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https://en.wikipedia.org/wiki/1990%20Cambodian%20League | Statistics of the Cambodian League for the 1990 season.
Overview
Ministry of Transports won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1991%20Cambodian%20League | Statistics of the Cambodian League for the 1991 season.
Overview
Municipal Constructions won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1992%20Cambodian%20League | Statistics of the Cambodian League for the 1992 season.
Overview
Municipal Constructions won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1993%20Cambodian%20League | Statistics of the Cambodian League for the 1993 season.
Overview
National Defense Ministry won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1994%20Cambodian%20League | Statistics of the Cambodian League for the 1994 season.
Overview
Civil Aviation won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1995%20Cambodian%20League | Statistics of the Cambodian League for the 1995 season.
Overview
Civil Aviation won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1996%20Cambodian%20League | Statistics of the Cambodian League for the 1996 season.
Overview
Body Guards Club won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1997%20Cambodian%20League | Statistics of the Cambodian League for the 1997 season.
Overview
Body Guards Club won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1998%20Cambodian%20League | Statistics of the Cambodian League for the 1998 season.
Overview
Royal Dolphins won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/1999%20Cambodian%20League | Statistics of the Cambodian League for the 1999 season.
Overview
Royal Dolphins won the championship.
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/2005%20Cambodian%20League | Statistics of the Cambodian League for the 2005 season.
Overview
Khemara Keila FC won the championship.
Teams
10 participants:
Hello United
Nagacorp
Khemara Keila FC
Royal Navy
Royal Cambodian Armed Force (RCAF)
Army Division of Logistics (General Logistics)
Military Police
Kandal Province
Koh Kong Province
Siem Reap Province
Top of table
The top four team qualified to championship play-off
Khemara Keila FC
Hello United
Military Police
Nagacorp
Championship play-off
Semi-finals
05 Oct 2005 Hello United 4-2 Nagacorp
06 Oct 2005 Khemara Keila 2-1 Military Police
Third place
Final
References
RSSSF
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/Characteristic%20exponent | In mathematics, characteristic exponent may refer to:
Characteristic exponent of a field, a number equal to 1 if the field has characteristic 0, and equal to p if the field has characteristic p > 0
Lyapunov characteristic exponent, a quantity that characterizes the rate of separation
Characteristic exponent of Stable distribution
The logarithm of a characteristic function
Logarithm of a characteristic multiplier in the Floquet theory
Solution of the indicial equation of the Frobenius method |
https://en.wikipedia.org/wiki/Temple%20University%20College%20of%20Science%20and%20Technology | Temple University's College of Science and Technology houses the departments of Biology, Chemistry, Computer & Information Sciences, Earth & Environmental Science, Mathematics, and Physics. It is one of the largest schools or colleges of its kind in the Philadelphia region with more than 200 faculty and 4000 undergraduate and graduate students. Michael L. Klein is dean of the college and Laura H. Carnell Professor.
Founded in 1998 from the science departments in what was then the College of Arts and Sciences, the College of Science and Technology offers bachelor's, master's, and doctoral degrees in all six departments as well as science with teaching bachelor's degrees through the TUteach program, based on the UTeach program.
Undergraduate Research Program
The College of Science and Technology offers the CST Undergraduate Research Program (URP). Students selected to participate work with a faculty sponsor to perform research in the faculty member's lab. It may also be possible for students to earn a stipend for additional work performed in the lab in excess of the required research course requirements. Students may be asked to participate in conferences, author papers or to showcase their research work in the department or at the URP Research Symposium.
Centers and Institutes for Advanced Research & Education
Center for Advanced Photonics Research
Center for Biophysics and Computational Biology
Center for Computational Genetics and Genomics
Center for Data Analytics and Biomedical Informatics
Center for Materials Theory
Institute for Computational Molecular Science
Sbarro Health Research Organization
Research Support Facilities
Research and Instructional Support Facility (RISF)
Solid Phase Peptide Synthesis and Analysis (SPPS)
Materials Research Facility
Notable faculty
Antonio Giordano, Biology
Michael L. Klein, Chemistry
Jie Wu, Computer & Information Sciences
Igor Rivin, Mathematics
Xiaoxing Xi, Physics
Notable alumni
F. Albert Cotton, chemist
Angelo DiGeorge, pediatric endocrinologist
Bernard Roizman, virologist
Herbert Scarf, mathematical economist
References
Temple University |
https://en.wikipedia.org/wiki/2010%20Rajnavy%20Rayong%20F.C.%20season | The 2010 season was Rajnavy's eighth season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.
24 October 2010: Rajnavy Rayong finished in 10th place in the Thai Premier League.
Current squad
Updated 7 January 2010
Results
Thai Premier League
League table
FA Cup
Third round
Fourth round
Quarter-final
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Third round
1st leg
2nd leg
Queen's Cup
References
2010
Thai football clubs 2010 season |
https://en.wikipedia.org/wiki/2010%20Police%20United%20F.C.%20season | The 2010 season was Police United's 8th season in the top division of Thai football. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Team kit
Chronological list of events
26 November 2009'': Police announce that they will re-locate to the Thammasat Stadium in Rangsit for the 2010 campaign10 November 2009: The Thai Premier League 2010 season first leg fixtures were announced.14 July 2010: Police United were knocked out by TTM Phichit in the FA Cup third round.22 September 2010: Police United were knocked out by Osotspa Saraburi in the League Cup second round.24 October 2010''': Police United finished in 11th place in the Thai Premier League.
Current squad
Results
Thai Premier League
League table
FA Cup
Third round
League Cup
First round
1st leg
2nd leg
Second round
1st leg
2nd leg
Queen's Cup
References
2010
Thai football clubs 2010 season |
https://en.wikipedia.org/wiki/1983%20Chinese%20Taipei%20National%20Football%20League | Statistics of the Chinese Taipei National Football League for the 1983 season.
Overview
Flying Camel won the championship.
References
RSSSF
1983
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1984%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1984 season.
Overview
Flying Camel won the championship.
References
RSSSF
1983
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1986%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1986 season.
Overview
Taipei City Bank won the championship.
References
RSSSF
1983
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1987%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1987 season.
Overview
Taipower won the championship.
References
RSSSF
1983
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1988%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1988 season.
Overview
Flying Camel won the championship.
References
RSSSF
1983
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1989%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1989 season.
Overview
Taipei City Bank won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1991%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1991 season.
Overview
Taipei City Bank won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1992%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1992 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1993%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1993 season.
Overview
Flying Camel won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1995%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1995 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1996%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1996 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1998%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1998 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1999%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 1999 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20Chinese%20Taipei%20National%20Football%20League | Statistics of the Chinese Taipei National Football League in the 2001–02 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
Chinese Taipei
1
1 |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20Chinese%20Taipei%20National%20Football%20League | Statistics of the Chinese Taipei National Football League for the 2002–03 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
Chinese Taipei
1
1 |
https://en.wikipedia.org/wiki/2004%20Chinese%20Taipei%20National%20Football%20League | Statistics of Chinese Taipei National Football League in the 2004 season.
Overview
Taipower won the championship.
References
RSSSF
Chinese Taipei National Football League seasons
Chinese Taipei
Chinese Taipei
1 |
https://en.wikipedia.org/wiki/1990%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1990 season.
Overview
University of Guam won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1991%20Guam%20Men%27s%20Soccer%20League | Statistics of the Guam League in the 1991 season.
Overview
University of Guam won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1992%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1992 season.
Overview
University of Guam won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1993%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1993 season.
Overview
University of Guam won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1994%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1994 season.
