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https://en.wikipedia.org/wiki/Michael%20Gelfond | Michael Gelfond is a Professor in Computer Sciences at Texas Tech University in the United States. He received a degree in mathematics from the Steklov Institute of Mathematics in Russia in 1974 and emigrated to the United States in 1978. Gelfond's research interests are in the areas of computational logic and knowledge representation. He is a Fellow of the Association for the Advancement of Artificial Intelligence, and an Area Editor (in Knowledge Representation and Nonmonotonic Reasoning) of the journal Theory and Practice of Logic Programming.
He, together with Vladimir Lifschitz, defined stable model semantics for logic programs, which later became the theoretical foundation for Answer Set Programming, a new declarative programming paradigm.
References
External links
Michael Gelfond's homepage at Texas Tech University
Michael Gelfond's publications on DBLP
Living people
American computer scientists
Texas Tech University faculty
Logic programming researchers
Fellows of the Association for the Advancement of Artificial Intelligence
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Kenya%20National%20Bureau%20of%20Statistics | The Kenya National Bureau of Statistics (KNBS) is a department in Kenya's Ministry of Planning which collects and compiles regular cross-sectoral data for the government. The Bureau was established through the Statistics Act of 2006 and initiated in February 2007. The office places staff in other departments of the Kenya Government to collect data. The headquarters is in Herufi House in Nairobi.
Programmes
Kenya Population and Housing Census
In part, the National Bureau of Statistics oversees the Kenya Population and Housing Census, which occurs every ten years. The most recent census was in 2019, and documents the population of country. An audit by the Auditor General's office published in 2021 found that half of the KSh 18.5 billion budget of the Kenya Population and Housing Census for the 2019 census, could not be accounted for.
National Sampling Survey and Evaluation Programme
The National Sampling Survey and Evaluation Programme is a household data collection program.
References
External links
Kenya National Bureau of Statistics homepage
Government agencies of Kenya |
https://en.wikipedia.org/wiki/R.%20H.%20Bruck | Richard Hubert Bruck (December 26, 1914 – December 18, 1991) was an American mathematician best known for his work in the field of algebra, especially in its relation to projective geometry and combinatorics.
Bruck studied at the University of Toronto, where he received his doctorate in 1940 under the supervision of Richard Brauer.
He spent most his career as a professor at University of Wisconsin–Madison, advising at least 31 doctoral students.
He is best known for his 1949 paper coauthored with H. J. Ryser, the results of which became known as the Bruck–Ryser theorem (now known in a generalized form as the Bruck-Ryser-Chowla theorem), concerning the possible orders of finite projective planes.
In 1946, he was awarded a Guggenheim Fellowship.
In 1956, he was awarded the Chauvenet Prize for his article Recent Advances in the Foundations of Euclidean Plane Geometry. In 1962, he was an invited speaker at the International Congress of Mathematicians in Stockholm.
In 1963, he was a Fulbright Lecturer at the University of Canberra.
In 1965 a Groups and Geometry conference was held at the University of Wisconsin in honor of Bruck's retirement.
Dick Bruck and his wife Helen were supporters of the fine arts. They were patrons of the regional American Players Theatre in Wisconsin.
Selected publications
(3rd ed. in 1971, )
Notes
External links
Biography at the University of Texas
Bruck–Ryser–Chowla Theorem at Mathworld
1914 births
1991 deaths
20th-century American mathematicians
University of Wisconsin–Madison faculty
University of Toronto alumni
Combinatorialists |
https://en.wikipedia.org/wiki/Estimation%20%28disambiguation%29 | Estimation is the process of finding a usable approximation of a result.
Estimation may also refer to:
Estimation theory, a field in statistics, also used in signal processing
Estimation statistics, a data analysis approach in frequentist statistics
Estimation (project management)
Approximation, finding estimates in the form of upper or lower bounds for a quantity that cannot readily be evaluated precisely
Forecasting
Prediction
See also
Guesstimate, an informal estimate when little information is available |
https://en.wikipedia.org/wiki/Rad%C3%B3%E2%80%93Kneser%E2%80%93Choquet%20theorem | In mathematics, the Radó–Kneser–Choquet theorem, named after Tibor Radó, Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism of the unit circle is a harmonic diffeomorphism of the open unit disk. The result was stated as a problem by Radó and solved shortly afterwards by Kneser in 1926. Choquet, unaware of the work of Radó and Kneser, rediscovered the result with a different proof in 1945. Choquet also generalized the result to the Poisson integral of a homeomorphism from the unit circle to a simple Jordan curve bounding a convex region.
Statement
Let f be an orientation-preserving homeomorphism of the unit circle |z| = 1 in C and define the Poisson integral of f by
for r < 1. Standard properties of the Poisson integral show that Ff is a harmonic function on |z| < 1 which extends by continuity to f on |z| = 1. With the additional assumption that f is orientation-preserving homeomorphism of this circle, Ff is an orientation preserving diffeomorphism of the open unit disk.
Proof
To prove that Ff is locally an orientation-preserving diffeomorphism, it suffices to show that the Jacobian at a point a in the unit disk is positive. This Jacobian is given by
On the other hand, that g is a Möbius transformation preserving the unit circle and the unit disk,
Taking g so that g(a) = 0 and taking the change of variable ζ = g(z), the chain rule gives
It follows that
It is therefore enough to prove positivity of the Jacobian when a = 0. In that case
where the an are the Fourier coefficients of f:
Following , the Jacobian at 0 can be expressed as a double integral
Writing
where h is a strictly increasing continuous function satisfying
the double integral can be rewritten as
Hence
where
This formula gives R as the sum of the sines of four non-negative angles with sum 2π, so it is always non-negative. But then the Jacobian at 0 is strictly positive and Ff is therefore locally a diffeomorphism.
It remains to deduce Ff is a homeomorphism. By continuity its image is compact so closed. The non-vanishing of the Jacobian, implies that Ff is an open mapping on the unit disk, so that the image of the open disk is open. Hence the image of the closed disk is an open and closed subset of the closed disk. By connectivity, it must be the whole disk. For |w| < 1, the inverse image of w is closed, so compact, and entirely contained in the open disk. Since Ff is locally a homeomorphism, it must be a finite set. The set of points w in the open disk with exactly n preimages is open. By connectivity every point has the same number N of preimages. Since the open disk is simply connected, N = 1. In fact taking any preimage of the origin, every radial line has a unique lifting to a preimage, and so there is an open subset of the unit disk mapping homeomorphically onto the open disk. If N > 1, its complement would also have to be open, contradicting connectivity.
Notes
References
Theorems in harmonic analysis |
https://en.wikipedia.org/wiki/Dive%20planning | Dive planning is the process of planning an underwater diving operation. The purpose of dive planning is to increase the probability that a dive will be completed safely and the goals achieved. Some form of planning is done for most underwater dives, but the complexity and detail considered may vary enormously.
Professional diving operations are usually formally planned and the plan documented as a legal record that due diligence has been done for health and safety purposes. Recreational dive planning may be less formal, but for complex technical dives, can be as formal, detailed and extensive as most professional dive plans. A professional diving contractor will be constrained by the code of practice, standing orders or regulatory legislation covering a project or specific operations within a project, and is responsible for ensuring that the scope of work to be done is within the scope of the rules relevant to that work. A recreational (including technical) diver or dive group is generally less constrained, but nevertheless is almost always restricted by some legislation, and often also the rules of the organisations to which the divers are affiliated.
The planning of a diving operation may be simple or complex. In some cases the processes may have to be repeated several times before a satisfactory plan is achieved, and even then the plan may have to be modified on site to suit changed circumstances. The final product of the planning process may be formally documented or, in the case of recreational divers, an agreement on how the dive will be conducted. A diving project may consist of a number of related diving operations.
A documented dive plan may contain elements from the following list:
Overview of Diving Activities
Schedule of Diving Operations
Specific Dive Plan Information
Budget
Objective
Commercial diving contractors will develop specifications for the operation in cooperation with the client, who will normally provide a specific objective. The client will generally specify what work is to be done, and the diving contractor will deal with the logistics of how to do it.
Other professional divers will usually plan their diving operations around an objective related to their primary occupation.
Recreational divers will generally choose an objective for entertainment value, or for training purposes.
It will generally be necessary to specify the following:
Work to be done, or the recreational equivalent
Equipment needed
Procedures to be used
Personnel required
Places
Times
Analysis of available information on the site
Expected surface conditions, such as sea state, air temperature, and wind chill factor
Expected underwater conditions, including water temperature, depth, type of bottom, tides and currents, visibility, extent of pollution, and other hazards
Assistance and emergency information, including location, status, and contact procedures for the nearest recompression chamber, evacuation and rescue facilities, and near |
https://en.wikipedia.org/wiki/Hamed%20Al-Balushi | Hamed Hamdan Al-Balushi (; born 2 March 1980), commonly known as Hamed Al-Balushi, is an Omani footballer who plays for Fanja SC.
Club career statistics
International career
Hamed was selected for the national team for the first time in 2007. He has represented the national team in the 2007 AFC Asian Cup.
References
External links
Hamed Al-Balushi at Goal.com
1980 births
Living people
Omani men's footballers
Oman men's international footballers
Men's association football forwards
2007 AFC Asian Cup players
Fanja SC players
Omani people of Baloch descent
Footballers at the 2006 Asian Games
Asian Games competitors for Oman |
https://en.wikipedia.org/wiki/Ivan%20Mili%C4%8Devi%C4%87 | Ivan Miličević (born 11 February 1988 in Osijek) is a Croatian football forward, who is currently playing for 1. FC Bad Kötzting in Germany.
Career statistics
References
External links
Ivan Miličević at Sportnet.hr
Ivan Miličević at FuPa
1988 births
Living people
Footballers from Osijek
Men's association football forwards
Croatian men's footballers
Croatia men's youth international footballers
Croatia men's under-21 international footballers
NK Osijek players
NK Olimpija Osijek players
San Antonio Scorpions players
NK Istra 1961 players
US Triestina Calcio 1918 players
Hong Kong Pegasus FC players
AZ Picerno players
ASD Sangiovannese 1927 players
USD Dro Alto Garda Calcio players
HNK Cibalia players
Croatian Football League players
North American Soccer League (2011–2017) players
Serie C players
Serie D players
Landesliga players
Croatian expatriate men's footballers
Expatriate men's soccer players in the United States
Croatian expatriate sportspeople in the United States
Expatriate men's footballers in Italy
Croatian expatriate sportspeople in Italy
Expatriate men's footballers in Hong Kong
Croatian expatriate sportspeople in Hong Kong
Expatriate men's footballers in Germany
Croatian expatriate sportspeople in Germany |
https://en.wikipedia.org/wiki/Cambridge%20Algebra%20System | Cambridge Algebra System (CAMAL) is a computer algebra system written in Cambridge University by David Barton, Steve Bourne, and John Fitch. It was initially used for computations in celestial mechanics and general relativity. The foundation code was written in Titan computer assembler,. In 1973, when Titan was replaced with an IBM370/85, it was rewritten in ALGOL 68C and then BCPL where it could run on IBM mainframes and assorted microcomputers.
References
Further reading
Computer algebra systems |
https://en.wikipedia.org/wiki/Sobolev%20spaces%20for%20planar%20domains | In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.
Sobolev spaces with boundary conditions
Let be a bounded domain with smooth boundary. Since is contained in a large square in , it can be regarded as a domain in by identifying opposite sides of the square. The theory of Sobolev spaces on can be found in , an account which is followed in several later textbooks such as and .
For an integer, the (restricted) Sobolev space is defined as the closure of in the standard Sobolev space .
.
Vanishing properties on boundary: For the elements of are referred to as " functions on which vanish with their first derivatives on ." In fact if agrees with a function in , then is in . Let be such that in the Sobolev norm, and set . Thus in . Hence for and ,
By Green's theorem this implies
where
with the unit normal to the boundary. Since such form a dense subspace of , it follows that on .
Support properties: Let be the complement of and define restricted Sobolev spaces analogously for . Both sets of spaces have a natural pairing with . The Sobolev space for is the annihilator in the Sobolev space for of and that for is the annihilator of . In fact this is proved by locally applying a small translation to move the domain inside itself and then smoothing by a smooth convolution operator.
Suppose in annihilates . By compactness, there are finitely many open sets covering such that the closure of is disjoint from and each is an open disc about a boundary point such that in small translations in the direction of the normal vector carry into . Add an open with closure in to produce a cover of and let be a partition of unity subordinate to this cover. If translation by is denoted by , then the functions
tend to as decreases to and still lie in the annihilator, indeed they are in the annihilator for a larger domain than , the complement of which lies in . Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in . These are necessarily smooth functions of compact support in .
Further vanishing properties on the boundary: The characterization in terms of annihilators shows that lies in if (and only if) it and its derivatives of order less than vanish on . In fact can be extended to by setting it to be on . This extension defines an element in using the formula for the norm
Moreover satisfies for g in .
Duality: For , define to be the orthogonal complement of in . Let be the orthogonal projection onto , so that |
https://en.wikipedia.org/wiki/Cantelli%27s%20inequality | In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. The inequality states that, for
where
is a real-valued random variable,
is the probability measure,
is the expected value of ,
is the variance of .
Applying the Cantelli inequality to gives a bound on the lower tail,
While the inequality is often attributed to Francesco Paolo Cantelli who published it in 1928, it originates in Chebyshev's work of 1874. When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. The Chebyshev inequality has "higher moments versions" and "vector versions", and so does the Cantelli inequality.
Comparison to Chebyshev's inequality
For one-sided tail bounds, Cantelli's inequality is better, since Chebyshev's inequality can only get
On the other hand, for two-sided tail bounds, Cantelli's inequality gives
which is always worse than Chebyshev's inequality (when ; otherwise, both inequalities bound a probability by a value greater than one, and so are trivial).
Proof
Let be a real-valued random variable with finite variance and expectation , and define (so that and ).
Then, for any , we have
the last inequality being a consequence of Markov's inequality. As the above holds for any choice of , we can choose to apply it with the value that minimizes the function . By differentiating, this can be seen to be , leading to
if
Generalizations
Various stronger inequalities can be shown.
He, Zhang, and Zhang showed (Corollary 2.3) when
and :
In the case this matches a bound in Berger's "The Fourth Moment Method",
This improves over Cantelli's inequality in that we can get a non-zero lower bound, even when .
See also
Chebyshev's inequality
Paley–Zygmund inequality
References
Probabilistic inequalities |
https://en.wikipedia.org/wiki/Frits%20Beukers | Frits Beukers () (born 1953, Ankara) is a Dutch mathematician, who works on number theory and hypergeometric functions.
In 1979 Beukers received his PhD at Leiden University under the direction of Robert Tijdeman with thesis The generalized Ramanujan–Nagell Equation, published in Acta Arithmetica, vol. 38, 1980/1981. From 1979 to 1980 he was a visiting scholar at the Institute for Advanced Study. He became a professor in Leiden and in the 2000s at Utrecht University.
Beukers works on questions of transcendence and irrationality in number theory, and on other topics. In connection with the famous proof by Roger Apéry (1978) on the irrationality of the values of the Riemann zeta function evaluated at the points 2 and 3, Beukers gave a much simpler alternate proof using Legendre polynomials. He also published on questions in mechanics about dynamical systems and their exact solvability.
Selected works
A rational approach to Pi, Nieuw Archief voor Wiskunde, 2000, Heft 4
References
External links
Homepage in Utrecht
1953 births
Living people
20th-century Dutch mathematicians
21st-century Dutch mathematicians
Leiden University alumni
Academic staff of Leiden University
Academic staff of Utrecht University
Institute for Advanced Study visiting scholars
People from Ankara |
https://en.wikipedia.org/wiki/Patrik%20Vass | Patrik Vass (born 17 January 1993) is a Hungarian football player who plays for Nyíregyháza.
Club statistics
Updated to games played as of 7 February 2022.
External links
Profile at HLSZ
Profile at MLSZ
1993 births
Living people
Footballers from Budapest
Hungarian men's footballers
Men's association football midfielders
MTK Budapest FC players
Vasas SC players
Gyirmót FC Győr players
Zalaegerszegi TE players
Nyíregyháza Spartacus FC players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Partial%20differential%20algebraic%20equation | In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.
Definition
A general PDAE is defined as:
where:
F is a set of arbitrary functions;
x is a set of independent variables;
y is a set of dependent variables for which partial derivatives are defined; and
z is a set of dependent variables for which no partial derivatives are defined.
The relationship between a PDAE and a partial differential equation (PDE) is analogous to the relationship between an ordinary differential equation (ODE) and a differential algebraic equation (DAE).
PDAEs of this general form are challenging to solve. Simplified forms are studied in more detail in the literature. Even as recently as 2000, the term "PDAE" has been handled as unfamiliar by those in related fields.
