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https://en.wikipedia.org/wiki/2012%E2%80%9313%20Wycombe%20Wanderers%20F.C.%20season | The 2012–13 Football League Two was Wycombe Wanderers' 125th season in existence and their nineteenth season in the Football League. This page shows the statistics of the club's players in the season, and also lists all matches that the club played during the season.
Wycombe Wanderers ended the season strongly and finished in 15th place in League Two, after a poor start to the campaign.
The end of the season also saw the retirement of Gareth Ainsworth (Wycombe's player-manager). Ainsworth's career had spanned 18 years and saw him play for 10 different clubs. He made his final appearance in Wycombe's 1–1 draw with Port Vale on 27 April 2013.
League Two
League table
Match results
Legend
Friendlies
Football League Two
FA Cup
League Cup
Football League Trophy
Squad statistics
Appearances and goals
|-
|colspan="14"|Players left the club before the end of the season:
|}
Top scorers
*Beavon left the club before the end of the season
Disciplinary record
Transfers
See also
2012–13 in English football
2012–13 Football League Two
Wycombe Wanderers F.C.
Gary Waddock
Gareth Ainsworth
References
External links
Wycombe Wanderers official website
Wycombe Wanderers F.C. seasons
Wycombe Wanderers |
https://en.wikipedia.org/wiki/List%20of%20HNK%20Rijeka%20records%20and%20statistics | HNK Rijeka are a Croatian professional association football club based in Rijeka, Croatia who compete in the Croatian First Football League. The club was formed in July 1946 as NK Kvarner, and played its first unofficial match on 7 August 1946 against Hajduk Split, winning 2–0. Rinaldo Petronio scored the first goal for the club. The first official game was played in the qualifiers for the 1946–47 Yugoslav First League against Unione Sportiva Operaia (Pula), losing 2–1 in Rijeka. Kvarner later won the return leg 4–1 in Pula and qualified for the Yugoslav championship. This article lists various records and statistics related to the club and individual players and managers.
All records and statistics accurate as of 29 October 2023.
Individual records and statistics
Current players and manager are in bold/italics.
Appearances
Most appearances:
All fixtures
684, Srećko Juričić (1974–85)
Official matches
351, Srećko Juričić (1974–85)
In Yugoslav First League
293, Srećko Juričić (1974–85)
In Croatian First Football League
196, Kristijan Čaval (1998–2005, 2010–13)
196, Damir Milinović (1994–2001, 2003–04)
In Yugoslav Cup
25, Srećko Juričić (1975–85)
In Croatian Cup
34, Dragan Tadić (1992–95, 2003–07)
In UEFA competitions
38, Zoran Kvržić (2013–15, 2017–20)
38, Ivan Tomečak (2013–15, 2019–21)
Most appearances in one season:
All official matches
52, Ivan Vargić (2013–14)
In Yugoslav First League
35, Tonči Gabrić (1990–91)
35, Dušan Kljajić (1990–91)
35, Fabijan Komljenović (1990–91)
In Croatian First Football League
36, Andrej Prskalo (2016–17)
36, Marin Tomasov (2015–16)
36, Nediljko Labrović (2022–23)
In Yugoslav Cup
8, Janko Janković (1986–87)
8, Igor Jelavić (1986–87)
8, Roberto Paliska (1986–87)
8, Davor Radmanović (1986–87)
8, Borče Sredojević (1986–87)
In Croatian Cup
10, Elvis Brajković (1993–94)
In UEFA competitions
12, Mato Jajalo (2014–15)
12, Vedran Jugović (2014–15)
12, Andrej Kramarić (2014–15)
12, Zoran Kvržić (2014–15)
12, Mate Maleš (2013–14)
12, Marko Vešović (2017–18)
12, Dario Župarić (2017–18)
Appearances in most seasons:
In top flight
12, Robert Rubčić
In UEFA competitions
6, Zoran Kvržić
6, Ivan Tomečak
Other records in the Croatian First Football League
Youngest player
16 years, 345 days, Filip Braut (25 May 2019 v Slaven Belupo)
Oldest player
35 years, 344 days, Mladen Romić (3 May 1998 v Mladost 127)
Oldest débutante
34 years, 290 days, Elvir Bolić (30 July 2006 v Cibalia)
Most minutes played
17,045 minutes, Damir Milinović (1994–2001, 2003–04)
Most minutes played (one season)
3,240 minutes, Nediljko Labrović (2022–23)
3,240 minutes, Andrej Prskalo (2016–17)
Most consecutive appearances
68, Đoni Tafra (1998–2000)
Most substituted player
56, Anas Sharbini (2005–09, 2013–15)
Most substituted player (one season)
20, Mario Gavranović (2016–17)
Most used substitute
61, Jasmin Samardžić (1992–97, 2003–04)
Most used s |
https://en.wikipedia.org/wiki/Bohlmann | Bohlmann is a German surname. Notable people with the surname include:
Frank Bohlmann (1917-1999), American footballer
Georg Bohlmann (1869–1928), German mathematician who specialized in probability theory and actuarial mathematics.
Georg Carl Bohlmann (1838–1920), Danish composer and organist
Hans-Joachim Bohlmann (1937–2009), German property vandal
Ralph Arthur Bohlmann (1932–2016), American theologian
Sabine Bohlmann (born 1969), German actress
See also
Theodor Bohlmann-Combrinck (1891–1956), German Wehrmacht general
Philip Bohlman (born 1952), American ethnomusicologist
German-language surnames |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20FK%20%C5%BDeljezni%C4%8Dar%20season |
Statistics 2012-13
Squad information
Total squad cost: €5,775,000
From the youth system
Disciplinary record
Includes all competitive matches. The list is sorted by position, and then shirt number.
Transfers
In
Total expenditure:
Out
Total income: €50,000
Competitions
Pre-season
Mid-season
Overall
League table
Results summary
Results by round
Matches
Kup Bosne i Hercegovine
Round of 32
Round of 16
Quarter-finals
Semi-finals
Final
UEFA Champions League
Second qualifying round
References
FK Željezničar Sarajevo seasons
Zeljeznicar |
https://en.wikipedia.org/wiki/Symmetric%20decreasing%20rearrangement | In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.
Definition for sets
Given a measurable set, in one defines the symmetric rearrangement of called as the ball centered at the origin, whose volume (Lebesgue measure) is the same as that of the set
An equivalent definition is
where is the volume of the unit ball and where is the volume of
Definition for functions
The rearrangement of a non-negative, measurable real-valued function whose level sets (for ) have finite measure is
where denotes the indicator function of the set
In words, the value of gives the height for which the radius of the symmetric
rearrangement of is equal to We have the following motivation for this definition. Because the identity
holds for any non-negative function the above definition is the unique definition that forces the identity to hold.
Properties
The function is a symmetric and decreasing function whose level sets have the same measure as the level sets of that is,
If is a function in then
The Hardy–Littlewood inequality holds, that is,
Further, the Pólya–Szegő inequality holds. This says that if and if then
The symmetric decreasing rearrangement is order preserving and decreases distance, that is,
and
Applications
The Pólya–Szegő inequality yields, in the limit case, with the isoperimetric inequality. Also, one can use some relations with harmonic functions to prove the Rayleigh–Faber–Krahn inequality.
Nonsymmetric decreasing rearrangement
We can also define as a function on the nonnegative real numbers rather than on all of Let be a σ-finite measure space, and let be a measurable function that takes only finite (that is, real) values μ-a.e. (where "-a.e." means except possibly on a set of -measure zero). We define the distribution function by the rule
We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement) of as the function by the rule
Note that this version of the decreasing rearrangement is not symmetric, as it is only defined on the nonnegative real numbers. However, it inherits many of the same properties listed above as the symmetric version, namely:
and are equimeasurable, that is, they have the same distribution function.
The Hardy-Littlewood inequality holds, that is,
-a.e. implies
for all real numbers
for all
-a.e. implies
for all positive real numbers
for all positive real numbers
The (nonsymmetric) decreasing rearrangement function arises often in the theory of rearrangement-invariant Banach function spaces. Especially important is the following:
Luxemburg Representation Theorem. Let be a rearrangement-invariant Banach function norm over a resonant measure space Then there exists a (possibly not unique) rearrangement-invariant function norm on such that for all nonnegative measurable fun |
https://en.wikipedia.org/wiki/James%20P.%20Crutchfield | James P. Crutchfield (born 1955) is an American mathematician and physicist. He received his B.A. summa cum laude in physics and mathematics from the University of California, Santa Cruz, in 1979 and his Ph.D. in physics there in 1983. He is currently a professor of physics at the University of California, Davis, where he is director of the Complexity Sciences Center—a new research and graduate program in complex systems. Prior to this, he was research professor at the Santa Fe Institute for many years, where he ran the Dynamics of Learning Group and SFI's Network Dynamics Program. From 1985 to 1997, he was a research physicist in the physics department at the University of California, Berkeley. He has been a visiting research professor at the Sloan Center for Theoretical Neurobiology, University of California, San Francisco; a postdoctoral fellow of the Miller Institute for Basic Research in Science at UCB; a UCB physics department IBM postdoctoral fellow in condensed matter physics; a distinguished visiting research professor of the Beckman Institute at the University of Illinois, Urbana-Champaign; and a Bernard Osher Fellow at the San Francisco Exploratorium.
Research
Over the last three decades, Crutchfield has worked in the areas of nonlinear dynamics, solid-state physics, astrophysics, fluid mechanics, critical phenomena and phase transitions, chaos, and pattern formation. His current research interests center on computational mechanics, the physics of complexity, statistical inference for nonlinear processes, genetic algorithms, evolutionary theory, machine learning, quantum dynamics, and distributed intelligence. He has published over 100 papers in these areas.
In 2022, Crutchfield and his graduate student Kyle Ray described a way to bring the heat production of conventional circuits below the theoretical limit of Landauer's principle by encoding information not as pulses of charge but in the momentum of moving particles.
Life
While a graduate student, Crutchfield and students from the University of California, Santa Cruz (including Doyne Farmer) built a series of computers that were capable of calculating the motion of a moving roulette ball, predicting which numbers could be excluded from the outcome. Equipped with hidden electronic equipment of the early days of "mobile" computing, trials in Las Vegas showed success. However, because of technical limitation and the infamous gambler's ruin, the success was only partial and it was not feasible to use to make large profits. A book written about this project (The Eudaemonic Pie / Newton's Casino: The Bizarre True Story of How a Band of Physicists and Computer Wizards Took on Las Vegas) describes Crutchfield in his early years as "hacker-in-residence": "Crutchfield surfs, snorkels, and backpacks. But what he really cares about in life are computers...".
Selected publications
Melanie Mitchell, Peter Hraber (Santa Fe Institute), James P. Crutchfield (University of California, Berkeley |
https://en.wikipedia.org/wiki/Exeter%20point | In geometry, the Exeter point is a special point associated with a plane triangle. It is a triangle center and is designated as X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986. This is one of the recent triangle centers, unlike the classical triangle centers like centroid, incenter, and Steiner point.
Definition
The Exeter point is defined as follows.
Let be any given triangle. Let the medians through the vertices meet the circumcircle of at respectively. Let be the triangle formed by the tangents at to the circumcircle of . (Let be the vertex opposite to the side formed by the tangent at the vertex , be the vertex opposite to the side formed by the tangent at the vertex , and be the vertex opposite to the side formed by the tangent at the vertex .) The lines through are concurrent. The point of concurrence is the Exeter point of .
Trilinear coordinates
The trilinear coordinates of the Exeter point are
Properties
The Exeter point of triangle ABC lies on the Euler line (the line passing through the centroid, the orthocenter , the de Longchamps point, the Euler centre and the circumcenter) of triangle ABC. So there are 6 points collinear over one only line.
References
Triangle centers |
https://en.wikipedia.org/wiki/Leon%20Petrosyan | Leon Petrosjan () (born December 18, 1940) is a professor of Applied Mathematics and the Head of the Department of Mathematical Game theory and Statistical Decision Theory at the St. Petersburg University, Russia.
