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https://en.wikipedia.org/wiki/Jorge%20Iv%C3%A1n%20Bocanegra | Jorge Iván Bocanegra born in Líbano, Tolima, Colombia on 23 March 1991 is a football player, who plays for Millonarios in the Categoría Primera A, as striker
.
Statistics (Official games/Colombian Ligue and Colombian Cup)
(As of November 14, 2010)
References
External links
1991 births
Living people
Colombian men's footballers
Deportes Tolima footballers
Millonarios F.C. players
Bogotá F.C. footballers
Men's association football forwards
Sportspeople from Tolima Department |
https://en.wikipedia.org/wiki/Wiriagar%20River | The Wiriagar or Aimau River is a river in southern West Papua province, Indonesia.{
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https://en.wikipedia.org/wiki/Herbert%20Enderton | Herbert Bruce Enderton (April 15, 1936 – October 20, 2010) was an American mathematician. He was a Professor Emeritus of Mathematics at UCLA and a former member of the faculties of Mathematics and of Logic and the Methodology of Science at the University of California, Berkeley.
Enderton also contributed to recursion theory, the theory of definability, models of analysis, computational complexity, and the history of logic.
He earned his Ph.D. at Harvard in 1962. He was a member of the American Mathematical Society from 1961 until his death.
Personal life
He lived in Santa Monica. He married his wife, Cathy, in 1961 and they had two sons; Eric and Bert.
Later years
From 1980 to 2002 he was coordinating editor of the reviews section of the Association for Symbolic Logic's Journal of Symbolic Logic.
Death
He died from leukemia in 2010.
Selected publications
References
External links
Herbert B. Enderton home page
Enderton publications
Herbert Enderton UCLA lectures on YouTube
1936 births
2010 deaths
20th-century American mathematicians
21st-century American mathematicians
American logicians
Mathematical logicians
Set theorists
Harvard University alumni
University of California, Berkeley College of Letters and Science faculty
University of California, Los Angeles faculty
Place of birth missing |
https://en.wikipedia.org/wiki/B7%20polytope | {{DISPLAYTITLE:B7 polytope}}
In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128 vertices respectively. The 7-demicube is added with half of the symmetry.
They can be visualized as symmetric orthographic projections in Coxeter planes of the B7 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 128 polytopes can be made in the B7, B6, B5, B4, B3, B2, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.
These 128 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Notes
7-polytopes |
https://en.wikipedia.org/wiki/A7%20polytope | {{DISPLAYTITLE:A7 polytope}}
In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A7 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 71 polytopes can be made in the A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has [k+1] symmetry. For even k and symmetrically ringed-diagrams, symmetry doubles to [2(k+1)].
These 71 polytopes are each shown in these 6 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Notes
7-polytopes |
https://en.wikipedia.org/wiki/Koreans%20in%20Sri%20Lanka | There are roughly 948 Koreans in Sri Lanka, according to the 2013 statistics of South Korea's Ministry of Foreign Affairs and Trade).
Demography
The number of Koreans in Sri Lanka was recorded at 854 in 2009, 948 in 2011, and 782 in 2013. Among them were 268 international students studying at Sri Lankan universities, and 514 with other types of visas. The vast majority (513) lived at Colombo, with another 96 in Western Province (namely Negombo and Gampaha), 124 in Central Province (Kandy and Nuwara Eliya), and 49 in other areas.
Religion
South Korean Christian missionaries are active in Sri Lanka. They have conducted charitable activities, such as constructing disaster assistance centres. However, their activities to promote religious conversion have also caused conflict with the predominantly Buddhist and Hindu local people.
Community organisations
There is a Korean Association of Sri Lanka, as well as a weekend school for Korean language. The former has organised various public events to inform the public about Korean culture and history, such as a special lecture commemorating the start of the Korean War.
References
Further reading
External links
Association of Koreans in Sri Lanka
Korean School of Sri Lanka
Sri Lanka
Asian diaspora in Sri Lanka
Sri Lanka |
https://en.wikipedia.org/wiki/Jordan%E2%80%93Schur%20theorem | In mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan. In that form, it states that there is a function ƒ(n) such that given a finite subgroup G of the group of invertible n-by-n complex matrices, there is a subgroup H of G with the following properties:
H is abelian.
H is a normal subgroup of G.
The index of H in G satisfies (G : H) ≤ ƒ(n).
Schur proved a more general result that applies when G is not assumed to be finite, but just periodic. Schur showed that ƒ(n) may be taken to be
((8n)1/2 + 1)2n2 − ((8n)1/2 − 1)2n2.
A tighter bound (for n ≥ 3) is due to Speiser, who showed that as long as G is finite, one can take
ƒ(n) = n! 12n(π(n+1)+1)
where π(n) is the prime-counting function. This was subsequently improved by Hans Frederick Blichfeldt who replaced the 12 with a 6. Unpublished work on the finite case was also done by Boris Weisfeiler. Subsequently, Michael Collins, using the classification of finite simple groups, showed that in the finite case, one can take ƒ(n) = (n + 1)! when n is at least 71, and gave near complete descriptions of the behavior for smaller n.
See also
Burnside's problem
References
Theorems in group theory |
https://en.wikipedia.org/wiki/Susan%20Morey | Susan Morey is an American mathematician and a professor and chair of the Mathematics department at Texas State University in San Marcos, Texas.
Education and career
Morey received a B.S. in mathematics with Honors from the University of Missouri in 1990 and a Ph.D. in mathematics from Rutgers University in 1995. Her dissertation The Equations of Rees Algebras of ideals of Low Codimension was supervised by Wolmer Vasconcelos. After receiving her Ph.D., Morey held a postdoctoral position at the University of Texas at Austin. She became an assistant professor at Texas State (then Southwest Texas State University) in 1997. Morey was awarded tenure and promotion to associate professor in 2001 and promotion to full professor in 2010. She became chair of the mathematics department in 2015. She received the Everette Swinney Excellence in Teaching Award from Texas State in 2016.
Morey is known for her work in commutative algebra, in particular, for work on normal rings and algebraic and combinatorial properties of edge ideals of graphs and hypergraphs. Her work is published in the Journal of Pure and Applied Algebra, the Journal of Algebraic Combinatorics, Communications in Algebra, Progress in Commutative Algebra, the Proceedings of the American Mathematical Society, and other journals.
Morey was selected a Fellow of the Association for Women in Mathematics in the Class of 2021 "for inspiring and mentoring several generations of women mathematicians, whom she has helped and encouraged to reach their full potential; and for support of graduate students through the Stokes Alliance for Minority Participation".
References
External links
Susan Morey's Author Profile Page on MathSciNet
Susan Morey's Faculty Profile at Texas State University
Official website
Living people
20th-century American mathematicians
21st-century American mathematicians
Women mathematicians
Fellows of the Association for Women in Mathematics
Rutgers University alumni
Year of birth missing (living people)
University of Missouri alumni |
https://en.wikipedia.org/wiki/Cornelia%20Dru%C8%9Bu | Cornelia Druțu is a Romanian mathematician notable for her contributions in the area of geometric group theory. She is Professor of mathematics at the University of Oxford and Fellow of Exeter College, Oxford.
Education and career
Druțu was born in Iaşi, Romania. She attended the Emil Racoviță High School (now the National College Emil Racoviță) in Iași. She earned a B.S. in Mathematics from the University of Iași, where besides attending the core courses she received extra curricular teaching in geometry and topology from Professor Liliana Răileanu.
Druțu earned a Ph.D. in Mathematics from University of Paris-Sud, with a thesis entitled Réseaux non uniformes des groupes de Lie semi-simple de rang supérieur et invariants de quasiisométrie, written under the supervision of Pierre Pansu. She then joined the University of Lille 1 as Maître de conférences (MCF). In 2004 she earned her Habilitation degree from the University of Lille 1.
In 2009 she became Professor of mathematics at the Mathematical Institute, University of Oxford.
She held visiting positions at the Max Planck Institute for Mathematics in Bonn, the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, the Mathematical Sciences Research Institute in Berkeley, California. She visited the Isaac Newton Institute in Cambridge as holder of a Simons Fellowship. From 2013 to 2020 she chaired the European Mathematical Society/European Women in Mathematics scientific panel of women mathematicians.
Awards
In 2009, Druțu was awarded the Whitehead Prize by the London Mathematical Society for her work in geometric group theory.
In 2017, Druțu was awarded a Simons Visiting Fellowship.
Publications
Selected contributions
The quasi-isometry invariance of relative hyperbolicity; a characterization of relatively hyperbolic groups using geodesic triangles, similar to the one of hyperbolic groups.
A classification of relatively hyperbolic groups up to quasi-isometry; the fact that a group with a quasi-isometric embedding in a relatively hyperbolic metric space, with image at infinite distance from any peripheral set, must be relatively hyperbolic.
The non-distortion of horospheres in symmetric spaces of non-compact type and in Euclidean buildings, with constants depending only on the Weyl group.
