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https://en.wikipedia.org/wiki/2007%20Yokohama%20FC%20season	2007 Yokohama FC season Competitions Domestic results J.League 1 Emperor's Cup J.League Cup Player statistics Other pages J. League official site Yokohama FC Yokohama FC seasons
https://en.wikipedia.org/wiki/2007%20Ventforet%20Kofu%20season	2007 Ventforet Kofu season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Ventforet Kofu Ventforet Kofu seasons
https://en.wikipedia.org/wiki/2007%20Albirex%20Niigata%20season	2007 Albirex Niigata season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Notes Other pages J. League official site Albirex Niigata Albirex Niigata seasons
https://en.wikipedia.org/wiki/2007%20J%C3%BAbilo%20Iwata%20season	2007 Júbilo Iwata season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Jubilo Iwata Júbilo Iwata seasons
https://en.wikipedia.org/wiki/2007%20Kyoto%20Sanga%20FC%20season	2007 Kyoto Sanga F.C. season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Kyoto Sanga F.C. Kyoto Sanga FC seasons
https://en.wikipedia.org/wiki/2007%20Gamba%20Osaka%20season	2007 Gamba Osaka season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Gamba Osaka Gamba Osaka seasons
https://en.wikipedia.org/wiki/2007%20Cerezo%20Osaka%20season	2007 Cerezo Osaka season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Cerezo Osaka Cerezo Osaka seasons
https://en.wikipedia.org/wiki/2007%20Vissel%20Kobe%20season	2007 Vissel Kobe season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Vissel Kobe Vissel Kobe seasons
https://en.wikipedia.org/wiki/2007%20Sanfrecce%20Hiroshima%20season	2007 Sanfrecce Hiroshima season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Sanfrecce Hiroshima Sanfrecce Hiroshima seasons
https://en.wikipedia.org/wiki/2007%20Tokushima%20Vortis%20season	2007 Tokushima Vortis season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Tokushima Vortis Tokushima Vortis seasons
https://en.wikipedia.org/wiki/2007%20Ehime%20FC%20season	2007 Ehime FC season Competitions Domestic results J.League 2 Emperor's Cup Player statistics Other pages J. League official site Ehime FC Ehime FC seasons
https://en.wikipedia.org/wiki/2007%20Avispa%20Fukuoka%20season	2007 Avispa Fukuoka season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Avispa Fukuoka Avispa Fukuoka seasons
https://en.wikipedia.org/wiki/2007%20Sagan%20Tosu%20season	2007 Sagan Tosu season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Sagan Tosu Sagan Tosu seasons
https://en.wikipedia.org/wiki/2007%20Oita%20Trinita%20season	2007 Oita Trinita season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Oita Trinita Oita Trinita seasons
https://en.wikipedia.org/wiki/Snub%2024-cell%20honeycomb	In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation (or honeycomb) by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra. It can be seen as an alternation of a truncated 24-cell honeycomb, and can be represented by Schläfli symbol s{3,4,3,3}, s{31,1,1,1}, and 3 other snub constructions. It is defined by an irregular decachoron vertex figure (10-celled 4-polytope), faceted by four snub 24-cells, one 16-cell, and five 5-cells. The vertex figure can be seen topologically as a modified tetrahedral prism, where one of the tetrahedra is subdivided at mid-edges into a central octahedron and four corner tetrahedra. Then the four side-facets of the prism, the triangular prisms become tridiminished icosahedra. Symmetry constructions There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored snub 24-cell, 16-cell, and 5-cell facets. In all cases, four snub 24-cells, five 5-cells, and one 16-cell meet at each vertex, but the vertex figures have different symmetry generators. See also Regular and uniform honeycombs in 4-space: Tesseractic honeycomb 16-cell honeycomb 24-cell honeycomb Truncated 24-cell honeycomb 5-cell honeycomb Truncated 5-cell honeycomb Omnitruncated 5-cell honeycomb References T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs 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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 133 , o4s3s3s4o, s3s3s b3s4o, s3s3s b3s *b3s, o3o3o4s3s, s3s3s4o3o - sadit - O133 5-polytopes Honeycombs (geometry)
https://en.wikipedia.org/wiki/Runcinated%206-cubes	In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube. There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-orthoplex. Runcinated 6-cube Alternate names Small prismated hexeract (spox) (Jonathan Bowers) Images Biruncinated 6-cube Alternate names Small biprismated hexeractihexacontatetrapeton (sobpoxog) (Jonathan Bowers) Images Runcitruncated 6-cube Alternate names Prismatotruncated hexeract (potax) (Jonathan Bowers) Images Biruncitruncated 6-cube Alternate names Biprismatotruncated hexeract (boprag) (Jonathan Bowers) Images Runcicantellated 6-cube Alternate names Prismatorhombated hexeract (prox) (Jonathan Bowers) Images Runcicantitruncated 6-cube Alternate names Great prismated hexeract (gippox) (Jonathan Bowers) Images Biruncitruncated 6-cube Alternate names Biprismatotruncated hexeract (boprag) (Jonathan Bowers) Images Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - boprag External links Polytopes of Various Dimensions Multi-dimensional Glossary 6-polytopes
https://en.wikipedia.org/wiki/Runcinated%206-orthoplexes	In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-orthoplex. There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-cube. Runcinated 6-orthoplex Alternate names Small prismatohexacontatetrapeton (spog) (Jonathan Bowers) Images Runcicantellated 6-orthoplex Alternate names Prismatorhombated hexacontatetrapeton (prog) (Jonathan Bowers) Images Runcitruncated 6-orthoplex Alternate names Prismatotruncated hexacontatetrapeton (potag) (Jonathan Bowers) Images Biruncicantellated 6-cube Alternate names Great biprismated hexeractihexacontatetrapeton (gobpoxog) (Jonathan Bowers) Images Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. x3o3o3x3o4o - spog, x3o3x3x3o4o - prog, x3x3o3x3o4o - potag, o3x3x3x3x4o - gobpoxog External links Polytopes of Various Dimensions Multi-dimensional Glossary 6-polytopes
https://en.wikipedia.org/wiki/Alexei%20Kostrikin	Alexei Ivanovich Kostrikin () (12 February 1929 – 22 September 2000) was a Russian mathematician, specializing in algebra and algebraic geometry. Life Kostrikin graduated from the Faculty of Mechanics and Mathematics at Moscow State University in 1952. In 1960, he earned a Doctor of Sciences degree under Igor Shafarevich at the Steklov Institute of Mathematics with a thesis on the Burnside problem. He became a faculty member at Moscow State University in 1963 and became a professor at the same university in 1976. In 1998, he became Honoured Professor of Moscow State University. Prizes Kostrikin was awarded the USSR State Prize in 1968 for his research on finite groups and Lie algebras and was elected the corresponding member of the USSR Academy of Sciences in 1976. Books Alexei Kostrikin published many scientific articles, books and textbooks, including a university textbook about algebra Introduction to algebra, translated into English and other languages. Selected publications Around Burnside, Springer Verlag 1990 2012 pbk reprint with Pham Huu Tiep: Orthogonal Decompositions and Integral Lattices, de Gruyter 1994 with Yuri Manin: Linear algebra and geometry, Gordon and Breach 1989; 1997 pbk edition Introduction to Algebra, Springer Verlag 1982 (Russian original 1977) Exercises in algebra: a collection of exercises in algebra, linear algebra and geometry, Gordon and Breach 1996 Further reading V. A. Artamonov, Yu. A .Bahturin, I. A. Chubarov, et al., Dedication: Alexei Ivanovich Kostrikin, Comm. Algebra 29 (9) (2001), ix–xiv. Preface: On the 70th birthday of Alexei Ivanovich Kostrikin, J. Math. Sci. (New York) 93 (6) (1999), 801–808. E. B. Vinberg, E. S. Golod, E. I. Zelmanov, et al., Aleksei Ivanovich Kostrikin [1929–2000] (Russian), Uspekhi Mat. Nauk 56 3(339) (2001), 143–145. E. B. Vinberg, E. S. Golod, E. I. Zelmanov, et al., Aleksei Ivanovich Kostrikin [1929–2000], Russian Math. Surveys 56 (3) (2001), 559–561. References External links 1929 births 2000 deaths 20th-century Russian mathematicians Soviet mathematicians Algebraists Algebraic geometers Corresponding Members of the Russian Academy of Sciences Moscow State University alumni Recipients of the USSR State Prize
https://en.wikipedia.org/wiki/Plus%E2%80%93minus%20%28disambiguation%29	Plus–minus is a sports statistic used to measure a player's impact on the game. Plus-minus, ±, or variants may also refer to: Plus–minus sign (±), a symbol used in mathematics and related fields Plus minus method, a geophysical method to interpret seismic refraction profiles Music Plus-Minus (Stockhausen), a 1963 composition by Karlheinz Stockhausen +/- (band), an American indietronic band formed 2001 +/- (Buke and Gase EP) (2008) +- Singles 1978-80, a 2010 Joy Division compilation album + -, a 2015 album by Mew See also Plus and minus signs, mathematical symbols Radical 33, a Chinese radical with the symbol "士" 士 (disambiguation), other uses of the symbol "士"
https://en.wikipedia.org/wiki/Winnipeg%20Blue%20Bombers%20all-time%20records%20and%20statistics	The following is a list of Winnipeg Blue Bombers all-time records and statistics current to the 2023 CFL season. Each category lists the top five players, where known, except for when the fifth place player is tied in which case all players with the same number are listed. Tenure Most Games Played 394 – Bob Cameron (1980–2002) 293 – Troy Westwood (1991–2007, 2009) 249 – Chris Walby (1981–1996) 220 – Stan Mikawos (1982–1996) 199 – Milt Stegall (1995–2008) Most Seasons Played 23 – Bob Cameron (1980–2002) 18 – Troy Westwood (1991–2007, 2009) 16 – Bill Ceretti (1933–1949) 16 – Chris Walby (1981–1996) 15 – Roger Savoie (1951–1965) 15 – Stan Mikawos (1982–1996) Most Consecutive Games Played 353 – Bob Cameron (1980–2000) 279 – Troy Westwood (1991–2007) Scoring Most Points – Career 2,748 – Troy Westwood (1991–2007, 2009) 1,840 – Trevor Kennerd (1980–1991) 890 – Milt Stegall (1995–2008) 802 – Justin Medlock (2016–2019) 673 – Bernie Ruoff (1975–1979) Most Points – Season 227 – Justin Medlock – 2016 226 – Justin Medlock – 2017 213 – Troy Westwood – 1994 209 – Troy Westwood – 1993 202 – Troy Westwood – 2002 Most Points – Game 36 – Bob McNamara – versus BC Lions, October 13, 1956 30 – Ernie Pitts – versus Saskatchewan Roughriders, August 29, 1959 24 – Seven players, nine times Most Touchdowns – Career 147 – Milt Stegall (1995–2008) 79 – Charles Roberts (2001–2008) 75 – Leo Lewis (1955–1966) 63 – Gerry James (1952–1963) 62 – James Murphy (1983–1990) Most Touchdowns – Season 23 – Milt Stegall – 2002 19 – Gerry James – 2002 17 – Milt Stegall – 2005 16 – Ernie Pitts – 1959 16 – Ronald Williams – 1997 16 – Charles Roberts – 2007 16 – Dalton Schoen – 2022 Most Touchdowns – Game 6 – Bob McNamara – versus BC Lions, October 13, 1956 5 – Ernie Pitts – versus Saskatchewan Roughriders, August 29, 1959 4 – Seven players, nine times Most Rushing Touchdowns – Career 64 – Charles Roberts (2001–2008) 57 – Gerry James (1952–1963) 48 – Leo Lewis (1955–1966) 44 – Willard Reaves (1983–1987) 30 – Jim Washington (1974–1979) Most Rushing Touchdowns – Season 18 – Gerry James – 1957 16 – Ronald Williams – 1997 16 – Charles Roberts – 2007 15 – Robert Mimbs – 1991 14 – Willard Reaves – 1984 Most Rushing Touchdowns – Game 4 – Bob McNamara – at BC Lions, October 13, 1956 4 – Willard Reaves – versus Hamilton Tiger-Cats, September 15, 1984 4 – Tim Jessie – at Ottawa Rough Riders, July 25, 1989 4 – Charles Roberts – versus BC Lions, September 8, 2002 4 – Charles Roberts – at Edmonton Eskimos, June 28, 2007 Most Receiving Touchdowns – Career 144 – Milt Stegall (1995–2008) 61 – James Murphy (1983–1990) 54 – Ernie Pitts (1957–1969) 48 – Joe Poplawski (1978–1986) 46 – Jeff Boyd (1983–1987) 46 – Rick House (1979–1991) 46 – Terrence Edwards (2007–2013) Most Receiving Touchdowns – Season 23 – Milt Stegall – 2002 18 – Gerald Alphin – 1994 17 – Milt Stegall – 2005 16 – Ernie Pitts – 1959 16 – Dalton Schoen – 2022 Most Receiving Touchdowns – Game 5 – Ernie Pitts – versus Saskatchewa