Fields of research
The research interests of Leon Petrosjan lie mostly in the fields of operations research, game theory, differential games, and control theory.
In particular, he contributed to the study of the following topics:
Solution of zero-sum simple pursuit games such as "lifeline game", "two pursuers – one evader game", "deadline game in a half plane". Proof of the existence of an epsilon-saddle point in piecewise open loop strategies in general dynamic zero-sum games with prescribed duration and independent motions. Method of solution of pursuit games based on the technique of invariant counter of pursuit (regular case).
Differential pursuit games with incomplete information including games with information delay about the state of the game. Finite search games and dynamic search games. Construction of saddle points using mixed piecewise open loop strategies. Solution of concrete games with incomplete information.
Investigation and refinement of the Nash equilibrium concept for multistage games with perfect information, on the bases of the so-called players preference functions. Proof of the uniqueness of such an equilibrium. Derivation of the system of the first order partial differential extremal equations for the payoffs in Nash equilibrium for differential games. Description of classes of Nash equilibrium in concrete differential games.
Statement and investigation of the time-consistency problem in n-person differential games. Analysis of classical optimality principles from cooperative and non cooperative game theory from the point of their time consistency. Proof of the time inconsistency of the most known optimality principles. Regularization methods (integral and differential) based upon the IDP (imputation distribution procedures) which gives the possibility of construction new time consistent optimality principles from the previously time inconsistent ones.
Applications to environmental protection. Methods of creation of time consistent policy in long range environmental planning based upon the considered approaches for cooperative and non cooperative differential games.
Academic activities
Leon Petrosjan is the Editor of the journal International Game Theory Review (W.S. Pbl., Singapore, London); the Editor of the international periodical Game Theory and Applications (Nova sci. Pbl. N.Y., USA); the Chief Editor of the Vestnik Peterburgskogo Universiteta, seria 10: Applied Mathematics, Control, Informatics; and the Chief Editor of the journal Mathematical Game Theory and Applications (Karelian Research Centre of RAS).
Two special issues of the International Game Theory Review were dedicated to Prof. Leon A. Petrosyan — one of the Founding Editors of the Review — on his 70th and 75th birthdays (Vol. 12, No. 4, |
https://en.wikipedia.org/wiki/Quillen%27s%20lemma | In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field k is algebraic over k. In contrast to a version of Schur's lemma due to Dixmier, it does not require k to be uncountable. Quillen's original short proof uses generic flatness.
References
Lemmas
Theorems about algebras
Lie algebras |
https://en.wikipedia.org/wiki/Thompson%20transitivity%20theorem | In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It originated in the proof of the odd order theorem by , where it was used to prove the Thompson uniqueness theorem.
Statement
Suppose that G is a finite group and p a prime such that all p-local subgroups are p-constrained. If A is a self-centralizing normal abelian subgroup of a p-Sylow subgroup such that A has rank at least 3, then the centralizer CG(A) act transitively on the maximal A-invariant q subgroups of G for any prime q ≠ p.
References
Theorems about finite groups |
https://en.wikipedia.org/wiki/Liao%20Shijun | Liao Shijun (; born September 15, 1963) is a fluid mechanics and applied mathematics expert working in homotopy analysis method (HAM), nonlinear waves, nonlinear dynamics, and applied mathematics. He was born in Wuhan, Hubei Province, China. Liao is a professor at Shanghai Jiao Tong University.
External links
Liao's homepage
Advances in the Homotopy Analysis Method
Academic staff of Shanghai Jiao Tong University
Physicists from Hubei
1963 births
Living people
People from Wuhan
Educators from Hubei
Mathematicians from Hubei |
https://en.wikipedia.org/wiki/Michal%20Filla | Michal Filla (born 11 October 1981) is a Czech motorcycle racer.
Career statistics
Grand Prix motorcycle racing
By season
Races by year
(key)
Supersport World Championship
Races by year
(key)
External links
Profile on MotoGP.com
Profile on WorldSBK.com
1981 births
Living people
Czech motorcycle racers
250cc World Championship riders
Supersport World Championship riders
Sportspeople from Brno |
https://en.wikipedia.org/wiki/Jo%C3%A3o%20Fernandes%20%28motorcyclist%29 | João Fernandes is a Grand Prix motorcycle racer from Portugal.
Career statistics
By season
World Endurance Championship:
2007 - Champion SuperProduction Class - 5th Overall
2007 - Le Mans 24H - 5th SuperProduction Class - 15th Overall
2007 - 6H Albacete - Winner SuperProduction Class - 8th Overall
2007 - 24H Orschesleben - 3rd SuperProduction Class - 10th Overall
2007 - 8H Suzuka - Winner SuperProduction Class - 14th Overall
2007 - 8H Qatar - 3rd SuperProduction Class - 9th Overall
Macau Grand Prix (Highlights):
2009 - 3rd - Supersport 600cc
2008 - 3rd - Supersport 600cc
2007 - 4th - Supersport 600cc
2005 - 4th - Supersport 600cc
2003 - 5th - Supersport 600cc
National Portuguese Championship
1998 - Champion - Honda CBR600 Trophy
1999 - 5th - Supersport 600cc
2000 - 6th - Supersport 600cc
2001 - Vice-Champion - Honda CBR600 Trophy
2002 - 6th - SuperStock 1000cc
National China Championship:
2005 - Champion - Superbikes 1000cc
2006 - Champion - SuperSport 600cc
2008 - Champion - SuperSport 600cc
2009 - 3rd - Superbikes 1000cc
2010 - 2nd - Superbikes 1000cc
References
External links
Profile on motogp.com
1977 births
Living people
Portuguese motorcycle racers
250cc World Championship riders
Sportspeople from Lisbon |
https://en.wikipedia.org/wiki/Jan%20Roelofs | Jan Roelofs is a Grand Prix motorcycle racer from the Netherlands.
Career statistics
By season
Races by year
(key)
References
External links
Racesport.nl Article (NED)
DueRuote.it Article (IT)
DueRuote.it Article (IT)
Shutteshock.com Article
1985 births
Living people
Dutch motorcycle racers
250cc World Championship riders
21st-century Dutch people |
https://en.wikipedia.org/wiki/Randy%20Gevers | Randy Gevers (born 3 January 1981) is a Dutch motorcycle racer. He won the Dutch 250cc Championship in 2007.
Career statistics
Grand Prix motorcycle racing
By season
Races by year
(key)
References
External links
Profile on MotoGP.com
1981 births
Living people
Dutch motorcycle racers
125cc World Championship riders
250cc World Championship riders
21st-century Dutch people |
https://en.wikipedia.org/wiki/Locally%20compact%20field | In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. These kinds of fields were originally introduced in p-adic analysis since the fields are locally compact topological spaces constructed from the norm on . The topology (and metric space structure) is essential because it allows one to construct analogues of algebraic number fields in the p-adic context.
Structure
Finite dimensional vector spaces
One of the useful structure theorems for vector spaces over locally compact fields is that the finite dimensional vector spaces have only an equivalence class of norm: the sup norm pg. 58-59.
Finite field extensions
Given a finite field extension over a locally compact field , there is at most one unique field norm on extending the field norm ; that is,for all which is in the image of . Note this follows from the previous theorem and the following trick: if are two equivalent norms, andthen for a fixed constant there exists an such thatfor all since the sequence generated from the powers of converge to .
Finite Galois extensions
If the index of the extension is of degree and is a Galois extension, (so all solutions to the minimal polynomial of any is also contained in ) then the unique field norm can be constructed using the field norm pg. 61. This is defined asNote the n-th root is required in order to have a well-defined field norm extending the one over since given any in the image of its norm issince it acts as scalar multiplication on the -vector space .
Examples
Finite fields
All finite fields are locally compact since they can be equipped with the discrete topology. In particular, any field with the discrete topology is locally compact since every point is the neighborhood of itself, and also the closure of the neighborhood, hence is compact.
Local fields
The main examples of locally compact fields are the p-adic rationals and finite extensions . Each of these are examples of local fields. Note the algebraic closure and its completion are not locally compact fields pg. 72 with their standard topology.
Field extensions of Qp
Field extensions can be found by using Hensel's lemma. For example, has no solutions in since only equals zero mod if , but has no solutions mod . Hence is a quadratic field extension.
See also
References
Topology
External links
Inequality trick https://math.stackexchange.com/a/2252625 |
https://en.wikipedia.org/wiki/Linear%20topology | In algebra, a linear topology on a left -module is a topology on that is invariant under translations and admits a fundamental system of neighborhood of that consists of submodules of If there is such a topology, is said to be linearly topologized. If is given a discrete topology, then becomes a topological -module with respect to a linear topology.
See also
References
Bourbaki, N. (1972). Commutative algebra (Vol. 8). Hermann.
Topology
Topological algebra
Topological groups |
https://en.wikipedia.org/wiki/Dubins%20path | In geometry, the term Dubins path typically refers to the shortest curve that connects two points in the two-dimensional Euclidean plane (i.e. x-y plane) with a constraint on the curvature of the path and with prescribed initial and terminal tangents to the path, and an assumption that the vehicle traveling the path can only travel forward. If the vehicle can also travel in reverse, then the path follows the Reeds–Shepp curve.
Lester Eli Dubins (1920–2010) proved using tools from analysis that any such path will consist of maximum curvature and/or straight line segments. In other words, the shortest path will be made by joining circular arcs of maximum curvature and straight lines.
Discussion
Dubins proved his result in 1957. In 1974 Harold H. Johnson proved Dubins' result by applying Pontryagin's maximum principle. In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. In 1992 the same result was shown again using Pontryagin's maximum principle. More recently, a geometric curve-theoretic proof has been provided by J. Ayala, D. Kirszenblat and J. Hyam Rubinstein. A proof characterizing Dubins paths in homotopy classes has been given by J. Ayala.
Applications
The Dubins path is commonly used in the fields of robotics and control theory as a way to plan paths for wheeled robots, airplanes and underwater vehicles. There are simple geometric and analytical methods to compute the optimal path.
For example, in the case of a wheeled robot, a simple kinematic car model (also known as Dubins' car) for the systems is:
where is the car's position, is the heading, the car is moving at a constant speed , and the turn rate control is bounded. In this case the maximum turning rate corresponds to some minimum turning radius (and equivalently maximum curvature). The prescribed initial and terminal tangents correspond to initial and terminal headings. The Dubins' path gives the shortest path joining two oriented points that is feasible for the wheeled-robot model.
The optimal path type can be described using an analogy with cars of making a 'right turn (R)' , 'left turn (L)' or driving 'straight (S).' An optimal path will always be at least one of the six types: RSR, RSL, LSR, LSL, RLR, LRL. For example, consider that for some given initial and final positions and tangents, the optimal path is shown to be of the type 'RSR.' Then this corresponds to a right-turn arc (R) followed by a straight line segment (S) followed by another right-turn arc (R). Moving along each segment in this sequence for the appropriate length will form the shortest curve that joins a starting point A to a terminal point B with the desired tangents at each endpoint and that does not exceed the given curvature.
Dubins Interval Problem
Dubins interval problem is a key variant of the Dubins path problem, where an |
https://en.wikipedia.org/wiki/Gerard%20Washnitzer | Gerard Washnitzer (1926 in New York City – April 2, 2017) was an American mathematician specializing in algebraic geometry.
Washnitzer studied at Princeton University under Emil Artin and in 1950 received a Ph.D. (A Dirichlet Principle for analytic functions of several complex variables) under the supervision of Salomon Bochner. In 1952 he was a C. L. E. Moore instructor at the Massachusetts Institute of Technology. After that, he was an associate professor at Johns Hopkins University and then a professor at Princeton University. From 1960 to 1961 and from 1967 to 1968 he was at the Institute for Advanced Study.
In 1968, together with Paul Monsky, he introduced the Monsky–Washnitzer cohomology, which is a p-adic cohomology theory for non-singular algebraic varieties.
Among his students was William Fulton.
References
The original article was the translation (yahoo) of the corresponding German article.