Overview
Tumon Taivon won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1995%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1995 season.
Overview
G-Force won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1996%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1996 season.
Overview
G-Force won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1997%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1997 season.
Overview
Tumon Soccer Club won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1998%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1998 season.
Overview
Anderson Soccer Club won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1999%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 1999 season.
Overview
Coors Light Silver Bullets won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2000%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 2000 season.
Overview
Coors Light Silver Bullets won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2001%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 2001 season.
Overview
Staywell Zoom won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2002%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 2002 season.
Overview
Guam Shipyard won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2003%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 2003 season.
Overview
Guam Shipyard won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2004%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 2004 season.
Overview
Guam U-18 won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2005%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League in the 2005 season.
Overview
Guam Shipyard won the championship.
References
RSSSF
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/1966%E2%80%9367%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1966–67 season.
Overview
Al Arabi Kuwait won the championship.
References
RSSSF
Kuwait Premier League seasons
Kuwait
football |
https://en.wikipedia.org/wiki/1967%E2%80%9368%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1967–68 season.
Overview
Al Kuwait Kaifan won the championship.
References
RSSSF
Kuwait Premier League seasons
Kuwait
football |
https://en.wikipedia.org/wiki/1968%E2%80%9369%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1968–69 season.
Overview
Al Qadisiya Kuwait won the championship.
References
RSSSF
Kuwait Premier League seasons
Kuwait
football |
https://en.wikipedia.org/wiki/1969%E2%80%9370%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1969–70 season.
Overview
Al Arabi Kuwait won the championship.
References
RSSSF
Kuwait Premier League seasons
Kuwait
football |
https://en.wikipedia.org/wiki/1970%E2%80%9371%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1970–71 season.
Overview
Al Qadisiya Kuwait won the championship.
References
RSSSF
1970–71
1970–71 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1972%E2%80%9373%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1972–73 season.
Overview
Al Qadisiya Kuwait won the championship.
References
RSSSF
1972–73
1972–73 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1973%E2%80%9374%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1973–74 season.
Overview
Al Kuwait Kaifan won the championship.
References
RSSSF
1973–74
1973–74 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1974%E2%80%9375%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1974–75 season.
Overview
Al Qadisiya Kuwait won the championship.
References
RSSSF
1974–75
1974–75 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1975%E2%80%9376%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1975–76 season.
Overview
Al Qadisiya Kuwait won the championship.
References
RSSSF
1975–76
1975–76 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1976%E2%80%9377%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1976–77 season.
Overview
Al Kuwait Kaifan won the championship.
References
RSSSF
1976–77
1976–77 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1977%E2%80%9378%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1977–78 season.
Overview
Al Qadisiya Kuwait won the championship.
References
RSSSF
1977–78
1977–78 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1978%E2%80%9379%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1978–79 season.
Overview
Al Kuwait Kaifan won the championship.
References
RSSSF
1978–79
1978–79 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1979%E2%80%9380%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1979–80 season.
Overview
Al Arabi Kuwait won the championship.
References
RSSSF
1979–80
1979–80 in Asian association football leagues
football |
https://en.wikipedia.org/wiki/1980%E2%80%9381%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1980–81 season.
Overview
Al Salmiya Club won the championship.
References
RSSSF
1980-81
1980–81 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1981%E2%80%9382%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1981–82 season.
Overview
Al Arabi Kuwait won the championship.
References
RSSSF
1981–82
1981–82 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/1982%E2%80%9383%20Kuwaiti%20Premier%20League | Statistics of Kuwaiti Premier League in the 1982–83 season.
Overview
Al Arabi Kuwait won the championship.
References
RSSSF
1982–83
1982–83 in Asian association football leagues
1 |
https://en.wikipedia.org/wiki/2000%20Lao%20League | Statistics of Lao League in the 2000 season.
Overview
Vientiane Municipality FC won the sixth national games, beating Champassak Province FC from the south of Laos. There is some debate as to whether this was the national championship. One source suggests that it was however, there is also evidence that National Bank, who played in the top division, won the National Trophy and that this is assumed to have been a cup competition and could have been the national title for the year.
Sixth National Games
The following teams took part in sixth National Games which may well have been the national championship for the year:
Institutional Teams
Banks
Industry Ministry
Social Welfare Ministry
Prime Ministers Office
Education Ministry
Interior Ministry
Public Health Ministry
Army
Information and Culture Ministry
Foreign Affairs Ministry
Provincial/City Teams
Vientiane
Xieng Khuang
Bolikhamsay
Sayaboury
Khammuan
Saravan
Luang Prabang
Sekong
Houaphan
Bokeo
Champassak
Savannakhet
Houaphan
Vientiane Municipality
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/2001%20Lao%20League | Statistics for the 2001 season of the Lao League.
Overview
Banks won the championship, which was arranged on a group stage. Two pools of six played each other with the top two qualifying for the semi-finals.
Group stage
Pool 1
Banks (Finished first in Group)
Interior Ministry (Finished second in Group)
Luang Prabang Province
Education Ministry
Army
Pool 2
Champassak Province (Finished first in Group)
Vientiane municipality (Finished first in Group)
Industry Ministry
Khammuan Province
Savannakhet Province
Semi-finals
Third-place match
Final
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/2003%20Lao%20League | Statistics for the 2003 season of the Lao League.
Overview
MCTPC FC (Ministry of Communication, Transportation, Post and Construction), described in the source as Telecom and Transportation, won the championship.
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/2006%20Lao%20League | Statistics of Lao League in the 2006 season.
Overview
Vientiane FC won the championship.
References
RSSSF
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/2007%20Lao%20League | Statistics of Lao League in the 2007 season.
Overview
Lao-American College FC won the championship.
References
RSSSF
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/2010%20Lao%20League | Statistics of Lao League in the 2010 season.
Clubs
Lao Army FC
Bank FC
City Copy Center FC
Ezra FC
Lao-American College FC
Ministry of Public Security (MPS)
Ministry of Public Works and Transport FC
Vientiane FC
The season ran from 27 February to 11 April and all matches were played on Saturdays and Sundays at the Chao Anouvong Stadium in Vientiane. Bank of Laos were champions.
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/1996%E2%80%9397%20Lebanese%20Premier%20League | Statistics of Lebanese Premier League for the 1996–97 season.
Overview
Al-Ansar won the championship.
League standings
References
RSSSF
Leb
1996–97 in Lebanese football
Lebanese Premier League seasons
1996–97 Lebanese Premier League |
https://en.wikipedia.org/wiki/Dodecagram | In geometry, a dodecagram () is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon (with Schläfli symbol and a turning number of 5). There are also 4 regular compounds and
Regular dodecagram
There is one regular form: {12/5}, containing 12 vertices, with a turning number of 5. A regular dodecagram has the same vertex arrangement as a regular dodecagon, which may be regarded as {12/1}.
Dodecagrams as regular compounds
There are four regular dodecagram star figures: {12/2}=2{6}, {12/3}=3{4}, {12/4}=4{3}, and {12/6}=6{2}. The first is a compound of two hexagons, the second is a compound of three squares, the third is a compound of four triangles, and the fourth is a compound of six straight-sided digons. The last two can be considered compounds of two compound hexagrams and the last as three compound tetragrams.
Dodecagrams as isotoxal figures
An isotoxal polygon has two vertices and one edge type within its symmetry class. There are 5 isotoxal dodecagram star with a degree of freedom of angles, which alternates vertices at two radii, one simple, 3 compounds, and 1 unicursal star.