The quadratic filling for certain linear solvable groups (with uniform constants for large classes of such groups).
A construction of a 2-generated recursively presented group with continuum many non-homeomorphic asymptotic cones. Under the continuum hypothesis, a finitely generated group may have at most continuum many non-homeomorphic asymptotic cones, hence the result is sharp.
A characterization of Kazhdan's property (T) and of the Haagerup property using affine isometric actions on median spaces.
A study of generalizations of Kazhdan's property (T) for uniformly convex Banach spaces.
A proof that random groups satisfy strengthened versions of Kazhdan's property (T) for high enough |
https://en.wikipedia.org/wiki/Tam%C3%A1s%20Hausel | Tamás Hausel (born 1972) is a Hungarian mathematician working in the areas of combinatorial, differential and algebraic geometry and topology. More specifically the global analysis, geometry, topology and arithmetic of hyperkähler manifolds, Yang–Mills instantons, non-Abelian Hodge theory, Geometric Langlands program, and representation theory of quivers and Kac–Moody algebras.
Hausel is currently associated with the Institute of Science and Technology Austria (IST) where he has been a full professor since 2016. Prior to joining IST he was a professor at École Polytechnique Fédérale de Lausanne (EPFL). He was previously at the University of Oxford, both a Royal Society University Research Fellow at the university's mathematical institute, and a Tutorial Fellow in Mathematics at Wadham College. Previous to that, Hausel was an assistant and then associate professor at the University of Texas at Austin.
Awards
In 2008, Hausel was awarded the Whitehead Prize by the London Mathematical Society for his investigations into hyperkähler geometry which have led him to prove deep results in fields as diverse as the representation theory of quivers, mirror symmetry and Yang–Mills instantons.
Publications
References
Academics of the University of Oxford
Living people
Differential geometers
Algebraic geometers
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Whitehead Prize winners
1972 births
Alumni of the University of Cambridge
Academic staff of the École Polytechnique Fédérale de Lausanne
University of Texas faculty
Eötvös Loránd University alumni |
https://en.wikipedia.org/wiki/Timothy%20Browning | Timothy Browning is a mathematician working in number theory, examining the interface of analytic number theory and Diophantine geometry.
Browning is currently a Professor of number theory at the Institute of Science and Technology Austria (ISTA) in Klosterneuburg, Austria.
Awards
In 2008, Browning was awarded the Whitehead Prize by the London Mathematical Society for his significant contributions on the interface of analytic number theory and arithmetic geometry concerning the number and distribution of rational and integral solutions to Diophantine equations.
In 2009, Browning won the Ferran Sunyer i Balaguer Prize. The prize is awarded for a mathematical monograph of an expository nature presenting the latest developments in an active area of research in Mathematics, in which the applicant has made important contributions. Browning won the prize for his monograph entitled Quantitative Arithmetic of Projective Varieties.
In 2010, Browning was awarded the Leverhulme Mathematics Prize for his work on number theory and diophantine geometry.
Publications
References
1976 births
Academics of the University of Bristol
Living people
Whitehead Prize winners
Alumni of the University of Oxford
20th-century British mathematicians
21st-century British mathematicians
Number theorists
Geometers |
https://en.wikipedia.org/wiki/Wolstenholme%20prime | In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century.
Interest in these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two.
The only two known Wolstenholme primes are 16843 and 2124679 . There are no other Wolstenholme primes less than 109.
Definition
Wolstenholme prime can be defined in a number of equivalent ways.
Definition via binomial coefficients
A Wolstenholme prime is a prime number p > 7 that satisfies the congruence
where the expression in left-hand side denotes a binomial coefficient.
In comparison, Wolstenholme's theorem states that for every prime p > 3 the following congruence holds:
Definition via Bernoulli numbers
A Wolstenholme prime is a prime p that divides the numerator of the Bernoulli number Bp−3. The Wolstenholme primes therefore form a subset of the irregular primes.
Definition via irregular pairs
A Wolstenholme prime is a prime p such that (p, p–3) is an irregular pair.
Definition via harmonic numbers
A Wolstenholme prime is a prime p such that
i.e. the numerator of the harmonic number expressed in lowest terms is divisible by p3.
Search and current status
The search for Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2007. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time. The 1964 discovery was later independently confirmed in the 1970s. This remained the only known example of such a prime for almost 20 years, until the discovery announcement of the second Wolstenholme prime 2124679 in 1993. Up to 1.2, no further Wolstenholme primes were found. This was later extended to 2 by McIntosh in 1995 and Trevisan & Weber were able to reach 2.5. The latest result as of 2007 is that there are only those two Wolstenholme primes up to .
Expected number of Wolstenholme primes
It is conjectured that infinitely many Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes ≤ x is about ln ln x, where ln denotes the natural logarithm. For each prime p ≥ 5, the Wolstenholme quotient is defined as
Clearly, p is a Wolstenholme prime if and only if Wp ≡ 0 (mod p). Empirically one may assume that the remainders of Wp modulo p are uniformly distributed in the set {0, 1, ..., p–1}. By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/p.
See also
Wieferich prime
Wall–Sun–Sun prime
Wilson prime
Table of congruences
Not |
https://en.wikipedia.org/wiki/Lantern%20relation | In geometric topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mapping class group of a surface. The most general version of the relation involves seven Dehn twists. The relation was discovered by Dennis Johnson in 1979.
General form
The general form of the lantern relation involves seven Dehn twists in the mapping class group of a disk with three holes, as shown in the figure on the right. According to the relation,
where , , and are the right-handed Dehn twists around the blue curves , , and , and , , , are the right-handed Dehn twists around the four red curves.
Note that the Dehn twists , , , on the right-hand side all commute (since the curves are disjoint, so the order in which they appear does not matter. However, the cyclic order of the three Dehn twists on the left does matter:
Also, note that the equalities written above are actually equality up to homotopy or isotopy, as is usual in the mapping class group.
General surfaces
Though we have stated the lantern relation for a disk with three holes, the relation appears in the mapping class group of any surface in which such a disk can be embedded in a nontrivial way. Depending on the setting, some of the Dehn twists appearing in the lantern relation may be homotopic to the identity function, in which case the relation involves fewer than seven Dehn twists.
The lantern relation is used in several different presentations for the mapping class groups of surfaces.
References
External links
Sketches of Topology – The Lantern Relation
Geometric topology
Homeomorphisms |
https://en.wikipedia.org/wiki/A6%20polytope | {{DISPLAYTITLE:A6 polytope}}
In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 35 polytopes can be made in the A6, A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetric ringed diagrams, symmetry doubles to [2(k+1)].
These 35 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
Notes
6-polytopes |
https://en.wikipedia.org/wiki/B6%20polytope | {{DISPLAYTITLE:B6 polytope}}
In 6-dimensional geometry, there are 64 uniform polytopes with B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices respectively. The 6-demicube is added with half the symmetry.
They can be visualized as symmetric orthographic projections in Coxeter planes of the B6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 64 polytopes can be made in the B6, B5, B4, B3, B2, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.
These 64 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Notes
6-polytopes |
https://en.wikipedia.org/wiki/Hopkins%E2%80%93Levitzki%20theorem | In abstract algebra, in particular ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in modules over semiprimary rings. A ring R (with 1) is called semiprimary if R/J(R) is semisimple and J(R) is a nilpotent ideal, where J(R) denotes the Jacobson radical. The theorem states that if R is a semiprimary ring and M is an R-module, the three module conditions Noetherian, Artinian and "has a composition series" are equivalent. Without the semiprimary condition, the only true implication is that if M has a composition series, then M is both Noetherian and Artinian.
The theorem takes its current form from a paper by Charles Hopkins and a paper by Jacob Levitzki, both in 1939. For this reason it is often cited as the Hopkins–Levitzki theorem. However Yasuo Akizuki is sometimes included since he proved the result for commutative rings a few years earlier, in 1935.
Since it is known that right Artinian rings are semiprimary, a direct corollary of the theorem is: a right Artinian ring is also right Noetherian. The analogous statement for left Artinian rings holds as well. This is not true in general for Artinian modules, because there are examples of Artinian modules which are not Noetherian.
Another direct corollary is that if R is right Artinian, then R is left Artinian if and only if it is left Noetherian.
Sketch of proof
Here is the proof of the following: Let R be a semiprimary ring and M a left R-module. If M is either Artinian or Noetherian, then M has a composition series. (The converse of this is true over any ring.)
Let J be the radical of R. Set . The R-module may then be viewed as an -module because J is contained in the annihilator of . Each is a semisimple -module, because is a semisimple ring. Furthermore, since J is nilpotent, only finitely many of the are nonzero. If M is Artinian (or Noetherian), then has a finite composition series. Stacking the composition series from the end to end, we obtain a composition series for M.