Solution methods
Semi-discretization is a common method for solving PDAEs whose independent variables are those of time and space, and has been used for decades. This method involves removing the spatial variables using a discretization method, such as the finite volume method, and incorporating the resulting linear equations as part of the algebraic relations. This reduces the system to a DAE, for which conventional solution methods can be employed.
References
Partial differential equations
Differential equations
Multivariable calculus
Numerical analysis |
https://en.wikipedia.org/wiki/Miami%20Heat%20accomplishments%20and%20records | This page details the all-time statistics, records, and other achievements pertaining to the Miami Heat. The Miami Heat is an American professional basketball team currently playing in the National Basketball Association.
Franchise accomplishments and awards
Individual awards
NBA Most Valuable Player
LeBron James – 2012, 2013
NBA Finals MVP
Dwyane Wade – 2006
LeBron James – 2012, 2013
NBA Conference Finals Most Valuable Player Award
Jimmy Butler – 2023
NBA All-Star Game MVP
Dwyane Wade – 2010
NBA Scoring Champion
Dwyane Wade – 2009
NBA Defensive Player of the Year
Alonzo Mourning – 1999, 2000
NBA Most Improved Player Award
Rony Seikaly – 1990
Isaac Austin – 1997
NBA Sixth Man of the Year
Tyler Herro – 2022
Best NBA Player ESPY Award
Dwyane Wade – 2006
LeBron James - 2012
NBA Coach of the Year
Pat Riley – 1997
NBA Executive of the Year
Pat Riley – 2011
J. Walter Kennedy Citizenship Award
P. J. Brown – 1997
Alonzo Mourning – 2002
All-NBA First Team
Dwyane Wade – 2009, 2010
Shaquille O'Neal – 2005, 2006
LeBron James – 2011–2014
Alonzo Mourning – 1999
Tim Hardaway – 1997
All-NBA Second Team
Dwyane Wade – 2005, 2006, 2011
Tim Hardaway – 1998, 1999
Alonzo Mourning – 2000
Jimmy Butler – 2023
All-NBA Third Team
Dwyane Wade – 2007, 2012, 2013
Jimmy Butler – 2020, 2021
NBA All-Defensive First Team
Alonzo Mourning – 1999, 2000
LeBron James – 2011, 2012, 2013
NBA All-Defensive Second Team
P.J. Brown – 1997, 1999
Bruce Bowen – 2001
Dwyane Wade – 2005, 2009, 2010
LeBron James – 2014
Hassan Whiteside – 2016
Bam Adebayo – 2020, 2021
Jimmy Butler – 2021
NBA All-Rookie First Team
Sherman Douglas – 1990
Steve Smith – 1992
Caron Butler – 2003
Dwyane Wade – 2004
Michael Beasley – 2009
Kendrick Nunn – 2020
NBA All-Rookie Second Team
Kevin Edwards – 1989
Glen Rice – 1990
Willie Burton – 1991
Udonis Haslem – 2004
Mario Chalmers – 2009
Justise Winslow– 2016
Tyler Herro – 2020
NBA All-Star Weekend
NBA All-Star Game MVP
Dwyane Wade - 2010
NBA All-Star Skills Challenge Champion
Dwyane Wade – 2006, 2007
Bam Adebayo – 2020
NBA All-Star Three-point Shootout Champion
Glen Rice – 1995
Jason Kapono – 2007
Daequan Cook – 2009
James Jones – 2011
NBA All-Star Slam Dunk Contest Champion
Harold Miner – 1993, 1995
Derrick Jones Jr. – 2020
NBA All-Star selections
Alonzo Mourning – 1996, 1997, 2000, 2001, 2002
Tim Hardaway – 1997, 1998
Anthony Mason – 2001
Shaquille O'Neal – 2005, 2006, 2007
Dwyane Wade – 2005–2016, 2019
LeBron James – 2011–2014
Chris Bosh – 2011–2016
Goran Dragic – 2018
Jimmy Butler – 2020, 2022
Bam Adebayo – 2020, 2023
Stan Van Gundy – 2005 (as Head Coach)
Erik Spoelstra - 2013, 2022 (as Head Coach)
Franchise records for regular season
Most points scored in a game
LeBron James – 61
Glen Rice – 56
Dwyane Wade – 55
LeBron James – 51
Dwyane Wade – 50
Highest points per game in a season
Dwyane Wade – 30.2
Dwyane Wade – 27.4
Dwyane Wade – 27.2
LeBron James – 27.1
LeBr |
https://en.wikipedia.org/wiki/Frank%20Fucarino | Frank A. Fucarino (July 24, 1920 – April 3, 2012) was an American professional basketball player for the Toronto Huskies. He played in the first ever NBA game.
BAA career statistics
Regular season
References
External links
1920 births
2012 deaths
American expatriate basketball people in Canada
American men's basketball players
Forwards (basketball)
LIU Brooklyn Blackbirds men's basketball players
Toronto Huskies players
Undrafted National Basketball Association players
Basketball players from Queens, New York |
https://en.wikipedia.org/wiki/%C3%89mile%20Cotton | Émile Clément Cotton (5 February 1872 – 14 March 1950) was a professor of mathematics at the University of Grenoble. His PhD thesis studied differential geometry in three dimensions, with the introduction of the Cotton tensor. He held the professorship from 1904 until his 1942 retirement. He was the brother of Aimé Cotton.
References
Differential geometers
French mathematicians
1872 births
1950 deaths |
https://en.wikipedia.org/wiki/Symmetrizable%20compact%20operator | In mathematics, a symmetrizable compact operator is a compact operator on a Hilbert space that can be composed with a positive operator with trivial kernel to produce a self-adjoint operator. Such operators arose naturally in the work on integral operators of Hilbert, Korn, Lichtenstein and Marty required to solve elliptic boundary value problems on bounded domains in Euclidean space. Between the late 1940s and early 1960s the techniques, previously developed as part of classical potential theory, were abstracted within operator theory by various mathematicians, including M. G. Krein, William T. Reid, Peter Lax and Jean Dieudonné. Fredholm theory already implies that any element of the spectrum is an eigenvalue. The main results assert that the spectral theory of these operators is similar to that of compact self-adjoint operators: any spectral value is real; they form a sequence tending to zero; any generalized eigenvector is an eigenvector; and the eigenvectors span a dense subspace of the Hilbert space.
Discussion
Let H be a Hilbert space. A compact operator K on H is symmetrizable if there is a bounded self-adjoint operator S on H such that S is positive with trivial kernel, i.e. (Sx,x) > 0 for all non-zero x, and SK is self-adjoint:
In many applications S is also compact. The operator S defines a new inner product on H
Let HS be the Hilbert space completion of H with respect to this inner product.
The operator K defines a formally self-adjoint operator on the dense subspace H of HS. As Krein (1947) and noted, the operator has the same operator norm as K. In fact the self-adjointness condition implies
It follows by induction that, if (x,x)S = 1, then
Hence
If K is only compact, Krein gave an argument, invoking Fredholm theory, to show that K defines a compact operator on HS. A shorter argument is available if K belongs to a Schatten class.
When K is a Hilbert–Schmidt operator, the argument proceeds as follows. Let R be the unique positive square root of S and for ε > 0 define
These are self-adjoint Hilbert–Schmidt operator on H which are uniformly bounded in the Hilbert–Schmidt norm:
Since the Hilbert–Schmidt operators form a Hilbert space, there is a subsequence converging weakly to s self-adjoint Hilbert–Schmidt operator A. Since Aε R tends to RK in Hilbert–Schmidt norm, it follows that
Thus if U is the unitary induced by R between HS and H, then the operator KS induced by the restriction of K corresponds to A on H:
The operators K − λI and K* − λI are Fredholm operators of index 0 for λ ≠ 0, so any spectral value of K or K* is an eigenvalue and the corresponding eigenspaces are finite-dimensional. On the other hand, by the special theorem for compact operators, H is the orthogonal direct sum of the eigenspaces of A, all finite-dimensional except possibly for the 0 eigenspace. Since RA = K* R, the image under R of the λ eigenspace of A lies in the λ eigenspace of K*.
Similarly R carries the λ eigenspace of K into the λ ei |
https://en.wikipedia.org/wiki/Oakes%20Park%2C%20Sheffield | {
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https://en.wikipedia.org/wiki/Davy%20Brouwers | Davy Brouwers (born 3 February 1988) is a Belgian footballer who currently plays for Patro Eisden in the Belgian First Amateur Division as a right winger.
Statistics
External links
Voetbal International profile
1988 births
Living people
Belgian men's footballers
Belgian expatriate men's footballers
K. Patro Eisden Maasmechelen players
Lommel S.K. players
Helmond Sport players
MVV Maastricht players
K.S.V. Roeselare players
Eerste Divisie players
Challenger Pro League players
Belgian Third Division players
Men's association football midfielders
Expatriate men's footballers in the Netherlands
Belgian expatriate sportspeople in the Netherlands
K.V.V. Thes Sport Tessenderlo players
Footballers from Genk |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20the%20Republic%20of%20Ireland | This page details football records in the Republic of Ireland.
League
Team records
Titles
Most top-flight League titles: 20, Shamrock Rovers
Most consecutive League titles: 4,
Shamrock Rovers (1983-84 to 1986-87 and 2020 to 2023)
Top-flight Appearances
Most appearances: 92 seasons, Bohemians (1921–present)
Points records
Most points in a season: 87, Dundalk (2018)
Goals records
Most goals in a season: 85, Dundalk (2018)
Runs
Longest unbeaten run ????
(2021)
Featuring in European Football in consecutive seasons: 12, Shelbourne FC
Unbeaten seasons: 2, Shamrock Rovers
Individual records
League Goalscorers
235, Brendan Bradley
Most deliveries in a season
Kieran Sadlier 117
Cork City 2017
Most appearances
634 , Al Finucane
Cup
Team records
Most final wins: 25, Shamrock Rovers
Most appearances: 33, Shamrock Rovers
Most appearances without winning: 2, joint record:
Brideville (1927, 1930)
Dolphin (1932, 1933)
Cork Celtic (1964, 1969)
Most final loses: 11, Shelbourne
Highest scoring game: 4-3, joint record
Bohemians 4 - 3 Dundalk
Derry City 4 - 3 St. Patrick's Athletic (aet)
League Cup
Team records
Most final wins: 11, Derry City
Most appearances: 13, Derry City
Most appearances without winning: 3, Finn Harps
Total titles won
Gold - current winners of competition (were active).
References
Ireland
Association football in the Republic of Ireland |
https://en.wikipedia.org/wiki/Quantum%20scar | Quantum scarring refers to a phenomenon where the eigenstates of a classically chaotic quantum system have enhanced probability density around the paths of unstable classical periodic orbits. The instability of the periodic orbit is a decisive point that differentiates quantum scars from the more trivial observation that the probability density is enhanced in the neighborhood of stable periodic orbits. The latter can be understood as a purely classical phenomenon, a manifestation of the Bohr correspondence principle, whereas in the former, quantum interference is essential. As such, scarring is both a visual example of quantum-classical correspondence, and simultaneously an example of a (local) quantum suppression of chaos.
A classically chaotic system is also ergodic, and therefore (almost) all of its trajectories eventually explore evenly the entire accessible phase space. Thus, it would be natural to expect that the eigenstates of the quantum counterpart would fill the quantum phase space in the uniform manner up to random fluctuations in the semiclassical limit. However, scars are a significant correction to this assumption. Scars can therefore be considered as an eigenstate counterpart of how short periodic orbits provide corrections to the universal spectral statistics of the random matrix theory. There are rigorous mathematical theorems on quantum nature of ergodicity, proving that the expectation value of an operator converges in the semiclassical limit to the corresponding microcanonical classical average. Nonetheless, the quantum ergodicity theorems do not exclude scarring if the quantum phase space volume of the scars gradually vanishes in the semiclassical limit.
On the classical side, there is no direct analogue of scars. On the quantum side, they can be interpreted as an eigenstate analogy to how short periodic orbits correct the universal random matrix theory eigenvalue statistics. Scars correspond to nonergodic states which are permitted by the quantum ergodicity theorems. In particular, scarred states provide a striking visual counterexample to the assumption that the eigenstates of a classically chaotic system would be without structure. In addition to conventional quantum scars, the field of quantum scarring has undergone its renaissance period, sparked by the discoveries of perturbation-induced scars and many-body scars (see below).
Scar theory
The existence of scarred states is rather unexpected based on the Gutzwiller trace formula, which connects the quantum mechanical density of states to the periodic orbits in the corresponding classical system. According to the trace formula, a quantum spectrum is not a result of a trace over all the positions, but it is determined by a trace over all the periodic orbits only. Furthermore, every periodic orbit contributes to an eigenvalue, although not exactly equally. It is even more unlikely that a particular periodic orbit would stand out in contributing to a particular eigenstat |
https://en.wikipedia.org/wiki/Wall%27s%20finiteness%20obstruction | In geometric topology, a field within mathematics, the obstruction to a finitely dominated space X being homotopy-equivalent to a finite CW-complex is its Wall finiteness obstruction w(X) which is an element in the reduced zeroth algebraic K-theory of the integral group ring . It is named after the mathematician C. T. C. Wall.
By work of John Milnor on finitely dominated spaces, no generality is lost in letting X be a CW-complex. A finite domination of X is a finite CW-complex K together with maps and such that . By a construction due to Milnor it is possible to extend r to a homotopy equivalence where is a CW-complex obtained from K by attaching cells to kill the relative homotopy groups .
The space will be finite if all relative homotopy groups are finitely generated. Wall showed that this will be the case if and only if his finiteness obstruction vanishes. More precisely, using covering space theory and the Hurewicz theorem one can identify with . Wall then showed that the cellular chain complex is chain-homotopy equivalent to a chain complex of finite type of projective -modules, and that will be finitely generated if and only if these modules are stably-free. Stably-free modules vanish in reduced K-theory. This motivates the definition
.
See also
Algebraic K-theory
Whitehead torsion
References
.
.
Geometric topology
Algebraic K-theory
Surgery theory |
https://en.wikipedia.org/wiki/Finiteness%20theorem | In mathematics, there are several finiteness theorems.
Ahlfors finiteness theorem
Finiteness theorem for a proper morphism
Finiteness theorem for formal schemes |
https://en.wikipedia.org/wiki/Torsion%20sheaf | In mathematics, a torsion sheaf is a sheaf of abelian groups on a site for which, for every object U, the space of sections is a torsion abelian group. Similarly, for a prime number p, we say a sheaf is p-torsion if every section over any object is killed by a power of p.
A torsion sheaf on an étale site is the union of its constructible subsheaves.
See also
Twisted sheaf
Notes
References
J. S. Milne, Étale Cohomology
Sheaf theory |
https://en.wikipedia.org/wiki/Foundations%20of%20Differential%20Geometry | Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publishers. Both were published again in 1996 as Wiley Classics Library.
The first volume considers manifolds, fiber bundles, tensor analysis, connections in bundles, and the role of Lie groups. It also covers holonomy, the de Rham decomposition theorem and the Hopf–Rinow theorem. According to the review of James Eells, it has a "fine expositional style" and consists of a "special blend of algebraic, analytic, and geometric concepts". Eells says it is "essentially a textbook (even though there are no exercises)". An advanced text, it has a "pace geared to a [one] term graduate course".
The second volume considers submanifolds of Riemannian manifolds, the Gauss map, and the second fundamental form. It continues with geodesics on Riemannian manifolds, Jacobi fields, the Morse index, the Rauch comparison theorems, and the Cartan–Hadamard theorem. Then it ascends to complex manifolds, Kähler manifolds, homogeneous spaces, and symmetric spaces. In a discussion of curvature representation of characteristic classes of principal bundles (Chern–Weil theory), it covers Euler classes, Chern classes, and Pontryagin classes. The second volume also received a favorable review by J. Eells in Mathematical Reviews.
References
Mathematics textbooks
1963 non-fiction books
1969 non-fiction books |
https://en.wikipedia.org/wiki/Neumann%E2%80%93Poincar%C3%A9%20operator | In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space. Within the language of potential theory it reduces the partial differential equation to an integral equation on the boundary to which the theory of Fredholm operators can be applied. The theory is particularly simple in two dimensions—the case treated in detail in this article—where it is related to complex function theory, the conjugate Beurling transform or complex Hilbert transform and the Fredholm eigenvalues of bounded planar domains.
Dirichlet and Neumann problems
Green's theorem for a bounded region Ω in the plane with smooth boundary ∂Ω states that
One direct way to prove this is as follows. By subtraction, it is sufficient to prove the theorem for a region bounded by a simple smooth curve. Any such is diffeomorphic to the closed unit disk. By change of variables it is enough to prove the result there. Separating the A and B terms, the right hand side can be written as a double integral starting in the x or y direction, to which the fundamental theorem of calculus can be applied. This converts the integral over the disk into the integral over its boundary.