Dodecagrams as isogonal figures
A regular dodecagram can be seen as a quasitruncated hexagon, t{6/5}={12/5}. Other isogonal (vertex-transitive) variations with equally spaced vertices can be constructed with two edge lengths.
Complete graph
Superimposing all the dodecagons and dodecagrams on each other – including the degenerate compound of six digons (line segments), {12/6} – produces the complete graph K12.
Regular dodecagrams in polyhedra
Dodecagrams can also be incorporated into uniform polyhedra. Below are the three prismatic uniform polyhedra containing regular dodecagrams (there are no other dodecagram-containing uniform polyhedra).
Dodecagrams can also be incorporated into star tessellations of the Euclidean plane.
Dodecagram Symbolism
Dodecagrams or twelve-pointed stars have been used as symbols for the following:
the twelve tribes of Israel, in Judaism
the twelve disciples, in Christianity
the twelve olympians, in Hellenic Polytheism
the twelve signs of the zodiac
the International Order of Twelve Knights and Daughters of Tabor, an African-American fraternal group
the fictional secret society Manus Sancti, in the Knights of Manus Sancti series by Bryn Donovan
The twelve tribes of Nauru on the national flag.
See also
Stellation
Star polygon
List of regular polytopes and compounds
References
Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), .
Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)
12 |
https://en.wikipedia.org/wiki/Zoran%20Ban | Zoran Ban (born 27 May 1973) is a Croatian retired professional footballer who played as a striker.
He played a match against Estonia in 1994 for the U-21 team of Croatia.
Career statistics
Club
Honours
Player
Club
Genk
Belgian Cup: 1999–2000
References
External links
Zoran Ban
1973 births
Living people
Footballers from Rijeka
Men's association football forwards
Yugoslav men's footballers
Croatian men's footballers
Croatia men's under-21 international footballers
HNK Rijeka players
Juventus FC players
C.F. Os Belenenses players
Boavista F.C. players
Delfino Pescara 1936 players
Royal Excel Mouscron players
K.R.C. Genk players
R.A.E.C. Mons (1910) players
Calcio Foggia 1920 players
Yugoslav First League players
Croatian Football League players
Serie A players
Primeira Liga players
Serie B players
Belgian Pro League players
Croatian expatriate men's footballers
Expatriate men's footballers in Italy
Expatriate men's footballers in Portugal
Expatriate men's footballers in Belgium
Croatian expatriate sportspeople in Italy
Croatian expatriate sportspeople in Portugal
Croatian expatriate sportspeople in Belgium |
https://en.wikipedia.org/wiki/Pentagram%20map | In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper though it seems that the special case, in which the map is defined for pentagons only, goes back to an 1871 paper of Alfred Clebsch and a 1945 paper of Theodore Motzkin. The pentagram map is similar in spirit to the constructions underlying Desargues' theorem and Poncelet's porism. It echoes the rationale and construction underlying a conjecture of Branko Grünbaum concerning the diagonals of a polygon.
Definition of the map
Basic construction
Suppose that the vertices of the polygon P are given by The image of P under the pentagram map is the polygon Q with vertices as shown in the figure. Here is the intersection of the diagonals and , and so on.
On a basic level, one can think of the pentagram map as an operation defined on convex polygons in the plane. From a more sophisticated point of view, the pentagram map is defined for a polygon contained in the projective plane over a field provided that the vertices are in sufficiently general position. The pentagram map commutes with projective transformations and thereby induces a mapping on the moduli space of projective equivalence classes of polygons.
Labeling conventions
The map is slightly problematic, in the sense that the indices of the P-vertices are naturally odd integers whereas the indices of Q-vertices are naturally even integers. A more conventional approach to the labeling would be to label the vertices of P and Q by integers of the same parity. One can arrange this either by adding or subtracting 1 from each of the indices of the Q-vertices. Either choice is equally canonical. An even more conventional choice would be to label the vertices of P and Q by consecutive integers, but again there are two natural choices for how to align these labellings: Either is just clockwise from or just counterclockwise. In most papers on the subject, some choice is made once and for all at the beginning of the paper and then the formulas are tuned to that choice.
There is a perfectly natural way to label the vertices of the second iterate of the pentagram map by consecutive integers. For this reason, the second iterate of the pentagram map is more naturally considered as an iteration defined on labeled polygons. See the figure.
Twisted polygons
The pentagram map is also defined on the larger space of twisted polygons.
A twisted N-gon is a bi-infinite sequence of points in the projective plane that is N-periodic modulo a projective transformation That is, some projective transformation M carries to for all k. The map M is called the monodromy of the twisted N-gon. When M is the identity, a twisted N-gon can be interpreted a |
https://en.wikipedia.org/wiki/Map%20folding | In the mathematics of paper folding, map folding and stamp folding are two problems of counting the number of ways that a piece of paper can be folded. In the stamp folding problem, the paper is a strip of stamps with creases between them, and the folds must lie on the creases. In the map folding problem, the paper is a map, divided by creases into rectangles, and the folds must again lie only along these creases.
credits the invention of the stamp folding problem to Émile Lemoine. provides several other early references.
Labeled stamps
In the stamp folding problem, the paper to be folded is a strip of square or rectangular stamps, separated by creases, and the stamps can only be folded along those creases.
In one commonly considered version of the problem, each stamp is considered to be distinguishable from each other stamp, so two foldings of a strip of stamps are considered equivalent only when they have the same vertical sequence of stamps.
For example, there are six ways to fold a strip of three different stamps:
These include all six permutations of the stamps, but for more than three stamps not all permutations are possible. If, for a permutation , there are two numbers and with the same parity such that the four numbers , , , and appear in in that cyclic order, then cannot be folded. The parity condition implies that the creases between stamps and , and between stamps and , appear on the same side of the stack of folded stamps, but the cyclic ordering condition implies that these two creases cross each other, a physical impossibility. For instance, the four-element permutation 1324 cannot be folded, because it has this forbidden pattern with and . All remaining permutations, without this pattern, can be folded.
The number of different ways to fold a strip of stamps is given by the sequence
1, 2, 6, 16, 50, 144, 462, 1392, 4536, 14060, 46310, 146376, 485914, 1557892, 5202690, ... .
These numbers are always divisible by (because a cyclic permutation of a foldable stamp sequence is always itself foldable), and the quotients of this division are
1, 1, 2, 4, 10, 24, 66, 174, 504, 1406, 4210, 12198, 37378, 111278, 346846, 1053874, ... ,
the number of topologically distinct ways that a half-infinite curve can make crossings with a line, called "semimeanders". These are closely related to meanders, ways for a closed curve to make the same number of crossings with a line. Meanders correspond to solutions of the stamp-folding problem in which the first and last stamp can be connected to each other to form a continuous loop of stamps.
In the 1960s, John E. Koehler and W. F. Lunnon implemented algorithms that, at that time, could calculate these numbers for up to 28 stamps.
Despite additional research, the known methods for calculating these numbers take exponential time as a function of .
Thus, there is no formula or efficient algorithm known that could extend this sequence to very large values of . Nevertheless, heuristic methods |
https://en.wikipedia.org/wiki/Jie-zhong%20Zou | Jie-zhong Zou (born October 15, 1947) is a mathematician known for his research on mathematical probability theory and its applications, in particular in topics such as homogeneous Markov chains, queuing theory and mathematical finance. He entered Changsha Railway Institute (Central South University now) in 1980 and received his Ph.D. at the Changsha Railway Institute in 1987 under advisor Zhen-ting Hou. Since 1987 Jie-zhong Zou has been on the faculty at Changsha Railway Institute (Central South University now).