1926 births
2017 deaths
20th-century American mathematicians
21st-century American mathematicians
Scientists from New York City
Mathematicians from New York (state)
Johns Hopkins University faculty
Princeton University faculty
Massachusetts Institute of Technology fellows
Institute for Advanced Study visiting scholars
Massachusetts Institute of Technology School of Science faculty |
https://en.wikipedia.org/wiki/Meik%20Kevin%20Minnerop | Meik Kevin Minnerop is a Grand Prix motorcycle racer from Germany.
Career statistics
By season
Races by year
(key)
References
External links
Profile on motogp.com
Profile on SpeedWeek.com
IDM Statistic
1990 births
Living people
Sportspeople from Siegen
German motorcycle racers
250cc World Championship riders |
https://en.wikipedia.org/wiki/Octadecahedron | In geometry, an octadecahedron (or octakaidecahedron) is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron.
In chemistry, "the octadecahedron" commonly refers to a specific structure with C2v symmetry, the edge-contracted icosahedron, formed from a regular icosahedron with one edge contracted. It is the shape of the closo-boranate ion [B11H11]2−.
Convex
There are 107,854,282,197,058 topologically distinct convex octadecahedra, excluding mirror images, having at least 11 vertices. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
Examples
The most familiar octadecahedra are the heptadecagonal pyramid, hexadecagonal prism, and the octagonal antiprism. The hexadecagonal prism and the octagonal antiprism are uniform polyhedra, with regular bases and square or equilateral triangular sides. Four more octadecahedra are also found among the Johnson solids: the square gyrobicupola, the square orthobicupola, the elongated square cupola (also known as the diminished rhombicuboctahedron), and the sphenomegacorona. Four Johnson solids have octadecahedral duals: the elongated triangular orthobicupola, the elongated triangular gyrobicupola, the gyroelongated triangular bicupola, and the triangular hebesphenorotunda.
In addition, some uniform star polyhedra are also octadecahedra:
References
Polyhedra |
https://en.wikipedia.org/wiki/Potential%20good%20reduction | In mathematics, potential good reduction is a property of the reduction modulo a prime or, more generally, prime ideal, of an algebraic variety.
Definitions
Good reduction refers to the reduced variety having the same properties as the original, for example, an algebraic curve having the same genus, or a smooth variety remaining smooth. Potential good reduction refers to the situation over a sufficiently large finite extension of the field of definition.
Equivalent formulations
For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer.
See also
Elliptic surface
References
Abelian varieties |
https://en.wikipedia.org/wiki/Gill%20Brown | Gill Moss (née Brown) (born 26 February 1965) is a British former field hockey player who competed in the 1988 Summer Olympics.
She is currently employed as a teacher of mathematics at Alcester Grammar School, Warwickshire, UK.
References
External links
1965 births
Living people
British female field hockey players
Olympic field hockey players for Great Britain
Field hockey players at the 1988 Summer Olympics |
https://en.wikipedia.org/wiki/Krull%E2%80%93Akizuki%20theorem | In commutative algebra, the Krull–Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring, K its total ring of fractions. Suppose L is a finite extension of K. If and B is reduced,
then B is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal of B, is finite over A.
Note that the theorem does not say that B is finite over A. The theorem does not extend to higher dimension. One important consequence of the theorem is that the integral closure of a Dedekind domain A in a finite extension of the field of fractions of A is again a Dedekind domain. This consequence does generalize to a higher dimension: the Mori–Nagata theorem states that the integral closure of a noetherian domain is a Krull domain.
Proof
First observe that and KB is a finite extension of K, so we may assume without loss of generality that .
Then for some .
Since each is integral over K, there exists such that is integral over A.
Let .
Then C is a one-dimensional noetherian ring, and , where denotes the total ring of fractions of C.
Thus we can substitute C for A and reduce to the case .
Let be minimal prime ideals of A; there are finitely many of them. Let be the field of fractions of and the kernel of the natural map . Then we have:
and .
Now, if the theorem holds when A is a domain, then this implies that B is a one-dimensional noetherian domain since each is and since . Hence, we reduced the proof to the case A is a domain. Let be an ideal and let a be a nonzero element in the nonzero ideal . Set . Since is a zero-dim noetherian ring; thus, artinian, there is an such that for all . We claim
Since it suffices to establish the inclusion locally, we may assume A is a local ring with the maximal ideal . Let x be a nonzero element in B. Then, since A is noetherian, there is an n such that and so . Thus,
Now, assume n is a minimum integer such that and the last inclusion holds. If , then we easily see that . But then the above inclusion holds for , contradiction. Hence, we have and this establishes the claim. It now follows:
Hence, has finite length as A-module. In particular, the image of there is finitely generated and so is finitely generated. The above shows that has dimension zero and so B has dimension one. Finally, the exact sequence of A-modules shows that is finite over A.
References
Theorems in algebra
Commutative algebra |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20Cagliari%20Calcio%20season | The 2012–13 Cagliari Calcio season is the 93rd season in club history.
Players
Current squad
Out on loan
Matches
Legend
Serie A
Coppa Italia
Squad statistics
Appearances and goals
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Goalkeepers
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Defenders
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Midfielders
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Forwards
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Players transferred out during the season
Top scorers
This includes all competitive matches. The list is sorted by shirt number when total goals are equal.
Last updated: 28 October 2012
Source: Competitions
Sources
Cagliari Calcio
Cagliari Calcio seasons |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20Calcio%20Catania%20season | The 2012–13 Calcio Catania season is the 81st season in club history.
Players
Current squad
On loan
Competitions
Legend
Serie A
League table
Matches
Coppa Italia
Squad statistics
Appearances and goals
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Goalkeepers
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Defenders
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Midfielders
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Forwards
|-
! colspan="10" style="background:#dcdcdc; text-align:center"| Players transferred out during the season
Top scorers
This includes all competitive matches. The list is sorted by shirt number when total goals are equal.
{| class="wikitable sortable" style="font-size: 95%; text-align: center;"
|-
!width=15|
!width=15|
!width=15|
!width=15|
!width=150|Name
!width=80|Serie A
!width=80|Coppa Italia
!width=80|Total
|-
|1
|9
|FW
|
|Gonzalo Bergessio
|13
|2
|15
|-
|2
|17
|MF
|
|Alejandro Gómez
|8
|1
|9
|-
|3
|10
|MF
|
|Francesco Lodi
|6
|2
|8
|-
|4
|28
|MF
|
|Pablo Barrientos
|5
|0
|5
|-
|5
|4
|MF
|
|Sergio Almirón
|4
|0
|4
|-
|=
|19
|MF
|
|Lucas Castro
|4
|0
|4
|-
|7
|6
|DF
|
|Nicola Legrottaglie
|3
|0
|3
|-
|8
|3
|DF
|
|Nicolás Spolli
|2
|0
|2
|-
|=
|12
|DF
|
|Giovanni Marchese
|2
|0
|2
|-
|10
|13
|MF
|
|Mariano Izco
|1
|0
|1
|-
|=
|16
|MF
|
|Mario Paglialunga
|1
|0
|1
|-
|=
|26
|MF
|
|Keko
|1
|0
|1
Sources
Calcio Catania
Catania FC seasons |
https://en.wikipedia.org/wiki/Topos%20%28disambiguation%29 | Topos may refer to:
Mathematics
Topos (plural topoi) – a type of category in mathematics
Classifying topos – a topos that categorifies the models of a structure in another topos
Effective topos – a topos that captures the idea of effectivity in mathematics
Étale topos – the category of étale sheaves
Philosophy and literature
Rhetoric topos – topoi in rhetorical invention
Literary topos – topoi in literary theory
Topical logic – reasoning from commonplace topoi
Topos hyperuranionos – Platonic realm of archetypes
Other
Los Topos – California theatre troupe
Oo-Topos – interactive science-fiction game
Topo (climbing) (plural topos) – description of a climbing route
Topos de Reynosa FC – a Mexican football club
Topos de Tlatelolco – a non-for-profit rescue organization based in Mexico
Topos V – a sculpture by Eduardo Chillida, displayed in Barcelona
See also
Toos (disambiguation)
Topo (disambiguation) |
https://en.wikipedia.org/wiki/Hong%20Jin-gi | Hong Jin-Gi (; born 20 October 1990) is a South Korean footballer who plays as a centre back for Busan IPark.
Club career statistics
External links
1990 births
Living people
Men's association football defenders
South Korean men's footballers
Jeonnam Dragons players
K League 1 players |
https://en.wikipedia.org/wiki/Kafr%20Laha | Kafr Laha () is a town in the Homs Governorate north of Homs in northern Syria. In 2004 it had a population of 20,041 according to the Central Bureau of Statistics of Syria. Its inhabitants are predominantly Sunni Muslims. It is the largest town in the Houla region. Nearby localities include Tallaf to the northeast, Tell Dahab to the north, Aqrab to the northwest, Qarmas and Maryamin to the west, al-Taybah al-Gharbiyah to the southwest and Taldou to the southeast.
History
Kafr Laha has been identified as the ancient Aramaean settlement of Byt'l also known as "Bethel".
Kafr Laha has been the site of demonstrations against the Assad government during the ongoing Syrian uprising which began in 2011.
References
Bibliography
Populated places in Homs District |
https://en.wikipedia.org/wiki/Hausdorff%20completion | In algebra, the Hausdorff completion of a group G with filtration is the inverse limit of the discrete group . A basic example is a profinite completion. The image of the canonical map is a Hausdorff topological group and its kernel is the intersection of all : i.e., the closure of the identity element. The canonical homomorphism is an isomorphism, where is a graded module associated to the filtration.
The concept is named after Felix Hausdorff.
References
Nicolas Bourbaki, Commutative algebra
Commutative algebra |
https://en.wikipedia.org/wiki/Deir%20al-Adas | Deir al-Adas ( ) is a village in southern Syria, administratively part of the Daraa Governorate. It is situated about 40 kilometers northwest of Daraa. According to the Central Bureau of Statistics (CBS), it had a population of 3,723.
The name literally means "Monastery () of the Lentils ()".
History
In 1838, Deir al-Adas was noted as a village in the el-Jeidur district.
Syrian Civil War
On 10 June 2022, eleven farmworkers were killed after a landmine exploded underneath their car in the village.
See also
Hauran
References
Bibliography
External links
http://www.discover-syria.com
Map of town, Google Maps
Sanameine-map, 19L
Populated places in Al-Sanamayn District |
https://en.wikipedia.org/wiki/Mori%E2%80%93Nagata%20theorem | In algebra, the Mori–Nagata theorem introduced by and , states the following: let A be a noetherian reduced commutative ring with the total ring of fractions K. Then the integral closure of A in K is a direct product of r Krull domains, where r is the number of minimal prime ideals of A.
The theorem is a partial generalization of the Krull–Akizuki theorem, which concerns a one-dimensional noetherian domain. A consequence of the theorem is that if R is a Nagata ring, then every R-subalgebra of finite type is again a Nagata ring .
The Mori–Nagata theorem follows from Matijevic's theorem.
References
Commutative algebra
Theorems in ring theory |
https://en.wikipedia.org/wiki/Mori%20domain | In algebra, a Mori domain, named after Yoshiro Mori by , is an integral domain satisfying the ascending chain condition on integral divisorial ideals. Noetherian domains and Krull domains both have this property. A commutative ring is a Krull domain if and only if it is a Mori domain and completely integrally closed. A polynomial ring over a Mori domain need not be a Mori domain. Also, the complete integral closure of a Mori domain need not be a Mori (or, equivalently, Krull) domain.
Notes
References
Commutative algebra |
https://en.wikipedia.org/wiki/Benjamin%20Peirce%20%28disambiguation%29 | Benjamin Peirce (1809–1880), Professor of Mathematics at Harvard University.