Let Ω be a region bounded by a simple closed curve. Given a smooth function f on the closure of Ω its normal derivative ∂nf at a boundary point is the directional derivative in the direction of the outward pointing normal vector.
Applying Green's theorem with A = vx u and B = vy u gives the first of Green's identities:
where the Laplacian Δ is given by
Swapping u and v and subtracting gives the second of Green's identities:
If now u is harmonic in Ω and v = 1, then this identity implies that
so the integral of the normal derivative of a harmonic function on the boundary of a region always vanishes.
A similar argument shows that the average of a harmonic function on the boundary of a disk equals its value at the centre. Translating the disk can be taken to be centred at 0. Green's identity can be applied to an annulus formed of the boundary of the disk and a small circle centred on 0 with v = z2: it follows that the average is independent of the circle. It tends to the value at its value at 0 as the radius of the smaller circle decreases. This result also follows easily using Fourier series and the Poisson integral.
For continuous functions f on the whole plane which are smooth in Ω and the complementary region Ωc, the first derivative can have a jump across the boundary of Ω. The value of the normal derivative at a boundary point can be computed from inside or outside Ω. The interior normal derivative will be denoted by ∂n− and the exterior normal derivative by ∂n+. With this terminology the four basic problems of classical potential theory are as follows:
Interior Dirichlet problem: ∆u = 0 in Ω, u = f on ∂ |
https://en.wikipedia.org/wiki/Locally%20acyclic%20morphism | In algebraic geometry, a morphism of schemes is said to be locally acyclic if, roughly, any sheaf on S and its restriction to X through f have the same étale cohomology, locally. For example, a smooth morphism is universally locally acyclic.
References
.
Morphisms of schemes |
https://en.wikipedia.org/wiki/Rayleigh%20mixture%20distribution | In probability theory and statistics a Rayleigh mixture distribution is a weighted mixture of multiple probability distributions where the weightings are equal to the weightings of a Rayleigh distribution. Since the probability density function for a (standard) Rayleigh distribution is given by
Rayleigh mixture distributions have probability density functions of the form
where is a well-defined probability density function or sampling distribution.
The Rayleigh mixture distribution is one of many types of compound distributions in which the appearance of a value in a sample or population might be interpreted as a function of other underlying random variables. Mixture distributions are often used in mixture models, which are used to express probabilities of sub-populations within a larger population.
See also
Mixture distribution
List of probability distributions
References
Continuous distributions
Compound probability distributions |
https://en.wikipedia.org/wiki/Jeremy%20Crowe | Jeremy Crowe (born 24 July 1985) is an Australian 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
Australian motorcycle racers
Living people
1985 births
125cc World Championship riders
Supersport World Championship riders |
https://en.wikipedia.org/wiki/Brendan%20Clarke%20%28motorcyclist%29 | Brendan Clarke is a Grand Prix motorcycle racer from Australia.
Career statistics
By season
References
External links
Profile on motogp.com
Motorcycle racers from Brisbane
Living people
1984 births
500cc World Championship riders
Supersport World Championship riders |
https://en.wikipedia.org/wiki/Paul%20Young%20%28motorcyclist%29 | Paul Young is a former Grand Prix motorcycle racer from Australia. He worked for and raced Triumph Motorcycles.
Career statistics
By season
References
External links
Profile on motogp.com
Australian motorcycle racers
Living people
1969 births
500cc World Championship riders
Supersport World Championship riders |
https://en.wikipedia.org/wiki/Craig%20Connell | Craig Connell is a Grand Prix motorcycle racer from Australia.
Career statistics
By season
References
External links
Profile on motogp.com
Australian motorcycle racers
Living people
1968 births
250cc World Championship riders
500cc World Championship riders
Superbike World Championship riders |
https://en.wikipedia.org/wiki/Exact%20C%2A-algebra | In mathematics, an exact C*-algebra is a C*-algebra that preserves exact sequences under the minimum tensor product.
Definition
A C*-algebra E is exact if, for any short exact sequence,
the sequence
where ⊗min denotes the minimum tensor product, is also exact.
Properties
Every nuclear C*-algebra is exact.
Every sub-C*-algebra and every quotient of an exact C*-algebra is exact. An extension of exact C*-algebras is not exact in general.
It follows that every sub-C*-algebra of a nuclear C*-algebra is exact.
Characterizations
Exact C*-algebras have the following equivalent characterizations:
A C*-algebra A is exact if and only if A is nuclearly embeddable into B(H), the C*-algebra of all bounded operators on a Hilbert space H.
A C*-algebra is exact if and only if every separable sub-C*-algebra is exact.
A separable C*-algebra A is exact if and only if it is isomorphic to a subalgebra of the Cuntz algebra .
References
C*-algebras |
https://en.wikipedia.org/wiki/List%20of%20National%20Basketball%20Association%20annual%20minutes%20leaders | In basketball, minutes of game time during which a player is on the court are recorded. The minutes played statistics are recorded as far back as the 1951–52 season when statistics on minutes were first compiled by the National Basketball Association (NBA). Fifteen times the average leader has played fewer than 40 minutes per game and eight times the leader has played more than 46 minutes per game. Wilt Chamberlain has the seven highest leading totals, while Nate "Tiny" Archibald is the only other single-season leader to average over 46 minutes per game. In one season, Chamberlain averaged over 48 minutes per game (meaning that he rested fewer minutes during the season than he played in overtime during the season).
To qualify as minutes leader, the player must appear in at least 58 games (out of 82). However, a player who appears in fewer than 58 games may qualify as annual minutes leader if his minute total would have given him the greatest average, had he appeared in 58 games. This has been the requirement since the 2013–14 season.
Chamberlain led the league in minutes played per game nine times, followed by Allen Iverson (7) and Michael Finley (3) times. Twelve other players have led the league in minutes per game twice, eight of them in consecutive years. Chamberlain holds the record for consecutive titles with five, followed by Iverson with three (two times). Fifteen times a member of the San Francisco/Philadelphia/Golden State Warriors has led the league in average minutes. Nine full seasons and parts of another the average leader played for the Philadelphia 76ers. Seventeen seasons and parts of another the leader played for either the Philadelphia Warriors or the Philadelphia 76ers.
In 16 of the 61 seasons since the statistic has been kept, the minutes per game leader was not the total minutes played leader. Larry Bird and Iverson are the only multiple leaders in average who were surpassed in total minutes multiple times. Five of Iverson's seven times and both of Bird's times as the average leader they were surpassed in total minutes. Kevin Durant is the only player to lead the league in total minutes without finishing in the top three in average minutes during the same season. He is also the only player to never lead in average minutes, but lead in total minutes multiple times. The first four times and five of the first seven times that the leader in average did not lead in total minutes, he finished second in total minutes. However, the last six times that the average leader did not lead the league in total minutes, he was outside of the top 10 and the last nine times, he was outside of the top 5.
Key
Annual leaders
Multiple-time leaders
Notes
References
General
Specific
National Basketball Association lists
National Basketball Association statistical leaders |
https://en.wikipedia.org/wiki/Tau%20%281/3%29 | Tau is a public artwork by American artist Tony Smith, located on the urban campus of Hunter College, in New York City, New York, United States. Fascinated by mathematics, biology and crystals, Smith designed Tau with geometry at its root.
Description
American artist Tony Smith created Tau (1/3) in 1961–62. Its title refers to the Greek letter 'T', which also describes the shape of the sculpture. Intended as one of three limited editions, its sister, Tau (AP), was fabricated at Lippincott Foundry, and installed in 2006 at Meadowlands Park, in his hometown of South Orange.
Many of Smith's sculptures were made up of a space lattice: groupings of simple platonic solids, in Tau'''s case two such solids, octahedrons and tetrahedrons. The original model for the sculpture was created by Smith in 1961-62 using his signature process of joining small cardboard tetrahedrons, a process he began while recuperating after a severe automobile accident in the spring of 1961.
During this period, Smith was transitioning from his 20-year career in architecture to focus on painting and making sculptures. Smith had also started teaching at Hunter College, New York, in 1962. It took over 20 years for the piece to be installed outside the upper east side college at the 6 train's 68th Street entrance in 1984. In 2004, Hunter College held an exhibition, "Tracing Tau", curated by William C. Agee that offered an insight into the sculpture and its beginnings through paper models, drawings and plans of the work.
Historical informationTau forms part of Smith's series of cast bronze and painted steel sculptures including Amaryllis (1965) and The Snake Is Out (1962), all evolution of his first titled sculpture, Throne (1956–57). Though Tau'' is one of Smith's less publicized works, it is part of a body of work inspired by his oft-cited, revelatory road trip to the unfinished New Jersey Turnpike in the early 1950s. "When I was teaching at Cooper Union in the first year or two of the '50s, someone told me how I could get on to the unfinished New Jersey Turnpike. I took three students and drove from somewhere in the Meadows to New Brunswick. It was a dark night and there were no lights or shoulder markers, lines, railings or anything at all except the dark pavement moving through the landscape of the flats, rimmed by hills in the distance, but punctuated by stacks, towers, fumes and colored lights. This drive was a revealing experience. The road and much of the landscape was artificial, and yet it couldn't be called a work of art. On the other hand, it did something for me that art had never done. At first I didn't know what it was, but its effect was to liberate me from many of the views I had had about art. It seemed that there had been a reality there which had not had any expression in art."
"The experience on the road was something mapped out but not socially recognized. I thought to myself, it ought to be clear that's the end of art. Most paintings look pretty pictorial |
https://en.wikipedia.org/wiki/Position%20and%20momentum%20spaces | In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with units of [mass][length][time]−1.
Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function.
These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Another quantity is useful to define in the context of waves. The wave vector k (or simply "k-vector") has dimensions of reciprocal length, making it an analogue of angular frequency ω which has dimensions of reciprocal time. The set of all wave vectors is k-space. Usually r is more intuitive and simpler than k, though the converse can also be true, such as in solid-state physics.
Quantum mechanics provides two fundamental examples of the duality between position and momentum, the Heisenberg uncertainty principle ΔxΔp ≥ ħ/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the de Broglie relation p = ħk which states the momentum and wavevector of a free particle are proportional to each other. In this context, when it is unambiguous, the terms "momentum" and "wavevector" are used interchangeably. However, the de Broglie relation is not true in a crystal.
Position and momentum spaces in classical mechanics
Lagrangian mechanics
Most often in Lagrangian mechanics, the Lagrangian L(q, dq/dt, t) is in configuration space, where q = (q1, q2,..., qn) is an n-tuple of the generalized coordinates. The Euler–Lagrange equations of motion are
(One overdot indicates one time derivative). Introducing the definition of canonical momentum for each generalized coordinate
the Euler–Lagrange equations take the form
The Lagrangian can be expressed in momentum space also, L′(p, dp/dt, t), where p = (p1, p2, ..., pn) is an n-tuple of the generalized momenta. A Legendre transformation is performed to change the variables in the total differential of the generalized coordinate space Lagrangian;
where the definition of generalized momentum and Euler–Lagrange equations have replaced the partial derivatives o |
https://en.wikipedia.org/wiki/Michael%20J.%20Larsen | Michael Jeffrey Larsen is an American mathematician, a distinguished professor of mathematics at Indiana University Bloomington.
Academic biography
In high school, Larsen tied with four other competitors for the top score in the 1977 International Mathematical Olympiad in Belgrade, winning a gold medal. As an undergraduate mathematics student at Harvard University, Larsen became a Putnam Fellow in 1981 and 1983. He graduated from Harvard in 1984, and earned his Ph.D. from Princeton University in 1988, under the supervision of Gerd Faltings. After working at the Institute for Advanced Study he joined the faculty of the University of Pennsylvania in 1990, and then moved to the University of Missouri in 1997. He joined the Indiana University faculty in 2001.
His wife Ayelet Lindenstrauss is also a mathematician and Indiana University professor. Their son Daniel at age 13 became the youngest person to publish a crossword in the New York Times.
Research
Larsen is known for his research in arithmetic algebraic geometry, combinatorial group theory, combinatorics, and number theory. He has written highly cited papers on domino tiling of Aztec diamonds, topological quantum computing, and on the representation theory of braid groups.
Awards and honors
In 2013 he became a fellow of the American Mathematical Society, for "contributions to group theory, number theory, topology, and algebraic geometry".
He received the E. H. Moore Research Article Prize of the AMS in 2013 (jointly with Richard Pink).
Selected publications
.
.
.
.
References
External links
Home page at Indiana University
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
Putnam Fellows
Harvard University alumni
Princeton University alumni
University of Pennsylvania faculty
University of Missouri faculty
Indiana University faculty
Fellows of the American Mathematical Society
International Mathematical Olympiad participants |
https://en.wikipedia.org/wiki/Matrix%20splitting | In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960.
Regular splittings
We seek to solve the matrix equation
where A is a given n × n non-singular matrix, and k is a given column vector with n components. We split the matrix A into
where B and C are n × n matrices. If, for an arbitrary n × n matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative entries, we write M1 ≥ M2.
Definition: A = B − C is a regular splitting of A if B−1 ≥ 0 and C ≥ 0.
We assume that matrix equations of the form
where g is a given column vector, can be solved directly for the vector x. If () represents a regular splitting of A, then the iterative method
where x(0) is an arbitrary vector, can be carried out. Equivalently, we write () in the form
The matrix D = B−1C has nonnegative entries if () represents a regular splitting of A.
It can be shown that if A−1 > 0, then < 1, where represents the spectral radius of D, and thus D is a convergent matrix. As a consequence, the iterative method () is necessarily convergent.
If, in addition, the splitting () is chosen so that the matrix B is a diagonal matrix (with the diagonal entries all non-zero, since B must be invertible), then B can be inverted in linear time (see Time complexity).
Matrix iterative methods
Many iterative methods can be described as a matrix splitting. If the diagonal entries of the matrix A are all nonzero, and we express the matrix A as the matrix sum
where D is the diagonal part of A, and U and L are respectively strictly upper and lower triangular n × n matrices, then we have the following.
The Jacobi method can be represented in matrix form as a splitting
The Gauss–Seidel method can be represented in matrix form as a splitting
The method of successive over-relaxation can be represented in matrix form as a splitting
Example
Regular splitting
In equation (), let
Let us apply the splitting () which is used in the Jacobi method: we split A in such a way that B consists of all of the diagonal elements of A, and C consists of all of the off-diagonal elements of A, negated. (Of course this is not the only useful way to split a matrix into two matrices.) We have
Since B−1 ≥ 0 and C ≥ 0, the splitting () is a regular splitting. Since A−1 > 0, the spectral radius < 1. (The approximate eigenvalues of D are ) Hence, the matrix D is convergent and the method () necessarily converges for the problem (). Note that the diagonal elements |
https://en.wikipedia.org/wiki/Kosta%20Runjai%C4%87 | Kosta Runjaić (born 4 June 1971) is a German football manager of Serbian descent who serves as the manager of the Polish side Legia Warsaw.
Managerial statistics
Honours
Manager
Legia Warsaw
Polish Cup: 2022–23
Individual
Ekstraklasa Coach of the Month: July 2019, December 2020, August 2022
References
External links
1971 births
Living people
Footballers from Vienna
Austrian emigrants to Germany
Naturalized citizens of Germany
German people of Croatian descent
Austrian people of Croatian descent
Men's association football players not categorized by position
German men's footballers
Austrian men's footballers
FSV Frankfurt players
German football managers
Austrian football managers
VfR Aalen managers
SV Darmstadt 98 managers
MSV Duisburg managers
1. FC Kaiserslautern managers
TSV 1860 Munich managers
Pogoń Szczecin managers
Legia Warsaw managers
2. Bundesliga managers
3. Liga managers
Ekstraklasa managers
German expatriate football managers
Austrian expatriate football managers
Expatriate football managers in Poland
German expatriate sportspeople in Poland
Austrian expatriate sportspeople in Poland
Sportspeople from Darmstadt (region) |
https://en.wikipedia.org/wiki/Riemann%27s%20minimal%20surface | In differential geometry, Riemann's minimal surface is a one-parameter family of minimal surfaces described by Bernhard Riemann in a posthumous paper published in 1867. Surfaces in the family are singly periodic minimal surfaces with an infinite number of ends asymptotic to parallel planes, each plane "shelf" connected with catenoid-like bridges to the neighbouring ones. Their intersections with horizontal planes are circles or lines; Riemann proved that they were the only minimal surfaces fibered by circles in parallel planes besides the catenoid, helicoid and plane. They are also the only nontrivial embedded minimal surfaces in Euclidean 3-space invariant under the group generated by a nontrivial translation. It is possible to attach extra handles to the surfaces, producing higher-genus minimal surface families.
