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https://en.wikipedia.org/wiki/Antonio%20Maria%20Bordoni | Antonio Maria Bordoni (19 July 1789 – 26 March 1860) was an Italian mathematician who did research on mathematical analysis, geometry, and mechanics. Joining the faculty of the University of Pavia in 1817, Bordoni is generally considered to be the founder of the mathematical school of Pavia. He was a member of various learned academies, notably the Accademia dei XL. Bordoni's famous students were Francesco Brioschi, Luigi Cremona, Eugenio Beltrami, Felice Casorati and Delfino Codazzi.
Biography
Antonio Bordoni was born in Mezzana Corti (province of Pavia) on 19 July 1788, and graduated in Mathematics from Pavia on 7 June 1807.
After just two months he was appointed teacher of mathematics at the military School of Pavia, established by Napoleon, and held such office until 1816 when the school was closed due to the political situation of the times.
On 1 November 1817 he became full professor of Elementary Pure mathematics at the University and in 1818 he held the chair of Infinitesimal Calculus, Geodesy and Hydrometry, a discipline he taught for 23 years.
In 1827 and 1828 he was dean of the University itself. In 1854, as the Faculty of Mathematics of the University of Pavia (it previously belonged to the one of the Philosophy) was established, he was elected Director of Mathematical Studies and held such office until his death, which occurred 26 March 1860, just a month after being appointed senator.
Works
Sopra l'equilibrio di un poligono qualunque. Memoria del signor Antonio Bordoni professore nella scuola militare di Pavia, Milano, Regia Cesarea Stamperia di Governo, 1814.
Nuovi teoremi di meccanica elementare memoria del sig. A. Bordoni, inserita nell'ottavo tomo del Giornale di Fisica Chimica ec. del Sig. Brugnatelli, Pavia: dalla tipografia eredi Galeazzi, 1815.
De' contorni delle ombre ordinarie trattato di A. Bordoni già prof. nella Scuola Militare di Pavia ed uno dei quaranta nella Società Italiana delle Scienze, Milano, Imperiale Regia Stamperia, 1816.
Trattato di geodesia elementare di Antonio Bordoni, con 17 tavole, Milano, per P.E. Giusti fonditore-tipografo, 1825.
Proposizioni teoriche e pratiche trattate in iscuola dal professore Antonio Bordoni e raccolte dal dottor Carlo Pasi, Pavia, dalla tipografia Bizzoni, 1829.
Lezioni di calcolo sublime, Milano, per P. E. Giusti, 1831.
Sulle svolte ordinarie delle strade in Opuscoli matematici e fisici di diversi autori, Milano, presso Emilio Giusti, 1834.
Trattato di geodesia elementare, Pavia, dalla Tip. di P. Bizzoni, 1843
References
External links
History of Faculty of Engineering – Università di Pavia
1789 births
1860 deaths
19th-century Italian mathematicians
Mathematical analysts
Differential geometers
Academic staff of the University of Pavia |
https://en.wikipedia.org/wiki/Usman%20Pribadi | Usman Pribadi (born September 19, 1983) is an Indonesian former footballer that last play for Madura in the 2018 Liga 2.
Club statistics
References
External links
1983 births
Living people
Indonesian men's footballers
PSMS Medan players
PSDS Deli Serdang players
Gresik United F.C. players
Deltras F.C. players
Putra Samarinda F.C. players
Persik Kediri players
Persita Tangerang players
Persijap Jepara players
PSS Sleman players
Persiwa Wamena players
Persika 1951 players
Madura F.C. players
Indonesian Premier Division players
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Isdiantono | Isdiantono (born November 8, 1978 in Banyuwangi) is an Indonesian former footballer that currently head coach of Persewangi Banyuwangi in the Liga 3.
Club statistics
References
External links
1978 births
Men's association football defenders
Living people
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Deltras F.C. players
Arema F.C. players
Persijap Jepara players
Putra Samarinda F.C. players
People from Banyuwangi Regency
Indonesian football managers
Footballers from East Java |
https://en.wikipedia.org/wiki/Arifki%20Eka%20Putra | Arifki Eka Putra (born February 19, 1987) is an Indonesian former footballer.
Club statistics
References
External links
1987 births
Men's association football defenders
Men's association football midfielders
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Bontang F.C. players
Putra Samarinda F.C. players
Persela Lamongan players
Persiba Balikpapan players
PS Mitra Kukar players
Celebest F.C. players
Badak Lampung F.C. players
Living people
People from Bengkulu |
https://en.wikipedia.org/wiki/Johan%20Yoga%20Utama | Johan Yoga Utama (born February 19, 1990 in Semarang) is an Indonesian professional footballer who plays as a striker.
Club statistics
Club
Honours
Individual
Indonesia Soccer Championship B Top Goalscorer: 2016 (14 goals)
References
External links
Johan Yoga Utama at Liga Indonesia
1990 births
Living people
Footballers from Semarang
Men's association football forwards
Javanese people
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
PSIS Semarang players
Persiba Balikpapan players
Persib Bandung players
Putra Samarinda F.C. players
Dewa United F.C. players
PSM Makassar players
Persis Solo players
Persiraja Banda Aceh players
Badak Lampung F.C. players
Semen Padang F.C. players
Persebaya Surabaya players
PSIM Yogyakarta players
Indonesia men's youth international footballers |
https://en.wikipedia.org/wiki/Fajar%20Legian%20Siswanto | Fajar Legian Siswanto (born August 27, 1987) is an Indonesian former footballer. He was born in Indonesia to a German father and Balinese mother.
Club statistics
References
External links
1987 births
Living people
Indo people
Indonesian people of German descent
Men's association football midfielders
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Persih Tembilahan players
Putra Samarinda F.C. players
Semen Padang F.C. players
Persepam Madura Utama players
Indonesian Premier Division players
Footballers from Jakarta |
https://en.wikipedia.org/wiki/Agung%20Prasetyo%20%28footballer%2C%20born%201978%29 | Agung Prasetyo (born January 16, 1978 in Surabaya) is an Indonesian former footballer.
Club statistics
References
External links
1978 births
Men's association football goalkeepers
Living people
Javanese people
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Arema F.C. players
Deltras F.C. players
PKT Bontang players
Persis Solo players
PS Mitra Kukar players
Putra Samarinda F.C. players
Gresik United F.C. players
Footballers from Surabaya |
https://en.wikipedia.org/wiki/Mick%20Heath | Michael Heath (born 9 January 1953) is an English former semi-professional footballer who played in the Football League as a forward for Brentford.
Career statistics
References
1953 births
Living people
Footballers from Hillingdon
English men's footballers
Men's association football forwards
Walton & Hersham F.C. players
Brentford F.C. players
Southall F.C. players
Wimbledon F.C. players
English Football League players |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20Saudi%20First%20Division | Statistics of the 1989–90 Saudi First Division.
External links
Saudi Arabia Football Federation
Saudi League Statistics
Al Jazirah 1 Feb 1987 issue 5239
Saudi First Division League seasons
Saudi Professional League
2 |
https://en.wikipedia.org/wiki/Kolmogorov%20automorphism | In mathematics, a Kolmogorov automorphism, K-automorphism, K-shift or K-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law. All Bernoulli automorphisms are K-automorphisms (one says they have the K''-property), but not vice versa. Many ergodic dynamical systems have been shown to have the K-property, although more recent research has shown that many of these are in fact Bernoulli automorphisms.
Although the definition of the K-property seems reasonably general, it stands in sharp distinction to the Bernoulli automorphism. In particular, the Ornstein isomorphism theorem does not apply to K-systems, and so the entropy is not sufficient to classify such systems – there exist uncountably many non-isomorphic K-systems with the same entropy. In essence, the collection of K-systems is large, messy and uncategorized; whereas the B-automorphisms are 'completely' described by Ornstein theory.
Formal definition
Let be a standard probability space, and let be an invertible, measure-preserving transformation. Then is called a K-automorphism, K-transform or K-shift, if there exists a sub-sigma algebra such that the following three properties hold:
Here, the symbol is the join of sigma algebras, while is set intersection. The equality should be understood as holding almost everywhere, that is, differing at most on a set of measure zero.
Properties
Assuming that the sigma algebra is not trivial, that is, if , then It follows that K-automorphisms are strong mixing.
All Bernoulli automorphisms are K-automorphisms, but not vice versa.
Kolmogorov automorphisms are precisely the natural extensions of exact endomorphisms, i.e. mappings for which consists of measure-zero sets or their complements, where is the sigma-algebra of measureable sets,.
References
Further reading
Christopher Hoffman, "A K counterexample machine", Trans. Amer. Math. Soc.'' 351 (1999), pp 4263–4280.
Ergodic theory |
https://en.wikipedia.org/wiki/K-transform | In mathematics, the K transform (also called the Kemp-Macdonald Transform or Single-Pixel X-ray Transform) is an integral transform introduced by R. Scott Kemp and Ruaridh Macdonald in 2016. The transform allows a 3-dimensional inhomogeneous object to be reconstructed from scalar point measurements taken in the volume external to the object.
Gunther Uhlmann proved that the K transform exhibits global uniqueness on , meaning that different objects will always have a different K transform. This uniqueness arises by the use of a monotonic, nonlinear transform of the X-ray transform. By selecting the exponential function for the transform function, which coincides with attenuation of particles in matter in accordance with the Beer–Lambert law, the K transform can be used to perform tomography of 3-dimensional objects using a low-resolution single-pixel detector. A numerical inversion using the BFGS optimization algorithm was explored by Fichtlscherer.
Definition
Let an object be a function of compact support that maps into the positive real numbers
The K-transform of the object is defined as
where is the set of all lines originating at a point and terminating on the single-pixel detector , and is the X-ray transform.
Applications
The K transform has been proposed as a means of performing a physical one-time pad encryption of a physical object.
References
Integral transforms |
https://en.wikipedia.org/wiki/Lattice%20%28module%29 | In mathematics, in the field of ring theory, a lattice is a module over a ring that is embedded in a vector space over a field, giving an algebraic generalisation of the way a lattice group is embedded in a real vector space.
Formal definition
Let R be an integral domain with field of fractions K. An R-submodule M of a K-vector space V is a lattice if M is finitely generated over R. It is full if .
Pure sublattices
An R-submodule N of M that is itself a lattice is an R-pure sublattice if M/N is R-torsion-free. There is a one-to-one correspondence between R-pure sublattices N of M and K-subspaces W of V, given by
See also
Lattice (group), for the case where M is a Z-module embedded in a vector space V over the field of real numbers R, and the Euclidean metric is used to describe the lattice structure
References
Module theory |
https://en.wikipedia.org/wiki/Ferenc%20Fodor | Ferenc Fodor (born 22 March 1991) is a Hungarian football player who plays for Tiszakécske.
Club career
On 16 June 2021 Fodor signed a two-year contract with Győri ETO.
Career statistics
References
External links
Profile at HLSZ
1991 births
Living people
Footballers from Pécs
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Men's association football defenders
Oldham Athletic A.F.C. players
Northwich Victoria F.C. players
Pécsi MFC players
Kozármisleny SE footballers
Nyíregyháza Spartacus FC players
Puskás Akadémia FC players
Kisvárda FC players
Győri ETO FC players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungarian expatriate men's footballers
Expatriate men's footballers in England
Hungarian expatriate sportspeople in England |
https://en.wikipedia.org/wiki/Ade%20Iwan%20Setiawan | Ade Iwan Setiawan (born March 19, 1984) is an Indonesian former footballer.
Club career statistics
References
External links
1984 births
Men's association football defenders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Indonesian Premier Division players
Persikabo Bogor players
Persih Tembilahan players
Persiwa Wamena players
Persebaya Surabaya players
Celebest F.C. players |
https://en.wikipedia.org/wiki/Hari%20Novian%20Caniago | Hari Novian Caniago (born on November 6, 1986) is an Indonesian former footballer.
