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https://en.wikipedia.org/wiki/Conditioning%20%28probability%29 | Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random result if the condition is completely specified; otherwise, if the condition is left random, the result of conditioning is also random.
Conditioning on the discrete level
Example: A fair coin is tossed 10 times; the random variable X is the number of heads in these 10 tosses, and Y is the number of heads in the first 3 tosses. In spite of the fact that Y emerges before X it may happen that someone knows X but not Y.
Conditional probability
Given that X = 1, the conditional probability of the event Y = 0 is
More generally,
One may also treat the conditional probability as a random variable, — a function of the random variable X, namely,
The expectation of this random variable is equal to the (unconditional) probability,
namely,
which is an instance of the law of total probability
Thus, may be treated as the value of the random variable corresponding to X = 1. On the other hand, is well-defined irrespective of other possible values of X.
Conditional expectation
Given that X = 1, the conditional expectation of the random variable Y is More generally,
(In this example it appears to be a linear function, but in general it is nonlinear.) One may also treat the conditional expectation as a random variable, — a function of the random variable X, namely,
The expectation of this random variable is equal to the (unconditional) expectation of Y,
namely,
or simply
which is an instance of the law of total expectation
The random variable is the best predictor of Y given X. That is, it minimizes the mean square error on the class of all random variables of the form f(X). This class of random variables remains intact if X is replaced, say, with 2X. Thus, It does not mean that rather, In particular, More generally, for every function g that is one-to-one on the set of all possible values of X. The values of X are irrelevant; what matters is the partition (denote it αX)
of the sample space Ω into disjoint sets {X = xn}. (Here are all possible values of X.) Given an arbitrary partition α of Ω, one may define the random variable E ( Y | α ). Still, E ( E ( Y | α)) = E ( Y ).
Conditional probability may be treated as a special case of conditional expectation. Namely, P ( A | X ) = E ( Y | X ) if Y is the indicator of A. Therefore the conditional probability also depends on the partition αX generated by X rather than on X itself; P ( A | g(X) ) = P (A | X) = P (A | α), α = αX = αg(X).
On the other hand, conditioning on an event B is well-defined, provided that irrespective of any partition that may contain B as one of several parts.
Conditional distribution
Given X = |
https://en.wikipedia.org/wiki/Koji%20Hirose | is a Japanese retired footballer.
Club career statistics
Updated to 23 February 2017.
References
External links
Profile at Tochigi SC
1984 births
Living people
Hannan University alumni
Association football people from Kyoto Prefecture
Japanese men's footballers
J2 League players
J3 League players
Sagan Tosu players
Tochigi SC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yu%20Eto | is a former Japanese footballer who last played for Kataller Toyama.
Club career statistics
Updated to 2 February 2018.
References
External links
Profile at Kataller Toyama
1983 births
Living people
Fukuoka University alumni
People from Ōnojō
Association football people from Fukuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Sagan Tosu players
Tokushima Vortis players
Kataller Toyama players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Toshimitsu%20Asai | is a Japanese football player. He plays for Nara Club.
Club statistics
References
External links
Profile at Nara Club
1983 births
Living people
Shizuoka Sangyo University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Sagan Tosu players
Blaublitz Akita players
Nara Club players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Takuma%20Hidaka | is a former Japanese football player.
Club statistics
Updated to 22 January 2016.
References
External links
1983 births
Living people
Meiji University alumni
Association football people from Hiroshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Sagan Tosu players
Hokkaido Consadole Sapporo players
Kataller Toyama players
Men's association football defenders |
https://en.wikipedia.org/wiki/Junya%20Yamashiro | is a Japanese football player.
Club statistics
References
External links
1985 births
Living people
Association football people from Kyoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Cerezo Osaka players
Sagan Tosu players
V-Varen Nagasaki players
Zweigen Kanazawa players
Japan Soccer College players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yasumichi%20Uchima | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Miyazaki Sangyo-keiei University alumni
Association football people from Okinawa Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Sagan Tosu players
Gainare Tottori players
Men's association football defenders |
https://en.wikipedia.org/wiki/Takuya%20Muro | is a former Japanese football player.
Club statistics
References
External links
1982 births
Living people
Kansai Gaidai University alumni
Association football people from Osaka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Tokyo Verdy players
Sagan Tosu players
Oita Trinita players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Luk%C3%A1%C5%A1%20Jani%C4%8D | Lukáš Janič (born 30 December 1986) is a professional Slovak football midfielder who plays for Tatran Oravské Veselé.