Benjamin Peirce may also refer to:
Benjamin Osgood Peirce (1854–1914), Hollis Professor of Mathematics and Natural Philosophy at Harvard University
USCS Benjamin Peirce, a survey ship in commission in the United States Coast Survey from 1855 to 1868
Benjamin Peirce (librarian) (1778–1831), librarian of the Harvard Library
See also
Benjamin Pierce (disambiguation)
Benjamin Pearse (1832–1902), Canadian public servant |
https://en.wikipedia.org/wiki/Isoperimetric%20point | In geometry, the isoperimetric point is a triangle center — a special point associated with a plane triangle. The term was originally introduced by G.R. Veldkamp in a paper published in the American Mathematical Monthly in 1985 to denote a point in the plane of a triangle having the property that the triangles have isoperimeters, that is, having the property that
Isoperimetric points in the sense of Veldkamp exist only for triangles satisfying certain conditions. The isoperimetric point of in the sense of Veldkamp, if it exists, has the following trilinear coordinates.
Given any triangle one can associate with it a point having trilinear coordinates as given above. This point is a triangle center and in Clark Kimberling's Encyclopedia of Triangle Centers (ETC) it is called the isoperimetric point of the triangle . It is designated as the triangle center X(175). The point X(175) need not be an isoperimetric point of triangle in the sense of Veldkamp. However, if isoperimetric point of triangle in the sense of Veldkamp exists, then it would be identical to the point X(175).
The point with the property that the triangles have equal perimeters has been studied as early as 1890 in an article by Emile Lemoine.
Existence of isoperimetric point in the sense of Veldkamp
Let be any triangle. Let the sidelengths of this triangle be . Let its circumradius be and inradius be . The necessary and sufficient condition for the existence of an isoperimetric point in the sense of Veldkamp can be stated as follows.
The triangle has an isoperimetric point in the sense of Veldkamp if and only if
For all acute angled triangles we have , and so all acute angled triangles have isoperimetric points in the sense of Veldkamp.
Properties
Let denote the triangle center X(175) of triangle .
lies on the line joining the incenter and the Gergonne point of .
If is an isoperimetric point of in the sense of Veldkamp, then the excircles of triangles are pairwise tangent to one another and is their radical center.
If is an isoperimetric point of in the sense of Veldkamp, then the perimeters of are equal to
where is the area, is the circumradius, is the inradius, and are the sidelengths of .
Soddy circles
Given a triangle one can draw circles in the plane of with centers at such that they are tangent to each other externally. In general, one can draw two new circles such that each of them is tangential to the three circles with as centers. (One of the circles may degenerate into a straight line.) These circles are the Soddy circles of . The circle with the smaller radius is the inner Soddy circle and its center is called the inner Soddy point or inner Soddy center of . The circle with the larger radius is the outer Soddy circle and its center is called the outer Soddy point or outer Soddy center of triangle .
The triangle center X(175), the isoperimetric point in the sense of Kimberling, is the outer Soddy point of .
References
Exte |
https://en.wikipedia.org/wiki/Administrative%20division%20codes%20of%20the%20People%27s%20Republic%20of%20China | The administrative division codes of the People's Republic of China identify the administrative divisions of China at county level and above. They are published by the National Bureau of Statistics of China with the latest version issued on September 30, 2015.
Coding scheme
Reading from left to right, administrative division codes contain the following information:
The first and second digits identify the highest level administrative division, which may be a province, autonomous region, municipality or Special Administrative Region (SAR).
Digits three and four show summary data for the associated prefecture-level city, prefecture (地区 dìqū), autonomous prefecture, Mongolian league, municipal city district or county. Codes 01 – 20 and 51 – 70 identify provincial level cities, codes 21 – 50 represent prefectures, autonomous prefectures and Mongolian leagues.
The fifth and sixth digits represent the county-level division – city district, county-level city, county and the banner area of Inner Mongolia. Codes 01 – 18 represent municipal districts or regions (autonomous prefectures and Mongolian leagues) under the jurisdiction of county-level cities. Codes 21 – 80 stand for counties and Mongolian banner areas while codes 81 – 99 represent county level cities directly administered by a province.
Division codes for statistical use
Division codes for statistical use consist of the administrative division codes and an additional 6 digits, identifying the administrative divisions of China at the village level and above.
For example, in the code 110102 007 003, 110102 refers to Xicheng District, Beijing, 007 refers to Yuetan Subdistrict and 003 refers to Yuetan Community.
See also
Administrative divisions of China
ISO 3166-2:CN
OKATO, a somewhat similar numeric code system used in Russia
References
National Standards of the People's Republic of China (Guobiao) GB/T 2260—1999
External links
China geography-related lists
Geocodes |
https://en.wikipedia.org/wiki/2012%20Dhivehi%20League%20Round%202 | In Round 2, all eight teams play against each other. A total of 28 matches will be played in this round.
League table
Matches
Round 2 statistics
Scorers
Assists
Hat-tricks
4 Player scored 4 goals
Clean sheets
Clean sheets by Club:
New Radiant SC (2)
Club All Youth Linkage (2)
Victory SC (2)
Maziya S&RC (2)
Club Eagles (1)
Vyansa (1)
VB Addu FC (1)
Club Valencia (0)
Clean sheets by goalkeepers:
Imran Mohamed (New Radiant SC) (2)
Ibrahim Siyad (Club All Youth Linkage) (2)
Lavent Vanli (Victory SC) (2)
Mohamed Imran (Maziya S&RC) (2)
Abdulla Ziyazan (VB Addu FC) (2)
Mohamed Yamaan (Club Eagles) (1)
Alexander Osei Domfeh (Vyansa) (1)
Ibrahim Ifrah Areef (Club Valencia) (0)
Hussain Habeeb (VB Addu FC) (0)
Mohamed Saruhan (Club Valencia) (0)
Mohamed Nishah (Victory SC) (0)
References
2 |
https://en.wikipedia.org/wiki/Parry%20point%20%28triangle%29 | In geometry, the Parry point is a special point associated with a plane triangle. It is the triangle center designated X(111) in Clark Kimberling's Encyclopedia of Triangle Centers. The Parry point and Parry circle are named in honor of the English geometer Cyril Parry, who studied them in the early 1990s.
Parry circle
Let be a plane triangle. The circle through the centroid and the two isodynamic points of is called the Parry circle of . The equation of the Parry circle in barycentric coordinates is
The center of the Parry circle is also a triangle center. It is the center designated as X(351) in the Encyclopedia of Triangle Centers. The trilinear coordinates of the center of the Parry circle are
Parry point
The Parry circle and the circumcircle of triangle intersect in two points. One of them is a focus of the Kiepert parabola of . The other point of intersection is called the Parry point of .
The trilinear coordinates of the Parry point are
The point of intersection of the Parry circle and the circumcircle of which is a focus of the Kiepert hyperbola of is also a triangle center and it is designated as X(110) in Encyclopedia of Triangle Centers. The trilinear coordinates of this triangle center are
See also
Lester circle
References
Triangle centers |
https://en.wikipedia.org/wiki/Indiana%20Academy%20%28disambiguation%29 | Indiana Academy may refer:
Indiana Academy, a private Seventh-Day Adventist high school in Cicero, Indiana.
Indiana Academy for Science, Mathematics, and Humanities, a public charter school located on the Ball State University campus.
Indiana E-Learning Academy is a joint program of the Intelenet Commission and the Indiana Department of Education. |
https://en.wikipedia.org/wiki/Daeyang%20Gallery%20and%20House | The Daeyang Gallery and House, designed by Steven Holl Architects, is located in the Kangbuk neighborhood of Seoul, South Korea.
The geometry of the roof plan was inspired by a 1967 sketch for a music score by Hungarian composer István Anhalt.
Three pavilion, one for entry, one event space, and one residential, are separated by a reflecting pool. Below, they are connected by a continuous art gallery space.
Skylights cut in the roof of the pavilions and in the base of the reflecting pool bring natural light to the spaces, and gallery level below.
The interiors of the pavilions have red and charcoal stained wood, and the exterior is a rain screen of patinated copper.
The 10703 sf house and gallery was completed in June 2012.
References
Art museums and galleries in Seoul
Steven Holl buildings |
https://en.wikipedia.org/wiki/Yoshiro%20Mori%20%28mathematician%29 | Yoshiro Mori is a Japanese mathematician working on commutative algebra who introduced the Mori–Nagata theorem and whose work led to Mori domains.
References
20th-century Japanese mathematicians
Year of birth missing
Possibly living people
Place of birth missing |
https://en.wikipedia.org/wiki/Gerg%C5%91%20G%C5%91cze | Gergő Gőcze (born 30 April 1990, in Szombathely) is a Hungarian football player who currently plays for Sárvári FC.
Club statistics
Updated to games played as of 22 January 2016.
References
External links
HLSZ
1990 births
Living people
Footballers from Szombathely
Hungarian men's footballers
Hungarian expatriate men's footballers
Men's association football goalkeepers
Szombathelyi Haladás footballers
Kozármisleny SE footballers
Pécsi MFC players
FC Ajka players
Kaposvári Rákóczi FC players
Puskás Akadémia FC players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungarian expatriate sportspeople in Austria
Expatriate men's footballers in Austria |
https://en.wikipedia.org/wiki/Attila%20Szak%C3%A1ly | Attila Szakály (born 30 June 1992) is a Hungarian football player who plays for Szombathelyi Haladás.
Club statistics
Updated to games played as of 27 June 2020.
References
External links
HLSZ
1992 births
Living people
People from Körmend
Hungarian men's footballers
Men's association football midfielders
Szombathelyi Haladás footballers
Zalaegerszegi TE players
Kaposvári Rákóczi FC players
BFC Siófok players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players
Hungary men's under-21 international footballers
Sportspeople from Vas County
21st-century Hungarian people |
https://en.wikipedia.org/wiki/M%C3%A1rk%20Jagodics | Márk Jagodics (born 10 April 1992) is a Hungarian football player who plays for Szombathelyi Haladás.
Club statistics
Updated to games played as of 15 May 2021.
References
External links
HLSZ
1992 births
Living people
Footballers from Szombathely
Hungarian men's footballers
Hungary men's under-21 international footballers
Men's association football defenders
Szombathelyi Haladás footballers
FC Ajka players
Mezőkövesdi SE footballers
Budafoki MTE footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
21st-century Hungarian people |
https://en.wikipedia.org/wiki/Seminormal%20ring | In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy , there is s with and . This definition was given by as a simplification of the original definition of .
A basic example is an integrally closed domain, i.e., a normal ring. For an example which is not normal, one can consider the non-integral ring , or the ring of a nodal curve.
In general, a reduced scheme can be said to be seminormal if every morphism which induces a homeomorphism of topological spaces, and an isomorphism on all residue fields, is an isomorphism of schemes.
A semigroup is said to be seminormal if its semigroup algebra is seminormal.
References
Charles Weibel, The K-book: An introduction to algebraic K-theory
Commutative algebra
Ring theory |
https://en.wikipedia.org/wiki/1976%E2%80%9377%20Seattle%20SuperSonics%20season | The 1976–77 NBA season was the SuperSonics' 10th season in the NBA.
Draft picks
Roster
Depth chart
Regular season
Season standings
Record vs. opponents
Game log
Player statistics
References
Seattle SuperSonics seasons
Seattle |
https://en.wikipedia.org/wiki/Almost%20commutative%20ring | In algebra, a filtered ring A is said to be almost commutative if the associated graded ring is commutative.
Basic examples of almost commutative rings involve differential operators. For example, the enveloping algebra of a complex Lie algebra is almost commutative by the PBW theorem. Similarly, a Weyl algebra is almost commutative.
See also
Ore condition
Gelfand–Kirillov dimension
References
Victor Ginzburg, Lectures on D-modules
Ring theory |
https://en.wikipedia.org/wiki/Gelfand%E2%80%93Kirillov%20dimension | In algebra, the Gelfand–Kirillov dimension (or GK dimension) of a right module M over a k-algebra A is:
where the supremum is taken over all finite-dimensional subspaces and .
An algebra is said to have polynomial growth if its Gelfand–Kirillov dimension is finite.
Basic facts
The Gelfand–Kirillov dimension of a finitely generated commutative algebra A over a field is the Krull dimension of A (or equivalently the transcendence degree of the field of fractions of A over the base field.)
In particular, the GK dimension of the polynomial ring Is n.