https://en.wikipedia.org/wiki/Internal%20category	In mathematics, more specifically in category theory, internal categories are a generalisation of the notion of small category, and are defined with respect to a fixed ambient category. If the ambient category is taken to be the category of sets then one recovers the theory of small categories. In general, internal categories consist of a pair of objects in the ambient category—thought of as the 'object of objects' and 'object of morphisms'—together with a collection of morphisms in the ambient category satisfying certain identities. Group objects, are common examples of internal categories. There are notions of internal functors and natural transformations that make the collection of internal categories in a fixed category into a 2-category. Definitions Let be a category with pullbacks. An internal category in consists of the following data: two -objects named "object of objects" and "object of morphisms" respectively and four -arrows subject to coherence conditions expressing the axioms of category theory. See . See also Enriched category References Category theory
https://en.wikipedia.org/wiki/Diego%20Alberto%20Morales	Diego Alberto Morales (born November 29, 1986 in Villa María, Córdoba) is an Argentine football striker playing for Cantolao. References External links Argentine Primera statistics at Fútbol XXI 1986 births Living people Argentine men's footballers Men's association football forwards People from Villa María Footballers from Córdoba Province, Argentina Argentine Primera División players Peruvian Primera División players Campeonato Brasileiro Série A players Saudi Pro League players Ecuadorian Serie A players Chacarita Juniors footballers Club Atlético Tigre footballers Al-Ahli Saudi FC players Clube Náutico Capibaribe players L.D.U. Quito footballers Club Atlético Colón footballers Academia Deportiva Cantolao players Argentine expatriate men's footballers Expatriate men's footballers in Saudi Arabia Expatriate men's footballers in Brazil Expatriate men's footballers in Ecuador Expatriate men's footballers in Peru Argentine expatriate sportspeople in Saudi Arabia Argentine expatriate sportspeople in Brazil Argentine expatriate sportspeople in Ecuador Argentine expatriate sportspeople in Peru
https://en.wikipedia.org/wiki/Garnir%20relations	In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules Vλ in terms of standard polytabloids. Specht modules in terms of polytabloids Given a partition λ of n, one has the Specht module Vλ. In characteristic 0, this is an irreducible representation of the symmetric group Sn. One can construct Vλ explicitly in terms of polytabloids as follows: Start with the permutation representation of Sn acting on all Young tableaux of shape λ, which are fillings of the Young diagram of λ with numbers 1, 2, ... n, each used once (note that we do not require the tableaux to be standard, there are no conditions imposed along rows or columns). The group Sn acts by permuting the positions in each tableau (for instance there is a cyclic permutation the cycles the entries of the first row one place forward). A Young tabloid is an orbit of Young tableaux under the action of the row permutations, the subgroup of Sn of permutations that permute the positions in each row separately (this "Young subgroup" is a product of symmetric groups, one for each row). The Young tabloid of T is denoted {T}. Now consider the free Abelian group of polytabloids, the formal linear combinations with integer coefficients of Young tabloids. To any Young tableau T one associates a polytabloid eT as follows. One first forms the orbit of T under the action of the group of column permutations (another Young subgroup, defined similarly to row permutations but permuting positions within individual columns only). Then, writing the result of action on a tableau T by a column permutation σ as Tσ, defines: does not imply , since the actions of row and column permutations do not commute in general. The Specht module Vλ is then the subspace of the space of all polytabloids spanned by the polytabloids eT for all Young tableaux T of shape λ. Straightening polytabloids and the Garnir elements The above construction gives an explicit description of the Specht module Vλ. However, the polytabloids associated to different Young tableaux are not necessarily linearly independent, indeed, the dimension of Vλ is exactly the number of standard Young tableaux of shape λ. In fact, the polytabloids associated to standard Young tableaux span Vλ; to express other polytabloids in terms of them, one uses a straightening algorithm. Given a Young tableau S, we construct the polytabloid eS as above. Without loss of generality, all columns of S are increasing, otherwise we could instead start with the modified Young tableau with increasing columns, whose polytabloid will differ at most by a sign. S is then said to not have any column descents. We want to express eS as a linear combination of standard polytabloids, i.e. polytabloids associated to standard Young tableaux. To do this, we would like permutations πi such that in all tableaux Sπi, a row descent has been eliminated, with . This then expresses S in terms of polytabloids that are closer to being standard. The permutations
https://en.wikipedia.org/wiki/E9%20honeycomb	{{DISPLAYTITLE:E9 honeycomb}} In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded. E10 is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram is a nonlinear graph, but there are three simplest ones, with a single ring at the end of its 3 branches: 621, 261, 162. 621 honeycomb The 621 honeycomb is constructed from alternating 9-simplex and 9-orthoplex facets within the symmetry of the E10 Coxeter group. This honeycomb is highly regular in the sense that its symmetry group (the affine E9 Weyl group) acts transitively on the k-faces for k ≤ 7. All of the k-faces for k ≤ 8 are simplices. This honeycomb is last in the series of k21 polytopes, enumerated by Thorold Gosset in 1900, listing polytopes and honeycombs constructed entirely of regular facets, although his list ended with the 8-dimensional the Euclidean honeycomb, 521. Construction It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 9-orthoplex, 711. Removing the node on the end of the 1-length branch leaves the 9-simplex. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 521 honeycomb. The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 421 polytope. The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 321 polytope. The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 221 polytope. Related polytopes and honeycombs The 621 is last in a dimensional series of semiregular polytopes and honeycombs, identified in 1900 by Thorold Gosset. Each member of the sequence has the previous member as its vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes. 261 honeycomb The 261 honeycomb is composed of 251 9-honeycomb and 9-simplex facets. It is the final figure in the 2k1 family. Construction It is created by a Wythoff construction upon a set of 10 hyperplane mirrors in 9-dimensional hyperbolic space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the short branch leaves the 9-simplex. Removing the node on the end of the 6-length branch leaves the 251 honeycomb. This is an infinite facet because E10 is a paracompact hyperbolic group. The vertex
https://en.wikipedia.org/wiki/Kurt%20Leichtweiss	Kurt Leichtweiß (March 2, 1927 in Villingen-Schwenningen – June 23, 2013) was a mathematician specializing in convex and differential geometry. In 1944, while still in high school Leichtweiß traveled to the Oberwolfach Research Institute for Mathematics where his mathematical interests were encouraged. He studied at the University of Freiburg and the ETH Zurich. He was a student of Emanuel Sperner and Wilhelm Süss. He was then a lecturer in Freiburg and in 1963 became a professor at TU Berlin. From 1970 until his retirement in 1995, he was a professor at the University of Stuttgart. Books (with Wilhelm Blaschke), Elementary Differential Geometry, Springer, 1973. Affine geometry of convex bodies, Wiley, 1998. Convex Geometry, Springer, 1980. Analytic Geometry, First Course, Teubner, 1972. References External links Prof. Dr. Kurt Leichtweiss, contact page. 1927 births 2013 deaths People from Villingen-Schwenningen People from the Republic of Baden 20th-century German mathematicians 21st-century German mathematicians Geometers University of Freiburg alumni Academic staff of the University of Freiburg ETH Zurich alumni Academic staff of the Technical University of Berlin
https://en.wikipedia.org/wiki/Rectified%209-cubes	In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube. There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-cube are located in the square face centers of the 9-cube. Vertices of the trirectified 9-orthoplex are located in the cube cell centers of the 9-cube. Vertices of the quadrirectified 9-cube are located in the tesseract centers of the 9-cube. These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry. Rectified 9-cube Alternate names Rectified enneract (Acronym ren) (Jonathan Bowers) Images Birectified 9-cube Alternate names Birectified enneract (Acronym barn) (Jonathan Bowers) Images Trirectified 9-cube Alternate names Trirectified enneract (Acronym tarn) (Jonathan Bowers) Images Quadrirectified 9-cube Alternate names Quadrirectified enneract (Acronym nav) (Jonathan Bowers) Images Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne External links Polytopes of Various Dimensions Multi-dimensional Glossary 9-polytopes
https://en.wikipedia.org/wiki/Ottawa%20Rough%20Riders%20all-time%20records%20and%20statistics	The following is a list of Ottawa Rough Riders all-time records and statistics over their existence from 1876 to 1996. Games played Most Games Played 201 – Moe Racine (1958–74) 186 – Gerry Organ (1971–77, 79–83) 169 – Bob Simpson (1950–62) 167 – Ron Stewart (1958–70) 166 – Russ Jackson (1958–69) Most Seasons Played 22 – Eddie Emerson (1912–15, 19–35, 37) 17 – Moe Racine (1958–74) 14 – Joe Tubman (1913–15, 19–29) 13 – Charlie Connell (1920–32) 13 – Bob Simpson (1950–62) 13 – Ron Stewart (1958–70) Scoring Most points – Career 1462 – Gerry Organ (1971–77, 79–83) 841 – Dean Dorsey (1984–87, 89–90) 772 – Terry Baker (1990–95) 402 – Ron Stewart (1958–70) 392 – Moe Racine (1958–74) Most Points – Season 202 – Terry Baker – 1991 184 – Terry Baker – 1992 178 – Terry Baker – 1994 176 – Dean Dorsey – 1990 148 – Dean Dorsey – 1989 Most Points – Game 24 – Dave Thelen – versus Toronto Argonauts, September 16, 1959 24 – Ron Stewart – at Montreal Alouettes, October 10, 1960 24 – Art Green – versus Hamilton Tiger-Cats, September 7, 1975 24 – Dean Dorsey – versus Saskatchewan Roughriders, September 24, 1989 Most Touchdowns – Career 70 – Bob Simpson (1950–62) 67 – Ron Stewart (1958–70) 59 – Tony Gabriel (1975–81) 55 – Russ Jackson (1958–69) 54 – Whit Tucker (1962–70) Most Touchdowns – Season 18 – Alvin Walker – 1982 16 – Ron Stewart – 1960 15 – Art Green – 1976 14 – Vic Washington – 1969 14 – Art Green – 1975 14 – Tony Gabriel – 1976 Most Touchdowns – Game 4 – Ken Charlton – versus Hamilton Tigers, November 9, 1946 4 – Dave Thelen – versus Toronto Argonauts, September 16, 1959 4 – Ron Stewart – at Montreal Alouettes, October 10, 1960 4 – Art Green – versus Hamilton Tiger-Cats, September 7, 1975 Most Receiving Touchdowns – Career 65 – Bob Simpson (1950–62) 61 – Tony Gabriel (1975–81) 54 – Whit Tucker (1962–70) 34 – Stephen Jones (1990–94) 33 – Hugh Oldham (1970–74) Most Receiving Touchdowns – Season 14 – Tony Gabriel – 1976 13 – Whit Tucker – 1968 13 – Hugh Oldham – 1970 12 – David Williams – 1990 11 – Tony Gabriel – 1978 11 – Stephen Jones – 1990 11 – Jock Climie – 1993 Most Receiving Touchdowns – Game 3 – Many Most Rushing Touchdowns – Career 54 – Russ Jackson (1958–69) 43 – Ron Stewart (1958–70) 39 – Dave Thelen (1958–64) 32 – Art Green (1973–76, 78) 24 – Alvin Walker (1982–84) Most Rushing Touchdowns – Season 15 – Ron Stewart – 1960 13 – Art Green – 1976 13 – Alvin Walker – 1982 11 – Art Green – 1975 10 – Alvin Walker – 1983 10 – Reggie Barnes – 1991 Passing Most Passing Yards – Career 24,593 – Russ Jackson (1958–69) 11,251 – Damon Allen (1989–91) 10,937 – J. C. Watts (1981–86) 10,288 – Tom Burgess (1986, 1992–93) 9663 – Tom Clements (1975–78) Most Passing Yards – Season 5063 – Tom Burgess – 1993 4275 – Damon Allen – 1991 4173 – Danny Barrett – 1994 4026 – Tom Burgess – 1992 3977 – David Archer – 1996 Most Passing Yards – Game 471 – Chris Isaac – versus Montreal Concordes, July 29, 1982 467 – David Archer – versus BC Lions, July 12, 19