He was awarded the Rollo Davidson Prize in 1987, and was elected a Fellow of the Chinese Mathematical Society.
He attended the International Congress of Mathematician in Beijing in 2002.
Papers
Jie-zhong Zou, "The oscillation problem for p-functions". D.Phil. Thesis, Changsha Railway Institute 1986 (Chinese).
Jie-zhong Zou, "Some New Inequalities for p-Functions ". Journal of the London Mathematical Society 1988 s2-38(2):356-366.
References
Institute of Probability and Statistics,CSU
External links
Rollo Davidson Prize
Jie-zhong Zou Home page
List of Participants in ICM-2002
1947 births
Mathematicians from Hunan
Living people
Probability theorists
Central South University alumni
Academic staff of the Central South University |
https://en.wikipedia.org/wiki/Biostatistics%20%28journal%29 | Biostatistics is a peer-reviewed scientific journal covering biostatistics, that is, statistics for biological and medical research.
The journals that had cited Biostatistics the most by 2008 were Biometrics, Journal of the American Statistical Association, Biometrika, Statistics in Medicine, and Journal of the Royal Statistical Society, Series B.
Scott Zeger and Peter Diggle were the founding editors of Biostatistics.
References
External links
IMPACT FACTOR AND RANKING
Biostatistics journals
Statistics journals
Academic journals established in 2000
Oxford University Press academic journals |
https://en.wikipedia.org/wiki/Gabriel%20Schacht | Gabriel Schacht, (born 8 May 1981) is a Brazilian former professional footballer who as a midfielder.
Career
Schacht joined Aboomoslem in 2010.
Career statistics
Notes
References
1981 births
Living people
Brazilian men's footballers
Men's association football midfielders
F.C. Aboomoslem players
Marília Atlético Clube players
Al-Sailiya SC players
Grêmio Esportivo Brasil players
Botafogo Futebol Clube (SP) players
Sport Club Internacional players
Associação Atlética Ponte Preta players
Mogi Mirim Esporte Clube players
Sociedade Esportiva e Recreativa Caxias do Sul players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Iran
Expatriate men's footballers in Iran
People from Carazinho
Footballers from Rio Grande do Sul |
https://en.wikipedia.org/wiki/Comparison%20of%20vector%20algebra%20and%20geometric%20algebra | Geometric algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations.
Vector algebra uses all dimensions and signatures, as does geometric algebra, notably 3+1 spacetime as well as 2 dimensions.
Basic concepts and operations
Geometric algebra (GA) is an extension or completion of vector algebra (VA). The reader is herein assumed to be familiar with the basic concepts and operations of VA and this article will mainly concern itself with operations in the GA of 3D space (nor is this article intended to be mathematically rigorous). In GA, vectors are not normally written boldface as the meaning is usually clear from the context.
The fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors: scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated .
The ungeneralized 3D vector form of the geometric product is:
that is the sum of the usual dot (inner) product and the outer (exterior) product (this last is closely related to the cross product and will be explained below).
In VA, entities such as pseudovectors and pseudoscalars need to be bolted on, whereas in GA the equivalent bivector and pseudovector respectively exist naturally as subspaces of the algebra.
For example, applying vector calculus in 2 dimensions, such as to compute torque or curl, requires adding an artificial 3rd dimension and extending the vector field to be constant in that dimension, or alternately considering these to be scalars. The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines these as a pseudoscalar field (a bivector), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product.
Translations between formalisms
Here are some comparisons between standard vector relations and their corresponding exterior product and geometric product equivalents. All the exterior and geometric product equivalents here are good for more than three dimensions, and some also for two. In two dimensions the cross product is undefined even if what it describes (like torque) is perfectly well defined in a plane without introducing an arbitrary normal vector outside of the space.
Many of these relationships only require the introduction of the exterior product to generalize, but since that may not be familiar to somebody with only a background in vector algebra and calculus, some examples are given.
Cross and exterior products
is perpendicular to the plane containing and .
is an oriented representation of the same plane.
We have the pseudoscalar (right handed orthonormal frame) and so
returns a bivector and
returns a vector p |
https://en.wikipedia.org/wiki/Harmonious%20set | In mathematics, a harmonious set is a subset of a locally compact abelian group on which every weak character may be uniformly approximated by strong characters. Equivalently, a suitably defined dual set is relatively dense in the Pontryagin dual of the group. This notion was introduced by Yves Meyer in 1970 and later turned out to play an important role in the mathematical theory of quasicrystals. Some related concepts are model sets, Meyer sets, and cut-and-project sets.
Definition
Let Λ be a subset of a locally compact abelian group G and Λd be the subgroup of G generated by Λ, with discrete topology. A weak character is a restriction to Λ of an algebraic homomorphism from Λd into the circle group:
A strong character is a restriction to Λ of a continuous homomorphism from G to T, that is an element of the Pontryagin dual of G.
A set Λ is harmonious if every weak character may be approximated by
strong characters uniformly on Λ. Thus for any ε > 0 and any weak character χ, there exists a strong character ξ such that
If the locally compact abelian group G is separable and metrizable (its topology may be defined by a translation-invariant metric) then harmonious sets admit another, related, description. Given a subset Λ of G and a positive ε, let Mε be the subset of the Pontryagin dual of G consisting of all characters that are almost trivial on Λ:
Then Λ is harmonious if the sets Mε are relatively dense in the sense of Besicovitch: for every ε > 0 there exists a compact subset Kε of the Pontryagin dual such that
Properties
A subset of a harmonious set is harmonious.
If Λ is a harmonious set and F is a finite set then the set Λ + F is also harmonious.
The next two properties show that the notion of a harmonious set is nontrivial only when the ambient group is neither compact nor discrete.
A finite set Λ is always harmonious. If the group G is compact then, conversely, every harmonious set is finite.
If G is a discrete group then every set is harmonious.
Examples
Interesting examples of multiplicatively closed harmonious sets of real numbers arise in the theory of diophantine approximation.
Let G be the additive group of real numbers, θ >1, and the set Λ consist of all finite sums of different powers of θ. Then Λ is harmonious if and only if θ is a Pisot number. In particular, the sequence of powers of a Pisot number is harmonious.
Let K be a real algebraic number field of degree n over Q and the set Λ consist of all Pisot or Salem numbers of degree n in K. Then Λ is contained in the open interval (1,∞), closed under multiplication, and harmonious. Conversely, any set of real numbers with these 3 properties consists of all Pisot or Salem numbers of degree n in some real algebraic number field K of degree n.
See also
Almost periodic function
References
Yves Meyer, Algebraic numbers and harmonic analysis, North-Holland Mathematical Library, vol.2, North-Holland, 1972
Harmonic analysis
Diophantine approximatio |
https://en.wikipedia.org/wiki/Meyer%20set | In mathematics, a Meyer set or almost lattice is a relatively dense set X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after Yves Meyer, who introduced and studied them in the context of diophantine approximation. Nowadays Meyer sets are best known as mathematical model for quasicrystals. However, Meyer's work precedes the discovery of quasicrystals by more than a decade and was entirely motivated by number theoretic questions.
Definition and characterizations
A subset X of a metric space is relatively dense if there exists a number r such that all points of X are within distance r of X, and it is uniformly discrete if there exists a number ε such that no two points of X are within distance ε of each other. A set that is both relatively dense and uniformly discrete is called a Delone set. When X is a subset of a vector space, its Minkowski difference X − X is the set {x − y | x, y in X} of differences of pairs of elements of X.
