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https://en.wikipedia.org/wiki/Snub%20octaoctagonal%20tiling | In geometry, the snub octaoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,8}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
A higher symmetry coloring can be constructed from [8,4] symmetry as s{8,4}, . In this construction there is only one color of octagon.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Snub%20tetraoctagonal%20tiling | In geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,4}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
The snub tetraoctagonal tiling is seventh in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Chiral figures
Hyperbolic tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/2003%E2%80%9304%20Galatasaray%20S.K.%20season | The 2003–04 season was Galatasaray's 100th in existence and the 46th 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 have played in the season.
Squad statistics
Transfers
In
Out
Süper Lig
Standings
Türkiye Kupası
Second round
Third round
UEFA Champions League
Third qualifying round
Group stage
UEFA Cup
Third round
Attendance
Sold season tickets: 38,000
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
2000s in Istanbul
Galatasaray Sports Club 2003–04 season |
https://en.wikipedia.org/wiki/Claudiu%20Pascariu | Claudiu Dumitru Pascariu (born 25 October 1988) is Romanian footballer who plays as a defender.
Club statistics
Updated to games played as of 3 March 2013.
References
External links
Claudiu Pascariu at fupa.net
1988 births
Living people
Footballers from Arad, Romania
Romanian men's footballers
Men's association football defenders
Liga I players
Liga II players
Nemzeti Bajnokság I players
FC UTA Arad players
ASA 2013 Târgu Mureș players
Budapest Honvéd FC players
FC Bihor Oradea (1958) players
CS Podgoria Pâncota players
CS Crișul Chișineu-Criș players
Romanian expatriate men's footballers
Romanian expatriate sportspeople in Hungary
Expatriate men's footballers in Hungary
Romanian expatriate sportspeople in Germany
Expatriate men's footballers in Germany |
https://en.wikipedia.org/wiki/Kirsti%20Andersen | Kirsti Andersen (born December 9, 1941, Copenhagen), published under the name Kirsti Pedersen, is a Danish historian of mathematics. She is an Associate Professor of the History of Science at Aarhus University, where she had her Candidate examination in 1967.
Work
Andersen has written on the early history of mathematical analysis (for example, Cavalieri and Roberval).
She has also written extensively on the history of graphical perspective. In a 1985 article she related the science of perspective as described by Simon Stevin, Frans van Schooten, Willem 's Gravesande, Brook Taylor, and Johann Heinrich Lambert. In a 1987 article she examined the ancient roots of linear perspective as found in Euclid's Optics and Ptolemy (Geography and Planisphaerium). In 1991 she recalled Desargues’ method of perspective. In 1992 her book on Brook Taylor appeared, and she wrote on the alternative "plan and elevation technique". In 2007 her The Geometry of an Art provided a comprehensive study. According to the publisher’s summary, the book is a "case study of the difficulties in bridging the gap between those with mathematical knowledge and the mathematically untrained practitioners who wish to use this knowledge." The book covers Leon Battista Alberti, Piero della Francesca, Albrecht Dürer, Leonardo da Vinci, Guidobaldo del Monte, and Gaspard Monge as well as the previously mentioned authors.
Andersen has also written about Danish history of mathematics, and has championed the use of mathematics in high school history classes.
In 2005 she was awarded a doctorate in Aarhus. She is married to Henk Bos.
Selected publications
Andersen, Kirsti (1985) "Cavalieri's method of indivisibles", Archive for History of Exact Sciences 31(4): 291–367.
Bos, H. J. M.; Bunn, R.; Dauben, Joseph W.; Grattan-Guinness, I.; Hawkins, Thomas W.; Pedersen, Kirsti Møller (1980) From the calculus to set theory, 1630–1910. An introductory history, edited by Ivor Grattan-Guinness, Gerald Duckworth and Company Ltd., London, .
Scholz, Erhard; Andersen, Kirsti; Bos, Henk J. M.; et al. (1990) Geschichte der Algebra. (German) [History of algebra] Eine Einführung. [An introduction] Lehrbücher und Monographien zur Didaktik der Mathematik [Textbooks and Monographs on the Didactics of Mathematics], 16. Bibliographisches Institut, Mannheim, .
Andersen, Kirsti (1980) "An impression of mathematics in Denmark in the period 1600–1800", Special issue dedicated to Olaf Pedersen on his sixtieth birthday. Centaurus 24: 316–334.
Andersen, Kirsti (2007), The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge, Springer
Andersen, Kirsti (2011) "One of Berkeley's arguments on compensating errors in the calculus", Historia Mathematica 38(2): 219–231.
Notes
External links
Homepage
Ekspertdatabasen
1941 births
Living people
Danish historians of mathematics
Danish women historians
20th-century Danish women writers
20th-century Danish historians
21st-century Dani |
https://en.wikipedia.org/wiki/2004%E2%80%9305%20Galatasaray%20S.K.%20season | The 2004–05 season was Galatasaray's 101st in existence and the 47th 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 have played in the season.
Squad statistics
Players in / out
In
Out
Süper Lig
Standings
Türkiye Kupası
Second round
Third round
Quarter final
Semi final
Final
Friendlies
Attendance
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
2004–05 in Turkish football
2000s in Istanbul
Galatasaray Sports Club 2004–05 season |
https://en.wikipedia.org/wiki/Matroid%20polytope | In mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid. Given a matroid , the matroid polytope is the convex hull of the indicator vectors of the bases of .
Definition
Let be a matroid on elements. Given a basis of , the indicator vector of is
where is the standard th unit vector in . The matroid polytope is the convex hull of the set
Examples
Let be the rank 2 matroid on 4 elements with bases
That is, all 2-element subsets of except . The corresponding indicator vectors of are
The matroid polytope of is
These points form four equilateral triangles at point , therefore its convex hull is the square pyramid by definition.
Let be the rank 2 matroid on 4 elements with bases that are all 2-element subsets of . The corresponding matroid polytope is the octahedron. Observe that the polytope from the previous example is contained in .
If is the uniform matroid of rank on elements, then the matroid polytope is the hypersimplex .
Properties
A matroid polytope is contained in the hypersimplex , where is the rank of the associated matroid and is the size of the ground set of the associated matroid. Moreover, the vertices of are a subset of the vertices of .
Every edge of a matroid polytope is a parallel translate of for some , the ground set of the associated matroid. In other words, the edges of correspond exactly to the pairs of bases that satisfy the basis exchange property: for some Because of this property, every edge length is the square root of two. More generally, the families of sets for which the convex hull of indicator vectors has edge lengths one or the square root of two are exactly the delta-matroids.
Matroid polytopes are members of the family of generalized permutohedra.
Let be the rank function of a matroid . The matroid polytope can be written uniquely as a signed Minkowski sum of simplices:
where is the ground set of the matroid and is the signed beta invariant of :
Related polytopes
Independence matroid polytope
The matroid independence polytope or independence matroid polytope is the convex hull of the set
The (basis) matroid polytope is a face of the independence matroid polytope. Given the rank of a matroid , the independence matroid polytope is equal to the polymatroid determined by .
Flag matroid polytope
The flag matroid polytope is another polytope constructed from the bases of matroids. A flag is a strictly increasing sequence
of finite sets. Let be the cardinality of the set . Two matroids and are said to be concordant if their rank functions satisfy
Given pairwise concordant matroids on the ground set with ranks , consider the collection of flags where is a basis of the matroid and . Such a collection of flags is a flag matroid . The matroids are called the constituents of .
For each |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Galatasaray%20S.K.%20season | The 2005–06 season was Galatasaray's 102nd in existence and the 48th 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 have played in the season.
The season also saw a first in Turkish football; for the first time in history the team that entered the last week first, Fenerbahçe, failed to win the title. Fenerbahçe and Galatasaray went into the last week deadlocked at 80 points and Fenerbahçe had a better head-to-head record. Fenerbahçe needed only a win to defend their title and win their third successive championship. However, a 1-1 draw to Denizlispor combined with a 3-0 Galatasaray win against Kayserispor gave Galatasaray their 16th league title.
Squad statistics
Players in / out
Süper Lig
Standings
Türkiye Kupası
Group stage
Quarter-final
UEFA Cup
First round
Friendlies
Attendance
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
2005–06 in Turkish football
Turkish football championship-winning seasons
2000s in Istanbul
Galatasaray Sports Club 2005–06 season |
https://en.wikipedia.org/wiki/Alternated%20octagonal%20tiling | In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)} or h{8,3}.
Geometry
Although a sequence of edges seem to represent straight lines (projected into curves), careful attention will show they are not straight, as can be seen by looking at it from different projective centers.
Dual tiling
In art
Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and then "fall back again whence they came". White curves within the figure, through the middle of each line of fish, divide the plane into squares and triangles in the pattern of the tritetragonal tiling. However, in the tritetragonal tiling, the corresponding curves are chains of hyperbolic line segments, with a slight angle at each vertex, while in Escher's woodcut they appear to be smooth hypercycles.
Related polyhedra and tiling
See also
Circle Limit III
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Douglas Dunham Department of Computer Science University of Minnesota, Duluth
Examples Based on Circle Limits III and IV, 2006:More “Circle Limit III” Patterns, 2007:A “Circle Limit III” Calculation, 2008:A “Circle Limit III” Backbone Arc Formula
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Uniform tilings
Octagonal tilings |
https://en.wikipedia.org/wiki/Cantic%20octagonal%20tiling | In geometry, the tritetratrigonal tiling or shieldotritetragonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2(4,3,3). It can also be named as a cantic octagonal tiling, h2{8,3}.
Dual tiling
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Uniform tilings
Octagonal tilings |
https://en.wikipedia.org/wiki/Snub%20order-8%20triangular%20tiling | In geometry, the snub tritetratrigonal tiling or snub order-8 triangular tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of s{(3,4,3)} and s{3,8}.
Images
Drawn in chiral pairs:
Symmetry
The alternated construction from the truncated order-8 triangular tiling has 2 colors of triangles and achiral symmetry. It has Schläfli symbol of s{3,8}.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-8 tilings
Snub tilings
Triangular tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Alternated%20order-4%20hexagonal%20tiling | In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}.
Uniform constructions
There are four uniform constructions, with some of lower ones which can be seen with two colors of triangles:
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hexagonal tilings
Hyperbolic tilings
Isogonal tilings
Order-4 tilings
Semiregular tilings |
https://en.wikipedia.org/wiki/Cantic%20order-4%20hexagonal%20tiling | In geometry, the cantic order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{(4,4,3)} or h2{6,4}.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hexagonal tilings
Hyperbolic tilings
Isogonal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Snub%20order-6%20square%20tiling | In geometry, the snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}.
Images
Symmetry
The symmetry is doubled as a snub order-6 square tiling, with only one color of square. It has Schläfli symbol of s{4,6}.
Related polyhedra and tiling
The vertex figure 3.3.3.4.3.4 does not uniquely generate a uniform hyperbolic tiling. Another with quadrilateral fundamental domain (3 2 2 2) and 2*32 symmetry is generated by :
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
Footnotes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Sharath%20Kuniyil | Sharath Kuniyil is a goalkeeper who played for Mumbai FC and Mohammedan in the I-League. He made his debut in a 1-0 defeat by Salgaoca.
Career statistics
Club
Statistics accurate as of 11 May 2013
References
Mumbai FC players
Indian men's footballers
Living people
1986 births
Men's association football goalkeepers
Footballers from Kerala |
https://en.wikipedia.org/wiki/Log5 | Log5 is a method of estimating the probability that team A will win a game against team B, based on the odds ratio between the estimated winning probability of Team A and Team B against a larger set of teams.