In Grothendieck categories
Several generalizations and extensions of the theorem exist. One concerns Grothendieck categories: if G is a Grothendieck category with an Artinian generator, then every Artinian object in G is Noetherian.
See also
Artinian module
Noetherian module
Composition series
References
Charles Hopkins (1939) Rings with minimal condition for left ideals, Ann. of Math. (2) 40, pages 712–730.
T. Y. Lam (2001) A first course in noncommutative rings, Springer-Verlag. page 55
Jakob Levitzki (1939) On rings which satisfy the minimum condition for the right-hand ideals, Compositio Mathematica, v. 7, pp. 214222.
Theorems in ring theory |
https://en.wikipedia.org/wiki/Trudi%20Lacey | Trudi Lacey (born December 12, 1958) is an American basketball head coach, most recently of the Washington Mystics of the Women's National Basketball Association (WNBA).
NC State statistics
Source
USA Basketball
Lacey was named to the team representing the US at the inaugural William Jones Cup competition in Taipei, Taiwan. In subsequent years, the teams would be primarily college age players, but in the inaugural event, eight of the twelve players, including Lacey, were in high school. The USA team had a record of 3–4, finishing in fifth place, although one of the wins was over South Korea, who would go on to win the gold medal.
Lacey was chosen to represent the USA on the USA Basketball team at the 1981 World University games, held in Bucharest, Romania. After winning the opening game, the USA was challenged by China, who held a halftime lead. The USA came back to win by two points, helped by 26 points from Denise Curry and 12 from Lacey. The USA was also challenged by Canada, who led at halftime, but the USA won by three points 79–76. The USA beat host team Romania to set up a match with undefeated Russia for the gold medal. The Russian team was too strong, and won the gold, leaving the US with the silver medal. Lacey averaged 6.4 points per game.
Lacey played on the 1983 World University games team, coached by Jill Hutchison. She helped the team win the gold medal.
In 1995, Lacey served as an assistant coach to the R. William Jones Cup Team. The competition was held in Taipei, Taiwan. The USA team won its first six games, but four of the six were won by single-digit margins. Their seventh game was against Russia, and they fell 100–84. The final game was against South Korea, and a victory would assure the gold medal, but the South Korean team won 80–76 to win the gold medal. The USA team won the bronze.
References
External links
Trudi Lacey biography from gopack.com
1958 births
Living people
People from Clifton Forge, Virginia
American women's basketball coaches
American women's basketball players
Charlotte Sting coaches
James Madison Dukes women's basketball coaches
Maryland Terrapins women's basketball coaches
NC State Wolfpack women's basketball players
FISU World University Games gold medalists for the United States
Universiade medalists in basketball
Washington Mystics coaches
Washington Mystics head coaches
Women's National Basketball Association general managers
Medalists at the 1981 Summer Universiade
Medalists at the 1983 Summer Universiade
South Florida Bulls women's basketball coaches
Francis Marion Patriots women's basketball coaches
Queens Royals women's basketball coaches |
https://en.wikipedia.org/wiki/KR-theory | In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by , motivated by applications to the Atiyah–Singer index theorem for real elliptic operators.
Definition
A real space is a defined to be a topological space with an involution. A real vector bundle over a real space X is defined to be a complex vector bundle E over X that is also a real space, such that the natural maps from E to X and from ×E to E commute with the involution, where the involution acts as complex conjugation on . (This differs from the notion of a complex vector bundle in the category of Z/2Z spaces, where the involution acts trivially on .)
The group KR(X) is the Grothendieck group of finite-dimensional real vector bundles over the real space X.
Periodicity
Similarly to Bott periodicity, the periodicity theorem for KR states that KRp,q = KRp+1,q+1, where KRp,q is suspension with respect to Rp,q =
Rq + iRp (with a switch in the order of p and q), given by
and Bp,q, Sp,q are the unit ball and sphere in Rp,q.
References
K-theory |
https://en.wikipedia.org/wiki/1908%E2%80%9309%20S.L.%20Benfica%20season | The 1908–09 season was Sport Lisboa e Benfica's fifth season in existence and the club's third competitive season.
Campeonato de Lisboa
Table
Matches
Player statistics
|}
References
S.L. Benfica seasons
S.L. Benfica season
S.L. Benfica season
1908–09 in Portuguese football |
https://en.wikipedia.org/wiki/Mesh%20parameterization | Given two surfaces with the same topology, a bijective mapping between them exists. On triangular mesh surfaces, the problem of computing this mapping is called mesh parameterization. The parameter domain is the surface that the mesh is mapped onto.
Parameterization was mainly used for mapping textures to surfaces. Recently, it has become a powerful tool for many applications in mesh processing. Various techniques are developed for different types of parameter domains with different parameterization properties.
Applications
Texture mapping
Normal mapping
Detail transfer
Morphing
Mesh completion
Mesh Editing
Mesh Databases
Remeshing
Surface fitting
Techniques
Barycentric Mappings
Differential Geometry Primer
Non-Linear Methods
Implementations
A fast and simple stretch-minimizing mesh parameterization
Graphite: ABF++, LSCM, Spectral LSCM
Linear discrete conformal parameterization
Discrete Exponential Map
Boundary First Flattening
Scalable Locally Injective Mappings
See also
Parametrization
Texture atlas
UV Mapping
External links
"Mesh Parameterization: theory and practice"
3D computer graphics |
https://en.wikipedia.org/wiki/Archimede%20construction%20systems | Archimede construction systems are construction techniques achieving rhombic dodecahedral shapes, a space-filling geometry. In America, most of these systems generate building envelopes made up of as little as two panel shapes and sizes, this feature allowing for maximum industrialization. These panels are generally pressure injected with a rigid structural insulating foam like polyurethane. In Europe and Asia, post and beam structure often create a dodecahedral shape that is later filled with different cladding materials. Although the basic geometry sometimes contains an entire house, most applications of the system are an agglomeration of cells each forming a rhombic dodecahedral living space that can be a room or a larger living area when combined with adjacent dodecahedral modules.
History
The peculiar three-dimensional structure of the honey bee comb has intrigued for thousands of years. It is quite possible that some isolated construction project used a scaled-up version of this zonohedron, however a largely visible use of these shapes only appeared in the early 1980s.
Les Systèmes Archimede Inc. of Tring Junction QC started to produce applying US4462191, a patent owned by J. Poirier who also co-founded the manufacturing firm with Placide Poulin .
References
External links
High-Tech Housing, Popular Science Magazine
Why Rhombic Dodecahedral shells are so strong, text and illustrations by J.B. Poirier,arch.
Arctic Refuges on Stilts, Forbes Magazine
Housing Industry Takes Some Tips From The Bees, The Montreal Gazette
Prefabs Doing Well, The Montreal Gazette
Architecture
Prefabricated buildings |
https://en.wikipedia.org/wiki/Felipe%20Silva%20%28footballer%29 | Felipe de Oliveira Silva (born 28 May 1990), known as Felipe Silva or simply Felipe, is a Brazilian footballer who plays as an attacking midfielder for Inter de Limeira.
Career statistics
References
External links
WebSoccerClub
1990 births
Living people
Footballers from Piracicaba
Brazilian men's footballers
Men's association football midfielders
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Sociedade Esportiva Palmeiras players
Rio Branco Esporte Clube players
Esporte Clube Bahia players
Olaria Atlético Clube players
Guarani FC players
Mogi Mirim Esporte Clube players
Club Athletico Paranaense players
Figueirense FC players
Associação Atlética Ponte Preta players
Ceará Sporting Club players
Associação Chapecoense de Futebol players
Associação Atlética Internacional (Limeira) players
J1 League players
Sanfrecce Hiroshima players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Japan
Expatriate men's footballers in Japan |
https://en.wikipedia.org/wiki/N-group | n-group (category theory), an N-category that possesses a group-like structure.
N-group (finite group theory), a finite group all of whose local subgroups are solvable. |
https://en.wikipedia.org/wiki/N-group%20%28finite%20group%20theory%29 | In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial p-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.
Simple N-groups
The simple N-groups were classified by in a series of 6 papers totaling about 400 pages.
The simple N-groups consist of the special linear groups PSL2(q), PSL3(3), the Suzuki groups Sz(22n+1), the unitary group U3(3), the alternating group A7, the Mathieu group M11, and the Tits group. (The Tits group was overlooked in Thomson's original announcement in 1968, but Hearn pointed out that it was also a simple N-group.) More generally Thompson showed that any non-solvable N-group is a subgroup of Aut(G) containing G for some simple N-group G.
generalized Thompson's theorem to the case of groups where all 2-local subgroups are solvable. The only extra simple groups that appear are the unitary groups U3(q).
Proof
gives a summary of Thompson's classification of N-groups.
The primes dividing the order of the group are divided into four classes π1, π2, π3, π4 as follows
π1 is the set of primes p such that a Sylow p-subgroup is nontrivial and cyclic.