With these definitions, a Meyer set may be defined as a relatively dense set X for which X − X is uniformly discrete. Equivalently, it is a Delone set for which X − X is Delone, or a Delone set X for which there exists a finite set F with X − X ⊂ X + F
Some additional equivalent characterizations involve the set
defined for a given X and ε, and approximating (as ε approaches zero) the definition of the reciprocal lattice of a lattice. A relatively dense set X is a Meyer set if and only if
For all ε > 0, Xε is relatively dense, or equivalently
There exists an ε with 0 < ε < 1/2 for which Xε is relatively dense.
A character of an additively closed subset of a vector space is a function that maps the set to the unit circle in the plane of complex numbers, such that the sum of any two elements is mapped to the product of their images. A set X is a harmonious set if, for every character χ on the additive closure of X and every ε > 0, there exists a continuous character on the whole space that ε-approximates χ. Then a relatively dense set X is a Meyer set if and only if it is harmonious.
Examples
Meyer sets include
The points of any lattice
The vertices of any rhombic Penrose tiling
The Minkowski sum of another Meyer set with any nonempty finite set
Any relatively dense subset of another Meyer set
References
Metric geometry
Crystallography
Lattice points |
https://en.wikipedia.org/wiki/Thomas%20Zaslavsky | Thomas Zaslavsky (born 1945) is an American mathematician specializing in combinatorics.
Zaslavsky's mother Claudia Zaslavsky was a high school mathematics teacher and an ethnomathematician in New York; his father Sam Zaslavsky (from Manhattan) was an electrical engineer. Thomas Zaslavsky graduated from the City College of New York. At M.I.T. he studied hyperplane arrangements with Curtis Greene and received a Ph.D. in 1974. In 1975 the American Mathematical Society published his doctoral thesis.
Zaslavsky has been a professor of mathematics at the Binghamton University, New York since 1985. He has published papers on matroid theory and hyperplane arrangements. He has also written on coding theory, lattice point counting, and Sperner theory. Zaslavsky has made available a bibliography on signed graphs and their applications.
Select publications
References
Thomas Zaslavsky's homepage
Microsoft academic search
Graph theorists
The Bronx High School of Science alumni
Massachusetts Institute of Technology School of Science alumni
Binghamton University faculty
1945 births
Living people |
https://en.wikipedia.org/wiki/Wolfgang%20L%C3%BCck | Wolfgang Lück (born 19 February 1957 in Herford) is a German mathematician who is an internationally recognized expert in algebraic topology.
Life and work
After receiving his Abitur from the Ravensberger Gymnasium in Herford in 1975, he studied at the University of Göttingen where he obtained his Diplom in 1981 and his doctoral degree under Tammo tom Dieck in 1984. His thesis was entitled Eine allgemeine Beschreibung für Faserungen auf projektiven Klassengruppen und Whiteheadgruppen.
From 1982 on he was research assistant and from 1985 on he was assistant in Göttingen. In 1989 Lück received his Habilitation. From 1990–91, he was associate professor at the University of Kentucky in Lexington. From 1991 until 1996, he was professor at the University of Mainz, and from 1996 until 2010 he taught at the University of Münster. Since 2010 he has been a professor at the University of Bonn. In 2003, he was awarded the Max Planck Research Award and in 2008 the Gottfried Wilhelm Leibniz Prize.
Lück has made significant contributions in topology; he and his coauthors resolved many cases of the Farrell-Jones conjecture and the Borel conjecture. He has also contributed to the development of the theory of L2-invariants (such as L2-Betti numbers and L2-cohomology) of manifolds, which were originally introduced by Michael Atiyah and are defined by means of operator algebras. These invariants have applications in group theory and geometry.
In 2009 and 2010 Lück was president of the German Mathematical Society, whose vice president he had been since 2006. From 2011 until 2017, he was Director of the Hausdorff Research Institute for Mathematics (HIM) in Bonn. In 2012, he became a fellow of the American Mathematical Society.
Selected publications
L2 Invarianten von Mannigfaltigkeiten und Gruppen, Jahresbericht DMV, Bd.99, 1997, Heft 3
Editor together with F. Thomas Farrell and Lothar Göttsche: Topology of high-dimensional manifolds, ICTP Lecture Notes, 2002
References
External links
1957 births
Living people
20th-century German mathematicians
Gottfried Wilhelm Leibniz Prize winners
Academic staff of the University of Münster
Fellows of the American Mathematical Society
People from Herford
University of Kentucky faculty
Academic staff of Johannes Gutenberg University Mainz
Academic staff of the University of Bonn
University of Göttingen alumni
Academic staff of the University of Göttingen
Topologists
21st-century German mathematicians
Presidents of the German Mathematical Society |
https://en.wikipedia.org/wiki/Afshin%20Kamaei | Afshin Kamaie (born September 16, 1974) is a retired Iranian footballer.
Club career
Club career statistics
Assist Goals
References
1974 births
Living people
Iranian men's footballers
Esteghlal Ahvaz F.C. players
Persian Gulf Pro League players
Men's association football midfielders
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Unit%20hyperbola | In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola to complement it in the plane. This pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the unit hyperbola is in use, the alternative radial length is
The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals
The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical complex numbers, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola.
Asymptotes
Generally asymptotic lines to a curve are said to converge toward the curve. In algebraic geometry and the theory of algebraic curves there is a different approach to asymptotes. The curve is first interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the projective curve at a point at infinity, thus circumventing any need for a distance concept and convergence. In a common framework (x, y, z) are homogeneous coordinates with the line at infinity determined by the equation z = 0. For instance, C. G. Gibson wrote:
For the standard rectangular hyperbola in ℝ2, the corresponding projective curve is which meets z = 0 at the points P = (1 : 1 : 0) and Q = (1 : −1 : 0). Both P and Q are simple on F, with tangents x + y = 0, x − y = 0; thus we recover the familiar 'asymptotes' of elementary geometry.
Minkowski diagram
The Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are
units of 30 centimetres length and nanoseconds, or
astronomical units and intervals of 8 minutes and 20 seconds, or
light years and years.
Each of these scales of coordinates results in photon connections of events along diagonal lines of slope plus or minus one.
Five elements constitute the diagram Hermann Minkowski used to describe the relativity transformations: the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the conjugate diameter.
The plane with the axes refers to a resting frame of reference. The diameter of th |
https://en.wikipedia.org/wiki/Pisg | Pisg or PISG may refer to:
pisg (Software) - Perl IRC Statistics Generator
Provisional Institutions of Self-Government established in Kosovo by the United Nations in 2003. |
https://en.wikipedia.org/wiki/Neighborly%20polytope | In geometry and polyhedral combinatorics, a -neighborly polytope is a convex polytope in which every set of or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an edge, forming a complete graph. 2-neighborly polytopes with more than four vertices may exist only in spaces of four or more dimensions, and in general a -neighborly polytope (other than a simplex) requires a dimension of or more. A -simplex is -neighborly. A polytope is said to be neighborly, without specifying , if it is -neighborly for . If we exclude simplices, this is the maximum possible : in fact, every polytope that is -neighborly for some is a simplex.
In a -neighborly polytope with , every 2-face must be a triangle, and in a -neighborly polytope with , every 3-face must be a tetrahedron. More generally, in any -neighborly polytope, all faces of dimension less than are simplices.
The cyclic polytopes formed as the convex hulls of finite sets of points on the moment curve in -dimensional space are automatically neighborly. Theodore Motzkin conjectured that all neighborly polytopes are combinatorially equivalent to cyclic polytopes. However, contrary to this conjecture, there are many neighborly polytopes that are not cyclic: the number of combinatorially distinct neighborly polytopes grows superexponentially, both in the number of vertices of the polytope and in the dimension.