Let and be the average winning probabilities of team A and B and let be the probability of team A winning over team B, then we have the following odds ratio equation
One can then solve
The name Log5 is due to Bill James but the method of using odds ratios in this way dates back much farther. This is in effect a logistic rating model and is therefore equivalent to the Bradley–Terry model used for paired comparisons, the Elo rating system used in chess and the Rasch model used in the analysis of categorical data.
The following notable properties exist:
If , Log5 will give A a 100% chance of victory.
If , Log5 will give A a 0% chance of victory.
If , Log5 will give each team a 50% chance of victory.
If , Log5 will give A a probability of victory.
If , Log5 will give A a chance of victory.
Additional applications
In addition to head-to-head winning probability, a general formula can be applied to calculate head-to-head probability of outcomes such as batting average in baseball.
Sticking with our batting average example, let be the batter's batting average (probability of getting a hit), and let be the pitcher's batting average against (probability of allowing a hit). Let be the league-wide batting average (probability of anyone getting a hit) and let be the probability of batter B getting a hit against pitcher P.
Or, simplified as
References
Bill James
Baseball terminology
Baseball statistics
Sports records and statistics |
https://en.wikipedia.org/wiki/Risk%20inclination%20formula | The risk inclination formula uses the principle of moments, or Varignon's theorem, to calculate the first factorial moment of probability in order to define this center point of balance among all confidence weights (i.e., the point of risk equilibration).
The formal derivation of the RIF is divided into three separate calculations: (1) calculation of 1st factorial moment, (2) calculation of inclination, and (3) calculation of the risk inclination score.
The RIF is a component of the risk inclination model.
References
Actuarial science
Applied psychology
Risk
Educational assessment and evaluation
Educational research
Psychometrics |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20Galatasaray%20S.K.%20season | The 2006–07 season was Galatasaray's 103rd in existence and the 49th 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 have played in the season.
Squad statistics
Players in / out
Süper Lig
Standings
Türkiye Kupası
Group stage
Quarter-final
UEFA Champions League
Third qualifying round
Group stage
Süper Kupa
Friendlies
Attendance
Notes
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
2000s in Istanbul
Galatasaray Sports Club 2006–07 season |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Galatasaray%20S.K.%20season | The 2007–08 season was Galatasaray's 104th in existence and the 50th 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 have played in the season.
Squad statistics
Players in / out
Süper Lig
Standings
Türkiye Kupası
Group stage
Quarter-final
Semi-final
UEFA Cup
Second qualifying round
First round
Group stage
Round of 32
Attendance
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
Turkish football championship-winning seasons
2000s in Istanbul
Galatasaray Sports Club 2007–08 season |
https://en.wikipedia.org/wiki/Akito%20Kawamoto | is a Japanese football player who plays for Nankatsu SC. He made his debut on 2 March 2013 in a 1–1 draw against Vegalta Sendai.
Club statistics
Updated to 23 February 2016.
References
External links
Profile at Ventforet Kofu
1990 births
Living people
Ryutsu Keizai University alumni
Association football people from Shiga Prefecture
Japanese men's footballers
J1 League players
J2 League players
Ventforet Kofu players
Tochigi SC players
Men's association football forwards
FISU World University Games gold medalists for Japan
Universiade medalists in football
Medalists at the 2011 Summer Universiade |
https://en.wikipedia.org/wiki/Kenneth%20Young%20%28physicist%29 | Kenneth Young (楊綱凱 1947) is a professor of physics at the Chinese University of Hong Kong (CUHK). He obtained his BSc in Physics in 1969, and his PhD in Physics and Mathematics at the California Institute of Technology, USA. He took a position at CUHK in 1973, and embarked on a highly regarded career as a theoretical physicist. He has produced extensive research in elementary particles, field theory, high energy phenomenology and dissipative systems. Young has contributed greatly to the development of higher education in Hong Kong, administering grants, educational program development, and worked to develop both Chinese and international professional associations by assuming various responsibilities during their development. In the later stages of his career Young has moved away from administration roles in universities, and toward direct teaching of students. He reflects that "one has to have passion in one’s subject. You cannot disguise it and it would help tremendously if the students could feel and see you have it in you. It makes teaching all the more effective." He is also a proponent of contextual teaching in physics.
Awards and honours
Founding Master-Designate of CW Chu College
Pro-Vice-Chancellor, CUHK
Trustee of the Croucher Foundation
Fellow of the American Physical Society
Vice-Chancellor’s Exemplary Teaching Award, CUHK
Chairman of the Hong Kong Research Grants Council
Member of the International Eurasian Academy of Sciences
Member of the University Grants Committee, Hong Kong Special Administrative Region
Secretary and then Vice-President of the Association of Asia Pacific Physical Societies
Vice Chairman of the Board of Adjudicators for the Shaw Prize
Selected publications
A. Maassen van den Brink, K. Young and M.H. Yung, "Eigenvector expansion and Petermann factor for ohmically damped oscillators", Journal of Physics A, Vol. 39, pp. 3725–3740 (2006).
T.S. Lo, S.S.M. Wong and K. Young, "Determination of a finite-range potential from discrete phase-shift data by inverse scattering", Physical Review Letters, Vol. 37, pp. 9501–9513 (2004).
E.S.C. Ching, P.T. Leung, A. Maassen van den Brink, W.M. Suen, S.S. Tong and K. Young, "Quasinormal-mode expansion for waves in open systems", Reviews of Modern Physics, Vol. 70, pp. 1545–1554 (1998).
P.T. Leung, Y.T. Liu, W.M. Suen, C.Y. Tam and K. Young, "Quasinormal Modes of Dirty Black Holes", Physical Review Letters, Vol. 78, pp. 2894–2897 (1997).
E.S.C. Ching, P.T. Leung, W.M. Suen and K. Young, "Late-Time Tail of Wave Propagation on Curved Space Time", Physical Review Letters, Vol. 74, pp. 2414–2417 (1995).
References
Living people
1947 births
Theoretical physicists
California Institute of Technology alumni
Academic staff of the Chinese University of Hong Kong
20th-century Chinese physicists
21st-century Chinese physicists
Alumni of St. Paul's Co-educational College
Fellows of the American Physical Society |
https://en.wikipedia.org/wiki/Rados%C5%82aw%20Murawski | Radosław Paweł Murawski (born 22 April 1994) is a Polish professional footballer who plays as a midfielder for Lech Poznań.
Career statistics
Honours
Lech Poznań
Ekstraklasa: 2021–22
References
External links
1994 births
Living people
Footballers from Gliwice
Polish men's footballers
Poland men's youth international footballers
Poland men's under-21 international footballers
Men's association football midfielders
Piast Gliwice players
Palermo FC players
Denizlispor footballers
Lech Poznań players
Lech Poznań II players
Ekstraklasa players
I liga players
II liga players
Serie B players
Süper Lig players
Polish expatriate men's footballers
Expatriate men's footballers in Italy
Expatriate men's footballers in Turkey
Polish expatriate sportspeople in Italy
Polish expatriate sportspeople in Turkey |
https://en.wikipedia.org/wiki/Order-3%20apeirogonal%20tiling | In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.
The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as {∞,2}.
Images
Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.
Uniform colorings
Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal tiling, each from different reflective triangle group domains:
Symmetry
The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,∞)] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as ∞∞2 symmetry by adding a mirror bisecting the fundamental domain. Dividing a fundamental domain by 3 mirrors creates a ∞32 symmetry.
A larger subgroup is constructed [(∞,∞,∞*)], index 8, as (∞*∞∞) with gyration points removed, becomes (*∞∞).
Related polyhedra and tilings
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.
See also
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
Hexagonal tiling honeycomb, similar {6,3,3} honeycomb in H3.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-3 tilings
Regular tilings |
https://en.wikipedia.org/wiki/Infinite-order%20triangular%20tiling | In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.
Symmetry
A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction.
Related polyhedra and tiling
This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.
Other infinite-order triangular tilings
A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:
See also
Infinite-order tetrahedral honeycomb
List of regular polytopes
List of uniform planar tilings
Tilings of regular polygons
Triangular tiling
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic tilings
Infinite-order tilings
Isogonal tilings
Isohedral tilings
Regular tilings
Triangular tilings |
https://en.wikipedia.org/wiki/Truncated%20order-3%20apeirogonal%20tiling | In geometry, the truncated order-3 apeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{∞,3}.
Dual tiling
The dual tiling, the infinite-order triakis triangular tiling, has face configuration V3.∞.∞.
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
See also
List of uniform planar tilings
Tilings of regular polygons
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Order-3 tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Triapeirogonal%20tiling | In geometry, the triapeirogonal tiling (or trigonal-horocyclic tiling) is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,3}.
Uniform colorings
The half-symmetry form, , has two colors of triangles:
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform quasiregular polyhedra with vertex configurations (3.n.3.n), and [n,3] Coxeter group symmetry.
See also
List of uniform planar tilings
Tilings of regular polygons
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
http://bendwavy.org/klitzing/incmats/o3xinfino.htm
o3x∞o
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Isotoxal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Truncated%20infinite-order%20triangular%20tiling | In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.
Symmetry
The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.
See also
List of uniform planar tilings
Tilings of regular polygons
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic tilings
Infinite-order tilings
Isogonal tilings
Triangular tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Rhombitriapeirogonal%20tiling | In geometry, the rhombtriapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of rr{∞,3}.
Symmetry
This tiling has [∞,3], (*∞32) symmetry. There is only one uniform coloring.
Similar to the Euclidean rhombitrihexagonal tiling, by edge-coloring there is a half symmetry form (3*∞) orbifold notation. The apeireogons can be considered as truncated, t{∞} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,∞}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an infinite-order triangular tiling results, constructed as a snub triapeirotrigonal tiling, .
Related polyhedra and tiling
Symmetry mutations
This hyperbolic tiling is topologically related as a part of sequence of uniform cantellated polyhedra with vertex configurations (3.4.n.4), and [n,3] Coxeter group symmetry.
See also
List of uniform planar tilings
Tilings of regular polygons
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Truncated%20triapeirogonal%20tiling | In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.
Symmetry
The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
A special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]+, (∞∞3), and semidirect subgroup [(∞,∞,3+)], (3*∞). Given [∞,3] with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.
An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).
Related polyhedra and tiling
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
See also
List of uniform planar tilings
Tilings of regular polygons
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Snub%20triapeirogonal%20tiling | In geometry, the snub triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of sr{∞,3}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
The dual tiling:
Related polyhedra and tiling
This hyperbolic tiling is topologically related as a part of sequence of uniform snub polyhedra with vertex configurations (3.3.3.3.n), and [n,3] Coxeter group symmetry.
See also
List of uniform planar tilings
Tilings of regular polygons
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Apeirogonal tilings
Chiral figures
Hyperbolic tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/D%C3%A1vid%20M%C3%A1rkv%C3%A1rt | Dávid Márkvárt (born 20 September 1994) is a Hungarian football player who plays for Szeged.
Club career
On 21 June 2021, Márkvárt signed with Vasas.
Club statistics
Updated to games played as of 20 May 2021.