https://en.wikipedia.org/wiki/Alex%20Cruz%20%28footballer%2C%20born%201985%29	Alex José da Cruz (born January 11, 1985) simply known as Alex Cruz a Brazilian attacking midfielder. He currently plays for Ferroviária. Career statistics (Correct ) Honours Ivinhema Campeonato Sul-Mato-Grossense: 2008 Flamengo Campeonato Brasileiro Série A: 2009 References External links ogol.com.br Flapédia 1985 births Living people Brazilian men's footballers Campeonato Brasileiro Série A players Campeonato Brasileiro Série D players Americano FC players Esporte Clube Comercial (MS) players Clube Esportivo Naviraiense players Ivinhema Futebol Clube players CR Flamengo footballers Guarani FC players Corumbaense Futebol Clube players Associação Ferroviária de Esportes players Men's association football midfielders Footballers from Mato Grosso do Sul People from Fátima do Sul
https://en.wikipedia.org/wiki/Chiral%20algebra	In mathematics, a chiral algebra is an algebraic structure introduced by as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. They give an 'coordinate independent' notion of vertex algebras, which are based on formal power series. Chiral algebras on curves are essentially conformal vertex algebras. Definition A chiral algebra on a smooth algebraic curve is a right D-module , equipped with a D-module homomorphism on and with an embedding , satisfying the following conditions (Skew-symmetry) (Jacobi identity) The unit map is compatible with the homomorphism ; that is, the following diagram commutes Where, for sheaves on , the sheaf is the sheaf on whose sections are sections of the external tensor product with arbitrary poles on the diagonal: is the canonical bundle, and the 'diagonal extension by delta-functions' is Relation to other algebras Vertex algebra The category of vertex algebras as defined by Borcherds or Kac is equivalent to the category of chiral algebras on equivariant with respect to the group of translations. Factorization algebra Chiral algebras can also be reformulated as factorization algebras. See also Chiral homology Chiral Lie algebra References Further reading Conformal field theory Representation theory
https://en.wikipedia.org/wiki/List%20of%20Stoke%20City%20F.C.%20records%20and%20statistics	Stoke City Football Club is an English professional association football club based in Stoke-on-Trent, Staffordshire. Founded as Stoke Ramblers in 1863 the club changed its name to Stoke in 1878 and then to Stoke City in 1925 after Stoke-on-Trent was granted city status. They are the second oldest professional football club in the world, after Notts County, and are one of the founding members of the Football League. They currently play in the Football League Championship, the second tier of English football. They have never been lower than the third tier. Their first, and to date only major trophy, the League Cup was won in 1972, when the team beat Chelsea 2–1. The club's highest league finish in the top division is 4th, which was achieved in the 1935–36 and 1946–47 seasons. Stoke played in the FA Cup Final in 2011, finishing runners-up to Manchester City and have reached three FA Cup semi-finals; in 1899 then consecutively in 1971 and 1972. Stoke have competed in European football on three occasions, firstly in 1972–73 then in 1974–75 and most recently in 2011–12. The club has won the Football League Trophy twice, in 1992 and in 2000. The club's record appearance maker is Eric Skeels, who made 597 appearances between 1959 and 1976, and the club's record goalscorer is John Ritchie, who scored 176 goals in 351 appearances from 1962 to 1975. Honours League Football League Championship Runners-up: 2007–08 Football League Second Division: 3 Champions: 1932–33, 1962–63, 1992–93 Runners-up: 1921–22 Third Place: (Promoted) 1978–79 Play-off Winners: 2001–02 Football League Third Division North: 1 Champions: 1926–27 Football Alliance: 1 Champions: 1890–91 Birmingham & District League: 1 Champions: 1910–11 Southern League Division Two: 2 Champions:1909–10, 1914–15 Runners-up: 1910–11 Cups FA Cup Runners-up: 2010–11 Semi-finalists: 1898–99, 1970–71 (3rd place), 1971–72 (4th place) League Cup: 1 Winners: 1971–72 Runners-up: 1963–64 Football League Trophy: 2 Winners: 1991–92, 1999–2000 Watney Cup: 1 Winners: 1973 Staffordshire Senior Cup: 15 Winners: 1877–78, 1878–79, 1903–04 (shared), 1913–14, 1933–34, 1964–65, 1968–69 (shared), 1970–71, 1974–75, 1975–76, 1981–82, 1992–93, 1994–95, 1998–99, 2016–17 Runners-up: 1882–83, 1885–86, 1894–95, 1900–01, 1902–03, 2002–03, 2005–06, 2010–11 Birmingham Senior Cup: 2 Winners: 1901, 1914 Runners-up: 1910, 1915, 1920, 1921 Isle of Man Trophy: 3 Winners: 1987, 1991, 1992 Runners-up: 1985 Bass Charity Vase: 5 Winners: 1980, 1991, 1992, 1995, 1998 Runners-up: 1890, 1894, 1990, 1996 Player records Appearances Most appearances in total (League & Cup) – 597 Eric Skeels (1959–76) Most League appearances – 507 Eric Skeels (1959–76) Most appearances in total (Including war-time) – 675 John McCue (1940–60) Most Consecutive Appearances – 148 Tony Allen (1960–63) Youngest Player – Emre Tezgel 16 years, 112 days v Leyton Orient 9 January 2022 Oldest Player – Stanley Matthews 50 years, 5 days v Fulham 6 February
https://en.wikipedia.org/wiki/Cantic%208-cube	In eight-dimensional geometry, a cantic 8-cube or truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube. Alternate names Truncated demiocteract Truncated hemiocteract (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations: (±1,±1,±3,±3,±3,±3,±3,±3) with an odd number of plus signs. Images Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. External links Polytopes of Various Dimensions Multi-dimensional Glossary 8-polytopes
https://en.wikipedia.org/wiki/Rectified%2010-cubes	In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube. There are 10 rectifications of the 10-cube, with the zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are located in the square face centers of the 10-cube. Vertices of the trirectified 10-cube are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the 10-orthoplex. These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry. Rectified 10-cube Alternate names Rectified dekeract (Acronym rade) (Jonathan Bowers) Cartesian coordinates Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,±1,±1,±1,0) Images Birectified 10-cube Alternate names Birectified dekeract (Acronym brade) (Jonathan Bowers) Cartesian coordinates Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,±1,±1,0,0) Images Trirectified 10-cube Alternate names Tririrectified dekeract (Acronym trade) (Jonathan Bowers) Cartesian coordinates Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,±1,0,0,0) Images Quadrirectified 10-cube Alternate names Quadrirectified dekeract Quadrirectified decacross (Acronym terade) (Jonathan Bowers) Cartesian coordinates Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,0,0,0,0) Images Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker External links Polytopes of Various Dimensions Multi-dimensional Glossary 10-polytopes
https://en.wikipedia.org/wiki/Rectified%2010-simplexes	In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex. These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry. There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex. Rectified 10-simplex The rectified 10-simplex is the vertex figure of the 11-demicube. Alternate names Rectified hendecaxennon (Acronym ru) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex. Images Birectified 10-simplex Alternate names Birectified hendecaxennon (Acronym bru) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex. Images Trirectified 10-simplex Alternate names Trirectified hendecaxennon (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex. Images Quadrirectified 10-simplex Alternate names Quadrirectified hendecaxennon (Acronym teru) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex. Images Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) x3o3o3o3o3o3o3o3o3o - ux, o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3
https://en.wikipedia.org/wiki/The%20Mathematics%20Enthusiast	The Mathematics Enthusiast is a triannual peer-reviewed open access academic journal covering undergraduate mathematics, mathematics education, including historical, philosophical, and cross-cultural perspectives on mathematics. It is hosted by ScholarWorks at the University of Montana. The journal was established in 2004 and its founding editor-in-chief is Bharath Sriraman. The journal exists as an independent entity in order to give authors full copyright over their articles, and is not affiliated with any commercial publishing companies. Abstracting and indexing The journal is abstracted and indexed in Academic Search Complete, Emerging Sources Citation Index, PsycINFO, and Scopus. References External links Triannual journals English-language journals Mathematics education journals Open access journals Academic journals established in 2004 University of Montana Academic journals associated with universities and colleges of the United States
https://en.wikipedia.org/wiki/IJB	IJB may refer to: International Journal of Biomathematics International Journal of Biometeorology International Journal of Biosciences
https://en.wikipedia.org/wiki/Mathematics%20%26%20Mechanics%20of%20Solids	Mathematics & Mechanics of Solids is a peer-reviewed academic journal that publishes papers in the fields of Mechanics and Mathematics. The journal's editor is David J Steigmann (University of California). It has been in publication since 1996 and is currently published by SAGE Publications. Scope Mathematics and Mechanics of Solids is an international journal which publishes original research in solid mechanics and materials science. The journal’s aim is to publish original, self-contained research that focuses on the mechanical behaviour of solids with particular emphasis on mathematical principles. Abstracting and indexing Mathematics & Mechanics of Solids is abstracted and indexed in, among other databases: SCOPUS, and the Social Sciences Citation Index. According to the Journal Citation Reports, its 2016 impact factor is 2.953, ranking it 72 out of 275 journals in the category ‘Materials Science, Multidisciplinary’. and 11 out of 100 journals in the category ‘Mathematics, Interdisciplinary Applications’. and 13 out of 133 journals in the category ‘Mechanics’. References External links Mathematics journals 8 times per year journals
https://en.wikipedia.org/wiki/Chang%20number	In mathematics, the Chang number of an irreducible representation of a simple complex Lie algebra is its dimension modulo 1 + h, where h is the Coxeter number. Chang numbers are named after , who rediscovered an element of order h + 1 found by . showed that there is a unique class of regular elements σ of order h + 1, in the complex points of the corresponding Chevalley group. He showed that the trace of σ on an irreducible representation is −1, 0, or +1, and if h + 1 is prime then the trace is congruent to the dimension mod h+1. This implies that the dimension of an irreducible representation is always −1, 0, or +1 mod h + 1 whenever h + 1 is prime. Examples In particular, for the exceptional compact Lie groups G2, F4, E6, E7, and E8 the number h + 1 = 7, 13, 13, 19, 31 is always prime, so the Chang number of an irreducible representation is always +1, 0, or −1. For example, the first few irreducible representations of G2 (with Coxeter number h = 6) have dimensions 1, 7, 14, 27, 64, 77, 182, 189, 273, 286,... These are congruent to 1, 0, 0, −1, 1, 0, 0, 0, 0, −1,... mod 7 = h + 1. References Representation theory
https://en.wikipedia.org/wiki/Runcic%206-cubes	In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube. Runcic 6-cube Alternate names Cantellated 6-demicube/demihexeract Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations: (±1,±1,±1,±3,±3,±3) with an odd number of plus signs. Images Related polytopes Runcicantic 6-cube Alternate names Cantitruncated 6-demicube/demihexeract Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations: (±1,±1,±3,±5,±5,±5) with an odd number of plus signs. Images Related polytopes This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family. There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique: Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. x3o3o b3x3o3o, x3x3o b3x3o3o External links Polytopes of Various Dimensions Multi-dimensional Glossary 6-polytopes