Club career statistics
References
External links
1986 births
Men's association football defenders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Indonesian Premier Division players
PS Pasbar West Pasaman players
PSP Padang players
PSCS Cilacap players
Persik Kediri players
Persiwa Wamena players
Persih Tembilahan players
Persita Tangerang players
Persiba Bantul players |
https://en.wikipedia.org/wiki/Yohanes%20L.%20G.%20Kabagaimu | Yohanes Kabagaimu (born December 16, 1980) is an Indonesian former footballer.
Club career statistics
References
External links
1980 births
Men's association football midfielders
Living people
Indonesian men's footballers
Indonesian Christians
Sportspeople from Papua
Liga 1 (Indonesia) players
Persiwa Wamena players
Indonesian Premier Division players
Perseman Manokwari players |
https://en.wikipedia.org/wiki/Fred%20Ferdinando%20Mote | Fred Ferdinando Mote or Fred Mote or just called Nando (born November 15, 1989 in Wamena) is an Indonesian former footballer.
Club career statistics
References
External links
1989 births
Men's association football midfielders
Living people
Indonesian men's footballers
Footballers from Papua
Liga 1 (Indonesia) players
Persiwa Wamena players
Perseru Serui players
People from Wamena |
https://en.wikipedia.org/wiki/Taufik%20Soleh | Taufik Soleh (born November 25, 1985) is an Indonesian former footballer.
Club career statistics
Honours
Club honors
Mojokerto Putra
First Division (1): 2008–09
References
External links
1985 births
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Persiwa Wamena players
Indonesian Premier Division players
Persebaya Surabaya players
PS Mojokerto Putra players |
https://en.wikipedia.org/wiki/Harmoko%20%28footballer%29 | Harmoko (born March 6, 1989) is an Indonesian former footballer.
Club career statistics
References
External links
1989 births
Men's association football forwards
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Persema Malang players
Persiwa Wamena players
Indonesian Premier Division players
Persekam Metro players
Perseta Tulungagung players
Indonesia men's youth international footballers
Footballers from Malang
Footballers from East Java |
https://en.wikipedia.org/wiki/Marchelino%20Mandagi | Marchelino Mandagi (born February 16, 1990 in Tomohon) is an Indonesian former footballer.
Club career statistics
Honours
Club honors
Persisam Putra Samarinda
Premier Division (1): 2008–09
References
External links
1990 births
Living people
People from Tomohon
Sportspeople from North Sulawesi
Men's association football defenders
Men's association football midfielders
Indonesian Christians
Indonesian men's footballers
Liga 1 (Indonesia) players
Bontang F.C. players
Persiwa Wamena players
Perseru Serui players
Indonesian Premier Division players
Putra Samarinda F.C. players |
https://en.wikipedia.org/wiki/Arip%20Kurniawan | Arip Kurniawan (born March 5, 1987) is an Indonesian former footballer.
Club statistics
References
External links
1987 births
Men's association football forwards
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
PSCS Cilacap players
Kalteng Putra F.C. players
Persiba Balikpapan players
Persiraja Banda Aceh players
PSAP Sigli players
Indonesian Premier Division players
Persibo Bojonegoro players
Persipasi Bekasi players
Perserang Serang players
People from Serang
Footballers from Banten |
https://en.wikipedia.org/wiki/Ichwani%20Hasanuddin | Ichwani Hasanuddin (born June 6, 1986) is an Indonesian former footballer.
Club statistics
References
External links
1986 births
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSAP Sigli players
Indonesian Premier Division players |
https://en.wikipedia.org/wiki/Sayuti | Sayuti (born June 30, 1980) is an Indonesian professional football coach and former player who is currently assistant coach of PS Pidie Jaya in the Liga 3.
Club statistics
Honours
Club
PS Pidie Jaya
2016 ISC Liga Nusantara Aceh zone: 2016
References
External links
1980 births
Men's association football forwards
Living people
Acehnese people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSAP Sigli players
Indonesian Premier Division players
People from Pidie Regency
Footballers from Aceh |
https://en.wikipedia.org/wiki/Nanda%20Zulmi | Nanda Zulmi (2 June 1989 – 6 July 2017) was an Indonesian former footballer who played as a midfielder. He played for PSAP Sigli in the Indonesia Super League at the 2011–12 season.
Club statistics
References
External links
1989 births
2017 deaths
Men's association football midfielders
Acehnese people
People from Bireuën Regency
Indonesian men's footballers
Liga 1 (Indonesia) players
PSAP Sigli players
Footballers from Aceh |
https://en.wikipedia.org/wiki/Abdul%20Faisal | Abdul Faisal (born March 27, 1986) is an Indonesian former footballer.
Club statistics
References
External links
1986 births
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSAP Sigli players
Indonesian Premier Division players
PSP Padang players
PSSB Bireuen players |
https://en.wikipedia.org/wiki/Indra%20Gunawan%20%28footballer%29 | Indra Gunawan (born October 12, 1982) is an Indonesian former footballer.
Club statistics
References
External links
1982 births
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSAP Sigli players
Indonesian Premier Division players
Persibat Batang players
PSIM Yogyakarta players |
https://en.wikipedia.org/wiki/Hendra%20Syahputra | Hendra Syahputra (born September 1, 1983) is an Indonesian former footballer.
Club statistics
References
External links
1983 births
Men's association football defenders
Living people
Acehnese people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSLS Lhokseumawe players
PSAP Sigli players
Indonesian Premier Division players
Persiraja Banda Aceh players
PSSB Bireuen players
People from Bireuën Regency
Footballers from Aceh |
https://en.wikipedia.org/wiki/Miriam%20Leiva | Miriam Almaguer Leiva is a Cuban-American mathematician and mathematics educator, the first American Hispanic woman to earn a doctorate in mathematics and mathematics education. She is the Bonnie Cone Distinguished Professor for Teaching Emerita in the Department of Mathematics at the University of North Carolina at Charlotte, and the founder of TODOS: Mathematics for All, an organization devoted to advocacy for and encouragement of Latinx students in mathematics. She is also an author of many secondary-school mathematics textbooks.
Education and career
Leiva moved from Cuba to the US as a teenager in the 1950s.
She did her undergraduate studies at Guilford College, graduating in 1961,
and was initially denied admission for graduate study in mathematics at the University of North Carolina for being a woman. Nevertheless, she persisted, and earned a master's degree there in 1966 under the mentorship of Alfred Brauer, with a thesis on Elementary estimates for the least positive primitive root modulo pr.
After finishing her master's degree, she became a secondary school mathematics teacher. Later, she obtained a teaching position at the University of North Carolina, and while teaching there completed her doctorate in mathematics and mathematics education through a distance education program at Union Institute & University.
Recognition
In 2008, TODOS gave Leiva their Iris Carl Equity and Leadership Award.
In 2013 the National Council of Teachers of Mathematics (NCTM) gave her the inaugural Kay Gilliland Equity Lecture Award for "contributions to equity in mathematics education".
In 2014 the NCTM gave her the Mathematics Education Trust Lifetime Achievement Award for Distinguished Service to Mathematics Education.
References
External links
TODOS: Mathematics for All
Year of birth missing (living people)
Living people
Cuban emigrants to the United States
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Mathematics educators
Guilford College alumni
University of North Carolina at Charlotte faculty
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Omnitruncated%206-simplex%20honeycomb | In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-simplex facets.
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A lattice
The A lattice (also called A) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.
∪
∪
∪
∪
∪
∪
= dual of
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 6-space:
6-cubic honeycomb
6-demicubic honeycomb
6-simplex honeycomb
Truncated 6-simplex honeycomb
222 honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
7-polytopes |
https://en.wikipedia.org/wiki/Omnitruncated%208-simplex%20honeycomb | In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-simplex facets.
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A lattice
The A lattice (also called A) is the union of nine A8 lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex
∪
∪
∪
∪
∪
∪
∪
∪
= dual of .
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 8-space:
8-cubic honeycomb
8-demicubic honeycomb
8-simplex honeycomb
Truncated 8-simplex honeycomb
521 honeycomb
251 honeycomb
152 honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
9-polytopes |
https://en.wikipedia.org/wiki/Omnitruncated%207-simplex%20honeycomb | In seven-dimensional Euclidean geometry, the omnitruncated 7-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 7-simplex facets.
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A7* lattice
The A lattice (also called A) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.
∪
∪
∪
∪
∪
∪
∪
= dual of .
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 7-space:
7-cubic honeycomb
7-demicubic honeycomb
7-simplex honeycomb
Truncated 7-simplex honeycomb
331 honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
8-polytopes |
https://en.wikipedia.org/wiki/Sukman%20Suaib | Sukman Suaib (born June 27, 1983) is an Indonesian former footballer.
Club statistics
References
External links
1983 births
Men's association football midfielders
Living people
Acehnese people
People from Pidie Regency
Indonesian men's footballers
Liga 1 (Indonesia) players
PSAP Sigli players
Indonesian Premier Division players
Footballers from Aceh |
https://en.wikipedia.org/wiki/Muhammad%20Ali%20%28footballer%2C%20born%201985%29 | Muhammad Ali (born 10 April 1985) is an Indonesian former footballer who played as a defender or midfielder.
Club statistics
References
External links
1985 births
Men's association football defenders
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSAP Sigli players
Indonesian Premier Division players
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Boris%20%C5%BDivanovi%C4%87 | Boris Živanović (; born 18 July 1989) is a Serbian footballer who most recently played for Szeged 2011. He has Serbian and Hungarian passport.
Club statistics
Updated to games played as of 11 November 2014.
References
HLSZ
External links
1989 births
Living people
Footballers from Belgrade
Serbian men's footballers
Men's association football midfielders
FK Zemun players
FK Rad players
FK Mačva Šabac players
FK Radnički 1923 players
Budapest Honvéd FC players
Nyíregyháza Spartacus FC players
Nemzeti Bajnokság I players
Pittsburgh Riverhounds SC players
FC Koper players
FC Dunărea Călărași players
FK Borac Čačak players
Serbian expatriate men's footballers
Expatriate men's footballers in Hungary
Expatriate men's soccer players in the United States
Serbian expatriate sportspeople in Hungary
USL Championship players
Slovenian PrvaLiga players
Liga II players
Expatriate men's footballers in Slovenia
Serbian expatriate sportspeople in Slovenia
Expatriate men's footballers in Romania
Serbian expatriate sportspeople in Romania |
https://en.wikipedia.org/wiki/Pernicious%20number | In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1s when it is written as a binary number.
Examples
The first pernicious number is 3, since 3 = 112 and 1 + 1 = 2, which is a prime. The next pernicious number is 5, since 5 = 1012, followed by 6 (1102), 7 (1112) and 9 (10012). The sequence of pernicious numbers begins
Properties
No power of two is a pernicious number. This is trivially true, because powers of two in binary form are represented as a one followed by zeros. So each power of two has a Hamming weight of one, and one is not considered to be a prime. On the other hand, every number of the form with , including every Fermat number, is a pernicious number. This is because the sum of the digits in binary form is 2, which is a prime number.
A Mersenne number has a binary representation consisting of ones, and is pernicious when is prime. Every Mersenne prime is a Mersenne number for prime , and is therefore pernicious. By the Euclid–Euler theorem, the even perfect numbers take the form for a Mersenne prime ; the binary representation of such a number consists of a prime number of ones, followed by zeros. Therefore, every even perfect number is pernicious.
Related numbers
Odious numbers are numbers with an odd number of 1s in their binary expansion ().
Evil numbers are numbers with an even number of 1s in their binary expansion ().
References
Base-dependent integer sequences |
https://en.wikipedia.org/wiki/Zhang%20Huikang | Zhang Huikang (born 22 July 1962) is a Chinese football goalkeeper who played for China in the 1988 Asian Cup. He also played for Shanghai city team and South China in Hong Kong.