References
External links
HLSZ
MLSZ
1994 births
Living people
Footballers from Szekszárd
Hungarian men's footballers
Men's association football midfielders
Hungary men's international footballers
Pécsi MFC players
Puskás Akadémia FC players
Diósgyőri VTK players
Vasas SC players
Szeged-Csanád Grosics Akadémia footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Tetraapeirogonal%20tiling | In geometry, the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.
Uniform constructions
There are 3 lower symmetry uniform construction, one with two colors of apeirogons, one with two colors of squares, and one with two colors of each:
Symmetry
The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating *∞∞2 and *∞44 symmetry.
Related polyhedra and tiling
See also
List of uniform planar tilings
Tilings of regular polygons
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, "The Hyperbolic Archimedean Tessellations")
External links
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Isotoxal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Infinite-order%20square%20tiling | In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.
Uniform colorings
There is a half symmetry form, , seen with alternating colors:
Symmetry
This tiling represents the mirror lines of *∞∞∞∞ symmetry. The dual to this tiling defines the fundamental domains of (*2∞) orbifold symmetry.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
References
External links
Hyperbolic and Spherical Tiling Gallery
Hyperbolic tilings
Infinite-order tilings
Isogonal tilings
Isohedral tilings
Regular tilings
Square tilings |
https://en.wikipedia.org/wiki/Truncated%20order-4%20apeirogonal%20tiling | In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}.
Uniform colorings
A half symmetry coloring is tr{∞,∞}, has two types of apeirogons, shown red and yellow here. If the apeirogonal curvature is too large, it doesn't converge to a single ideal point, like the right image, red apeirogons below. Coxeter diagram are shown with dotted lines for these divergent, ultraparallel mirrors.
Symmetry
From [∞,∞] symmetry, there are 15 small index subgroup by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as ∞42 symmetry by adding a mirror bisecting the fundamental domain. The subgroup index-8 group, [1+,∞,1+,∞,1+] (∞∞∞∞) is the commutator subgroup of [∞,∞].
Related polyhedra and tiling
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Order-4 tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Order-4%20apeirogonal%20tiling | In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.
Symmetry
This tiling represents the mirror lines of *2∞ symmetry. It dual to this tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices.
Uniform colorings
Like the Euclidean square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r{∞,∞}, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations.
Related polyhedra and tiling
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
See also
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-4 tilings
Regular tilings |
https://en.wikipedia.org/wiki/Truncated%20infinite-order%20square%20tiling | In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.
Uniform color
In (*∞44) symmetry this tiling has 3 colors. Bisecting the isosceles triangle domains can double the symmetry to *∞42 symmetry.
Symmetry
The dual of the tiling represents the fundamental domains of (*∞44) orbifold symmetry. From [(∞,4,4)] (*∞44) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to *∞42 by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,∞,1+,4,1+,4)] (∞22∞22) is the commutator subgroup of [(∞,4,4)].
Related polyhedra and tiling
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
Hyperbolic tilings
Infinite-order tilings
Isogonal tilings
Square tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Truncated%20tetraapeirogonal%20tiling | In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.
Related polyhedra and tilings
Symmetry
The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,∞,1+,4,1+] (∞2∞2) is the commutator subgroup of [∞,4].
A larger subgroup is constructed as [∞,4*], index 8, as [∞,4+], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞4), and another [∞*,4], index ∞ as [∞+,4], (∞*2) with gyration points removed as (*2∞). And their direct subgroups [∞,4*]+, [∞*,4]+, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2∞).
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
Apeirogonal tilings
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings |
https://en.wikipedia.org/wiki/Rhombitetraapeirogonal%20tiling | In geometry, the rhombitetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{∞,4}.
Constructions
There are two uniform constructions of this tiling, one from [∞,4] or (*∞42) symmetry, and secondly removing the mirror middle, [∞,1+,4], gives a rectangular fundamental domain [∞,∞,∞], (*∞222).
Symmetry
The dual of this tiling, called a deltoidal tetraapeirogonal tiling represents the fundamental domains of (*∞222) orbifold symmetry. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.
Related polyhedra and tiling
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Snub%20tetraapeirogonal%20tiling | In geometry, the snub tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{∞,4}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
Chiral figures
Hyperbolic tilings
Infinite-order tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Apeirogonal%20tiling | In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include:
Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces
Order-3 apeirogonal tiling, hyperbolic tiling with 3 apeirogons around a vertex
Order-4 apeirogonal tiling, hyperbolic tiling with 4 apeirogons around a vertex
Order-5 apeirogonal tiling, hyperbolic tiling with 5 apeirogons around a vertex
Infinite-order apeirogonal tiling, hyperbolic tiling with an infinite number of apeirogons around a vertex
See also
Apeirogonal antiprism
Apeirogonal prism
Apeirohedron
Apeirogonal tilings |
https://en.wikipedia.org/wiki/Georgia%20Benkart | Georgia McClure Benkart (December 30, 1947 – April 29, 2022) was an American mathematician who was known for her work in the structure and representation theory of Lie algebras and related algebraic structures.
She published over 130 journal articles and co-authored three American Mathematical Society memoirs in four broad categories: modular Lie algebras; combinatorics of Lie algebra representations; graded algebras and superalgebras; and quantum groups and related structures.
Education and career
Benkart received her BS degree summa cum laude from the Ohio State University in 1970 and an MPhil in mathematics from Yale University in 1973. She completed her doctoral work at Yale under Nathan Jacobson and wrote a dissertation entitled Inner Ideals and the Structure of Lie Algebras. She was awarded a PhD in mathematics from the Yale University in 1974.
Upon completing her doctoral degree, Benkart began her long career at the University of Wisconsin–Madison, first as a MacDuffee Instructor and eventually as a E. B. Van Vleck Professor of Mathematics until she retired from teaching in 2006.
She held visiting positions at the Mathematical Sciences Research Institute in Berkeley, California, the Institute for Advanced Study in Princeton, New Jersey, the Aspen Center for Physics, and the University of Virginia. During her career, Benkart delivered over 350 invited talks including 3 plenary lectures at the Joint Mathematics Meetings and the Emmy Noether Lecture at the International Congress of Mathematicians in Seoul, South Korea in 2014.
Personal life
Benkart was born on December 30, 1947, in Youngstown, Ohio, to George Benkart II and Florence K. Benkart. Her father served in the Army Corps of Engineers and her mother was a teacher in Youngstown's "ethnically rich south side."
Benkart died on April 29, 2022, aged 74, from undisclosed causes in Madison, Wisconsin. Her survivors included her sister, Paula Kaye Benkart.
Research
Benkart made a contribution to the classification of simple modular Lie algebras. Her work with J. Marshall Osborn on toroidal rank-one Lie algebras became one of the building blocks of the classification. The complete description of Hamiltonian Lie Algebras (with Gregory, Osborn, Strade, Wilson) can stand alone, and also has applications in the theory of pro-p groups.
In 2009, she published, jointly with Thomas Gregory and Alexander Premet, the first complete proof of the recognition theorem for graded Lie algebras in characteristics at least 5.
In the early 1990s, Benkart and Efim Zelmanov started to work on classification of root-graded Lie algebras and intersection matrix algebras. The latter were introduced by Peter Slodowy in his work on singularities. Berman and Moody recognized that these algebras (generalizations of affine Kac–Moody algebras) are universal root graded Lie algebras and classified them for simply laced root systems. Benkart and Zelmanov tackled the remaining cases involving the Freudenthal magic |
https://en.wikipedia.org/wiki/Samuela%20Kautoga | Samuela Kautoga is a Fijian footballer who lives in New Zealand and plays as a defender for Manukau United.
Career statistics
Scores and results list Fiji's goal tally first, score column indicates score after each Kautoga goal.
Honours
2017 Fiji Football Association Cup Tournament Player of the Tournament
References
Living people
1987 births
I-Taukei Fijian people
Fijian men's footballers
Men's association football defenders
Fiji men's international footballers
Labasa F.C. players
Hekari United F.C. players
Lautoka F.C. players
Ba F.C. players
Amicale F.C. players
2008 OFC Nations Cup players
2012 OFC Nations Cup players
2016 OFC Nations Cup players
Fijian expatriate sportspeople in New Zealand
Expatriate men's association footballers in New Zealand
Fijian expatriate sportspeople in Papua New Guinea
Expatriate men's footballers in Papua New Guinea
Fijian expatriate sportspeople in Vanuatu
Expatriate men's footballers in Vanuatu |
https://en.wikipedia.org/wiki/Shafarevich%E2%80%93Weil%20theorem | In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois groups. It was introduced by for local fields and by for global fields.
Statement
Suppose that F is a global field, K is a normal extension of F, and L is an abelian extension of K. Then the Galois group Gal(L/F) is an extension of the group Gal(K/F) by the abelian group Gal(L/K), and this extension corresponds to an element of the cohomology group H2(Gal(K/F), Gal(L/K)). On the other hand, class field theory gives a fundamental class in H2(Gal(K/F),IK) and a reciprocity law map from IK to Gal(L/K). The Shafarevich–Weil theorem states that the class of the extension Gal(L/F) is the image of the fundamental class under the homomorphism of cohomology groups induced by the reciprocity law map .
Shafarevich stated his theorem for local fields in terms of division algebras rather than the fundamental class . In this case, with L the maximal abelian extension of K, the extension Gal(L/F) corresponds under the reciprocity map to the normalizer of K in a division algebra of degree [K:F] over F, and Shafarevich's theorem states that the Hasse invariant of this division algebra is 1/[K:F]. The relation to the previous version of the theorem is that division algebras correspond to elements of a second cohomology group (the Brauer group) and under this correspondence the division algebra with Hasse invariant 1/[K:F] corresponds to the fundamental class.
References
Reprinted in his collected works, pages 4–5
, reprinted in volume I of his collected papers,
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Shafarevich%20theorem | In mathematics, the Shafarevich theorem, named for Igor Shafarevich, may refer to:
Néron–Ogg–Shafarevich criterion
Golod–Shafarevich theorem about class field towers
Grothendieck–Ogg–Shafarevich formula
Shafarevich's theorem on solvable Galois groups
Shafarevich–Weil theorem about the fundamental class in class field theory
Shafarevich's theorem on elliptic curves with good reduction outside a given set. |
https://en.wikipedia.org/wiki/Onsager%E2%80%93Machlup%20function | The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to the Lagrangian of a dynamical system. It is named after Lars Onsager and who were the first to consider such probability densities.
The dynamics of a continuous stochastic process from time to in one dimension, satisfying a stochastic differential equation
where is a Wiener process, can in approximation be described by the probability density function of its value at a finite number of points in time :
where
and , and . A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes , but in the limit the probability density function becomes ill defined, one reason being that the product of terms
diverges to infinity. In order to nevertheless define a density for the continuous stochastic process , ratios of probabilities of lying within a small distance from smooth curves and are considered:
as , where is the Onsager–Machlup function.
Definition
Consider a -dimensional Riemannian manifold and a diffusion process on with infinitesimal generator , where is the Laplace–Beltrami operator and is a vector field. For any two smooth curves ,
where is the Riemannian distance, denote the first derivatives of , and is called the Onsager–Machlup function.
The Onsager–Machlup function is given by
where is the Riemannian norm in the tangent space at , is the divergence of at , and is the scalar curvature at .
Examples
The following examples give explicit expressions for the Onsager–Machlup function of a continuous stochastic processes.
