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https://en.wikipedia.org/wiki/1957%E2%80%9358%20American%20Soccer%20League | Statistics of American Soccer League II in season 1957–58.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1957-58 |
https://en.wikipedia.org/wiki/1958%E2%80%9359%20American%20Soccer%20League | Statistics of American Soccer League II in season 1958–59.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1958-59 |
https://en.wikipedia.org/wiki/1959%E2%80%9360%20American%20Soccer%20League | Statistics of American Soccer League II in season 1959–60.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1959-60 |
https://en.wikipedia.org/wiki/1960%E2%80%9361%20American%20Soccer%20League | Statistics of American Soccer League II in season 1960–61.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1960-61 |
https://en.wikipedia.org/wiki/1961%20International%20Soccer%20League | Statistics of International Soccer League II in season 1961.
League standings
Section I
Section II
Matches
Championship finals
First leg
Second leg
Dukla Prague won 9–2 on aggregate.
References
Series on "Pitch Invasion" by Tom Dunmore from 2011:
Part 3: Expanded Dreamsn
Part 4: Struggling Towards Orbit
International Soccer League seasons
International Soccer League, 1961 |
https://en.wikipedia.org/wiki/1962%20International%20Soccer%20League | Statistics of International Soccer League III in season 1962.
League standings
Section I
Section II
Championship finals
First leg
Second leg
América won 3–1 on aggregate.
American Challenge Cup
FK Dukla Prague defeated America-RJ, 1–1 and 2–1, on goal aggregate.
References
International Soccer League seasons
International Soccer League, 1962 |
https://en.wikipedia.org/wiki/1962%E2%80%9363%20American%20Soccer%20League | Statistics of American Soccer League II in season 1962–63.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1962-63 |
https://en.wikipedia.org/wiki/1963%E2%80%9364%20American%20Soccer%20League | Statistics of American Soccer League II in season 1963–64.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1963-64 |
https://en.wikipedia.org/wiki/1964%20International%20Soccer%20League | Statistics of International Soccer League in season 1964.
League standings
Section I
Section II
Championship finals
First leg
Second leg
Zagłębie Sosnowiec won 5–0 on aggregate.
American Challenge Cup
FK Dukla Prague defeated Zagłębie Sosnowiec 3–1 and 1–1, on goal aggregate.
References
International Soccer League seasons
International Soccer League, 1964 |
https://en.wikipedia.org/wiki/1964%E2%80%9365%20American%20Soccer%20League | Statistics of American Soccer League II in season 1964–65.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1964-65 |
https://en.wikipedia.org/wiki/1964%E2%80%9365%20Eastern%20Professional%20Soccer%20Conference%20season | Statistics of Eastern Professional Soccer Conference in season 1964/1965.
League standings
G W T L GF GA PTS
North Division
BW Gottschee 16 9 1 6 27 22 19
German-Hungarians 14 8 2 4 37 27 18
Giuliana 15 6 5 4 31 26 17
New York Hungaria 15 7 3 5 34 29 17
Boston Metros 13 5 3 5 24 21 13
Newark Ukrainians 15 1 2 12 13 42 4
South Division
New York Ukrainians 15 9 3 3 33 12 21
New York Inter 16 8 5 3 31 28 21
Ukrainian Nationals 14 7 5 2 30 14 19
New York Hota 15 6 5 4 21 17 17
New York Americans 16 4 4 8 21 34 12
Greek-Americans 13 3 3 7 21 30 9
Minerva-Pfuelzer 17 3 1 13 15 36 7
References
EASTERN PROFESSIONAL SOCCER CONFERENCE (RSSSF)
Eastern |
https://en.wikipedia.org/wiki/1965%20International%20Soccer%20League | Statistics of International Soccer League in season 1965.
League standings
Section I
Section II
Championship finals
First leg
Second leg
Polonia Bytom won 5–1 on aggregate.
American Challenge Cup
Polonia Bytom defeated FK Dukla Prague, 2–0 and 1–1, on goal aggregate.
References
International Soccer League seasons
International Soccer League, 1965 |
https://en.wikipedia.org/wiki/1963%20International%20Soccer%20League | Statistics of International Soccer League in season 1963.
League standings
Section I
Section II
Championship finals
First leg
Second leg
West Ham United won 2–1 on aggregate.
American Challenge Cup
FK Dukla Prague defeated West Ham United, 1–0 and 1–1, on goal aggregate.
References
International Soccer League seasons
International Soccer League, 1963 |
https://en.wikipedia.org/wiki/1965%E2%80%9366%20American%20Soccer%20League | Statistics of American Soccer League II in season 1965–66.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
American Soccer League, 1965-66 |
https://en.wikipedia.org/wiki/1966%E2%80%9367%20American%20Soccer%20League | Statistics of American Soccer League II in season 1966–67.
League standings
Championship final
References
American Soccer League II (RSSSF)
American Soccer League 1967
American Soccer League (1933–1983) seasons
1965–66 in American soccer
2 |
https://en.wikipedia.org/wiki/1967%E2%80%9368%20American%20Soccer%20League | Statistics of American Soccer League II in season 1967–68.
Overview
For the 1967–68 season the league set up a two division system to work as a pro/reg tandem, with the First Division being the higher tier. Ultimately pro/reg wound up occurring only on paper because Hartford S.C withdrew after five games, and by season's end Boston, Baltimore, Newark Portuguese, Patterson Roma, and New Brunswick all folded.
Nearly all of the season was played in the later part of 1967, with a handful of make-up games and the playoffs scheduled for early 1968. Weather issues, a slate of regional cup and National Challenge Cup matches, friendlies against both NASL and international sides, and the apparent disorganization of the league itself conspired to continually reschedule those few remaining regular season matches until April. Individually, notching his 15th goal, Rochester's Nelson Bergamo overtook Ivan Paletta of Philadelphia's Ukrainian Nationals for the ASL scoring title in the Lancers' final match of the season on April 28.
The finals were played more than a month later in late May and early June. Newspaper stories at the time indicate that because the Boston Tigers had already closed up shop, the league quickly adapted on the fly and matched First Division winners, Ukrainian Nationals of Philadelphia with Premier Division champions, New York Inter, instead playing semifinal matches, as was originally planned.
The Uke-Nats won the two-legged final, 8–4, on aggregate.
League standings
Championship final
First leg
Second leg
1969 ASL Champions: Ukrainian Nationals (8–4 aggregate)
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2
2 |
https://en.wikipedia.org/wiki/1968%20American%20Soccer%20League | Statistics of American Soccer League II in season 1968.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1969%20American%20Soccer%20League | Statistics of the American Soccer League II for the 1969 season.
League standings
ASL All-Stars
Playoffs
Bracket
Northern Division playoff
Championship final
First leg
Second leg
1969 ASL Champions: Washington Darts (4-0 aggregate)
Season awards
Most Valuable Player: Jim Lefkos, Syracuse
Coach of the Year: Lincoln Phillips, Washington
Manager of the Year: Walter Bahr, Philadelphia Spartans
Rookie of the Year: Bob DiLuca, Rochester
Most Improved Player: Jerry Kliveka, Philadelphia Spartans
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1970%20American%20Soccer%20League | Statistics of American Soccer League II in season 1970.
League standings
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1972%20American%20Soccer%20League | Statistics of the American Soccer League II for the 1972 season.
League standings
Playoffs
Bracket
First round
Eastern playoff
Championship final
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1973%20American%20Soccer%20League | Statistics of the American Soccer League II for the 1973 season.
League standings
Playoffs
Bracket
Semifinals
Final
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1974%20American%20Soccer%20League | Statistics of the American Soccer League II for the 1974 season.
League standings
Playoffs
Bracket
#Play suspended after extra time because of weather.
Semifinals
*Boston Astros forfeit for refusing to play overtime.
ASL Championship Series
The ASL championship was set as a two-match aggregate, with overtime to be played after the second leg to break a tie. Game 2 finished regulation with the teams tied on aggregate, 3–3. With the first overtime completed, a violent thunderstorm flooded the field and knocked out the stadium lights, effectively ending the match. A week later the match was replayed. After the Oceaneers' Rich Kratzer tied it late in regulation, the teams again went to extra time. In the second half of extra time Rhode Island got goals from Mohammad Attiah and Charlie McCully for the win, before New York added a final tally late.
#Play suspended after first overtime because of weather.
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1975%20American%20Soccer%20League | Statistics of American Soccer League II in season 1975.
League standings
Playoffs
Bracket
Semifinals
Championship
*Play suspended after 9 sudden-death overtimes; teams declared co-champions by commissioner Bob Cousy.
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1976%20American%20Soccer%20League | Statistics of American Soccer League II in season 1976.
League standings
Scoring system is as follows: Teams are awarded five points for a win and two points for a draw. Teams earn a bonus point for each goal scored up to three.
ASL All-Stars
Playoffs
Bracket
Division semifinals
Division finals
Championship final
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1977%20American%20Soccer%20League | Statistics of American Soccer League II in season 1977.
League standings
ASL All-Stars
Playoffs
Bracket
Division semifinals
Division finals
Championship final
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1978%20American%20Soccer%20League | Statistics of American Soccer League II in season 1978.
League standings
ASL All-Stars
Playoffs
Bracket
Division semifinals
Division finals
Championship final
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1979%20American%20Soccer%20League | Statistics of American Soccer League II in season 1979.
League standings
ASL All-Stars
Playoffs
Bracket
Division semifinals
Division finals
Championship final
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2
1979 in American soccer |
https://en.wikipedia.org/wiki/1980%20American%20Soccer%20League | Statistics of American Soccer League II in season 1980.
League standings
ASL All-Stars
Playoffs
Bracket
Conference finals
Championship final
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1981%20American%20Soccer%20League | Statistics of American Soccer League II in season 1981.
League standings
ASL All-Stars
Playoffs
Bracket
1st Round
Semifinals
Championship final
Post season awards
Most Valuable Player: Billy Boljevic, NY Eagles
Coach of the year: Jimmy McGeough, NY United
Rookie of the year: Tony Suarez, Carolina
Executive of the Year: Robert Benson, Carolina
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1982%20American%20Soccer%20League | Statistics of the American Soccer League II for the 1982 season.
League standings
Playoffs
Bracket
1st Round
Semi-finals
ASL Championship
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1983%20American%20Soccer%20League | Statistics of the American Soccer League II for the 1983 season.
League standings
Playoffs
Bracket
Semi-finals
ASL Championship
References
American Soccer League II (RSSSF)
American Soccer League (1933–1983) seasons
2 |
https://en.wikipedia.org/wiki/1992%20American%20Professional%20Soccer%20League | Final league standings for the 1992 American Professional Soccer League season.
Regular season
Playoffs
Bracket
Semifinal 1
Semifinal 2
Final
Match statistics
Professional Cup
All five APSL teams took part in the Professional Cup, along with two teams from the Canadian Soccer League and one from the National Professional Soccer League. Just as they would in the APSL Final a week later, Colorado defeated Tampa Bay. This combination, along with winning the 1992 APSL regular season, gave the Foxes a treble.
Professional Cup Final
Points leaders
Honors
MVP: Taifour Diané
Leading goal scorer: Jean Harbor
Leading goalkeeper: Mark Dodd
Rookie of the Year: Taifour Diané
Coach of the Year: Ricky Hill
All-League Best XI
References
External links
The Year in American Soccer - 1992
USA - A-League (American Professional Soccer League) (RSSSF)
APSL/A-League seasons
1
1992 |
https://en.wikipedia.org/wiki/1993%20American%20Professional%20Soccer%20League | Statistics of American Professional Soccer League in season 1993.
History
In 1993, the league added three teams from Canada. The Canadian Soccer League had collapsed at the end of the 1992 season and the Vancouver 86ers and Toronto Blizzard along with a new club the Montreal Impact moved to the APSL. Vancouver topped the regular season standings, but fell in the playoff semifinals to the Los Angeles Salsa. In the other semifinal, the Colorado Foxes defeated the Tampa Bay Rowdies. Although the Foxes had a better record than the Salsa, the championship game took place in Los Angeles because the Foxes home stadium of Englewood High School had a homecoming football game the night of the championship.
In 1993 before the USSF chose MLS as Division 1, a couple teams had significant capital backing, had local TV and radio deals, and many of the players were US national team hopefuls or Canadian internationals.
Regular season
The competition was a single table on the league principle with a balanced schedule home and away where each of the seven teams plays the other six four times. The league`s regular season was played over twenty weeks, beginning April 30 and concluding Sept. 12. The top four in the table qualified for a single-elimination tournament held in September. The league was a generally close competition, given the points system adopted all teams were still in the playoff race into early August or about 70% of the season. The points system included 6pts for a win, 4pts for a shootout win, 2pts for a shootout loss, and bonus points for goals to a maximum of three. If the game was tied, then instead of following FIFA rules of two 15-minute extra halves followed by penalty kicks, the APSL did two 7.5 minute extra halves followed by the NASL shootout. The shootout consisted of the player starting 35 yards from the net, goalkeeper in net, and five seconds for the player to score (essentially a timed five second break-away skills competition). Game day rosters had to have eleven of the eighteen as domestic players.