References
External links
http://www.math.indiana.edu/gallery/minimalSurface.phtml
http://www.indiana.edu/~minimal/essays/riemann/index.html
http://virtualmathmuseum.org/Surface/riemann/riemann.html
Differential geometry
Minimal surfaces
Bernhard Riemann |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Russia | This page details football records in Russia.
Team records
Most championships won
Overall
10, Spartak Moscow (1992, 1993, 1994, 1996, 1997, 1998, 1999, 2000, 2001, 2017)
Consecutives
6, Spartak Moscow (1996–2001)
Highest points total
72, Zenit St. Petersburg (2020), Spartak Moscow (1999).
Most seasons in Russian Premier League
24, CSKA Moscow
24, Lokomotiv Moscow
24, Spartak Moscow
23, Dynamo Moscow
23, Krylia Sovetov Samara
Individual records
League Appearances
549, Igor Akinfeev
League Goalscorers
158, Artem Dzyuba
Most successful clubs overall
References
Russia
Football in Russia |
https://en.wikipedia.org/wiki/NASU%20Institute%20of%20Mathematics | Institute of Mathematics of the National Academy of Sciences of Ukraine () is a government-owned research institute in Ukraine that carries out basic research and trains highly qualified professionals in the field of mathematics. It was founded on 13 February 1934.
Notable research results
The perturbation theory of toroidal invariant manifolds of dynamical systems was developed here by academician M. M. Bogolyubov, Yu. O. Mitropolsky, academician of the NAS of Ukraine and the Russian Academy of Sciences, and A. M. Samoilenko, academician of the NAS of Ukraine. The theory's methods are used to investigate oscillation processes in broad classes of applied problems, in particular, the phenomena of passing through resonance and various bifurcations and synchronizations.
Sharkovsky's order theorem was devised by its author while he worked for the institute. It became the basis for the theory of one-dimensional dynamical systems that enabled the study of chaotic evolutions in deterministic systems, and, in particular, of ‘ideal turbulence’.
The school of the NAS academician Yu. M. Berezansky constructed the theory of generalized functions of infinitely many variables on the basis of spectral approach and operators of generalized translation.
The school of the NAS academician A. V. Skorokhod investigated a broad range of problems related to random processes and stochastic differential equations.
Heuristic methods of phase lumping of complex systems were validated, important results in queuing theory and reliability theory were obtained, and a series of limit theorems for semi-Markov processes were proved by V. S. Korolyuk, academician of the NAS of Ukraine. He has also constructed the Poisson approximation for stochastic homogeneous additive functional with semi-Markov switching.
Directors
1934 — 1939 Dmitry Grave
1939 — 1941 Mikhail Lavrentyev
1941 — 1944 Yurii Pfeiffer, united institute of mathematics and physics
1944 — 1948 Mikhail Lavrentyev
1948 — 1955 Aleksandr Ishlinskiy
1955 — 1958 Boris Gnedenko
1958 — 1988 Yurii Mitropolskiy
1988 — 2020 Anatoly Samoilenko
2021 — Alexander Timokha
Scientific departments
Algebra
Analytical mechanics
Applied researches
Approximation theory
Complex analysis and potential theory
Differential equations and oscillation theory
Dynamics and stability of multi-dimensional systems
Fractal Analysis
Functional Analysis
Mathematical physics
Nonlinear analysis
Numerical mathematics
Partial differential equations
Theory of dynamical systems
Theory of functions
Theory of random processes
Topology
Publications
The Institute publishes several scientific journals:
Methods of Functional Analysis and Topology
Nonlinear Oscillations
Symmetry, integrability and Geometry: Methods and Applications (SIGMA)
Ukrainian Mathematical Journal
References
External links
Official Facebook Page
Page on Math-Net.Ru
Institutes of the National Academy of Sciences of Ukraine
Research institutes in the |
https://en.wikipedia.org/wiki/Zacatelco | Zacatelco () is a city and capital of Zacatelco municipality located south of the state of Tlaxcala. According to the population census conducted by the National Institute of Statistics and Geography 2010, the city has a population of 38.466 people, it is the sixth most populous city in the state and is part of the Metropolitan area of Puebla. The city is also head of the third electoral district of Tlaxcala.
The city was founded on December 1, 1529, by Agustín de Castañeda. In 1723 it is very important because it forms Zacatelco republic; this is achieved by joining the towns of San Juan, San Lorenzo, Santo Toribio, Santa Catarina, San Marcos and San Antonio, which depended on Tepeyanco. The most important historical figures of the city were the brothers Arenas, prominent revolutionaries. Domingo Arenas took the first land committee of Mexico in 1915, was one reason for coining the phrase battle: Zacatelco; the heart of south.
Zacatelco is located in the southern part of the valley of Tlaxcala, on the border with the state of Puebla, is at an altitude of 2,210 meters above sea level, making it one of the highest cities in Mexico. It is located just 11.9 km from the state capital, 27.1 km from the city of Puebla and 121.8 km from Mexico City.
History
Prehispanic Period
The first human settlements discovered in Zacatelco correspond to culture or cultural Tlatempa phase, between 1200 B. C. and 800 B. C. These settlements are a scattered village and a town, probably the same Zacatelco, where high architectural structure in the boundary is observed with the present municipality of Tepeyanco.
Between 800 and 350 B. C., the Texoloc phase and develops according to the mapping done by the Puebla-Tlaxcala Archaeological Project (PAPT), in the town of Zacatelco a large town or city is located adjacent to Cholula. For Tezoquipan phase, in the area now occupied by the town of Zacatelco, you can locate a town bordering Cholula, also it is known that there was contact with two villages and a people more than the municipalities of Tepeyanco and Tetlatlahuca. Although there is no evidence of the existence of sites in what is now Zacatelco, if it is known that during this phase was under the influence of Cholula and stay there apparently.
Despite this, most historians refer only to the four domains that form the Republic of Tlaxcala and the arrival of the Spaniards their chiefs are the lords of Tlaxcala. According to this division, Zacatelco belonged to the lordship of Ocotelulco until the arrival of Hispanics on tlaxcaltecas land
Modern Age
On January 18, General Oscar Aguilar returned to the state capital, from Zacatelco; where he disarmed a group that had taken up arms in order to prevent Ignacio Mendoza assumed the governorship of the state. The general Aguilar could seize 56 firearms political sides of Zacatelco; several members however were the state of Puebla. To ensure order in the region, a federal garrison of 40 men who avoided the more serious |
https://en.wikipedia.org/wiki/Mil%C3%A1n%20Kal%C3%A1sz | Milán Kalász (born 30 April 1992 in Ajka) is a Hungarian football player. He is currently a free agent.
Club statistics
Updated to games played as of 22 September 2013.
References
Haladas FC
Illes Academia
1992 births
Living people
People from Ajka
Hungarian men's footballers
Men's association football midfielders
Szombathelyi Haladás footballers
Nemzeti Bajnokság I players
Footballers from Veszprém County |
https://en.wikipedia.org/wiki/Merkuryev | Merkuryev () is a Russian surname.
Alexander Merkurjev (born 1955) is a Russian-born American mathematician, who has made major contributions to the field of algebra;
Andrey Merkurjev (born 1977) is a Russian ballet dancer;
Evgeny Merkurjev (1936—2007) was a Soviet and Russian theatre actor, worked in the Na Liteinom Theatre;
Pyotr Merkuryev (1943–2010) was a Soviet and Russian actor;
Stanislav Merkuryev (1945—1993) was a Soviet and Russian mathematical physicist;
Vasili Merkuryev (1904–1978) was a Soviet actor, he appeared in 46 films between 1935 and 1974;
Russian-language surnames |
https://en.wikipedia.org/wiki/Rigidity%20matroid | In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In a rigidity matroid for a graph with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph.
Definition
A framework is an undirected graph, embedded into d-dimensional Euclidean space by providing a d-tuple of Cartesian coordinates for each vertex of the graph. From a framework with n vertices and m edges, one can define a matrix with m rows and nd columns, an expanded version of the incidence matrix of the graph called the rigidity matrix. In this matrix, the entry in row e and column (v,i) is zero if v is not an endpoint of edge e. If, on the other hand, edge e has vertices u and v as endpoints, then the value of the entry is the difference between the ith coordinates of v and u.
The rigidity matroid of the given framework is a linear matroid that has as its elements the edges of the graph. A set of edges is independent, in the matroid, if it corresponds to a set of rows of the rigidity matrix that is linearly independent. A framework is called generic if the coordinates of its vertices are algebraically independent real numbers. Any two generic frameworks on the same graph G determine the same rigidity matroid, regardless of their specific coordinates. This is the (d-dimensional) rigidity matroid of G.
Statics
A load on a framework is a system of forces on the vertices (represented as vectors). A stress is a special case of a load, in which equal and opposite forces are applied to the two endpoints of each edge (which may be imagined as a spring) and the forces formed in this way are added at each vertex. Every stress is an equilibrium load, a load that does not impose any translational force on the whole system (the sum of its force vectors is zero) nor any rotational force. A linear dependence among the rows of the rigidity matrix may be represented as a self-stress, an assignment of equal and opposite forces to the endpoints of each edge that is not identically zero but that adds to zero at every vertex. Thus, a set of edges forms an independent set in the rigidity matroid if and only if it has no self-stress.
The vector space of all possible loads, on a system of n vertices, has dimension dn, among which the equilibrium loads form a subspace of dimension
. An independent set in the rigidity matroid has a system of equilibrium loads whose dimension equals the cardinality of the set, so the maximum rank that any set in the matroid can have is . If a set has this rank, it follows that its set of stresses is the same as the space of equilibrium loads. Alternatively and equivalently, in this case eve |
https://en.wikipedia.org/wiki/Behrens%E2%80%93Fisher%20distribution | In statistics, the Behrens–Fisher distribution, named after Ronald Fisher and Walter Behrens, is a parameterized family of probability distributions arising from the solution of the Behrens–Fisher problem proposed first by Behrens and several years later by Fisher. The Behrens–Fisher problem is that of statistical inference concerning the difference between the means of two normally distributed populations when the ratio of their variances is not known (and in particular, it is not known that their variances are equal).
Definition
The Behrens–Fisher distribution is the distribution of a random variable of the form
where T1 and T2 are independent random variables each with a Student's t-distribution, with respective degrees of freedom ν1 = n1 − 1 and ν2 = n2 − 1, and θ is a constant. Thus the family of Behrens–Fisher distributions is parametrized by ν1, ν2, and θ.
Derivation
Suppose it were known that the two population variances are equal, and samples of sizes n1 and n2 are taken from the two populations:
where "i.i.d" are independent and identically distributed random variables and N denotes the normal distribution. The two sample means are
The usual "pooled" unbiased estimate of the common variance σ2 is then
where S12 and S22 are the usual unbiased (Bessel-corrected) estimates of the two population variances.
Under these assumptions, the pivotal quantity
has a t-distribution with n1 + n2 − 2 degrees of freedom. Accordingly, one can find a confidence interval for μ2 − μ1 whose endpoints are
where A is an appropriate quantile of the t-distribution.
However, in the Behrens–Fisher problem, the two population variances are not known to be equal, nor is their ratio known. Fisher considered the pivotal quantity
This can be written as
where
are the usual one-sample t-statistics and
and one takes θ to be in the first quadrant. The algebraic details are as follows:
The fact that the sum of the squares of the expressions in parentheses above is 1 implies that they are the squared cosine and squared sine of some angle.
The Behren–Fisher distribution is actually the conditional distribution of the quantity (1) above, given the values of the quantities labeled cos θ and sin θ. In effect, Fisher conditions on ancillary information.
Fisher then found the "fiducial interval" whose endpoints are
where A is the appropriate percentage point of the Behrens–Fisher distribution. Fisher claimed that the probability that μ2 − μ1 is in this interval, given the data (ultimately the Xs) is the probability that a Behrens–Fisher-distributed random variable is between −A and A.
Fiducial intervals versus confidence intervals
Bartlett showed that this "fiducial interval" is not a confidence interval because it does not have a constant coverage rate. Fisher did not consider that a cogent objection to the use of the fiducial interval.
Further reading
Kendall, Maurice G., Stuart, Alan (1973) The Advanced Theo |
https://en.wikipedia.org/wiki/Zero%20order | Zero order reaction
Zero-order process (statistics), a sequence of random variables, each independent of the previous ones
Zero order process (chemistry), a chemical reaction in which the rate of change of concentration is independent of the concentrations
Zeroth-order approximation, an approximation of a function by a constant
Zeroth-order logic, a form of logic without quantifiers
See also
Zero (disambiguation)
0O (disambiguation) |
https://en.wikipedia.org/wiki/Outline%20of%20forgery | The following outline is provided as an overview and topical guide to forgery:
Forgery – process of making, adapting, or imitating objects, statistics, or documents with the intent to deceive.
Types of forgery
Archaeological forgery
Art forgery
Black propaganda — false information and material that purports to be from a source on one side of a conflict, but is actually from the opposing side
Counterfeiting
Counterfeit money — types of counterfeit coins include the cliché forgery, the fourrée and the slug
Counterfeit consumer goods
Counterfeit medication
Counterfeit watches
Unapproved aircraft parts
Watered stock
False documents
Forgery as a covert operation
Identity document forgery
Fake passport
Literary forgery
Fake memoirs
Pseudopigraphy — the false attribution of a work, not always as an act of forgery
Musical forgery — music allegedly written by composers of past eras, but actually composed later by someone else
Philatelic forgery — fake stamps produced to defraud stamp collectors
Signature forgery
Legality of forgery
Kenya
Forgery of Foreign Bills Act 1803
Forgery Act 1830
Forgery, Abolition of Punishment of Death Act 1832
Forgery Act 1837
Forgery Act 1861
Forgery Act 1870
Forgery Act 1913
Forgery and Counterfeiting Act 1981
International
Anti-Counterfeiting Trade Agreement
Council of Europe Convention on the Counterfeiting of Medical Products
Related offences
Phishing — impersonating a reputable organization via electronic media, which often involves creating a replica of a trustworthy website
Uttering — knowingly passing on a forgery with the intent to defraud
Detection and prevention of forgery
Anti-counterfeiting agencies and organisations
Authentics Foundation — an international non-governmental organization that raises public awareness of counterfeits
Central Bank Counterfeit Deterrence Group — an international group of central banks that investigates emerging threats to the security of banknotes
Counterfeit Coin Bulletin — a now-defunct publication of the American Numismatic Association
Alliance Against Counterfeit Spirits — the trade association for the worldwide spirit industry's protection against counterfeit produce
Philatelic Foundation — a major source of authentication for rare and valuable postage stamps
United States Secret Service — the agency responsible for the prevention and investigation of counterfeit U.S. currency
Verified-Accredited Wholesale Distributors — a program that offers accreditation to wholesale pharmaceutical distribution facilities
Tools and techniques
Authentication — the act of confirming the truth of an attribute of a single piece of data claimed to be true by an entity.
Counterfeit banknote detection pen — uses an iodine-based ink that reacts with the starch found in counterfeit banknotes
EURion constellation — a pattern of symbols incorporated into banknote designs, which can be detected by imaging software
Geometric lathe — a 19th-century lathe use |
https://en.wikipedia.org/wiki/Schwarz%20minimal%20surface | In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.
In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces. They were later named by Alan Schoen in his seminal report that described the gyroid and other triply periodic minimal surfaces.
The surfaces were generated using symmetry arguments: given a solution to Plateau's problem for a polygon, reflections of the surface across the boundary lines also produce valid minimal surfaces that can be continuously joined to the original solution. If a minimal surface meets a plane at right angles, then the mirror image in the plane can also be joined to the surface. Hence given a suitable initial polygon inscribed in a unit cell periodic surfaces can be constructed.
The Schwarz surfaces have topological genus 3, the minimal genus of triply periodic minimal surfaces.
They have been considered as models for periodic nanostructures in block copolymers, electrostatic equipotential surfaces in crystals, and hypothetical negatively curved graphite phases.
Schwarz P ("Primitive")
Schoen named this surface 'primitive' because it has two intertwined congruent labyrinths, each with the shape of an inflated tubular version of the simple cubic lattice. While the standard P surface has cubic symmetry the unit cell can be any rectangular box, producing a family of minimal surfaces with the same topology.
It can be approximated by the implicit surface
.
The P surface has been considered for prototyping tissue scaffolds with a high surface-to-volume ratio and porosity.
Schwarz D ("Diamond")
Schoen named this surface 'diamond' because it has two intertwined congruent labyrinths, each having the shape of an inflated tubular version of the diamond bond structure. It is sometimes called the F surface in the literature.