Wiener process on the real line
The Onsager–Machlup function of a Wiener process on the real line is given by
Proof: Let be a Wiener process on and let be a twice differentiable curve such that . Define another process by and a measure by
For every , the probability that for every satisfies
By Girsanov's theorem, the distribution of under equals the distribution of under , hence the latter can be substituted by the former:
By Itō's lemma it holds that
where is the second derivative of , and so this term is of order on the event where for every and will disappear in the limit , hence
Diffusion processes with constant diffusion coefficient on Euclidean space
The Onsager–Machlup function in the one-dimensional case with constant diffusion coefficient is given by
In the -dimensional case, with equal to the unit matrix, it is given by
where is the Euclidean norm and
Generalizations
Generalizations have been obtained by weakening the differentiability condition on the curve . Rather than taking the maximum distance between the stochastic process and the curve over a time interval, other conditions have been considered such as distances based on completely convex norms and Hölder, Besov and Sobolev type no |
https://en.wikipedia.org/wiki/Givi%20Ioseliani | Givi Ioseliani (born 25 October 1990 in Tbilisi) is a Georgian football player who currently plays for Samtredia.
Club statistics
Updated to games played as of 12 May 2013.
References
Sources
MLSZ
1990 births
Living people
Footballers from Tbilisi
Men's footballers from Georgia (country)
Georgia (country) men's under-21 international footballers
Men's association football midfielders
FC Torpedo Kutaisi players
Kecskeméti TE players
FC Sioni Bolnisi players
FC Tskhinvali players
FC Samtredia players
Dinamo Zugdidi players
Erovnuli Liga players
Nemzeti Bajnokság I players
Expatriate men's footballers from Georgia (country)
Expatriate men's footballers in Hungary
Expatriate sportspeople from Georgia (country) in Hungary |
https://en.wikipedia.org/wiki/Tangent%20space%20to%20a%20functor | In algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction is based on the following observation. Let X be a scheme over a field k.
To give a -point of X is the same thing as to give a k-rational point p of X (i.e., the residue field of p is k) together with an element of ; i.e., a tangent vector at p.
(To see this, use the fact that any local homomorphism must be of the form
)
Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point , the fiber of over p is called the tangent space to F at p.
If the functor F preserves fibered products (e.g. if it is a scheme), the tangent space may be given the structure of a vector space over k. If F is a scheme X over k (i.e., ), then each v as above may be identified with a derivation at p and this gives the identification of with the space of derivations at p and we recover the usual construction.
The construction may be thought of as defining an analog of the tangent bundle in the following way. Let . Then, for any morphism of schemes over k, one sees ; this shows that the map that f induces is precisely the differential of f under the above identification.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Signy%20Arctander | Signy Arctander (26 October 1895 – 23 September 1971) was a Norwegian statistician and economist. She was born in Bergen, a daughter of politician Sofus Arctander. She was appointed at the Statistics Norway from 1920, and worked for this institution until her retirement in 1965; from 1960 to 1963 as acting director. Among her research works are Miljøundersøkelse for Oslo from 1928, two reports on the situation of children, and the study Arbeidsvilkår for hushjelp from 1937. She was decorated Knight, First Class of the Order of St. Olav in 1966.
References
1895 births
1971 deaths
Scientists from Bergen
Norwegian economists
Norwegian women economists
Norwegian statisticians
Women statisticians |
https://en.wikipedia.org/wiki/Infinite-order%20apeirogonal%20tiling | In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.
Symmetry
This tiling represents the fundamental domains of *∞∞ symmetry.
Uniform colorings
This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.
Related polyhedra and tiling
The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain.
a{∞,∞} or = ∪
See also
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Apeirogonal tilings
Hyperbolic tilings
Infinite-order tilings
Isogonal tilings
Isohedral tilings
Regular tilings
Self-dual tilings |
https://en.wikipedia.org/wiki/Snub%20apeiroapeirogonal%20tiling | In geometry, the snub apeiroapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{∞,∞}. It has 3 equilateral triangles and 2 apeirogons around every vertex, with vertex figure 3.3.∞.3.∞.
Dual tiling
Related polyhedra and tiling
The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.
See also
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
Hyperbolic tilings
Infinite-order tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Circular%20analysis | In statistics, circular analysis is the selection of the details of a data analysis using the data that is being analysed. It is often referred to as double dipping, as one uses the same data twice. Circular analysis unjustifiably inflates the apparent statistical strength of any results reported and, at the most extreme, can lead to the apparently significant result being found in data that consists only of noise. In particular, where an experiment is implemented to study a postulated effect, it is a misuse of statistics to initially reduce the complete dataset by selecting a subset of data in ways that are aligned to the effects being studied. A second misuse occurs where the performance of a fitted model or classification rule is reported as a raw result, without allowing for the effects of model-selection and the tuning of parameters based on the data being analyzed.
Examples
At its most simple, it can include the decision to remove outliers, after noticing this might help improve the analysis of an experiment. The effect can be more subtle. In functional magnetic resonance imaging (fMRI) data, for example, considerable amounts of pre-processing is often needed. These might be applied incrementally until the analysis 'works'. Similarly, the classifiers used in a multivoxel pattern analysis of fMRI data require parameters, which could be tuned to maximise the classification accuracy.
In geology, the potential for circular analysis has been noted in the case of maps of geological faults, where these may be drawn on the basis of an assumption that faults develop and propagate in a particular way, with those maps being later used as evidence that faults do actually develop in that way.
Solutions
Careful design of the analysis one plans to perform, prior to collecting the data, means the analysis choice is not affected by the data collected. Alternatively, one might decide to perfect the classification on one or two participants, and then use the analysis on the remaining participant data. Regarding the selection of classification parameters, a common method is to divide the data into two sets, and find the optimum parameter using one set and then test using this parameter value on the second set. This is a standard technique used (for example) by the princeton MVPA classification library.
Notes
References
Model selection
Misuse of statistics |
https://en.wikipedia.org/wiki/Influential%20observation | In statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates.
Assessment
Various methods have been proposed for measuring influence. Assume an estimated regression , where is an n×1 column vector for the response variable, is the n×k design matrix of explanatory variables (including a constant), is the n×1 residual vector, and is a k×1 vector of estimates of some population parameter . Also define , the projection matrix of . Then we have the following measures of influence:
, where denotes the coefficients estimated with the i-th row of deleted, denotes the i-th value of matrix's main diagonal. Thus DFBETA measures the difference in each parameter estimate with and without the influential point. There is a DFBETA for each variable and each observation (if there are N observations and k variables there are N·k DFBETAs). Table shows DFBETAs for the third dataset from Anscombe's quartet (bottom left chart in the figure):
Outliers, leverage and influence
An outlier may be defined as a data point that differs significantly from other observations.
A high-leverage point are observations made at extreme values of independent variables.
Both types of atypical observations will force the regression line to be close to the point.
In Anscombe's quartet, the bottom right image has a point with high leverage and the bottom left image has an outlying point.
See also
Influence function (statistics)
Outlier
Leverage
Partial leverage
Regression analysis
Anomaly detection
References
Further reading
Actuarial science
Regression diagnostics
Robust statistics |
https://en.wikipedia.org/wiki/Gender%20gaps%20in%20mathematics%20and%20reading | The gender gaps in mathematics and reading achievement refer to the finding that, on average, the two sexes perform differently in mathematics and reading skills on tests. On average, boys and men exceed in mathematics, while girls and women exceed in reading skills.
Mathematics and reading gaps by country
The Programme for International Student Assessment assesses the performance of 15-year-olds in mathematics and reading in OECD and OECD partner countries. The table below lists the scores of the PISA 2009 assessment in mathematics and reading by country, as well as the difference between boys and girls. Gaps in bold font mean that the gender gap is statistically significant (p<0.05). A positive mathematics gap means that boys outperform girls, a negative mathematics gap means that girls outperform boys. A positive reading gap means that girls outperform boys (no country has a negative reading gap). There is a negative correlation between the mathematics and reading gender gaps, that is, nations with a larger mathematics gap have a smaller reading gap and vice versa.
References
Gender equality
Gender and education |
https://en.wikipedia.org/wiki/List%20of%20East%20Stirlingshire%20F.C.%20records%20and%20statistics | East Stirlingshire F.C. is a Scottish association football club from Falkirk. The club was founded in 1881 joined the Scottish Football League in 1900.
This list encompasses the major honours won by East Stirlingshire, records set by the club, its managers and its players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by East Stirlingshire players in international tournaments, and the highest transfer fees paid and received by the club. Attendance records at Firs Park, the club's home ground between 1921 and 2008, are also included in the list.
Honours
League
Division Two (before 1975) and First Division (after 1975):
Winners (1): 1931–32
Runners-up (1): 1962–63
Division Three (1923 to 1926), Division C (1946 to 1949) and Second Division (after 1975):
Winners (1): 1947–48
Runners-up (2): 1923–24, 1979–80
Runners-up (3) : 2016–17 in Lowland Football League.
Player records
Most appearances
Scottish Football League appearances from 1948 to 2013 only.
Goalscorers
Most league goals in one season: 41, Andy Rodgers in the 2016–17 Lowland Football League.
International caps
First capped player: Humphrey Jones for Wales against England on 23 February 1889.
First capped player for Scotland: Archibald Ritchie against Wales on 21 March 1891.
Most capped player: Humphrey Jones with 5 caps for Wales as an East Stirlingshire player.
Most capped player for Scotland: David Alexander, 2 caps as an East Stirlingshire player.
Transfers
Record transfer fee paid: £6,000 for Colin McKinnon from Falkirk, 1991.
Record transfer fee received: £35,000 for Jim Docherty to Chelsea, 1978.
Club records
Goals
Most league goals scored in a season: 111 (in 38 matches in the 1931–32 season, Division Two).
Fewest league goals scored in a season: 21 (in 22 matches in the 1911–12 season, Division Two).
Most league goals conceded in a season: 121 (in 36 matches in the 1956–57 season, Division Two).
Fewest league goals conceded in a season: 26 (in 22 matches in the 1947–48 season, Division C).
Points
Most points in a season:
Two points for a win: 55 (in 38 matches in the 1931–32 season, Division Two).
Three points for a win: 67 (in 30 matches in the 2016–17 season, Scottish Lowland Football League).
Fewest points in a season:
Two points for a win:
12 (in 22 matches in the 1905–06 season, Division Two).
12 (in 34 matches in the 1963–64 season, Division One).
Three points for a win:
8 (in 36 matches in the 2003–04 season, Third Division).
Matches
Firsts
First match as Britannia: Britannia 0–7 Falkirk 2nd XI, 4 December 1880.
First match as East Stirlingshire: Falkirk 5–0 East Stirlingshire, 27 August 1881.
First league match: East Stirlingshire 2–3 Airdrieonians, 18 August 1900.
First Lowland Football League match: East Stirlingshire 2-2 Vale of Leithen, 6 August 2016.
First Scottish Cup match: Milngavie Thistle 2–1 East Stirlingshire, S |
https://en.wikipedia.org/wiki/Shih%20Su-mei | Shih Su-mei (; born 20 July 1952) is a Taiwanese politician. She was the Minister of the Directorate General of Budget, Accounting and Statistics of the Executive Yuan from 2008 to 2016.
Education
Shih earned her bachelor's degree in business administration from National Taiwan University.
References
1952 births
Living people
Kuomintang politicians in Taiwan
National Taiwan University alumni
Women government ministers of Taiwan
21st-century Taiwanese women politicians
21st-century Taiwanese politicians
Government ministers of Taiwan
Politicians of the Republic of China on Taiwan from Taipei |
https://en.wikipedia.org/wiki/Adam%20Deja | Adam Deja (born 24 June 1993) is a Polish professional footballer who plays as a midfielder for Górnik Łęczna.