π2 is the set of primes p such that a Sylow p-subgroup P is non-cyclic but SCN3(P) is empty
π3 is the set of primes p such that a Sylow p-subgroup P has SCN3(P) nonempty and normalizes a nontrivial abelian subgroup of order prime to p.
π4 is the set of primes p such that a Sylow p-subgroup P has SCN3(P) nonempty but does not normalize a nontrivial abelian subgroup of order prime to p.
The proof is subdivided into several cases depending on which of these four classes the prime 2 belongs to, and also on an integer e, which is the largest integer for which there is an elementary abelian subgroup of rank e normalized by a nontrivial 2-subgroup intersecting it trivially.
Gives a general introduction, stating the main theorem and proving many preliminary lemmas.
characterizes the groups E2(3) and S4(3) (in Thompson's notation; these are the exceptional group G2(3) and the symplectic group Sp4(3)) which are not N-groups but whose characterizations are needed in the proof of the main theorem.
covers the case where 2∉π4. Theorem 11.2 shows that if 2∈π2 then the group is PSL2(q), M11, A7, U3(3), or PSL3(3). The possibility that 2∈π3 is ruled out by showing that any such group must be a C-group and using Suzuki's classification of C-groups to check that none of the groups found by Suzuki satisfy this condition.
and cover the cases when 2∈π4 and e≥3, or e=2. He shows that either G is a C-group so a Suzuki group, or satisfies his characterization of the groups E2(3) and S4(3) in his second paper, which are not N-groups.
covers the case when 2∈π4 and e=1, where the only possibilities are that G is a C-group or the Tits group.
Consequences
A minimal simple group is a non-cyclic simple group all of whose proper su |
https://en.wikipedia.org/wiki/Sagitta%20%28disambiguation%29 | Sagitta is a constellation, named after the Latin word for arrow.
Sagitta may also refer to:
Sagitta (arrowworm), a genus of chaetognaths in the class Sagittoidea
Sagitta (geometry), the depth of an arc
Sagitta (optics), a measure of the glass removed to yield an optical curve
Ligularia sagitta, a plant species
Sagitta, one of the otoliths, a structure in the inner ear of some fish
N.V. Vliegtuigbouw 013 Sagitta, a Dutch-designed glider that first flew in 1960
USNS Sagitta (T-AK-87) (1944–1959)
Versine, a trigonometric function
Sagitta, pseudonym of Scottish-German author John Henry Mackay (1864–1993)
See also
Sagittal, in mathematics
Sagittaria. a genus of aquatic plants
Sagittarius (disambiguation)
Sagit (disambiguation) |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20Adelaide%20United%20FC%20%28W-League%29%20season | The Adelaide United W-League 2010 season was Adelaide United's third season in the W-League.
Technical staff
Players
Squad
Transfers
In
Out
Squad statistics
Last updated 20 November 2010
Regular season
Standings
Fixtures
Adelaide United played 10 games during the 2010–11 W-League season:
Leading scorers
No player scored more than once during the season.
References
External links
Official club website
2010-11
2010–11 W-League (Australia) by team |
https://en.wikipedia.org/wiki/Group%20of%20rational%20points%20on%20the%20unit%20circle | In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side lengths a, b, c, with c the hypotenuse, such that the sides have no common factor larger than 1. Then on the unit circle there exists the rational point (a/c, b/c), which, in the complex plane, is just a/c + ib/c, where i is the imaginary unit. Conversely, if (x, y) is a rational point on the unit circle in the 1st quadrant of the coordinate system (i.e. x > 0, y > 0), then there exists a primitive right triangle with sides xc, yc, c, with c being the least common multiple of the denominators of x and y. There is a correspondence between points (a, b) in the x-y plane and points a + ib in the complex plane which is used below.
Group operation
The set of rational points on the unit circle, shortened G in this article, forms an infinite abelian group under rotations. The identity element is the point (1, 0) = 1 + i0 = 1. The group operation, or "product" is (x, y) * (t, u) = (xt − uy, xu + yt). This product is angle addition since x = cos(A) and y = sin(A), where A is the angle that the vector (x, y) makes with the vector (1,0), measured counter-clockwise. So with (x, y) and (t, u) forming angles A and B with (1, 0) respectively, their product (xt − uy, xu + yt) is just the rational point on the unit circle forming the angle A + B with (1, 0). The group operation is expressed more easily with complex numbers: identifying the points (x, y) and (t, u) with x + iy and t + iu respectively, the group product above is just the ordinary complex number multiplication (x + iy)(t + iu) = xt − yu + i(xu + yt), which corresponds to the point (xt − uy, xu + yt) as above.
Example
3/5 + 4/5i and 5/13 + 12/13i (which correspond to the two most famous Pythagorean triples (3,4,5) and (5,12,13)) are rational points on the unit circle in the complex plane, and thus are elements of G. Their group product is −33/65 + 56/65i, which corresponds to the Pythagorean triple (33,56,65). The sum of the squares of the numerators 33 and 56 is 1089 + 3136 = 4225, which is the square of the denominator 65.
Other ways to describe the group
The set of all 2×2 rotation matrices with rational entries coincides with G. This follows from the fact that the circle group is isomorphic to , and the fact that their rational points coincide.
Group structure
The structure of G is an infinite sum of cyclic groups. Let G2 denote the subgroup of G generated by the point . G2 is a cyclic subgroup of order 4. For a prime p of form 4k + 1, let Gp denote the subgroup of elements with denominator pn where n is a non-negative integer. Gp is an infinite cyclic group, and the point (a2 − b2)/p + (2ab/p)i is a generator of Gp. Furthermore, by factoring the denominators of an eleme |
https://en.wikipedia.org/wiki/Raza%20Razai | Raza Razai (born February 16, 1980) is an Afghan football player. He has played for Afghanistan national team.
National team statistics
External links
1980 births
Living people
Afghan men's footballers
Men's association football forwards
Afghanistan men's international footballers |
https://en.wikipedia.org/wiki/Ahmad%20Zia%20Azimi | Ahmed Zia Azimi (born 1977) is an Afghan football player. He has played for Afghanistan national team.
National team statistics
External links
1977 births
Living people
Afghan men's footballers
Footballers at the 2002 Asian Games
Men's association football defenders
Asian Games competitors for Afghanistan
Afghanistan men's international footballers |
https://en.wikipedia.org/wiki/Najiballah%20Karimi | Najiballah Karimi (born 1979) is an Afghan football player. He has played for Afghanistan national team.
National team statistics
External links
1979 births
Living people
Afghan men's footballers
Men's association football midfielders
Place of birth missing (living people)
Afghanistan men's international footballers
Footballers at the 2002 Asian Games
Asian Games competitors for Afghanistan |
https://en.wikipedia.org/wiki/Rahil%20Ahmad%20Fourmoli | Ahmad Rahil Fourmoli (born 30 November 1971) is an Afghan football player. He has played for Afghanistan national team.
National team statistics
External links
1971 births
Living people
Afghan men's footballers
Footballers at the 2002 Asian Games
Men's association football defenders
Asian Games competitors for Afghanistan
Afghanistan men's international footballers |
https://en.wikipedia.org/wiki/Abdul%20Maroof%20Gullestani | Abdul Maroof Gullestani (born 8 June 1986) is an Afghan football player. He has played for Afghanistan national team.
National team statistics
External links
Living people
Afghan men's footballers
1986 births
Men's association football defenders
Afghanistan men's international footballers |
https://en.wikipedia.org/wiki/Semi-infinite | In mathematics, semi-infinite objects are objects which are infinite or unbounded in some but not all possible ways.
In ordered structures and Euclidean spaces
Generally, a semi-infinite set is bounded in one direction, and unbounded in another. For instance, the natural numbers are semi-infinite considered as a subset of the integers; similarly, the intervals and and their closed counterparts are semi-infinite subsets of . Half-spaces and half-lines are sometimes described as semi-infinite regions.
Semi-infinite regions occur frequently in the study of differential equations. For instance, one might study solutions of the heat equation in an idealised semi-infinite metal bar.
A semi-infinite integral is an improper integral over a semi-infinite interval. More generally, objects indexed or parametrised by semi-infinite sets may be described as semi-infinite.
Most forms of semi-infiniteness are boundedness properties, not cardinality or measure properties: semi-infinite sets are typically infinite in cardinality and measure.
In optimization
Many optimization problems involve some set of variables and some set of constraints. A problem is called semi-infinite if one (but not both) of these sets is finite. The study of such problems is known as semi-infinite programming.
References
Infinity |
https://en.wikipedia.org/wiki/Strongly%20embedded%20subgroup | In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever g is not in H.
The Bender–Suzuki theorem, proved by extending work of , classifies the groups G with a strongly embedded subgroup H. It states that either
G has cyclic or generalized quaternion Sylow 2-subgroups and H contains the centralizer of an involution
or G/O(G) has a normal subgroup of odd index isomorphic to one of the simple groups PSL2(q), Sz(q) or PSU3(q) where q≥4 is a power of 2 and H is O(G)NG(S) for some Sylow 2-subgroup S.
revised Suzuki's part of the proof.
extended Bender's classification to groups with a proper 2-generated core.