The convex hull of a set of random points, drawn from a Gaussian distribution with the number of points proportional to the dimension, is with high probability -neighborly for a value that is also proportional to the dimension.
The number of faces of all dimensions of a neighborly polytope in an even number of dimensions is determined solely from its dimension and its number of vertices by the Dehn–Sommerville equations: the number of -dimensional faces, , satisfies the inequality
where the asterisk means that the sums ends at and final term of the sum should be halved if is even. According to the upper bound theorem of , neighborly polytopes achieve the maximum possible number of faces of any -vertex -dimensional convex polytope.
A generalized version of the happy ending problem applies to higher-dimensional point sets, and implies that for every dimension and every there exists a number with the property that every points in general position in -dimensional space contain a subset of points that form the vertices of a neighborly polytope.
References
Polyhedral combinatorics |
https://en.wikipedia.org/wiki/Mehdi%20Kheiri | Mehdi Mir Kheiri (born February 11, 1983 is an Iranian footballer who currently plays for Padideh in the Iranian Premier League.
Club career
Kheiri joined Saipa F.C. in 2009.
Club career statistics
Assist Goals
References
1983 births
Living people
Iranian men's footballers
Saipa F.C. players
F.C. Aboomoslem players
PAS Hamedan F.C. players
Saba Qom F.C. players
Naft Tehran F.C. players
Shahr Khodro F.C. players
Esteghlal Khuzestan F.C. players
Sportspeople from Babol
Men's association football midfielders
Footballers from Mazandaran province |
https://en.wikipedia.org/wiki/Arthur%20Dacres | Arthur Dacres (1624–1678) was an English physician and academic, briefly Gresham Professor of Geometry.
Life
He was sixth son of Sir Thomas Dacres, knight, of Cheshunt, and was born in that parish, where he was baptised on 18 April 1624. He entered Magdalene College, Cambridge, in December 1642, and graduated B.A. in 1645. He was elected a fellow of his college on 22 July 1646, and took the degree of M.D. on 28 July 1654.
He settled in London and was elected a fellow of the College of Physicians on 26 June 1665, and assistant-physician to Sir John Micklethwaite at St. Bartholomew's Hospital on the resignation of Dr. Christopher Terne, 13 May 1653. On 20 May 1664 he was appointed professor of geometry at Gresham College, but only held office for ten months. He was censor at the College of Physicians in 1672, and died in September 1678, being still assistant-physician at St. Bartholomew's.
Notes
References
1624 births
1678 deaths
17th-century English medical doctors
Alumni of Magdalene College, Cambridge
People from Cheshunt
Fellows of the Royal College of Physicians |
https://en.wikipedia.org/wiki/Secondary%20vector%20bundle%20structure | In mathematics, particularly differential topology, the secondary vector bundle structure
refers to the natural vector bundle structure on the total space TE of the tangent bundle of a smooth vector bundle , induced by the push-forward of the original projection map .
This gives rise to a double vector bundle structure .
In the special case , where is the double tangent bundle, the secondary vector bundle is isomorphic to the tangent bundle
of through the canonical flip.
Construction of the secondary vector bundle structure
Let be a smooth vector bundle of rank . Then the preimage of any tangent vector in in the push-forward of the canonical projection is a smooth submanifold of dimension , and it becomes a vector space with the push-forwards
of the original addition and scalar multiplication
as its vector space operations. The triple becomes a smooth vector bundle with these vector space operations on its fibres.
Proof
Let be a local coordinate system on the base manifold with and let
be a coordinate system on adapted to it. Then
so the fiber of the secondary vector bundle structure at in is of the form
Now it turns out that
gives a local trivialization for , and the push-forwards of the original vector space operations read in the adapted coordinates as
and
so each fibre is a vector space and the triple is a smooth vector bundle.
Linearity of connections on vector bundles
The general Ehresmann connection on a vector bundle can be characterized in terms of the connector map
where is the vertical lift, and is the vertical projection. The mapping
induced by an Ehresmann connection is a covariant derivative on in the sense that
if and only if the connector map is linear with respect to the secondary vector bundle structure on . Then the connection is called linear. Note that the connector map is automatically linear with respect to the tangent bundle structure .
See also
Connection (vector bundle)
Double tangent bundle
Ehresmann connection
Vector bundle
References
P.Michor. Topics in Differential Geometry, American Mathematical Society (2008).
Differential geometry
Topology
Differential topology |
https://en.wikipedia.org/wiki/Fred%20S.%20Roberts | Fred Stephen Roberts (born June 19, 1943) is an American mathematician, a professor of mathematics at Rutgers University, and a former director of DIMACS.
Biography
Roberts did his undergraduate studies at Dartmouth College, and received his Ph.D. from Stanford University in 1968; his doctoral advisor was Dana Scott. After holding positions at the University of Pennsylvania, RAND, and the Institute for Advanced Study, he joined the Rutgers faculty in 1972.
He has been vice president of the Society for Industrial and Applied Mathematics twice, in 1984 and 1986, and has been director of DIMACS since 1996.
Research
Roberts' research concerns graph theory and combinatorics, and their applications in modeling problems in the social sciences and biology. Among his contributions to pure mathematics, he is known for introducing the concept of boxicity, the minimum dimension needed to represent a given undirected graph as an intersection graph of axis-parallel boxes.
Books
Roberts is the author or co-author of the following books:
Discrete Mathematical Models, with Applications to Social, Biological and Environmental Problems, Prentice-Hall, 1976, . Russian translation, Nauka, 1986.
Graph Theory and its Applications to the Problems of Society, CBMS-NSF Regional Conference Series in Applied Mathematics 29, SIAM, 1987, .
Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences, Encyclopedia of Mathematics and its Applications 7, Addison-Wesley, 1979, . Reprinted by Cambridge University Press, 2009.
Applied Combinatorics, Prentice-Hall, 1984. 2nd edition (with B. Tesman), 2004, . 3rd edition, Chapman & Hall, 2009. Chinese translation, Pearson Education Asia, 2005 and 2007.
He is also the editor of nearly 20 edited volumes.
Awards and honors
Roberts received the ACM SIGACT Distinguished Service Prize in 1999. In 2001, he won the National Science Foundation Science and Technology Centers Pioneer Award for "pioneering the science and technology center concept". In 2003, DIMACS held a Conference on Applications of Discrete Mathematics and Theoretical Computer Science, in honor of Roberts' 60th birthday. In 2012 he became a fellow of the American Mathematical Society.
References
External links
Roberts' web site at DIMACS.
1943 births
Living people
Dartmouth College alumni
Stanford University alumni
Rutgers University faculty
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/MathOverflow | MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network, but distinct from math.stackexchange.com.
It is primarily for asking questions on mathematics research – i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known – and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative.
Origin and history
The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David Zureick-Brown, and Scott Morrison on 28 September 2009. The hosting was supported by Ravi Vakil. The site originally ran on a separate installation of the StackExchange 1.0 software engine; on June 25, 2013, it was integrated in the regular Stack Exchange Network, running SE 2.0.
Naming
According to MathOverflow FAQ, the proper spelling is "MathOverflow" rather than "Math Overflow".
Use of mathematical formulas
The original version of the website did not support LaTeX markup for mathematical formulas. To support most of the functionality of LaTeX, MathJax was added in order for the site to transform math equations into their appropriate forms. In its current state, any post including "Math Mode" (text between $'s) will translate into proper mathematical notation.