Career statistics
Club
References
1993 births
Living people
People from Olesno
Footballers from Opole Voivodeship
Polish men's footballers
Men's association football midfielders
Ekstraklasa players
I liga players
II liga players
Górnik Zabrze players
MKS Kluczbork players
Podbeskidzie Bielsko-Biała players
MKS Cracovia players
Arka Gdynia players
Korona Kielce players
GKS Górnik Łęczna players |
https://en.wikipedia.org/wiki/Architectonic%20and%20catoptric%20tessellation | In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space with prime space groups and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of the corresponding catoptric tessellation, and vice versa. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as gyrations or prismatic stacks (and their duals) which are excluded from these categories.
Enumeration
The pairs of architectonic and catoptric tessellations are listed below with their symmetry group. These tessellations only represent four symmetry space groups, and also all within the cubic crystal system. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed.
Vertex Figures
The vertex figures of all architectonic honeycombs, and the dual cells of all catoptric honeycombs are shown below, at the same scale and the same orientation:
Symmetry
These four symmetry groups are labeled as:
References
Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi (2009), p. 54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry
Further reading
Branko Grünbaum, (1994) Uniform tilings of 3-space. Geombinatorics 4, 49 - 56.
Norman Johnson (1991) Uniform Polytopes, Manuscript
A. Andreini, (1905) Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 75–129. PDF
George Olshevsky, (2006) Uniform Panoploid Tetracombs, Manuscript PDF
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] See p318
Honeycombs (geometry)
John Horton Conway |
https://en.wikipedia.org/wiki/Furug%20Qodirov | Furug Qodirov is a Tajikistani footballer who plays as a defender for CSKA Pomir Dushanbe.
Career statistics
International
Statistics accurate as of match played 23 July 2011
References
External links
1992 births
Living people
Tajikistani men's footballers
Tajikistan men's international footballers
Men's association football defenders
Place of birth missing (living people)
Tajikistan men's youth international footballers |
https://en.wikipedia.org/wiki/Arthur%20Stanley%20Ramsey | Arthur Stanley Ramsey (9 September 1867 – 31 December 1954) was a British mathematician and author of mathematics and physics textbooks. He was Fellow of Magdalene College, Cambridge, and its President from 1915–52.
Biography
The son of Rev. Adam Averell Ramsey of Dewsbury, a Congregational minister, and his wife Hephzibah, Ramsey was educated at Batley Grammar School and Magdalene College, Cambridge where he read Mathematics (B.A. (6th Wrangler) 1889; M.A. 1893).
He was Assistant Master at Fettes College from 1890 to 1897, moving into academia as Fellow of Magdalene in 1897. He was Bursar of the college, 1904–13 and University Lecturer in Mathematics, 1926-32. As a tutor, he supervised the maths work of William Empson, who would go on to apply path-breaking tools of analytical logic to the criticism of literature.
In 1902 Ramsey married (Mary) Agnes (1875-1927), daughter of Rev. Plumpton Stravenson Wilson, vicar of Horbling, Lincs. and sister to the cricketer and footballer Geoffrey Plumpton Wilson. Mary herself was academically accomplished, having earned a Class II Honours Certificate in Modern History from St Hugh's College, Oxford. In April 1913, Mary stood for election to the Cambridge Board of Guardians in Bridge Ward, and was elected with 321 votes.
Ramsey and his wife had two daughters, Bridget and Margaret, and two sons, philosopher and mathematician Frank Plumpton Ramsey (1903–1930) and Michael Ramsey (1904–1988) who was the Archbishop of Canterbury for thirteen years. Mary Agnes was killed in 1927 in a road traffic accident.
He is buried in the Ascension Parish Burial Ground in Cambridge; his son Frank and wife Mary are buried in the same plot. His home, Howfield, Buckingham Road, is now part of Cambridge Blackfriars.
Publications
1913: (with W. H. Besant) A Treatise on Hydromechanics from Google Books.
1956: An Introduction to the Theory of Newtonian Attraction, Cambridge University Press
References
External links
1867 births
Fellows of Magdalene College, Cambridge
1954 deaths
19th-century British mathematicians
20th-century British mathematicians
British science writers
People educated at Batley Grammar School
Alumni of Magdalen College, Oxford
Place of birth missing |
https://en.wikipedia.org/wiki/Roshdi%20Rashed | Roshdi Rashed (Arabic: رشدي راشد), born in Cairo in 1936, is a mathematician, philosopher and historian of science, whose work focuses largely on mathematics and physics of the medieval Arab world. His work explores and illuminates the unrecognized Arab scientific tradition, being one of the first historians to study in detail the ancient and medieval texts, their journey through the Eastern schools and courses, their immense contributions to Western science, particularly in regarding the development of algebra and the first formalization of physics.
Biography
Roshdi Rashed is the author of several books and scientific articles in History of Science. He is currently Emeritus Director of Research (special class) at CNRS (France). He was director of the Centre for History of Arab and Medieval Science and Philosophies (until 2001) Paris, and also director of the doctoral formation in epistemology and history of science, Paris Diderot University (until 2001). He is Emeritus Professor at Tokyo University, and at the Mansoura University, and also at the Paris Diderot University. He was a founder (1984) and Director (until May 1993) of the REHSEIS (Research Epistemology and History of Science and Scientific Institutions) research team, CNRS, Paris. He had several distinctions including: the CNRS Bronze medal (1977), Knight of the Honour Legion (1989), the Alexandre Koyré medal of the International Academy for History of Science (1990), the history of science medal and award of the Academy of Sciences for the Developing World (1990), medal and award of Kuwait Foundation for the Advancement of Sciences (1999), Avicenna gold medal of UNESCO (1999), medal of CNRS (2001), medal of the Arab World Institute (2004). He had several honorary positions as Vice-President of the International Academy of the History of Science (1997), member of the Royal Belgian Academy of Sciences (2002), member of the Tunisian Academy "Bayt al-Hikma" (2012).
Editorial activity
Editor of "Arabic Sciences and Philosophy: a historical journal", Cambridge University Press (UK), .
Editor of the series "History of the Arab Sciences", Beirut (Lebanon).
Director of Collections "Arab Science and Philosophy. Studies and times" and "Arab Science and Philosophy. Texts and Studies", Les Belles Lettres (France).
Editor of the series "Science in history", Blanchard (France).
Committee member reading of the "Revue de synthèse" (Springer Verlag), and the "Historia Scientiarum" (WHSO record number 30031), and also the "Journal de l’histoire des mathématiques" published by the "Société Mathématique de France" (Journal of the History of Mathematics published by the Mathematical Society of France).
Books
"Introduction to the History of Science," co-author.
"Vol. 1: Elements and Instruments", Hachette, Paris, 1971.
"Vol. 2: Purpose and methods. Examples", Hachette, Paris, 1972.
"Al-Bahir in Algebra As-Samaw'al", with S. Ahmad, University Press of Damascus, Damascus, 1972.
"Condorcet: Mat |
https://en.wikipedia.org/wiki/Non-autonomous%20system%20%28mathematics%29 | In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is a dynamic equation on a smooth fiber bundle over . For instance, this is the case of non-autonomous mechanics.
An r-order differential equation on a fiber bundle is represented by a closed subbundle of a jet bundle of . A dynamic equation on is a differential equation which is algebraically solved for a higher-order derivatives.
In particular, a first-order dynamic equation on a fiber bundle is a kernel of the covariant differential of some connection on . Given bundle coordinates on and the adapted coordinates on a first-order jet manifold , a first-order dynamic equation reads
For instance, this is the case of Hamiltonian non-autonomous mechanics.
A second-order dynamic equation
on is defined as a holonomic
connection on a jet bundle . This
equation also is represented by a connection on an affine jet bundle . Due to the canonical
embedding , it is equivalent to a geodesic equation
on the tangent bundle of . A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation.
See also
Autonomous system (mathematics)
Non-autonomous mechanics
Free motion equation
Relativistic system (mathematics)
References
De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989).
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ().
Differential equations
Classical mechanics
Dynamical systems |
https://en.wikipedia.org/wiki/Midpoint%20%28disambiguation%29 | A midpoint is the middle point of a line segment in geometry.
Midpoint may also refer to:
Midpoint (astrology)
Midpoint (company)
Midpoint (screenwriting)
Midpoint (album), a 2022 album by Tom Chaplin
Midpoint Café, a restaurant, souvenir and antique shop on US Route 66 in Adrian, Texas
Midpoint Memorial Bridge, connects Fort Myers and Cape Cora in Florida
Midpoint method, in numerical analysis
MidPoint Music Festival, held in Cincinnati, Ohio
Midpoint Trade Books, book sales, distribution, and marketing company founded in 1996
See also
Start Point (disambiguation)
Endpoint (disambiguation)
Bullet (typography) (•) and interpunct ( · ) are both occasionally known as "mid point" or "mid dot". |
https://en.wikipedia.org/wiki/Montserrat%20Teixidor%20i%20Bigas | Montserrat Teixidor i Bigas (born February 25, 1958) is a Spanish-American academic who is a professor of mathematics at Tufts University in Medford, Massachusetts. She specializes in algebraic geometry, especially Moduli of Vector Bundles on curves.
Education
Teixidor i Bigas was born in Barcelona in 1958. She earned a bachelor's degree and PhD from the University of Barcelona, where she wrote her dissertation, "Geometry of linear systems on algebraic curves", under the supervision of Gerard Eryk Welters.
Career
She worked in the department of pure mathematics at the University of Liverpool, where she wrote "The divisor of curves with a vanishing theta-null", for Compositio Mathematica in 1988.
In 1997, she proved Lange's conjecture for the generic curve, with Barbara Russo, which states that "If , then there exist stable vector bundles with ." They also clarified what happens in the interval using a degeneration argument to a reducible curve.
She took up an appointment as an Associate Professor of Mathematics at Tufts University, and has been on the faculty of Tufts since 1989. She has been a reviewer for several journals, including the Bulletin of the American Mathematical Society, the Duke Mathematical Journal, and the journal of algebraic geometry. She has held visiting positions at Brown University and the University of Cambridge. She was also a co-organizer of the Clay Institute's workgroup on Vector Bundles on Curves.
In 2004, she spent a year at Radcliffe College as a Vera M. Schuyler Fellow, devoting her time to study of "the interplay between the geometry of curves and the equations defining them."
Selected publications
Montserrat Teixidor i Bigas, "Brill-Noether theory for vector bundles," Duke Math. J. Volume 62, Number 2 (1991), 385-400.
Montserrat Teixidor i Bigas Curves in Grassmannians, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1597–1603
Montserrat Teixidor i Bigas "Green's conjecture for the generic -gonal curve of genus ," Duke Math. J. 111 (2002), no. 2, 195–222.
Montserrat Teixidor i Bigas Existence of coherent systems, Internat. J. Math. 19 (2008), no. 4, 449–454.
Ivona Grzegorczyk, Montserrat Teixidor i Bigas, Brill-Noether theory for stable vector bundles, Moduli spaces and vector bundles, 29–50, London Math. Soc. Lecture Note Ser., 359, CUP, Cambridge (2009)
Montserrat Teixidor i Bigas, Vector bundles on reducible curves and applications, Clay Mathematics Proceedings (2011)
Tawanda Gwena, Montserrat Teixidor i Bigas, Maps between moduli spaces of vector bundles and the base locus of the theta divisor
Brian Osserman, Montserrat Teixidor i Bigas Linked alternating forms and linked symplectic Grassmannians, Int. Math. Res. Not. IMRN 2014, no. 3, 720–744.