Career statistics
International statistics
External links
Team China Stats
1962 births
1988 AFC Asian Cup players
Footballers at the 1988 Summer Olympics
Olympic footballers for China
China men's international footballers
Chinese men's footballers
Living people
Footballers at the 1990 Asian Games
Men's association football goalkeepers
Asian Games competitors for China
Shanghai Shenhua F.C. players
south China AA players |
https://en.wikipedia.org/wiki/Hans%20Michael%20Maitzen | Hans Michael Maitzen (born 28 March 1943 in Graz) is an Austrian astronomer.
After elementary (1949-1953) and secondary school attendance (1953-1961) in Graz, he began studying astronomy, mathematics and physics at the University of Graz in 1961, completing a PhD in astronomy in July 1967. During the last three years of his studies he was also a scientific assistant at the Institute for Astronomy at the University of Graz.
Thereafter he was a scientific assistant at the Institute of Astronomy of the University of Bochum from 1969 until 1976. From then till 1985 he was scientific assistant in the Institute of Astronomy at the University of Vienna. In March 1980 Maitzen received his Doctor habilitatus there in astronomy. Since 1985 he has been a scientific advisor at the institute. Author of more than 200 scientific publications, he has since 1986 represented the Austrian astronomical community in its relations with the European Southern Observatory (ESO). In March 1989, Kurt Waldheim, the federal president of Austria, granted him the title Außerordentlicher Professor.
Maitzen developed a filter-photometric tool for discovering chemically peculiar magnetic stars, which detects a distinctive flux depression in the radiation of these stars around the 520-nanometre wavelength.
Maitzen is married and has four children.
Esperanto activities
Maitzen is a professor of the International Academy of Sciences San Marino, known by its Esperanto initials as AIS, and has contributed to many of its study sessions. For several years he has served as a member of the AIS Academic Senate; from 2012 to 2015 he is serving as the Academy's president.
Maitzen is an active Esperantist. From 1968 to 1969 he was president of the World Esperanto Youth Organization. Several times he helped organize contributions to the Internacia Kongresa Universitato, which offers specialized lecture programs coinciding with the annual World Congress of Esperanto of the UEA. He collaborated on astronomical terminology for the second edition of Plena Ilustrita Vortaro de Esperanto, a very extensive monolingual Esperanto dictionary often abbreviated as PIV.
In Vienna he has made significant contributions to Esperanto life and culture through a 1987 international scientific symposium for the 100th anniversary of the Esperanto language and with a 2001 seminar on the occasion of the European Year of Languages. Maitzen is the chief representative of the Universal Esperanto Association (UEA) to the United Nations Office at Vienna (UNOV).
Selected publications
Pri la astronomio – koncize ("Concerning astronomy, concisely", Esperanto), with Géza Felső, in Sciencaj Komunikaĵoj, SEC Budapest, No. 9, 55, 1985.
La reprezentiĝo de faka terminologio en enciklopedioj, demonstre de astronomio en Plena Ilustrita Vortaro ("The representation of subject-matter terminology en encyclopædias, demonstrated for astronomy in Plena Ilustrita Vortaro", Esperanto) in Modernaj rimedoj de komunikado, KAEST 1998 |
https://en.wikipedia.org/wiki/Li%20Xiao%20%28footballer%29 | Li Xiao is a Chinese football forward who played for China in the 1992 Asian Cup. He also played for Shanghai.
Career statistics
International statistics
External links
Team China Stats
1967 births
Living people
Chinese men's footballers
Chinese football managers
Shanghai Shenhua F.C. players
Wuhan Optics Valley F.C. players
China men's international footballers
Footballers from Shanghai
1992 AFC Asian Cup players
Shanghai Shenxin F.C. managers
Asian Games silver medalists for China
Medalists at the 1994 Asian Games
Asian Games medalists in football
Men's association football forwards
Wuhan Yangtze River F.C. managers
Footballers at the 1994 Asian Games |
https://en.wikipedia.org/wiki/2012%20Lao%20League | Statistics of Lao League in the 2012 season.
Clubs
Eastern Star FC
Ezra FC
Lao Airlines FC
Lao-American College FC
Lao Army FC
Lao Lane Xang FC
Lao Police Club
Pheuanphatthana FC
Vientiane FC
Yotha FC (previously Ministry of Public Works and Transport FC)
League table
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/Karen%20Parshall | Karen Hunger Parshall (born 1955, Virginia; née Karen Virginia Hunger) is an American historian of mathematics. She is the Commonwealth Professor of History and Mathematics at the University of Virginia with a joint appointment in the Corcoran Department of History and Department of Mathematics. From 2009 to 2012, Parshall was the Associate Dean for the Social Sciences in the College of Arts in Sciences at UVA, and from 2016 to 2019 she was the chair of the Corcoran Department of History.
Education and career
Parshall double-majored in French and mathematics at the University of Virginia, where she earned her master's degree in mathematics in 1978. She earned her PhD in 1982 in history from the University of Chicago under the direction of the historian Allen G. Debus (1926–2009) and the mathematician Israel Herstein. The subject of her dissertation was the history of the theory of algebras, especially the work of Joseph Wedderburn (The contributions of J. H. M. Wedderburn to the theory of algebras, 1900–1910).
From 1982 to 1987, Parshall was an assistant professor at Sweet Briar College and in 1987/88 at the University of Illinois at Urbana-Champaign. Since 1988 she has taught the mathematics, the history of mathematics, and the history of science at the University of Virginia, where she became in 1988 an assistant professor, in 1993 an associate professor and in 1999 a professor. She was a visiting professor at the Australian National University in Canberra, at the École des Hautes Etudes en Sciences Sociales (1985 and 2010) and at the Pierre and Marie Curie University in Paris (2016).
Work
Parshall's academic specialty is the development of mathematics in the US in the late 19th century and early 20th century (particularly the Chicago School). As one example, she has studied the work of Leonard Dickson, who was greatly influenced by contact with German mathematicians such as Felix Klein at the time of the Columbian Exposition of 1893. She has also focused on the history of algebra. She edited the correspondence of James Joseph Sylvester published by Oxford University Press and wrote a biography of Sylvester.
Recognition
In the academic year 1996/97 Parshall was a Guggenheim Fellow. In 1994 she was an invited speaker at the International Congress of Mathematicians (ICM) in Zürich (Mathematics in National Contexts (1875–1900): An International Overview). Since 2002 she has been a corresponding member of the Académie internationale d’histoire des sciences in Paris. From 1996 to 1999, she was editor of the journal Historia Mathematica. Parshall was in the governing body of the History of Science Society and from 1998 to 2001 of the American Mathematical Society (AMS).
In 2012, she became an inaugural fellow of the American Mathematical Society. She is the 2018 winner of the Albert Leon Whiteman Memorial Prize of the American Mathematical Society "for her outstanding work in the history of mathematics, and in particular, for her work on the |
https://en.wikipedia.org/wiki/Super%20Minkowski%20space | In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super Poincaré algebra.
Construction
Abstract construction
Abstractly, super Minkowski space is the space of (right) cosets within the Super Poincaré group of Lorentz group, that is,
.
This is analogous to the way ordinary Minkowski spacetime can be identified with the (right) cosets within the Poincaré group of the Lorentz group, that is,
.
The coset space is naturally affine, and the nilpotent, anti-commuting behavior of the fermionic directions arises naturally from the Clifford algebra associated with the Lorentz group.
Direct sum construction
For this section, the dimension of the Minkowski space under consideration is .
Super Minkowski space can be concretely realized as the direct sum of Minkowski space, which has coordinates , with 'spin space'. The dimension of 'spin space' depends on the number of supercharges in the associated super Poincaré algebra to the super Minkowski space under consideration. In the simplest case, , the 'spin space' has 'spin coordinates' with , where each component is a Grassmann number. In total this forms 4 spin coordinates.
The notation for super Minkowski space is then .
There are theories which admit supercharges. Such cases have extended supersymmetry. For such theories, super Minkowski space is labelled , with coordinates with .
Definition
The underlying supermanifold of super Minkowski space is isomorphic to a super vector space given by the direct sum of ordinary Minkowski spacetime in d dimensions (often taken to be 4) and a number of real spinor representations of the Lorentz algebra. (When this is slightly ambiguous because there are 2 different real spin representations, so one needs to replace by a pair of integers , though some authors use a different convention and take copies of both spin representations.)
However this construction is misleading for two reasons: first, super Minkowski space is really an affine space over a group rather than a group, or in other words it has no distinguished "origin", and second, the underlying supergroup of translations is not a super vector space but a nilpotent supergroup of nilpotent length 2.
This supergroup has the following Lie superalgebra. Suppose that is Minkowski space (of dimension ), and is a finite sum of irreducible real spinor representations for -dimensional Minkowski space.
Then there is an invariant, symmetric bilinear map . It is positive definite in the sense that, for any , the element is in the closed positive cone of , and if . This bilinear map is unique up to isomorphism.
The Lie superalgebra has as its even part, and as its odd (fermionic) part. The invariant bilinear map is extended to the whole superalgebra to define the (graded) Lie bracket , where the Lie bracket of anythi |
https://en.wikipedia.org/wiki/History%20of%20the%20function%20concept | The mathematical concept of a function dates from the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.
Mathematicians of the 18th century typically regarded a function as being defined by an analytic expression. In the 19th century, the demands of the rigorous development of analysis by Weierstrass and others, the reformulation of geometry in terms of analysis, and the invention of set theory by Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set to another.
Functions before the 17th century
Already in the 12th century, mathematician Sharaf al-Din al-Tusi analyzed the equation in the form stating that the left hand side must at least equal the value of for the equation to have a solution. He then determined the maximum value of this expression. It is arguable that the isolation of this expression is an early approach to the notion of a "function". A value less than means no positive solution; a value equal to corresponds to one solution, while a value greater than corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in Islamic mathematics, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe.
According to Dieudonné and Ponte, the concept of a function emerged in the 17th century as a result of the development of analytic geometry and the infinitesimal calculus. Nevertheless, Medvedev suggests that the implicit concept of a function is one with an ancient lineage. Ponte also sees more explicit approaches to the concept in the Middle Ages:
Historically, some mathematicians can be regarded as having foreseen and come close to a modern formulation of the concept of function. Among them is Oresme (1323–1382) In his theory, some general ideas about independent and dependent variable quantities seem to be present.
The development of analytical geometry around 1640 allowed mathematicians to go between geometric problems about curves and algebraic relations between "variable coordinates x and y." Calculus was developed using the notion of variables, with their associated geometric meaning, which persisted well into the eighteenth century. However, the terminology of "function" came to be used in interactions between Leibniz and Bernoulli towards the end of the 17th century.
The notion of "function" in analysis
The term "function" was literally introduced by Gottfried Leibniz, in a 1673 letter, to describe a quantity related to points of a curve, such as a coordinate or curve's slope. Johann Bernoulli started calling expressions made of a single variable "functions." In 1698, he agreed with |
https://en.wikipedia.org/wiki/List%20of%20Gabala%20FC%20records%20and%20statistics | Gabala FK is an Azerbaijani professional football club based in Qabala.
This list encompasses the major records set by the club and their players in the Azerbaijan Premier League. The player records section includes details of the club's goalscorers and those who have made more than 50 appearances in first-team competitions.
Player
Most appearances
Players played over 50 competitive, professional matches only. Appearances as substitute (goals in parentheses) included in total.
Goal scorers
Competitive, professional matches only, appearances including substitutes appear in brackets.