https://en.wikipedia.org/wiki/Steric%206-cubes	In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube. Steric 6-cube Alternate names Runcinated demihexeract/6-demicube Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations: (±1,±1,±1,±1,±1,±3) with an odd number of plus signs. Images Related polytopes Stericantic 6-cube Alternate names Runcitruncated demihexeract/6-demicube Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations: (±1,±1,±1,±3,±3,±5) with an odd number of plus signs. Images Steriruncic 6-cube Alternate names Runcicantellated demihexeract/6-demicube Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations: (±1,±1,±1,±1,±3,±5) with an odd number of plus signs. Images Steriruncicantic 6-cube Alternate names Runcicantitruncated demihexeract/6-demicube Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations: (±1,±1,±1,±3,±5,±7) with an odd number of plus signs. Images Related polytopes There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique: Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. x3o3o b3o3x3o - sophax, x3x3o b3o3x3o - pithax, x3o3o b3x3x3o - prohax, x3x3o b3x3x3o - gophax External links Polytopes of Various Dimensions Multi-dimensional Glossary 6-polytopes
https://en.wikipedia.org/wiki/Canadian%20Australians	Canadian Australians () refers to Australians who identify as being of Canadian descent. It may also refer to Canadian immigrants and expatriates residing in Australia. According to statistics from 2006, there were as many as 21,000 Australians who have Canadian ancestors. Many Canadian Australians have immigrated from mainland Canada, as well as from the United States of America and from the United Kingdom. According to the 2016 Australian Census, there were 43,049 Canadian born Australians in Australia in 2016, which is an increase from 38,871 persons according to the 2011 Australian Census. The number of immigrants stem from both countries being former British colonies and both being parliamentary democracies in the Westminster tradition (Collins, 2015). History The first Canadian Australians were immigrants from both upper (now Ontario) and lower (now Quebec) mainland Canada who came to New South Wales and Tasmania because they were in exile during the 1837-38 Canadian civil war (High Commission of Canada, 2011). There were 154 prisoners that were sent to Australia (High Commission of Canada, 2011). Fifty eight of those were French Canadians that were sent to New South Wales without speaking English(High Commission of Canada, 2011). Of the convicts that settled in New South Wales, all were initially assigned as labourers and eventually allowed to be free settlers. These workers helped to build the foundation of Sydney's infrastructure that we still have today including Parramatta Road, Canada Bay, Exile Bay and a monument in Cabarita Park in Concord (High Commission of Canada, 2011). The second wave of Canadian Australians came in 1851 in search of gold (Museums Victoria, 2017). The Australian Gold Rush saw people from around the world flock to Australia in search of gold, which included hundreds of Canadians (Museums Victoria, 2017). They made an impact in the popular town of Ballarat, where the ‘Canadian Gully’ was the name given to a gully after a Canadian gold miner found success and a large gold nugget was also named ‘The Canadian’ (Museums Victoria, 2017). Canadian Australians helped to develop the foundations of modern-day Australian society in the years during and after the gold rush. In 1854 Samuel McGowan created Australia's first telegraph line which stretched from Melbourne to Williamstown (Museums Victoria, 2017). Around the same time, George and William Chaffe made multiple irrigation schemes throughout the Murray River (Museums Victoria, 2017). This allowed for water to be transported further away than before, which led to greater farming success and eventually the beginning of the dried fruit industry in South Australia and Victoria. Australian-Canadian similar experiences during WWI On the 4th of August in 1914 Great Britain declared war on Germany. Both Canada and Australia's prime ministers accepted that their armies would join the imperial armies due to Australia and Canada being British Dominions at the time. Canad
https://en.wikipedia.org/wiki/Martin%20T.%20Barlow	Martin Thomas Barlow FRS FRSC (born 16 June 1953 in London) is a British mathematician who is professor of mathematics at the University of British Columbia in Canada since 1992. History Barlow is the son of Andrew Dalmahoy Barlow (1916–2006) and his wife Yvonne. He is thus the grandson of Alan Barlow, and his wife Nora (née Darwin), through whom he is a great-great-grandson of Charles Darwin. He is the nephew of Horace Barlow (also FRS and Fellow of Trinity). In 1994 he married Colleen McLaughlin. He was educated Sussex House School, St Paul's School, London, Trinity College, Cambridge (BA 1975, Diploma 1976, ScD 1993); University College of Swansea (PhD). Barlow worked as a research fellow of the University of Liverpool 1978–1980. He was a Fellow of Trinity College, Cambridge, 1979–1992. He worked in the Statistical Laboratory, University of Cambridge 1981–1985 and was a Royal Society University Research Fellow 1985–1992. Work His mathematical interests include probability, Brownian motion and fractal sets. His doctoral students include Steven N. Evans. Recognition He was awarded the Rollo Davidson Prize in 1984. He was elected a Fellow of the Royal Society of Canada in 1998. He was elected a Fellow of the Royal Society in 2005. In 2012 he became a fellow of the American Mathematical Society. In 2018 the Canadian Mathematical Society listed him in their inaugural class of fellows. References 'BARLOW, Prof. Martin Thomas', Who's Who 2011, A & C Black, 2011; online edn, Oxford University Press, Dec 2010 ; online edn, Oct 2010 accessed 21 May 2011 External links Academic homepage 1953 births Fellows of the American Mathematical Society Fellows of the Canadian Mathematical Society Fellows of the Royal Society Fellows of the Royal Society of Canada Fellows of Trinity College, Cambridge Living people People educated at St Paul's School, London Alumni of Trinity College, Cambridge Alumni of Swansea University 20th-century British mathematicians 21st-century British mathematicians Academic staff of the University of British Columbia Faculty of Science Probability theorists Canadian Fellows of the Royal Society People educated at Sussex House School
https://en.wikipedia.org/wiki/Greg%20Hjorth	Greg Hjorth (14 June 1963 – 13 January 2011) was an Australian Professor of Mathematics, chess International Master (1984) and joint (with Ian Rogers) Commonwealth Champion in 1983. He worked in the field of mathematical logic. Chess career Hjorth came second in the 1980 Australian Chess Championship, at the age of 16. He won the Doeberl Cup in Canberra in 1982, 1985 and 1987, and played for Australia in the Chess Olympiads of 1982, 1984 and 1986. According to Chessmetrics, his best single performance was at the 1984 British Chess Championship, where he scored 4/7 against 2551-rated opposition, for a performance rating of 2570. Hjorth retired from most chess in the 1980s. Mathematical career Hjorth earned his PhD in 1993, under the direction of W. Hugh Woodin, with a dissertation entitled On the influence of second uniform indiscernible. He held faculty positions at the University of California, Los Angeles and the University of Melbourne. University Of Melbourne has Greg Hjorth Memorial Prize (biennial scholarship is awarded to the most outstanding postgraduate thesis in mathematics, with preference given to areas of logic, set theory, measure theory or related topics passed by examiners within the previous two calendar years) named after him. Among his most important contributions to set theory was the so-called theory of turbulence, used in the theory of Borel equivalence relations. In 1998, he was an Invited Speaker of the International Congress of Mathematicians in Berlin. Death Hjorth died suddenly of a heart attack in Melbourne, on 13 January 2011. Book G. Hjorth: Classification and Orbit Equivalence Relations, Mathematical Surveys and Monographs, 75, American Mathematical Society, Providence, Rhode Island, 2000. References External links Greg Hjorth's webpage at UCLA Chessmetrics Player Profile: Greg Hjorth IM Greg Hjorth interviewed by FM Grant Szuveges 1963 births 2011 deaths Set theorists Mathematicians from Melbourne Australian chess players Chess International Masters Chess Olympiad competitors University of California, Los Angeles faculty Mathematical logicians
https://en.wikipedia.org/wiki/Stericated%206-cubes	In six-dimensional geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-cube. There are 8 unique sterications for the 6-cube with permutations of truncations, cantellations, and runcinations. Stericated 6-cube Alternate names Small cellated hexeract (Acronym: scox) (Jonathan Bowers) Images Steritruncated 6-cube Alternate names Cellirhombated hexeract (Acronym: catax) (Jonathan Bowers) Images Stericantellated 6-cube Alternate names Cellirhombated hexeract (Acronym: crax) (Jonathan Bowers) Images Stericantitruncated 6-cube Alternate names Celligreatorhombated hexeract (Acronym: cagorx) (Jonathan Bowers) Images Steriruncinated 6-cube Alternate names Celliprismated hexeract (Acronym: copox) (Jonathan Bowers) Images Steriruncitruncated 6-cube Alternate names Celliprismatotruncated hexeract (Acronym: captix) (Jonathan Bowers) Images Steriruncicantellated 6-cube Alternate names Celliprismatorhombated hexeract (Acronym: coprix) (Jonathan Bowers) Images Steriruncicantitruncated 6-cube Alternate names Great cellated hexeract (Acronym: gocax) (Jonathan Bowers) Images Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. External links Polytopes of Various Dimensions Multi-dimensional Glossary 6-polytopes
https://en.wikipedia.org/wiki/Stericated%206-orthoplexes	In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex. There are 16 unique sterications for the 6-orthoplex with permutations of truncations, cantellations, and runcinations. Eight are better represented from the stericated 6-cube. Stericated 6-orthoplex Alternate names Small cellated hexacontatetrapeton (Acronym: scag) (Jonathan Bowers) Images Steritruncated 6-orthoplex Alternate names Cellitruncated hexacontatetrapeton (Acronym: catog) (Jonathan Bowers) Images Stericantellated 6-orthoplex Alternate names Cellirhombated hexacontatetrapeton (Acronym: crag) (Jonathan Bowers) Images Stericantitruncated 6-orthoplex Alternate names Celligreatorhombated hexacontatetrapeton (Acronym: cagorg) (Jonathan Bowers) Images Steriruncinated 6-orthoplex Alternate names Celliprismated hexacontatetrapeton (Acronym: copog) (Jonathan Bowers) Images Steriruncitruncated 6-orthoplex Alternate names Celliprismatotruncated hexacontatetrapeton (Acronym: captog) (Jonathan Bowers) Images Steriruncicantellated 6-orthoplex Alternate names Celliprismatorhombated hexacontatetrapeton (Acronym: coprag) (Jonathan Bowers) Images Steriruncicantitruncated 6-orthoplex Alternate names Great cellated hexacontatetrapeton (Acronym: gocog) (Jonathan Bowers) Images Snub 6-demicube The snub 6-demicube defined as an alternation of the omnitruncated 6-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1,1]+ or [4,(3,3,3,3)+], and constructed from 12 snub 5-demicubes, 64 snub 5-simplexes, 60 snub 24-cell antiprisms, 160 3-s{3,4} duoantiprisms, 240 2-sr{3,3} duoantiprisms, and 11520 irregular 5-simplexes filling the gaps at the deleted vertices. Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-orthoplex or 6-orthoplex. Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. External links Polytopes of Various Dimensions Multi-dimensional Glossary 6-polytopes