References
Living people
Algebraic geometers
Tufts University faculty
20th-century Spanish mathematicians
21st-century Spanish mathematicians
University of Barcelona alumni
20th-century women mathematicians
21st-century women mathematicians
Radcliffe Coll |
https://en.wikipedia.org/wiki/Stirling%20permutation | In combinatorial mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value from 1 to k) with the additional property that, for each value i appearing in the permutation, the values between the two copies of i are larger than i. For instance, the 15 Stirling permutations of order three are
1,1,2,2,3,3; 1,2,2,1,3,3; 2,2,1,1,3,3;
1,1,2,3,3,2; 1,2,2,3,3,1; 2,2,1,3,3,1;
1,1,3,3,2,2; 1,2,3,3,2,1; 2,2,3,3,1,1;
1,3,3,1,2,2; 1,3,3,2,2,1; 2,3,3,2,1,1;
3,3,1,1,2,2; 3,3,1,2,2,1; 3,3,2,2,1,1.
The number of Stirling permutations of order k is given by the double factorial (2k − 1)!!. Stirling permutations were introduced by in order to show that certain numbers (the numbers of Stirling permutations with a fixed number of descents) are non-negative. They chose the name because of a connection to certain polynomials defined from the Stirling numbers, which are in turn named after 18th-century Scottish mathematician James Stirling.
Stirling permutations may be used to describe the sequences by which it is possible to construct a rooted plane tree with k edges by adding leaves one by one to the tree. For, if the edges are numbered by the order in which they were inserted, then the sequence of numbers in an Euler tour of the tree (formed by doubling the edges of the tree and traversing the children of each node in left to right order) is a Stirling permutation. Conversely every Stirling permutation describes a tree construction sequence, in which the next edge closer to the root from an edge labeled i is the one whose pair of values most closely surrounds the pair of i values in the permutation.
Stirling permutations have been generalized to the permutations of a multiset with more than two copies of each value. Researchers have also studied the number of Stirling permutations that avoid certain patterns.
See also
Langford pairing, a different type of permutation of the same multiset
References
Permutations
Combinatorics |
https://en.wikipedia.org/wiki/Robert%20Sinclair%20MacKay | Robert Sinclair MacKay (born 1956) is a British mathematician and professor at the University of Warwick. He researches dynamical systems, the calculus of variations, Hamiltonian dynamics and applications to complex systems in physics, engineering, chemistry, biology and economics.
Education
MacKay was educated at Newcastle High School, leaving in 1974. He completed his Bachelor of Arts degree with first class honours in mathematics at Trinity College, Cambridge in 1977, and completed Part III of the tripos with distinction in 1978. He obtained his PhD in astrophysical sciences in 1982 from the Plasma Physics Laboratory at Princeton University for research supervised by John M. Greene and Martin David Kruskal.
Career and research
Between 1982 and 1995, MacKay held postdoctoral research positions at Queen Mary College, London, the Institut des Hautes Etudes Scientifiques, and the University of Warwick. From 1995 to 2000 he was Professor of Nonlinear Dynamics in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, Director of the Nonlinear Centre, and Fellow of Trinity College. In 2000 he returned to Warwick as Professor of Mathematics and Director of Mathematical Interdisciplinary Research.
Awards and honours
MacKay was awarded the Stefanos Pnevmatikos International Award in 1992. He was elected a Fellow of the Royal Society (FRS) in 2000. In 2012 he was elected President of the Institute of Mathematics and its Applications.
Personal life
MacKay was born to Donald MacCrimmon MacKay and Valerie MacKay (née Wood) in 1956. His younger brother David J. C. MacKay FRS was the Regius Professor of Engineering at the University of Cambridge.
References
Living people
20th-century British mathematicians
21st-century British mathematicians
Fellows of the Royal Society
Fellows of the Institute of Physics
People educated at Newcastle-under-Lyme School
Fellows of Trinity College, Cambridge
Academics of the University of Cambridge
Princeton University alumni
1956 births |
https://en.wikipedia.org/wiki/Relativistic%20system%20%28mathematics%29 | In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle over . For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold whose fibration over is not fixed. Such a system admits transformations of a coordinate on depending on other coordinates on . Therefore, it is called the relativistic system. In particular, Special Relativity on the
Minkowski space is of this type.
Since a configuration space of a relativistic system has no
preferable fibration over , a
velocity space of relativistic system is a first order jet
manifold of one-dimensional submanifolds of . The notion of jets of submanifolds
generalizes that of jets of sections
of fiber bundles which are utilized in covariant classical field theory and
non-autonomous mechanics. A first order jet bundle is projective and, following the terminology of Special Relativity, one can think of its fibers as being spaces
of the absolute velocities of a relativistic system. Given coordinates on , a first order jet manifold is provided with the adapted coordinates
possessing transition functions
The relativistic velocities of a relativistic system are represented by
elements of a fibre bundle , coordinated by , where is the tangent bundle of . Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads
For instance, if is the Minkowski space with a Minkowski metric , this is an equation of a relativistic charge in the presence of an electromagnetic field.
See also
Non-autonomous system (mathematics)
Non-autonomous mechanics
Relativistic mechanics
Special relativity
References
Krasil'shchik, I. S., Vinogradov, A. M., [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, .
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ().
Differential equations
Classical mechanics
Theory of relativity |
https://en.wikipedia.org/wiki/Susan%20Montgomery | M. Susan Montgomery (born 2 April 1943 in Lansing, MI) is a distinguished American mathematician whose current research interests concern noncommutative algebras: in particular, Hopf algebras, their structure and representations, and their actions on other algebras. Her early research was on group actions on rings.
Education
Montgomery received her B.A. in 1965 from the University of Michigan and her Ph.D. in Mathematics from the University of Chicago in 1969 under the supervision of I. N. Herstein.
Career
Upon receiving her Ph.D. from Chicago, Montgomery spent one year on the faculty at DePaul University. Montgomery joined the faculty of the University of Southern California (USC) in 1970 and was promoted to the rank of Professor in 1982. She was chair of the Department of Mathematics at USC
from 1996 to 1999. Montgomery has spent sabbaticals at the Hebrew University of Jerusalem, the University of Leeds, the University of Wisconsin, the University of Munich, the University of New South Wales, the Mittag-Leffler Institute, and the Mathematical Sciences Research Institute.
Montgomery wrote about a hundred research articles and several books, of which Hopf algebras and their actions on rings is her most cited work. This book includes a discussion of Hopf-Galois theory, an area to which Montgomery has significantly contributed, and an introduction to quantum group theory.
Honors
Montgomery was awarded a Guggenheim Foundation Fellowship in 1984 and a Raubenheimer Outstanding Faculty Award by USC in 1987.
She gave an American Mathematical Society (AMS) Invited Address at the Joint Mathematics Meetings in 1984. In 1995 she gave an Invited Address at the Joint AMS-Israel Math Union Meeting in Jerusalem.
In 2009, she gave a plenary lecture at the summer meeting of the Canadian Mathematical Society. She has also given numerous lectures at meetings and universities around the world.
Montgomery was the Principal Lecturer at the Conference Board of the Mathematical Sciences (CBMS) 1992 Conference on Hopf Algebras. Her CBMS monograph Hopf Algebras and their Actions on Rings is highly cited. She has written one other book and has edited five collections of research articles.
She served as an editor for the Journal of Algebra for over 20 years. She was also an editor for the AMS Proceedings, AMS Mathematical Surveys and Monographs, and Advances in Mathematics, and currently is on the editorial boards of Algebras and Representation Theory and of Algebra and Number Theory.
Montgomery has been very active in the American Mathematical Society, serving on the Board of Trustees from 1986–1996. She has also served on the Council, the Policy Committee on Publications, and on the Nominating Committee.
In 2013 she was elected to a 3-year term as a Vice-President of the American Mathematical Society. She was also a member of the National Research Council's Board on Mathematical Sciences and Their Applications (BMSA), serving one year on the Executive |
https://en.wikipedia.org/wiki/Federal%20Statistical%20System%20of%20the%20United%20States | The Federal Statistical System of the United States is the decentralized network of federal agencies which produce data and official statistics about the people, economy, natural resources, and infrastructure of the United States.
Background
In contrast to many other countries, the United States does not have a primary statistical agency. Instead, the statistical system is decentralized, with 13 statistical agencies, two of which are independent agencies and the remaining 11 generally located in different government departments. This structure keeps statistical work in close proximity to the various cabinet-level departments that use the information. In addition, three other statistical units of government agencies are recognized by the OMB as having statistical work as part of their mission.
As of fiscal year 2013 (FY13), the 13 principal statistical agencies have statistical activities as their core mission and conduct much of the government’s statistical work. A further 89 federal agencies were appropriated at least $500,000 of statistical work in FY11, FY12, or FY13 in conjunction with their primary missions. All together, the total budget allocated to the Federal Statistical System is estimated to be $6.7 billion for FY13.
The Federal Statistical System is coordinated through the Office of Management and Budget (OMB). OMB establishes and enforces statistical policies and standards, ensures that resources are proposed for priority statistical programs, and approves statistical surveys conducted by the Federal government under the Paperwork Reduction Act. The Chief Statistician of the United States, also housed within OMB, provides oversight, coordination, and guidance for Federal statistical activities, working in collaboration with leaders of statistical agencies.
Centralization efforts
To streamline operations and reduce costs, several proposals have been made to consolidate the federal statistical system into fewer agencies, or even a single agency. In 2011, President Barack Obama's proposal to reorganize the U.S. Department of Commerce included placing several statistical agencies under one umbrella.
Principal statistical agencies
Statistical units
These are subcomponents of agencies recognized by the OMB as having statistical work as part of their mission:
Microeconomic Surveys Unit (Federal Reserve Board of Governors)
Center for Behavioral Health Statistics and Quality (Substance Abuse and Mental Health Services Administration, Department of Health and Human Services)
National Animal Health Monitoring System (Animal and Plant Health Inspection Service, Department of Agriculture)
See also
Federal Statistical Research Data Centers
References
External links
Statistical Programs and Standards, U.S. Office of Management and Budget: obamawhitehouse.archives.gov
Statistical organizations in the United States
Agencies of the United States government |
https://en.wikipedia.org/wiki/Igor%20Fernandes | Igor Fernandes da Silva Araújo (born 6 June 1992), known as Igor Fernandes, is a Brazilian footballer who plays as a left back for São Bernardo.
Career statistics
Honours
Corinthians
Campeonato Paulista: 2013
Recopa Sudamericana: 2013
Sport Recife
Copa do Nordeste: 2014
Avaí
Campeonato Catarinense: 2019
Remo
Copa Verde: 2021
References
External links
Living people
1992 births
Footballers from São Paulo
Brazilian men's footballers
Men's association football defenders
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Campeonato Brasileiro Série C players
Campeonato Brasileiro Série D players
Sport Club Corinthians Paulista players
Associação Atlética Flamengo players
Sport Club do Recife players
Clube Atlético Linense players
Esporte Clube Tigres do Brasil players
Grêmio Novorizontino players
Red Bull Bragantino II players
ABC Futebol Clube players
Avaí FC players |
https://en.wikipedia.org/wiki/Adra%2C%20Syria | Adra () is a town in southern Syria, administratively part of the Rif Dimashq Governorate, located northeast of Damascus. According to the Syria Central Bureau of Statistics, the town had a population of 20,559 in the 2004 census. The Hujr ibn Adi Mosque is located in the town.