Internationals
Team
Record wins
Record win: 8–0; v Neftchi Baku, 2016–17 Azerbaijan Premier League, 10 September 2016, & v Mil-Muğan, 2017-18 Azerbaijan Cup, 29 November 2017
Record League win: 8–0 v Neftchi Baku, 2016–17 Azerbaijan Premier League, 10 September 2016
Record Azerbaijan Cup win: 8–0 v Mil-Muğan, 2017-18 Azerbaijan Cup, 29 November 2017
Record away win: 8–0 v Neftchi Baku, 2016–17 Azerbaijan Premier League, 10 September 2016
Record home win: 8–0 v Mil-Muğan, 2017-18 Azerbaijan Cup, 29 November 2017
Record defeats
Record defeat: 0–4
v Khazar Lankaran, 2006-07 Azerbaijan Premier League, 9 December 2006
v Inter Baku, 2007-08 Azerbaijan Premier League, 28 May 2008
Record League defeat: 0–4
v Khazar Lankaran, 2006-07 Azerbaijan Premier League, 9 December 2006
v Inter Baku, 2007-08 Azerbaijan Premier League, 28 May 2008
Record away defeat: 0–4
v Khazar Lankaran, 2006-07 Azerbaijan Premier League, 9 December 2006
v Inter Baku, 2007-08 Azerbaijan Premier League, 28 May 2008
Record Azerbaijan Cup defeat: 0–5
v Inter Baku, Semi-Final 1st leg, 29 April 2009
Record home defeat: 1–4
v Karvan, Azerbaijan Premier League, 17 September 2006
v Neftchi Baku, Azerbaijan Premier League, 2 April 2007
Wins/draws/losses in a season
Most wins in a league season: 22 – 2005–06
Most draws in a league season: 14 – 2009–10
Most defeats in a league season: 16 – 2006-07
Fewest wins in a league season: 4 – 2006-07
Fewest draws in a league season: 4 – 2006-07
Fewest defeats in a league season: 2 – 2005–06
Goals
Most League goals scored in a season: 72 – 2005–06
Most Premier League goals scored in a season: 43 – 2011–12
Fewest League goals scored in a season: 17 – 2006-07
Most League goals conceded in a season: 48 – 2009–10
Fewest League goals conceded in a season: 14 – 2005–06
Points
Most points in a season:
72 in 30 matches, First Division,2005–06
Fewest points in a season:
16 in 24 matches, Azerbaijan Premier League, 2006-07
References
http://gabalafc.az/?mod=news&id=363&lang=en
External links
Official Website
FK Qəbələ at AFFA.AZ
FK Qəbələ at UEFA.COM
FK Qəbələ at EUFO.DE
FK Qəbələ at Weltfussball.de
FK Qəbələ at Football-Lineups.com
Gabala FC
Gabala SC |
https://en.wikipedia.org/wiki/Ledi%20Utomo | Ledi Utomo (born 13 June 1983) is an Indonesian former footballer who plays as a defender.
Club statistics
References
External links
1983 births
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Persita Tangerang players
Persikota Tangerang players
Persitara Jakarta Utara players
PSMS Medan players
Persiba Balikpapan players
PSCS Cilacap players
Indonesian Premier Division players
Indonesia men's international footballers
Men's association football defenders
Sportspeople from Tangerang
Footballers from Banten |
https://en.wikipedia.org/wiki/Zainal%20Anwar | Zainal Anwar (born January 26, 1980) is an Indonesian former footballer plays as a full-back or midfielder.
Club statistics
References
External links
1980 births
Men's association football defenders
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Persita Tangerang players
Persikota Tangerang players
Persikabo Bogor players
Persipasi Bekasi players
PSMS Medan players
Persebaya Surabaya players
Celebest F.C. players
Indonesian Premier Division players
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Zulkarnain%20%28footballer%29 | Zulkarnain (born on September 14, 1982, in Medan) is an Indonesian former footballer.
Club statistics
References
External links
1982 births
Living people
Footballers from Medan
Men's association football midfielders
Indonesian men's footballers
Liga 1 (Indonesia) players
PSMS Medan players
Indonesian Premier Division players
Persiraja Banda Aceh players
PSDS Deli Serdang players
PSPS Riau players
PSSB Bireuen players
PS Kwarta Deli Serdang players |
https://en.wikipedia.org/wiki/Denny%20Rumba | Denny Rumba (born 16 May 1985) is an Indonesian former professional footballer who plays as a full-back.
Club statistics
References
External links
1985 births
Living people
Footballers from Semarang
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Persiba Bantul players
PSIS Semarang players
PSMS Medan players
Persepam Madura Utama players
PSS Sleman players
PSIR Rembang players
Indonesian Premier Division players
Men's association football fullbacks |
https://en.wikipedia.org/wiki/Spinor%20genus | In mathematics, the spinor genus is a classification of quadratic forms and lattices over the ring of integers, introduced by Martin Eichler. It refines the genus but may be coarser than proper equivalence.
Definitions
We define two Z-lattices L and M in a quadratic space V over Q to be spinor equivalent if there exists a transformation g in the proper orthogonal group O+(V) and for every prime p there exists a local transformation fp of Vp of spinor norm 1 such that M = g fpLp.
A spinor genus is an equivalence class for this equivalence relation. Properly equivalent lattices are in the same spinor genus, and lattices in the same spinor genus are in the same genus. The number of spinor genera in a genus is a power of two, and can be determined effectively.
Results
An important result is that for indefinite forms of dimension at least three, each spinor genus contains exactly one proper equivalence class.
See also
Genus of a quadratic form
References
Quadratic forms |
https://en.wikipedia.org/wiki/Arie%20Supriyatna | Arie Supriyatna (born December 25, 1984) is an Indonesian former footballer who plays as a striker.
Club statistics
References
External links
1984 births
Men's association football forwards
Living people
Sportspeople from Tangerang
Footballers from Banten
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Indonesian Premier Division players
Persipasi Bekasi players
PSMS Medan players
Bhayangkara Presisi Indonesia F.C. players
Borneo F.C. Samarinda players
Persikabo Bogor players |
https://en.wikipedia.org/wiki/Alamsyah%20Nasution | Sulaiman Alamsyah Nasution (born June 11, 1981) is an Indonesian former footballer.
Club statistics
Domestic league
International cups
Hounors
Clubs
Sriwijaya FC :
Liga Indonesia Premier Division champions : 1 (2007)
Piala Indonesia champions : 3 (2007, 2009, 2010)
References
External links
1981 births
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSMS Medan players
PSPS Riau players
Sriwijaya F.C. players
People of Batak descent
Indonesian Premier Division players
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Ramadhan%20Saputra | Ramadhan Saputra (born on May 5, 1986) is an Indonesian former footballer who plays as a defender.
Club statistics
References
External links
Ramadhan Saputra at Liga Indonesia
1986 births
Men's association football central defenders
Sportspeople from Tangerang
Footballers from Banten
Living people
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
Persita Tangerang
Semen Padang F.C. players
PSMS Medan players
Persiwa Wamena players
Persiba Bantul players
Persik Kediri players
Persela Lamongan players
Persik Kendal players
RANS Nusantara F.C. players
Perserang Serang players |
https://en.wikipedia.org/wiki/Genus%20of%20a%20quadratic%20form | In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently a free Z-module of finite rank. Two such forms are in the same genus if they are equivalent over the local rings Zp for each prime p and also equivalent over R.
Equivalent forms are in the same genus, but the converse does not hold. For example, x2 + 82y2 and 2x2 + 41y2 are in the same genus but not equivalent over Z. Forms in the same genus have equal discriminant and hence there are only finitely many equivalence classes in a genus.
The Smith–Minkowski–Siegel mass formula gives the weight or mass of the quadratic forms in a genus, the count of equivalence classes weighted by the reciprocals of the orders of their automorphism groups.
Binary quadratic forms
For binary quadratic forms there is a group structure on the set C of equivalence classes of forms with given discriminant. The genera are defined by the generic characters. The principal genus, the genus containing the principal form, is precisely the subgroup C2 and the genera are the cosets of C2: so in this case all genera contain the same number of classes of forms.
See also
Spinor genus
References
External links
Quadratic forms |
https://en.wikipedia.org/wiki/Novi%20Handriawan | Thomas Novi Handriawan (born November 4, 1986) is an Indonesian former footballer.
Club statistics
References
External links
1986 births
Men's association football defenders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSMS Medan players
Indonesian Premier Division players
PSDS Deli Serdang players
Semen Padang F.C. players
People from Deli Serdang Regency
Footballers from North Sumatra |
https://en.wikipedia.org/wiki/Anton%20Samba | Anton Samba (born April 5, 1982) is an Indonesian former footballer.
Club statistics
Hounors
Clubs
Persiwa Wamena :
Indonesia Super League runner-up : 1 (2008-09)
References
External links
1982 births
Men's association football midfielders
Living people
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Persim Maros players
Persiba Balikpapan players
Arema F.C. players
Persiwa Wamena players
Persikad Depok players
Persidafon Dafonsoro players
Persikabo Bogor players
PSMS Medan players
Persepam Madura Utama players
Persatu Tuban players
Persika 1951 players
Yahukimo F.C. players
Footballers from South Sulawesi |
https://en.wikipedia.org/wiki/Wawan%20Widiantoro | Wawan Widiantoro (born January 20, 1977) is an Indonesian former footballer.
Club statistics
Hounors
Clubs
Persik Kediri :
Liga Indonesia Premier Division champions : 2 (2003), (2006)
References
External links
1977 births
Men's association football defenders
Living people
Javanese people
People from Kediri (city)
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Arema F.C. players
Persik Kediri players
PKT Bontang players
PSMS Medan players
Perseta Tulungagung players
PSS Sleman players
Footballers from East Java |
https://en.wikipedia.org/wiki/Agus%20Cima | Agus Cima (born November 8, 1983) is an Indonesian former professional footballer who plays as a full-back.
Club career statistics
References
External links
1983 births
Men's association football defenders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
PSPS Riau players
Indonesian Premier Division players
PSMS Medan players
PS Barito Putera players
Dewa United F.C. players |
https://en.wikipedia.org/wiki/Gusripen%20Efendi | Gusripen Efendi (born August 14, 1986 in Muara Mahat, Kampar Regency) is an Indonesian former footballer.
Club statistics
References
External links
1986 births
Men's association football midfielders
Living people
Sportspeople from Riau
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Liga 2 (Indonesia) players
PSPS Riau players
Semen Padang F.C. players
Persita Tangerang players
Persis Solo players
PSP Padang players
RANS Nusantara F.C. players |
https://en.wikipedia.org/wiki/2013%20Lao%20League | Statistics of Lao League in the 2013 season. League started on 16 March 2013.
Clubs
SHB Champasak(New additional)
Ezra
Friends Development
Hoang Anh Attapeu(New additional)
Lao Lane Xang FC
Lao Police Club
Eastern Star FC
Yotha FC
Stadium capacities
* Stadium use for home team maybe unconfirmed
League table
Top scorer
References
Lao Premier League seasons
1
Laos
Laos |
https://en.wikipedia.org/wiki/1981%E2%80%9382%20Galatasaray%20S.K.%20season | The 1981–82 season was Galatasaray's 78th in existence and the 24th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
2nd leg Galatasaray SK – Bursa SK squad has not been added
Players in / out
In
Out
1. Lig
Standings
Matches
Kick-off listed in local time (EET)
Türkiye Kupası
Kick-off listed in local time (EET)
5th Round
6th Round
1/4 Final
1/2 Final
Final
Süper Kupa-Cumhurbaşkanlığı Kupası
Kick-off listed in local time (EET)
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Donanma Kupası
Polis Vakfı Kupası
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1981–82 season
1980s in Istanbul |
https://en.wikipedia.org/wiki/Encyclopedia%20of%20Mathematics%20%28James%20Tanton%29 | Encyclopedia of Mathematics is a 2005 encyclopedic reference work by American author James Tanton that was published by Facts-on-File of New York.
Synopsis
The book has over 1000 entries, which discuss various concepts, definitions, people, and theorems that pertain to mathematics. The book also contains six essays that discuss the various branches of mathematics and their history.