Playoffs
Bracket
Semifinal 1
Semifinal 2
Final
Points leaders
Honors
MVP: Paulinho
Leading goal scorer: Paulinho
Leading goalkeeper: Jim St. Andre
Rookie of the Year: Jason De Vos
Coach of the Year: Ken Fogarty
First Team All League
Goalkeeper: Paul Dolan
Defenders: Robin Fraser, Danny Pena, Mark Watson, Patrice Ferri
Midfielders: Paulinho, Ivor Evans, Paul Dougherty, Ted Eck
Forwards: Taifour Diané, Paul Wright
Second Team All League
Goalkeeper: Ian Feuer
Defenders: Jason De Vos, Steve MacDonald, Kim Roentved, Mark Santel
Midfielders: Chad Ashton, Bryan Haynes, Pierre Morice, Steve Trittschuh
Forwards: Domenic Mobilio, Hector Marinaro
References
External links
The Year in American Soccer - 1993
USA - A-League (American Professional Soccer League) (RSSSF)
APSL/A-League seasons
1
1993 in Canadian soccer |
https://en.wikipedia.org/wiki/2000%20USL%20D3%20Pro%20League | Statistics of USL D3 Pro League in season 2000.
League standings
GP W L D GF GA BP Pts
Northern Division
New Jersey Stallions 18 14 3 1 43 21 7 64
South Jersey Barons 18 12 6 0 46 33 9 57
New Hampshire Phantoms 18 10 6 2 30 20 5 47
Reading Rage 18 9 9 0 28 29 4 40
Western Mass Pioneers 18 8 8 2 30 28 4 38
Cape Cod Crusaders 18 7 6 5 29 23 4 37
Delaware Wizards 18 5 12 1 28 46 5 26
Rhode Island Stingrays 18 5 11 2 22 41 2 24
Southern Division
Texas Rattlers 18 14 4 0 43 29 8 64
Wilmington Hammerheads 18 13 4 1 47 19 7 60
Charlotte Eagles 18 11 4 3 47 22 7 54
Carolina Dynamo 18 9 8 1 38 31 5 42
Houston Hurricanes 18 8 9 1 42 41 9 42
Roanoke Wrath 18 6 11 1 24 45 4 29
Northern Virginia Royals 18 6 12 0 30 53 4 28
Austin Lone Stars 18 2 15 1 18 45 2 11
Western Division
Chico Rooks 18 12 6 0 39 28 8 56
Utah Blitzz 18 11 4 3 34 20 7 54
Tucson Fireballs 18 11 5 2 36 27 7 53
Stanislaus United Cruisers 18 7 10 1 35 32 5 34
Riverside County Elite 18 7 10 1 35 22 5 34
Arizona Sahuaros 18 5 12 1 46 53 7 28
Conference Quarterfinals: Western Massachusetts defeated South Jersey 3-2(OT)
Reading defeated New Hampshire 3–2.
Carolina defeated Texas 4–2.
Charlotte defeated Wilmington 3–1.
Chico defeated Stanislaus County 3–2.
Utah defeated Tulsa 1–0.
Conference Semifinals: Utah defeated Chico 1-0 (OT)
Western Massachusetts defeated Reading 4–0.
Charlotte defeated Carolina 4–1.
Conference Finals: Charlotte defeated Utah 4–2.
New Jersey defeated Western Massachusetts 1–0.
CHAMPIONSHIP: Charlotte defeated New Jersey 5–0.
Playoffs
Conference semifinals
Conference Finals
League Semi-Finals
D3 Pro League Championship Game
References
2000
3 |
https://en.wikipedia.org/wiki/2001%20USL%20D3%20Pro%20League | Statistics of USL D3 Pro League in season 2001.
League standings
Playoffs
Conference semi-finals
Boston defeated Reading, 6-0
New Jersey defeated New Hampshire 4-2
Greenville defeated Carolina 2-1
Stanlslaus defeated Arizona 2-1
Chico defeated Tucson 5-1
Conference Finals/Quarterfinals
Boston defeated New Jersey 2-0
Greenville defeated Wilmington 3-3 (4-3 PK)
Stanislaus United defeated Chico 2-1
Semifinals
Greeneville defeated Boston 2-1 (OT)
Utah defeated Stanislaus United 3-0
USL D3 Pro League Championship
Utah defeated Greenville 1-0
References
3
2001
Sports events affected by the September 11 attacks |
https://en.wikipedia.org/wiki/2002%20USL%20D3%20Pro%20League | Statistics of the USL D3 Pro League for the 2002 season.
League standings
Northern Conference
Atlantic Conference
Southern Conference
Western Conference
Playoffs
First Round
San Diego defeated Arizona, 3-0.
Carolina defeated Greenville, 1-0.
Connecticut defeated Western Massachusetts 0-2, 3-2
Connecticut Wolves advance win a 5–4 point difference in a Best-of-2 Series.
Conference Finals
Long Island defeated New York 2-2 (5-4 PK)
Wilmington defeated Carolina 2-1
Utah defeated San Diego 1-0
Semifinals
Wilmington defeated Utah 3-1
Long Island defeated Connecticut 3-0
USL D3 Pro League Championship Game
Long Island defeated Wilmington 2-1
References
3
2002 |
https://en.wikipedia.org/wiki/Fibrifold | In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by , who introduced a system of notation for 3-dimensional fibrifolds and used this to assign names to the 219 affine space group types. 184 of these are considered reducible, and 35 irreducible.
Irreducible cubic space groups
The 35 irreducible space groups correspond to the cubic space group.
Irreducible group symbols (indexed 195−230) in Hermann–Mauguin notation, Fibrifold notation, geometric notation, and Coxeter notation:
References
Symmetry
Finite groups
Discrete groups |
https://en.wikipedia.org/wiki/List%20of%20Major%20League%20Baseball%20career%20ERA%20leaders | In baseball statistics, earned run average (ERA) is the mean of earned runs given up by a pitcher per nine innings pitched (i.e. the traditional length of a game). It is determined by dividing the number of earned runs allowed by the number of innings pitched and multiplying by nine. Runs resulting from defensive errors (including pitchers' defensive errors) are recorded as unearned runs and are not used to determine ERA.
This is a list of the top 100 players in career earned run average, who have thrown at least 1,000 innings.
Ed Walsh holds the MLB earned run average record with a 1.816. Addie Joss (1.887) and Jim Devlin (1.896) are the only other pitchers with a career earned run average under 2.000.
Key
List
Stats updated as of September 30, 2023.
See also
Baseball statistics
List of Major League Baseball annual ERA leaders
Notes
References
External links
ERA
ERA in career |
https://en.wikipedia.org/wiki/Distance%20from%20a%20point%20to%20a%20line | In Euclidean geometry, the distance from a point to a line is the shortest distance from a given point to any point on an infinite straight line. It is the perpendicular distance of the point to the line, the length of the line segment which joins the point to nearest point on the line. The algebraic expression for calculating it can be derived and expressed in several ways.
Knowing the distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on a graph, etc. In Deming regression, a type of linear curve fitting, if the dependent and independent variables have equal variance this results in orthogonal regression in which the degree of imperfection of the fit is measured for each data point as the perpendicular distance of the point from the regression line.
Line defined by an equation
In the case of a line in the plane given by the equation , where , and are real constants with and not both zero, the distance from the line to a point is
The point on this line which is closest to has coordinates:
Horizontal and vertical lines
In the general equation of a line, , and cannot both be zero unless is also zero, in which case the equation does not define a line. If and , the line is horizontal and has equation . The distance from to this line is measured along a vertical line segment of length in accordance with the formula. Similarly, for vertical lines (b = 0) the distance between the same point and the line is , as measured along a horizontal line segment.
Line defined by two points
If the line passes through two points and then the distance of from the line is:
The denominator of this expression is the distance between and . The numerator is twice the area of the triangle with its vertices at the three points, , and . See: . The expression is equivalent to , which can be obtained by rearranging the standard formula for the area of a triangle: , where is the length of a side, and is the perpendicular height from the opposite vertex.
Line defined by point and angle
If the line passes through the point with angle , then the distance of some point to the line is
Proofs
An algebraic proof
This proof is valid only if the line is neither vertical nor horizontal, that is, we assume that neither nor in the equation of the line is zero.
The line with equation has slope , so any line perpendicular to it will have slope (the negative reciprocal). Let be the point of intersection of the line and the line perpendicular to it which passes through the point (, ). The line through these two points is perpendicular to the original line, so
Thus,
and by squaring this equation we obtain:
Now consider,
using the above squared equation. But we also have,
since is on .
Thus,
and we obtain the length of the line segment determined by these two points,
A geometric proof
This proof is valid only if the line is not horizontal or ver |
https://en.wikipedia.org/wiki/Andr%C3%A9%E2%80%93Quillen%20cohomology | In commutative algebra, André–Quillen cohomology is a theory of cohomology for commutative rings which is closely related to the cotangent complex. The first three cohomology groups were introduced by and are sometimes called Lichtenbaum–Schlessinger functors T0, T1, T2, and the higher groups were defined independently by and using methods of homotopy theory. It comes with a parallel homology theory called André–Quillen homology.
Motivation
Let A be a commutative ring, B be an A-algebra, and M be a B-module. The André–Quillen cohomology groups are the derived functors of the derivation functor DerA(B, M). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings and a C-module M, there is a three-term exact sequence of derivation modules:
This term can be extended to a six-term exact sequence using the functor Exalcomm of extensions of commutative algebras and a nine-term exact sequence using the Lichtenbaum–Schlessinger functors. André–Quillen cohomology extends this exact sequence even further. In the zeroth degree, it is the module of derivations; in the first degree, it is Exalcomm; and in the second degree, it is the second degree Lichtenbaum–Schlessinger functor.
Definition
Let B be an A-algebra, and let M be a B-module. Let P be a simplicial cofibrant A-algebra resolution of B. André notates the qth cohomology group of B over A with coefficients in M by , while Quillen notates the same group as . The qth André–Quillen cohomology group is:
Let denote the relative cotangent complex of B over A. Then we have the formulas:
See also
Cotangent complex
Deformation Theory
Exalcomm
References
Generalizations
André–Quillen cohomology of commutative S-algebras
Homology and Cohomology of E-infinity ring spectra
Commutative algebra
Homotopy theory
Cohomology theories |
https://en.wikipedia.org/wiki/2009%20Air%20New%20Zealand%20Cup%20statistics | The 2009 Air New Zealand Cup ran from 30 July to 7 November. This page includes all statistics from the 14 teams during the 13 rounds of the round-robin.
There were 3953 points scored with an average of 43.4 points per game; there also were 406 tries scored.
Hawke's Bay scored the most points with 372 and, along with Canterbury and Wellington, the most tries with 40, while Southland had the best defensive record through the competition with only 189 points scored against them through 13 rounds.
Matt Berquist, from Hawke's Bay, scored the most points out of every player in the competition - with 156 points and an average of 14.2 points through his 11 matches. Zac Guildford scored the most tries this season with 13.
Overall
Team
The lists showing all statistics for all teams about points, tries and disciplinary cards.
Points
The table showing how many points scored by each team (white) and how many points each team was scored against them (grey) in the 2009 Air New Zealand Cup Round Robin. Hawke's Bay scored the most points so far with 372 (28.6 points a game), while Southland had the fewest points scored against them with 189 (14.5 points a game).
Tries
The list of how many tries each team scored in the round robin. Canterbury, Wellington and Hawke's Bay scored the most tries this season with 40 each while Taranaki, North Harbour and Northland scored the fewest with 20 each.
Competition Points
List of how many competitions points each team scored with overall and week totals.
Disciplinary cards
List of teams whose players received yellow and/or red cards. Bay of Plenty were issued the most cards with 5 while North Harbour were issued the only red card of the season.
Player
The list of the top players who have scored the most points and tries in the 2009 Air New Zealand Cup. There were 3,953 points including 406 tries scored with a total of 218 players scoring them, Matt Berquist leads them all with 156 points and an average of 14.2 points per game. There have also been 33 yellow cards and 1 red card issued.
Top Ten Points Scorers
A total of 218 players from each team scored points in the round robin, Matt Berquist has scored the most with 156 and average of 14.2 points per game.
Top Try Scorers
A total of 218 players have scored a total of 406 tries. Zac Guildford has scored the most tries by a player this season with 13.
Top Goal Kickers
Mathew Berquist leads all goal kickers this season with 82.2% success rate.
Disciplinary Cards
The list of all players who have received a yellow or red card in the 2009 Air New Zealand Cup. Luke Braid and Colin Bourke, both from Bay of Plenty are the players who received the most disciplinary cards with two yellows each.
Individual Team Statistics
The lists showing each teams; points scorers, try scorers, goal kickers and disciplinary card recipients where available.
Auckland
Auckland scored 272 total points this season including 29 tries. Ash Moeke led them with 75 points scored through 13 games |
https://en.wikipedia.org/wiki/Murod%20Zukhurov | Murod Zukhurov is an Uzbekistani retired goalkeeper. He was born on 23 February 1983 in Tashkent.
Zukhurov played for Bunyodkor in the 2009 AFC Champions League group stage.