Usage
As of April 4, 2012, there were 16,496 registered users on MathOverflow, most of whom were located in the United States (35%), India (12%), and the United Kingdom (6%). By December 11, 2018, the number of registered users had grown to 87,850. As of June 2019, 123,448 questions have been posted.
In 2011, questions were answered an average of 3.9 hours after they were posted, and "Acceptable" answers took an average of 5.01 hours.
Reception
Terence Tao compared it to "the venerable newsgroup sci.math, but with more modern, 'Web 2.0' features."
John C. Baez writes that "website 'Math Overflow' has become a universal clearinghouse for math questions".
According to Gil Kalai, MathOverflow "is ran by an energetic and impressive group of very (very very) young people".
Jordan Ellenberg comments that the website "offers a constantly changing array of new questions" and is "addictive" in a "particularly pure form", as he compares it to the Polymath Project.
Jared Keller in The Atlantic writes, "Math Overflow is almost an anti-social network, focused solely on productively addressing the problems posed by its users." He quotes Scott Morrison saying "Mathematicians as a whole are surprisingly skeptical of many as |
https://en.wikipedia.org/wiki/Annette%20Dobson | Annette Jane Dobson (born 4 September 1945) is a Professor of Biostatistics in the University of Queensland's Australian Women and Girl's Health Research (AWaGHR) Centre in the School of Public Health. Dobson was Director of the Australian Longitudinal Study on Women's Health from 1995 to 2013. She is a highly cited publication author, a book author, and has received an Australia Day award.
Qualification
Dobson earned a Bachelor of Science degree from the University of Adelaide in 1966. Moving on to James Cook University, she completed a Master of Science in 1970 and a PhD (Doctor of Philosophy) in 1974. She was recognised as an Accredited Statistician in 1998 by the Statistical Society of Australia, and received a Graduate Certificate of Management in 2001 from the University of New England (Australia).
Research interests
Her research interests lie in the fields of biostatistics, epidemiology, longitudinal studies, and social determinants of health. In biostatistics, she is specifically interested in generalized linear modeling, clinical biostatistics, and statistical methods in longitudinal studies. Dobson's topics in epidemiology include tobacco control, diabetes, cardiovascular disease, obesity and health care service use.
Positions
Dobson is the founding Director of the Australian Longitudinal Study on Women's Health (ALSWH) and was director of the Centre for Longitudinal and Life Course Research from 2012 - 2021.
She was the inaugural chair of the BCA Master of Biostatistics at its inception in 2000.
Awards
Dobson was made a Member of the Order of Australia in 2010 for her service to public health and biostatistics as a research and academic, particularly through the collection and analysis of data relating to cardiovascular disease and women's and veterans' health, which provided a basis for public health interventions and policies to reduce disease burden in the population.
Dobson won the Sidney Sax medal in 2003, the pre-eminent prize awarded by the Public Health Association of Australia. Dobson received the 2012 Moyal Medal for her contributions to statistics and in 2015 she was elected Fellow of the Australian Academy of Health and Medical Sciences (FAHMS).
She is also an elected member of the International Statistical Institute.
An introduction to GLM
She wrote the book An introduction to generalized linear models.
Most highly cited publications
Kuulasmaa K, Tunstall-Pedoe H, Dobson A, Fortmann S, Sans S, Tolonen H, Evans A, Ferrario M, Tuomilehto J. Estimation of contribution of changes in classic risk factors to trends in coronary-event rates across the WHO MONICA Project populations.
Brown WJ, Bryson L, Byles JE, Dobson AJ, Lee C, Mishra G, Schofield M. Women's Health Australia: Recruitment for a national longitudinal cohort study.
References
External links
Annette Dobson page at Queensland University
1945 births
Living people
Members of the Order of Australia
Academic staff of the University of Queensland
Wo |
https://en.wikipedia.org/wiki/Statistical%20risk | Statistical risk is a quantification of a situation's risk using statistical methods. These methods can be used to estimate a probability distribution for the outcome of a specific variable, or at least one or more key parameters of that distribution, and from that estimated distribution a risk function can be used to obtain a single non-negative number representing a particular conception of the risk of the situation.
Statistical risk is taken account of in a variety of contexts including finance and economics, and there are many risk functions that can be used depending on the context.
One measure of the statistical risk of a continuous variable, such as the return on an investment, is simply the estimated variance of the variable, or equivalently the square root of the variance, called the standard deviation. Another measure in finance, one which views upside risk as unimportant compared to downside risk, is the downside beta. In the context of a binary variable, a simple statistical measure of risk is simply the probability that a variable will take on the lower of two values.
There is a sense in which one risk A can be said to be unambiguously greater than another risk B (that is, greater for any reasonable risk function): namely, if A is a mean-preserving spread of B. This means that the probability density function of A can be formed, roughly speaking, by "spreading out" that of B. However, this is only a partial ordering: most pairs of risks cannot be unambiguously ranked in this way, and different risk functions applied to the estimated distributions of two such unordered risky variables will give different answers as to which is riskier.
In the context of statistical estimation itself, the risk involved in estimating a particular parameter is a measure of the degree to which the estimate is likely to be inaccurate.
See also
Risk analysis
Applied probability |
https://en.wikipedia.org/wiki/Wilhelmus%20Luxemburg | Wilhelmus Anthonius Josephus Luxemburg (11 April 1929 – 2 October 2018) was a Dutch American mathematician who was a professor of mathematics at the California Institute of Technology.
He received his B.A. from the University of Leiden in 1950; his M.A., in 1953; his Ph.D., from the Delft Institute of Technology, in 1955. He was assistant professor at Caltech during 1958–60; Associate Professor, during 1960–62; Professor, during 1962–2000; Professor Emeritus, from 2000. He was the Executive Officer for Mathematics during 1970–85. In 2012 he became a fellow of the American Mathematical Society. Luxemburg became a corresponding member of the Royal Netherlands Academy of Arts and Sciences in 1974.
Luxemburg contributed to the development of non-standard analysis by popularizing the construction of hyperreal numbers in the 1960s. Though Edwin Hewitt had shown the construction in 1948, the formalization of non-standard analysis is generally associated with Abraham Robinson.
Other notable work he did was in the theory of Riesz spaces (partially ordered vector spaces where the order structure is a lattice).
Selected publications
1955: Banach function spaces. Thesis, Technische Hogeschool te Delft, 1955.
1969: "A general theory of monads", in Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967) pp. 18–86 Holt, Rinehart and Winston
1971: (with Zaanen, A. C.) Riesz Spaces. Vol. I. North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York.
1976: (with Stroyan, K. D.) Introduction to the Theory of Infinitesimals. Pure and Applied Mathematics, No. 72. Academic Press
1978: (with Schep, A. R.) "A Radon-Nikodym type theorem for positive operators and a dual", Nederl. Akad. Wetensch. Indag. Math. 40, no. 3, 357–375.
1979: Some Aspects of the Theory of Riesz Spaces, University of Arkansas Lecture Notes in Mathematics, 4. University of Arkansas, Fayetteville, Ark.
References
External links
See also
Influence of non-standard analysis
20th-century Dutch mathematicians
21st-century Dutch mathematicians
American people of Dutch descent
20th-century American mathematicians
21st-century American mathematicians
California Institute of Technology faculty
Delft University of Technology alumni
Fellows of the American Mathematical Society
Members of the Royal Netherlands Academy of Arts and Sciences
People from Delft
1929 births
2018 deaths
Functional analysts |
https://en.wikipedia.org/wiki/Carol%20Walker | Carol Lee Walker (born 1935) is a retired American mathematician and mathematics textbook author. Walker's early mathematical research, in the 1960s and 1970s, concerned the theory of abelian groups. In the 1990s, her interests shifted to fuzzy logic and fuzzy control systems.