Career statistics
Last updated: 16 May 2010
External links
MFK Košice profile
1986 births
Living people
Footballers from Prešov
Slovak men's footballers
Slovak expatriate men's footballers
Slovakia men's under-21 international footballers
1. FC Tatran Prešov players
FC VSS Košice players
Korona Kielce players
Sandecja Nowy Sącz players
Podbeskidzie Bielsko-Biała players
FC ViOn Zlaté Moravce players
FK Teplice players
MFK Zemplín Michalovce players
ŠKF Sereď players
MFK Ružomberok players
ŠK Odeva Lipany players
TJ Tatran Oravské Veselé players
Men's association football midfielders
Slovak First Football League players
2. Liga (Slovakia) players
3. Liga (Slovakia) players
Ekstraklasa players
I liga players
Czech First League players
Slovak expatriate sportspeople in Poland
Slovak expatriate sportspeople in the Czech Republic
Slovak expatriate sportspeople in Austria
Expatriate men's footballers in Poland
Expatriate men's footballers in the Czech Republic
Expatriate men's footballers in Austria |
https://en.wikipedia.org/wiki/Andr%C3%A9%20Haefliger | André Haefliger (; 22 May 19297 March 2023) was a Swiss mathematician who worked primarily on topology.
Education and career
Haefliger went to school in Nyon and then attended his final years at Collège de Genève in Geneva. He studied mathematics at the University of Lausanne from 1948 to 1952. He worked for two years as a teaching assistant at École Polytechnique de l'Université de Lausanne. He then moved to University of Strasbourg, then he followed Charles Ehresmann in Paris, where he received his Ph.D. degree in 1958. His thesis was entitled "Structures feuilletées et cohomologie à valeurs dans un faisceau de groupoïdes" and was written under the supervision of Charles Ehresmann.
Haefliger got a research fellowship for one year at the University of Paris, where he participated in the seminar of Henri Cartan, and then from 1959 to 1961 he worked at the Institute for Advanced Study in Princeton, New Jersey. Since 1962 he has been a full professor at the University of Geneva until his retirement in 1996.
In 1966 Haefliger was invited speaker at the International Congress of Mathematicians in Moscow. In 1974–75, he was president of the Swiss Mathematical Society.
Haefliger obtained a Doctorate honoris causa from the ETH Zurich in 1992 and from the University of Dijon in 1997. In 2020 Haefliger and Martin Bridson were awarded the American Mathematical Society's Leroy P. Steele Prize for Mathematical Exposition, for their book Metric Spaces of Non-Positive Curvature (Springer Verlag, 1999).
Haefliger died on 7 March 2023, at the age of 93.
Research
Haefliger's main research interests were differential topology and geometry.
Haefliger found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold. In two papers in the Annals he studied various embedding of spheres in relations to knot theory. He has also made important contributions in the theory of foliations, introducing the notion of Haefliger structures.
He wrote more than 80 papers in peer review journals and had 20 Ph.D. students, including Augustin Banyaga and the future Field Medalist Vaughan Jones.
Selected works
"Travaux de Novikov sur les feuilletages." Séminaire Bourbaki 10 (1966-1968): 433-444.
"Sur les classes caractéristiques des feuilletages." Séminaire Bourbaki 14 (1971-1972): 239-260.
"Sphères d'homotopie nouées." Séminaire Bourbaki 9 (1964-1966): 57-68.
"Feuilletages riemanniens." Séminaire Bourbaki 31 (1988-1989): 183-197.
"Plongements de variétés dans le domaine stable." Séminaire Bourbaki 8 (1962-1964): 63-77.
(Ph.D. Thesis)
References
External links
1929 births
2023 deaths
Swiss mathematicians
University of Strasbourg alumni
University of Lausanne alumni
Academic staff of the University of Geneva
Topologists
Institute for Advanced Study people
People from Nyon
University of Paris alumni |
https://en.wikipedia.org/wiki/Christophe%20Breuil | Christophe Breuil (; born 1968) is a French mathematician, who works in arithmetic geometry and algebraic number theory.
Work
With Fred Diamond, Richard Taylor and Brian Conrad in 1999, he proved the Taniyama–Shimura conjecture, which previously had only been proved for semistable elliptic curves by Andrew Wiles and Taylor in their proof of Fermat's Last Theorem. Later, he worked on the p-adic Langlands conjecture.