Career statistics
Club
International
Statistics accurate as of match played 10 September 2013
References
1983 births
Living people
Uzbekistani men's footballers
PFK Metallurg Bekabad players
Navbahor Namangan players
FC Nasaf players
FC Bunyodkor players
2011 AFC Asian Cup players
Men's association football goalkeepers
Uzbekistan Super League players
Uzbekistan men's international footballers |
https://en.wikipedia.org/wiki/Bonse%27s%20inequality | In number theory, Bonse's inequality, named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that if p1, ..., pn, pn+1 are the smallest n + 1 prime numbers and n ≥ 4, then
(the middle product is short-hand for the primorial of pn)
Mathematician Denis Hanson showed an upper bound where .
See also
Primorial prime
Notes
References
Theorems about prime numbers
Inequalities |
https://en.wikipedia.org/wiki/Chord%20function | The term chord function may refer to:
Diatonic function – in music, the role of a chord in relation to a diatonic key;
In mathematics, the length of a chord of a circle as a trigonometric function of the length of the corresponding arc; see in particular Ptolemy's table of chords. |
https://en.wikipedia.org/wiki/Exponential-logarithmic%20distribution | In probability theory and statistics, the Exponential-Logarithmic (EL) distribution is a family of lifetime distributions with
decreasing failure rate, defined on the interval [0, ∞). This distribution is parameterized by two parameters and .
Introduction
The study of lengths of the lives of organisms, devices, materials, etc., is of major importance in the biological and engineering sciences. In general, the lifetime of a device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering terms) or 'immunity' (in biological terms).
The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008).
This model is obtained under the concept of population heterogeneity (through the process of
compounding).
Properties of the distribution
Distribution
The probability density function (pdf) of the EL distribution is given by Tahmasbi and Rezaei (2008)
where and . This function is strictly decreasing in and tends to zero as . The EL distribution has its modal value of the density at x=0, given by
The EL reduces to the exponential distribution with rate parameter , as .
The cumulative distribution function is given by
and hence, the median is given by
.
Moments
The moment generating function of can be determined from the pdf by direct integration and is given by
where is a hypergeometric function. This function is also known as Barnes's extended hypergeometric function. The definition of is
where and .
The moments of can be derived from . For
, the raw moments are given by
where is the polylogarithm function which is defined as
follows:
Hence the mean and variance of the EL distribution
are given, respectively, by
The survival, hazard and mean residual life functions
The survival function (also known as the reliability
function) and hazard function (also known as the failure rate
function) of the EL distribution are given, respectively, by
The mean residual lifetime of the EL distribution is given by
where is the dilogarithm function
Random number generation
Let U be a random variate from the standard uniform distribution.
Then the following transformation of U has the EL distribution with
parameters p and β:
Estimation of the parameters
To estimate the parameters, the EM algorithm is used. This method is discussed by Tahmasbi and Rezaei (2008). The EM iteration is given by
Related distributions
The EL distribution has been generalized to form the Weibull-logarithmic distribution.
If X is defined to be the random variable which is the minimum of N independent realisations from an exponential distribution with rate parameter β, and if N is a realisation from a logarithmic distribution (where the parameter p in the usual parameterisation is replaced by ), then X has the exponential-logarithmic distribution in the parameterisation used above.
References
Continuous distributions
Su |
https://en.wikipedia.org/wiki/Jacobi%20form | In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp(n;R) and the Heisenberg group . The theory was first systematically studied by .
Definition
A Jacobi form of level 1, weight k and index m is a function of two complex variables (with τ in the upper half plane) such that
for all integers λ, μ.
has a Fourier expansion
Examples
Examples in two variables include Jacobi theta functions, the Weierstrass ℘ function, and Fourier–Jacobi coefficients of Siegel modular forms of genus 2. Examples with more than two variables include characters of some irreducible highest-weight representations of affine Kac–Moody algebras. Meromorphic Jacobi forms appear in the theory of Mock modular forms.
References
Modular forms
Theta functions |
https://en.wikipedia.org/wiki/Juan%20Carlos%20Ferrero%20career%20statistics | This is a list of the main career statistics of professional tennis player Juan Carlos Ferrero.
Major finals
Grand Slam finals
Singles: 3 (1 title, 2 runners-up)
Masters Series finals
Singles: 6 (4 titles, 2 runners-up)
Masters Cup final
Singles: 1 (1 runners-up)
ATP career finals
Singles: 34 (16 titles, 18 runner-ups)
ATP Challengers & ITF Futures Singles finals
Performance timelines
Singles
Doubles
ITF Futures Doubles finals
Head-to-head against other players
Ferrero's win-loss record against certain players who have been ATPranked World No. 10 or better is as follows:
Players who have been ranked world No. 1 are in boldface.
ATP Tour career earnings
* As of 19 Nov 2012.
Top-10 wins per season
Wins over top-10 players per season
Notes
References
External links
Ferrero, Juan Carlos |
https://en.wikipedia.org/wiki/Lleyton%20Hewitt%20career%20statistics | This is a list of the main career statistics of Australian tennis player, Lleyton Hewitt. To date, Hewitt has won thirty ATP singles titles including two grand slam singles titles, two ATP Masters 1000 singles titles and two year-ending championships. He was also the runner-up at the 2004 Tennis Masters Cup, 2004 US Open and 2005 Australian Open. Hewitt was first ranked World No. 1 by the Association of Tennis Professionals (ATP) on November 19, 2001.
Records and career milestones
In 1997, aged 15 years and 11 months, Hewitt qualified for the Australian Open, becoming the youngest qualifier in the event's history. The following year, Hewitt (ranked World No. 550 at the time) upset Andre Agassi en route to winning his first ATP singles title at the Next Generation Adelaide International, becoming the third youngest player to win an ATP singles title after Aaron Krickstein and Michael Chang and the lowest ranked ATP singles champion in history. In 2000, Hewitt became the first teenager since Pete Sampras to claim four singles titles in the same season when he won titles in Adelaide, Sydney, Scottsdale and Queen's. His victory at the latter event also meant that he had now won at least one singles title on each playing surface (hard, clay and grass). In September, Hewitt reached his first grand slam semi-final at the US Open, losing to Sampras in straight sets but won his first grand slam title of any sort by winning the doubles event with Max Mirnyi, thus becoming the youngest player (at 19 years and 6 months) to win a grand slam doubles title in the Open era. In November, he reached his first ATP Masters 1000 final in Stuttgart before finishing his season with a round robin loss at the year-ending Tennis Masters Cup, an event which he had qualified for the first time in his career. Hewitt finished the year ranked World No. 7, marking his first finish in the year-end top ten.
In June 2001, Hewitt reached his first quarterfinal at the French Open, losing to Juan Carlos Ferrero in straight sets before going on to win his first grand slam singles title at the US Open, defeating Pete Sampras in the final in straight sets. In November, he won his first year-end championship at the Tennis Masters Cup, becoming the first Australian player to do so and as a result, became the World No. 1 for the first time in his career. Aged 20 years and 8 months at the time, Hewitt was the youngest male to have reached the summit of the ATP Singles Rankings until Carlos Alcaraz achieved this at age 19 in 2022. He finished the year with a tour leading win-loss record of 80–18; six singles titles (tied with Gustavo Kuerten for most titles won this season) and the year-end No. 1 ranking, which was another first for a male Australian player.
After a disappointing start to the 2002 season, Hewitt embarked on a 15-match winning streak, collecting titles in San Jose and Indian Wells, defeating Andre Agassi and Tim Henman respectively before losing in the semi-finals of the |
https://en.wikipedia.org/wiki/List%20of%20derivatives%20and%20integrals%20in%20alternative%20calculi | There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea.
The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi.
Table
In the following table is the digamma function, is the K-function, is subfactorial, are the generalized to real numbers Bernoulli polynomials.
See also
Indefinite product
Product integral
Fractal derivative
References
External links
Non-Newtonian calculus website
Non-Newtonian calculus
Mathematics-related lists
Mathematical tables |
https://en.wikipedia.org/wiki/Difference%20algebra | Difference algebra is a branch of mathematics concerned with the study of difference (or functional) equations from the algebraic point of view. Difference algebra is analogous to differential algebra but concerned with difference equations rather than differential equations. As an independent subject it was initiated by Joseph Ritt and his student Richard Cohn.
Difference rings, difference fields and difference algebras
A difference ring is a commutative ring together with a ring endomorphism . Often it is assumed that is injective. When is a field one speaks of a difference field. A classical example of a difference field is the field of rational functions with the difference operator given by . The role of difference rings in difference algebra is similar to the role of commutative rings in commutative algebra and algebraic geometry. A morphism of difference rings is a morphism of rings that commutes with . A difference algebra over a difference field is a difference ring with a -algebra structure such that is a morphism of difference rings, i.e. extends . A difference algebra that is a field is called a difference field extension.
Algebraic difference equations
The difference polynomial ring over a difference field in the (difference) variables is the polynomial ring over in the infinitely many variables . It becomes a difference algebra over by extending from to as suggested by the naming of the variables.
By a system of algebraic difference equations over one means any subset of . If is a difference algebra over the solutions of in are
Classically one is mainly interested in solutions in difference field extensions of . For example, if and is the field of meromorphic functions on with difference operator given by , then the fact that the gamma function satisfies the functional equation can be restated abstractly as .
Difference varieties
Intuitively, a difference variety over a difference field is the set of solutions of a system of algebraic difference equations over . This definition has to be made more precise by specifying where one is looking for the solutions. Usually one is looking for solutions in the so-called universal family of difference field extensions of . Alternatively, one may define a difference variety as a functor from the category of difference field extensions of to the category of sets, which is of the form for some .
There is a one-to-one correspondence between the difference varieties defined by algebraic difference equations in the variables and certain ideals in , namely the perfect difference ideals of . One of the basic theorems in difference algebra asserts that every ascending chain of perfect difference ideals in is finite. This result can be seen as a difference analog of Hilbert's basis theorem.
Applications
Difference algebra is related to many other mathematical areas, such as discrete dynamical systems, combinatorics, number theory, or model theory. While |
https://en.wikipedia.org/wiki/Mulberry%20Stepney%20Green%20Maths%2C%20Computing%20and%20Science%20College | Mulberry Stepney Green Maths, Computing and Science College is a coeducational comprehensive secondary school and sixth form. It is situated in Stepney, in the heart of the historic East End of London and adjacent to the developments in Docklands, it serves the local community, which is mainly Bangladeshi in origin. It has a well-equipped library, including 300 computers and a good range of fiction and reference books/material.
Previously a community school administered by Tower Hamlets London Borough Council, in March 2018 Stepney Green Maths, Computing and Science College converted to academy status. The school was sponsored by The Tower Trust.
Mulberry Stepney Green Computing, Maths and Science College joined the Mulberry Schools Trust on 1 September 2021.
It was an all-boys school but became coeducational from September 2020.
The curriculum is broad, there is a wide range of extra-curricular activities offered before, during and after school. The PE department organise football and table tennis at lunchtime, after school and on Saturdays. These clubs are especially organised to encourage children of the local community to come and experience sports with qualified PE teachers.
The school operates a sixth form provision in consortium with Bow School, Langdon Park School and St Paul's Way Trust School. The sixth form consortium is known as Sixth Form East.
References
SchoolsNet
External links
Mulberry Stepney Green Maths, Computing & Science College
Secondary schools in the London Borough of Tower Hamlets
Academies in the London Borough of Tower Hamlets
Stepney |
https://en.wikipedia.org/wiki/Kostant%20partition%20function | In representation theory, a branch of mathematics, the Kostant partition function, introduced by , of a root system is the number of ways one can represent a vector (weight) as a non-negative integer linear combination of the positive roots . Kostant used it to rewrite the Weyl character formula as a formula (the Kostant multiplicity formula) for the multiplicity of a weight of an irreducible representation of a semisimple Lie algebra. An alternative formula, that is more computationally efficient in some cases, is Freudenthal's formula.
The Kostant partition function can also be defined for Kac–Moody algebras and has similar properties.
Examples
A2
Consider the A2 root system, with positive roots , , and . If an element can be expressed as a non-negative integer linear combination of , , and , then since , it can also be expressed as a non-negative integer linear combination of the positive simple roots and :
with and being non-negative integers. This expression gives one way to write as a non-negative integer combination of positive roots; other expressions can be obtained by replacing with some number of times. We can do the replacement times, where . Thus, if the Kostant partition function is denoted by , we obtain the formula
.
This result is shown graphically in the image at right. If an element is not of the form , then .
B2
The partition function for the other rank 2 root systems are more complicated but are known explicitly.
For B2, the positive simple roots are , and the positive roots are the simple roots together with and . The partition function can be viewed as a function of two non-negative integers and , which represent the element . Then the partition function can be defined piecewise with the help of two auxiliary functions.
If , then . If , then . If , then . The auxiliary functions are defined for and are given by and for even, for odd.
G2
For G2, the positive roots are and , with denoting the short simple root and denoting the long simple root.
The partition function is defined piecewise with the domain divided into five regions, with the help of two auxiliary functions.
Relation to the Weyl character formula
Inverting the Weyl denominator
For each root and each , we can formally apply the formula for the sum of a geometric series to obtain
where we do not worry about convergence—that is, the equality is understood at the level of formal power series. Using Weyl's denominator formula
we obtain a formal expression for the reciprocal of the Weyl denominator:
Here, the first equality is by taking a product over the positive roots of the geometric series formula and the second equality is by counting all the ways a given exponential can occur in the product. The function is zero if the argument is a rotation and one if the argument is a reflection.