https://en.wikipedia.org/wiki/Pentic%206-cubes	In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope. There are 8 pentic forms of the 6-cube. Pentic 6-cube The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, . Alternate names Stericated 6-demicube/demihexeract Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±1,±1,±1,±3) with an odd number of plus signs. Images Penticantic 6-cube The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, . Alternate names Steritruncated 6-demicube/demihexeract cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±3,±3,±3,±5) with an odd number of plus signs. Images Pentiruncic 6-cube The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), . Alternate names Stericantellated 6-demicube/demihexeract cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±1,±3,±3,±5) with an odd number of plus signs. Images Pentiruncicantic 6-cube The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex), Alternate names Stericantitruncated demihexeract, stericantitruncated 7-demicube Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±3,±3,±5,±7) with an odd number of plus signs. Images Pentisteric 6-cube The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex), Alternate names Steriruncinated 6-demicube/demihexeract Small cellipriamated hemihexeract (Acronym: cophix) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±1,±1,±3,±5) with an odd number of plus signs. Images Pentistericantic 6-cube The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), . Alternate names Steriruncitruncated demihexeract/7-demicube cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers) Cartesian coordinates The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations: (±1,±1,±3,±3,±5,±7) with an odd number of plus signs. Images Pentisteriruncic 6-cube The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), . Alternate names Steriruncicantellated 6-demicube/demihexeract Celliprismatorhombated
https://en.wikipedia.org/wiki/Worcester%20County%20Mathematics%20League	The Worcester County Mathematics League (WOCOMAL) is a high school mathematics league composed of 32 high schools, most of which are in Worcester County, Massachusetts. It organizes seven mathematics competitions per year, four at the "varsity" level (up to grade 12) and three at the "freshman" level (up to grade nine, including middle school students). In the 2013–14 school year, WOCOMAL began allowing older students to compete in the freshman level competitions, calling this level of participation "junior varsity." Top schools from the varsity competition are selected to attend the Massachusetts Association of Math Leagues state competition. Contest format A competition consists of four, or nine rounds at the Freshman level or five rounds at the Varsity level. The team round consists of eight problems at the Freshman level and nine at the Varsity level. Regardless of level, each student competes in three of the individual rounds. In each individual round, competing students have ten minutes to answer three questions, worth one, two, and three points. The maximum meet score for a student is eighteen points. History The Worcester County Mathematics League was originally formed in 1963 as the Southern Worcester County Mathematics League (Sowocomal). The winningest school in league history is St. John's High School, with twelve league championships in the fourteen-year span between 1983–84 and 1996–97. Algonquin Regional High School won six consecutive league championships from 1998–99 to 2003–04. Current events The league currently has members from Western Middlesex Counties. In the past, it has had members from Hampshire County, Massachusetts, and Windham County, Connecticut. In the 2015–16 season, the champion of both the varsity division and the freshman division was the Advanced Math and Science Academy Charter School. League members AMSA Charter, Worcester Academy, and Mass Academy and took first, second, and third place among small-sized schools at the 2016 Massachusetts state championship math meet. St. John's High School took fifth place among medium-sized schools. WOCOMAL schools have taken first place in the state among small schools for thirteen consecutive years. At the 2016 New England championship math meet, league members AMSA Charter, Worcester Academy, and Mass Academy took first, second, and fifth place respectively among small schools. WOCOMAL schools have taken first or second place in New England among small schools for twelve consecutive years. External links WOCOMAL website 2016-17 varsity team rankings All-time varsity rankings 2016-17 freshman team rankings 1967-68 league bylaws 2008-09 league bylaws 2001-2011 Varsity meet problems 2012-2014 Varsity and Freshman meet problems and solutions 2001-2011 Freshman meet problems References Mathematics competitions Education in Worcester County, Massachusetts
https://en.wikipedia.org/wiki/Lindstr%C3%B6m%E2%80%93Gessel%E2%80%93Viennot%20lemma	In Mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the number of tuples of non-intersecting lattice paths, or, more generally, paths on a directed graph. It was proved by Gessel–Viennot in 1985, based on previous work of Lindström published in 1973. Statement Let G be a locally finite directed acyclic graph. This means that each vertex has finite degree, and that G contains no directed cycles. Consider base vertices and destination vertices , and also assign a weight to each directed edge e. These edge weights are assumed to belong to some commutative ring. For each directed path P between two vertices, let be the product of the weights of the edges of the path. For any two vertices a and b, write e(a,b) for the sum over all paths from a to b. This is well-defined if between any two points there are only finitely many paths; but even in the general case, this can be well-defined under some circumstances (such as all edge weights being pairwise distinct formal indeterminates, and being regarded as a formal power series). If one assigns the weight 1 to each edge, then e(a,b) counts the number of paths from a to b. With this setup, write . An n-tuple of non-intersecting paths from A to B means an n-tuple (P1, ..., Pn) of paths in G with the following properties: There exists a permutation of such that, for every i, the path Pi is a path from to . Whenever , the paths Pi and Pj have no two vertices in common (not even endpoints). Given such an n-tuple (P1, ..., Pn), we denote by the permutation of from the first condition. The Lindström–Gessel–Viennot lemma then states that the determinant of M is the signed sum over all n-tuples P = (P1, ..., Pn) of non-intersecting paths from A to B: That is, the determinant of M counts the weights of all n-tuples of non-intersecting paths starting at A and ending at B, each affected with the sign of the corresponding permutation of , given by taking to . In particular, if the only permutation possible is the identity (i.e., every n-tuple of non-intersecting paths from A to B takes ai to bi for each i) and we take the weights to be 1, then det(M) is exactly the number of non-intersecting n-tuples of paths starting at A and ending at B. Proof To prove the Lindström–Gessel–Viennot lemma, we first introduce some notation. An n-path from an n-tuple of vertices of G to an n-tuple of vertices of G will mean an n-tuple of paths in G, with each leading from to . This n-path will be called non-intersecting just in case the paths Pi and Pj have no two vertices in common (including endpoints) whenever . Otherwise, it will be called entangled. Given an n-path , the weight of this n-path is defined as the product . A twisted n-path from an n-tuple of vertices of G to an n-tuple of vertices of G will mean an n-path from to for some permutation in the symmetric group . This permutation will be called the twist of this twisted n-path, and denoted by (where P is the
https://en.wikipedia.org/wiki/Naoto%20Ishikawa	is a former Japanese football player. Club statistics References External links 1989 births Living people Association football people from Ibaraki Prefecture Japanese men's footballers J2 League players Mito HollyHock players Men's association football midfielders
https://en.wikipedia.org/wiki/Pentellated%206-cubes	In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube. There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex. Pentellated 6-cube Alternate names Pentellated 6-orthoplex Expanded 6-cube, expanded 6-orthoplex Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers) Images Pentitruncated 6-cube Alternate names Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers) Images Penticantellated 6-cube Alternate names Terirhombated hexeract (Acronym: topag) (Jonathan Bowers) Images Penticantitruncated 6-cube Alternate names Terigreatorhombated hexeract (Acronym: togrix) (Jonathan Bowers) Images Pentiruncitruncated 6-cube Alternate names Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (Jonathan Bowers) Images Pentiruncicantellated 6-cube Alternate names Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers) Images Pentiruncicantitruncated 6-cube Alternate names Terigreatoprismated hexeract (Acronym: tagpox) (Jonathan Bowers) Images Pentisteritruncated 6-cube Alternate names Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers) Images Pentistericantitruncated 6-cube Alternate names Tericelligreatorhombated hexeract (Acronym: tocagrax) (Jonathan Bowers) Images Omnitruncated 6-cube The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube. Alternate names Pentisteriruncicantituncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes) Omnitruncated hexeract Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers) Images Full snub 6-cube The full snub 6-cube or omnisnub 6-cube, defined as an alternation of the omnitruncated 6-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3,3]+, and constructed from 12 snub 5-cubes, 64 snub 5-simplexes, 60 snub tesseract antiprisms, 192 snub 5-cell antiprisms, 160 3-sr{4,3} duoantiprisms, 240 4-s{3,4} duoantiprisms, and 23040 irregular 5-simplexes filling the gaps at the deleted vertices. Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 197
https://en.wikipedia.org/wiki/Pentellated%206-orthoplexes	In six-dimensional geometry, a pentellated 6-orthoplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-orthoplex. There are unique 16 degrees of pentellations of the 6-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. Ten are shown, with the other 6 more easily constructed as a pentellated 6-cube. The simple pentellated 6-orthoplex (Same as pentellated 5-cube) is also called an expanded 6-orthoplex, constructed by an expansion operation applied to the regular 6-orthoplex. The highest form, the pentisteriruncicantitruncated 6-orthoplex, is called an omnitruncated 6-orthoplex with all of the nodes ringed. Pentitruncated 6-orthoplex Alternate names Teritruncated hexacontatetrapeton (Acronym: tacox) (Jonathan Bowers) Images Penticantellated 6-orthoplex Alternate names Terirhombated hexacontitetrapeton (Acronym: tapox) (Jonathan Bowers) Images Penticantitruncated 6-orthoplex Alternate names Terigreatorhombated hexacontitetrapeton (Acronym: togrig) (Jonathan Bowers) Images Pentiruncitruncated 6-orthoplex Alternate names Teriprismatotruncated hexacontitetrapeton (Acronym: tocrax) (Jonathan Bowers) Images Pentiruncicantitruncated 6-orthoplex Alternate names Terigreatoprismated hexacontitetrapeton (Acronym: tagpog) (Jonathan Bowers) Images Pentistericantitruncated 6-orthoplex Alternate names Tericelligreatorhombated hexacontitetrapeton (Acronym: tecagorg) (Jonathan Bowers) Images Related polytopes These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex. Notes References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 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] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. x4o3o3o3x3x - tacox, x4o3o3x3o3x - tapox, x4o3o3x3x3x - togrig, x4o3x3o3x3x - tocrax, x4x3o3x3x3x - tagpog, x4x3o3x3x3x - tecagorg External links Polytopes of Various Dimensions Multi-dimensional Glossary 6-polytopes