Academic life
Breuil attended schools in Brive-la-Gaillarde and Toulouse and studied from 1990 to 1992 at the École Polytechnique. In 1993, he obtained his DEA degree at the Paris-Sud 11 University located in Orsay. From 1993 to 1996 he conducted research at the École Polytechnique and taught simultaneously at the University of Paris-Sud, Orsay, and in 1996 received his PhD from the École Polytechnique, supervised by Jean-Marc Fontaine with the thesis "Cohomologie log-cristalline et représentations galoisiennes p -adiques". In 1997, he gave the Cours Peccot at the Collège de France. In 2001 he obtained a habilitation degree entitled "Aspects entiers de la théorie de Hodge p-adique et applications" at Paris-Sud 11 University. Between 2002 and 2010 he was at the IHES. From 2010 he has been in the Mathematics Department of University of Paris-Sud as Director of Research with the CNRS. In 2007–2008 he was a visiting professor at Columbia University.
Awards and recognition
In 1993 he was awarded the Prix Gaston Julia at the École Polytechnique.
In 2002 he received the of the French Academy of Sciences and the 2006 Prix Dargelos Anciens Élèves of the École Polytechnique.
He was an invited speaker in International Congress of Mathematicians 2010, Hyderabad on the topic of "Number Theory."
References
External links
Living people
20th-century French mathematicians
21st-century French mathematicians
Paris-Sud University alumni
1968 births
Arithmetic geometers
École Polytechnique alumni
Fermat's Last Theorem
Research directors of the French National Centre for Scientific Research |
https://en.wikipedia.org/wiki/Aztec%20diamond | In combinatorial mathematics, an Aztec diamond of order n consists of all squares of a square lattice whose centers (x,y) satisfy |x| + |y| ≤ n. Here n is a fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both x and y are half-integers.
The Aztec diamond theorem states that the number of domino tilings of the Aztec diamond of order n is 2n(n+1)/2. The Arctic Circle theorem says that a random tiling of a large Aztec diamond tends to be frozen outside a certain circle.
It is common to color the tiles in the following fashion. First consider a checkerboard coloring
of the diamond. Each tile will cover exactly one black square. Vertical tiles where the top square covers a black square,
is colored in one color, and the other vertical tiles in a second. Similarly for horizontal tiles.
Knuth has also defined Aztec diamonds of order n + 1/2. They are identical with the polyominoes associated with the centered square numbers.
Non-intersecting paths
Something that is very useful for counting tilings is looking at the non-intersecting paths through its corresponding directed graph. If we define our movements through a tiling (domino tiling) to be
(1,1) when we are the bottom of a vertical tiling
(1,0) where we are the end of a horizontal tiling
(1,-1) when we are at the top of a vertical tiling
Then through any tiling we can have these paths from our sources to our sinks. These movements are similar to Schröder paths. For example, consider an Aztec Diamond of order 2, and after drawing its directed graph we can label its sources and sinks. are our sources and are our sinks. On its directed graph, we can draw a path from to , this gives us a path matrix, ,
where all the paths from to . The number of tilings for order 2 is
det
According to Lindstrom-Gessel-Viennot, if we let S be the set of all our sources and T be the set of all our sinks of our directed graph then
detnumber of non-intersecting paths from S to T.
Considering the directed graph of the Aztec Diamond, it has also been shown by Eu and Fu that Schröder paths and the tilings of the Aztec diamond are in bijection. Hence, finding the determinant of the path matrix, , will give us the number of tilings for the Aztec Diamond of order n.
Another way to determine the number of tilings of an Aztec Diamond is using Hankel matrices of large and small Schröder numbers, using the method from Lindstrom-Gessel-Viennot again. Finding the determinant of these matrices gives us the number of non-intersecting paths of small and large Schröder numbers, which is in bijection with the tilings. The small Schröder numbers are and the large Schröder numbers are , and in general our two Hankel matrices will be
and
where det and det where (It also true that det where this is the Hankel matrix like , but started with instead of for the first entry of the matrix in the top left corner).
Other tiling problems
Consider the shape, blo |
https://en.wikipedia.org/wiki/FKG%20inequality | In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), due to . Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated. It was obtained by studying the random cluster model.
An earlier version, for the special case of i.i.d. variables, called Harris inequality, is due to , see below. One generalization of the FKG inequality is the Holley inequality (1974) below, and an even further generalization is the Ahlswede–Daykin "four functions" theorem (1978). Furthermore, it has the same conclusion as the Griffiths inequalities, but the hypotheses are different.
The inequality
Let be a finite distributive lattice, and μ a nonnegative function on it, that is assumed to satisfy the (FKG) lattice condition (sometimes a function satisfying this condition is called log supermodular) i.e.,
for all x, y in the lattice .