References
Finite groups |
https://en.wikipedia.org/wiki/Higman%27s%20theorem | Higman's theorem may refer to:
Hall–Higman theorem in group theory, proved in 1956 by Philip Hall and Graham Higman
Higman's embedding theorem in group theory, by Graham Higman
See also
Higman's lemma
Mathematics disambiguation pages |
https://en.wikipedia.org/wiki/Sma%C3%AFl%20Khaled | Smaïl Khaled (born September 8, 1975) is an Algerian football player. He has played for the Algeria national football team.
National team statistics
References
1975 births
Living people
Algerian men's footballers
Men's association football defenders
Algeria men's international footballers
21st-century Algerian people |
https://en.wikipedia.org/wiki/Kamel%20Bouacida | Kamel Bouacida (born August 6, 1976) is an Algerian footballer. He has played for the Algeria national team.
National team statistics
References
1976 births
Living people
Algerian men's footballers
Algeria men's under-23 international footballers
Men's association football defenders
Algeria men's international footballers
21st-century Algerian people |
https://en.wikipedia.org/wiki/Tarek%20Ghoul | Tarek Ghoul (born January 6, 1975 in El Harrach, Algiers Province) is a retired Algerian football player. He played for Algeria national team.
National team statistics
References
DZFoot Profile
1975 births
Living people
People from El Harrach
Algerian men's footballers
Algeria men's international footballers
USM Alger players
USM Blida players
USM El Harrach players
MC Oran players
MO Constantine players
1996 African Cup of Nations players
1998 African Cup of Nations players
Algeria men's under-23 international footballers
Competitors at the 1997 Mediterranean Games
Men's association football defenders
Mediterranean Games competitors for Algeria
21st-century Algerian people |
https://en.wikipedia.org/wiki/Samir%20Kherbouche | Samir Kherbouche (born January 28, 1976) is an Algerian football player. He has played for Algeria national team.
National team statistics
References
1976 births
Algeria men's international footballers
Algerian men's footballers
Algeria men's under-23 international footballers
CA Batna players
JSM Béjaïa players
Living people
MO Béjaïa players
OMR El Annasser players
People from Tlemcen
WA Tlemcen players
Amal Bou Saâda players
Competitors at the 1997 Mediterranean Games
Men's association football defenders
Mediterranean Games competitors for Algeria
21st-century Algerian people
20th-century Algerian people |
https://en.wikipedia.org/wiki/Roberto%20Jonas | Roberto Jonas Alonso (born 7 June 1967) is an Andorran footballer. He has played for the Andorra national team.
National team statistics
International goal
Scores and results list Andorra's goal tally first.
References
1967 births
Living people
Andorran men's footballers
Andorra men's international footballers
Men's association football defenders |
https://en.wikipedia.org/wiki/Dani%20Ferron | Daniel Ferrón Pérez (born 13 March 1980) is an Andorran football player. He has played for Andorra national team.
National team statistics
References
1980 births
Living people
Andorran men's footballers
Men's association football defenders
Andorra men's international footballers |
https://en.wikipedia.org/wiki/Txema%20Garc%C3%ADa | José Manuel García Luena (born 4 December 1974) is a footballer who plays for FC Encamp as a defender. Born in Spain, he has represented Andorra internationally.
National team statistics
References
1974 births
Living people
Footballers from San Sebastián
Spanish men's footballers
Spanish emigrants to Andorra
FC Andorra players
FC Rànger's players
FC Santa Coloma players
Segunda División B players
Tercera División players
Andorran men's footballers
Andorra men's international footballers
Men's association football fullbacks
Naturalised citizens of Andorra
Primera Divisió players |
https://en.wikipedia.org/wiki/Alfonso%20S%C3%A1nchez%20%28Andorran%20footballer%29 | Alfonso Sanchez (born 27 July 1974) is an Andorran football player. He has played for Andorra national team.
National team statistics
References
1974 births
Living people
Andorran men's footballers
Men's association football goalkeepers
Andorra men's international footballers |
https://en.wikipedia.org/wiki/Jordi%20Benet | Jordi Benet (born 15 July 1980) is an Andorran football player. He has played twice for the Andorra national team.
National team statistics
References
1980 births
Living people
Andorran men's footballers
Men's association football midfielders
Andorra men's international footballers |
https://en.wikipedia.org/wiki/David%20Buxo | David Buxo (born 16 June 1981) is an Andorran football player. He has played for Andorra national team.
National team statistics
References
1981 births
Living people
Andorran men's footballers
Men's association football midfielders
Andorra men's international footballers |
https://en.wikipedia.org/wiki/Lyapunov%20vector | In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system. They have been used in predictability analysis and as initial perturbations for ensemble forecasting in numerical weather prediction. In modern practice they are often replaced by bred vectors for this purpose.
Mathematical description
Lyapunov vectors are defined along the trajectories of a dynamical system. If the system can be described by a d-dimensional state vector the Lyapunov vectors , point in the directions in which an infinitesimal perturbation will grow asymptotically, exponentially at an average rate given by the Lyapunov exponents .
When expanded in terms of Lyapunov vectors a perturbation asymptotically aligns with the Lyapunov vector in that expansion corresponding to the largest Lyapunov exponent as this direction outgrows all others. Therefore, almost all perturbations align asymptotically with the Lyapunov vector corresponding to the largest Lyapunov exponent in the system.
In some cases Lyapunov vectors may not exist.
Lyapunov vectors are not necessarily orthogonal.
Lyapunov vectors are not identical with the local principal expanding and contracting directions, i.e. the eigenvectors of the Jacobian. While the latter require only local knowledge of the system, the Lyapunov vectors are influenced by all Jacobians along a trajectory.
The Lyapunov vectors for a periodic orbit are the Floquet vectors of this orbit.
Numerical method
If the dynamical system is differentiable and the Lyapunov vectors exist, they can be found by forward and backward iterations of the linearized system along a trajectory. Let map the system with state vector at time to the state at time . The linearization of this map, i.e. the Jacobian matrix describes the change of an infinitesimal perturbation . That is
Starting with an identity matrix the iterations
where is given by the Gram-Schmidt QR decomposition of , will asymptotically converge to matrices that depend only on the points of a trajectory but not on the initial choice of . The rows of the orthogonal matrices define a local orthogonal reference frame at each point and the first rows span the same space as the Lyapunov vectors corresponding to the largest Lyapunov exponents. The upper triangular matrices describe the change of an infinitesimal perturbation from one local orthogonal frame to the next. The diagonal entries of are local growth factors in the directions of the Lyapunov vectors. The Lyapunov exponents are given by the average growth rates
and by virtue of stretching, rotating and Gram-Schmidt orthogonalization the Lyapunov exponents are ordered as . When iterated forward in time a random vector contained in the space spanned by the first columns of will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In |
https://en.wikipedia.org/wiki/C-group | In mathematical group theory, a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection.
The simple C-groups were determined by , and his classification is summarized by . The classification of C-groups was used in Thompson's classification of N-groups.
The simple C-groups are
the projective special linear groups PSL2(p) for p a Fermat or Mersenne prime
the projective special linear groups PSL2(9)
the projective special linear groups PSL2(2n) for n≥2
the projective special linear groups PSL3(q) for q a prime power
the Suzuki groups Sz(22n+1) for n≥1
the projective unitary groups PU3(q) for q a prime power
CIT-groups
The C-groups include as special cases the CIT-groups, that are groups in which the centralizer of any involution is a 2-group. These were classified by , and the simple ones consist of the C-groups other than PU3(q) and PSL3(q). The ones whose Sylow 2-subgroups are elementary abelian were classified in a paper of , which was forgotten for many years until rediscovered by Feit in 1970.
TI-groups
The C-groups include as special cases the TI-groups (trivial intersection groups), that are groups in which any two Sylow 2-subgroups have trivial intersection. These were classified by , and the simple ones are of the form PSL2(q), PU3(q), Sz(q) for q a power of 2.
References
Finite groups |
https://en.wikipedia.org/wiki/C%20group | C group or variation, may refer to:
C-group, in mathematics group theory
C-Group culture (2400 BCE - 1550 BCE) an archaeological culture of Lower Nubia
cgroups (control groups) in Linux kernel namespace
CGroup, a subsidiary of Li & Fung
See also
Group C (disambiguation)
Group 3 (disambiguation)
Group (disambiguation)
C (disambiguation) |
https://en.wikipedia.org/wiki/Alex%20Rodriguez%20%28Andorran%20footballer%29 | Alex Rodriguez (born 15 October 1980) is an Andorran football player. He has played for Andorra national team and Santa Coloma.