Summary
Adra is the site of Syria's largest industrial city, located immediately east of the town. The industrial zone's area is around 7,000 hectares, with half being designated for services and the other half for industries. Its total estimated cost $570 million US. In 2008 there were 90 operating factories while 1,125 factories were under construction. The investment value for active factories was $646 million US. By 2010, Adra's industrial city contained the largest number of active factories and factories under construction in Syria, with a total of 1,952, a few more than in Shaykh Najjar. The city's director-general in 2010 was Ziad Badour.
The town is the site of the Adra Prison. Nearby localities include al-Rihan and Douma to the west, al-Shafuniyah and Hawsh Nasri to the southwest, Midaa to the south, Dumeir to the east, al-Qutayfah to the north, Hufayr al-Tahta to the northwest.
History
Adra is identified with the Biblical town of "Hadrach" mentioned by Zechariah who noted the city, which was defended by Damascus, was condemned.
The Ghassanids, who were Arab Christians, dominated Adra and fought against the Arab Muslim invaders commanded by Khalid ibn al-Walid in the Battle of Marj Rahit in July, 634 CE. The battle resulted in a decisive Rashidun victory and largescale Islamization took place in the area soon after.
Adra contains several graves of sahaba ("companions" of Islamic prophet Muhammad), including most notably that of Hujr ibn Adi. The Zengid ruler Imad al-Din Zengi encamped at Adra in early 1135 before attempting to besiege Damascus which was controlled by the Burids.
In the early 19th-century a ruined khan ("caravansary") was reported by Western travelers to be near the village of Adra.
In 2013, it was the site of the Adra massacre conducted by Islamist rebels against Syrian minorities.
References
Bibliography
Populated places in Douma District |
https://en.wikipedia.org/wiki/Order-5%20hexagonal%20tiling | In geometry, the order-5 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,5}.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-5 vertices with Schläfli symbol {n,5}, and Coxeter diagram , progressing to infinity.
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hexagonal tilings
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-5 tilings
Regular tilings |
https://en.wikipedia.org/wiki/Order-6%20pentagonal%20tiling | In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.
Uniform coloring
This regular tiling can also be constructed from [(5,5,3)] symmetry alternating two colors of pentagons, represented by t1(5,5,3).
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain, and 5 mirrors meeting at a point. This symmetry by orbifold notation is called *33333 with 5 order-3 mirror intersections.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-6 tilings
Pentagonal tilings
Regular tilings |
https://en.wikipedia.org/wiki/Truncated%20pentahexagonal%20tiling | In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.
Dual tiling
Symmetry
There are four small index subgroup from [6,5] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-5 hexagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [6,5] symmetry, and 3 with subsymmetry.
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings |
https://en.wikipedia.org/wiki/James%20Oxley | James G. Oxley is an Australian–American mathematician, Boyd Professor of Mathematics at Louisiana State University. He is known for his expertise in matroid theory and graph theory.
Oxley did his undergraduate studies in Australia, and earned a doctorate from the University of Oxford in 1978, under the supervision of Dominic Welsh. He joined the Louisiana State University faculty in 1982.
Oxley is the author of the book Matroid Theory (Oxford University Press, 1992).
In 2012 he became a fellow of the American Mathematical Society.
References
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
Australian mathematicians
Graph theorists
Alumni of the University of Oxford
Louisiana State University faculty
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Truncated%20order-5%20hexagonal%20tiling | In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{6,5}.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hexagonal tilings
Hyperbolic tilings
Isogonal tilings
Order-5 tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Pentahexagonal%20tiling | In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t1{6,5}.
Uniform colorings
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isotoxal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Truncated%20order-6%20pentagonal%20tiling | In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}.
Uniform colorings
Symmetry
The dual of this tiling represents the fundamental domains of the *553 symmetry. There are no mirror removal subgroups of [(5,5,3)], but this symmetry group can be doubled to 652 symmetry by adding a bisecting mirror to the fundamental domains.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-6 tilings
Pentagonal tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Rhombipentahexagonal%20tiling | In geometry, the rhombipentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{6,5}.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Snub%20pentahexagonal%20tiling | In geometry, the snub pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,5}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Chiral figures
Hyperbolic tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Order-6%20octagonal%20tiling | In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *33333333 with 8 order-3 mirror intersections. In Coxeter notation can be represented as [8*,6], removing two of three mirrors (passing through the octagon center) in the [8,6] symmetry.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [8,6,1+], gives [(8,8,3)], (*883). Removing two mirrors as [8,6*], leaves remaining mirrors (*444444).
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-6 tilings
Regular tilings
Octagonal tilings |
https://en.wikipedia.org/wiki/Truncated%20hexaoctagonal%20tiling | In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.
Dual tiling
Symmetry
There are six reflective subgroup kaleidoscopic constructed from [8,6] by removing one or two of three mirrors. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,8,1+,6,1+] (4343) is the commutator subgroup of [8,6].
A radical subgroup is constructed as [8,6*], index 12, as [8,6+], (6*4) with gyration points removed, becomes (*444444), and another [8*,6], index 16 as [8+,6], (8*3) with gyration points removed as (*33333333).
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings |
https://en.wikipedia.org/wiki/Order-8%20hexagonal%20tiling | In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,8}.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [6,8,1+], gives [(6,6,4)], (*664). Removing the mirror between the order 8 and 6 points, [6,1+,8], gives (*4232). Removing two mirrors as [6,8*], leaves remaining mirrors (*33333333).
Symmetry
This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*444444) with 6 order-4 mirror intersections. In Coxeter notation can be represented as [8,6*], removing two of three mirrors (passing through the square center) in the [8,6] symmetry.
Related polyhedra and tiling
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hexagonal tilings
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-8 tilings
Regular tilings |
https://en.wikipedia.org/wiki/Truncated%20order-6%20octagonal%20tiling | In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.
Uniform colorings
A secondary construction t{(8,8,3)} is called a truncated trioctaoctagonal tiling:
Symmetry
The dual to this tiling represent the fundamental domains of [(8,8,3)] (*883) symmetry. There are 3 small index subgroup symmetries constructed from [(8,8,3)] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
The symmetry can be doubled as 862 symmetry by adding a mirror bisecting the fundamental domain.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-6 tilings
Truncated tilings
Uniform tilings
Octagonal tilings |
https://en.wikipedia.org/wiki/Hexaoctagonal%20tiling | In geometry, the hexaoctagonal tiling is a uniform tiling of the hyperbolic plane.
Constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,6,1+], gives [(8,8,3)], (*883). Removing the mirror between the order 2 and 8 points, [1+,8,6], gives [(4,6,6)], (*664). Removing two mirrors as [8,1+,6,1+], leaves remaining mirrors (*4343).
Symmetry
The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of [8,6].
Related polyhedra and tiling
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isotoxal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Rhombihexaoctagonal%20tiling | In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,6}.
Symmetry
The dual tiling, called a deltoidal hexaoctagonal tiling represent the fundamental domains of *4232 symmetry, a half symmetry of [8,6], (*862) as [8,1+,6].
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Semiregular tilings |
https://en.wikipedia.org/wiki/Truncated%20order-8%20hexagonal%20tiling | In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.
Uniform colorings
This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
Symmetry
The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,6,1+,6,1+,4)] (332332) is the commutator subgroup of [(6,6,4)].
A large subgroup is constructed [(6,6,4*)], index 8, as (4*33) with gyration points removed, becomes (*38), and another large subgroup is constructed [(6,6*,4)], index 12, as (6*32) with gyration points removed, becomes (*(32)6).
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hexagonal tilings
Hyperbolic tilings
Isogonal tilings
Order-8 tilings
Semiregular tilings
Truncated tilings |
https://en.wikipedia.org/wiki/Snub%20hexaoctagonal%20tiling | In geometry, the snub hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are three triangles, one hexagon, and one octagon on each vertex. It has Schläfli symbol of sr{8,6}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Chiral figures
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Snub tilings |
https://en.wikipedia.org/wiki/Moderation%20%28disambiguation%29 | Moderation is the process of eliminating or lessening extremes.
Moderation can also refer to:
Moderation (statistics), when the relationship between two variables depends on a third variable
Moderation (Internet), the practice of managing discussion on an online forum
Moderation (game), the practice of refereeing multiplayer role-playing games
"Moderation" (song), by Florence and the Machine
The role of a neutron moderator in a nuclear reactor
See also
Moderate, a middle position in a left/right political system |
https://en.wikipedia.org/wiki/HOSVD-based%20canonical%20form%20of%20TP%20functions%20and%20qLPV%20models | Based on the key idea of higher-order singular value decomposition (HOSVD) in tensor algebra, Baranyi and Yam proposed the concept of HOSVD-based canonical form of TP functions and quasi-LPV system models. Szeidl et al. proved that the TP model transformation is capable of numerically reconstructing this canonical form.
Related definitions (on TP functions, finite element TP functions, and TP models) can be found here. Details on the control theoretical background (i.e., the TP type polytopic Linear Parameter-Varying state-space model) can be found here.
A free MATLAB implementation of the TP model transformation can be downloaded at or at MATLAB Central .
Existence of the HOSVD-based canonical form
Assume a given finite element TP function:
where . Assume that, the weighting functions in are othonormal (or we transform to) for . Then, the execution of the HOSVD on the core tensor leads to:
Then,
that is:
where weighting functions of are orthonormed (as both the and where orthonormed) and core tensor contains the higher-order singular values.
Definition
HOSVD-based canonical form of TP function
Singular functions of : The weighting functions , (termed as the -th singular function on the -th dimension, ) in vector form an orthonormal set:
where is the Kronecker delta function (, if and , if ).
The subtensors have the properties of
all-orthogonality: two sub tensors and are orthogonal for all possible values of and when ,
&* ordering: for all possible values of .
-mode singular values of : The Frobenius-norm , symbolized by , are -mode singular values of and, hence, the given TP function.
is termed core tensor.
The -mode rank of : The rank in dimension denoted by equals the number of non-zero singular values in dimension .
References
Multilinear algebra |
https://en.wikipedia.org/wiki/Fbsp%20wavelet | In applied mathematics, fbsp wavelets are frequency B-spline wavelets.
fbsp m-fb-fc
These frequency B-spline wavelets are complex wavelets whose spectrum are spline.
where sinc function that appears in Shannon sampling theorem.
m > 1 is the order of the spline
fb is a bandwidth parameter
fc is the wavelet center frequency
Clearly, Shannon wavelet (sinc wavelet) is a particular case of fbsp.
References
S.G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999,
C.S. Burrus, R.A. Gopinath, H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice-Hall, 1988, .
O. Cho, M-J. Lai, A Class of Compactly Supported Orthonormal B-Spline Wavelets in: Splines and Wavelets, Athens 2005, G Chen and M-J Lai Editors pp. 123–151.
M. Unser, Ten Good Reasons for Using Spline Wavelets, Proc. SPIE, Vol.3169, Wavelets Applications in Signal and Image Processing, 1997, pp. 422–431.