It can be approximated by the implicit surface
.
An exact expression exists in terms of elliptic integrals, based on the Weierstrass representation.
Schwarz H ("Hexagonal")
The H surface is similar to a catenoid with a triangular boundary, allowing it to tile space.
Schwarz CLP ("Crossed layers of parallels")
Illustrations
Susquehanna University - Triply Periodic Minimal Surfaces (Archived)
Indiana University - Triply Periodic Surfaces of Genus 3 (Archived)
Ruprecht-Karls-Universität Heidelberg - Bicontinuous cubic phases based on triply periodic minimal surfaces
Université libre de Bruxelles - Schwartz's Surface (Archived)
Virtual Math Museum - 3DXM Minimal Surface Gallery
References
Differential geometry
Minimal surfaces |
https://en.wikipedia.org/wiki/Saddle%20tower | In differential geometry, a saddle tower is a minimal surface family generalizing the singly periodic Scherk's second surface so that it has N-fold (N > 2) symmetry around one axis.
These surfaces are the only properly embedded singly periodic minimal surfaces in R3 with genus zero and finitely many Scherk-type ends in the quotient.
References
External links
Images of The Saddle Tower Surface Families
Minimal surfaces |
https://en.wikipedia.org/wiki/Tom%20Sanders%20%28mathematician%29 | Tom Sanders is an English mathematician, working on problems in additive combinatorics at the interface of harmonic analysis and analytic number theory.
Education
Sanders studied mathematics at the University of Cambridge, where he was awarded a PhD in 2007 for research on arithmetic combinatorics supervised by Timothy Gowers.
Career and research
He held a Junior Research Fellowship at Christ's College, Cambridge from 2006 until 2011, in addition to visiting fellowships at the Institute for Advanced Study in 2007, the MSRI in 2008, and the Mittag-Leffler Institute in 2009. Since 2011, he has held a Royal Society University Research Fellowship (URF) at the University of Oxford, where he is also a senior research fellow at the Mathematical Institute, and a Tutorial Fellow at St Hugh's College, Oxford.
Among other results, he has improved the theorem of Klaus Friedrich Roth on three-term arithmetic progressions, coming close to breaking the so-called logarithmic barrier. More precisely, he has shown that any subset of {1, 2, ..., N} of maximal cardinality containing no non-trivial three-term arithmetic progression is of size
Awards and honours
In February 2011, he was awarded the Adams Prize (jointly with Harald Helfgott) for having "employed deep harmonic analysis to understand arithmetic progressions and answer long-standing conjectures in number theory". In July 2012, he was awarded a Prize of the European Mathematical Society for his "fundamental results in additive combinatorics and harmonic analysis, which combine in a masterful way deep known techniques with the invention of new methods to achieve spectacular results." In July 2013, he was awarded the Whitehead Prize of the London Mathematical Society for his "spectacular results in additive combinatorics and related areas", in particular "for his paper obtaining the best known upper bounds for sets of integers containing no 3-term arithmetic progressions, for his work dramatically improving bounds connected with Freiman's theorem on sets with small doubling, and for other results in additive combinatorics and harmonic analysis." In September 2013, he was awarded the European Prize in Combinatorics. Although Sanders was known for improving the theorem of Klaus Friedrich Roth on three-term arithmetic progressions, in late 2013 he was awarded the foundation of Alaskan Ice Fishermans Gauntlet for eating the most snow crab legs in the last twenty years.
References
Year of birth missing (living people)
Living people
Place of birth missing (living people)
Alumni of the University of Cambridge
21st-century English mathematicians
Fellows of St Hugh's College, Oxford |
https://en.wikipedia.org/wiki/Free%20category | In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.
More precisely, the objects of the category are the vertices of the quiver, and the morphisms are paths between objects. Here, a path is defined as a finite sequence
where is a vertex of the quiver, is an edge of the quiver, and n ranges over the non-negative integers. For every vertex of the quiver, there is an "empty path" which constitutes the identity morphisms of the category.
The composition operation is concatenation of paths. Given paths
their composition is
.
Note that the result of the composition starts with the right operand of the composition, and ends with its left operand.
Examples
If is the quiver with one vertex and one edge from that object to itself, then the free category on has as arrows , , ∘,∘∘, etc.
Let be the quiver with two vertices , and two edges , from to and to , respectively. Then the free category on has two identity arrows and an arrow for every finite sequence of alternating s and s, including: , , ∘, ∘, ∘∘, ∘∘, etc.
If is the quiver , then the free category on has (in addition to three identity arrows), arrows , , and ∘.
If a quiver has only one vertex, then the free category on has only one object, and corresponds to the free monoid on the edges of .
Properties
The category of small categories Cat has a forgetful functor into the quiver category Quiv:
: Cat → Quiv
which takes objects to vertices and morphisms to arrows. Intuitively, "[forgets] which arrows are composites and which are identities". This forgetful functor is right adjoint to the functor sending a quiver to the corresponding free category.
Universal property
The free category on a quiver can be described up to isomorphism by a universal property. Let : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let be the forgetful functor defined above, and let be any quiver. Then there is a graph homomorphism : → (()) and given any category D and any graph homomorphism : → , there is a unique functor : () → D such that ()∘=, i.e. the following diagram commutes:
The functor is left adjoint to the forgetful functor .
See also
Free strict monoidal category
Free object
Adjoint functors
References
Free algebraic structures |
https://en.wikipedia.org/wiki/Poles%20in%20Iceland | Poles make up the largest group of immigrants in Iceland. On 1 January 2021, Statistics Iceland recorded 20,553 Polish-born people living in Iceland. Although small compared to the size of migrant groups in other countries, that makes them the biggest minority ethnic group in Iceland.
History
There have been several different migratory movements of Poles to Iceland. The first major migration occurred at the turn of the 19th century after Poland lost its statehood. However, for much of the Cold War period, most of the Polish population was restricted in their ability to travel outside of communist Poland at all.
More recently in 2004, an influx occurred after Poland joined the European Union, thereby easing restrictions on Polish citizens' eligibility to work in other European Economic Area states. In 2006, Iceland's construction industry boomed and Polish workers were increasingly hired to fulfill work demands. Within a year, the number of Polish migrants in the country increased by 81%. Poland also joined Iceland in the Schengen Zone in 2007. As a result, Poles do not need work or resident permits to live and work in Iceland. The global financial crisis of 2008 decreased the levels of migration drastically and more Poles repatriated than arrived in Iceland during this year.
Life in Iceland
The demographic is largely endogamous and insular. Poles in Iceland typically speak Polish, watch Polish television, continue to practice Catholicism and have opened Polish restaurants.
A study conducted in 2012 suggested that most Polish Icelanders used the English language more often than Icelandic in their daily lives, found English more useful and often learned it before learning Icelandic.
Politics
Poles living in Iceland can cast their vote in Polish elections. During the Polish presidential election in 2020 roughly 80% of Poles in Iceland voted for Rafał Trzaskowski (candidate of the Civic Platform) while only 20% voted for Andrzej Duda (candidate of the Law and Justice party). This contrasts with Poland where Andrzej Duda won the majority.
See also
Demographics of Iceland
Iceland–Poland relations
References
Ethnic groups in Iceland
Iceland
Iceland |
https://en.wikipedia.org/wiki/UBOS | UBOS may refer to:
Ultimate Book of Spells, a Canadian animated television series
Uganda Bureau of Statistics, a government agency |
https://en.wikipedia.org/wiki/Sigma%20Zeta | Sigma Zeta () is a national honor society founded in 1925 to recognize undergraduate excellence in the natural sciences, computer science, and mathematics. The society's purpose is to encourage and foster the attainment of knowledge in the natural and computer sciences and mathematics.
History
Sigma Zeta was founded in the fall of 1925 at the now defunct Shurtleff College as a local organization to provide recognition for Shurtleff science and mathematics students. Soon after that other local campuses took an interest in the group, and following the approval of the petition by McKendree College to start a chapter in June 1926 Sigma Zeta began its growth into a national collegiate honor society. In a letter that appeared in the correspondence section of the American Chemical Society Journal of Chemical Education Sigma Zeta was offered as an alternative for small colleges to the existing Sigma Xi honor society which often passed over small colleges for membership as they focused on larger Universities.
Sigma Zeta's annual convention has been held every year since 1926 except for 1943, 1944, and 1945 where it was canceled due to World War II, and in 2020 when it was canceled due to the COVID-19 Pandemic. The first three gatherings were held at the Shurtleff College campus which is now the site of the Southern Illinois University School of Dental Medicine.
Activities
The annual convention is Sigma Zeta's primary meeting where student members present papers and individual and chapter awards are presented. Individual chapters often undertake activities including the hosting of speakers and service projects to benefit their local communities. These activities can include programs for younger students at local schools helping to promote science education at early ages. Sigma Zeta's interdisciplinary nature has been described as a benefit for smaller colleges and universities because, "It brings together students from all areas of science and mathematics, including computer science, so they can all work together on projects."
Qualifications for Membership
Chapters select students for membership that have met the following eligibility criteria:
Major studies in at least one of the Natural Sciences, Computer Science, or Mathematics
Completion of 25 semester hours of coursework with 15 hours in the Natural Sciences, Computer Science, or Mathematics
A Grade Point Average of 3.0 out of 4.0 both in Science and Mathematics as well as cumulatively among all classes taken
Chapters
Chapters in existence are listed in the order they joined Sigma Zeta.
Beta, McKendree University, 1926
Gamma, Medical College of Virginia, 1927
Epsilon, Otterbein College, 1929
Lambda, Mansfield University, 1936
Mu, Minnesota State University, Mankato, 1937
Xi, Ball State University, 1938
Pi, Millikin University, 1943
Rho, University of Indianapolis, 1943
Sigma, Our Lady of the Lake University, 1944
Tau, East Stroudsburg University of Pennsylvania, 1947
Upsilon, An |
https://en.wikipedia.org/wiki/Hemiperfect%20number | In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the divisor function, the sum of all positive divisors of n.
The first few hemiperfect numbers are:
2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, ...
Example
24 is a hemiperfect number because the sum of the divisors of 24 is
1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = × 24.
The abundancy index is 5/2 which is a half-integer.
Smallest hemiperfect numbers of abundancy k/2
The following table gives an overview of the smallest hemiperfect numbers of abundancy k/2 for k ≤ 13 :
The current best known upper bounds for the smallest numbers of abundancy 15/2 and 17/2 were found by Michel Marcus.
The smallest known number of abundancy 15/2 is ≈ , and the smallest known number of abundancy 17/2 is ≈ .
There are no known numbers of abundancy 19/2.
See also
Semiperfect number
Perfect number
Multiply perfect number
References
Integer sequences
Perfect numbers |
https://en.wikipedia.org/wiki/List%20of%20Mexican%20states%20by%20population%20growth%20rate | The following table shows the 32 federal entities of Mexico, ranked in order by population growth from the 2020 to the 2010 National Census Population from the National Institute of Statistics and Geography.
See also
Mexico
States of Mexico
Geography of Mexico
List of Mexican states by population
List of Mexican states by area
Ranked list of Mexican states
List of Mexican states by HDI
List of Mexican states by GDP
List of Mexican states by GDP per capita
References
Population
Mexico, population growth rate |
https://en.wikipedia.org/wiki/Partition%20matroid | In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a capacity constraint - a maximum number of allowed elements from this category. The independent sets of a partition matroid are exactly the sets in which, for each category, the number of elements from this category is at most the category capacity.
Formal definition
Let be a collection of disjoint sets ("categories"). Let be integers with ("capacities"). Define a subset to be "independent" when, for every index , . The sets satisfying this condition form the independent sets of a matroid, called a partition matroid.
The sets are called the categories or the blocks of the partition matroid.
A basis of the partition matroid is a set whose intersection with every block has size exactly . A circuit of the matroid is a subset of a single block with size exactly . The rank of the matroid is .
Every uniform matroid is a partition matroid, with a single block of elements and with . Every partition matroid is the direct sum of a collection of uniform matroids, one for each of its blocks.
In some publications, the notion of a partition matroid is defined more restrictively, with every . The partitions that obey this more restrictive definition are the transversal matroids of the family of disjoint sets given by their blocks.
Properties
As with the uniform matroids they are formed from, the dual matroid of a partition matroid is also a partition matroid, and every minor of a partition matroid is also a partition matroid. Direct sums of partition matroids are partition matroids as well.
Matching
A maximum matching in a graph is a set of edges that is as large as possible subject to the condition that no two edges share an endpoint. In a bipartite graph with bipartition , the sets of edges satisfying the condition that no two edges share an endpoint in are the independent sets of a partition matroid with one block per vertex in and with each of the numbers equal to one. The sets of edges satisfying the condition that no two edges share an endpoint in are the independent sets of a second partition matroid. Therefore, the bipartite maximum matching problem can be represented as a matroid intersection of these two matroids.
More generally the matchings of a graph may be represented as an intersection of two matroids if and only if every odd cycle in the graph is a triangle containing two or more degree-two vertices.
Clique complexes
A clique complex is a family of sets of vertices of a graph that induce complete subgraphs of . A clique complex forms a matroid if and only if is a complete multipartite graph, and in this case the resulting matroid is a partition matroid. The clique complexes are exactly the set systems that can be formed as intersections of families of partition matroids for which every |
https://en.wikipedia.org/wiki/Heavy%20traffic%20approximation | In queueing theory, a discipline within the mathematical theory of probability, a heavy traffic approximation (sometimes heavy traffic limit theorem or diffusion approximation) is the matching of a queueing model with a diffusion process under some limiting conditions on the model's parameters. The first such result was published by John Kingman who showed that when the utilisation parameter of an M/M/1 queue is near 1 a scaled version of the queue length process can be accurately approximated by a reflected Brownian motion.
Heavy traffic condition
Heavy traffic approximations are typically stated for the process X(t) describing the number of customers in the system at time t. They are arrived at by considering the model under the limiting values of some model parameters and therefore for the result to be finite the model must be rescaled by a factor n, denoted
and the limit of this process is considered as n → ∞.
There are three classes of regime under which such approximations are generally considered.
The number of servers is fixed and the traffic intensity (utilization) is increased to 1 (from below). The queue length approximation is a reflected Brownian motion.
Traffic intensity is fixed and the number of servers and arrival rate are increased to infinity. Here the queue length limit converges to the normal distribution.
A quantity β is fixed where
with ρ representing the traffic intensity and s the number of servers. Traffic intensity and the number of servers are increased to infinity and the limiting process is a hybrid of the above results. This case, first published by Halfin and Whitt is often known as the Halfin–Whitt regime or quality-and-efficiency-driven (QED) regime.
Results for a G/G/1 queue
Theorem 1. Consider a sequence of G/G/1 queues indexed by .
For queue let denote the random inter-arrival time, denote the random service time;
let denote the traffic intensity with and ;
let denote the waiting time in queue for a customer in steady state;
Let and
Suppose that , , and . then
provided that:
(a)
(b) for some , and are both less than some constant for all .
Heuristic argument
Waiting time in queue
Let be the difference between the nth service time and the nth inter-arrival time;
Let be the waiting time in queue of the nth customer;
Then by definition:
After recursive calculation, we have:
Random walk
Let , with are i.i.d;
Define and ;
Then we have
we get by taking limit over .
Thus the waiting time in queue of the nth customer is the supremum of a random walk with a negative drift.
Brownian motion approximation
Random walk can be approximated by a Brownian motion when the jump sizes approach 0 and the times between the jump approach 0.
We have and has independent and stationary increments. When the traffic intensity approaches 1 and tends to , we have after replaced with continuous value according to functional central limit theorem. Thus the waiting time in q |
https://en.wikipedia.org/wiki/ATMP%20%28disambiguation%29 | ATMP is a chemical compound used as a chelator.
ATMP may also refer to:
Advances in Theoretical and Mathematical Physics, a peer-reviewed mathematics journal
All Terrain Mobility Platform
All Things Must Pass, an album by George Harrison
Advanced Therapy Medicinal Product, a classification for medicinal products that are based on genes, cells or tissues |
https://en.wikipedia.org/wiki/Distribution%20%28number%20theory%29 | In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function.
The original examples of distributions occur, unnamed, as functions φ on Q/Z satisfying
Such distributions are called ordinary distributions. They also occur in p-adic integration theory in Iwasawa theory.
Let ... → Xn+1 → Xn → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let X be their projective limit. We give each Xn the discrete topology, so that X is compact. Let φ = (φn) be a family of functions on Xn taking values in an abelian group V and compatible with the projective system:
for some weight function w. The family φ is then a distribution on the projective system X.