Education and career
Walker was born in Martinez, California on August 19, 1935, and went to high school in Montrose, Colorado. She studied music education at the University of Colorado Boulder, with a year off to work as a primary-school music teacher in Colorado, and graduated in 1957. Next, she went to the University of Denver for graduate study in mathematics, but after one year transferred to New Mexico State University, where she earned a master's degree in 1961 and completed her PhD in 1963. Her dissertation, On -pure sequences of abelian groups, was supervised by David Kent Harrison.
After postdoctoral research at the Institute for Advanced Study, she returned to Mexico State University as an assistant professor in 1964, and quickly earned tenure as an associate professor in 1966. She was promoted to full professor in 1972. She chaired the Department of Mathematical Sciences from 1979 to 1993, and served as associate dean of arts and sciences from 1993 until her retirement in 1996.
Books
Walker is the coauthor of books including:
Mathematics for the Liberal Arts Student (with Fred Richman and Robert J. Wisner, Brooks-Cole, 1967; 2nd ed., 1973; 3rd ed., with James Brewer, Prentice-Hall, 2000; 4th ed., 2003)
Doing Mathematics with Scientific WorkPlace (with Darel Hardy, Brooks-Cole, 1995; multiple editions)
A First Course in Fuzzy and Neural Control (with Hung T. Nguyen, Radipuram Prasad, and Elbert Walker, CRC Press, 2003)
Applied Algebra: Codes, Ciphers, and Discrete Algorithms (with Darel Hardy, Prentice-Hall, 2003)
Calculus: Understanding Its Concepts and Methods (with Darel Hardy, Fred Richman, and Robert J. Wisner, MacKichan Software, 2006)
Recognition
The New Mexico State University alumni gave Walker their Distinguished Alumni Award in 2001.
Personal life
Walker was married to Elbert Walker (1930–2018), another mathematician who joined the New Mexico State University faculty in 1957.
References
1935 births
Living people
People from Martinez, California
American mathematicians
American women mathematicians
Group theorists
University of Colorado Boulder alumni
New Mexico State University alumni
New Mexico State University faculty |
https://en.wikipedia.org/wiki/Whitechurch%2C%20Ontario | {
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"features": [
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"geometry": {
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-81.40547275543213,
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}Whitechurch is a small residential village located in south-western Ontario.
Location
Whitechurch is located along the border of Huron and Bruce County. It is approximately 10 km west of Wingham, and 8 km east of Lucknow. The coordinates to Whitechurch are 43°55'01.9"N 81°24'20.6"W.
History
Originally named Ulster, the name was changed when surveying for the former Ontario Highway 86, now Huron/Bruce County Road 86.
Churches
The Whitechurch United Church closed on June 24, 2007.
The Whitechurch Presbyterian Church closed in the summer of 2016.
The Whitechurch Amish Mennonite Church was established in 1999 as a daughter congregation of the Cedar Grove Amish Mennonite Church. In 2018 the church had 35 members and was a member of the Maranatha Amish Mennonite Churches. The ministerial team included Bishop Larry Ropp, Minister Charles Jantzi, and Deacon Jeffrey Kuepfer.
Points of Interest
The following points of interest are located within village limits.
Whitechurch Community Hall
Whitechurch Community Park
Whitechurch Community Baseball Park
References
Communities in Huron County, Ontario |
https://en.wikipedia.org/wiki/2010%20CD%20Universidad%20San%20Mart%C3%ADn%20season | The 2010 season is the 7th season of competitive football by University of San Martín de Porres.
Statistics
Appearances and goals
Competition Overload
Copa Sudamericana 2010
Second stage
First stage
Primera División Peruana 2010
Final Nacional
Liguilla Final – Group A
Regular season
Club Deportivo Universidad de San Martín de Porres seasons
2010 in Peruvian football |
https://en.wikipedia.org/wiki/Omnitruncated%20polyhedron | In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra.
All omnitruncated polyhedra are zonohedra. They have Wythoff symbol p q r | and vertex figures as 2p.2q.2r.
More generally an omnitruncated polyhedron is a bevel operator in Conway polyhedron notation.
List of convex omnitruncated polyhedra
There are three convex forms. They can be seen as red faces of one regular polyhedron, yellow or green faces of the dual polyhedron, and blue faces at the truncated vertices of the quasiregular polyhedron.
List of nonconvex omnitruncated polyhedra
There are 5 nonconvex uniform omnitruncated polyhedra.
Other even-sided nonconvex polyhedra
There are 8 nonconvex forms with mixed Wythoff symbols p q (r s) |, and bow-tie shaped vertex figures, 2p.2q.-2q.-2p. They are not true omnitruncated polyhedra: the true omnitruncates p q r | or p q s | have coinciding 2r-gonal or 2s-gonal faces respectively that must be removed to form a proper polyhedron. All these polyhedra are one-sided, i.e. non-orientable. The p q r | degenerate Wythoff symbols are listed first, followed by the actual mixed Wythoff symbols.
General omnitruncations (bevel)
Omnitruncations are also called cantitruncations or truncated rectifications (tr), and Conway's bevel (b) operator. When applied to nonregular polyhedra, new polyhedra can be generated, for example these 2-uniform polyhedra:
See also
Uniform polyhedron
References
Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.
Polyhedra |
https://en.wikipedia.org/wiki/Fourier%E2%80%93Deligne%20transform | In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category of ℓ-adic sheaves over the affine line. It was introduced by Pierre Deligne on November 29, 1976 in a letter to David Kazhdan as an analogue of the usual Fourier transform. It was used by Gérard Laumon to simplify Deligne's proof of the Weil conjectures.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Plasma%20transferred%20wire%20arc%20thermal%20spraying | Plasma transferred wire arc (PTWA) thermal spraying is a thermal spraying process that deposits a coating on the internal surface of a cylindrical surface, or external surface of any geometry. It is predominantly known for its use in coating the cylinder bores of an internal combustion engine, enabling the construction of aluminium engine blocks without cast iron cylinder sleeves.
The inventors of PTWA received the 2009 IPO National Inventor of the Year award. This technology was initially patented and developed by Flame-Spray Industries, and subsequently improved upon by Flame-Spray and Ford.
Process
A single conductive wire is used as feedstock for the system. A supersonic plasma jet—formed by a transferred arc between a non-consumable cathode and the wire—melts and atomizes the wire. A stream of air transports the atomized metal onto the substrate. The particles flatten upon striking the surface of the substrate due to their high kinetic energy. The particles rapidly solidify upon contact and can assume both crystalline and amorphous phases. There is also the possibility of producing multi-layer coatings via stacked layers of particles, increasing wear resistance. All conductive wires up to and including can be used as feedstock material, including "cored" wires. Refractory metals, as well as low melt materials, are easily deposited.
Applications
PTWA can be used to apply a coating to wear surfaces of engine or transmission components, serving as a plain bearing. For the cylinder bores of hypoeutectic aluminum-silicon alloy blocks, PTWA's main advantages over cast iron liners are reduced weight and cost. The thinner bore surface also allows for more compact bore spacing, and can potentially provide better heat transfer.
Automotive engines that use PTWA include the BMW B58, Nissan VR38DETT, and Ford Coyote. Caterpillar and Ford also use PTWA to remanufacture engines.
References
External links
PTWA internal coating system
http://www.sae.org/mags/aei/manuf/7624
Metallurgical processes
Coatings
spraying |
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