National team statistics
Updated 28 September 2014
References
External links
Andorra - Record International Players
1980 births
Living people
Andorran men's footballers
Andorra men's international footballers
FC Santa Coloma players
Men's association football defenders |
https://en.wikipedia.org/wiki/Alain%20Montwani | Alain Motwani Marina (born 12 January 1984) is a retired Andorran footballer, who has played for Andorra national team, and a manager.
National team statistics
References
1984 births
Living people
Footballers from Andorra la Vella
Andorran men's footballers
Andorra men's international footballers
Andorran football managers
Men's association football forwards |
https://en.wikipedia.org/wiki/Xavier%20Soria | Xavier Soria (born 2 June 1972) is an Andorran football player. He has played for Andorra national team.
National team statistics
References
1972 births
Living people
Andorran men's footballers
Men's association football midfielders
Andorra men's international footballers |
https://en.wikipedia.org/wiki/Gen%C3%ADs%20Garc%C3%ADa | Genís García Iscla (born 18 May 1978) is an Andorran footballer. He currently plays for the Andorra national team and FC Santa Coloma.
National team statistics
Updated 28 September 2014
References
External links
1978 births
Living people
Andorran men's footballers
FC Santa Coloma players
Men's association football midfielders
Andorra men's international footballers |
https://en.wikipedia.org/wiki/Moment%20curve | In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form
In the Euclidean plane, the moment curve is a parabola, and in three-dimensional space it is a twisted cubic. Its closure in projective space is the rational normal curve.
Moment curves have been used for several applications in discrete geometry including cyclic polytopes, the no-three-in-line problem, and a geometric proof of the chromatic number of Kneser graphs.
Properties
Every hyperplane intersects the moment curve in a finite set of at most d points. If a hyperplane intersects the curve in exactly d points, then the curve crosses the hyperplane at each intersection point. Thus, every finite point set on the moment curve is in affine general position.
Applications
The convex hull of any finite set of points on the moment curve is a cyclic polytope. Cyclic polytopes have the largest possible number of faces for a given number of vertices, and in dimensions four or more have the property that their edges form a complete graph. More strongly, they are neighborly polytopes, meaning that each set of at most d/2 vertices of the polytope forms one of its faces. Sets of points on the moment curve also realize the maximum possible number of simplices, , among all possible Delaunay triangulations of sets of n points in d dimensions.
In the Euclidean plane, it is possible to divide any area or measure into four equal subsets, using the ham sandwich theorem. Similarly but more complicatedly, any volume or measure in three dimensions may be partitioned into eight equal subsets by three planes. However, this result does not generalize to five or more dimensions, as the moment curve provides examples of sets that cannot be partitioned into 2d subsets by d hyperplanes. In particular, in five dimensions, sets of five hyperplanes can partition segments of the moment curve into at most 26 pieces. It is not known whether four-dimensional partitions into 16 equal subsets by four hyperplanes are always possible, but it is possible to partition 16 points on the four-dimensional moment curve into the 16 orthants of a set of four hyperplanes.
A construction based on the moment curve can be used to prove a lemma of Gale, according to which, for any positive integers k and d, it is possible to place 2k + d points on a d-dimensional sphere in such a way that every open hemisphere contains at least k points. This lemma, in turn, can be used to calculate the chromatic number of the Kneser graphs, a problem first solved in a different way by László Lovász.
The moment curve has also been used in graph drawing, to show that all n-vertex graphs may be drawn with their vertices in a three-dimensional integer grid of side length O(n) and with no two edges crossing. The main idea is to choose a prime number p larger than n and to place vertex i of the graph at coordinates
(i, i 2 mod p, i 3 mod p).
Then a plane can only c |
https://en.wikipedia.org/wiki/Avelino%20Lopes%20%28footballer%29 | Avelino Lopes (born March 4, 1974) is an Angolan football player. He has played for Angola national team.
National team statistics
References
1974 births
Living people
Angolan men's footballers
Men's association football forwards
Angola men's international footballers |
https://en.wikipedia.org/wiki/Chinguila | Chinguila (born August 14, 1978) is an Angolan football player. He has played for Angola national team.
National team statistics
References
1978 births
Living people
Angolan men's footballers
Men's association football defenders
Angola men's international footballers |
https://en.wikipedia.org/wiki/Didi%20%28Angolan%20footballer%29 | Didi is an Angolan football player. He has played for Angola national team.
National team statistics
References
External links
Living people
Angolan men's footballers
Men's association football defenders
Year of birth missing (living people)
Angola men's international footballers |
https://en.wikipedia.org/wiki/Manuel%20Sala | Manuel Sala (born May 5, 1982) is an Angolan football player. He has played for Angola national team.
National team statistics
References
1982 births
Living people
Angolan men's footballers
Men's association football defenders
Angola men's international footballers |
https://en.wikipedia.org/wiki/Dias%20Caires | Yahenda Joaquim Caires Fernandes, also known as Dias Caires (born April 18, 1978) is a former Angolan football player. He has played for Angola national team.
National team statistics
References
1978 births
Living people
Angolan men's footballers
Atlético Petróleos de Luanda players
Atlético Sport Aviação players
G.D. Sagrada Esperança players
G.D. Interclube players
Girabola players
Angola men's international footballers
Men's association football defenders |
https://en.wikipedia.org/wiki/Renato%20Campos | Renato Baptista Campos (born September 5, 1980) is a retired Angolan football player. He has played for the Angolan national team.
National team statistics
References
External links
Biography at Jornal dos Desportos
1980 births
Living people
Angolan men's footballers
Atlético Petróleos de Luanda players
Atlético Sport Aviação players
Men's association football defenders
Angola men's international footballers |
https://en.wikipedia.org/wiki/Elisio | Elísio Muondo Dala (born October 3, 1980) is a retired Angolan football player. He has played for Angola national team.
He is the older brother of Gelson Dala.
National team statistics
References
1980 births
Living people
Angolan men's footballers
C.D. Primeiro de Agosto players
Angola men's international footballers
Men's association football defenders |
https://en.wikipedia.org/wiki/Marito | Mário André Rodrigues João best known as Marito (born August 30, 1977) is a retired Angolan football goalkeeper. He has played for Angola national team.
National team statistics
References
External links
1977 births
Living people
Footballers from Luanda
Angolan men's footballers
Angola men's international footballers
1998 African Cup of Nations players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Fernando%20Pereira%20%28Angolan%20footballer%29 | Fernando Pereira (born December 16, 1973) is an Angolan football player. He has played for Angola national team.
National team statistics
See also
List of Angola international footballers
References
1973 births
Living people
Angolan men's footballers
Men's association football goalkeepers
Angola men's international footballers |
https://en.wikipedia.org/wiki/Sim%C3%A3o%20%28Angolan%20footballer%29 | Simão (born July 10, 1976) is an Angolan football player. He has played for Angola national team.
National team statistics
References
1976 births
Living people
Angolan men's footballers
Men's association football midfielders
Angola men's international footballers |
https://en.wikipedia.org/wiki/Fofana%20%28Angolan%20footballer%29 | Pedro Cassunda Domingos known as Fofana (born April 3, 1982) is a former Angolan football player. He has played for Angola national team.
National team statistics
References
Atlético Sport Aviação players
S.L. Benfica (Luanda) players
Girabola players
1982 births
Living people
Angolan men's footballers
Angola men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Jonas%20%28footballer%2C%20born%201972%29 | Carlos Emanuel Romeu Lima (22 September 1972 – 16 October 2018), better known as Jonas, was an Angolan footballer who played for the Angola national team.
National team statistics
References
Living people
1972 births
Angolan men's footballers
Men's association football midfielders
Angola men's international footballers
Atlético Petróleos de Luanda players
Estrela Clube Primeiro de Maio players |
https://en.wikipedia.org/wiki/Filipe%20Nzanza | Filipe Nzanza (born April 21, 1970) is a retired Angolan football player. He has played for Angola national team and for Clube Desportivo Primeiro de Agosto.
National team statistics
References
1970 births
Living people
Angolan men's footballers
Men's association football midfielders
Angola men's international footballers |
https://en.wikipedia.org/wiki/Rats%20%28footballer%29 | Ambrósio Amaro Manuel Pascoal best known as Rats, (born 5 May 1977) is a retired Angolan football player. He has played for Angola national team.
National team statistics
References
1977 births
Living people
Angolan men's footballers
Men's association football midfielders
Académica Petróleos do Lobito players
Girabola players
Angola men's international footballers |
https://en.wikipedia.org/wiki/Ralston%20Phoenix | Ralston Phoenix is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Antigua Barracuda FC players
USL Championship players
Men's association football forwards
1981 births |
https://en.wikipedia.org/wiki/Steveroy%20Anthony | Steveroy Anthony (born 7 July 1971) is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
1971 births
Living people
Antigua and Barbuda men's footballers
Place of birth missing (living people)
Antigua and Barbuda men's international footballers
Men's association football defenders |
https://en.wikipedia.org/wiki/Aisam-ul-Haq%20Qureshi%20career%20statistics | Aisam-ul-Haq Qureshi is a professional tennis player who is the current Pakistani number one doubles player. He has reached two major finals in total: (1 Doubles, 1 Mixed), both at the 2010 US Open. Qureshi has been ranked as high as world No. 8 in the ATP doubles rankings.