Reception
Booklist praised the book as a useful "basic resource for students who wish to have a better understanding of simple or not-so--simple mathematical concepts" and the School Library Journal called the encyclopedia "comprehensive".
References
External links
2005 non-fiction books
Encyclopedias of mathematics |
https://en.wikipedia.org/wiki/Cyclotruncated%206-simplex%20honeycomb | In six-dimensional Euclidean geometry, the cyclotruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, truncated 6-simplex, bitruncated 6-simplex, and tritruncated 6-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.
Structure
It can be constructed by seven sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-simplex honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 6-space:
6-cubic honeycomb
6-demicubic honeycomb
6-simplex honeycomb
Omnitruncated 6-simplex honeycomb
222 honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
7-polytopes |
https://en.wikipedia.org/wiki/Windu%20Hanggono%20Putra | Windu Hanggoro Putra (born August 21, 1988 in Jakarta) is an Indonesian former footballer.
Club statistics
References
External links
1988 births
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
Indonesian Premier Division players
Persija Jakarta players
PSPS Riau players
Persita Tangerang players
Footballers from Jakarta |
https://en.wikipedia.org/wiki/Fance%20Hariyanto | Fance Hariyanto (born October 1, 1983, in Pekanbaru) is an Indonesian former footballer.
Club statistics
References
External links
1983 births
Men's association football goalkeepers
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSPS Riau players
Indonesian Premier Division players
Persijap Jepara players
People from Pekanbaru
Sportspeople from Riau |
https://en.wikipedia.org/wiki/Thomas%20Weber%20%28footballer%29 | Thomas Weber (born 29 May 1993) is an Austrian footballer who plays for SV Stripfing.
Club statistics
Updated to games played as of 16 June 2014.
References
External links
1993 births
Living people
Footballers from Vienna
Austrian men's footballers
Austria men's under-21 international footballers
Men's association football defenders
FC Admira Wacker Mödling players
Floridsdorfer AC players
Austrian Football Bundesliga players |
https://en.wikipedia.org/wiki/1982%E2%80%9383%20Galatasaray%20S.K.%20season | The 1982–83 season was Galatasaray's 79th in existence and the 25th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Kick-off listed in local time (EET)
Türkiye Kupası
Kick-off listed in local time (EET)
6th Round
European Cup Winners' Cup
1st round
2nd round
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Donanma Kupası
Akıl Hastalar Vakfı tournament
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1982–83 season
1980s in Istanbul
Galatasaray Sports Club 1982–83 season |
https://en.wikipedia.org/wiki/Mariano%20Giaquinta | Mariano Giaquinta (born Caltagirone, 1947), is an Italian mathematician mainly known for his contributions to the fields of calculus of variations and regularity theory of partial differential equation. He is currently professor of Mathematics at the Scuola Normale Superiore di Pisa and he is the director of De Giorgi center at Pisa.
Career
Giaquinta is well known for his basic work in elliptic regularity theory, and especially in the setting of vectorial variational problems. Together with Enrico Giusti he has obtained innovative results on the regularity of minima of variational integrals and related singular sets. The main novelty is in the fact that, for the first time, the regularity of minimizers is obtained using directly the minimality properties without appealing to Euler–Lagrange equation of the functionals, which in general is not supposed to exist in the cases considered. His work with Giuseppe Modica on the local higher integrability properties of solutions to elliptic systems has had an influence on the development of partial regularity theory. Many of these results are summarized in his 1983 book.
Giaquinta has been one of the founders, and for many years the managing editor, of the journal "Calculus of Variations and PDE".
Awards
Giaquinta won the Bartolozzi Prize of the Italian Mathematical Union in 1979, in 1990 he was awarded with Humboldt research award and in 2006 with the Amerio Prize. He has been an invited speaker at the 1986 International congress of mathematicians. Giaquinta belongs to the ISI list of highly cited researchers in Mathematics and he is a member of the German Academy of Sciences.
Selected publications
.
.
.
.
.
References
21st-century Italian mathematicians
Living people
1947 births
PDE theorists
Variational analysts |
https://en.wikipedia.org/wiki/Frederik%20Schuh | Frederik Schuh (7 February 1875, Amsterdam – 6 January 1966, The Hague) was a Dutch mathematician.
Career
He completed his PhD in algebraic geometry from Amsterdam University in 1905, where his advisor was Diederik Johannes Korteweg. He taught at the Technische Hoogeschool at Delft (1907–1909 and 1916–1945) and at Groningen (1909–1916).
Works
He was the inventor of the Chomp game and wrote The Master Book of Mathematical Recreations (1943).
References
External links
Mathematics Genealogy
1875 births
1966 deaths
20th-century Dutch mathematicians
Game theorists
Scientists from Amsterdam
University of Amsterdam alumni
Academic staff of the Delft University of Technology
Academic staff of the University of Groningen |
https://en.wikipedia.org/wiki/Potato%20paradox | The potato paradox is a mathematical calculation that has a counter-intuitive result. The Universal Book of Mathematics states the problem as such:
Then reveals the answer:
In Quine's classification of paradoxes, the potato paradox is a veridical paradox.
If the potatoes are 99% water, the dry mass is 1%. This means that the 100 kg of potatoes contains 1 kg of dry mass, which does not change, as only the water evaporates.
In order to make the potatoes be 98% water, the dry mass must become 2% of the total weight—double what it was before. The amount of dry mass, 1 kg, remains unchanged, so this can only be achieved by reducing the total mass of the potatoes. Since the proportion that is dry mass must be doubled, the total mass of the potatoes must be halved, giving the answer 50 kg.
Mathematical proofs
Let x be the new total mass of the potatoes (dry + water).
Let d be the dry mass of the potatoes and w, the mass of water within the potatoes.
Recall w is 98% of the total mass, that is, w = 0.98x.
Therefore, x = d + w = d + 0.98x, i.e., x = d / 0.02 = 50 kg.
In our case, d = 1 kg so the new mass of the potatoes will indeed be 50 kg.
Let X be the mass lost. Since the solid (non-water) mass remains constant, then
X = initial water content – final water content
X = 99% 100 kg – 98% (100 kg – X)
X = 99 kg – 98 kg + 0.98X
1 kg = 0.02X
X = (1 kg)/0.02 = 50 kg
In popular culture
The potato paradox has made its way into popular culture. In one instance, it was the "Puzzler" on the Car Talk radio show. It was subsequently featured on Neatorama. It was also named one of the "Five Famous Paradoxes".
References
External links
Mathematical paradoxes
Potatoes |
https://en.wikipedia.org/wiki/Gregory%20Garibian | Gregory Markari (Markarovich) Garibian (December 13, 1924 – June 8, 1991) was a Soviet Armenian physicist, academician-secretary of the Department of Physics and Mathematics of the Armenian Academy of Sciences (AS)(1973–1991). He is known for developing the Theory of Transition Radiation and showing the feasibility of functional transition radiation detectors (TRDs). [8] [9]
Biography
G.M.Garibian was born in 1924 in Tiflis (now - Tbilisi, Georgia) in a family of a Medical Doctor and a homemaker. Eventually the family moved to Baku (Azerbaijan) where Garibian got his general education. In 1943 he graduated from school in Baku and went to Moscow. Physical science was Garibian's passion in life. Even at a very young age he followed news in the world of physics and was very excited when in 1942 he learned about the Alikhanian brothers’ expedition to Mount Aragats (Armenia) in order to search for protons in cosmic rays. Garibian was accepted into the Department of Physics and Mathematics of Moscow State University which he graduated from in 1948 immediately to leave for Yerevan and join the Yerevan Physics Institute, which was founded by Artyom Alikhanian in 1943. After that time Garibian dedicated himself to scientific research in Theoretical physics, in the fields of Quantum electrodynamics, Cosmic rays, and High energy particles. All his life he worked at the Yerevan Physics Institute consecutively as researcher, scientific secretary of the institute, deputy director and head of laboratory. He actively participated in the creation of the Yerevan Synchrotron and also in the establishment of high-altitude cosmic ray stations on Mount Aragats.
Garibian's main scientific achievement was the discovery of X-Ray Transition Radiation and the development of the Theory of Transition Radiation. He also showed the feasibility of a functional Transition Raditation Detector (TRD) - a tool for identification of high energy ultrarelativistic particles.
In the end of the 1940s and the beginning of the 1950s, the main points of interest of the researchers at the Yerevan Institute of Physics were cosmic rays and the physics of elementary particles. One of the problems bothering experimental physicists working with cosmic rays at that time were measurements of very high energies of the relativistic particles in the cosmic radiation, as with the increase of energy levels of the particles the available methods of registration of their energies were becoming less and less effective.
Garibian tried to develop methods that would help to resolve that problem. As a starting point of his research, he used results published in the article written by Ginzburg and Frank in 1946, where the theoretical existence of transition radiation was predicted. The new type of radiation appearing as a result of charged particle passing through the boundary between two layers of matter. In 1959, Garibyan discovered x-ray transition radiation, the intensity of which had a linear depende |
https://en.wikipedia.org/wiki/Dudley%20triangle | Dudley Triangle may also refer to a neighborhood of Boston, Massachusetts.
In mathematics, the Dudley triangle is a triangular array of integers that was defined by . It consists of the numbers
.
Dudley exhibited several rows of this triangle, and challenged readers to find the next row; the challenge was met by J. G. Mauldon, who proposed two different solutions. In one of Mauldon's solutions, the number at the intersection of the mth and nth diagonals (counting the top of the triangle as having m = n = 1) is given by the formula
Notes
References
Triangles of numbers |
https://en.wikipedia.org/wiki/Kheyrollah%20Veisi | Kheyrollah Veisi () is an Iranian footballer.
Club career
Veisi joined Foolad F.C. in summer 2011.
Club career statistics
References
Kheyrollah Veisi at PersianLeague.com
1988 births
Living people
Esteghlal Ahvaz F.C. players
Foolad F.C. players
Iranian men's footballers
Men's association football defenders
Footballers from Khuzestan province |
https://en.wikipedia.org/wiki/M.%20Zahrul%20Bin%20Azhar | M. Zahrul Bin Azhar (born September 5, 1982) is an Indonesian former footballer who plays as a midfielder.
Club statistics
References
External links
1982 births
Men's association football midfielders
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSPS Riau players
Persih Tembilahan players
Indonesian Premier Division players |
https://en.wikipedia.org/wiki/Susanto%20%28footballer%29 | Susanto (born August 21, 1987) is an Indonesian former footballer who plays as a goalkeeper.
Club statistics
References
External links
1987 births
Men's association football goalkeepers
Living people
Indonesian men's footballers
Liga 1 (Indonesia) players
PSPS Riau players
People from Pekanbaru
Sportspeople from Riau |
https://en.wikipedia.org/wiki/Victory%20Yendra | Victory Yendra (born September 10, 1989 in Kuantan Singingi Regency) is an Indonesian former footballer who plays as a striker.
Club statistics
References
External links
1989 births
Men's association football forwards
Living people
Indonesian men's footballers
Indonesian Premier Division players
Liga 1 (Indonesia) players
Persih Tembilahan players
PSPS Riau players
Persiks Kuantan Singingi players
Sportspeople from Riau |
https://en.wikipedia.org/wiki/Aples%20Tecuari | Aples Gideon Tecuari (born 21 April 1973) is an Indonesian former footballer who previously played as a defender.
Clubs career
Club statistics
International career
In 1996 Aples's international career began.