https://en.wikipedia.org/wiki/Principle%20of%20marginality	In statistics, the principle of marginality is the fact that the average (or main) effects of variables in an analysis are marginal to their interaction effect—that is, the main effect of one explanatory variable captures the effect of that variable averaged over all values of a second explanatory variable whose value influences the first variable's effect. The principle of marginality implies that, in general, it is wrong to test, estimate, or interpret main effects of explanatory variables where the variables interact or, similarly, to model interaction effects but delete main effects that are marginal to them. While such models are interpretable, they lack applicability, as they ignore the dependence of a variable's effect upon another variable's value. Nelder and Venables have argued strongly for the importance of this principle in regression analysis. Regression form If two independent continuous variables, say x and z, both influence a dependent variable y, and if the extent of the effect of each independent variable depends on the level of the other independent variable then the regression equation can be written as: where i indexes observations, a is the intercept term, b, c, and d are effect size parameters to be estimated, and e is the error term. If this is the correct model, then the omission of any of the right-side terms would be incorrect, resulting in misleading interpretation of the regression results. With this model, the effect of x upon y is given by the partial derivative of y with respect to x; this is , which depends on the specific value at which the partial derivative is being evaluated. Hence, the main effect of x – the effect averaged over all values of z – is meaningless as it depends on the design of the experiment (specifically on the relative frequencies of the various values of z) and not just on the underlying relationships. Hence: In the case of interaction, it is wrong to try to test, estimate, or interpret a "main effect" coefficient b or c, omitting the interaction term. In addition: In the case of interaction, it is wrong to not include b or c, because this will give incorrect estimates of the interaction. See also General linear model Analysis of variance References Marginality Regression analysis Analysis of variance
https://en.wikipedia.org/wiki/Rodion%20Kuzmin	Rodion Osievich Kuzmin (, 9 November 1891, Riabye village in the Haradok district – 24 March 1949, Leningrad) was a Soviet mathematician, known for his works in number theory and analysis. His name is sometimes transliterated as Kusmin. He was an Invited Speaker of the ICM in 1928 in Bologna. Selected results In 1928, Kuzmin solved the following problem due to Gauss (see Gauss–Kuzmin distribution): if x is a random number chosen uniformly in (0, 1), and is its continued fraction expansion, find a bound for where Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy. In 1930, Kuzmin proved that numbers of the form ab, where a is algebraic and b is a real quadratic irrational, are transcendental. In particular, this result implies that Gelfond–Schneider constant is transcendental. See Gelfond–Schneider theorem for later developments. He is also known for the Kusmin-Landau inequality: If is continuously differentiable with monotonic derivative satisfying (where denotes the Nearest integer function) on a finite interval , then Notes External links (The chronology there is apparently wrong, since J. V. Uspensky lived in USA from 1929.) 1891 births 1949 deaths People from Gorodoksky Uyezd Soviet mathematicians Number theorists Mathematical analysts Academic staff of Perm State University
https://en.wikipedia.org/wiki/Prismatic%20uniform%204-polytope	In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons. The prismatic uniform 4-polytopes consist of two infinite families: Polyhedral prisms: products of a line segment and a uniform polyhedron. This family is infinite because it includes prisms built on 3-dimensional prisms and antiprisms. Duoprisms: product of two regular polygons. Convex polyhedral prisms The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytope are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract). There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms. The symmetry number of a polyhedral prism is twice that of the base polyhedron. Tetrahedral prisms: A3 × A1 Octahedral prisms: BC3 × A1 Icosahedral prisms: H3 × A1 Duoprisms: [p] × [q] The second is the infinite family of uniform duoprisms, products of two regular polygons. Their Coxeter diagram is of the form This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism. The elements of a p,q-duoprism (p ≥ 3, q ≥ 3) are: Cells: p q-gonal prisms, q p-gonal prisms Faces: pq squares, p q-gons, q p-gons Edges: 2pq Vertices: pq There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms with the exception of the great duoantiprism. Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms: 3-3 duoprism - - 6 triangular prisms 3-4 duoprism - - 3 cubes, 4 triangular prisms 4-4 duoprism - - 8 cubes (same as tesseract) 3-5 duoprism - - 3 pentagonal prisms, 5 triangular prisms 4-5 duoprism - - 4 pentagonal prisms, 5 cubes 5-5 duoprism - - 10 pentagonal prisms 3-6 duoprism - - 3 hexagonal prisms, 6 triangular prisms 4-6 duoprism - - 4 hexagonal prisms, 6 cubes 5-6 duoprism - - 5 hexagonal prisms, 6 pentagonal prisms 6-6 duoprism - - 12 hexagonal prisms ... Polygonal prismatic prisms The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - - p cubes and 4 p-gonal prisms - (All
https://en.wikipedia.org/wiki/Kentaro%20Nakata	is a former Japanese football player. Club statistics References External links 1989 births Living people Association football people from Kumamoto Prefecture Japanese men's footballers J2 League players Yokohama FC players Matsumoto Yamaga FC players FC Kariya players Men's association football defenders
https://en.wikipedia.org/wiki/Daurenbek%20Tazhimbetov	Daurenbek Tazhimbetov (; born 2 July 1985 is a Kazakh football player who plays for FC Akzhayik. Career In July 2013, Tazhimbetov joined FC Astana on loan. Career statistics Club career statistics Last update: 28 October 2012 International goals References External links 1985 births Living people Kazakhstani men's footballers Kazakhstan men's international footballers Kazakhstan Premier League players FC Shakhter Karagandy players FC Astana players FC Ordabasy players FC Kaisar players FC Taraz players Men's association football forwards People from Kyzylorda Region
https://en.wikipedia.org/wiki/Metric%20Structures%20for%20Riemannian%20and%20Non-Riemannian%20Spaces	Metric Structures for Riemannian and Non-Riemannian Spaces is a book in geometry by Mikhail Gromov. It was originally published in French in 1981 under the title Structures métriques pour les variétés riemanniennes, by CEDIC (Paris). History The 1981 edition was edited by Jacques Lafontaine and Pierre Pansu. The English version, considerably expanded, was published in 1999 by Birkhäuser Verlag, with appendices by Pierre Pansu, Stephen Semmes, and Mikhail Katz. The book was well received and has been reprinted several times. References Riemannian geometry Mathematics books Systolic geometry Birkhäuser books
https://en.wikipedia.org/wiki/Vitali%20Yevstigneyev	Vitali Yevstigneyev (; born 8 August 1985) is a Kazakh football player, who plays for FC Taraz in the Kazakhstan Premier League. Career statistics International Statistics accurate as of match played 11 October 2011 International goals References External links Player's profile on the club website 1985 births Living people Kazakhstani men's footballers Kazakhstan men's international footballers Kazakhstan Premier League players FC Aktobe players FC Ordabasy players FC Taraz players Men's association football midfielders Place of birth missing (living people)
https://en.wikipedia.org/wiki/Stephen%20Semmes	Stephen William Semmes (born 26 May 1962) is the Noah Harding Professor of Mathematics at Rice University. He is known for contributions to analysis on metric spaces, as well as harmonic analysis, complex variables, partial differential equations, and differential geometry. He received his B.S. at the age of 18, a Ph.D. at 21 from Washington University in St. Louis and became a full professor at Rice at 25. Awards Semmes was awarded a Sloan Fellowship in 1987. In 1994, he gave an invited talk at the International Congress of Mathematicians. Publications Coifman, R.; Lions, P.-L.; Meyer, Y.; Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286. David, Guy; Semmes, Stephen: Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs, 38. American Mathematical Society, Providence, RI, 1993. David, G.; Journé, J.-L.; Semmes, S.: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. [Calderón-Zygmund operators, para-accretive functions and interpolation] Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56. David, Guy; Semmes, Stephen: Fractured fractals and broken dreams. Self-similar geometry through metric and measure. Oxford Lecture Series in Mathematics and its Applications, 7. The Clarendon Press, Oxford University Press, New York, 1997. (97j:46033) Semmes, S. Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Math. (N.S.) 2 (1996), no. 2, 155–295. Stephen Semmes. "Appendix B: Metric spaces and mappings seen at many scales" (pp. 401–518). In Gromov, Misha: Metric Structures for Riemannian and Non-Riemannian Spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc., Boston, MA, 1999. xx+585 pp. References External links Living people 20th-century American mathematicians 21st-century American mathematicians Washington University in St. Louis mathematicians Mathematical analysts Washington University in St. Louis alumni Rice University faculty 1962 births
https://en.wikipedia.org/wiki/Regular%20element%20of%20a%20Lie%20algebra	In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element is regular if its centralizer in has dimension equal to the rank of , which in turn equals the dimension of some Cartan subalgebra (note that in earlier papers, an element of a complex semisimple Lie algebra was termed regular if it is semisimple and the kernel of its adjoint representation is a Cartan subalgebra). An element a Lie group is regular if its centralizer has dimension equal to the rank of . Basic case In the specific case of , the Lie algebra of matrices over an algebraically closed field (such as the complex numbers), a regular element is an element whose Jordan normal form contains a single Jordan block for each eigenvalue (in other words, the geometric multiplicity of each eigenvalue is 1). The centralizer of a regular element is the set of polynomials of degree less than evaluated at the matrix , and therefore the centralizer has dimension (which equals the rank of , but is not necessarily an algebraic torus). If the matrix is diagonalisable, then it is regular if and only if there are different eigenvalues. To see this, notice that will commute with any matrix that stabilises each of its eigenspaces. If there are different eigenvalues, then this happens only if is diagonalisable on the same basis as ; in fact is a linear combination of the first powers of , and the centralizer is an algebraic torus of complex dimension (real dimension ); since this is the smallest possible dimension of a centralizer, the matrix is regular. However if there are equal eigenvalues, then the centralizer is the product of the general linear groups of the eigenspaces of , and has strictly larger dimension, so that is not regular. For a connected compact Lie group , the regular elements form an open dense subset, made up of -conjugacy classes of the elements in a maximal torus which are regular in . The regular elements of are themselves explicitly given as the complement of a set in , a set of codimension-one subtori corresponding to the root system of . Similarly, in the Lie algebra of , the regular elements form an open dense subset which can be described explicitly as adjoint -orbits of regular elements of the Lie algebra of , the elements outside the hyperplanes corresponding to the root system. Definition Let be a finite-dimensional Lie algebra over an infinite field. For each , let be the characteristic polynomial of the adjoint endomorphism of . Then, by definition, the rank of is the least integer such that for some and is denoted by . For example, since for every x, is nilpotent (i.e., each is nilpotent by Engel's theorem) if and only if . Let . By definition, a regular element of is an element of the set . Since is a polynomial function on , with respect to the Zariski topology, the set is an open subset of .