The FKG inequality then says that for any two monotonically increasing functions ƒ and g on , the following positive correlation inequality holds:
The same inequality (positive correlation) is true when both ƒ and g are decreasing. If one is increasing and the other is decreasing, then they are negatively correlated and the above inequality is reversed.
Similar statements hold more generally, when is not necessarily finite, not even countable. In that case, μ has to be a finite measure, and the lattice condition has to be defined using cylinder events; see, e.g., Section 2.2 of .
For proofs, see or the Ahlswede–Daykin inequality (1978). Also, a rough sketch is given below, due to , using a Markov chain coupling argument.
Variations on terminology
The lattice condition for μ is also called multivariate total positivity, and sometimes the strong FKG condition; the term (multiplicative) FKG condition is also used in older literature.
The property of μ that increasing functions are positively correlated is also called having positive associations, or the weak FKG condition.
Thus, the FKG theorem can be rephrased as "the strong FKG condition implies the weak FKG condition".
A special case: the Harris inequality
If the lattice is totally ordered, then the lattice condition is satisfied trivially for any measure μ. In case the measure μ is uniform, the FKG inequality is Chebyshev's sum inequality: if the two increasing functions take on values and , then
More generally, for any probability measure μ on and increasing functions ƒ and g,
which follows immediately from
The lattice condition is trivially satisfied also when the lattice is the product of totally ordered lattices, , and is a product measure. Often all the factors (both the lattices and the measures) are identical, i.e., μ is the probability distribution of i.i.d. random variables.
The FKG i |
https://en.wikipedia.org/wiki/Structure%20constants | In mathematics, the structure constants or structure coefficients of an algebra over a field are the coefficients of the basis expansion (into linear combination of basis vectors) of the products of basis vectors.
Because the product operation in the algebra is bilinear, by linearity knowing the product of basis vectors allows to compute the product of any elements (just like a matrix allows to compute the action of the linear operator on any vector by providing the action of the operator on basis vectors).
Therefore, the structure constants can be used to specify the product operation of the algebra (just like a matrix defines a linear operator). Given the structure constants, the resulting product is obtained by bilinearity and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra.
Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used when discussing Lie algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles (recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket, usually defined via the commutator).
Definition
Given a set of basis vectors for the underlying vector space of the algebra, the product operation is uniquely defined by the products of basis vectors:
.
The structure constants or structure coefficients are just the coeffiecients of in the same basis:
.
Otherwise said they are the coefficients that express as linear combination of the basis vectors .
The upper and lower indices are frequently not distinguished, unless the algebra is endowed with some other structure that would require this (for example, a pseudo-Riemannian metric, on the algebra of the indefinite orthogonal group so(p,q)). That is, structure constants are often written with all-upper, or all-lower indexes. The distinction between upper and lower is then a convention, reminding the reader that lower indices behave like the components of a dual vector, i.e. are covariant under a change of basis, while upper indices are contravariant.
The structure constants obviously depend on the chosen basis. For Lie algebras, one frequently used convention for the basis is in terms of the ladder operators defined by the Cartan subalgebra; this is presented further down in the article, after some preliminary examples.
Example: Lie algebras
For a Lie algebra, the basis vectors are termed the generators of the algebra, and the product rather called the Lie bracket (often the Lie bracket is an additional product operation beyond the already existing product, thus necessitating a separate name). For two vectors and in the algebra, the Lie bracket is denoted .
Again, there is no particular need to distinguish the upper and lower indices; they can be written all up or all down. In physics, it is common to use the notation for |
https://en.wikipedia.org/wiki/Kodai%20Watanabe | is a Japanese footballer who plays for Vonds Ichihara from 2023.
Career
On 23 December 2022, Watanabe joined to Kantō club part of JRL, Vonds Ichihara for upcoming 2023 season.
Career statistics
Updated to the end 2022 season.
Club
References
External links
Profile at Renofa Yamaguchi
1986 births
Living people
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Vegalta Sendai players
Montedio Yamagata players
Renofa Yamaguchi FC players
Thespakusatsu Gunma players
Vonds Ichihara players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yugo%20Ichiyanagi | is a Japanese football player who plays for Taiwanese football club AC Taipei.
Club statistics
Updated to 6 November 2020.