Qureshi made his professional tennis debut on the main tour at the Chennai Open in 2001. So far in his career, Qureshi has won a total of 16 doubles titles.
Below is a list of career achievements and titles won by Aisam-ul-Haq Qureshi.
Significant finals
Grand Slam tournament finals
Doubles: 1 (1 runner-up)
Mixed doubles: 1 (1 runner-up)
Masters 1000 finals
Doubles: 3 (2 titles, 1 runner-up)
ATP career finals
Doubles: 42 (18 titles, 24 runners-up)
Other career finals
Singles
Doubles
Challengers and Futures finals
Singles: 22 (16–6)
Doubles: 86 (47–39)
Career performance timeline
Source for the following tables:
Singles
Doubles
Mixed doubles
ATP ranking
Singles
Doubles
Grand Slam seedings
The advances into finals by Qureshi are in italics.
Men's doubles
Mixed doubles
References
General
Career finals, Grand Slam tournament seedings, information for the performance timelines, top 10 wins and national participation information have been taken from these sources:
Specific
Qureshi, Aisam-Ul-Haq
Tennis in Pakistan |
https://en.wikipedia.org/wiki/Kevin%20Roberts%20%28Antigua%20and%20Barbudan%20footballer%29 | Kevin Roberts is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football defenders
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Kenroy%20Smith | Kenroy Smith is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football defenders
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Desmond%20Bleau | Desmond Bleau (born September 2, 1982) is an Antigua and Barbudan football player. He played for Antigua and Barbuda national team.
National team statistics
International goals
Scores and results list Antigua and Barbuda's goal tally first.
References
1982 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Verton%20Harris | Verton Harris (born 15 July 1980) is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
1980 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Old Road F.C. players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Rackley%20Thomas | Rackley Thomas is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
1980 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football defenders |
https://en.wikipedia.org/wiki/Rowan%20Isaac | Rowan Isaac (born March 5, 1977) is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
1977 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Elvis%20Anthony | Elvis Anthony (born July 4, 1970) is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
1970 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Neil%20Schyan%20Jeffers | Neil Schyan Jeffers (born March 22, 1977) is an Antigua and Barbudan football player. He plays for Antigua and Barbuda national team.
National team statistics
References
1977 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football defenders
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Winston%20Roberts | Winston Roberts (born January 28, 1976) is an Antigua and Barbudan football player. He plays for Antigua and Barbuda national team.
National team statistics
International goals
Scores and results list Antigua and Barbuda's goal tally first.
References
External links
1976 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kevin%20Watts | Kevin Watts (born April 24, 1975) is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
1975 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Quentin%20Clarke | Quentin Clarke (born April 8, 1969) is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
International goals
Scores and results list Antigua and Barbuda's goal tally first.
References
1969 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Arnold%20James | Arnold James (born April 30, 1974) is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team.
National team statistics
References
1974 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Taj%20Charles | Taj Charles is an Antigua and Barbudan football player. He has played for Antigua and Barbuda national team. He was born on 23 August 1977 in Antigua and Barbuda.
National team statistics
References
External links
1977 births
Living people
Antigua and Barbuda men's footballers
Antigua and Barbuda men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kelly%20Frederick | Kelly Roberto Rafique Frederick (born October 7, 1978) is an Antigua and Barbudan former football player.
Career statistics
Club
Notes
International
International goals
Scores and results list Antigua and Barbuda's goal tally first.
References
1978 births
Living people
University of Nevada, Las Vegas alumni
Antigua and Barbuda men's footballers
Antigua and Barbuda expatriate men's footballers
Antigua and Barbuda men's international footballers
Men's association football forwards
C.F. União de Coimbra players
Expatriate men's footballers in Portugal
Expatriate men's soccer players in the United States |
https://en.wikipedia.org/wiki/William%20I.%20McLaughlin | William I. McLaughlin (born 1935) is a retired American space scientist.
After being awarded his PhD in mathematics at the University of California at Berkeley in 1968 (with a thesis on celestial mechanics), he worked at Bellcomm, Inc. in Washington D.C. on the Apollo lunar-landing program. In 1971 he joined the Jet Propulsion Laboratory, where he remained until his retirement in 1999. He participated in a number of projects, including Viking, SEASAT, and the Infrared Astronomical Satellite (IRAS). He served as the JPL deputy directory of astrophysics, the manager of the Voyager 2 flight engineering office during the spacecraft's encounter with Uranus, and manager of the Mission Profiling and Sequencing Section.
In 1977 he suggested that widely observable celestial events such as nova explosions may be used by hypothetical extraterrestrial intelligences for scheduling the transmission of signals. This would allow receivers in other stellar systems to estimate when the signals would arrive. To test this concept, during a six-month period in 1988, the star Epsilon Eridani was observed using a 40-foot radio telescope at the National Radio Astronomy Observatory in Green Bank, West Virginia, with Nova Cygni 1975 being used as the timer. However, no anomalous radio signals were observed. This does not, of course, preclude the idea from being confirmed at a later date.
He received the NASA Exceptional Service Medal in 1984 for his work on the IRAS and the NASA Outstanding Leadership Medal for his work on the Voyager encounter with Uranus. Asteroid 4838 Billmclaughlin is named after him.
References
Living people
1935 births
Space scientists |
https://en.wikipedia.org/wiki/Oppermann%27s%20conjecture | Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers. It is closely related to but stronger than Legendre's conjecture, Andrica's conjecture, and Brocard's conjecture. It is named after Danish mathematician Ludvig Oppermann, who announced it in an unpublished lecture in March 1877.
Statement
The conjecture states that, for every integer x > 1, there is at least one prime number between
x(x − 1) and x2,
and at least another prime between
x2 and x(x + 1).
It can also be phrased equivalently as stating that the prime-counting function must take unequal values at the endpoints of each range. That is:
π(x2 − x) < π(x2) < π(x2 + x) for x > 1
with π(x) being the number of prime numbers less than or equal to x.
The end points of these two ranges are a square between two pronic numbers, with each of the pronic numbers being twice a pair triangular number. The sum of the pair of triangular numbers is the square.
Consequences
If the conjecture is true, then the gap size would be on the order of
.
This also means there would be at least two primes between x2 and (x + 1)2 (one in the range from x2 to x(x + 1) and the second in the range from x(x + 1) to (x + 1)2), strengthening Legendre's conjecture that there is at least one prime in this range. Because there is at least one non-prime between any two odd primes it would also imply Brocard's conjecture that there are at least four primes between the squares of consecutive odd primes. Additionally, it would imply that the largest possible gaps between two consecutive prime numbers could be at most proportional to twice the square root of the numbers, as Andrica's conjecture states.
The conjecture also implies that at least one prime can be found in every quarter revolution of the Ulam spiral.
Status
Even for small values of x, the numbers of primes in the ranges given by the conjecture are much larger than 1, providing strong evidence that the conjecture is true. However, Oppermann's conjecture has not been proved
See also
Bertrand's postulate
Firoozbakht's conjecture
Prime number theorem
References
Conjectures about prime numbers
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Aleksandr%20Kotelnikov | Aleksandr Petrovich Kotelnikov (; October 20, 1865 – March 6, 1944) was a Russian and Soviet mathematician specializing in geometry and kinematics.
Biography
Aleksandr was the son of , a colleague of Nikolai Lobachevsky. The subject of hyperbolic geometry was non-Euclidean geometry, a departure from tradition. The early exposure to Lobachevsky's work eventually led to Aleksandr undertaking the job of editing Lobachevsky's works.
Kotelnikov studied at Kazan University, graduating in 1884. He began teaching at a gymnasium.
Having an interest in mechanics, he did graduate study. His thesis was The Cross-Product Calculus and Certain of its Applications in Geometry and Mechanics. His work contributed to the development of screw theory and kinematics. Kotelnikov began instructing at the university in 1893. His habilitation thesis was The Projective Theory of Vectors (1899).
In Kiev, Kotelnikov was professor and head of the department of pure mathematics until 1904. Returning to Kazan, he headed the mathematics department until 1914. He was at the Kyiv Polytechnic Institute directing the department of Theoretical Mechanics until 1924, when he moved to Moscow and took up teaching at Bauman Technical University.
In addition to the Works of Lobachevsky, Kotelnikov was also the editor of the collected works of Nikolay Zhukovsky, the father of Russian aerodynamics.
One reviewer put Kotelnikov at the head of a chain of investigations of Spaces over Algebras. Successive researchers included D.N. Zeiliger, A.P. Norden, and B. A. Rosenfel'd.