https://en.wikipedia.org/wiki/Principal%20subalgebra	In mathematics, a principal subalgebra of a complex simple Lie algebra is a 3-dimensional simple subalgebra whose non-zero elements are regular. A finite-dimensional complex simple Lie algebra has a unique conjugacy class of principal subalgebras, each of which is the span of an sl2-triple. References https://aiolatest.com/application-of-abstract-algebra-in-real-life/ Lie algebras Representation theory
https://en.wikipedia.org/wiki/CC%20system	In computational geometry, a CC system or counterclockwise system is a ternary relation introduced by Donald Knuth to model the clockwise ordering of triples of points in general position in the Euclidean plane. Axioms A CC system is required to satisfy the following axioms, for all distinct points p, q, r, s, and t: Cyclic symmetry: If then . Antisymmetry: If then not . Nondegeneracy: Either or . Interiority: If and and , then . Transitivity: If and and , and and , then . Triples of points that are not distinct are not considered as part of the relation. Construction from planar point sets A CC system may be defined from any set of points in the Euclidean plane, with no three of the points collinear, by including in the relation a triple of distinct points whenever the triple lists these three points in counterclockwise order around the triangle that they form. Using the Cartesian coordinates of the points, the triple pqr is included in the relation exactly when The condition that the points are in general position is equivalent to the requirement that this matrix determinant is never zero for distinct points p, q, and r. However, not every CC system comes from a Euclidean point set in this way. Equivalent notions CC systems can also be defined from pseudoline arrangements, or from sorting networks in which the compare-exchange operations only compare adjacent pairs of elements (as in for instance bubble sort), and every CC system can be defined in this way. This relation is not one-to-one, but the numbers of nonisomorphic CC systems on n points, of pseudoline arrangements with n lines, and of sorting networks on n values, are within polynomial factors of each other. There exists a two-to-one correspondence between CC systems and uniform acyclic oriented matroids of rank 3. These matroids in turn have a 1-1 correspondence to topological equivalence classes of pseudoline arrangements with one marked cell. Algorithmic applications The information given by a CC system is sufficient to define a notion of a convex hull within a CC system. The convex hull is the set of ordered pairs pq of distinct points with the property that, for every third distinct point r, pqr belongs to the system. It forms a cycle, with the property that every three points of the cycle, in the same cyclic order, belong to the system. By adding points one at a time to a CC system, and maintaining the convex hull of the points added so far in its cyclic order using a binary search tree, it is possible to construct the convex hull in time O(n log n), matching the known time bounds for convex hull algorithms for Euclidean points. It is also possible to find a single convex hull vertex, as well as the combinatorial equivalent of a bisecting line through a system of points, from a CC system in linear time. The construction of an extreme vertex allows the Graham scan algorithm for convex hulls to be generalized from point sets to CC systems, with a number of q
https://en.wikipedia.org/wiki/CumFreq	In statistics and data analysis the application software CumFreq is a tool for cumulative frequency analysis of a single variable and for probability distribution fitting. Originally the method was developed for the analysis of hydrological measurements of spatially varying magnitudes (e.g. hydraulic conductivity of the soil) and of magnitudes varying in time (e.g. rainfall, river discharge) to find their return periods. However, it can be used for many other types of phenomena, including those that contain negative values. Software features CumFreq uses the plotting position approach to estimate the cumulative frequency of each of the observed magnitudes in a data series of the variable. The computer program allows determination of the best fitting probability distribution. Alternatively it provides the user with the option to select the probability distribution to be fitted. The following probability distributions are included: normal, lognormal, logistic, loglogistic, exponential, Cauchy, Fréchet, Gumbel, Pareto, Weibull, Generalized extreme value distribution, Laplace distribution, Burr distribution (Dagum mirrored), Dagum distribution (Burr mirrored), Gompertz distribution, Student distribution and other. Another characteristic of CumFreq is that it provides the option to use two different probability distributions, one for the lower data range, and one for the higher. The ranges are separated by a break-point. The use of such composite (discontinuous) probability distributions can be useful when the data of the phenomenon studied were obtained under different conditions. During the input phase, the user can select the number of intervals needed to determine the histogram. He may also define a threshold to obtain a truncated distribution. The output section provides a calculator to facilitate interpolation and extrapolation. Further it gives the option to see the Q–Q plot in terms of calculated and observed cumulative frequencies. ILRI provides examples of application to magnitudes like crop yield, watertable depth, soil salinity, hydraulic conductivity, rainfall, and river discharge. Generalizing distributions The program can produce generalizations of the normal, logistic, and other distributions by transforming the data using an exponent that is optimized to obtain the best fit. This feature is not common in other distribution-fitting software which normally include only a logarithmic transformation of data obtaining distributions like the lognormal and loglogistic. Generalization of symmetrical distributions (like the normal and the logistic) makes them applicable to data obeying a distribution that is skewed to the right (using an exponent <1) as well as to data obeying a distribution that is skewed to the left (using an exponent >1). This enhances the versatility of symmetrical distributions. Inverting distributions Skew distributions can be mirrored by distribution inversion (see survival function, or complementary d
https://en.wikipedia.org/wiki/2008%20Consadole%20Sapporo%20season	2008 Consadole Sapporo season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Consadole Sapporo Hokkaido Consadole Sapporo seasons
https://en.wikipedia.org/wiki/Random%20group	In mathematics, random groups are certain groups obtained by a probabilistic construction. They were introduced by Misha Gromov to answer questions such as "What does a typical group look like?" It so happens that, once a precise definition is given, random groups satisfy some properties with very high probability, whereas other properties fail with very high probability. For instance, very probably random groups are hyperbolic groups. In this sense, one can say that "most groups are hyperbolic". Definition The definition of random groups depends on a probabilistic model on the set of possible groups. Various such probabilistic models yield different (but related) notions of random groups. Any group can be defined by a group presentation involving generators and relations. For instance, the Abelian group has a presentation with two generators and , and the relation , or equivalently . The main idea of random groups is to start with a fixed number of group generators , and imposing relations of the form where each is a random word involving the letters and their formal inverses . To specify a model of random groups is to specify a precise way in which , and the random relations are chosen. Once the random relations have been chosen, the resulting random group is defined in the standard way for group presentations, namely: is the quotient of the free group with generators , by the normal subgroup generated by the relations seen as elements of : The few-relator model of random groups The simplest model of random groups is the few-relator model. In this model, a number of generators and a number of relations are fixed. Fix an additional parameter (the length of the relations), which is typically taken very large. Then, the model consists in choosing the relations at random, uniformly and independently among all possible reduced words of length at most involving the letters and their formal inverses . This model is especially interesting when the relation length tends to infinity: with probability tending to as a random group in this model is hyperbolic and satisfies other nice properties. Further remarks More refined models of random groups have been defined. For instance, in the density model, the number of relations is allowed to grow with the length of the relations. Then there is a sharp "phase transition" phenomenon: if the number of relations is larger than some threshold, the random group "collapses" (because the relations allow to show that any word is equal to any other), whereas below the threshold the resulting random group is infinite and hyperbolic. Constructions of random groups can also be twisted in specific ways to build group with particular properties. For instance, Gromov used this technique to build new groups that are counter-examples to an extension of the Baum–Connes conjecture. References Mikhail Gromov. Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8,
https://en.wikipedia.org/wiki/2008%20Montedio%20Yamagata%20season	2008 Montedio Yamagata season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Montedio Yamagata Montedio Yamagata seasons
https://en.wikipedia.org/wiki/2008%20Kashima%20Antlers%20season	2008 Kashima Antlers season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Kashima Antlers Kashima Antlers seasons
https://en.wikipedia.org/wiki/2008%20Mito%20HollyHock%20season	2008 Mito HollyHock season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Mito HollyHock Mito HollyHock seasons
https://en.wikipedia.org/wiki/2008%20Thespa%20Kusatsu%20season	2008 Thespa Kusatsu season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Thespa Kusatsu Thespakusatsu Gunma seasons
https://en.wikipedia.org/wiki/2008%20Urawa%20Red%20Diamonds%20season	2008 Urawa Red Diamonds season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Urawa Red Diamonds Urawa Red Diamonds seasons
https://en.wikipedia.org/wiki/2008%20Omiya%20Ardija%20season	2008 Omiya Ardija season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Omiya Ardija Omiya Ardija seasons
https://en.wikipedia.org/wiki/2008%20JEF%20United%20Chiba%20season	2008 JEF United Ichihara Chiba season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site JEF United Ichihara Chiba JEF United Chiba seasons
https://en.wikipedia.org/wiki/2008%20Kashiwa%20Reysol%20season	2008 Kashiwa Reysol season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Kashiwa Reysol Kashiwa Reysol seasons
https://en.wikipedia.org/wiki/2008%20FC%20Tokyo%20season	The 2008 FC Tokyo season was the team's 10th season as a member of J.League Division 1. Competitions Domestic results J.League 1 Emperor's Cup J.League Cup Player statistics Other pages J. League official site Tokyo 2008
https://en.wikipedia.org/wiki/2008%20Tokyo%20Verdy%20season	2008 Tokyo Verdy season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Tokyo Verdy Tokyo Verdy seasons
https://en.wikipedia.org/wiki/2008%20Kawasaki%20Frontale%20season	2008 Kawasaki Frontale season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Kawasaki Frontale Kawasaki Frontale seasons
https://en.wikipedia.org/wiki/2008%20Yokohama%20F.%20Marinos%20season	2008 Yokohama F. Marinos season Competitions Domestic results J.League 1 Emperor's Cup J.League Cup Player statistics Other pages J.League official site Yokohama F. Marinos Yokohama F. Marinos seasons
https://en.wikipedia.org/wiki/2008%20Yokohama%20FC%20season	2008 Yokohama FC season Competitions Domestic results J.League 2 Emperor's Cup Player statistics Other pages J. League official site Yokohama FC Yokohama FC seasons
https://en.wikipedia.org/wiki/2008%20Ventforet%20Kofu%20season	2008 Ventforet Kofu season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Ventforet Kofu Ventforet Kofu seasons
https://en.wikipedia.org/wiki/2008%20Albirex%20Niigata%20season	2008 Albirex Niigata season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Albirex Niigata Albirex Niigata seasons
https://en.wikipedia.org/wiki/2008%20J%C3%BAbilo%20Iwata%20season	2008 Júbilo Iwata season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Jubilo Iwata Júbilo Iwata seasons
https://en.wikipedia.org/wiki/2008%20FC%20Gifu%20season	2008 F.C. Gifu season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Gifu 2008
https://en.wikipedia.org/wiki/2008%20Kyoto%20Sanga%20FC%20season	2008 Kyoto Sanga F.C. season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Kyoto Sanga F.C. Kyoto Sanga FC seasons
https://en.wikipedia.org/wiki/2008%20Gamba%20Osaka%20season	2008 Gamba Osaka season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Gamba Osaka Gamba Osaka seasons
https://en.wikipedia.org/wiki/2008%20Cerezo%20Osaka%20season	The 2008 Cerezo Osaka season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Cerezo Osaka Cerezo Osaka seasons
https://en.wikipedia.org/wiki/2008%20Vissel%20Kobe%20season	2008 Vissel Kobe season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Vissel Kobe Vissel Kobe seasons
https://en.wikipedia.org/wiki/2008%20Sanfrecce%20Hiroshima%20season	2008 Sanfrecce Hiroshima season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Sanfrecce Hiroshima Sanfrecce Hiroshima seasons
https://en.wikipedia.org/wiki/2008%20Tokushima%20Vortis%20season	2008 Tokushima Vortis season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Tokushima Vortis Tokushima Vortis seasons
https://en.wikipedia.org/wiki/2008%20Avispa%20Fukuoka%20season	2008 Avispa Fukuoka season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Avispa Fukuoka Avispa Fukuoka seasons
https://en.wikipedia.org/wiki/2008%20Oita%20Trinita%20season	2008 Oita Trinita season Competitions Domestic results J. League 1 Emperor's Cup J. League Cup Player statistics Other pages J. League official site Oita Trinita Oita Trinita seasons