References
External links
1985 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Taiwan Football Premier League players
Tokyo Verdy players
Sagan Tosu players
Vegalta Sendai players
Fagiano Okayama players
Matsumoto Yamaga FC players
FC Ryukyu players
Thespakusatsu Gunma players
Taichung Futuro F.C. players
Footballers at the 2006 Asian Games
Men's association football defenders
Asian Games competitors for Japan |
https://en.wikipedia.org/wiki/Keita%20Isozaki | is a former Japanese football player.
Playing career
Club statistics
References
External links
1980 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Shonan Bellmare players
Mito HollyHock players
Vegalta Sendai players
Sagan Tosu players
Men's association football defenders |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20China | These are the Records for the Chinese Football League since its inception in 1994.
Scorers
All-time top scorers
The following is a cumulative record of all time goals scored by single players in Chinese Top Level Professional League, since its inception in 1994 as the Chinese Jia-A League. Players with at least 50 goals are listed.
Top scoring foreign players
Appearances
All-time top appearances
The following table lists players who have played the most games within the top tier of Chinese football. Zou Yougen is the first player to make 300 or more appearances within the Jia-A or CSL League.
Top Appearance foreigner players
This section listed the foreign players who had most appearances in Chinese Top Level professional football leagues
International performances by players
Scorer record in International club competition
Appearance record in International club competition
Manager
Most matches in Charge
League winning manager
this table listed manager of Chinese top level professional league Champions
All-time League Table
The All-Time League Table is a cumulative record of all match results, points and goals of every team that has played in the League since its inception in 1994. The table that follows is accurate as of the end of the Chinese Super League 2009 season. Teams in bold are part of the Chinese Super League 2010. Numbers in bold are the record (highest) numbers in each column.
League attendance
The attendance figures are announced by the league department of CFA. 1994–2001 Jia-A figures was corrected by Titan Sports., also on Blog.Sina.Com, the figures after 2002 was calculated from the record of the Official CFA site
Awards
China used to have three main award series, they are:
Mr. China Football Award, started from 1994, voted by fans all over the country, to award the player of the year.
Three Golden Awards, organized by China Sports Daily, started from 1983, to award the best performance players in all kinds of tournament, not only in the leagues, the awards includes:
Player of the year – Golden Ball awards
Best Scorer of the year – Golden Boot awards
Referee of the year – Golden Whistle awards
Official CFA League awards, Is the Official League awards, Awarded by CFA, includes:
Most Valuable players of the league awards (MVP)
League Top Scorer awards
Manager of the year awards
Referee of the year awards
in 1999 season, three other items added as regular awards
Youth Player of the year awards (MYP)
Club of the year awards
Fair play club of the year awards
The league oscar also involve some occasional awards
In the year 2002, the three series above was merged into the Official CFA Annual Awards. It now includes:
Mr. Football League Golden Ball awards, also called MVP awards or Mr. China Football awards by some media
Best Scorer Golden Boot awards
Ref'e'ree of the year Golden Whistle awards
Manager of the year awards
Youth player of the year awards
Best player awards
Most Valuable Player of the League
M |
https://en.wikipedia.org/wiki/Akihiro%20Sato%20%28footballer%2C%20born%20August%201986%29 | is a Japanese footballer who plays for Montedio Yamagata in the J2 League.
Club career
After four season with Kashima Antlers, he was released by the club in November 2015.
Career statistics
Updated to end of 2018 season.
References
External links
Profile at Roasso Kumamoto
1986 births
Living people
Association football people from Mie Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
Ehime FC players
Kashima Antlers players
Roasso Kumamoto players
Montedio Yamagata players
Footballers at the 2006 Asian Games
Men's association football goalkeepers
Asian Games competitors for Japan |
https://en.wikipedia.org/wiki/Kenta%20Uchida | is a Japanese football player who plays for Ventforet Kofu.
Career
On 22 December 2016, Uchida signed for Nagoya Grampus after a two years-period at Ehime FC.
Career statistics
Club
References
External links
Profile at Nagoya Grampus
1989 births
Living people
Association football people from Mie Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
Ehime FC players
Shimizu S-Pulse players
Kataller Toyama players
Nagoya Grampus players
Montedio Yamagata players
Ventforet Kofu players
Men's association football defenders |
https://en.wikipedia.org/wiki/Takuya%20Marutani | is a Japanese retired football player who last played for Oita Trinita.
Career
Marutani retired in December 2019.
Career statistics
Updated to 25 February 2019.
1Includes Japanese Super Cup, J. League Championship and FIFA Club World Cup.