(Warfield) For any real number r ≥ 2, there exists a finitely generated algebra whose GK dimension is r.
In the theory of D-Modules
Given a right module M over the Weyl algebra , the Gelfand–Kirillov dimension of M over the Weyl algebra coincides with the dimension of M, which is by definition the degree of the Hilbert polynomial of M. This enables to prove additivity in short exact sequences for the Gelfand–Kirillov dimension and finally to prove Bernstein's inequality, which states that the dimension of M must be at least n. This leads to the definition of holonomic D-modules as those with the minimal dimension n, and these modules play a great role in the geometric Langlands program.
References
Coutinho: A primer of algebraic D-modules. Cambridge, 1995
Further reading
Algebra
Dimension |
https://en.wikipedia.org/wiki/Distribution%20algebra | In algebra, the distribution algebra of a p-adic Lie group G is the K-algebra of K-valued distributions on G. (See the reference for a more precise definition.)
References
Algebra |
https://en.wikipedia.org/wiki/Rademacher%20system | In mathematics, in particular in functional analysis, the Rademacher system, named after Hans Rademacher, is an incomplete orthogonal system of functions on the unit interval of the following form:
The Rademacher system is stochastically independent, and is closely related to the Walsh system. Specifically, the Walsh system can be constructed as a product of Rademacher functions.
References
External links
Rademacher system in the Encyclopedia of Mathematics
Functional analysis |
https://en.wikipedia.org/wiki/Nikolas%20Proesmans | Nikolas Proesmans (born 11 May 1992 in Tongeren) is a Belgian midfielder who plays for A. C. Sangiustese.
Career statistics
Club
Honours
Újpest
Magyar Kupa: 2013–14
References
External links
Player profile at HLSZ
Player profile at MLSZ
1992 births
Living people
Sportspeople from Tongeren
Belgian men's footballers
Belgian expatriate men's footballers
Men's association football midfielders
Sint-Truidense V.V. players
Újpest FC players
FC Ararat Yerevan players
AC Ancona players
Belgian Pro League players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Armenian Premier League players
Serie D players
Expatriate men's footballers in Hungary
Expatriate men's footballers in Armenia
Expatriate men's footballers in Italy
Belgian expatriate sportspeople in Hungary
Belgian expatriate sportspeople in Italy
Footballers from Limburg (Belgium)
AC Sangiustese players |
https://en.wikipedia.org/wiki/R%C3%B3bert%20Litauszki | Róbert Litauszki (born 15 March 1990) is a Hungarian football player who plays for Vasas.
Club statistics
Updated to games played as of 14 March 2020.
Honours
Újpest
Magyar Kupa (1): 2013–14
References
External links
Player profile at HLSZ
Player profile at MLSZ
1990 births
Living people
Footballers from Budapest
Hungarian men's footballers
Hungary men's under-21 international footballers
Men's association football defenders
Újpest FC players
MKS Cracovia players
Vasas SC players
Nemzeti Bajnokság II players
Nemzeti Bajnokság I players
Ekstraklasa players
Hungarian expatriate men's footballers
Expatriate men's footballers in Poland
Hungarian expatriate sportspeople in Poland |
https://en.wikipedia.org/wiki/Voter%20model | In the mathematical theory of probability, the voter model is an interacting particle system introduced by Richard A. Holley and Thomas M. Liggett in 1975.
One can imagine that there is a "voter" at each point on a connected graph, where the connections indicate that there is some form of interaction between a pair of voters (nodes). The opinions of any given voter on some issue changes at random times under the influence of opinions of his neighbours. A voter's opinion at any given time can take one of two values, labelled 0 and 1. At random times, a random individual is selected and that voter's opinion is changed according to a stochastic rule. Specifically, one of the chosen voter's neighbors is chosen according to a given set of probabilities and that neighbor’s opinion is transferred to the chosen voter.
An alternative interpretation is in terms of spatial conflict. Suppose two nations control the areas (sets of nodes) labelled 0 or 1. A flip from 0 to 1 at a given location indicates an invasion of that site by the other nation.
Note that only one flip happens each time. Problems involving the voter model will often be recast in terms of the dual system of coalescing Markov chains. Frequently, these problems will then be reduced to others involving independent Markov chains.
Definition
A voter model is a (continuous time) Markov process with state space and transition rates function , where is a d-dimensional integer lattice, and •,• is assumed to be nonnegative, uniformly bounded and continuous as a function of in the product topology on . Each component is called a configuration. To make it clear that stands for the value of a site x in configuration ; while means the value of a site x in configuration at time .
The dynamic of the process are specified by the collection of transition rates. For voter models, the rate at which there is a flip at from 0 to 1 or vice versa is given by a function of site . It has the following properties:
for every if or if
for every if for all
if and
is invariant under shifts in
Property (1) says that and are fixed points for the evolution. (2) indicates that the evolution is unchanged by interchanging the roles of 0's and 1's. In property (3), means , and implies if , and implies if .
Clustering and coexistence
The interest in is the limiting behavior of the models. Since the flip rates of a site depends on its neighbours, it is obvious that when all sites take the same value, the whole system stops changing forever. Therefore, a voter model has two trivial extremal stationary distributions, the point-masses and on or respectively, which represent consensus. The main question to be discussed is whether or not there are others, which would then represent coexistence of different opinions in equilibrium. It is said coexistence occurs if there is a stationary distribution that concentrates on configurations with infinitely many 0's and 1's. On the other hand, if fo |
https://en.wikipedia.org/wiki/Relator%20%28disambiguation%29 | Relator may refer to:
Relator, a concept in group theory
Relator (law)
"Relator" (song) |
https://en.wikipedia.org/wiki/Poisson%20scatter%20theorem | In probability theory, The Poisson scatter theorem describes a probability model of random scattering. It implies that the number of points in a fixed region will follow a Poisson distribution.
Statement
Let there exist a chance process realized by a set of points (called hits) over a bounded region such that:
1) There are only a finite number of hits over the entire region K.
2) There are no multiple hits at a single point.
3) There is homogeneity and independence among the hits. i.e. For any non-overlapping subregions , , the numbers of hits in these regions are independent.
In any region B, let NB be the number of hits in B. Then there exists a positive constant such that for each subregion , NB has a Poisson distribution with parameter , where is the area of B (remember that this is , in other measure spaces, could mean different things, i.e. length in ). In addition, for any non-overlapping regions , the random variables are independent from one another.
The positive constant is called the intensity parameter, and is equivalent to the number of hits in a unit area of K.
Proof:
Also,
While the statement of the theorem here is limited to , the theorem can be generalized to any-dimensional space. Some calculations change depending on the space that the points are scattered in (as is mentioned above), but the general assumptions and outcomes still hold.
Example
Consider raindrops falling on a rooftop. The rooftop is the region , while the raindrops can be considered the hits of our system. It is reasonable to assume that the number of raindrops that fall in any particular region of the rooftop follows a poisson distribution. The Poisson Scatter Theorem, states that if one was to subdivide the rooftops into k disjoint sub-regions, then the number of raindrops that hits a particular region with intensity of the rooftop is independent from the number of raindrops that hit any other subregion. Suppose that 2000 raindrops fall in 1000 subregions of the rooftop, randomly. The expected number of raindrops per subregion would be 2. So the distribution of the number of raindrops on the whole rooftop is Poisson with intensity parameter 2. The distribution of the number of raindrops falling on 1/5 of the rooftop is Poisson with intensity parameter 2/5.
Due to the reproductive property of the Poisson distribution, k independent random scatters on the same region can superimpose to produce a random scatter that follows a poisson distribution with parameter .
Notes
^ Pitman 2003, p. 230.
References
Pitman, Jim (2003). Probability. Springer.
Probability theorems |
https://en.wikipedia.org/wiki/K-graph%20C%2A-algebra | For C*-algebra in mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category with domain and codomain maps and , together with a functor which satisfies the following factorisation property: if then there are unique with such that .
Aside from its category theory definition, one can think of k-graphs as a higher-dimensional analogue of directed graphs (digraphs). k- here signifies the number of "colors" of edges that are involved in the graph.
If k=1, a k-graph is just an ordinary directed graph.
If k=2, there are two different colors of edges involved in the graph and additional factorization rules of 2-color equivalent classes should be defined. The factorization rule on k-graph skeleton is what distinguishes one k-graph defined on the same skeleton from another k-graph. k can be any natural number greater than or equal to 1.
The reason k-graphs were first introduced by Kumjian, Pask et al. was to create examples of C*-algebras from them. k-graphs consist of two parts: skeleton and factorization rules defined on the given skeleton. Once k-graph is well-defined, one can define functions called 2-cocycles on each graph, and C*-algebras can be built from k-graphs and 2-cocycles. k-graphs are relatively simple to understand from a graph theory perspective, yet just complicated enough to reveal different interesting properties at the C*-algebra level. The properties such as homotopy and cohomology on the 2-cocycles defined on k-graphs have implications to C*-algebra and K-theory research efforts. No other known use of k-graphs exist to this day; k-graphs are studied solely for the purpose of creating C*-algebras from them.
Background
The finite graph theory in a directed graph form a category under concatenation called the free object category (generated by the graph). The length of a path in gives a
functor from this category into the natural numbers .
A k-graph is a natural generalisation of this concept which was introduced in 2000 by Alex Kumjian and David Pask.
Examples
It can be shown that a 1-graph is precisely the path category of a directed graph.
The category consisting of a single object and k commuting morphisms , together with the map defined by is a k-graph.
Let , then is a k-graph when gifted with the structure maps , , and .
Notation
The notation for k-graphs is borrowed extensively from the corresponding notation for categories:
For let .
By the factorisation property it follows that .
For and we have , and .
If for all and then is said to be row-finite with no sources.
Visualisation - Skeletons
A k-graph is best visualized by drawing its 1-skeleton as a k-coloured graph where
, , inherited
from
and defined by
if and only if where are the canonical
generators for . The factorisation property in for elements
of degree where gives rise to relations between the edges of
.
C*-algebra
As with graph-algebras one may associate a C*-algebra to a k-graph:
Let be |
https://en.wikipedia.org/wiki/The%20Annals%20of%20Applied%20Statistics | The Annals of Applied Statistics is a peer-reviewed scientific journal published by the Institute of Mathematical Statistics, covering all areas of statistics, featuring papers in the applied half of this range. It was established in 2007, with Bradley Efron as founding editor-in-chief. According to the Journal Citation Reports, the journal has a 2022 impact factor of 1.8.
References
External links
Statistics journals
Academic journals established in 2007
Quarterly journals
English-language journals
Institute of Mathematical Statistics academic journals |
https://en.wikipedia.org/wiki/3-step%20group | In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different.
CN groups
In the theory of CN groups, a 3-step group (for some prime p) is a group such that:
is a Frobenius group with kernel
is a Frobenius group with kernel
Any 3-step group is a solvable CN-group, and conversely any solvable CN-group is either nilpotent, or a Frobenius group, or a 3-step group.
Example: the symmetric group S4 is a 3-step group for the prime .
Odd order groups
defined a three-step group to be a group G satisfying the following conditions:
The derived group of G is a Hall subgroup with a cyclic complement Q.
If H is the maximal normal nilpotent Hall subgroup of G, then G⊆HCG(H)⊆G and HCG is nilpotent and H is noncyclic.
For q∈Q nontrivial, CG(q) is cyclic and non-trivial and independent of q.
References
Finite groups |
https://en.wikipedia.org/wiki/Yff%20center%20of%20congruence | In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and Peter Yff initiated the study of this triangle center in 1987.
Isoscelizer
An isoscelizer of an angle in a triangle is a line through points , where lies on and on , such that the triangle is an isosceles triangle. An isoscelizer of angle is a line perpendicular to the bisector of angle . Isoscelizers were invented by Peter Yff in 1963.
Yff central triangle
Let be any triangle. Let be an isoscelizer of angle , be an isoscelizer of angle , and be an isoscelizer of angle . Let be the triangle formed by the three isoscelizers. The four triangles and are always similar.
There is a unique set of three isoscelizers such that the four triangles and are congruent. In this special case formed by the three isoscelizers is called the Yff central triangle of .