International goals
Honours
International
Indonesia
AFF Championship runner-up: 2002
References
External links
1973 births
Men's association football defenders
Living people
People from Jayapura Regency
Indonesian men's footballers
Indonesia men's international footballers
Footballers from Papua
Indonesian Premier Division players
Liga 1 (Indonesia) players
Madura United F.C. players
Perseman Manokwari players
Persija Jakarta players
PSPS Riau players
Footballers from Jayapura |
https://en.wikipedia.org/wiki/Edm%C3%A9-Gilles%20Guyot | Edmé-Gilles Guyot (1706–1786) was a French mail clerk, physician, postmaster, cartographer, inventor and author on the subject of mathematics, physics and magic. He experimented with optical illusions and with the theory behind performance magic. His developments into the apparent appearance of ghosts, using the projection of a figure into smoke, helped to create the technology and techniques used in phantasmagoria.
Mathematics, science, and magic
Manufacturer of conjuring apparatus and scientific instruments, Guyot was accused of exploiting and revealing the tricks used at the time by magicians and science populizers like Nicolas-Philippe Ledru and François Pelletier. He created "magic theatres" for the aristocracy – small boxes that use lanterns and slides to create an animated story. Guyot's work was influential in the development of magic lanterns and their use in phantasmagoria. In 1770 he detailed a method of simultaneously using two different slides in this early projection device. His example was a sea that would become increasingly stormy, throwing around the ships that were sailing on it. He advised that the slides would need to be very carefully painted in order to create a realistic and beautiful animation. His writings on the subject were translated into English and German and were widely circulated around Europe. His experiments led to the technique of projecting images onto smoke to create the appearance of ghostly apparitions. In 1779 Guyot described the use of transformation slides in magic lanterns to create simple animations.
Nouvelles recreations physiques et mathematiques
Guyot's four part book Nouvelles recreations physiques et mathematiques featured descriptions of experiments and examples of how various innovative mathematical and magical tricks could be performed. The book was first published in 1769 and included an explanation of Hooper's paradox, It also includes detailed, illustrated techniques for the performance of the cups and balls trick that is regarded as being greatly influential.
The book was adapted into English by William Hooper, under the title Rational Recreations being released in 1774 without credit to Guyot.
Medicine
Guyot is credited with describing in 1724 the catheterization of the Eustachian tube, one of the first means of middle ear inflation.
Publications
References
External links
Nouvelles récréations physiques et mathématiques/Neue physikalische und mathematische Belustigungen From the Harry Houdini Collection in the Rare Book and Special Collection Division at the Library of Congress
La France littéraire, Duchesne, Paris 1759, p. 72.
Correspondance littéraire, philosophique et critique, January 1770, pp. 444-5.
1706 births
1786 deaths
18th-century French physicians
French cartographers
18th-century French mathematicians
18th-century French inventors |
https://en.wikipedia.org/wiki/BCS%20statistics | The Bowl Championship Series (BCS) is a selection system that began in the 1998 season. It creates match-ups in five bowl games between ten of the top ranked teams in the NCAA Division I Football Bowl Subdivision (FBS), including the BCS National Championship Game. These are relevant team and individual statistics of BCS games and ranking system.
BCS National Championship Game Statistics
The team leading at halftime is 13–0 in BCS National Championship Games. (In the 2009 game, Florida and Oklahoma were tied at halftime.)
Fourteen Heisman Trophy winners have appeared in BCS Bowl Games, with twelve competing for the national championship. (Eleven appeared in the year that they won). Two Heisman Trophy winners have appeared in two BCS bowl games (Jason White in 2004 and 2005, and Matt Leinart in 2005 and 2006). Their teams have gone 6–8 in those games.
BCS Regular Season Record Statistics
While numerous teams have gone undefeated and appeared in a BCS bowl game, the 1999 Florida State team was the first team ranked No. 1 in the preseason Associated Press Poll to go undefeated and win the National Championship. USC was also ranked No.1 in preseason and went on to win the National Championship, but that championship was vacated due to rules violations by the school. No team has won a BCS title ranked No. 3 or No. 4 in preseason.
Only one team not ranked in the AP Top 20 in preseason has ever gone on to win a BCS National Championship. In 2010, The Auburn Tigers began the year ranked No. 22 in the nation. They were also the first team outside of the top 10 since 1990 to even clinch a share of the title.
The worst record for any team to qualify for a BCS bowl berth are the 2012 Wisconsin Badgers at 8–5. Five 8-4 teams have earned a BCS bowl berth: Syracuse in 1998, Stanford in 1999, Purdue in 2000, Pittsburgh in 2004, and The University of Connecticut in 2010. Those five teams lost their BCS Bowl game.
References
Bowl Championship Series |
https://en.wikipedia.org/wiki/Mojtaba%20Shiri%20%28footballer%2C%20born%201990%29 | Mojtaba Shiri (; born 13 February 1990) is an Iranian footballer.
Club career
Shiri joined Rah Ahan in summer 2011 after success in technical test by the coach, Ali Daei.
Club career statistics
Last Update 14 June 2019
References
1990 births
Living people
Sportspeople from Qom
Iranian men's footballers
Saba Qom F.C. players
Rah Ahan Tehran F.C. players
Naft Tehran F.C. players
Paykan F.C. players
Persian Gulf Pro League players
Azadegan League players
Men's association football wingers |
https://en.wikipedia.org/wiki/Szabolcs%20Csorba | Szabolcs Csorba (born 24 October 1991) is a Hungarian football player currently playing for the Bangladeshi team Dhaka Abahani as a forward.
Club statistics
Updated to games played as of 4 August 2012.
External links
MLSZ
1991 births
Living people
Footballers from Debrecen
Hungarian men's footballers
Men's association football midfielders
Debreceni VSC players |
https://en.wikipedia.org/wiki/M%C3%A1rk%20Sz%C3%A9csi | Márk Szécsi (born 22 May 1994 in Eger) is a Hungarian football player currently playing for the Hungarian team Debreceni VSC as a midfielder.
Club statistics
Updated to games played as of 15 May 2022.
External links
MLSZ
References
1994 births
Living people
Footballers from Eger
Hungarian men's footballers
Men's association football forwards
Debreceni VSC players
Kecskeméti TE players
Nyíregyháza Spartacus FC players
Puskás Akadémia FC players
Nemzeti Bajnokság I players |
https://en.wikipedia.org/wiki/Equidissection | In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triangles. In fact, most polygons cannot be equidissected at all.
Much of the literature is aimed at generalizing Monsky's theorem to broader classes of polygons. The general question is: Which polygons can be equidissected into how many pieces? Particular attention has been given to trapezoids, kites, regular polygons, centrally symmetric polygons, polyominos, and hypercubes.
Equidissections do not have many direct applications. They are considered interesting because the results are counterintuitive at first, and for a geometry problem with such a simple definition, the theory requires some surprisingly sophisticated algebraic tools. Many of the results rely upon extending p-adic valuations to the real numbers and extending Sperner's lemma to more general colored graphs.
Overview
Definitions
A dissection of a polygon P is a finite set of triangles that do not overlap and whose union is all of P. A dissection into n triangles is called an n-dissection, and it is classified as an even dissection or an odd dissection according to whether n is even or odd.
An equidissection is a dissection in which every triangle has the same area. For a polygon P, the set of all n for which an n-equidissection of P exists is called the spectrum of P and denoted S(P). A general theoretical goal is to compute the spectrum of a given polygon.
A dissection is called simplicial if the triangles meet only along common edges. Some authors restrict their attention to simplicial dissections, especially in the secondary literature, since they are easier to work with. For example, the usual statement of Sperner's lemma applies only to simplicial dissections. Often simplicial dissections are called triangulations, although the vertices of the triangles are not restricted to the vertices or edges of the polygon. Simplicial equidissections are therefore also called equal-area triangulations.
The terms can be extended to higher-dimensional polytopes: an equidissection is set of simplexes having the same n-volume.
Preliminaries
It is easy to find an n-equidissection of a triangle for all n. As a result, if a polygon has an m-equidissection, then it also has an mn-equidissection for all n. In fact, often a polygon's spectrum consists precisely of the multiples of some number m; in this case, both the spectrum and the polygon are called principal and the spectrum is denoted . For example, the spectrum of a triangle is . A simple example of a non-principal polygon is the quadrilateral with vertices (0, 0), (1, 0), (0, 1), (3/2, 3/2); its spectrum includes 2 and 3 but not 1.
Affine transformations of the plane are useful for studying equidissections, including translations, uniform and non-uniform scaling, reflections, rotations, shears, an |
https://en.wikipedia.org/wiki/False%20coverage%20rate | In statistics, a false coverage rate (FCR) is the average rate of false coverage, i.e. not covering the true parameters, among the selected intervals.
The FCR gives a simultaneous coverage at a (1 − α)×100% level for all of the parameters considered in the problem. The FCR has a strong connection to the false discovery rate (FDR). Both methods address the problem of multiple comparisons, FCR from confidence intervals (CIs) and FDR from P-value's point of view.
FCR was needed because of dangers caused by selective inference. Researchers and scientists tend to report or highlight only the portion of data that is considered significant without clearly indicating the various hypothesis that were considered. It is therefore necessary to understand how the data is falsely covered. There are many FCR procedures which can be used depending on the length of the CI – Bonferroni-selected–Bonferroni-adjusted, Adjusted BH-Selected CIs (Benjamini and Yekutieli 2005). The incentive of choosing one procedure over another is to ensure that the CI is as narrow as possible and to keep the FCR. For microarray experiments and other modern applications, there are a huge number of parameters, often tens of thousands or more and it is very important to choose the most powerful procedure.
The FCR was first introduced by Daniel Yekutieli in his PhD thesis in 2001.
Definitions
Not keeping the FCR means when , where is the number of true null hypotheses, is the number of rejected hypothesis, is the number of false positives, and is the significance level. Intervals with simultaneous coverage probability can control the FCR to be bounded by .
Classification of multiple hypothesis tests
The problems addressed by FCR
Selection
Selection causes reduced average coverage. Selection can be presented as conditioning on an event defined by the data and may affect the coverage probability of a CI for a single parameter. Equivalently, the problem of selection changes the basic sense of P-values. FCR procedures consider that the goal of conditional coverage following any selection rule for any set of (unknown) values for the parameters is impossible to achieve. A weaker property when it comes to selective CIs is possible and will avoid false coverage statements. FCR is a measure of interval coverage following selection. Therefore, even though a 1 − α CI does not offer selective (conditional) coverage, the probability of constructing a no covering CI is at most α, where
Selection and multiplicity
When facing both multiplicity (inference about multiple parameters) and selection, not only is the expected proportion of coverage over selected parameters at 1−α not equivalent to the expected proportion of no coverage at α, but also the latter can no longer be ensured by constructing marginal CIs for each selected parameter. FCR procedures solve this by taking the expected proportion of parameters not covered by their CIs among the selected parameters, where the proportion is 0 |
https://en.wikipedia.org/wiki/Cohen%20ring | In algebra, a Cohen ring is a field or a complete discrete valuation ring of mixed characteristic whose maximal ideal is generated by p. Cohen rings are used in the Cohen structure theorem for complete Noetherian local rings.
See also
Norm field
References
Cohen's paper was written when "local ring" meant what is now called a "Noetherian local ring".
Commutative algebra |
https://en.wikipedia.org/wiki/Alireza%20Ghadiri | Alireza Ghadiri (, born 8 October 1979) is an Iranian football goalkeeper, currently playing for Saba.
Club career
Club career statistics
References
External links
Alireza Ghadiri at Persian League
Living people
Iranian men's footballers
1979 births
Saba Qom F.C. players
Shahid Ghandi Yazd F.C. players
Sanat Mes Kerman F.C. players
Place of birth missing (living people)
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Alewyn%20Burger | Alewyn Burger (born 1951) is a South African banker and former Chief Operating Officer of Standard Bank. He has a PhD in mathematics from the University of South Africa and attended Harvard Business School's six-week Advanced Management Program (1991).
He semi-retired from his position at Standard Bank in 2011 and currently acts as an advisor for PwC and SAP. He also teaches as a visiting professor at the University of Stellenbosch Business School.