A function f on X is "locally constant", or a "step function" if it factors through some Xn. We can define an integral of a step function against φ as
The definition extends to more general projective systems, such as those indexed by the positive integers ordered by divisibility. As an important special case consider the projective system Z/nZ indexed by positive integers ordered by divisibility. We identify this with the system (1/n)Z/Z with limit Q/Z.
For x in R we let ⟨x⟩ denote the fractional part of x normalised to 0 ≤ ⟨x⟩ < 1, and let {x} denote the fractional part normalised to 0 < {x} ≤ 1.
Examples
Hurwitz zeta function
The multiplication theorem for the Hurwitz zeta function
gives a distribution relation
Hence for given s, the map is a distribution on Q/Z.
Bernoulli distribution
Recall that the Bernoulli polynomials Bn are defined by
for n ≥ 0, where bk are the Bernoulli numbers, with generating function
They satisfy the distribution relation
Thus the map
defined by
is a distribution.
Cyclotomic units
The cyclotomic units satisfy distribution relations. Let a be an element of Q/Z prime to p and let ga denote exp(2πia)−1. Then for a≠ 0 we have
Universal distribution
One considers the distributions on Z with values in some abelian group V and seek the "universal" or most general distribution possible.
Stickelberger distributions
Let h be an ordinary distribution on Q/Z taking values in a field F. Let G(N) denote the multiplicative group of Z/NZ, and for any function f on G(N) we extend f to a function on Z/NZ by taking f to be zero off G(N). Define an element of the group algebra F[G(N)] by
The group algebras form a projective system with limit X. Then the functions gN form a distribution on Q/Z with values in X, the Stickelberger distribution associated with h.
p-adic measures
Consider the special case when the value group V of a distribution φ on X takes values in a local field K, finite over Qp, or more generally, in a finite-dimensional
p-adic Banach space W over K, with valuation |·|. We call φ a measure if |φ| is bounded on compact open subsets |
https://en.wikipedia.org/wiki/M%C3%A1rton%20Czuczi | Márton Czuczi (born 20 May 1992 in Budapest) is a Hungarian football player who currently plays for FC Dabas.
Career
In 2019, Czuczi returned to Pénzügyőr SE.
Club statistics
Updated to games played as of 27 April 2014.
References
External links
1992 births
Living people
Footballers from Budapest
Hungarian men's footballers
Men's association football goalkeepers
Budapest Honvéd FC players
Békéscsaba 1912 Előre footballers
Szigetszentmiklósi TK footballers
Mezőkövesdi SE footballers
Pénzügyőr SE footballers
Monori SE players
FC Dabas footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
21st-century Hungarian people |
https://en.wikipedia.org/wiki/Firoozbakht%27s%20conjecture | In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture) is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht who stated it first in 1982.
The conjecture states that (where is the nth prime) is a strictly decreasing function of n, i.e.,
Equivalently:
see , .
By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444. Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 264 ≈ .
If the conjecture were true, then the prime gap function would satisfy:
Moreover:
see also . This is among the strongest upper bounds conjectured for prime gaps, even somewhat stronger than the Cramér and Shanks conjectures. It implies a strong form of Cramér's conjecture and is hence inconsistent with the heuristics of Granville and Pintz and of Maier which suggest that
occurs infinitely often for any where denotes the Euler–Mascheroni constant.
Two related conjectures (see the comments of ) are
which is weaker, and
which is stronger.
See also
Prime number theorem
Andrica's conjecture
Legendre's conjecture
Oppermann's conjecture
Cramér's conjecture
Notes
References
Conjectures about prime numbers
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/List%20of%20FC%20Midtjylland%20players |
List of notable players
Statistics are up to date as of 22 August 2012.
Please help to expand this list.
Note: the source for the Career dates and Total apps/goals for a number of the below players is unclear and the reference used should be added.
FC Midtjylland
Midtjylland
Association football player non-biographical articles |
https://en.wikipedia.org/wiki/Finite%20subdivision%20rule | In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.
Definition
A subdivision rule takes a tiling of the plane by polygons and turns it into a new tiling by subdividing each polygon into smaller polygons. It is finite if there are only finitely many ways that every polygon can subdivide. Each way of subdividing a tile is called a tile type. Each tile type is represented by a label (usually a letter). Every tile type subdivides into smaller tile types. Each edge also gets subdivided according to finitely many edge types. Finite subdivision rules can only subdivide tilings that are made up of polygons labelled by tile types. Such tilings are called subdivision complexes for the subdivision rule. Given any subdivision complex for a subdivision rule, we can subdivide it over and over again to get a sequence of tilings.
For instance, binary subdivision has one tile type and one edge type:
Since the only tile type is a quadrilateral, binary subdivision can only subdivide tilings made up of quadrilaterals. This means that the only subdivision complexes are tilings by quadrilaterals. The tiling can be regular, but doesn't have to be:
Here we start with a complex made of four quadrilaterals and subdivide it twice. All quadrilaterals are type A tiles.
Examples of finite subdivision rules
Barycentric subdivision is an example of a subdivision rule with one edge type (that gets subdivided into two edges) and one tile type (a triangle that gets subdivided into 6 smaller triangles). Any triangulated surface is a barycentric subdivision complex.
The Penrose tiling can be generated by a subdivision rule on a set of four tile types (the curved lines in the table below only help to show how the tiles fit together):
Certain rational maps give rise to finite subdivision rules. This includes most Lattès maps.
Every prime, non-split alternating knot or link complement has a subdivision rule, with some tiles that do not subdivide, corresponding to the boundary of the link complement. The subdivision rules show what the night sky would look like to someone living in a knot complement; because the universe wraps around itself (i.e. is not simply connected), an observer would see the visible universe repeat itself in an infinite pattern. The subdivision rule describes that pattern.
The subdivision rule looks different for different geometries. This is a subdivision rule for the trefoil knot, which is not a hyperbolic knot: |
https://en.wikipedia.org/wiki/Zero-order%20process%20%28statistics%29 | In probability theory and statistics, a zero-order process is a stochastic process in which each observation is independent of all previous observations. For example, a zero-order process in marketing would be one in which the brands purchased next do not depend on the brands purchased before, implying a fixed probability of purchase since it is zero order in regards to probability.
References
Stochastic processes |
https://en.wikipedia.org/wiki/Affine-regular%20polygon | In geometry, an affine-regular polygon or affinely regular polygon is a polygon that is related to a regular polygon by an affine transformation. Affine transformations include translations, uniform and non-uniform scaling, reflections, rotations, shears, and other similarities and some, but not all linear maps.
Examples
All triangles are affine-regular. In other words, all triangles can be generated by applying affine transformations to an equilateral triangle. A quadrilateral is affine-regular if and only if it is a parallelogram, which includes rectangles and rhombuses as well as squares. In fact, affine-regular polygons may be considered a natural generalization of parallelograms.
Properties
Many properties of regular polygons are invariant under affine transformations, and affine-regular polygons share the same properties. For instance,
an affine-regular quadrilateral can be equidissected into equal-area triangles if and only if is even, by affine invariance of equidissection and Monsky's theorem on equidissections of squares. More generally an -gon with may be equidissected into equal-area triangles if and only if is a multiple of .
References
Affine geometry
Types of polygons |
https://en.wikipedia.org/wiki/Conditional%20dependence | In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs. For example, if and are two events that individually increase the probability of a third event and do not directly affect each other, then initially (when it has not been observed whether or not the event occurs)
( are independent).
But suppose that now is observed to occur. If event occurs then the probability of occurrence of the event will decrease because its positive relation to is less necessary as an explanation for the occurrence of (similarly, event occurring will decrease the probability of occurrence of ). Hence, now the two events and are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have
Conditional dependence of A and B given C is the logical negation of conditional independence . In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.
Example
In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event be 'I have a new phone'; event be 'I have a new watch'; and event be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy. Let us assume that the event has occurred – meaning 'I am happy'. Now if another person sees my new watch, he/she will reason that my likelihood of being happy was increased by my new watch, so there is less need to attribute my happiness to a new phone.
To make the example more numerically specific, suppose that there are four possible states given in the middle four columns of the following table, in which the occurrence of event is signified by a in row and its non-occurrence is signified by a and likewise for and That is, and The probability of is for every
and so
In this example, occurs if and only if at least one of occurs. Unconditionally (that is, without reference to ), and are independent of each other because —the sum of the probabilities associated with a in row —is while
But conditional on having occurred (the last three columns in the table), we have
while
Since in the presence of the probability of is affected by the presence or absence of and are mutually dependent conditional on
See also
References
Independence (probability theory) |
https://en.wikipedia.org/wiki/Peter%20H.%20Haynes | Peter Howard Haynes (born 23 July 1958) is a British applied mathematician in the Faculty of Mathematics at the University of Cambridge. He is a Fellow of Queens' College, Cambridge and served as Head of the Department of Applied Mathematics and Theoretical Physics (DAMTP) from 2005 to 2015.
He was educated at the Royal Grammar School, Guildford, and Queens' College, Cambridge (B.A. 1979, M.A. 1983, Ph.D. 1984).
Research
His research includes fluid dynamics and atmospheric dynamics.
References
Living people
20th-century British mathematicians
21st-century British mathematicians
Fluid dynamicists
Cambridge mathematicians
Fellows of Queens' College, Cambridge
1958 births
People educated at Royal Grammar School, Guildford
Alumni of Queens' College, Cambridge |
https://en.wikipedia.org/wiki/Tele%20%28footballer%29 | Marcio Alves dos Santos (born February 2, 1990), known as Tele, is a Brazilian football player.
Club statistics
References
External links
1990 births
Living people
Brazilian men's footballers
J1 League players
J2 League players
Japan Football League players
Hokkaido Consadole Sapporo players
FC Machida Zelvia players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football forwards
People from Dom Pedrito
Footballers from Rio Grande do Sul |
https://en.wikipedia.org/wiki/1983%E2%80%9384%20Galatasaray%20S.K.%20season | The 1983–84 season was Galatasaray's 80th in existence and the 26th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Kick-off listed in local time (EET)
Türkiye Kupası
Kick-off listed in local time (EET)
5th Round
6th Round
1/4 Final
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Donanma Kupası
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1983–84 season
1980s in Istanbul
Galatasaray Sports Club 1983–84 season |
https://en.wikipedia.org/wiki/Pythagoras%20number | In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p(K) of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares.
A Pythagorean field is a field with Pythagoras number 1: that is, every sum of squares is already a square.
Examples
Every non-negative real number is a square, so p(R) = 1.
For a finite field of odd characteristic, not every element is a square, but all are the sum of two squares, so p = 2.
By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are sums of three squares, so p(Q) = 4.
Properties
Every positive integer occurs as the Pythagoras number of some formally real field.
The Pythagoras number is related to the Stufe by p(F) ≤ s(F) + 1. If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1, and both cases are possible: for F = C we have s = p = 1, whereas for F = F5 we have s = 1, p = 2.
As a consequence, the Pythagoras number of a non-formally-real field is either a power of 2, or 1 more than a power of 2. All such cases occur: i.e., for each pair (s,p) of the form (2k,2k) or (2k,2k + 1), there exists a field F such that (s(F),p(F)) = (s,p). For example, quadratically closed fields (e.g., C) and fields of characteristic 2 (e.g., F2) give (s(F),p(F)) = (1,1); for primes p ≡ 1 (mod 4), Fp and the p-adic field Qp give (1,2); for primes p ≡ 3 (mod 4), Fp gives (2,2), and Qp gives (2,3); Q2 gives (4,4), and the function field Q2(X) gives (4,5).
The Pythagoras number is related to the height of a field F: if F is formally real then h(F) is the smallest power of 2 which is not less than p(F); if F is not formally real then h(F) = 2s(F).
Notes
References
Field (mathematics)
Sumsets |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Uruguay | This page details football records in Uruguay.
Most successful clubs overall
Football in Uruguay
Uruguay |
https://en.wikipedia.org/wiki/Absolute%20value%20%28disambiguation%29 | The absolute value is a value of a real number.
Absolute value may also refer to:
Absolute value (algebra), a generalization of the absolute value of a real number
Absolute value theorem in mathematics, also known as the "squeeze theorem"
Absolute Value (album), the second full-length album by rapper Akrobatik
Absolute value (ethics), a philosophical absolute independent of individual and cultural views |
https://en.wikipedia.org/wiki/U-invariant | In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field.
The universal invariant u(F) of a field F is the largest dimension of an anisotropic quadratic space over F, or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that u is the smallest number such that every form of dimension greater than u is isotropic, or that every form of dimension at least u is universal.
Examples
For the complex numbers, u(C) = 1.
If F is quadratically closed then u(F) = 1.
The function field of an algebraic curve over an algebraically closed field has u ≤ 2; this follows from Tsen's theorem that such a field is quasi-algebraically closed.
If F is a non-real global or local field, or more generally a linked field, then u(F) = 1, 2, 4 or 8.
Properties
If F is not formally real and the characteristic of F is not 2 then u(F) is at most , the index of the squares in the multiplicative group of F.
u(F) cannot take the values 3, 5, or 7. Fields exist with u = 6 and u = 9.
Merkurjev has shown that every even integer occurs as the value of u(F) for some F.
Alexander Vishik proved that there are fields with u-invariant for all .
The u-invariant is bounded under finite-degree field extensions. If E/F is a field extension of degree n then
In the case of quadratic extensions, the u-invariant is bounded by
and all values in this range are achieved.
The general u-invariant
Since the u-invariant is of little interest in the case of formally real fields, we define a general u-invariant to be the maximum dimension of an anisotropic form in the torsion subgroup of the Witt ring of F, or ∞ if this does not exist. For non-formally-real fields, the Witt ring is torsion, so this agrees with the previous definition. For a formally real field, the general u-invariant is either even or ∞.
Properties
u(F) ≤ 1 if and only if F is a Pythagorean field.
References
Field (mathematics)
Quadratic forms |
https://en.wikipedia.org/wiki/Quadratically%20closed%20field | In mathematics, a quadratically closed field is a field in which every element has a square root.
Examples
The field of complex numbers is quadratically closed; more generally, any algebraically closed field is quadratically closed.
The field of real numbers is not quadratically closed as it does not contain a square root of −1.
The union of the finite fields for n ≥ 0 is quadratically closed but not algebraically closed.
The field of constructible numbers is quadratically closed but not algebraically closed.
Properties
A field is quadratically closed if and only if it has universal invariant equal to 1.
Every quadratically closed field is a Pythagorean field but not conversely (for example, R is Pythagorean); however, every non-formally real Pythagorean field is quadratically closed.
A field is quadratically closed if and only if its Witt–Grothendieck ring is isomorphic to Z under the dimension mapping.
A formally real Euclidean field E is not quadratically closed (as −1 is not a square in E) but the quadratic extension E() is quadratically closed.
Let E/F be a finite extension where E is quadratically closed. Either −1 is a square in F and F is quadratically closed, or −1 is not a square in F and F is Euclidean. This "going-down theorem" may be deduced from the Diller–Dress theorem.
Quadratic closure
A quadratic closure of a field F is a quadratically closed field containing F which embeds in any quadratically closed field containing F. A quadratic closure for any given F may be constructed as a subfield of the algebraic closure Falg of F, as the union of all iterated quadratic extensions of F in Falg.
Examples
The quadratic closure of R is C.
The quadratic closure of is the union of the .
The quadratic closure of Q is the field of complex constructible numbers.
References
Field (mathematics) |
https://en.wikipedia.org/wiki/First%20Nations%20and%20diabetes | There are high rates of diabetes in First Nation people compared to the general Canadian population. Statistics from 2011 showed that 17.2% of First Nations people living on reserves had type 2 diabetes.
Contributing factors to the high prevalence of type 2 diabetes between First Nation and the general population include a combination of environmental (lifestyle, diet, poverty), and genetic and biological factors (e.g. thrifty genotype hypothesis, thrifty phenotype). To what extent each factor plays a role is not clear.
Diabetes mellitus Type 2
Rates of obesity and type 2 diabetes (T2D) in First Nation communities were non-existent 20 years ago, but increased steeply. Age-standardized rates of T2D show 17.2% prevalence of T2D among First Nations individuals living on reserves, compared to 5.0% in the non-Aboriginal population;
Statistics indicate that the T2D prevalence rate in First Nations people is 3 to 5 times higher than the general Canadian population. As well as having a higher rate of T2D than the general population, there are also differences in the disease pattern in First Nations T2D patients compared to the general population, especially in terms of age of onset and gestational diabetes.
Diabetes in youth
Diabetes in First Nations has increasingly become a disease of the younger population, who thus experience a high burden of disease, diabetes-related complications and co-morbidity. To illustrate, in the general population type 2 diabetes is an old-age associated disease: New diabetes cases peaked in First Nations people between ages 40–49 compared with a non-First Nations peak of age 70+.