Continuous wavelets |
https://en.wikipedia.org/wiki/Superscripts%20and%20Subscripts%20%28Unicode%20block%29 | Superscripts and Subscripts is a Unicode block containing superscript and subscript numerals, mathematical operators, and letters used in mathematics and phonetics. The use of subscripts and superscripts in Unicode allows any polynomial, chemical and certain other equations to be represented in plain text without using any form of markup like HTML or TeX. Other superscript letters can be found in the Spacing Modifier Letters, Phonetic Extensions and Phonetic Extensions Supplement blocks, while the superscript 1, 2, and 3, inherited from ISO 8859-1, were included in the Latin-1 Supplement block.
Block
History
The following Unicode-related documents record the purpose and process of defining specific characters in the Superscripts and Subscripts block:
See also
Unicode superscripts and subscripts
Phonetic symbols in Unicode
Latin script in Unicode
References
Unicode blocks |
https://en.wikipedia.org/wiki/Lokutu | Lokutu (sometimes spelled Lukutu) is a town in Tshopo province, Democratic Republic of the Congo. It is on the west bank of the Congo River, and was formerly named Elisabetha.
Local Statistics
Lokutu has 747 kilometres of operational roads. It also have 1,985 houses, 6000 schools, two hospitals, 64 gun stores, eight dispensaries, and two health centers.
Feronia's Oil Palm Plantations'
The Feronia Plantation in Lokutu is the companies largest plantation, 63,560 hectares, and was established in 1920. The plantation is situated alongside the Congo River in the Orientale Province of the DRC.
Although the plantation is 63,000+ hectares, wildlife buffer zones make up a large amount of the area, so it will never be planted on.
Conflict
On March 16, 2019, within the Lokutu oil palm plantation concession area of the Canadian company Feronia Inc, in the Tshopo Province, in the Democratic Republic of the Congo, military forces fired live bullets at local Bolombo and Wamba villagers in the municipality of Mwingi, and from Bokala-wamba. This incident follows the weeks of tension that have been growing between the locals and the companies at Lokutu and Boteka plantations. Complaints have been filed by the communities with the international complaints mechanism of the German development bank (DEG) denouncing the company's illegal occupation of their territories.
The communities blocked the company vehicles from transporting palm nuts from the plantation sites and state it is the only way for them to be heard. The communities have also been protesting unpaid or underpaid wages for local workers on the plantations. According to some local sources, Feronia claimed to have resumed payments, but many locals stated they only received partial payments.
The villagers took action and began harvesting and processing palm oil from the plantation themselves and destroyed multiple bridges used by the company for transport. This is when the military action and shots fired began to take place.
National and international civil society organizations are actively supporting the affected communities. The long-standing occupation of Lokutu territory by Feronia was a direct cause of these conflicts.
Between 12 and 16 September 2019, the national police arrested numerous villagers from communities involved in an international mediation process regarding the occupation of their land by the Canadian palm oil company Feronia Inc (PHC-Feronia in Lokutu). Following a meeting between community leaders and members of the German development bank's complaint mechanism panel, these arrests were violently made in the middle of the night.
The Support and Information Network for National Organizations (RIAO-RDC) denounces the threats and attacks by PHC-Feronia targeting members of the communities that mandated RIAO-RDC to file a complaint on their behalf with the complaint mechanism of the DEG (German development bank) so that ultimately the Congolese communities can have their claim |
https://en.wikipedia.org/wiki/Parent%20function | In mathematics, a parent function is the core representation of a function type without manipulations such as translation and dilation. For example, for the family of quadratic functions having the general form
the simplest function is
.
This is therefore the parent function of the family of quadratic equations.
For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes. For example, the graph of y = x − 4x + 7 can be obtained from the graph of y = x by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2).
For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = ), then stretching it parallel to the Y axis using a stretch factor R, where R = A + B. This is because A sin(x) + B cos(x) can be written as R sin(x−α) (see List of trigonometric identities).
The concept of parent function is less clear for polynomials of higher power because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as x, or, to simplify further, x when n is even and x for odd n. Turning points may be established by differentiation to provide more detail of the graph.
See also
Curve sketching
References
External links
Video explanation at VirtualNerd.com
Curves
Elementary algebra
Functions and mappings |
https://en.wikipedia.org/wiki/Mohammed%20Huwaidi%20Al-Hooti | Mohammed Huwaidi Al-Hooti (; born 19 January 1986), commonly known as Mohammed Huwaidi, is an Omani footballer who plays as a goalkeeper for Al-Nahda Club.
Club career statistics
International career
Mohammed was selected for the national team for the first time in 2007. He has made appearances in the 2011 AFC Asian Cup qualification and the 2010 Gulf Cup of Nations and has represented the national team in the 2010 FIFA World Cup qualification and the 2014 FIFA World Cup qualification.
Honours
Club
Omani Super Cup (1): 2011
References
External links
1986 births
Living people
Omani men's footballers
Oman men's international footballers
Men's association football goalkeepers
Muscat Club players
Al-Orouba SC players
Ahli Sidab Club players
Al-Nahda Club (Oman) players |
https://en.wikipedia.org/wiki/Mansoor%20Al-Nuaimi | Mansoor Ghamil Al-Nuaimi (; born 13 February 1989), commonly known as Mansoor Al-Nuaimi, is an Omani footballer who plays for Al-Nahda Club.
Club career statistics
International career
Mansoor is part of the first team squad of the Oman national football team. He was selected for the national team for the first time in 2009. He made his first appearance for Oman on 17 November 2009 in a friendly match against Brazil. He has made an appearance in the 2011 AFC Asian Cup qualification.
Honours
Club
Oman Professional League (1): 2013-14
Sultan Qaboos Cup (0): Runner-up 2012, 2013
Oman Super Cup (2): 2009, 2014
References
External links
1989 births
Living people
Omani men's footballers
Oman men's international footballers
Men's association football forwards
Al-Nahda Club (Oman) players
Oman Professional League players |
https://en.wikipedia.org/wiki/Drew%20Cannon | Drew Cannon (born April 21, 1990) is an American statistician and sports writer who currently works on the Boston Celtics staff.
As a child, Cannon was fascinated by sports statistics and, after reading the work of Bill James, began to design his own statistical projects to analyze sports. At age 15, he got an internship with well-known basketball scout Dave Telep. Over the next seven years, his research helped improve Telep's recruiting, while Telep worked to round out Cannon's personality. Cannon developed writing skills during college and his research was published by Basketball Prospectus, ESPN, and Kenpom.com. He graduated from Duke in 2012, and was hired by Brad Stevens to do statistical analysis for the Butler basketball team. Cannon produced regular reports on how to increase the team's efficiency. The success of his recommendations won over doubters and led to multiple reporters describing Cannon as Butler's "secret weapon".
In July 2013, Stevens was hired by the Boston Celtics and brought Cannon with him.
Early life
As a young child growing up in Raleigh, North Carolina, Drew Cannon was attracted to numbers, in particular sports statistics. At age eight, he read through his father's Bill James baseball books. By thirteen, he was designing his own statistical projects to analyze sports - for example, comparing Negro league baseball players to Major Leaguers from the same time period. Cannon "probably had about 25 [different] projects going on" by age 15, recalled his father Jim Cannon.
Cannon played sports as a child, but was not athletically gifted. He was the sixth man for his junior high team, but did not play for his high school team. Cannon has a younger sister, Maria, and a younger brother, Chris.
At a casual lunch with friends in 2004, Jim Cannon met recruiting specialist Dave Telep. Soon the conversation turned to basketball. Telep had just finished reading Moneyball and was intrigued with the idea of bringing advanced statistics to basketball. "My mind (was) wide open about how we can apply [Moneyball] to basketball", Telep recalled. After lunch, Cannon approached Telep: "There's this kid in my house that I don't know what to do with. Can you help me?" Telep met with Drew Cannon, then a 15-year-old sophomore in high school, and soon offered him an internship, paying $600 for the summer.
Internship and college
Over the next seven years, Telep mentored Cannon. Cannon's analytical strengths were obvious, but he lacked the personality to be successful in the sporting world. His first report was like "straight out of a scientific journal", Telep recalls. Telep worked, teaching Cannon how to "communicating to the common man". He forced Cannon to interact with coaches and players, rather than just crunch numbers behind the scenes. "If he was going to do this, he couldn't do it with a lab coat on," Telep explained. Cannon spent so much time with Telep that his kids thought Cannon lived there. He joined Telep on rec |
https://en.wikipedia.org/wiki/Restricted%20root%20system | In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its dual can be identified. For symmetric spaces of noncompact type arising as homogeneous spaces of a semisimple Lie group, the restricted root system and its Weyl group are related to the Iwasawa decomposition of the Lie group.
See also
Satake diagram
References
Lie groups
Lie algebras |
https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani%20representation%20theorem | In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuous functions on the unit interval, who extended the result to some non-compact spaces, and who extended the result to compact Hausdorff spaces.
There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures.
The representation theorem for positive linear functionals on Cc(X)
The following theorem represents positive linear functionals on Cc(X), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X. The Borel sets in the following statement refer to the σ-algebra generated by the open sets.
A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is a Radon measure if and only if
for every compact ;
For every Borel set , (outer regularity);
The relation holds whenever is open or when is Borel and (inner regularity).
Theorem. Let be a locally compact Hausdorff space. For any positive linear functional on Cc(X), there is a unique Radon measure μ on X such that
One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on Cc(X). This is the way adopted by Bourbaki; it does of course assume that X starts life as a topological space, rather than simply as a set. For locally compact spaces an integration theory is then recovered.
Without the condition of regularity the Borel measure need not be unique. For example, let X be the set of ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by "open intervals". The linear functional taking a continuous function to its value at Ω corresponds to the regular Borel measure with a point mass at Ω. However it also corresponds to the (non-regular) Borel measure that assigns measure 1 to any Borel set if there is closed and unbounded set with , and assigns measure 0 to other Borel sets. (In particular the singleton {Ω} gets measure 0, contrary to the point mass measure.)
Historical remark
In its original form by F. Riesz (1909) the theorem states that every continuous linear functional A[f] over the space C([0, 1]) of continuous functions in the interval [0,1] can be represented in the form
where α(x) is a function of bounded variation on the interval [0, 1], and the integral is a Riemann–Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation (t |
https://en.wikipedia.org/wiki/Dahan%20%28surname%29 | Dahan is a surname. Notable people with the surname include:
Amy Dahan, French mathematician and historian of mathematics and climate change
Dudu Dahan, former Israeli football player
Mor Dahan, Israeli football player
Nissim Dahan, Israeli politician
Olivier Dahan, French film director and screenwriter
Theodosius V Dahan, 18th-century patriarch of Melkite Greek Catholic Church
Hebrew-language surnames |
https://en.wikipedia.org/wiki/Mabinogion%20sheep%20problem | In probability theory, the Mabinogion sheep problem or Mabinogian urn is a problem in stochastic control introduced by , who named it after a herd of magic sheep in the Welsh collection of tales, the Mabinogion.
Statement
At time t = 0 there is a herd of sheep each of which is black or white. At each time t = 1, 2, ... a sheep is selected at random, and a sheep of the opposite color (if one exists) is changed to be the same color as the selected sheep. At any time one may remove as many sheep (of either color) as one wishes from the flock. The problem is to do this in such a way as to maximize the expected final number of black sheep.
The optimal solution at each step is to remove just enough white sheep so that there are more black sheep than white sheep.
References
Probability problems
Stochastic control
Optimal decisions |
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