Rewriting the character formula
This argument shows that we can convert the Weyl character formula for the irreducible representation with |
https://en.wikipedia.org/wiki/Differential%20of%20a%20function | In calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by
where is the derivative of with respect to , and is an additional real variable (so that is a function of and ). The notation is such that the equation
holds, where the derivative is represented in the Leibniz notation , and this is consistent with regarding the derivative as the quotient of the differentials. One also writes
The precise meaning of the variables and depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. Traditionally, the variables and are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis.
History and usage
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential as an infinitely small (or infinitesimal) change in the value of the function, corresponding to an infinitely small change in the function's argument . For that reason, the instantaneous rate of change of with respect to , which is the value of the derivative of the function, is denoted by the fraction
in what is called the Leibniz notation for derivatives. The quotient is not infinitely small; rather it is a real number.
The use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy (1823) defined the differential without appeal to the atomism of Leibniz's infinitesimals. Instead, Cauchy, following d'Alembert, inverted the logical order of Leibniz and his successors: the derivative itself became the fundamental object, defined as a limit of difference quotients, and the differentials were then defined in terms of it. That is, one was free to define the differential by an expression
in which and are simply new variables taking finite real values, not fixed infinitesimals as they had been for Leibniz.
According to , Cauchy's approach was a significant logical improvement over the infinitesimal approach of Leibniz because, instead of invoking the metaphysical notion of infinitesimals, the quantities and could now be manipulated in exactly the same manner as any other real quantities
in a meaningful way. Cauchy's overall conceptual approach to differentials remains the standard one in modern analytical treatments, although the final word on rigor, a fully modern notion of the limit, was ultimately due to Karl Weierstrass.
In physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still |
https://en.wikipedia.org/wiki/George%20C.%20Papanicolaou | George C. Papanicolaou (; born January 23, 1943) is a Greek-American mathematician who specializes in applied and computational mathematics, partial differential equations, and stochastic processes. He is currently the Robert Grimmett Professor in Mathematics at Stanford University.
Biography
Papanicolaou was born on January 23, 1943, in Athens, Greece. He received his B.E.E. from Union College and his M.S. and Ph.D. from New York University (NYU) in 1969. His PhD thesis, performed under the supervision of Joseph Bishop Keller was entitled "On Stochastic Differential Equations and Applications". At NYU, he started out as an assistant professor in 1969 before moving up to associate professor in 1973 and finally professor in 1976. Later, in 1993, he relocated to Stanford.
He has had 42 doctoral students and 220 descendants. He is married, with three children.
Publications
Papanicolaou has more than 250 publications on a wide range of topics, including imaging, communications and time reversal, waves in random media, convection-diffusion, nonlinear waves, high contrast materials, mathematical finance, and homogenization.
Recognition
George Papanicolaou is a member of the National Academy of Sciences, and he is a Fellow of the American Academy of Arts and Sciences, the American Mathematical Society (AMS), and the Society for Industrial and Applied Mathematics (SIAM). He was a plenary speaker at the International Congress of Mathematicians in 1998 and the International Congress of Industrial and Applied Mathematics (ICIAM) in 2003. He was awarded a Sloan Research Fellowship (1974), a Guggenheim Fellowship (1983), the von Neumann Lectureship from SIAM (2006), the William Benter Prize in Applied Mathematics (2010), the Gibbs Lectureship of the AMS (2011), and the Lagrange Prize from ICIAM (2019). He received an Honorary Doctor of Science, University of Athens in 1987 and a Doctor Honoris Causa, University of Paris VII in 2011.
Books
"Asymptotic Analysis for Periodic Structures", Alain Bensoussan, Jacques-Louis Lions and George Papanicolaou, North Holland (1978), Reprinted by the American Mathematical Society (2011).
"Derivatives in Financial Markets with Stochastic Volatility", Jean-Pierre Fouque, George Papanicolaou and K. Ronnie Sircar, Cambridge University Press (2000).
"Wave Propagation and Time Reversal in Randomly Layered Media", Jean-Pierre Fouque, Josselin Garnier, George Papanicolaou and Knut Solna, Springer (2007).
"Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives", Jean-Pierre Fouque, George Papanicolaou , K. Ronnie Sircar and Knut Solna, Cambridge University Press (2011).
"Passive Imaging with Ambient Noise", Josselin Garnier and George Papanicolaou, Cambridge University Press (2016).
Notes
External links
Website
1943 births
Living people
Fellows of the Society for Industrial and Applied Mathematics
Sloan Research Fellows
Stanford University Department of Mathematics faculty
Courant Institute |
https://en.wikipedia.org/wiki/Mechanical%20similarity | In classical mechanics, a branch overlapping in physics and applied mathematics, mechanical similarity occurs when the potential energy is a homogeneous function of the positions of the particles, with the result that the trajectories of the particles in the system are geometrically similar paths, differing in size but retaining shape.
Consider a system of any number of particles and assume that the interaction energy between any pair of particles has the form
where r is the distance between the two particles. In such a case the solutions to the equations of motion are a series of geometrically similar paths, and the times of motion t at corresponding points on the paths are related to the linear size l of the path by
Examples
The period of small oscillations (k = 2) is independent of their amplitude.
The time of free fall under gravity (k = 1) is proportional to the square root of the initial altitude.
The square of the time of revolution of the planets (k = −1) is proportional to the cube of the orbital size.
See also
Virial theorem
References
Landau LD and Lifshitz EM (1976) Mechanics §10, 3rd. ed., Pergamon Press. (hardcover) and (softcover).
Classical mechanics |
https://en.wikipedia.org/wiki/Hamiltonian%20decomposition | In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs.
In the undirected case a Hamiltonian decomposition can also be described as a 2-factorization of the graph such that each factor is connected.
Necessary conditions
For a Hamiltonian decomposition to exist in an undirected graph, the graph must be connected and regular of even degree.
A directed graph with such a decomposition must be strongly connected and all vertices must have the same in-degree and out-degree as each other, but this degree does not need to be even.
Special classes of graphs
Complete graphs
Every complete graph with an odd number of vertices has a Hamiltonian decomposition. This result, which is a special case of the Oberwolfach problem of decomposing complete graphs into isomorphic 2-factors, was attributed to Walecki by Édouard Lucas in 1892.
Walecki's construction places of the vertices into a regular polygon, and covers the complete graph in this subset of vertices with Hamiltonian paths that zigzag across the polygon, with each path rotated from each other path by a multiple of .
The paths can then all be completed to Hamiltonian cycles by connecting their ends through the remaining vertex.
Expanding a vertex of a -regular graph into a clique of vertices, one for each endpoint of an edge at the replaced vertex, cannot change whether the graph has a Hamiltonian decomposition. The reverse of this expansion process, collapsing a clique to a single vertex, will transform any Hamiltonian decomposition in the larger graph into a Hamiltonian decomposition in the original graph. Conversely, Walecki's construction can be applied to the clique to expand any Hamiltonian decomposition of the smaller graph into a Hamiltonian decomposition of the expanded graph.
One kind of analogue of a complete graph, in the case of directed graphs, is a tournament. This is a graph in which every pair of distinct vertices is connected by a single directed edge, from one to the other; for instance, such a graph may describe the outcome of a round-robin tournament in sports, where each competitor in the tournament plays each other competitor, and edges are directed from the loser of each game to the winner. Answering a conjecture by Paul Kelly from 1968, Daniela Kühn and Deryk Osthus proved in 2012 that every sufficiently large regular tournament has a Hamiltonian decomposition.
Planar graphs
For 4-regular planar graphs, additional necessary conditions can be derived from Grinberg's theorem. An example of a 4-regular planar graph that does not meet these conditions, and does not have a Hamiltonian decomposition, is given by the medial graph of the Herschel graph.
Prisms
The prism over a graph is its Cartesian product with the two-vertex complete graph. For instance, the prism over a cycle graph |
https://en.wikipedia.org/wiki/Institute%20of%20Mathematics%20and%20Applications%2C%20Bhubaneswar | The Institute of Mathematics and Applications (IMA), located in Bhubaneswar, Odisha, in India, is a research and education institution that was established by the Government of Odisha in 1999. Its dual purposes are to conduct advanced research in pure and applied mathematics and to provide postgraduate education leading to master's and Ph.D. degrees in mathematics, computation, computational finance, and data science. The institute also runs training programs in schools aimed at increasing mathematics awareness and leading to competitions such as the Mathematics Olympiads. The UG and PG courses are currently affiliated to Utkal University, which is the largest affiliating university in the country.
History
The institute was established by the Government of Odisha in the year 1999, vide the Resolution of the Government of Odisha, Science and Technology Department letter no. 368-ST-I (SC) – 159/98 dated 31 May 1999 with wide-ranging aims and objectives as notified in the Gazette No. 18, 23 July 1999 / SRAVANA, 1, 1921. The institute has been registered on 28 March 2000 under the provision of Registration of Societies Act 1860 with Registration No. 20851 /187 of 1999–2000. The institute began functioning in 1999 inside a room of the Pathani Samanta Planetarium in Bhubaneswar. On 23 July 2008, IMA acquired its residential campus, built by Tata Steel in Bhubaneswar city, when it was established as a full-fledged, degree-granting academic institution. The new institute was inaugurated by Sj. Naveen Patnaik was then the honorable chief minister.
On 2022 Government decided to convert the institute into a full-fledged university and a centre of excellence.
Academics
B.Sc. Honors in Mathematics and Computing
The institute offers a Bachelor of Science (Honors) Degree in Mathematics and Computing. The course includes the study of pure mathematics as well as mathematical modeling and the use of abstract methods to solve concrete problems. Computer Science is major and compulsory for every student and it includes the study of Algorithms design, Analysis, Cellular Automata, and other branches in Theoretical Computer Science. Rigorous training and a lot of exposure to pure mathematics are given to the students. In the final semester, each student must submit a dissertation.
M.A./M.Sc. in Computational Finance
Finance as a subject has emerged to be an extremely involved branch of knowledge with a growing number of stock exchanges and investors in the field. Risk and return that constitute cardinal aspects of the subject remain as a major concern for every investor, individual as well as institutional. Volumes of data encountered in the process make investment decisions difficult. In apparently erratic behaviors of the market, however, the mathematicians try to capture patterns vis-a-vis predictability, which may help investors decide on their investment. Computational finance otherwise called financial engineering deals with portfolio selection, options, |
https://en.wikipedia.org/wiki/Institute%20of%20Mathematics%20and%20Applications | Institute of Mathematics and Applications may refer to:
Institute for Mathematics and its Applications, Minneapolis, US
Institute of Mathematics and Applications, Bhubaneswar, India
Institute of Mathematics and its Applications, London, UK
See also
Institute of Mathematics (disambiguation) |
https://en.wikipedia.org/wiki/Inverse%20bundle | In mathematics, the inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation.
Let be a fibre bundle. A bundle is called the inverse bundle of if their Whitney sum is a trivial bundle, namely if
Any vector bundle over a compact Hausdorff base has an inverse bundle.
References
Differential topology
Algebraic topology |
https://en.wikipedia.org/wiki/Baranyai%27s%20theorem | In combinatorial mathematics, Baranyai's theorem (proved by and named after Zsolt Baranyai) deals with the decompositions of complete hypergraphs.
Statement of the theorem
The statement of the result is that if are integers and r divides k, then the complete hypergraph decomposes into 1-factors. is a hypergraph with k vertices, in which every subset of r vertices forms a hyperedge; a 1-factor of this hypergraph is a set of hyperedges that touches each vertex exactly once, or equivalently a partition of the vertices into subsets of size r. Thus, the theorem states that the k vertices of the hypergraph may be partitioned into subsets of r vertices in different ways, in such a way that each r-element subset appears in exactly one of the partitions.
The case
In the special case , we have a complete graph on vertices, and we wish to color the edges with colors so that the edges of each color form a perfect matching. Baranyai's theorem says that we can do this whenever is even.
History
The r = 2 case can be rephrased as stating that every complete graph with an even number of vertices has an edge coloring whose number of colors equals its degree, or equivalently that its edges may be partitioned into perfect matchings. It may be used to schedule round-robin tournaments, and its solution was already known in the 19th century. The case that k = 2r is also easy.
The r = 3 case was established by R. Peltesohn in 1936. The general case was proved by Zsolt Baranyai in 1975.
References
.
.
.
Hypergraphs
Theorems in combinatorics |
https://en.wikipedia.org/wiki/Levitzky%27s%20theorem | In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indeed provided a solution to one of Köthe's questions as described in . The result was originally submitted in 1939 as , and a particularly simple proof was given in .
Proof
This is Utumi's argument as it appears in
Lemma
Assume that R satisfies the ascending chain condition on annihilators of the form where a is in R. Then
Any nil one-sided ideal is contained in the lower nil radical Nil*(R);
Every nonzero nil right ideal contains a nonzero nilpotent right ideal.
Every nonzero nil left ideal contains a nonzero nilpotent left ideal.