Academic life and early career
His studies started at the Rand Afrikaans University in Johannesburg where he received an MSc degree in mathematical statistics in 1974 and subsequently earned a PhD in mathematical statistics at the University of South Africa in 1981.
In 1991 he completed an Advanced Management Program (AMP) at Harvard Business School.
Advanced Management Program at Harvard Business School 1991
Advanced Executive program (AEP) at University of South Africa 1986
Received a MSc at the Rand Afrikaans University 1974
ABSA Bank merger
Burger had a key position during the Absa Bank Group merger between 1992 and 1997, which included amalgamating United Bank, Allied Bank, the Volkskas Group as well as the Sage Group. Burger was instrumental in integrating the IT systems into a single integrated operation, and transforming the organization from a product silo structure to a more customer centric.
Current and previous directorships
1999-2001 Frankling (Pty) Ltd Investment company.
1997-1998 Absa Bank Group Member of group board. Responsible for cards and electronic banking, Commercial banks, ABSA Direct, Group operations, Electronic Commerce and Group R&D
United Building Society 1979-1991
United Group Executive Director, accountable for all operations, research and technology services.
United Building Society. General Manager for IT, Research and Management Services divisions
Deputy General Manager, IT and Research Department
Assistant General Manager for the IT Division
Head Office Manager of Applications Programming, IT Division
Manager of the Research department
1974-1980 SA Defence Force in the DISA unit. Work relating to Statistics and Operations Research.
Standard Bank
Board and advisory positions
Absa Bank Group
Commercial Bank Divisional Board
Corporate Bank Divisional Board
Financial Services
Private Banking and Investment Services
Bankfin
Cortal Direct
SA Banking Council (representing ABSA)
Global Access SA
Destiny Electronic Commerce (JV with Naspers: Chairman)
Smartec Technonologies (JV with Franklin: Chairman)
Payments Association of SA (Chairman 1996-1998)
Transwitch Services, Easypay (JV with Pick ‘n Pay: Chairman 1996-1998)
Visa International Regional Board (Representing ABSA)
MasterCard Regional Board for Middle East and Africa (representing ABSA)
Maestro Global Board (representing ABSA)
Awards and recognitions
ICT Industry Leader of the year 2005
IT personality of the year in South Africa 2003
Finalist in South African National Bo |
https://en.wikipedia.org/wiki/Hossein%20Hosseini%20%28footballer%2C%20born%201988%29 | Seyyed Hossein Hosseini () is an Iranian football defender who last played for Ararat in Armenian Premier League.
Career
Hosseini joined Paykan in Summer 2011.
Club career statistics
References
,
Iranian men's footballers
Men's association football defenders
Saipa F.C. players
Paykan F.C. players
Iranian expatriate men's footballers
1988 births
Living people
Footballers from Tehran |
https://en.wikipedia.org/wiki/Stress%20resultants | Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells. The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions. As a consequence the three traction components that vary from point to point in a cross-section can be replaced with a set of resultant forces and resultant moments. These are the stress resultants (also called membrane forces, shear forces, and bending moment) that may be used to determine the detailed stress state in the structural element. A three-dimensional problem can then be reduced to a one-dimensional problem (for beams) or a two-dimensional problem (for plates and shells).
Stress resultants are defined as integrals of stress over the thickness of a structural element. The integrals are weighted by integer powers the thickness coordinate z (or x3). Stress resultants are so defined to represent the effect of stress as a membrane force N (zero power in z), bending moment M (power 1) on a beam or shell (structure). Stress resultants are necessary to eliminate the z dependency of the stress from the equations of the theory of plates and shells.
Stress resultants in beams
Consider the element shown in the adjacent figure. Assume that the thickness direction is x3. If the element has been extracted from a beam, the width and thickness are comparable in size. Let x2 be the width direction. Then x1 is the length direction.
Membrane and shear forces
The resultant force vector due to the traction in the cross-section (A) perpendicular to the x1 axis is
where e1, e2, e3 are the unit vectors along x1, x2, and x3, respectively. We define the stress resultants such that
where N11 is the membrane force and V2, V3 are the shear forces. More explicitly, for a beam of height t and width b,
Similarly the shear force resultants are
Bending moments
The bending moment vector due to stresses in the cross-section A perpendicular to the x1-axis is given by
Expanding this expression we have,
We can write the bending moment resultant components as
Stress resultants in plates and shells
For plates and shells, the x1 and x2 dimensions are much larger than the size in the x3 direction. Integration over the area of cross-section would have to include one of the larger dimensions and would lead to a model that is too simple for practical calculations. For this reason the stresses are only integrated through the thickness and the stress resultants are typically expressed in units of force per unit length (or moment per unit length) instead of the true force and moment as is the case for beams.
Membrane and shear forces
For plates and shells we have to consider two cross-sections. The first is perpendicular to the x1 axis and the second is perpendicular to the x2 axis. Following the same proce |
https://en.wikipedia.org/wiki/Runyenjes | {
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Runyenjes is the second largest town in Embu County, Kenya. It is located about 150 km from Nairobi, and 75 km from Mount Kenya, at an altitude of 1500m.The population is estimated at 58,000 people, the majority of which are of the Embu People who speak Kiembu, a Bantu language, as well as Kiswahili and English, the two official languages in Kenya. The area offers scenic views, with densely wooded hills, gentle valleys, flowing streams and rivers, waterfalls, as well as small-scale farms.
Etymology
Runyenjes derives its name from a popular Chief Runyenje of the colonial era who ruled from Thuci River to Sagana.
Geography
Runyenjes is located at an altitude of 1495.62 m (4906.89 ft) and lies on the windward side of Mt. Kenya. It is about 150 km from Nairobi city and about 25 km from Embu town along the Nairobi-Meru-Isiolo road. The road from Runyenjes to Nairobi is tarmacked and well-maintained. A drive along this road from Runyenjes town towards Embu-Nairobi offers scenic views of the Karue and Kirimiri hills, nearby waterfalls and, on a clear sky, a great view of Mt. Kenya. The section of this road from Runyenjes town to Meru town via Chuka town is also quite scenic offering views ranging from densely wooded hills, gentle valleys, flowing streams and rivers, as well as, tea and coffee small-scale farms.
According to Google maps, you can walk non-stop from Runyenjes town to the top of Mt. Kenya in approximately 18 hr 31 min, a distance of about 74 km.
Towns surrounding or nearby Runyenjes include Kianjokoma, Karurumo, Kigumo, Kathageri, Mukuuri, Kanja, Mufu, Rukuriri, Ena, Gichiche, Kathanjuri, Kyeni, Makutano, Kiaragana and Nthagaiya.
Demographics
The population of Runyenjes is estimated to be 58,000 people, majority of which are of the Embu People heritage and speak Kiembu, a Bantu language, as well as Kiswahili and English, the two official languages in Kenya.
Government
Runyenjes serves as constituency and municipality.
Religion
Similar to the rest of Embu county, most of the residents identify as Christians. There are several major Christian denominations churches located in or around the municipality, including but not limited to, the Anglican Church of Kenya, The Roman Catholic Church of Kenya, SDA Church and the Salvation Army Church. Well known church buildings include ACK St. Philips Rukuriri, St. Joseph Mukasa Catholic Church Mbiruri among many others.
References
Embu County
Populated places in Kenya |
https://en.wikipedia.org/wiki/Arman%20Ghasemi | Arman Ghasemi () is an Iranian football defender, who currently plays and captains for Paykan in Persian Gulf Pro League.
Career
Ghasemi joined Gahar Zagros in summer 2012.
Club Career Statistics
Last Update: 7 August 2014
References
External links
Arman Ghasemi at PersianLeague.com
Iranian men's footballers
Men's association football defenders
Paykan F.C. players
Rah Ahan Tehran F.C. players
Gahar Zagros F.C. players
Naft Masjed Soleyman F.C. players
Footballers from Tehran
1989 births
Living people |
https://en.wikipedia.org/wiki/Trilinear%20polarity | In Euclidean geometry, trilinear polarity is a certain correspondence between the points in the plane of a triangle not lying on the sides of the triangle and lines in the plane of the triangle not passing through the vertices of the triangle. "Although it is called a polarity, it is not really a polarity at all, for poles of concurrent lines are not collinear points." It was Jean-Victor Poncelet (1788–1867), a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.
Definitions
Let be a plane triangle and let be any point in the plane of the triangle not lying
on the sides of the triangle. Briefly, the trilinear polar of is the axis of perspectivity of the cevian triangle of and the triangle .
In detail, let the line meet the sidelines at respectively. Triangle is the cevian triangle of with reference to triangle . Let the pairs of line intersect at respectively. By Desargues' theorem, the points are collinear. The line of collinearity is the axis of perspectivity of triangle and triangle . The line is the trilinear polar of the point .
The points can also be obtained as the harmonic conjugates of with respect to the pairs of points respectively. Poncelet used this idea to define the concept of trilinear polars.
If the line is the trilinear polar of the point with respect to the reference triangle then is called the trilinear pole of the line with respect to the reference triangle .
Trilinear equation
Let the trilinear coordinates of the point be . Then the trilinear equation of the trilinear polar of is
Construction of the trilinear pole
Let the line meet the sides of triangle at respectively. Let the pairs of lines meet at . Triangles and are in perspective and let be the center of perspectivity. is the trilinear pole of the line .
Some trilinear polars
Some of the trilinear polars are well known.
The trilinear polar of the centroid of triangle is the line at infinity.
The trilinear polar of the symmedian point is the Lemoine axis of triangle .
The trilinear polar of the orthocenter is the orthic axis.
Trilinear polars are not defined for points coinciding with the vertices of triangle .
Poles of pencils of lines
Let with trilinear coordinates be the pole of a line passing through a fixed point with trilinear coordinates . Equation of the line is
Since this passes through ,
Thus the locus of is
This is a circumconic of the triangle of reference . Thus the locus of the poles of a pencil of lines passing through a fixed point is a circumconic of the triangle of reference.
It can be shown that is the perspector of , namely, where and the polar triangle with respect to are perspective. The polar triangle is bounded by the tangents to at the vertices of . For example, the Trilinear polar of a point on the circumcircle must pass through its perspector, the Symmedian point X(6).
References
External links
Geometrikon page : Trilinear |
https://en.wikipedia.org/wiki/Amjad%20Shokouh%20Magham | Amjad Shokouh Maghahm () is an Iranian football midfielder.
Career
Shokouh Magham joined Aluminium in summer 2012.
Club career statistics
References
External links
Amjad Shokouh Magham at PersianLeague.com
People from Sanandaj
Iranian men's footballers
Men's association football midfielders
Saba Qom F.C. players
F.C. Aboomoslem players
Shahin Bushehr F.C. players
Sanat Naft Abadan F.C. players
Aluminium Hormozgan F.C. players
Shahr Khodro F.C. players
1983 births
Living people |
https://en.wikipedia.org/wiki/Mih%C3%A1ly%20Nagy%20%28footballer%2C%20born%201992%29 | Krisztián Nagy (born 20 June 1992) is a Hungarian football player who plays for Kozármisleny.
Club career
On 16 June 2021, Nagy signed with Kisvárda.
Career statistics
.
References
External links
HLSZ
1992 births
Living people
Sportspeople from Baranya County
Hungarian men's footballers
Men's association football forwards
Budapest Honvéd FC players
Kazincbarcikai SC footballers
Kozármisleny SE footballers
Szentlőrinci SE footballers
FC Ajka players
Kisvárda FC players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Tarabai%20%28footballer%29 | Edison Luiz dos Santos (born 9 December 1985 in Osasco), also known as Tarabai, is a Brazilian football player who plays as a forward.
Career statistics
Note: "Other" includes Maltese First Division title decider (2012–13), Maltese Super Cup (2013–14), and K League Challenge promotion play-offs (2015).