This earlier onset of disease in First Nation population has serious health implications for the women, especially during her reproductive life-years: it increases the chance of her children to develop diabetes, contributing to diabetes prevalence and incidence in the future generations.
Diabetes in women
First Nations women in particular are at risk of developing diabetes, especially between ages 20–49. They have a four times higher incidence of diabetes than non-First Nation women as well as experiencing higher rates of gestational diabetes than non-Aboriginal females, 8–18% compared to 2–4%.
Gestational diabetes
A third type of diabetes, other than type 1 and type 2, is gestational diabetes mellitus. This is a temporary type of diabetes that occurs during pregnancy. Most women with gestational diabetes will return to normal glucose levels after delivery of the baby; if a woman does not return to normal glucose levels she will be re-diagnosed with type 2 diabetes and is no longer considered to have gestational diabetes.
Gestational diabetes carries risks for both the mother and the baby. It increases the likelihood of the infant developing T2D, and giving birth to high body-weight baby. High body-weight increases risk of the child developing diabetes even if mother does not have it.
Screening programs
The Review of Guidelines for Sc |
https://en.wikipedia.org/wiki/Jigme%20Tshering%20Dorji | Jigme Tshering Dorji (also spelled as Dorjee) is a Bhutanese international footballer. He made his first appearance for the Bhutan national football team in 2011.
Career statistics
International goals
References
1995 births
Living people
Bhutanese men's footballers
Bhutan men's international footballers
Men's association football defenders
Bhutanese expatriate men's footballers
Expatriate men's footballers in India
Bhutanese expatriate sportspeople in India |
https://en.wikipedia.org/wiki/Hurwitz%20problem | In mathematics, the Hurwitz problem (named after Adolf Hurwitz) is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables.
Description
There are well-known multiplicative relationships between sums of squares in two variables
(known as the Brahmagupta–Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity. These may be interpreted as multiplicativity for the norms on the complex numbers ), quaternions (), and octonions (), respectively.
The Hurwitz problem for the field is to find general relations of the form
with the being bilinear forms in the and : that is, each is a -linear combination of terms of the form .
We call a triple admissible for if such an identity exists. Trivial cases of admissible triples include The problem is uninteresting for of characteristic 2, since over such fields every sum of squares is a square, and we exclude this case. It is believed that otherwise admissibility is independent of the field of definition.
The Hurwitz–Radon theorem
Hurwitz posed the problem in 1898 in the special case and showed that, when coefficients are taken in , the only admissible values were His proof extends to a field of any characteristic except 2.
The "Hurwitz–Radon" problem is that of finding admissible triples of the form Obviously is admissible. The Hurwitz–Radon theorem states that is admissible over any field where is the function defined for odd, with and
Other admissible triples include and
See also
Composition algebra
Hurwitz's theorem (normed division algebras)
Radon–Hurwitz number
References
Field (mathematics)
Quadratic forms
Mathematical problems |
https://en.wikipedia.org/wiki/Theorem%20on%20formal%20functions | In algebraic geometry, the theorem on formal functions states the following:
Let be a proper morphism of noetherian schemes with a coherent sheaf on X. Let be a closed subscheme of S defined by and formal completions with respect to and . Then for each the canonical (continuous) map:
is an isomorphism of (topological) -modules, where
The left term is .
The canonical map is one obtained by passage to limit.
The theorem is used to deduce some other important theorems: Stein factorization and a version of Zariski's main theorem that says that a proper birational morphism into a normal variety is an isomorphism. Some other corollaries (with the notations as above) are:
Corollary: For any , topologically,
where the completion on the left is with respect to .
Corollary: Let r be such that for all . Then
Corollay: For each , there exists an open neighborhood U of s such that
Corollary: If , then is connected for all .
The theorem also leads to the Grothendieck existence theorem, which gives an equivalence between the category of coherent sheaves on a scheme and the category of coherent sheaves on its formal completion (in particular, it yields algebralizability.)
Finally, it is possible to weaken the hypothesis in the theorem; cf. Illusie. According to Illusie (pg. 204), the proof given in EGA III is due to Serre. The original proof (due to Grothendieck) was never published.
The construction of the canonical map
Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map.
Let be the canonical maps. Then we have the base change map of -modules
.
where is induced by . Since is coherent, we can identify with . Since is also coherent (as f is proper), doing the same identification, the above reads:
.
Using where and , one also obtains (after passing to limit):
where are as before. One can verify that the composition of the two maps is the same map in the lede. (cf. EGA III-1, section 4)
Notes
References
Further reading
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Igor%20Rivin |
Igor Rivin (born 1961 in Moscow, USSR) is a Russian-Canadian mathematician,
working in various fields of pure and applied mathematics, computer science,
and materials science. He was the Regius Professor of Mathematics at the University of St. Andrews from 2015 to 2017, and was the chief research officer at Cryptos Fund until 2019. He is doing research for Edgestream LP, in addition to his academic work.
Career
He received his B.Sc. (Hon) in mathematics from the University of Toronto in 1981, and his Ph.D. in 1986 from Princeton University under the direction of William Thurston. Following his doctorate, Rivin directed development of QLISP and the Mathematica kernel, before returning to academia in 1992, where he held positions at the Institut des Hautes Études Scientifiques, the Institute for Advanced Study, the University of Melbourne, Warwick, and Caltech. Since 1999, Rivin has been professor of mathematics at Temple University. Between 2015 and 2017 he was Regius Professor of Mathematics at the University of St. Andrews.
Major accomplishments
Rivin's PhD thesis and a series of extensions characterized hyperbolic 3-dimensional polyhedra in terms of their dihedral angles, resolving a long-standing open question of Jakob Steiner on the inscribable combinatorial types. These, and some related results in convex geometry, have been used in 3-manifold topology, theoretical physics, computational geometry, and the recently developed field of discrete differential geometry.
Rivin has also made advances in counting geodesics on surfaces, the study of generic elements of discrete subgroups of Lie groups, and in the theory of dynamical systems.
Rivin is also active in applied areas, having written large parts of the Mathematica 2.0 kernel, and he developed a database of hypothetical zeolites in collaboration with M. M. J. Treacy.
Rivin is a frequent contributor to MathOverflow.
Igor Rivin is the co-creator, with economist Carlo Scevola, of Cryptocurrencies Index 30 (CCi30), an index of the top 30 cryptocurrencies weighted by market capitalization. CCi30 is sometimes used by academic economists as a market index when comparing the cryptocurrency trading market as a whole with individual currencies.
Honors
First prize, Canadian Mathematical Olympiad, 1977
Whitehead prize of the London Mathematical Society, 1998
Advanced Research Fellowship of the EPSRC, 1998
Lady Davis Fellowship at the Hebrew University, 2006
Berlin Mathematical School professorship, 2011.
Fellow of the American Mathematical Society, 2014.
References
External links
Igor Rivin's author profile at MathSciNet
Igor Rivin's Google Scholar profile
Igor Rivin at Math Overflow
Canadian mathematicians
Jewish American scientists
University of Toronto alumni
Geometers
20th-century American mathematicians
21st-century American mathematicians
Living people
1961 births
Fellows of the American Mathematical Society
21st-century American Jews |
https://en.wikipedia.org/wiki/David%20Conlon | David Conlon (born 1982) is an Irish mathematician who is a Professor of Mathematics at the California Institute of Technology. His research interests are in Hungarian-style combinatorics, particularly Ramsey theory, extremal graph theory, combinatorial number theory, and probabilistic methods in combinatorics. He proved the first superpolynomial improvement on the Erdős–Szekeres bound on diagonal Ramsey numbers. He won the European Prize in Combinatorics in 2011 for his work in Ramsey theory and for his progress on Sidorenko's conjecture, and the Whitehead Prize in 2019.
Life
Conlon represented Ireland in the International Mathematical Olympiad in 1998 and 1999. He was an undergraduate in Trinity College Dublin, where he was elected a Scholar in 2001 and graduated in 2003. He earned a PhD from Cambridge University in 2009.
In 2019 he moved to California Institute of Technology, having been a fellow of Wadham College, Oxford and Professor of Discrete Mathematics in the Mathematics Institute at the University of Oxford.
Conlon has worked in Ramsey theory, and he proved the first superpolynomial improvement on the Erdős–Szekeres bound on diagonal Ramsey numbers. He won the European Prize in Combinatorics in 2011, for his work in Ramsey theory and for his progress on Sidorenko's conjecture that, for any bipartite graph H, uniformly random graphons have the fewest subgraphs isomorphic to H when the edge density is fixed. He was awarded the Whitehead Prize in 2019 "in recognition of his many contributions to combinatorics".
References
External links
Home page at Caltech
Home page at Oxford
1982 births
Living people
Alumni of Trinity College Dublin
Alumni of the University of Cambridge
California Institute of Technology faculty
Combinatorialists
International Mathematical Olympiad participants
21st-century Irish mathematicians
Scholars of Trinity College Dublin
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Uganda%20Bureau%20of%20Statistics | The Uganda Bureau of Statistics ("UBOS") is an agency of the Ugandan government. Formed by the Uganda Bureau of Statistics Act, 1998, the agency is mandated to "coordinate, monitor and supervise Uganda's National Statistical System".
Location
The headquarters of UBOS are located in Statistics House, at Plot 9 Colville Street on Nakasero Hill, in Kampala, Uganda's capital and largest city. This is at the corner of Colville Street and Nile Avenue. The coordinates of Statistics House are 0°18'58.0"N, 32°35'05.0"E (Latitude:0.316111; Longitude:32.584722).
Overview
The agency is supervised by the Uganda Ministry of Finance, Planning and Economic Development. UBOS is governed by a seven-person board of directors. Its scope of work includes conducting a national population census at least once every 10 years or so. The last national census was conducted in August 2014. The exercise cost an estimated UGX:75 billion and created an estimated 150,000 temporary jobs. The agency also publishes regular economic surveys and forecasts, including the monthly inflation figures for the country.
See also
Bank of Uganda
Uganda Revenue Authority
Economy of Uganda
References
External links
Website of Uganda Bureau of Statistics
Government agencies of Uganda
Economy of Uganda
Organizations established in 1998
Organisations based in Kampala
1998 establishments in Uganda
National statistical services |
https://en.wikipedia.org/wiki/2012%20Mongolian%20Premier%20League | Statistics of Niislel Lig in the 2012 season. The title was won by Erchim which was their seventh title.
League standings
All the teams played each other twice. the team which had the most points in two rounds became the winner.
Mongolian Football Federation Cup
Ten teams participated in the 2012 Mongolian Football Federation Cup. The final was played on 15 September 2012.
Erchim defeated Khasiin Khulguud in a penalty shoot-out, after the match ended in extra time with a 1-1 draw.
Super Cup
The 2012 Super Cup was played on 30 September 2012 between the league winner Erchim and the cup runner-up Khasiin Khulguud. It is usually played between the winners of the Niislel League and the Mongolian Football Federation Cup, but Erchim won both titles this year. Erchim defeated Khasiin Khulguud 7-2. As the winner of the Super Cup, they will participate in the 2013 AFC President's Cup representing Mongolia.
References
External links
FIFA.com
Soccerway.com
RSSSF.com
Mongolia Premier League seasons
Mongolia
Mongolia
football |
https://en.wikipedia.org/wiki/1984%E2%80%9385%20Galatasaray%20S.K.%20season | The 1984–85 season was Galatasaray's 81st in existence and the 27th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Kick-off listed in local time (EET)
Türkiye Kupası
Kick-off listed in local time (EET)
3rd Round
4th Round
1/4 Final
1/2 Final
Final
Süper Kupa-Cumhurbaşkanlığı Kupası
Kick-off listed in local time (EET)
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Donanma Kupası
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1984–85 season
1980s in Istanbul
Galatasaray Sports Club 1984–85 season |
https://en.wikipedia.org/wiki/Brauer%E2%80%93Wall%20group | In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field F is a group BW(F) classifying finite-dimensional graded central division algebras over the field. It was first defined by as a generalization of the Brauer group.
The Brauer group of a field F is the set of the similarity classes of finite dimensional central simple algebras over F under the operation of tensor product, where two algebras are called similar if the commutants of their simple modules are isomorphic. Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite dimensional central division algebras. The analogous construction for Z/2Z-graded algebras defines the Brauer–Wall group BW(F).
Properties
The Brauer group B(F) injects into BW(F) by mapping a CSA A to the graded algebra which is A in grade zero.
showed that there is an exact sequence
0 → B(F) → BW(F) → Q(F) → 0
where Q(F) is the group of graded quadratic extensions of F, defined as an extension of Z/2 by F*/F*2 with multiplication (e,x)(f,y) = (e + f, (−1)efxy). The map from BW(F) to Q(F) is the Clifford invariant defined by mapping an algebra to the pair consisting of its grade and determinant.
There is a map from the additive group of the Witt–Grothendieck ring to the Brauer–Wall group obtained by sending a quadratic space to its Clifford algebra. The map factors through the Witt group, which has kernel I3, where I is the fundamental ideal of W(F).
Examples
BW(C) is isomorphic to Z/2Z. This is an algebraic aspect of Bott periodicity of period 2 for the unitary group. The 2 super division algebras are C, C[γ] where γ is an odd element of square 1 commuting with C.
BW(R) is isomorphic to Z/8Z. This is an algebraic aspect of Bott periodicity of period 8 for the orthogonal group. The 8 super division algebras are R, R[ε], C[ε], H[δ], H, H[ε], C[δ], R[δ] where δ and ε are odd elements of square –1 and 1, such that conjugation by them on complex numbers is complex conjugation.
Notes
References
Field (mathematics)
Quadratic forms
Super linear algebra |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20Sporting%20CP%20season | The 2012–13 season is Sporting Clube de Portugal's 80th season in the top flight, the Primeira Liga, known as the Liga ZON Sagres for sponsorship purposes. This article shows player statistics and all matches that the club plays during the 2012–13 season.
It is considered to be the worst season ever in Sporting's history. The club ended in the seventh place in the league table, the lowest position in the club's history, thus failing to qualify for the following season's UEFA Champions League and UEFA Europa League. Sporting were also eliminated from the Europa League group stage for the first time ever, ending in fourth place.
Competitions
Legend
Primeira Liga
League table
Results by round
Matches
Taça de Portugal
Taça da Liga
Group stage
UEFA Europa League
Play-off round
Group stage
Players
Transfers
In
Adrien Silva → Académica → Loan return
Cédric → Académica → Loan return
Marcos Rojo → Spartak Moscow → Undisclosed fee
Valentín Viola → Racing Club → Undisclosed fee
Danijel Pranjić → Bayern Munich → Free transfer
Zakaria Labyad → PSV → Free transfer
Khalid Boulahrouz → VfB Stuttgart → Free transfer
Gelson Fernandes → Saint-Étienne → Free transfer
Miguel Lopes → Porto → Free transfer
Hugo Ventura → Porto → Free transfer
Joãozinho → Beira-Mar → Loan
Out
Agostinho Cá → Barcelona B → Undisclosed fee
Amido Baldé → Vitória de Guimarães → Contract termination
André Marques → Sion → End of contract
Diogo Rosado → Blackburn Rovers → End of contract
Edgar Ié → Barcelona B → Undisclosed fee
João Pereira → Valencia → €3,684,210 + €526,320
Mateus Fonseca → Chiasso → Free transfer
Ricardo Batista → ? → End of contract
Rodolfo Simões → Académico de Viseu → Free transfer
Tiago Ferreira → End of career
Marco Torsiglieri → Metalist Kharkiv → Undisclosed fee
Ânderson Polga → São José → End of contract
Celsinho → Târgu Mureș → End of contract
Jaime Valdés → Parma → Undisclosed fee
Matías Fernández → Fiorentina → €3,136,842 + €1,500,000
Florent Sinama Pongolle → Rostov → Contract termination
Mexer → Nacional → End of contract
Alberto Rodríguez → Rio Ave → Contract termination
Luis Aguiar → San Lorenzo de Almagro → Contract termination
Sebastián Ribas → Genoa → Loan return
Daniel Carriço → Reading → €750,000
Marat Izmailov → Porto → € Free transfer
Emiliano Insúa → Atlético Madrid → €3,500,000
Bruno Pereirinha → Lazio → €2,000,000
Xandão → Kuban Krasnodar → Loan return
Out on loan
References
External links
Official club website
Zerozero
2012-13
Portuguese football clubs 2012–13 season
Sporting CP |
https://en.wikipedia.org/wiki/Compound%20matrix | In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to exterior algebras, and their computation appears in a wide array of problems, such as in the analysis of nonlinear time-varying dynamical systems and generalizations of positive systems, cooperative systems and contracting systems.