Levitzki's Theorem
Let R be a right Noetherian ring. Then every nil one-sided ideal of R is nilpotent. In this case, the upper and lower nilradicals are equal, and moreover this ideal is the largest nilpotent ideal among nilpotent right ideals and among nilpotent left ideals.
Proof: In view of the previous lemma, it is sufficient to show that the lower nilradical of R is nilpotent. Because R is right Noetherian, a maximal nilpotent ideal N exists. By maximality of N, the quotient ring R/N has no nonzero nilpotent ideals, so R/N is a semiprime ring. As a result, N contains the lower nilradical of R. Since the lower nilradical contains all nilpotent ideals, it also contains N, and so N is equal to the lower nilradical. Q.E.D.
See also
Nilpotent ideal
Köthe conjecture
Jacobson radical
Notes
References
Theorems in ring theory |
https://en.wikipedia.org/wiki/Theta%20function%20of%20a%20lattice | In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm.
Definition
One can associate to any (positive-definite) lattice Λ a theta function given by
The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of an even unimodular lattice of rank n is actually a modular form of weight n/2. The theta function of an integral lattice is often written as a power series in so that the coefficient of qn gives the number of lattice vectors of norm 2n.
References
Theta functions |
https://en.wikipedia.org/wiki/National%20Institute%20for%20Mathematical%20and%20Biological%20Synthesis | The National Institute for Mathematical and Biological Synthesis is a research institute focused on the science of mathematics and biology, located on the University of Tennessee, Knoxville, campus. Known by its acronym NIMBioS (pronounced NIM-bus), the Institute is a National Science Foundation (NSF) Synthesis Center supported through NSF's Biological Sciences Directorate via a Cooperative Agreement with UT-Knoxville, totaling more than $35 million over ten years.
Background
The Institute opened in September 2008, with additional support from the U.S. Department of Homeland Security and the U.S. Department of Agriculture.
Since March 2009 when NIMBioS programs officially began, more than 5,000+ individuals from more than 50 countries and every U.S. state have participated in various research and educational activities.
Goals
Primary goals of NIMBios are:
to address key biological questions using cross-disciplinary approaches in mathematical biology
to foster the development of a cadre of researchers who are capable of conceiving and engaging in creative and collaborative connections across disciplines.
To achieve its goals, NIMBioS advances a wide variety of research and outreach/education activities designed to facilitate interaction between mathematicians and biologists to arrive at innovative solutions to environmental problems. Two primary mechanisms for research are Working Groups and Investigative Workshops. Working Groups are composed of 10-15 invited participants focusing on specific questions related to mathematical biology. Each group typically meets at the Institute two to three times over the course of two years. Investigative workshops may include 30-40 participants with some invited by organizers and others accepted through an open application process. Workshops are more general in focus and may lead to working group formation. NIMBioS also provides support for post-doctoral and sabbatical fellows, short-term visitors, graduate research assistants, and faculty collaborators at UT.
Function
One area of particular emphasis at NIMBioS has been modeling animal infectious diseases, such as white-nose syndrome in bats, pseudo-rabies virus in feral swine, Toxoplasma gondii in cats, and malaria from mosquitoes. As a leading international center for animal infectious disease modeling, NIMBioS has contributed significantly to global needs in analyzing the potential spread, impact and control of diseases that can move from animals to humans, such as West Nile virus, anthrax, swine flu and mad cow disease. NIMBioS also collaborates with the Great Smoky Mountains National Park to develop methods of particular interest for natural area management that are transferable to numerous U.S. locations.
NIMBioS encourages multidisciplinary participation in all its activities. Participants at NIMBioS have included behavioral biologists, ecologists, evolutionary biologists, computational scientists, anthropologists, geneticists, psychologist |
https://en.wikipedia.org/wiki/S.%20Ramanan | Sundararaman Ramanan (born 20 July 1937) is an Indian mathematician who works in the area of algebraic geometry, moduli spaces and Lie groups. He is one of India's leading mathematicians and recognised as an expert in algebraic geometry, especially in the area of moduli problems. He has also worked in differential geometry: his joint paper with MS Narasimhan on universal connections has been influential. It enabled SS Chern and B Simons to introduce what is known as the Chern-Simons invariant, which has proved useful in theoretical physics.
Education and career
He is an alumnus of the Ramakrishna Mission School in Chennai and the Vivekananda College in Chennai, where he completed a BA Honours in mathematics. He completed his PhD at the Tata Institute of Fundamental Research, under the direction of MS Narasimhan. He did his post-doctoral studies at the University of Oxford, Harvard University and ETH Zurich.
He later pursued a career at TIFR. He picked up the methods of modern differential geometry from the French mathematician Jean-Louis Koszul, and later successfully applied it for his research centred on algebraic geometry. He has also made contributions to the topics of Abelian variety and vector bundles.
Collaborations and influences
He collaborated with Raoul Bott, who was at Harvard University. He has been a visiting professor at Harvard University, University of California at Berkeley, the Institute of Advanced Study in Princeton, UCLA, University of Oxford, Cambridge University, the Max Planck Institute and University of Paris. In 1978, he gave one of the invited talks at the International Congress of Mathematicians in Helsinki. In 1999, he spoke about some aspects of the work of André Weil on the occasion of his being awarded the Inamouri Prize.
Ramanan discovered and encouraged Vijay Kumar Patodi, who proved part of the Atiyah-Singer index theorem, Patodi did his PhD under the combined direction of Narasimhan and Ramanan. Ramanan was MS Raghunathan's senior colleague and influenced him considerably.
Ramanan wrote the book Moduli of Abelian Varieties with Allan Adler, published by Springer-Verlag, and a graduate-level book on algebraic geometry called Global Calculus, published by the American Mathematical Society. He continues his contributions via teaching and mentoring at the Chennai Mathematical Institute, where he is an adjunct professor, and at the Institute of Mathematical Sciences, Chennai.
Awards
Ramanan received the Shanti Swarup Bhatnagar Prize, India's highest science prize, in 1979; the TWAS Prize for Mathematics in 2001 and the Ramanujan Medal in 2008.
Personal life
He is married to Anuradha Ramanan, a translator and former librarian. They have two daughters, Sumana Ramanan, a journalist, and Kavita Ramanan, a noted mathematician who is a professor of applied mathematics at Brown University in Providence, Rhode Island.
Selected publications
References
1937 births
Living people
Algebraic geometers
20th-century |
https://en.wikipedia.org/wiki/Equipollence%20%28geometry%29 | In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment AB from point A to point B has the opposite direction to line segment BA. Two parallel line segments are equipollent when they have the same length and direction.
Parallelogram property
A property of Euclidean spaces is the parallelogram property of vectors:
If two segments are equipollent, then they form two sides of a parallelogram:
History
The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently, the term vector was adopted for a class of equipollent line segments. Bellavitis's use of the idea of a relation to compare different but similar objects has become a common mathematical technique, particularly in the use of equivalence relations. Bellavitis used a special notation for the equipollence of segments AB and CD:
The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts:
Equipollences continue to hold when one substitutes for the lines in them, other lines which are respectively equipollent to them, however they may be situated in space. From this it can be understood how any number and any kind of lines may be summed, and that in whatever order these lines are taken, the same equipollent-sum will be obtained...
In equipollences, just as in equations, a line may be transferred from one side to the other, provided that the sign is changed...
Thus oppositely directed segments are negatives of each other:
The equipollence where n stands for a positive number, indicates that AB is both parallel to and has the same direction as CD, and that their lengths have the relation expressed by AB = n.CD.
The segment from A to B is a bound vector, while the class of segments equipollent to it is a free vector, in the parlance of Euclidean vectors.
Extension
Geometric equipollence is also used on the sphere:
To appreciate Hamilton's method, let us first recall the much simpler case of the Abelian group of translations in Euclidean three-dimensional space. Each translation is representable as a vector in space, only the direction and magnitude being significant, and the location irrelevant. The composition of two translations is given by the head-to-tail parallelogram rule of vector addition; and taking the inverse amounts to reversing direction. In Hamilton's theory of turns, we have a generalization of such a picture from the Abelian translation group to the non-Abelian SU(2). Instead of vectors in space, we deal with directed great circle arcs, of length < π on a unit sphere S2 in a Euclidean three-dimensional space. Two such arcs are deemed equivalent if by sliding one along its great circle it can be made to coincide with the other.
On a great circle of a sphere, two directed circular arcs are equipollent when they agree in direction and arc length. An equivalence class of such arcs is associated with a quaternion versor
where a is arc len |
https://en.wikipedia.org/wiki/G%C3%A1bor%20Gy%C3%B6mb%C3%A9r | Gábor Gyömbér (born 27 February 1988) is a Hungarian former footballer.
Club statistics
Updated to games played as of 2 December 2014.
Honours
Ferencváros
Hungarian League Cup: 2012–13
External links
Gábor Gyömbér profile at magyarfutball.hu
HLSZ
UEFA Official Website
1988 births
People from Makó
Sportspeople from Csongrád-Csanád County
Living people
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Hungary men's international footballers
Men's association football midfielders
Clube Náutico Capibaribe players
FC Sopron players
Pápai FC footballers
Ferencvárosi TC footballers
Puskás Akadémia FC players
Soroksár SC players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungarian expatriate men's footballers
Expatriate men's footballers in Brazil
Hungarian expatriate sportspeople in Brazil |
https://en.wikipedia.org/wiki/Bhatia%E2%80%93Davis%20inequality | In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ2 of any bounded probability distribution on the real line.
Statement
Let m and M be the lower and upper bounds, respectively, for a set of real numbers a1, ..., an , with a particular probability distribution. Let μ be the expected value of this distribution.
Then the Bhatia–Davis inequality states:
Equality holds if and only if every aj in the set of values is equal either to M or to m.
Proof
Since ,
.
Thus,
.
Extensions of the Bhatia–Davis inequality
If is a positive and unital linear mapping of a C* -algebra into a C* -algebra , and A is a self-adjoint element of satisfying m A M, then:
.
If is a discrete random variable such that
where , then:
,
where and .
Comparisons to other inequalities
The Bhatia–Davis inequality is stronger than Popoviciu's inequality on variances (note, however, that Popoviciu's inequality does not require knowledge of the expectation or mean), as can be seen from the conditions for equality. Equality holds in Popoviciu's inequality if and only if half of the aj are equal to the upper bounds and half of the aj are equal to the lower bounds. Additionally, Sharma has made further refinements on the Bhatia–Davis inequality.
See also
Cramér–Rao bound
Chapman–Robbins bound
Popoviciu's inequality on variances
References
Statistical inequalities
Theory of probability distributions |
https://en.wikipedia.org/wiki/D.%20J.%20Bartholomew | David John Bartholomew (6 August 1931 – 16 October 2017) was a British statistician who was president of the Royal Statistical Society between 1993 and 1995. He was professor of statistics at the London School of Economics between 1973 and 1996.
Career
Bartholomew was born 6 August 1931, the son of Albert and Joyce Bartholomew in Oakley, Bedfordshire. He was educated at Bedford Modern School and University College London, where he earned his BSc and PhD.
Bartholomew began his career as a scientist at the National Coal Board in 1955. In 1957 he became a lecturer in statistics at the University of Keele, before his appointment as a senior lecturer at the University College of Wales, Aberystwyth.
Bartholomew was appointed professor of statistics at the University of Kent in 1967 before being made professor of statistics at the London School of Economics in 1973, a position he held until 1996. He was emeritus professor (pro-director) between 1988 and 1991.
Bartholomew was president of the Royal Statistical Society, 1993–95 (honorary secretary, 1976–82; treasurer, 1989–93). He was vice-president of the Manpower Society (1987–95) and was chairman of the Science and Religion Forum between 1997 and 2000.
In 1955, Bartholomew married Marian Elsie Lake, and they have two daughters. Bartholomew was elected a Fellow of the British Academy in 1987.
Religious views
Bartholomew authored several books defending the existence of the Christian God from a Biblical and statistical basis, God of Chance (1984), Uncertain Belief: Is It Rational to Be a Christian? (1996) and God, Chance and Purpose (2008).
In his 1984 book God of Chance, Bartholomew argued that the universe is "designed in such a way that chance had a role to play... Chance was God's idea and... he uses it to ensure the variety, resilience and freedom necessary to achieve his purposes." Similarly, his 2008 book God, Chance and Purpose argues that chance is part of the means by which God governs the world.
Bartholomew debated atheist physicist Victor Stenger on whether or not God is a failed hypothesis.
Books
1961: & S.E. Finer and H.B. Berrington Backbench Opinion in the House of Commons, 1955–1959. Oxford: Pergamon Press.
1967: Stochastic Models for Social Processes, New York and London: John Wiley and Sons. (German translation 1970).
1971: & E.E. Bassett, Let's Look at the Figures: the quantitative approach to human affairs), Harmondsworth Middlesex: Penguin books (Dutch translation, 1971).
1971: & B.R.Morris (eds.), Aspects of Manpower Planning, London: English Universities Press.
1971: & A.R.Smith (eds.) Manpower and Management Science, London: English Universities Press.
1972: & R.E. Barlow, H.D. Brunk and J.M. Bremner, Statistical Inference under Order Restrictions, Chichester: John Wiley and Sons.
1973: Stochastic Models for Social Processes, (revised and enlarged) Chichester 2nd edition: John Wiley and Sons.