References
External links
1985 births
Living people
Footballers from Osasco
Brazilian men's footballers
Brazilian expatriate men's footballers
Men's association football forwards
Rio Preto Esporte Clube players
Vittoriosa Stars F.C. players
Hibernians F.C. players
Kecskeméti TE players
Seoul E-Land FC players
K League 2 players
Al Batin FC players
Al Raed FC players
Al-Shoulla FC players
Birkirkara F.C. players
Saudi Pro League players
Saudi First Division League players
Maltese Premier League players
Nemzeti Bajnokság I players
Expatriate men's footballers in Malta
Expatriate men's footballers in Hungary
Expatriate men's footballers in South Korea
Expatriate men's footballers in Saudi Arabia
Brazilian expatriate sportspeople in Malta
Brazilian expatriate sportspeople in Hungary
Brazilian expatriate sportspeople in South Korea
Brazilian expatriate sportspeople in Saudi Arabia |
https://en.wikipedia.org/wiki/Vahid%20Asgari | Vahid Asgari (; born 20 March 1985) is an Iranian football defender.
Career
Asgari is a part of Aboomoslem squad since 2006.
Club Career Statistics
References
External links
Vahid Asgari at PersianLeague.com
Footballers from Mashhad
Iranian men's footballers
Men's association football defenders
F.C. Aboomoslem players
Aluminium Hormozgan F.C. players
Shahr Khodro F.C. players
1985 births
Living people |
https://en.wikipedia.org/wiki/Bunce%E2%80%93Deddens%20algebra | In mathematics, a Bunce–Deddens algebra, named after John W. Bunce and James A. Deddens, is a certain type of AT algebra, a direct limit of matrix algebras over the continuous functions on the circle, in which the connecting maps are given by embeddings between families of shift operators with periodic weights.
Each inductive system defining a Bunce–Deddens algebra is associated with a supernatural number, which is a complete invariant for these algebras. In the language of K-theory, the supernatural number correspond to the group of the algebra. Also, Bunce–Deddens algebras can be expressed as the -crossed product of the Cantor set with a certain natural minimal action known as an odometer action. They also admit a unique tracial state. Together with the fact that they are AT, this implies they have real rank zero.
In a broader context of the classification program for simple separable nuclear C*-algebras, AT-algebras of real rank zero were shown to be completely classified by their K-theory, the Choquet simplex of tracial states, and the natural pairing between and traces. The classification of Bunce–Deddens algebras is thus a precursor to the general result.
It is also known that, in general, crossed products arising from minimal homeomorphism on the Cantor set are simple AT-algebras of real rank zero.
Definition and basic properties
Definition
Let denote continuous functions on the circle and be the -algebra of matrices with entries in . For a supernatural number , the corresponding Bunce–Deddens algebra is the direct limit:
One needs to define the embeddings
These imbedding maps arise from the natural embeddings between -algebras generated by shifts with periodic weights. For integers and , we define an embedding as follows. On a separable Hilbert space , consider the -algebra generated by weighted shifts of fixed period with respect to a fixed basis. embeds into in the obvious way; any -periodic weighted shift is also a -periodic weighted shift. is isomorphic to , where ) denotes the Toeplitz algebra. Therefore, contains the compact operators as an ideal, and modulo this ideal it is . Because the map from into preserves the compact operators, it descends into an embedding . It is this embedding that is used in the definition of Bunce–Deddens algebras.
The connecting maps
The 's can be computed more explicitly and we now sketch this computation. This will be useful in obtaining an alternative characterization description of the Bunce–Deddens algebras, and also the classification of these algebras.
The -algebra is in fact singly generated. A particular generator of is the weighted shift of period with periodic weights . In the appropriate basis of , is represented by the operator matrix
where is the unilateral shift. A direct calculation using functional calculus shows that the -algebra generated by is , where denotes the Toeplitz algebra, the -algebra generated by the unilateral shift. Since it is |
https://en.wikipedia.org/wiki/Douady%E2%80%93Earle%20extension | In mathematics, the Douady–Earle extension, named after Adrien Douady and Clifford Earle, is a way of extending homeomorphisms of the unit circle in the complex plane to homeomorphisms of the closed unit disk, such that the extension is a diffeomorphism of the open disk. The extension is analytic on the open disk. The extension has an important equivariance property: if the homeomorphism is composed on either side with a Möbius transformation preserving the unit circle the extension is also obtained by composition with the same Möbius transformation. If the homeomorphism is quasisymmetric, the diffeomorphism is quasiconformal. An extension for quasisymmetric homeomorphisms had previously been given by Lars Ahlfors and Arne Beurling; a different equivariant construction had been given in 1985 by Pekka Tukia. Equivariant extensions have important applications in Teichmüller theory; for example, they lead to a quick proof of the contractibility of the Teichmüller space of a Fuchsian group.
Definition
By the Radó–Kneser–Choquet theorem, the Poisson integral
of a homeomorphism f of the circle defines a harmonic diffeomorphism of the unit disk extending f. If f is quasisymmetric, the extension is not necessarily quasiconformal, i.e. the complex dilatation
does not necessarily satisfy
However F can be used to define another analytic extension Hf of f−1 which does satisfy this condition. It follows that
is the required extension.
For |a| < 1 define the Möbius transformation
It preserves the unit circle and unit disk sending a to 0.
If g is any Möbius transformation preserving the unit circle and disk, then
For |a| < 1 define
to be the unique w with |w| < 1 and
For |a| =1 set
Properties
Compatibility with Möbius transformations. By construction
for any Möbius transformations g and h preserving the unit circle and disk.
Functional equation. If |a|, |b| < 1 and
then
Continuity. If |a|, |b| < 1, define
If zn and wn lie in the unit disk and tend to z and w and homeomorphisms of the circle are defined by
then fn tends almost everywhere to
gz ∘ f ∘ g−w if |z|, |w| < 1;
gz ∘ f (w) if |z| < 1 and |w| = 1;
−z if |z| = 1 and |w| ≤ 1 with w ≠ f−1(z).
By the dominated convergence theorem, it follows that Φ(zn,wn) has a non-zero limit if w ≠ Hf(z). This implies that Hf is continuous on the closed unit disk. Indeed otherwise, by compactness, there would be a sequence zn tending to z in the closed disk, with wn = Hf(zn) tending to a limit w ≠ Hf(z). But then Φ(zn,wn) = 0 so has limit zero, a contradiction, since w ≠ Hf(z).
Smoothness and non-vanishing Jacobian on open disk. Hf is smooth with nowhere vanishing Jacobian on |z| < 1. In fact, because of the compatibility with Möbius transformations, it suffices to check that Hf is smooth near 0 and has non-vanishing derivative at 0.
If f has Fourier series
then the derivatives of Ff at 0 are given by
Thus the Jacobian of Ff at 0 is given by
Since Ff is an orientation-preserving diffeomorphism |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20AC%20Ajaccio%20season | The 2012–13 season was AC Ajaccio's 95th season.
Transfers
In
Out
Current squad and statistics
|}
Friendly matches
Competitions
Ligue 2
League table
Results summary
Results by round
Matches
Coupe de France
Coupe de la Ligue
Statistics
Top scorers
References
2012–13 season
Ajaccio |
https://en.wikipedia.org/wiki/M/M/%E2%88%9E%20queue | In queueing theory, a discipline within the mathematical theory of probability, the M/M/∞ queue is a multi-server queueing model where every arrival experiences immediate service and does not wait. In Kendall's notation it describes a system where arrivals are governed by a Poisson process, there are infinitely many servers, so jobs do not need to wait for a server. Each job has an exponentially distributed service time. It is a limit of the M/M/c queue model where the number of servers c becomes very large.
The model can be used to model bound lazy deletion performance.
Model definition
An M/M/∞ queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers currently being served. Since, the number of servers in parallel is infinite, there is no queue and the number of customers in the systems coincides with the number of customers being served at any moment.
Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
Service times have an exponential distribution with parameter μ and there are always sufficient servers such that every arriving job is served immediately. Transitions from state i to i − 1 are at rate iμ
The model has transition rate matrix
The state space diagram for this chain is as below.
Transient solution
Assuming the system starts in state 0 at time 0, then the probability the system is in state j at time t can be written as
from which the mean queue length at time t can be computed (writing N(t) for the number of customers in the system at time t given the system is empty at time zero)
Response time
The response time for each arriving job is a single exponential distribution with parameter μ. The average response time is therefore 1/μ.
Maximum number of customers in the system
Given the system is in equilibrium at time 0, we can compute the cumulative distribution function of the process maximum over a finite time horizon T in terms of Charlier polynomials.
Congestion period
The congestion period is the length of time the process spends above a fixed level c, starting timing from the instant the process transitions to state c + 1. This period has mean value
and the Laplace transform can be expressed in terms of Kummer's function.
Stationary analysis
The stationary probability mass function is a Poisson distribution
so the mean number of jobs in the system is λ/μ.
The stationary distribution of the M/G/∞ queue is the same as that of the M/M/∞ queue.
Heavy traffic
Writing Nt for the number of customers in the system at time t as ρ → ∞ the scaled process
converges to an Ornstein–Uhlenbeck process with normal distribution and correlation parameter 1, defined by the Itō calculus as
where W is a standard Brownian motion.
References
Single queueing nodes |
https://en.wikipedia.org/wiki/G-expectation | In probability theory, the g-expectation is a nonlinear expectation based on a backwards stochastic differential equation (BSDE) originally developed by Shige Peng.
Definition
Given a probability space with is a (d-dimensional) Wiener process (on that space). Given the filtration generated by , i.e. , let be measurable. Consider the BSDE given by:
Then the g-expectation for is given by . Note that if is an m-dimensional vector, then (for each time ) is an m-dimensional vector and is an matrix.
In fact the conditional expectation is given by and much like the formal definition for conditional expectation it follows that for any (and the function is the indicator function).
Existence and uniqueness
Let satisfy:
is an -adapted process for every
the L2 space (where is a norm in )
is Lipschitz continuous in , i.e. for every and it follows that for some constant
Then for any random variable there exists a unique pair of -adapted processes which satisfy the stochastic differential equation.
In particular, if additionally satisfies:
is continuous in time ()
for all
then for the terminal random variable it follows that the solution processes are square integrable. Therefore is square integrable for all times .
See also
Expected value
Choquet expectation
Risk measure – almost any time consistent convex risk measure can be written as
References
Wiener process |
https://en.wikipedia.org/wiki/Nonlinear%20expectation | In probability theory, a nonlinear expectation is a nonlinear generalization of the expectation. Nonlinear expectations are useful in utility theory as they more closely match human behavior than traditional expectations. The common use of nonlinear expectations is in assessing risks under uncertainty. Generally, nonlinear expectations are categorized into sub-linear and super-linear expectations dependent on the additive properties of the given sets. Much of the study of nonlinear expectation is attributed to work of mathematicians within the past two decades.
Definition
A functional (where is a vector lattice on a probability space) is a nonlinear expectation if it satisfies:
Monotonicity: if such that then
Preserving of constants: if then
The complete consideration of the given set, the linear space for the functions given that set, and the nonlinear expectation value is called the nonlinear expectation space.
Often other properties are also desirable, for instance convexity, subadditivity, positive homogeneity, and translative of constants. For a nonlinear expectation to be further classified as a sublinear expectation, the following two conditions must also be met:
Subadditivity: for then
Positive homogeneity: for then
For a nonlinear expectation to instead be classified as a superlinear expectation, the subadditivity condition above is instead replaced by the condition:
Superadditivity: for then
Examples
Choquet expectation: a subadditive or superadditive integral that is used in image processing and behavioral decision theory.
g-expectation via nonlinear BSDE's: frequently used to model financial drift uncertainty.
If is a risk measure then defines a nonlinear expectation.
Markov Chains: for the prediction of events undergoing model uncertainties.
References
Expected utility |
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