The circumcircle of the Yff central triangle is called the Yff central circle of the triangle.
Yff center of congruence
Let be any triangle. Let be the isoscelizers of the angles such that the triangle formed by them is the Yff central triangle of . The three isoscelizers are continuously parallel-shifted such that the three triangles are always congruent to each other until formed by the intersections of the isoscelizers reduces to a point. The point to which reduces to is called the Yff center of congruence of .
Properties
The trilinear coordinates of the Yff center of congruence are
Any triangle is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of .
Let be the incenter of . Let be the point on side such that , a point on side such that , and a point on side such that . Then the lines are concurrent at the Yff center of congruence. This fact gives a geometrical construction for locating the Yff center of congruence.
A computer assisted search of the properties of the Yff central triangle has generated several interesting results relating to properties of the Yff central triangle.
Generalization
The geometrical construction for locating the Yff center of congruence has an interesting generalization. The generalisation begins with an arbitrary point in the plane of a triangle . Then points are taken on the sides such that
The generalization asserts that the lines are concurrent.
See also
Congruent isoscelizers point
Central triangle
References
Triangle centers |
https://en.wikipedia.org/wiki/D%C3%A1vid%20G%C3%B6rg%C3%A9nyi | Dávid Görgényi (born 16 August 1990) is a professional Hungarian footballer who plays for Ajka.
Club statistics
Updated to games played as of 1 March 2014.
External links
HLSZ
MLSZ
1990 births
People from Mór
Footballers from Fejér County
Living people
Hungarian men's footballers
Hungary men's youth international footballers
Men's association football defenders
Vasas SC players
Vác FC players
FC Ajka players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Champernowne%20distribution | In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by D. G. Champernowne. Champernowne developed the distribution to describe the logarithm of income.
Definition
The Champernowne distribution has a probability density function given by
where are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as
using the fact that
Properties
The density f(y) defines a symmetric distribution with median y0, which has tails somewhat heavier than a normal distribution.
Special cases
In the special case it is the Burr Type XII density.
When ,
which is the density of the standard logistic distribution.
Distribution of income
If the distribution of Y, the logarithm of income, has a Champernowne distribution, then the density function of the income X = exp(Y) is
where x0 = exp(y0) is the median income. If λ = 1, this distribution is often called the Fisk distribution, which has density
See also
Generalized logistic distribution
References
Continuous distributions |
https://en.wikipedia.org/wiki/Gamma/Gompertz%20distribution | In probability and statistics, the Gamma/Gompertz distribution is a continuous probability distribution. It has been used as an aggregate-level model of customer lifetime and a model of mortality risks.
Specification
Probability density function
The probability density function of the Gamma/Gompertz distribution is:
where is the scale parameter and are the shape parameters of the Gamma/Gompertz distribution.
Cumulative distribution function
The cumulative distribution function of the Gamma/Gompertz distribution is:
Moment generating function
The moment generating function is given by:
where is a Hypergeometric function.
Properties
The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left.
Related distributions
When β = 1, this reduces to an Exponential distribution with parameter sb.
The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known, scale parameter
When the shape parameter of a Gompertz distribution varies according to a gamma distribution with shape parameter and scale parameter (mean = ), the distribution of is Gamma/Gompertz.
See also
Gompertz distribution
Customer lifetime value
Notes
References
Continuous distributions
hu:Gompertz-eloszlás |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20New%20Jersey%20Nets%20season | The 1986–87 New Jersey Nets season was the Nets' 11th season in the NBA.
Draft picks
Roster
Regular season
Season standings
Record vs. opponents
Game log
Player statistics
Season
Awards and records
Transactions
References
See also
1986–87 NBA season
New Jersey Nets season
New Jersey Nets seasons
New Jersey Nets
New Jersey Nets
20th century in East Rutherford, New Jersey
Meadowlands Sports Complex |
https://en.wikipedia.org/wiki/Congruent%20isoscelizers%20point | In geometry, the congruent isoscelizers point is a special point associated with a plane triangle. It is a triangle center and it is listed as X(173) in Clark Kimberling's Encyclopedia of Triangle Centers. This point was introduced to the study of triangle geometry by Peter Yff in 1989.
Definition
An isoscelizer of an angle in a triangle is a line through points and , where lies on and on , such that the triangle is an isosceles triangle. An isoscelizer of angle is a line perpendicular to the bisector of angle .
Let be any triangle. Let be the isoscelizers of the angles respectively such that they all have the same length. Then, for a unique configuration, the three isoscelizers are concurrent. The point of concurrence is the congruent isoscelizers point of triangle .
Properties
The trilinear coordinates of the congruent isoscelizers point of triangle are
The intouch triangle of the intouch triangle of triangle is perspective to , and the congruent isoscelizers point is the perspector. This fact can be used to locate by geometrical constructions the congruent isoscelizers point of any given .
See also
Yff center of congruence
Equal parallelians point
References
Triangle centers |
https://en.wikipedia.org/wiki/Abel%20equation%20of%20the%20first%20kind | In mathematics, an Abel equation of the first kind, named after Niels Henrik Abel, is any ordinary differential equation that is cubic in the unknown function. In other words, it is an equation of the form
where . If and , or and , the equation reduces to a Bernoulli equation, while if the equation reduces to a Riccati equation.
Properties
The substitution brings the Abel equation of the first kind to the "Abel equation of the second kind" of the form
The substitution
brings the Abel equation of the first kind to the canonical form
Dimitrios E. Panayotounakos and Theodoros I. Zarmpoutis discovered an analytic method to solve the above equation in an implicit form.
Notes
References
. (Old link: On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term)
Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)
Mancas, Stefan C., Rosu, Haret C., Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations. Physics Letters A 377 (2013) 1434–1438. [arXiv.org:1212.3636v3]
Ordinary differential equations |
https://en.wikipedia.org/wiki/Lester%20Dubins | Lester Dubins (April 27, 1920 – February 11, 2010) was an American mathematician noted primarily for his research in probability theory. He was a faculty member at the University of California at Berkeley from 1962 through 2004, and in retirement was Professor Emeritus of Mathematics and Statistics.
It has been thought that, since classic red-and-black casino roulette is a game in which the house on average wins more than the gambler, that "bold play", i.e. betting one's whole purse on a single trial, is a uniquely optimal strategy. While a graduate student at the University of Chicago, Dubins surprised his teacher Leonard Jimmie Savage with a mathematical demonstration that this is not true. Dubins and Savage wrote a book that appeared in 1965 titled How to Gamble if You Must (Inequalities for Stochastic Processes) which presented a mathematical theory of gambling processes and optimal behavior in gambling situations, pointing out their relevance to traditional approaches to probability. Under the influence of the work of Bruno de Finetti, Dubins and Savage worked in the context of finitely additive rather than countably additive probability theory, thereby bypassing some technical difficulties.
Dubins was the author of nearly a hundred scholarly publications. Besides probability, some of these were on curves of minimal length under constraints on curvature and initial and final tangents (see Dubins path), Tarski's circle squaring problem, convex analysis, and geometry.
His doctoral students include Theodore Hill. Together with Gideon E. Schwarz he proved the Dubins–Schwarz theorem.
Publications
Dubins–Spanier theorems
References
External links
20th-century American mathematicians
21st-century American mathematicians
1920 births
2010 deaths
University of California, Berkeley College of Letters and Science faculty
Scientists from New York City
University of Chicago alumni
Probability theorists
Mathematicians from New York (state)
Fair division researchers |
https://en.wikipedia.org/wiki/Stephen%20Barker | Stephen Barker may refer to:
Stephen Barker (politician) (1846–1924), English-born Australian politician
Stephen F. Barker (1927–2019), American philosopher of mathematics
Stephen Barker, surveyor who built the Stephen Barker House, Methuen, Massachusetts |
https://en.wikipedia.org/wiki/Congruence%20coefficient | In multivariate statistics, the congruence coefficient is an index of the similarity between factors that have been derived in a factor analysis. It was introduced in 1948 by Cyril Burt who referred to it as unadjusted correlation. It is also called Tucker's congruence coefficient after Ledyard Tucker who popularized the technique. Its values range between -1 and +1. It can be used to study the similarity of extracted factors across different samples of, for example, test takers who have taken the same test.
Definition
Let X and Y be column vectors of factor loadings for two different samples. The formula for the congruence coefficient, or rc, is then
Interpretation
Generally, a congruence coefficient of 0.90 is interpreted as indicating a high degree of factor similarity, while a coefficient of 0.95 or higher indicates that the factors are virtually identical. Alternatively, a value in the range 0.85–0.94 has been seen as corresponding to a fair similarity, with values higher than 0.95 indicating that the factors can be considered to be equal.
The congruence coefficient can also be defined as the cosine of the angle between factor axes based on the same set of variables (e.g., tests) obtained for two samples (see Cosine similarity). For example, with perfect congruence the angle between the factor axes is 0 degrees, and the cosine of 0 is 1.
Comparison with Pearson's r
The congruence coefficient is preferred to Pearson's r as a measure of factor similarity, because the latter may produce misleading results. The computation of the congruence coefficient is based on the deviations of factor loadings from zero, whereas r is based on the deviations from the mean of the factor loadings.
See also
RV coefficient
References
Factor analysis |
https://en.wikipedia.org/wiki/Flexible%20algebra | In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity:
for any two elements a and b of the set. A magma (that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a nonassociative algebra is flexible if its multiplication operator is flexible.
Every commutative or associative operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the multiplication of sedenions, which are not even alternative.
In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed that they satisfy the flexible identity.
Examples
Besides associative algebras, the following classes of nonassociative algebras are flexible:
Alternative algebras
Lie algebras
Jordan algebras (which are commutative)
Okubo algebras
Similarly, the following classes of nonassociative magmas are flexible:
Alternative magmas
Semigroups (which are associative magmas, and which are also alternative)
The sedenions, and all algebras constructed from these by iterating the Cayley–Dickson construction, are also flexible.
See also
Zorn ring
Maltsev algebra
References
Non-associative algebras
Properties of binary operations |
https://en.wikipedia.org/wiki/Ahmad%20Al%20Douni | Ahmad Al Douni (; born 4 February 1989) is a Syrian footballer who plays as a forward.
Career statistics
International career
International goals
Syria's score listed first; score column indicates score after each Al Douni goal.
References
External links
Career stats at goalzz.com
1989 births
Living people
Syrian men's footballers
Al-Shorta SC (Syria) players
Al-Ramtha SC players
Al-Quwa Al-Jawiya players
Duhok SC players
Al-Zawraa SC players
Al-Najaf SC players
Al-Riffa SC players
Mesaimeer SC players
Al-Markhiya SC players
Al-Jeel Club players
Qatari Second Division players
Men's association football forwards
Syria men's international footballers
Expatriate men's footballers in Bahrain
Expatriate men's footballers in Iraq
Expatriate men's footballers in Qatar
Expatriate men's footballers in Jordan
Expatriate men's footballers in Saudi Arabia
Syrian expatriate sportspeople in Bahrain
Syrian expatriate sportspeople in Iraq
Syrian expatriate sportspeople in Jordan
Syrian expatriate sportspeople in Qatar
Syrian expatriate sportspeople in Saudi Arabia
Syrian expatriate men's footballers
People from Baniyas
Syrian Premier League players |
https://en.wikipedia.org/wiki/1968%E2%80%9369%20Chicago%20Bulls%20season | The 1968–69 NBA season was the Bulls' third season in the NBA.
Draft picks
Roster
Regular season
Season standings
x – clinched playoff spot
Record vs. opponents
Game log
Player statistics
Awards and records
Jerry Sloan, NBA All-Defensive First Team
Jerry Sloan, NBA All-Star Game
References
Chicago
Chicago Bulls seasons
Chicago Bulls
Chicago Bulls |
https://en.wikipedia.org/wiki/Duopyramid | In geometry of 4 dimensions or higher, a double pyramid or duopyramid or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term duopyramid was used by George Olshevsky, as the dual of a duoprism.