https://en.wikipedia.org/wiki/2008%20Sagan%20Tosu%20season	2008 Sagan Tosu season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Sagan Tosu Sagan Tosu seasons
https://en.wikipedia.org/wiki/2008%20Roasso%20Kumamoto%20season	2008 Roasso Kumamoto season Competitions Domestic results J. League 2 Emperor's Cup Player statistics Other pages J. League official site Roasso Kumamoto Roasso Kumamoto seasons
https://en.wikipedia.org/wiki/Zinovy%20Reichstein	Zinovy Reichstein (born 1961) is a Russian-born American mathematician. He is a professor at the University of British Columbia in Vancouver. He studies mainly algebra, algebraic geometry and algebraic groups. He introduced (with Joe P. Buhler) the concept of essential dimension. Early life and education In high school, Reichstein participated in the national mathematics olympiad in Russia and was the third highest scorer in 1977 and second highest scorer in 1978. Because of the Antisemitism in the Soviet Union at the time, Reichstein was not accepted to Moscow University, even though he had passed the special math entrance exams. He attended a semester of college at Russian University of Transport instead. His family then decided to emigrate, arriving in Vienna, Austria, in August 1979 and New York, United States in the fall of 1980. Reichstein worked as a delivery boy for a short period of time in New York. He was then accepted to and attended California Institute of Technology for his undergraduate studies. Reichstein received his PhD degree in 1988 from Harvard University under the supervision of Michael Artin. Parts of his thesis entitled "The Behavior of Stability under Equivariant Maps" were published in the journal Inventiones Mathematicae. Career As of 2011, he is on the editorial board of the mathematics journal Transformation groups. Awards Winner of the 2013 Jeffery-Williams Prize awarded by the Canadian Mathematical Society Fellow of the American Mathematical Society, 2012 Invited Speaker to the International Congress of Mathematicians (Hyderabad, India 2010) References External links Algebraists Harvard University alumni 20th-century American mathematicians 21st-century American mathematicians Living people Academic staff of the University of British Columbia Place of birth missing (living people) Fellows of the American Mathematical Society 1961 births
https://en.wikipedia.org/wiki/Syntomic%20topology	In algebraic geometry, the syntomic topology is a Grothendieck topology introduced by . Mazur defined a morphism to be syntomic if it is flat and locally a complete intersection. The syntomic topology is generated by surjective syntomic morphisms of affine schemes. References External links Explanation of the word "syntomic" by Barry Mazur. Algebraic geometry
https://en.wikipedia.org/wiki/2011%E2%80%9312%20Galatasaray%20S.K.%20season	The 2011–12 season was Galatasarays 108th in existence and the club's 54th consecutive season in the Süper Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club played in during the season. Season overview Ünal Aysal was elected as the new president of Galatasaray in May 2011. After his election, he made an agreement with coach Fatih Terim for the 2011–12 season. The club signed Swedish international striker Johan Elmander for three seasons from Bolton Wanderers on a free transfer. Club Coaching staff Board of directors Medical staff Other personnel Grounds Kit Uniform Manufacturer: Nike Chest Advertising's: Türk Telekom Back Advertising's: Ülker Arm Advertising's: Avea Short Advertising's: Nikon Sponsorship Companies that Galatasaray S.K. currently has sponsorship deals with include. Players Squad information Transfers In Total spending: €23.80 million * - 1 player will leave the club at the end of the 2011-2012 season. Out Total income: €14.1M Expenditure: €9.70M * - if Atletico Madrid finishes La Liga as a UEFA Champions League participant: €1M bonus fee. If they finishes La Liga as a UEFA Europa League participant: €0.5M bonus fee. Friendly matches Pre-season Galatasaray start the 2011-12 season with a training session to be held in Florya on Monday June 27, 2011. On July 2, Galatasaray will be leaving for Austria in order to camp near the town of Wörgl until July 12. The second summer camp of Galatasaray is planned to be in Germany. Kickoff times are in CET. Other friendlies Competitions Overview Süper Lig Standings Results summary Results by round Matches Championship play-offs Playoff table Results summary Results by round Matches Turkish Cup Statistics Squad statistics \| Goals Includes all competitive matches. Last updated on 12 May 2012 Clean sheets Disciplinary record Overall Attendance Sold season tickets: 27,900 See also 2011–12 Süper Lig 2011–12 Turkish Cup References External links Galatasaray Sports Club Official Website Turkish Football Federation - Galatasaray A.Ş. uefa.com - Galatasaray AŞ 2011-12 Turkish football clubs 2011–12 season 2011-12 2011 in Istanbul 2012 in Istanbul Galatasaray Sports Club 2011–12 season
https://en.wikipedia.org/wiki/Kai%20Behrend	Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada. His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, Gromov–Witten invariants and Donaldson–Thomas theory (cf. Behrend function.) He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz trace formula to algebraic stacks. He is the recipient of the 2001 Coxeter–James Prize, the 2011 Jeffery–Williams Prize, and the 2015 CRM-Fields-PIMS Prize. He was elected to the 2018 class of fellows of the American Mathematical Society. Selected publications References External links The personal web page of Kai Behrend Algebraic geometers Geometers University of California, Berkeley alumni 20th-century German mathematicians 21st-century German mathematicians Living people Academic staff of the University of British Columbia Faculty of Science Year of birth missing (living people) Scientists from Hamburg Fellows of the American Mathematical Society Harvard University alumni
https://en.wikipedia.org/wiki/Larry%20Guth	Lawrence David Guth (born 1977) is a professor of mathematics at the Massachusetts Institute of Technology. Education and career Guth graduated from Yale in 2000, with BS in mathematics. In 2005, he got his PhD in mathematics from the Massachusetts Institute of Technology, where he studied geometry of objects with random shapes under the supervision of Tomasz Mrowka. After MIT, Guth went to Stanford as a postdoc, and later to the University of Toronto as an Assistant Professor. In 2011, New York University's Courant Institute of Mathematical Sciences hired Guth as a professor, listing his areas of interest as "metric geometry, harmonic analysis, and geometric combinatorics." In 2012, Guth moved to MIT, where he is Claude Shannon Professor of Mathematics. Research In his research, Guth has strengthened Gromov's systolic inequality for essential manifolds and, along with Nets Katz, found a solution to the Erdős distinct distances problem. His wide-ranging interests include the Kakeya conjecture and the systolic inequality. Recognition Guth won an Alfred P. Sloan Fellowship in 2010. He was an invited speaker at the International Congress of Mathematicians in India in 2010, where he spoke about systolic geometry. In 2013, the American Mathematical Society awarded Guth its annual Salem Prize, citing his "major contributions to geometry and combinatorics." In 2014 he received a Simons Investigator Award. In 2015, he received the Clay Research Award. He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to harmonic analysis, combinatorics and geometry, and for exposition of high level mathematics". On February 20, 2020, the National Academy of Sciences announced that Guth is the first winner of their new $20,000 Maryam Mirzakhani Prize in Mathematics for mid-career mathematicians. The citation states that his award is "for developing surprising, original, and deep connections between geometry, analysis, topology, and combinatorics, which have led to the solution of, or major advances on, many outstanding problems in these fields." In 2021, he was elected member of the US National Academy of Sciences. Personal He is the son of Alan Guth, a theoretical physicist known for the theory of inflation in cosmology. Work Metaphors in systolic geometry: the video . . . . . . . References Living people 20th-century American mathematicians Massachusetts Institute of Technology School of Science alumni Academic staff of the University of Toronto Geometers Courant Institute of Mathematical Sciences faculty 1977 births Simons Investigator Place of birth missing (living people) Fellows of the American Mathematical Society Yale College alumni 21st-century American mathematicians Members of the United States National Academy of Sciences
https://en.wikipedia.org/wiki/Ido%20Levy%20%28footballer%29	Ido Levy (born 31 July 1990) is an Israeli professional footballer who plays as a centre-back for Hapoel Hadera. Career statistics Honours Liga Leumit Winner (1): 2013–14 Israel State Cup Runner-up (1): 2013–14 References 1990 births Living people Men's association football defenders Israeli men's footballers Maccabi Netanya F.C. players Hapoel Herzliya F.C. players Hapoel Ra'anana A.F.C. players Hapoel Haifa F.C. players Hapoel Hadera F.C. players Israeli Premier League players Liga Leumit players Footballers from Hadera Israel men's under-21 international footballers
https://en.wikipedia.org/wiki/Rouhollah%20Arab	Rouhollah Arab (, born February 1, 1984) is an Iranian footballer who plays as a striker for Aluminium Arak in the Azadegan League. Club career Club Career Statistics Last Update: 10 May 2013 Assists Honours Club Azadegan League Runner up: 1 2009–10 with Sanat Naft References External links Arab's Interview at Navad Rouhollah Arab at Persian League 1984 births Living people Sportspeople from Sari, Iran Footballers from Mazandaran province Iranian men's footballers F.C. Nassaji Mazandaran players Sanat Naft Abadan F.C. players Zob Ahan Esfahan F.C. players Malavan F.C. players Persian Gulf Pro League players Azadegan League players Men's association football forwards
https://en.wikipedia.org/wiki/Differential%20poset	In mathematics, a differential poset is a partially ordered set (or poset for short) satisfying certain local properties. (The formal definition is given below.) This family of posets was introduced by as a generalization of Young's lattice (the poset of integer partitions ordered by inclusion), many of whose combinatorial properties are shared by all differential posets. In addition to Young's lattice, the other most significant example of a differential poset is the Young–Fibonacci lattice. Definitions A poset P is said to be a differential poset, and in particular to be r-differential (where r is a positive integer), if it satisfies the following conditions: P is graded and locally finite with a unique minimal element; for every two distinct elements x, y of P, the number of elements covering both x and y is the same as the number of elements covered by both x and y; and for every element x of P, the number of elements covering x is exactly r more than the number of elements covered by x. These basic properties may be restated in various ways. For example, Stanley shows that the number of elements covering two distinct elements x and y of a differential poset is always either 0 or 1, so the second defining property could be altered accordingly. The defining properties may also be restated in the following linear algebraic setting: taking the elements of the poset P to be formal basis vectors of an (infinite-dimensional) vector space, let D and U be the operators defined so that D x is equal to the sum of the elements covered by x, and U x is equal to the sum of the elements covering x. (The operators D and U are called the down and up operator, for obvious reasons.) Then the second and third conditions may be replaced by the statement that DU − UD = r&hairsp;I (where I is the identity). This latter reformulation makes a differential poset into a combinatorial realization of a Weyl algebra, and in particular explains the name differential: the operators "d/dx" and "multiplication by x" on the vector space of polynomials obey the same commutation relation as U and D/r. Examples The canonical examples of differential posets are Young's lattice, the poset of integer partitions ordered by inclusion, and the Young–Fibonacci lattice. Stanley's initial paper established that Young's lattice is the only distributive lattice, while showed that these are the only lattices. There is a canonical construction (called "reflection") of a differential poset given a finite poset that obeys all of the defining axioms below its top rank. (The Young–Fibonacci lattice is the poset that arises by applying this construction beginning with a single point.) This can be used to show that there are infinitely many differential posets. includes a remark that "[David] Wagner described a very general method for constructing differential posets which make it unlikely that [they can be classified]." This is made precise in , where it is shown that t
https://en.wikipedia.org/wiki/Compositio%20Mathematica	Compositio Mathematica is a monthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935. It is owned by the Foundation Compositio Mathematica, and since 2004 it has been published on behalf of the Foundation by the London Mathematical Society in partnership with Cambridge University Press. According to the Journal Citation Reports, the journal has a 2020 2-year impact factor of 1.456 and a 2020 5-year impact factor of 1.696. The editors-in-chief are Fabrizio Andreatta, David Holmes, Bruno Klingler, and Éric Vasserot. Early history The journal was established by L. E. J. Brouwer in response to his dismissal from Mathematische Annalen in 1928. An announcement of the new journal was made in a 1934 issue of the American Mathematical Monthly. In 1940, the publication of the journal was suspended due to the German occupation of the Netherlands. References External links Online archive (1935-1996) Academic journals associated with learned and professional societies of the United Kingdom Cambridge University Press academic journals Bimonthly journals English-language journals Mathematics education in the United Kingdom Mathematics journals Academic journals established in 1935 London Mathematical Society

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