1987) is an Emarati footballer who plays as a striker .
External links
Nasser Statistics At Goalzz.com
References
1987 births
Living people
Place of birth missing (living people)
Emirati men's footballers
Al Ain FC players
Al-Shaab CSC players
Dubai CSC players
Al Dhaid SC players
Masfout Club players
Hatta Club players
Ajman Club players
Al Urooba Club players
UAE First Division League players
UAE Pro League players
Men's association football forwards |
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