Polygonal forms
The lowest dimensional forms are 4 dimensional and connect two polygons. A p-q duopyramid or p-q'' fusil, represented by a composite Schläfli symbol {p} + {q}, and Coxeter-Dynkin diagram . The regular 16-cell can be seen as a 4-4 duopyramid or 4-4 fusil, , symmetry , order 128.
A p-q duopyramid or p-q fusil has Coxeter group symmetry [p,2,q], order 4pq. When p and q are identical, the symmetry in Coxeter notation is doubled as or [2p,2+,2q], order 8p2.
Edges exist on all pairs of vertices between the p-gon and q-gon. The 1-skeleton of a p-q duopyramid represents edges of each p and q polygon and pq complete bipartite graph between them.
Geometry
A p-q duopyramid can be seen as two regular planar polygons of p and q sides with the same center and orthogonal orientations in 4 dimensions. Along with the p and q edges of the two polygons, all permutations of vertices in one polygon to vertices in the other form edges. All faces are triangular, with one edge of one polygon connected to one vertex of the other polygon. The p and q sided polygons are hollow, passing through the polytope center and not defining faces. Cells are tetrahedra constructed as all permutations of edge pairs between each polygon.
It can be understood by analogy to the relation of the 3D prisms and their dual bipyramids with Schläfli symbol { } + {p}, and a rhombus in 2D as { } + { }. A bipyramid can be seen as a 3D degenerated duopyramid, by adding an edge across the digon { } on the inner axis, and adding intersecting interior triangles and tetrahedra connecting that new edge to p-gon vertices and edges.
Other nonuniform polychora can be called duopyramids by the same construction, as two orthogonal and co-centered polygons, connected with edges with all combinations of vertex pairs between the polygons. The symmetry will be the product of the symmetry of the two polygons. So a rectangle-rectangle duopyramid would be topologically identical to the uniform 4-4 duopyramid, but a lower symmetry [2,2,2], order 16, possibly doubled to 32 if the two rectangles are identical.
Coordinates
The coordinates of a p-q duopyramid (on a unit 3-sphere) can be given as:
All pairs of vertices are connected by edges.
Perspective projections
Orthogonal projections
The 2n vertices of a n-n duopyramid'' can be orthogonally projected into two regular n-gons with edges between all vertices of each n-gon.
The regular 16-cell can be seen as a 4-4 duopyramid, being dual to the 4-4 duoprism, which is the tesseract. As a 4-4 duopyramid, the 16-cell's symmetry is [4,2,4], order 64, and doubled to , order 128 with the 2 central squares interchangeab |
https://en.wikipedia.org/wiki/Stephen%20Quake | Stephen Ronald Quake (born 1969) is an American physicist, inventor, and entrepreneur.
Education and Career
Quake earned his B.S. in physics and M.S. in mathematics from Stanford in 1991 and his D.Phil. in theoretical physics from Oxford University in 1994 as a Marshall Scholar. His thesis research was in statistical mechanics and the effects of knots on polymers. He did his postdoctoral work at Stanford in single-molecule biophysics with Steven Chu. Quake joined the faculty of the California Institute of Technology at the age of 26, where he rose through the ranks and was ultimately appointed the Thomas and Doris Everhart Professor of Applied Physics and Physics. He moved back to Stanford University in 2005 to help launch a new department in Bioengineering, where he is now the Lee Otterson Professor of Bioengineering and Applied Physics. From 2006 to 2016 he was an Investigator of the Howard Hughes Medical Institute. He is an Andrew D. White Professor-at-Large at Cornell University.
Quake was elected a member of the National Academy of Engineering in 2013 for achievements in single-cell analysis and large-scale integration of microfluidic devices. He has also been elected to the National Academy of Sciences, the Institute of Medicine, the American Physical Society, the American Institute for Medical and Biological Engineering and the American Academy of Arts and Sciences.
He is the recipient of numerous international awards, including the Human Frontiers of Science Nakasone Prize, the Jacob Heskel Gabbay Award (2015), the MIT-Lemelson Prize for Innovation, the Raymond and Beverly Sackler International Prize in Biophysics, the NIH Director’s Pioneer Award, the American Society of Microbiology’s Promega Biotechnology Award, and the Royal Society of Chemistry Publishing’s Pioneer of Miniaturization Award. He has founded or co-founded several companies, including Fluidigm, Helicos Biosciences, Verinata Health, Quanticel Pharmaceuticals, Moleculo, Cellular Research and Immumetrix.
Quake is known for his new approaches to biological measurement. He has made contributions to the field of microfluidics, including the invention of microfluidic large scale integration, and developed applications of microfluidics to structural biology, drug discovery, and molecular affinity measurements. He has also made contributions to the field of genomics, including single molecule DNA sequencing, techniques to perform single cell gene expression and genome sequencing, the development of non-invasive prenatal diagnostics to replace amniocentesis, prenatal genome sequencing, non-invasive tests for heart transplant rejection, and the development of approaches to sequence and analyze an individual's immune system. His genome was the subject of clinical annotation by a large team in the Stanford Hospital.
Since 2022, Quake has been the head of the Chan Zuckerberg Institute Science division.
Relationship with He Jiankui
Quake is also known as a former postdoct |
https://en.wikipedia.org/wiki/Nobuyuki%20Abe%20%28footballer%29 | is a Japanese football player for Nagano Parceiro.
Club career statistics
Updated to 23 February 2020.
References
External links
Profile at Nagano Parceiro
Profile at Giravanz Kitakyushu
1984 births
Living people
Ryutsu Keizai University alumni
People from Higashiyamato, Tokyo
Association football people from Tokyo Metropolis
Japanese men's footballers
J1 League players
J2 League players
J3 League players
FC Tokyo players
Shonan Bellmare players
Giravanz Kitakyushu players
AC Nagano Parceiro players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Shota%20Sakaki | is a Japanese football player who plays for AC Nagano Parceiro.
Club statistics
Updated to 22 February 2018.
1Includes Emperor's Cup.
2Includes J.League Cup.
References
External links
J. League (#33)
1993 births
Living people
Association football people from Hokkaido
Japanese men's footballers
J1 League players
J2 League players
J3 League players
2. Liga (Austria) players
Hokkaido Consadole Sapporo players
J.League U-22 Selection players
SV Horn players
Tochigi SC players
AC Nagano Parceiro players
Japanese expatriate men's footballers
Expatriate men's footballers in Austria
Japanese expatriate sportspeople in Austria
Men's association football forwards |
https://en.wikipedia.org/wiki/Takahide%20Umebachi | is a Japanese football player, who curently plays as a defender for the Sutherland Sharks in the National Premier Leagues NSW competition.
Career statistics
Club
Updated to end of 2018 season.
1Includes Suruga Bank Championship.
Honours
Club
Kashima Antlers
J. League Cup (3) : 2011, 2012, 2015
Suruga Bank Championship (2) : 2012, 2013
References
External links
Profile at Kashima Antlers
Profile at Zweigen Kanazawa
1992 births
Living people
Association football people from Osaka Prefecture
People from Takatsuki, Osaka
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Kashima Antlers players
Montedio Yamagata players
Zweigen Kanazawa players
SC Sagamihara players
Kansai University alumni
Men's association football midfielders |
https://en.wikipedia.org/wiki/Takaya%20Osanai | is a Japanese football player for ReinMeer Aomori.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Fukushima United FC
Profile at ReinMeer Aomori
1993 births
Living people
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Hokkaido Consadole Sapporo players
AC Nagano Parceiro players
Fukushima United FC players
J.League U-22 Selection players
ReinMeer Aomori players
Men's association football defenders
Association football people from Sapporo |
https://en.wikipedia.org/wiki/Masaya%20Nozaki | is a Japanese football player who plays for ReinMeer Aomori.
Career statistics
Updated to 8 March 2018.
References
External links
Profile at YSCC Yokohama
Profile at Nagano Parceiro
1993 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Urawa Red Diamonds players
Avispa Fukuoka players
Gainare Tottori players
YSCC Yokohama players
ReinMeer Aomori players
AC Nagano Parceiro players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yohei%20Hayashi | is a Japanese football player for Blaublitz Akita.
Club statistics
Updated to 31 December 2020.
Honours
Blaublitz Akita
J3 League (1): 2020
References
External links
Profile at Oita Trinita
Profile at FC Tokyo
Profile at Akita
1989 births
Living people
Chuo University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
FC Tokyo players
FC Tokyo U-23 players
Fagiano Okayama players
Oita Trinita players
Blaublitz Akita players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yosuke%20Kawai | Yosuke Kawai (河井 陽介, born 4 August 1989) is a Japanese football player for Fagiano Okayama.
Career statistics
Club
Updated to 18 February 2019.
1Includes Emperor's Cup.
2Includes J. League Cup.
References
External links
Profile at Shimizu S-Pulse
1989 births
Living people
Keio University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Shimizu S-Pulse players
Fagiano Okayama players
Men's association football midfielders
FISU World University Games gold medalists for Japan
Universiade medalists in football
People from Fujieda, Shizuoka
Medalists at the 2011 Summer Universiade |
https://en.wikipedia.org/wiki/Ry%C5%8Dhei%20Shirasaki | is a Japanese football player for Shimizu S-Pulse.
Career statistics
Club
References
External links
Profile at Shimizu S-Pulse
1993 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Shimizu S-Pulse players
Kataller Toyama players
J.League U-22 Selection players
Kashima Antlers players
Sagan Tosu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Hiroyuki%20Abe%20%28footballer%29 | is a Japanese professional footballer who plays as an attacking midfielder or a winger for Shonan Bellmare in the J1 League.
Club statistics
Updated to 9 July 2022.
1 includes J. League Championship, Japanese Super Cup and Suruga Bank Championship appearances.
National team statistics
Honours
Gamba Osaka
J1 League – 2014
J2 League – 2013
Emperor's Cup – 2014, 2015
J.League Cup – 2014
Japanese Super Cup – 2015
Kawasaki Frontale
J1 League – 2017, 2018
J.League Cup – 2019
Japanese Super Cup – 2019
References
External links
Profile at Kawasaki Frontale
1989 births
Living people
Kwansei Gakuin University alumni
Association football people from Nara Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Gamba Osaka players
Gamba Osaka U-23 players
Kawasaki Frontale players
Nagoya Grampus players
Shonan Bellmare players
Japan men's international footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Takamitsu%20Yoshino | Takamitsu Yoshino (吉野 峻光, born April 24, 1989) is a Japanese football player.
Club statistics
Updated to 23 February 2016.
References
External links
1989 births
Living people
Kokushikan University alumni
Association football people from Kyoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
Cerezo Osaka players
Ventforet Kofu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Lee%20Kwang-seon | Lee Kwang-Seon (; born September 6, 1989) is a South Korean football player who plays for Gyeongnam FC.
Club statistics
References
External links
1989 births
Living people
Men's association football defenders
South Korean men's footballers
South Korean expatriate men's footballers
J1 League players
J2 League players
Vissel Kobe players
Avispa Fukuoka players
Jeju United FC players
Gimcheon Sangmu FC players
Gyeongnam FC players
K League 1 players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan |
https://en.wikipedia.org/wiki/Sena%20Inami | is a former Japanese football player.
Club statistics
References
External links
J. League (#26)
1992 births
Living people
Association football people from Ishikawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
V-Varen Nagasaki players
Júbilo Iwata players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Tatsuya%20Sakai | is a Japanese football player.
Club statistics
Updated to end of 2018 season.
National team statistics
References
External links
Profile at Oita Trinita
Profile at V-Varen Nagasaki
Japan National Football Team Database
1990 births
Living people
National Institute of Fitness and Sports in Kanoya alumni
Japanese men's footballers
Japan men's international footballers
J1 League players
J2 League players
Sagan Tosu players
Matsumoto Yamaga FC players
V-Varen Nagasaki players
Oita Trinita players
Montedio Yamagata players
Men's association football defenders
Tatsuya Sakai
Tatsuya Sakai
Japanese expatriate sportspeople in Thailand
Expatriate men's footballers in Thailand
Association football people from Fukuoka (city) |
https://en.wikipedia.org/wiki/Koki%20Kiyotake | is a Japanese football player for FC Ryukyu.