1976: (ed.) Manpower Planning, Harmondsworth, Middlesex: Pe |
https://en.wikipedia.org/wiki/2007%20FC%20Seoul%20season |
Pre-season
Pre-season match results
Competitions
Overview
K League
FA Cup
League Cup
Match reports and match highlights
Fixtures and Results at FC Seoul Official Website
Season statistics
K League records
All competitions records
Attendance records
Season total attendance is K League Regular Season, League Cup, FA Cup, AFC Champions League in the aggregate and friendly match attendance is not included.
K League season total attendance is K League Regular Season and League Cup in the aggregate.
Squad statistics
Goals
Assists
Coaching staff
Players
Team squad
All players registered for the 2007 season are listed.
(Out)
(Conscripted)
(Out)
(Conscripted)
(Conscripted)
(Conscripted)
(In)
Out on loan & military service
In : Transferred from other teams in the middle of season.
Out : Transferred to other teams in the middle of season.
Discharged : Transferred from Gwangju Sangmu and Police FC for military service after end of season. (Not registered in 2007 season.)
Conscripted : Transferred to Gwangju Sangmu and Police FC for military service after end of season.
Transfers
In
Rookie Draft
Out
Loan & Military service
Tactics
Tactical analysis
Starting eleven and formation
This section shows the most used players for each position considering a 4-4-2 formation.
Substitutes
See also
FC Seoul
References
FC Seoul 2007 Matchday Magazines
External links
FC Seoul Official Website
2007
Seoul |
https://en.wikipedia.org/wiki/1996%E2%80%932003%20Anyang%20LG%20Cheetahs%20seasons | Anyang LG Cheetahs was a South Korean professional football club based in Seoul.
Seasons statistics
All competitions records
※ K-League Championship results are not counted.
※ 1998, 1999, 2000 seasons had PSO and blows results are that PSO results are counted by drawn.
※ A: Adidas Cup, P: Prospecs Cup, PM: Philip Morris Korea Cup, D: Daehan Fire Insurance Cup
[1] In 2000, Tournament name was 1999–2000 Asian Cup Winners' Cup
[2] In 2002, Tournament name was 2001-02 Asian Club Championship
K League Championship records
Seasons Summary
1996 season summary
1997 season summary
1998 season summary
1999 season summary
2000 season summary
2001 season summary
2002 season summary
2003 season summary
Kits
First Kit
Second Kit
※ Notes
(2) 2002 1st kit and 2003 1st kit are same but colour of adidas logo and 3 stripes on shoulder are different
Transfers
1996 season
In
Rookie Draft
Out
Loan & Military service
1997 season
In
Rookie Draft
Out
Loan & Military service
1998 season
In
Rookie Draft
Out
Loan & Military service
1999 season
In
Rookie Draft
Out
Loan & Military service
2000 season
In
Rookie Draft
Out
Loan & Military service
2001 season
In
Rookie Draft
Out
Loan & Military service
2002 season
In
Rookie Free Agent
Out
Loan & Military service
2003 season
In
Rookie Free Agent
Out
Loan & Military service
See also
FC Seoul
References
FC Seoul Matchday Magazines
The K League history at K League official website
External links
FC Seoul Official Website
1996-2003 |
https://en.wikipedia.org/wiki/1991%E2%80%931995%20LG%20Cheetahs%20seasons | LG Cheetahs was a South Korean professional football club based in Seoul.
Seasons Statistics
All competitions records
※ 1993 season had PSO and blows results are that PSO results are counted by drawn.
※ A: Adidas Cup
Season Summary
1991 season summary
1992 season summary
1993 season summary
1994 season summary
1995 season summary
Kits
First Kit
Second Kit
Transfers
1991 season
In
Rookie Draft
Out
Loan & Military service
1992 season
In
Rookie Draft
Out
Loan & Military service
1993 season
In
Rookie Draft
Out
Loan & Military service
1994 season
In
Rookie Draft
Out
Loan & Military service
In
Rookie Draft
Out
Loan & Military service
See also
FC Seoul
References
FC Seoul Matchday Magazines
The K League history at K League official website
External links
FC Seoul Official Website
1991–1995 |
https://en.wikipedia.org/wiki/1984%E2%80%931990%20Lucky-Goldstar%20FC%20seasons | Lucky-Goldstar FC was a South Korean professional football club based in Seoul.
Seasons statistics
All competitions records
[1] In 1986, Tournament name was Korea Professional Football Championship
[2] In 1988 and 1989, Tournament name was National Football Championship
K League Championship records
Seasoans Summary
1984 season summary
1985 season summary
1986 season summary
1987 season summary
1988 season summary
1989 season summary
1990 season summary
Kits
First Kit
Second Kit
Third Kit
※ Notes
(1) In only 1987 season, All K League clubs wore white jerseys in home match, coloured jersey in away match like Major League Baseball.
Transfers
1984 season
Founding members
In
1984 season
In
Rookie Free Agent
Out
Loan & Military service
1986 season
In
Rookie Free Agent
Out
Loan & Military service
1987 season
In
Rookie Free Agent
Out
Loan & Military service
1988 season
In
Rookie Draft
Out
Loan & Military service
1989 season
In
Rookie Draft
Out
Loan & Military service
1990 season
In
Rookie Draft
Out
Loan & Military service
See also
FC Seoul
References
FC Seoul Matchday Magazines
The K League history at K League official website
External links
FC Seoul Official Website
1984–1990 |
https://en.wikipedia.org/wiki/List%20of%20FC%20Vaslui%20records%20and%20statistics | This is a list of statistics and records of the Romanian professional football club FC Vaslui. It is based in Vaslui, Romania, and currently plays in Liga IV. The club was founded in 2002 and its home ground is the Municipal Stadium.
Honours
FC Vaslui succeeded three league runners-up, two in the lower divisions and one in Liga I in the 2011–12 season. In 2010, Vaslui reached the Romanian Cup final, lost at the penalty shootout. Vaslui's highest European performance is winning the UEFA Intertoto Cup in 2008.
Domestic
Liga I:
Runner-up (1): 2011–12
Liga II:
Winners (1): 2004–05
Runner-up (1): 2003–04
Liga III:
Runner-up (1): 2002–03
Romanian Cup:
Runner-up (1): 2009–10
European
UEFA Intertoto Cup:
Winners (1): 2008
Player records
Appearances
Mike Temwanjera holds FC Vaslui's appearance record, having played 201 times over the course of 6 seasons from 2007 until now. He also holds the records for most European appearances, having made 17, but also the record for the most league appearances, with 173. Also, Wesley Lopes holds Romanian Cup appearances, with 14.
Most appearances in all competitions: Mike Temwanjera, 202.
Most league appearances: Mike Temwanjera, 174.
Most Romanian Cup appearances: Wesley Lopes, 14.
Most European appearances: Mike Temwanjera 17.
Youngest first-team player: Rareș Lazăr, 15 years and 47 days (against Ceahlăul Piatra Neamţ, 17 May 2014).
Oldest first-team player: Dorinel Munteanu, 39 years and 158 days (against Rapid București, 1 December 2007).
Oldest debutant: Dorinel Munteanu, 39 years and 32 days (against UTA Arad, 27 July 2007).
Most consecutive league appearances: Cristian Hăisan, 59 (from 30 September 2006 to 7 May 2007).
Most seasons as an ever-present: Bogdan Buhuş, 5 (from 2005–06 to 2009–10).
Longest-serving player: Cristian Hăisan, 8 years (from 2002 to 2010).
Most appearances
Most appearances in Liga I
Goalscorers
FC Vaslui's all-time leading scorer is Wesley Lopes, who scored 77 goals between 2009 until 2012. He also holds the record for most league goals with 61, for the most goals in European competition with 6, for the most goals in Romanian Cup with 10, but also the record for the most goals in a season with 37 in all competitions in the 2011–12 season.
Most goals in all competitions: Wesley Lopes, 77.
Most league goals: Wesley Lopes, 61.
Most Romanian Cup goals: Wesley Lopes, 10.
Most European goals: Wesley Lopes, 6.
Most goals in a season: Wesley Lopes, 37 (during the 2011–12 season).
Most hat-tricks in a season: Wesley Lopes, 3 (during the 2011–12 season).
Most hat-tricks: Wesley Lopes, 4.
Fastest hat-trick: Cătălin Andruş, 5 minutes (against CFR Paşcani, 12 October 2002).
Most consecutive league goals scored at Municipal: Marko Ljubinković, 5 during the 2007–08 season.
Highest-scoring substitute: Valentin Badea, 3.
Most penalties scored: Wesley Lopes, 15.
Most games without scoring for an outfield player: Bogdan Buhuş, 125.
Youngest goalscorer: Sorin Ungurianu, 18 years, 328 days (aga |
https://en.wikipedia.org/wiki/Ciro%20D%C3%ADaz | Ciro Díaz is composer, lead guitarist of the band Porno para Ricardo, and leader of the alternative rock band La Babosa Azul. Ciro earned a Bachelor in Mathematics from the University of Havana and learned to play the guitar on his own.
References
External links
La Babosa Azul
Cuban guitarists
Cuban male guitarists
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/2006%20Jeju%20United%20FC%20season | 2006 Season is first year of new franchise Jeju
Jeju United FC season 2006 statistics
{|class="wikitable"
|-bgcolor="#efefef"
! Season
! K-League
! Played
! W
! D
! L
! F
! A
! PTS
! K-League Cup
! FA Cup
! Manager
|-
|2006
|align=right|13th
|align=right|26
|align=right|5
|align=right|10
|align=right|11
|align=right|23
|align=right|30
|align=right|25
|align=right|8th
|align=right|Round of 32
|Jung Hae-Seong
|-
|}
Jeju United FC seasons
South Korean football clubs 2006 season |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20Maine%20Black%20Bears%20women%27s%20ice%20hockey%20season | The following is a list of events and statistics from the 2009–2010 Maine Black Bears women's ice hockey season. The Black Bears are an ice hockey team which represent the University of Maine. The head coach is Dan Lichterman. Assisting him are Karine Senecal, Sara Simard, and Meghan MacDonald.
Offseason
July 16: The 2009 Battle of the Bears trophy goes to the Women's Ice Hockey team. They topped all other Maine Black Bears teams with 2,594 points. The women's ice hockey team earned the win by earning the highest GPA of all sports teams for the academic year. The team was also awarded the Black Bear Citizenship Award this spring for their commitment to service throughout the year.
July 27: Head coach Dan Lichterman has announced that the program has received an additional commitment for the 2009–10 season. Darcia Leimgruber will join the Black Bears in the fall of 2009. Leimgruber is a 5-foot 3-inch forward from Basel, Switzerland.
Regular season
December: Jenna Ouellette played in four games and accumulated nine points (5 goals, 4 assists) leading the Black Bears to three victories. She recorded a point in all four games and notched multiple points in three games. She played a part in nine of the 12 Black Bear goals in December and assisted on all four goals vs. Vermont on Dec. 4. She scored the game-winning goal while on the power play in a 1–0 win over Vermont (12/5). In addition, she notched a pair of goals against Union (12/11) and another pair in the finale at Union (12/12).
January 26: Maine freshman forward Darcia Leimgruber (Basel, Switzerland) has been named to the Team Switzerland Olympic Women's Ice Hockey team that will represent Switzerland at the 2010 Winter Olympic Games in Vancouver, B.C. Leimgruber is the third Black Bear to have ties to the Women's Ice Hockey Olympic games. Raffi Wolf represented the Black Bears when she played on Team Germany at the 2002 games in Salt Lake City as well as the 2006 Torino games. Former club hockey student-athlete, Stacey Livingston, officiated the gold medal game at the 1998 Olympics in Nagano, Japan.
Standings
Roster
Schedule
Player stats
Skaters
Goaltenders
Postseason
NCAA hockey tournament
Awards and honors
Abby Barton, Hockey East Association Academic All-Star Team
Brittany Ott, 2010 WHEA All-Rookie Team
Jenna Ouellette – Maine, WHEA Player of the Month, December 2009
Jenna Ouellette, Hockey East Association Academic All-Star Team
Amy Stech, Runner up, Hockey East Sportsmanship Award
Hockey East all-academic team
Abby Barton
Jenna Ouellette
Amy Stech
Danielle Cyr
Jennie Gallo
Jessica Bond
Candice Currier
Melissa Gagnon
Ashley Norum
Dawn Sullivan
Kylie Smith
Chloe Tinker
References
External links
Official site
Maine Black Bears women's ice hockey seasons
Maine
Black
Black |
https://en.wikipedia.org/wiki/1931%20American%20Soccer%20League | Statistics of American Soccer League in season 1931.
Overview
At the start of this season the American Soccer League was in decline, suffering from the effects of the Great Depression. Clubs had begun to fold, merge and disappear. The eventual champions, New Bedford Whalers, had been formed by Sam Mark following the merger of Fall River F.C. and New York Yankees. Long time ASL members Brooklyn Wanderers folded before the season. However New York Americans, later to become a perennial contender in the second ASL, made their debut.