Definition
Let be an matrix with real or complex entries. If is a subset of size of and is a subset of size of , then the -submatrix of , written  , is the submatrix formed from by retaining only those rows indexed by and those columns indexed by . If , then is the -minor of .
The r th compound matrix of is a matrix, denoted , is defined as follows. If , then is the unique matrix. Otherwise, has size . Its rows and columns are indexed by -element subsets of and , respectively, in their lexicographic order. The entry corresponding to subsets and is the minor .
In some applications of compound matrices, the precise ordering of the rows and columns is unimportant. For this reason, some authors do not specify how the rows and columns are to be ordered.
For example, consider the matrix
The rows are indexed by and the columns by . Therefore, the rows of are indexed by the sets
and the columns are indexed by
Using absolute value bars to denote determinants, the second compound matrix is
Properties
Let be a scalar, be an matrix, and be an matrix. For a positive integer, let denote the identity matrix. The transpose of a matrix will be written , and the conjugate transpose by . Then:
, a identity matrix.
.
.
If , then .
If , then .
If , then .
If , then .
, which is closely related to Cauchy–Binet formula.
Assume in addition that is a square matrix of size . Then:
.
If has one of the following properties, then so does :
Upper triangular,
Lower triangular,
Diagonal,
Orthogonal,
Unitary,
Symmetric,
Hermitian,
Skew-symmetric,
Skew-hermitian,
Positive definite,
Positive semi-definite,
Normal.
If is invertible, then so is , and .
(Sylvester–Franke theorem) If , then .
Relation to exterior powers
Give the standard coordinate basis . The  th exterior power of is the vector space
whose basis consists of the formal symbols
where
Suppose that is an matrix. Then corresponds to a linear transformation
Taking the  th exterior power of this linear transformation determines a linear transformation
The matrix corresponding to this linear transformation (with respect to the above bases of the exterior powers) is . Taking exterior powers is a functor, which means that
This corresponds to the formula . It is closely related to, and is a strengthening of, the Cauchy–Binet formula.
Relation to adjugate matrices
Let be an matrix. Recall that its  th higher adjugate matrix is the matrix whose entry is
where, for an |
https://en.wikipedia.org/wiki/Minimal%20surface%20of%20revolution | In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface. It is generated by a curve that lies in the half-plane and connects the two points; among all the surfaces that can be generated in this way, it is the one that minimizes the surface area. A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.
Relation to minimal surfaces
A minimal surface of revolution is a subtype of minimal surface. A minimal surface is defined not as a surface of minimal area, but as a surface with a mean curvature of 0. Since a mean curvature of 0 is a necessary condition of a surface of minimal area, all minimal surfaces of revolution are minimal surfaces, but not all minimal surfaces are minimal surfaces of revolution. As a point forms a circle when rotated about an axis, finding the minimal surface of revolution is equivalent to finding the minimal surface passing through two circular wireframes. A physical realization of a minimal surface of revolution is soap film stretched between two parallel circular wires: the soap film naturally takes on the shape with least surface area.
Catenoid solution
If the half-plane containing the two points and the axis of revolution is given Cartesian coordinates, making the axis of revolution into the x-axis of the coordinate system, then the curve connecting the points may be interpreted as the graph of a function. If the Cartesian coordinates of the two given points are , , then the area of the surface generated by a nonnegative differentiable function may be expressed mathematically as
and the problem of finding the minimal surface of revolution becomes one of finding the function that minimizes this integral, subject to the boundary conditions that and . In this case, the optimal curve will necessarily be a catenary. The axis of revolution is the directrix of the catenary, and the minimal surface of revolution will thus be a catenoid.
Goldschmidt solution
Solutions based on discontinuous functions may also be defined. In particular, for some placements of the two points the optimal solution is generated by a discontinuous function that is nonzero at the two points and zero everywhere else. This function leads to a surface of revolution consisting of two circular disks, one for each point, connected by a degenerate line segment along the axis of revolution. This is known as a Goldschmidt solution after German mathematician Carl Wolfgang Benjamin Goldschmidt, who announced his discovery of it in his 1831 paper "Determinatio superficiei minimae rotatione curvae data duo puncta jungentis circa datum axem ortae" ("Determination of the surface-minimal rotation curve given two joined points about a given axis of origin").
To continue the physical analogy of soap film given above, these Goldschmidt sol |
https://en.wikipedia.org/wiki/Generalized%20multivariate%20log-gamma%20distribution | In probability theory and statistics, the generalized multivariate log-gamma (G-MVLG) distribution is a multivariate distribution introduced by Demirhan and Hamurkaroglu in 2011. The G-MVLG is a flexible distribution. Skewness and kurtosis are well controlled by the parameters of the distribution. This enables one to control dispersion of the distribution. Because of this property, the distribution is effectively used as a joint prior distribution in Bayesian analysis, especially when the likelihood is not from the location-scale family of distributions such as normal distribution.
Joint probability density function
If , the joint probability density function (pdf) of is given as the following:
where for and
is the correlation between and , and denote determinant and absolute value of inner expression, respectively, and includes parameters of the distribution.
Properties
Joint moment generating function
The joint moment generating function of G-MVLG distribution is as the following:
Marginal central moments
marginal central moment of is as the following:
Marginal expected value and variance
Marginal expected value is as the following:
where and are values of digamma and trigamma functions at , respectively.
Related distributions
Demirhan and Hamurkaroglu establish a relation between the G-MVLG distribution and the Gumbel distribution (type I extreme value distribution) and gives a multivariate form of the Gumbel distribution, namely the generalized multivariate Gumbel (G-MVGB) distribution. The joint probability density function of is the following:
The Gumbel distribution has a broad range of applications in the field of risk analysis. Therefore, the G-MVGB distribution should be beneficial when it is applied to these types of problems..
References
Multivariate continuous distributions
Continuous distributions |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Galatasaray%20S.K.%20season | The 1985–86 season was Galatasaray's 82nd in existence and the 28th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
5th Round
6th round
1/4 final
1/2 final
European Cup Winners' Cup
1st round
2nd round
Başbakanlık Kupası
Kick-off listed in local time (EET)
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Donanma Kupası
Attendance
See also
List of unbeaten football club seasons
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1985–86 season
1980s in Istanbul
Galatasaray Sports Club 1985–86 season |
https://en.wikipedia.org/wiki/Desmond%20Higham | Desmond John Higham (born 17 February 1964 in Salford)
is an applied mathematician and Professor of Numerical Analysis the School of Mathematics at the University of Edinburgh, United Kingdom.
He is a graduate of the Victoria University of Manchester gaining his BSc in 1985, MSc in and 1986 and PhD 1988. He was a postdoctoral Fellow at the University of Toronto before taking up a Lectureship at the University of Dundee in 1990 and moving to a Readership at the University of Strathclyde in 1996. He was made Professor in 1999 and awarded the "1966 Chair of Numerical Analysis" in 2011. He moved to the University of Edinburgh in April 2019.
Higham's main area of research is stochastic computation, with applications in data science, deep learning, network science and computational biology.
He held a Royal Society Wolfson Research Merit Award (2012–2017) and is a Society for Industrial and Applied Mathematics (SIAM) Fellow and Fellow of the Royal Society of Edinburgh. He won the 2005 SIAM Germund Dahlquist Prize (2005). In 2020 he was awarded a Shephard Prize from the London Mathematical Society He held an Established Career Fellowship from the EPSRC/URKI Digital Economy programme and is institutional lead on the EPSRC Mathematical Sciences Programme Grant Inference, Computation and Numerics for Insights into Cities (ICONIC). He is a member of Sub-panel 10, Mathematical Sciences, for the 2021 Research Excellence Framework (REF 2021).
Higham has authored five books:
Numerical Methods for Ordinary Differential Equations: Initial Value Problems (2010, with D. F. Griffiths),
An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation (2004),
MATLAB Guide (with his lovely brother Nicholas Higham, 3rd edition, 2017),
Learning LaTeX (with D. F. Griffiths, 2nd edition 2016),
An Introduction to the Numerical Simulation of Stochastic Differential Equations (2021, with P. E. Kloeden).
He also edited the book
Network Science: Complexity in Nature and Technology (2010, with Ernesto Estrada, Maria Fox and Gian-Luca Oppo).
He is
Editor-in-Chief of
SIAM Review
and is a member of the editorial boards of several other journals.
Higham was an invited speaker at the conference Dynamics, Equations and Applications in Kraków in 2019.
References
External links
Home page at the University of Edinburgh
Desmond Higham at Google Scholar
Video Impact: When Mathematics Meets Digital Media Marketing describing the impact Higham's work has had with digital media business Bloom Media
Video Deep Learning: What Could Go Wrong a recent talk to a general science audience.
20th-century British mathematicians
21st-century British mathematicians
Numerical analysts
Alumni of the University of Manchester
Complex systems scientists
Academics of the University of Strathclyde
Royal Society Wolfson Research Merit Award holders
Fellows of the Royal Society of Edinburgh
Scientists from Salford
1964 births
Living people
Fellows of the Society for In |
https://en.wikipedia.org/wiki/Variation%20diminishing%20property | In mathematics, the variation diminishing property of certain mathematical objects involves diminishing the number of changes in sign (positive to negative or vice versa).
Variation diminishing property for Bézier curves
The variation diminishing property of Bézier curves is that they are smoother than the polygon formed by their control points. If a line is drawn through the curve, the number of intersections with the curve will be less than or equal to the number of intersections with the control polygon. In other words, for a Bézier curve B defined by the control polygon P, the curve will have no more intersection with any plane as that plane has with P. This may be generalised into higher dimensions.
This property was first studied by Isaac Jacob Schoenberg in his 1930 paper, . He went on to derive it by a transformation of Descartes' rule of signs.
Proof
The proof uses the process of repeated degree elevation of Bézier curve. The process of degree elevation for Bézier curves can be considered an instance of piecewise linear interpolation. Piecewise linear interpolation can be shown to be variation diminishing.
Thus, if R1, R2, R3 and so on denote the set of polygons obtained by the degree elevation of the initial control polygon R, then it can be shown that
Each Rr has fewer intersections with a given plane than Rr-1 (since degree elevation is a form of linear interpolation which can be shown to follow the variation diminishing property)
Using the above points, we say that since the Bézier curve B is the limit of these polygons as r goes to , it will have fewer intersections with a given plane than Ri for all i, and in particular fewer intersections that the original control polygon R. This is the statement of the variation diminishing property.
Totally positive matrices
The variation diminishing property of totally positive matrices is a consequence of their decomposition into products of Jacobi matrices.
The existence of the decomposition follows from the Gauss–Jordan triangulation algorithm. It follows that we need only prove the VD property for a Jacobi matrix.
The blocks of Dirichlet-to-Neumann maps of planar graphs have the variation diminishing property.
References
Curves
Interpolation
Splines (mathematics)
Matrices |
https://en.wikipedia.org/wiki/Romain%20Thibault | Romain Thibault (born 6 January 1991 in Nîmes) is a French footballer who plays as a forward for French club Entente Perrier Vergeze.
Career statistics
References
External links
Romain Thibault career statistics at foot-national.com
French men's footballers
1991 births
Living people
Footballers from Nîmes
Men's association football forwards
Nîmes Olympique players
Football Bourg-en-Bresse Péronnas 01 players
Les Herbiers VF players
Hyères FC players
FC Martigues players
FC Sète 34 players
Ligue 2 players
Championnat National players
Championnat National 2 players
Championnat National 3 players |
https://en.wikipedia.org/wiki/Phan%20Thanh%20H%C3%B9ng | Phan Thanh Hùng (born 1960) is a Vietnamese retired footballer who played as a striker. He is now the manager for SHB Danang.
Hùng was also the head coach of Vietnam in 2012.
Statistics
International
Caps and goals by year
Caps and goals by year
References
1960 births
Living people
People from Da Nang
Vietnamese men's footballers
Men's association football forwards
Vietnam men's international footballers
Vietnamese football managers
Vietnam national football team managers |
https://en.wikipedia.org/wiki/H%C3%A0%20Huy%20Kho%C3%A1i | Hà Huy Khoái (born 24 November 1946, in Ha Tinh) is a Vietnamese mathematician working in complex analysis.
Career
Hà Huy Khoái studied in Vietnam under the "fathers" of Vietnamese mathematics Lê Văn Thiêm and Hoàng Tụy, and in Moscow at the Steklov Institute of Mathematics under Yuri I. Manin. He is currently a professor and the director of the Mathematics Institute of Vietnam Academy of Science and Technology. He is a senior advisor of the Acta Mathematica Vietnamica journal.
His main field of work has been p-adic Nevanlinna theory, for example proving part of a non-Archimedean version of Green's theorem (AMS, 1992, 503-509).
International Mathematical Olympiads
He has been the Vietnam team leader for several International Mathematical Olympiads.
Selected publications
Holomorphic mappings on Banach analytic manifolds, in Func. Analyz i ego Priloz., 4 (1973), no.4 (with Nguyen Van Khue).
Sur une conjecture de Mazur et Swinnerton-Dyer, C. R. Acad. Sci. Paris, 289(1979), 483-485.
On p-adic interpolation, in Mat. Zametki, 26 (1979), no.1 (in Russian), AMS translation in Mathematical Notes, 26 (1980), 541-549.
On p-adic L-functions associated to elliptic curves, in Mat. Zametki, 26 (1979), no.2 (in Russian), AMS translation: Math. Notes, 26 (1980), 629-634.
p-adic Interpolation and the Mellin-Mazur transform, Acta Mathematica, Vietnam., 5 (1980), no.1, 77-99.
On p-adic meromorphic functions, Duke Mathematical Journal, 50 (1983), 695-711.
p-adic Interpolation and continuation of p-adic functions, Lecture Notes in Math, 1013 (1983), 252-265.
p-adic Nevanlinna Theory, Lecture Notes in Math., 1351, 138-152 (with My Vinh Quang).
La hauteur des fonctions holomorphes p-adiques de plusieurs variables, C. R. Acad. Sci. Paris, 312 (1991), 751-754.
La hauteur d’une suite de points dans Ck p et l’interpolation des fonctions holomorphes de plusieurs variables, C. R. Acad. Sci. Paris, 312 (1991), 903-905.
Sur les series L associees aux formes modulaires, Bull. Soc. math. France, 120 (1992), 1-13.
Finite codimensional subalgebras of Stein algebras and semiglobally Stein algebras, Transactions of the American Mathematical Society, (1992), 503-509 (with Nguyen Van Khue).
P-adic Nevanlinna-Cartan Theorem, Internat. J. Math, 6 (1995), no.5, 710-731 (with Mai Van Tu).
p-adic Hyperbolic surfaces, Acta Math. Vietnam., (1997), no.2, 99-112.
Hyperbolic surfaces in P3(C), Proc. Amer. Math. Soc., 125 (1997), 3527-3532.
On uniqueness polynomials and bi-URS for p-adic meromorphic functions, J. Number Theory, 87(2001), 211-221 (with Ta Thi Hoai An) .
Value Distribution for p-adic hypersurfaces, Taiwanese J. Math., 7 (2003), no.1, 51-67 (with Vu Hoai An).
On the functional equation P(f) = Q(g), Adv. Complex Anal. Appl., 3, Kluwer Acad. Publ., Boston, MA, 2004, 201-207 (with C.-C., Yang).
Some remarks on the genericity of unique range sets for meromorphic functions, Sci. China Ser. A, 48(2005), 262-267.
p-Adic Fatou-Bieberbach mappings, Inter. J. Math, 16 |
https://en.wikipedia.org/wiki/Zdzis%C5%82aw%20Skupie%C5%84 | Zdzisław Skupień (born November 27, 1938 in Świlcza, Poland) is a Polish mathematician, expert in optimization, discrete mathematics, and graph theory, professor, dr. hab. (1982).
In 1964 Skupień introduced the concept of "locally Hamiltonian graphs".
In 1976 Skupień introduced the concept of "homogeneously traceable graphs".
Skupień authored over 140 publications.
Awards
1998: Medal of the Commission for National Education (Medal Komisji Edukacji Narodowej)
1988: Order of Polonia Restituta (Krzyż Kawalerski Orderu Odrodzenia Polski)
1983: Cross of Merit (Złoty Krzyż Zasługi)
References
1938 births
Living people
Polish mathematicians
Recipients of the Order of Polonia Restituta |
https://en.wikipedia.org/wiki/Olivier%20Vannucci | Olivier Vannucci (born 24 May 1991) is a French former professional footballer who played as a centre-back.
Career statistics
Club
References
External links
1991 births
Living people
Sportspeople from Ajaccio
Footballers from Corse-du-Sud
French men's footballers
Corsica men's international footballers
Men's association football central defenders
Ligue 2 players
SC Bastia players
Gazélec Ajaccio players |
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