His older brother, Hiroshi, is also a football player.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at JEF United Chiba
Profile at Roasso Kumamoto
1991 births
Living people
Fukuoka University alumni
Association football people from Ōita Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sagan Tosu players
Roasso Kumamoto players
JEF United Chiba players
Tokushima Vortis players
FC Ryukyu players
Men's association football midfielders
Sportspeople from Ōita (city) |
https://en.wikipedia.org/wiki/Masaki%20Miyasaka | Masaki Miyasaka (宮阪 政樹, born July 15, 1989) is a Japanese football player for Thespakusatsu Gunma.
Club statistics
Updated to 24 February 2019.
1Includes Promotion Playoffs to J1.
References
External links
Profile at Oita Trinita
Profile at Matsumoto Yamaga
1989 births
Living people
Meiji University alumni
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Montedio Yamagata players
Matsumoto Yamaga FC players
Oita Trinita players
Thespakusatsu Gunma players
Men's association football midfielders
FISU World University Games gold medalists for Japan
Universiade medalists in football
Medalists at the 2011 Summer Universiade |
https://en.wikipedia.org/wiki/Sai%20Kanakubo | Sai Kanakubo (金久保 彩, born January 11, 1989) is a Japanese football player who currently plays for Nara Club.
Club statistics
Updated to 18 November 2018.
References
External links
Profile at Vanraure Hachinohe
1989 births
Living people
Komazawa University alumni
Association football people from Ibaraki Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Mito HollyHock players
V-Varen Nagasaki players
AC Nagano Parceiro players
Kagoshima United FC players
Vanraure Hachinohe players
Nara Club players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kosei%20Ishigami | is a Japanese former footballer.
Club statistics
Updated to 23 February 2017.
References
External links
Profile at Mito HollyHock
1990 births
Living people
University of Tsukuba alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
J3 League players
Mito HollyHock players
Gainare Tottori players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yuto%20Suzuki | Yuto Suzuki (鈴木 雄斗, born December 7, 1993) is a Japanese professional footballer who plays as a winger for J.League club Júbilo Iwata.
Club statistics
References
External links
Profile at Júbilo Iwata
1993 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Mito HollyHock players
Montedio Yamagata players
Kawasaki Frontale players
Gamba Osaka players
Matsumoto Yamaga FC players
Júbilo Iwata players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Cha%20Young-hwan | Cha Young-Hwan (; born July 16, 1990) is a South Korean football player who plays as a defensive midfielder or centre-back for Yangju Citizen FC.
Club statistics
As of 3 December 2017
References
External links
Living people
South Korean men's footballers
South Korean expatriate men's footballers
J2 League players
Tochigi SC players
Zweigen Kanazawa players
Busan IPark players
Gimcheon Sangmu FC players
K League 1 players
K League 2 players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
1990 births
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kazuki%20Oiwa | Kazuki Oiwa (大岩 一貴, born 17 August 1989) is a Japanese football player for Shonan Bellmare.
Career statistics
Club
Updated to 7 August 2022.
1Includes Emperor's Cup.
2Includes J. League Cup.
References
External links
Profile at Vegalta Sendai
Profile at JEF United Chiba
1989 births
Living people
Chuo University alumni
Association football people from Aichi Prefecture
Japanese men's footballers
J1 League players
J2 League players
JEF United Chiba players
Vegalta Sendai players
Shonan Bellmare players
Men's association football defenders
FISU World University Games gold medalists for Japan
Universiade medalists in football
Medalists at the 2011 Summer Universiade |
https://en.wikipedia.org/wiki/Takahiro%20Tanaka%20%28footballer%29 | is a Japanese football player. He plays for FC Kariya.
Club career
Tanjong Pagar United
Tanaka signed for Tanjong Pagar United FC for the 2020 Singapore Premier League.
Club statistics
Updated to 02 February 2020.
References
External links
Profile at Briobecca Urayasu
J. League (#28)
1993 births
Living people
People from Hachiōji, Tokyo
Association football people from Tokyo
Japanese men's footballers
J2 League players
Japan Football League players
Tokyo Verdy players
FC Machida Zelvia players
Briobecca Urayasu players
Renofa Yamaguchi FC players
Suzuka Point Getters players
Men's association football defenders |
https://en.wikipedia.org/wiki/Marcell%20Matolcsi | Marcell Matolcsi (born 1 February 1991, in Budapest) is a professional Hungarian footballer who currently plays for Vasas SC.
Club statistics
Updated to games played as of 20 May 2012.
External links
HLSZ
MLSZ
1991 births
Living people
Footballers from Budapest
Hungarian men's footballers
Men's association football defenders
Vasas SC players |
https://en.wikipedia.org/wiki/Salem%20Reidan | Salem Reidan (born 13 June 1991 in Budapest) is a professional Hungarian footballer currently plays for Vasas SC.
Club statistics
Updated to games played as of 21 November 2012.
External links
HLSZ
MLSZ
1991 births
Living people
Footballers from Budapest
Hungarian men's footballers
Men's association football midfielders
Vasas SC players |
https://en.wikipedia.org/wiki/Michael%20Lin%20%28mathematician%29 | Michael Lin () (born June 8, 1942) is an Israeli mathematician, who has published scientific articles in the field of probability concentrating on Markov chains and ergodic theory. He serves as professor emeritus at the Department of Mathematics in Ben-Gurion University of the Negev (BGU). Additionally, he is a member of the academic board and serves as the academic coordinator at Achva Academic College. Professor Lin is considered a Zionist, as he gave up a position at Ohio State University in order to promote the field of mathematics in Israel.
Biography
Michael Lin was born in Israel. He holds a Bachelor of Science in Mathematics and Physics from The Hebrew University of Jerusalem (1963), Master of Science in Mathematics (1967) and a PhD in Mathematics also from The Hebrew University of Jerusalem (1971). In 1971 he was appointed as an assistant professor in Ohio State University. In 1976 he returned to Israel and became a senior lecturer in the Department of Mathematics at Ben-Gurion University of the Negev. Only 4 years later, at 1979, he became an associate professor and in 1984 he became a full professor. In 2011, Professor Lin retired and nowadays he serves as professor emeritus. During his career at Ben-Gurion University of the Negev he acted as:
Computer Science Coordinator, Department of Mathematics and Computer Science, BGU.
Member of BGU Computer Policy committee.
Chairman and Computer Science Coordinator, Department of Mathematics and Computer Science, BGU.
Senate representative to Executive Committee of Board of Trustees of BGU.
Senate representative to the BGU Executive Committee's subcommittee for student affairs.
President, Israel Mathematical Union.
Head of the Ethical Code Committee of BGU.
In 2004 Professor Lin also acted as a member of the committee electing the recipients of the Israel Prize in mathematics.
In addition to his academic activities, Professor Lin made a social-academic contribution, as he took a part in the 'Kamea Program'. The program helped immigrant scientists to continue working in their profession in the academy in Israel. Professor Lin assisted in the absorption of these immigrants in Ben-Gurion University of the Negev, specifically in the Department of Mathematics. Until his retirement, he was listed as an absorbing researcher of two scientists in his department. He was also responsible for the immigrants’ employment terms and insisted that they will be members of the Academic Staff Union. Additionally, Professor Lin was the university representative in a discussion regarding this program in the Israeli Parliament (knesset) and acted as an advisor regarding the newcomers’ academic seniority.
Research and publications
Professor Lin's published work focuses on two main areas of research in the field of probability: Ergodic theory and Markov chain. More specifically, he researched in several areas: mean and individual Ergodic theory, Central limit theorem and functional analysis.
Professor Lin h |
https://en.wikipedia.org/wiki/Jo-Wilfried%20Tsonga%20career%20statistics | This is a list of the main career statistics of French former professional tennis player, Jo-Wilfried Tsonga. Tsonga has won 18 ATP titles in singles, including 2 Masters titles at the 2008 Paris Masters and the 2014 Canada Masters. He was also the runner-up at the 2008 Australian Open and 2011 ATP World Tour Finals in singles. In addition, he was a silver medalist in men's doubles with Michaël Llodra at the 2012 London Olympics.
Career achievements
Tsonga reached his first career singles final and first Grand Slam singles final at the 2008 Australian Open. In the first round, Tsonga upset 9th seed Andy Murray in four sets and eventually reached the final after upsetting then world No. 2 Rafael Nadal in straight sets in the semifinals. In the final, Tsonga lost to the world No. 3 Novak Djokovic in four sets, after winning the first set, which was the only set which Djokovic dropped during the entire tournament. Following the event, Tsonga entered the Top 20 of the ATP rankings for the first time in his career, rising to world No. 18. In September of the same year, Tsonga avenged his Australian Open loss to Djokovic by defeating the Serb in the final of the PTT Thailand Open to win his first career singles title. Two months later, Tsonga defeated David Nalbandian in the final of the BNP Paribas Masters in Paris to win his first ATP Masters Series (later ATP World Tour Masters 1000) singles title, along with 3 Top 10 wins en route to the title, including a third round victory over Djokovic. Tsonga thus became the first home player to win it since Sébastien Grosjean in 2001 and remains the last home player to win it to date. Though he only played in a few tournaments, Tsonga's results throughout the year allowed him to qualify for the year-end ATP World Tour Finals for the first time in his career. However, he lost in the round robin stage after winning one of his three matches, which was his 3rd victory of the year against Djokovic. Tsonga finished the year at a then career-high singles ranking of world No. 6.
Since 2009, the highlights of Tsonga's career have been runner-up appearances at the 2011 BNP Paribas Masters and 2011 ATP World Tour Finals and semifinal appearances at the 2010 Australian Open, 2011 and 2012 Wimbledon Championships, along with 2013 and 2015 French Open.
In July 2011, Tsonga became the first player to have defeated each member of the "Big Four" at Grand Slam tournaments, after defeating Roger Federer at the 2011 Wimbledon Championships from 2 sets down. This feat was not repeated until 4 years later. He defeated Andy Murray and Rafael Nadal at the 2008 Australian Open, Novak Djokovic at the 2010 Australian Open and Roger Federer at the 2011 Wimbledon Championships and later, at the 2013 French Open.
In February 2012, Tsonga achieved a new career high singles ranking of world No. 5.
In August 2014, Tsonga won another Masters title in Toronto, becoming the first French player to win the title. He also became the 2nd pla |
https://en.wikipedia.org/wiki/Basic%20solution%20%28linear%20programming%29 | In linear programming, a discipline within applied mathematics, a basic solution is any solution of a linear programming problem satisfying certain specified technical conditions.
For a polyhedron and a vector , is a basic solution if:
All the equality constraints defining are active at
Of all the constraints that are active at that vector, at least of them must be linearly independent. Note that this also means that at least constraints must be active at that vector.
A constraint is active for a particular solution if it is satisfied at equality for that solution.
A basic solution that satisfies all the constraints defining (or, in other words, one that lies within ) is called a basic feasible solution.
References
Linear programming |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20PFC%20CSKA%20Sofia%20season | The 2012–13 season was PFC CSKA Sofia's 65th consecutive season in A Group. This article shows player statistics and all matches (official and friendly) that the club will play during the 2012–13 season.
Players
Squad stats
Appearances for competitive matches only
|-
|colspan="14"|Players sold or loaned out after the start of the season:
|}
As of 25 May 2013
Players in/out
Summer transfers
In:
Out:
Winter transfers
In:
Out:
Pre-season and friendlies
Pre-season
On-season (autumn)
Mid-season
On-season (spring)
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
Europa League
By ending as runner-up from A Grupa 2011/12, CSKA Sofia qualified for the Europa League. They started in the second qualifying round.
Second qualifying round
UEFA Club Rankings
This is the current UEFA Club Rankings, including season 2011–12.
See also
PFC CSKA Sofia
References
External links
CSKA Official Site
CSKA Fan Page with up-to-date information
Bulgarian A Professional Football Group
UEFA Profile
PFC CSKA Sofia seasons
Cska Sofia |
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