The season began on February 29, 1931. The first half ended on May 31, 1931. The second half began September 19, 1931, and the season concluded on December 27, 1931. The season saw strong performances by New Bedford Whalers, New York Giants and a resurgent Pawtucket Rangers. Hakoah All-Stars improved markedly to take fourth place. Although the Whalers won this season, they lost the playoff series for the overall 1931 champion, being defeated by Spring 1931 champions New York Giants in a play-off. Whalers took the first game at home 8-3 before the Giants came back to win 6-0, taking the series on a 9-8 aggregate score.
League standings
The percentage is a percentage of points won to points available, not a win-lose percentage.
First half
Second half
Playoff
First leg
Second leg
New York Giants won, 9–8, on aggregate.
Top goalscorers
External links
The Year in American Soccer - 1931
The Year in American Soccer - 1932
References
American
1931 |
https://en.wikipedia.org/wiki/1932%E2%80%9333%20American%20Soccer%20League | Statistics of American Soccer League in the 1932-33 season.
Overview
This was the last season for the first American Soccer League. After the disastrous 1932 season, the league began a 1932–1933 season in October 1932. Although it began well, with nine teams competing, the turmoil from earlier in the year continued as several teams withdrew during the first half and others during the mid-season break. At that point, the historical record becomes contradictory. The league may have intended the season to run from the fall of 1932 into the spring of 1933. However, a May 29, 1933, New York Times story has this headline: "Brookhattan Beats Americans At Soccer: Triumphs by 2-1 to Capture Honors for First Half of League Competition". This implies the possibility the league had abandoned the 1932–1933 season and tried to begin again in May 1933 with a full spring–fall season. Regardless, the league collapsed in the summer of 1933, to be replaced by the semi-professional American Soccer League that fall.
League standings
First half
Second half
External links
The Year in American Soccer - 1933
References
American Soccer League (1921–1933) seasons
American |
https://en.wikipedia.org/wiki/Extender%20%28set%20theory%29 | In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender.
A (κ, λ)-extender can be defined as an elementary embedding of some model of ZFC− (ZFC minus the power set axiom) having critical point κ ε M, and which maps κ to an ordinal at least equal to λ. It can also be defined as a collection of ultrafilters, one for each -tuple drawn from λ.
Formal definition of an extender
Let κ and λ be cardinals with κ≤λ. Then, a set is called a (κ,λ)-extender if the following properties are satisfied:
each is a κ-complete nonprincipal ultrafilter on [κ]<ω and furthermore
at least one is not κ+-complete,
for each at least one contains the set
(Coherence) The are coherent (so that the ultrapowers Ult(V,Ea) form a directed system).
(Normality) If is such that then for some
(Wellfoundedness) The limit ultrapower Ult(V,E) is wellfounded (where Ult(V,E) is the direct limit of the ultrapowers Ult(V,Ea)).
By coherence, one means that if and are finite subsets of λ such that is a superset of then if is an element of the ultrafilter and one chooses the right way to project down to a set of sequences of length then is an element of More formally, for where and where and for the are pairwise distinct and at most we define the projection
Then and cohere if
Defining an extender from an elementary embedding
Given an elementary embedding which maps the set-theoretic universe into a transitive inner model with critical point κ, and a cardinal λ, κ≤λ≤j(κ), one defines as follows:
One can then show that has all the properties stated above in the definition and therefore is a (κ,λ)-extender.
References
Inner model theory
Mathematical logic
Model theory
Large cardinals
Set theory |
https://en.wikipedia.org/wiki/Potential%20flow%20around%20a%20circular%20cylinder | In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.
Mathematical solution
A cylinder (or disk) of radius is placed in a two-dimensional, incompressible, inviscid flow. The goal is to find the steady velocity vector and pressure in a plane, subject to the condition that far from the cylinder the velocity vector (relative to unit vectors and ) is:
where is a constant, and at the boundary of the cylinder
where is the vector normal to the cylinder surface. The upstream flow is uniform and has no vorticity. The flow is inviscid, incompressible and has constant mass density . The flow therefore remains without vorticity, or is said to be irrotational, with everywhere. Being irrotational, there must exist a velocity potential :
Being incompressible, , so must satisfy Laplace's equation:
The solution for is obtained most easily in polar coordinates and , related to conventional Cartesian coordinates by and . In polar coordinates, Laplace's equation is (see Del in cylindrical and spherical coordinates):
The solution that satisfies the boundary conditions is
The velocity components in polar coordinates are obtained from the components of in polar coordinates:
and
Being inviscid and irrotational, Bernoulli's equation allows the solution for pressure field to be obtained directly from the velocity field:
where the constants and appear so that far from the cylinder, where . Using ,
In the figures, the colorized field referred to as "pressure" is a plot of
On the surface of the cylinder, or , pressure varies from a maximum of 1 (shown in the diagram in ) at the stagnation points at and to a minimum of −3 (shown in ) on the sides of the cylinder, at and . Likewise, varies from at the stagnation points to on the sides, in the low pressure.
Stream function
The flow being incompressible, a stream function can be found such that
It follows from this definition, using vector identities,
Therefore, a contour of a constant value of will also be a streamline, a line tangent to . For the flow past a cylinder, we find:
Physical interpretation
Laplace's equation is linear, and is one of the most elementary partial differential equations. This simple equation yields the entire solution for both and because of the constraint of irrotationality and incompressibility. Having obtained the solution for and , the consistency of the pressure gradient with the accelerations can be noted.
The dynamic pressure at the upstream stagnation point has value of . a value needed to decelerate the free stream flow of speed . This same |
https://en.wikipedia.org/wiki/G%C3%A1bor%20T%C3%B3th | Gábor Tóth (born 26 March 1987) is a Hungarian football player who plays for Dunaújváros.
Club statistics
Updated to games played as of 6 December 2014.
References
MLSZ
Lombard FC Papa Official Website
1987 births
People from Kiskunhalas
Footballers from Bács-Kiskun County
Living people
Hungarian men's footballers
Men's association football midfielders
Dunaújváros FC players
Pápai FC footballers
Szeged-Csanád Grosics Akadémia footballers
Dunaújváros PASE players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players |
https://en.wikipedia.org/wiki/Trindade%2C%20Pernambuco | Trindade is a municipality in the state of Pernambuco, Brazil. The estimated population in 2021, according to the Brazilian Institute of Geography and Statistics (IBGE) was 31,103 inhabitants and the total area is 295.77 km².
Geography
State - Pernambuco
Region - Sertão Pernambucano
Boundaries - Araripina (N and W); Ouricuri (S); Ouricuri and Ipubi (E).
Area - 229.57 km²
Elevation - 518 m
Hydrography - Brigida River
Vegetation - Caatinga hiperxerófila
Climate - semi arid - (Sertão) hot
Annual average temperature - 24.9 c
Distance to Recife - 645 km
Population at last census (2010) - 26,116
Economy
The main economic activities in Trindade are based in no metallic (gypsum) industry, commerce and agribusiness, especially creation of cattle, goats, sheep, pigs, chickens; and plantations of manioc. Trindade is located in the micro region of Araripina, which contains 95% of the Brazilian reserves of gypsum.
Economic Indicators
Economy by Sector
2006
Health Indicators
References
Municipalities in Pernambuco |
https://en.wikipedia.org/wiki/Samuelson%27s%20inequality | In statistics, Samuelson's inequality, named after the economist Paul Samuelson, also called the Laguerre–Samuelson inequality, after the mathematician Edmond Laguerre, states that every one of any collection x1, ..., xn, is within uncorrected sample standard deviations of their sample mean.
Statement of the inequality
If we let
be the sample mean and
be the standard deviation of the sample, then
Equality holds on the left (or right) for if and only if all the n − 1 s other than are equal to each other and greater (smaller) than
If you instead define then the inequality becomes
Comparison to Chebyshev's inequality
Chebyshev's inequality locates a certain fraction of the data within certain bounds, while Samuelson's inequality locates all the data points within certain bounds.
The bounds given by Chebyshev's inequality are unaffected by the number of data points, while for Samuelson's inequality the bounds loosen as the sample size increases. Thus for large enough data sets, Chebychev's inequality is more useful.
Applications
Samuelson's inequality may be considered a reason why studentization of residuals should be done externally.
Relationship to polynomials
Samuelson was not the first to describe this relationship: the first was probably Laguerre in 1880 while investigating the roots (zeros) of polynomials.
Consider a polynomial with all roots real:
Without loss of generality let and let
and
Then
and
In terms of the coefficients
Laguerre showed that the roots of this polynomial were bounded by
where
Inspection shows that is the mean of the roots and that b is the standard deviation of the roots.
Laguerre failed to notice this relationship with the means and standard deviations of the roots, being more interested in the bounds themselves. This relationship permits a rapid estimate of the bounds of the roots and may be of use in their location.
When the coefficients and are both zero no information can be obtained about the location of the roots, because not all roots are real (as can be seen from Descartes' rule of signs) unless the constant term is also zero.
References
Statistical inequalities |
https://en.wikipedia.org/wiki/%C3%89lton%20%28footballer%2C%20born%201%20August%201985%29 | Élton Rodrigues Brandão (born 1 August 1985), simply known as Élton, is a Brazilian professional footballer who plays as a striker Juventude.
Career statistics
Honours
Club
Santo André
Campeonato Paulista Série A2: 2008
Vasco
Campeonato Brasileiro Série B: 2009
Copa do Brasil: 2011
Corinthians
Copa Libertadores: 2012
Al-Nassr
Saudi Crown Prince Cup: 2013–14
Saudi Professional League: 2013–14
Cuiabá
Campeonato Mato-Grossense: 2021, 2022
References
External links
CBF profile
1985 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Legia Warsaw players
Iraty Sport Club players
Esporte Clube Santo André players
Associação Desportiva São Caetano players
CR Vasco da Gama players
Sport Club Corinthians Paulista players
Esporte Clube Vitória players
Clube Náutico Capibaribe players
S.C. Braga players
Al Nassr FC players
CR Flamengo footballers
JEF United Chiba players
Ceará Sporting Club players
Figueirense FC players
Sport Club do Recife players
Cuiabá Esporte Clube players
Centro Sportivo Alagoano players
Esporte Clube Juventude players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Ekstraklasa players
Primeira Liga players
J2 League players
Brazilian expatriate sportspeople in Poland
Expatriate men's footballers in Poland
Brazilian expatriate sportspeople in Portugal
Expatriate men's footballers in Portugal
Brazilian expatriate sportspeople in Saudi Arabia
Expatriate men's footballers in Saudi Arabia
Expatriate men's footballers in Japan
Saudi Pro League players
Men's association football forwards |
https://en.wikipedia.org/wiki/Al-Manshiyya%2C%20Jenin | Al-Manshiyya () is a Palestinian village in the West Bank governorate of Jenin. According to the Palestinian Central Bureau of Statistics, the village had a population of 156 inhabitants in mid-year 2006.
Footnotes
External links
Welcome To al-Manshiyya
Survey of Western Palestine, Map 8: IAA, Wikimedia commons
Riwaq Registry of Historic Buildings in Palestine - Al-Manshiyya
Villages in the West Bank
Jenin Governorate
Municipalities of the State of Palestine |
https://en.wikipedia.org/wiki/Noncommutative%20ring | In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.
Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.
Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ring is used as a shorthand for commutative ring.
Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.
Examples
Some examples of noncommutative rings:
The matrix ring of n-by-n matrices over the real numbers, where
Hamilton's quaternions
Any group ring constructed from a group that is not abelian
Some examples of rings that are not typically commutative (but may be commutative in simple cases):
The free ring generated by a finite set, an example of two non-equal elements being
The Weyl algebra , being the ring of polynomial differential operators defined over affine space; for example, , where the ideal corresponds to the commutator
The quotient ring , called a quantum plane, where
Any Clifford algebra can be described explicitly using an algebra presentation: given an -vector space of dimension with a quadratic form , the associated Clifford algebra has the presentation for any basis of ,
Superalgebras are another example of noncommutative rings; they can be presented as
There are finite noncommutative rings: for example, the -by- matrices over a finite field, for . The smallest noncommutative ring is the ring of the upper triangular matrices over the field with two elements; it has eight elements and all noncommutative rings with eight elements are isomorphic to it or to its opposite.
History
Beginning with division rings arising from geometry, the study of noncommutative rings has grown into a major area of modern algebra. The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors. An incomplete list of such contributors includes E. Artin, Richard Brauer, P. M. Cohn, W. R. Hamilton, I. N. Herstein, N. Jacobson, K. Morita, E. Noether, Ø. Ore, J. Wedderburn and others.
Differences between commutative and noncommutative algebra
Because noncommutative rings of scientific interest are more complicated than commutative rings, their structure, properties and behavior are less well understood. A great deal of work has been done successfully generalizing some results from commutative rings to noncommutative rings. A major difference between rings which are and are not commutat |
https://en.wikipedia.org/wiki/Noncommutative%20harmonic%20analysis | In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier transforms, the major business of non-commutative harmonic analysis is usually taken to be the extension of the theory to all groups G that are locally compact. The case of compact groups is understood, qualitatively and after the Peter–Weyl theorem from the 1920s, as being generally analogous to that of finite groups and their character theory.
The main task is therefore the case of G that is locally compact, not compact and not commutative. The interesting examples include many Lie groups, and also algebraic groups over p-adic fields. These examples are of interest and frequently applied in mathematical physics, and contemporary number theory, particularly automorphic representations.