Dual quaternions
Kotelnikov advanced an algebraic method of representing Euclidean motions that had been introduced by William Kingdon Clifford. Though developed to render motions in three-dimensional space, an eight-dimensional algebra of doubled quaternions was used. Clifford had shown that a space of rotations entailed elliptic space described by versors in his four-dimensional quaternions. According to Wilhelm Blaschke, it was Kotelnikov who initiated a "conversion principle" to take a dual rotation acting on elliptic space to a motion of , three-dimensional Euclidean space:
If r is one of the square roots of minus one in , then an underline () represents the elliptic line in the plane perpendicular to r (Blaschke: the united elliptic line). Using the inner product on formed by taking the product of a quaternion with its conjugate, the condition
is equivalent to
and implies that elliptic lines are perpendicular. Under these conditions, the Kotelnikov conversion to Euclidean motion is represented as
and where is the screw axis.
Other works
1925: Introduction to Theoretical Mechanics, Moscow-Leningrad
1927: The Principle of Relativity and Lobachevsky's Geometry, Kazan
1950: The Theory of Vectors and Complex Numbers, Moscow-Leningrad
References
Literature
A.T. Grigorian (1976) "Aleksandr Petrovich Kotelnikov", Dictionary of Scientific Biography.
B.L. Laptev & B.A. Rozenfel'd (1996) Mathematics |
https://en.wikipedia.org/wiki/Guerrero%20Amuzgo%20language | The Guerrero Amuzgo language is an Amuzgo language spoken in southwest Guerrero state in Mexico.
Statistics and history
There are 23,000 speakers, 10,000 that are monolingual. It is also known as Nomndaa or Ñomndaa. It belongs to the Oto-Manguean language family and the Amuzgoan subfamily. The use of the language is widespread and it is learned as a second language by Spanish and Nahuatl speakers living with the Guerrero speakers.
There is a positive cultural affinity toward the tongue and it is used in business, religion, and taught bilingually with Spanish until 6th grade. 10% of adults and 15% of children are literate in Amuzgo Guerrero. There are media such as videos, a dictionary and radio broadcasts in the language that propagate its use.
Phonology
Vowels
Sounds /æ, æ̃, æ̰, æ̰̃/ can also fluctuate to more mid sounds [ɛ, ɛ̃, ɛ̰, ɛ̰̃].
Consonants
Sounds [p, ᵐb, r] only appear in a few words.
Notes
Amuzgos
Guerrero
Indigenous languages of Mexico
Mesoamerican languages
Oto-Pamean languages |
https://en.wikipedia.org/wiki/Fermat%27s%20right%20triangle%20theorem | Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci. In its geometric forms, it states:
A right triangle in the Euclidean plane for which all three side lengths are rational numbers cannot have an area that is the square of a rational number. The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square.
A right triangle and a square with equal areas cannot have all sides commensurate with each other.
There do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle.
More abstractly, as a result about Diophantine equations (integer or rational-number solutions to polynomial equations), it is equivalent to the statements that:
If three square numbers form an arithmetic progression, then the gap between consecutive numbers in the progression (called a congruum) cannot itself be square.
The only rational points on the elliptic curve are the three trivial points with and .
The quartic equation has no nonzero integer solution.
An immediate consequence of the last of these formulations is that Fermat's Last Theorem is true in the special case that its exponent is 4.
Formulation
Squares in arithmetic progression
In 1225, Emperor Frederick II challenged the mathematician Fibonacci to take part in a mathematical contest against several other mathematicians, with three problems set by his court philosopher John of Palermo. The first of these problems asked for three rational numbers whose squares were equally spaced five units apart, solved by Fibonacci with the three numbers , , and .
In The Book of Squares, published later the same year by Fibonacci, he solved the more general problem of finding triples of square numbers that are equally spaced from each other, forming an arithmetic progression. Fibonacci called the gap between these numbers a congruum. One way of describing Fibonacci's solution is that the numbers to be squared are the difference of legs, hypotenuse, and sum of legs of a Pythagorean triangle, and that the congruum is four times the area of the same triangle. Fibonacci observed that it is impossible for a congruum to be a square number itself, but did not present a satisfactory proof of this fact.
If three squares , , and could form an arithmetic progression whose congruum was also a square , then these numbers would satisfy the Diophantine equations
That is, by the Pythagorean theorem, they would form two integer-sided right triangles in which the pair gives one leg and the hypotenuse of the smaller triangle and the same pair also forms the two legs of the larger triangle. But if (as Fibonacci asserted) no square congruum can exist, then there can be no two integ |
https://en.wikipedia.org/wiki/Hayk%20Hakobyan | Hayk Hakobyan (; born 26 December 1980) is an Armenian football player. He has played for Armenia national team.
National team statistics
References
1980 births
Living people
Armenian men's footballers
Armenia men's international footballers
Armenian Premier League players
Men's association football forwards |
https://en.wikipedia.org/wiki/Karen%20Asatryan | Karen Asatryan (born 21 December 1974) is an Armenian football player. He has played for Armenia national team.
National team statistics
References
1974 births
Living people
Armenian men's footballers
Men's association football midfielders
Armenia men's international footballers |
https://en.wikipedia.org/wiki/Arkady%20Dokhoyan | Arkady Dokhoyan (born 12 August 1977) is an Armenian football player. He has played for Armenia national team.
National team statistics
References
1977 births
Living people
Armenian men's footballers
Armenia men's international footballers
FC Urartu players
FC Mika players
Armenian Premier League players
Men's association football defenders |
https://en.wikipedia.org/wiki/Armen%20Petikyan | Armen Petikyan (born 19 February 1972) is an Armenian football player. He has played for Armenia national team.
National team statistics
References
1972 births
Living people
Armenian men's footballers
Men's association football defenders
Armenia men's international footballers
Soviet men's footballers |
https://en.wikipedia.org/wiki/Oleg%20Lotov | Oleg Lotov (; born 21 November 1975) is a former Kazakhstan international football defender.
Club career statistics
Last update: 8 November 2012
International matches
Honours
with Tobol
Intertoto Cup Winner: 2007
Kazakhstan League Champion: 2010
Kazakhstan League Runner-up: 2003, 2005
Kazakhstan Cup Winner: 2007
Kazakhstan Cup Runner-up: 2003
with Astana
Kazakhstan League Champion: 2000
Kazakhstan Cup Winner: 2000-2001
with Vostok
Kazakhstan Cup Runner-up: 1996
References
1975 births
Living people
Men's association football defenders
Kazakhstani men's footballers
Kazakhstan men's international footballers
Kazakhstan Premier League players
FC Vostok players
FC Zhenis players
FC Tobol players
FC Shakhter Karagandy players
Footballers at the 1998 Asian Games
Asian Games competitors for Kazakhstan
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Romeo%20Jenebyan | Romeo Jenebyan (born 10 September 1979) is a former Armenian football player. He has played for Armenia national team.
National team statistics
References
1979 births
Living people
Armenian men's footballers
Armenia men's international footballers
FC Urartu players
FC Mika players
Impuls FC players
Armenian Premier League players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Balep%20Ba%20Ndoumbouk | Balep Ba Ndoumbouk (born October 24, 1987) is an Armenian football player of Cameroonian ethnic origin. He has played for Armenia national team.
National team statistics
References
1987 births
Living people
Cameroonian men's footballers
Armenian men's footballers
Armenia men's international footballers
FC Pyunik players
Armenian Premier League players
Cameroonian emigrants to Armenia
Naturalized citizens of Armenia
Men's association football midfielders |
https://en.wikipedia.org/wiki/Roman%20Aparicio | Roman Aparicio (born December 3, 1980) is an Aruban football player. He has appeared for the Aruba national team in 2000, 2002, 2011, and 2016.
National team statistics
References
1980 births
Living people
Aruban men's footballers
Aruba men's international footballers
SV Racing Club Aruba players
SV Britannia players
People from Oranjestad, Aruba
Men's association football defenders |
https://en.wikipedia.org/wiki/Ikel%20Lopez | Ikel Lopez (born February 16, 1979) is an Aruban footballer. He formerly played for the Aruba national team.
National team statistics
References
1979 births
Living people
Aruban men's footballers
SV Estrella players
Men's association football defenders
Aruba men's international footballers |
https://en.wikipedia.org/wiki/Ryan%20Tromp | Ryan Tromp (born October 5, 1973) is an Aruban football player. He has played for Aruba national team.
National team statistics
References
1973 births
Living people
Aruban men's footballers
Men's association football goalkeepers
SV Jong Aruba players
Aruba men's international footballers |
https://en.wikipedia.org/wiki/Geoland%20Pantophlet | Geoland Pantophlet (born June 14, 1976 ) is an Aruban football player. He played for the Aruba national team in 1996 and 2004.
National team statistics
References
1976 births
Living people
Aruban men's footballers
Men's association football goalkeepers
SV Racing Club Aruba players
Aruba men's international footballers |
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