What to expect is known as the result of basic work of John von Neumann. He showed that if the von Neumann group algebra of G is of type I, then L2(G) as a unitary representation of G is a direct integral of irreducible representations. It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology. The analogue of the Plancherel theorem is abstractly given by identifying a measure on the unitary dual, the Plancherel measure, with respect to which the direct integral is taken. (For Pontryagin duality the Plancherel measure is some Haar measure on the dual group to G, the only issue therefore being its normalization.) For general locally compact groups, or even countable discrete groups, the von Neumann group algebra need not be of type I and the regular representation of G cannot be written in terms of irreducible representations, even though it is unitary and completely reducible. An example where this happens is the infinite symmetric group, where the von Neumann group algebra is the hyperfinite type II1 factor. The further theory divides up the Plancherel measure into a discrete and a continuous part. For semisimple groups, and classes of solvable Lie groups, a very detailed theory is available.
See also
Selberg trace formula
Langlands program
Kirillov orbit theory
Discrete series representation
Zonal spherical function
References
"Noncommutative harmonic analysis: in honor of Jacques Carmona", Jacques Carmona, Patrick Delorme, Michèle Vergne; Publisher Springer, 2004
Yurii I. Lyubich. Introduction to the Theory of Banach Representations of Groups. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.
Notes
Topological groups
Duality theories |
https://en.wikipedia.org/wiki/Neumann%20polynomial | In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case , are a sequence of polynomials in used to expand functions in term of Bessel functions.
The first few polynomials are
A general form for the polynomial is
and they have the "generating function"
where J are Bessel functions.
To expand a function f in the form
for , compute
where and c is the distance of the nearest singularity of from .
Examples
An example is the extension
or the more general Sonine formula
where is Gegenbauer's polynomial. Then,
the confluent hypergeometric function
and in particular
the index shift formula
the Taylor expansion (addition formula)
(cf.) and the expansion of the integral of the Bessel function,
are of the same type.
See also
Bessel function
Bessel polynomial
Lommel polynomial
Hankel transform
Fourier–Bessel series
Schläfli-polynomial
Notes
Polynomials
Special functions |
https://en.wikipedia.org/wiki/Yehuda%20Cohen | Yehuda Cohen (; 23 January 1914 – 8 August 2009) was an Israeli judge. He was appointed to the Israeli Supreme Court in 1982.
Biography
Yehuda Cohen was born in Safed. He studied mathematics at the American University of Beirut and returned to Mandate Palestine to study law. Cohen enlisted in the British Army and continued to serve as a major in the Israel Defense Forces.
Cohen served in the Jerusalem District Court from 1954, and became its president in 1980. In 1982, he was appointed to the Israeli Supreme Court, where he served until retirement.
Cohen was also active in public life, as chairman of HaMutzar HaYerushalmi organization for promotion of arts (1964–1993), vice president of Israeli chapter of Bnei Brit (1968–1970) and chairman of the board of Ezrat Nashim (later Herzog) hospital (1968–1993). In 1995 he was selected to head a national inquiry commission on the disappearance of children of Yemenite origin during the 1950s.
Cohen died on 8 August 2009, aged 95. He was buried in Rishon LeZion.
See also
Israeli judicial system
References
1914 births
2009 deaths
American University of Beirut alumni
Judges of the Supreme Court of Israel
People from Safed
British Army soldiers
Jews from Mandatory Palestine
Israeli military personnel |
https://en.wikipedia.org/wiki/Walter%20Warwick%20Sawyer | Walter Warwick Sawyer (or W. W. Sawyer) (April 5, 1911– February 15, 2008) was a mathematician,
mathematics educator and author, who taught on several continents.
Life and career
Walter Warwick Sawyer was born in St. Ives, Hunts, England on April 5, 1911. He attended
Highgate School in London. He was an undergraduate at St. John's College, Cambridge, obtaining a BA in 1933 and specializing in quantum theory and relativity. He was an assistant lecturer in mathematics from 1933 to 1937 at University College, Dundee and from 1937 to 1944 at University of Manchester. From 1945 to 1947, he was the head of mathematics at Leicester College of Technology.
In 1948 Sawyer became the first head of the mathematics department of what is now the University of Ghana. From 1951 to 1956, he was at Canterbury College (now the University of Canterbury in New Zealand). He left Canterbury College to become an associate professor at the University of Illinois, where he worked from winter 1957 through June 1958. While there, he criticized the New Math movement, which included criticism of the people who had hired him. From 1958 to 1965, he was a professor of mathematics at Wesleyan University, where he edited Mathematics Student Journal. In the fall of 1965 he became a professor at the University of Toronto, appointed to both the College of Education and the Department of Mathematics. He retired in 1976.
Sawyer was the author of some 11 books. He is probably best known for his semi-popular works Mathematician's Delight and Prelude to Mathematics. Both of these have been translated into many languages. Mathematician's Delight was still in print 65 years after it was written. Some mathematicians have credited these books with helping to inspire their choice of a career.
Sawyer died on February 15, 2008, at the age of 96. He is survived by a daughter, Anne León (Artist) and granddaughter, Anita León (Educator).
Partial bibliography
Mathematician's Delight, (Penguin, 1943), is probably his best known book.
Mathematics in Theory and Practice, (Odhams, 1952)
Prelude to Mathematics (Penguin, 1955)
Designing and Making, (Blackwell, 1957)
A Concrete Approach to Abstract Algebra, (Freeman, 1959)
What Is Calculus About?, (Yale University, 1961)
Vision in Elementary Mathematics, (Penguin, 1964)
A Path to Modern Mathematics, (Penguin, 1966)
Search for Pattern, (Penguin, 1970)
An Engineering Approach to Linear Algebra, (Cambridge University Press, 1972)
A First Look at Numerical Functional Analysis, (Oxford University Press, 1978)
Notes
External links
Walter Warwick Sawyer information and materials
1911 births
2008 deaths
20th-century English mathematicians
People educated at Highgate School
Alumni of St John's College, Cambridge
Wesleyan University faculty
Academic staff of the University of Toronto
British expatriates in Ghana
British expatriates in New Zealand
British expatriates in the United States
British expatriates in Canada |
https://en.wikipedia.org/wiki/Thomas%20Howarth%20%28footballer%29 | Thomas Grimshaw Howarth (1879–1959) was an English professional footballer who played as a wing half for Burnley in the early 1900s.
Career statistics
References
English men's footballers
Men's association football wing halves
Burnley F.C. players
English Football League players
1959 deaths
1879 births
Brentford F.C. players
Southern Football League players
People from Nelson, Lancashire
Footballers from Lancashire |
https://en.wikipedia.org/wiki/Hendecagonal%20prism | In geometry, the hendecagonal prism is one in an infinite set of convex prisms formed by square sides and two regular polygon caps, in this case two hendecagons.
So, it has 2 hendecagons and 11 squares as its faces.
Related polyhedra
External links
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Great%20triambic%20icosahedron | In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical dual uniform polyhedra. The exterior surface also represents the De2f2 stellation of the icosahedron. These figures can be differentiated by marking which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas. Alternatively, if the faces are filled with the even–odd rule, the internal structure of both shapes will differ.
The 12 vertices of the convex hull matches the vertex arrangement of an icosahedron.
Great triambic icosahedron
The great triambic icosahedron is the dual of the great ditrigonal icosidodecahedron, U47. It has 20 inverted-hexagonal (triambus) faces, shaped like a three-bladed propeller. It has 32 vertices: 12 exterior points, and 20 hidden inside. It has 60 edges.
The faces have alternating angles of and . The sum of the six angles is , and not as might be expected for a hexagon, because the polygon turns around its center twice. The dihedral angle equals .
Medial triambic icosahedron
The medial triambic icosahedron is the dual of the ditrigonal dodecadodecahedron, U41. It has 20 faces, each being simple concave isotoxal hexagons or triambi. It has 24 vertices: 12 exterior points, and 12 hidden inside. It has 60 edges.
The faces have alternating angles of and . The dihedral angle equals .
Unlike the great triambic icosahedron, the medial triambic icosahedron is topologically a regular polyhedron of index two. By distorting the triambi into regular hexagons, one obtains a quotient space of the hyperbolic order-5 hexagonal tiling:
As a stellation
It is Wenninger's 34th model as his 9th stellation of the icosahedron
See also
Triakis icosahedron
Small triambic icosahedron
Medial rhombic triacontahedron
References
H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , 3.6 6.2 Stellating the Platonic solids, pp.96-104
External links
gratrix.net Uniform polyhedra and duals
bulatov.org Medial triambic icosahedron Great triambic icosahedron
Polyhedra
Polyhedral stellation
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Small%20rhombihexacron | In geometry, the small rhombihexacron (or small dipteral disdodecahedron) is the dual of the small rhombihexahedron. It is visually identical to the small hexacronic icositetrahedron. Its faces are antiparallelograms formed by pairs of coplanar triangles.
Proportions
Each antiparallelogram has two angles of and two angles of . The diagonals of each antiparallelogram intersect at an angle of . The dihedral angle equals . The ratio between the lengths of the long edges and the short ones equals .
References
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Small%20hexacronic%20icositetrahedron | In geometry, the small hexacronic icositetrahedron is the dual of the small cubicuboctahedron. It is visually identical to the small rhombihexacron. A part of each dart lies inside the solid, hence is invisible in solid models.
Proportions
Its faces are darts, having two angles of , one of and one of . Its dihedral angles equal . The ratio between the lengths of the long edges and the short ones equals .
References
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Small%20dodecahemidodecacron | In geometry, the small dodecahemidodecacron is the dual of the small dodecahemidodecahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small icosihemidodecacron.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
The small dodecahemidodecahedron has six decagonal faces passing through the model center, the small dodecahemidodecacron can be seen as having six vertices at infinity.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Small%20icosihemidodecacron | In geometry, the small icosihemidodecacron is the dual of the small icosihemidodecahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small dodecahemidodecacron.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
The small icosihemidodecahedron has six decagonal faces passing through the model center, the small icosihemidodecacron has six vertices at infinity.
See also
Hemi-dodecahedron - The six vertices at infinity correspond directionally to the six vertices of this abstract polyhedron.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Small%20dodecicosacron | In geometry, the small dodecicosacron (or small dipteral trisicosahedron) is the dual of the small dodecicosahedron (U50). It is visually identical to the Small ditrigonal dodecacronic hexecontahedron. It has 60 intersecting bow-tie-shaped faces.
Proportions
Each face has two angles of and two angles of . The diagonals of each antiparallelogram intersect at an angle of . The dihedral angle equals . The ratio between the lengths of the long edges and the short ones equals , which is the golden ratio. Part of each face lies inside the solid, hence is invisible in solid models.
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Icositetragon | In geometry, an icositetragon (or icosikaitetragon) or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.
Regular icositetragon
The regular icositetragon is represented by Schläfli symbol {24} and can also be constructed as a truncated dodecagon, t{12}, or a twice-truncated hexagon, tt{6}, or thrice-truncated triangle, ttt{3}.
One interior angle in a regular icositetragon is 165°, meaning that one exterior angle would be 15°.
The area of a regular icositetragon is: (with t = edge length)
The icositetragon appeared in Archimedes' polygon approximation of pi, along with the hexagon (6-gon), dodecagon (12-gon), tetracontaoctagon (48-gon), and enneacontahexagon (96-gon).
Construction
As 24 = 23 × 3, a regular icositetragon is constructible using an angle trisector. As a truncated dodecagon, it can be constructed by an edge-bisection of a regular dodecagon.
Symmetry
The regular icositetragon has Dih24 symmetry, order 48. There are 7 subgroup dihedral symmetries: (Dih12, Dih6, Dih3), and (Dih8, Dih4, Dih2 Dih1), and 8 cyclic group symmetries: (Z24, Z12, Z6, Z3), and (Z8, Z4, Z2, Z1).
These 16 symmetries can be seen in 22 distinct symmetries on the icositetragon. John Conway labels these by a letter and group order. The full symmetry of the regular form is r48 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g24 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icositetragon, m=12, and it can be divided into 66: 6 squares and 5 sets of 12 rhombs. This decomposition is based on a Petrie polygon projection of a 12-cube.
Related polygons
A regular triangle, octagon, and icositetragon can completely fill a plane vertex.
An icositetragram is a 24-sided star polygon. There are 3 regular forms given by Schläfli symbols: {24/5}, {24/7}, and {24/11}. There are also 7 regular star figures using the same vertex arrangement: 2{12}, 3{8}, 4{6}, 6{4}, 8{3}, 3{8/3}, and 2{12/5}.
There are also isogonal icositetragrams constructed as deeper truncations of the regular dodecagon {12} and dodecagram {12/5}. These also generate two quasitruncations: t{12/11}={24/11}, and t{12/7}={24/7}.
Skew icositetragon
A skew icositetragon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an icositetragon |
https://en.wikipedia.org/wiki/Small%20rhombidodecacron | In geometry, the small rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces.
Proportions
Each face has two angles of and two angles of . The diagonals of each antiparallelogram intersect at an angle of . The ratio between the lengths of the long edges and the short ones equals , which is the golden ratio. The dihedral angle equals .
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
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