References
External links
Profile at Oita Trinita
Profile at Sanfrecce Hiroshima
1989 births
Living people
Association football people from Tottori Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
Oita Trinita players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kohei%20Shimizu | is a former Japanese professional footballer who played as a winger.
Career statistics
Club
Updated to 22 February 2019.
1Includes Emperor's Cup.
2Includes J. League Cup.
3Includes AFC Champions League.
4Includes FIFA Club World Cup, Japanese Super Cup and J. League Championship.
Honours
Club
Sanfrecce Hiroshima
J1 League (3) : 2012, 2013, 2015
J2 League (1) : 2008
Japanese Super Cup (4) : 2008, 2013, 2014, 2016
References
External links
Profile at Shimizu S-Pulse
Profile at Sanfrecce Hiroshima
1989 births
Living people
People from Munakata, Fukuoka
Association football people from Fukuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
Shimizu S-Pulse players
Ventforet Kofu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Stanislav%20Ki%C5%A1%C5%A1 | Stanislav Kišš (born 16 December 1978 in Prešov) is a professional Slovak football defender who currently plays for TJ Rozvoj Pušovce.
Career statistics
Last updated: 28 December 2009
External links
Player profile at mfkkosice.sk
1978 births
Living people
Slovak men's footballers
FC Steel Trans Ličartovce players
FC VSS Košice players
Slovak First Football League players
MFK Vranov nad Topľou players
FK Železiarne Podbrezová players
Men's association football fullbacks
Footballers from Prešov |
https://en.wikipedia.org/wiki/Erez%20Lieberman%20Aiden | Erez Lieberman Aiden (born 1980, né Erez Lieberman) is an American research scientist active in multiple fields related to applied mathematics. He is an associate professor at the Baylor College of Medicine, and formerly a fellow at the Harvard Society of Fellows and visiting faculty member at Google. He is an adjunct assistant professor of computer science at Rice University. Using mathematical and computational approaches, he has studied evolution in a range of contexts, including that of networks through evolutionary graph theory and languages in the field of culturomics. He has published scientific articles in a variety of disciplines.
Lieberman Aiden has won awards including the Lemelson–MIT Student Prize and the American Physical Society's Award for Outstanding Doctoral Thesis Research in Biological Physics. In 2009, Lieberman Aiden was named as one of 35 top innovators under 35 by Technology Review and in 2011 he was one of the recipients of the Presidential Early Career Award for Scientists and Engineers.
Early life and education
Lieberman grew up in Brooklyn with three siblings. He began computer programming at the age of seven. His father, Aharon Lieberman, was a technology entrepreneur and owned a factory in New Jersey. As a child Lieberman Aiden spoke Hebrew and Hungarian, making English his third language.
Lieberman Aiden studied mathematics, physics, and philosophy at Princeton, and earned a master's degree in History at Yeshiva University. He proceeded to complete a joint PhD in mathematics and bioengineering at the Harvard–MIT Division of Health Sciences and Technology, where he was advised by Eric Lander and Martin Nowak.
Research and career
Lieberman Aiden contributed to the founding of evolutionary graph theory along with his PhD supervisor Martin Nowak. He has since been involved in researching the three dimensional structure of the human genome and the field of culturomics.
3D genome structure
Lieberman Aiden was part of a team of scientists from the University of Massachusetts Medical School and MIT that first suggested human DNA folds into a fractal globule rather than an equilibrium globule. This finding explains how each cell's genome is able to be heavily compacted without forming a knot. Lieberman Aiden and coworkers invented a variant of chromosome conformation capture called "Hi-C" which produces a genome-wide measure of contact probabilities that point to a 3-dimensional genome structure. This technique combines existing chromosome capture methodology with next-generation sequencing, enabling an all-versus-all measure of chromatin contacts.
In 2009 this work was published in the journal Science and was featured as a cover illustration. Following the publication, Lieberman Aiden was quoted as saying:
In 2014, he served as a senior author on an article in Cell which described a refined method of Hi-C which his team used to describe the fundamental organization of DNA.
Culturomics
Lieberman Aiden was involved |
https://en.wikipedia.org/wiki/Gyrovector%20space | A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups. Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities (also called boosts – "boosts" are aspects of relative velocities, and should not be conflated with "translations"). This is achieved by introducing "gyro operators"; two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity.
Name
Gyrogroups are weakly associative group-like structures. Ungar proposed the term gyrogroup for what he called a gyrocommutative-gyrogroup, with the term gyrogroup being reserved for the non-gyrocommutative case, in analogy with groups vs. abelian groups. Gyrogroups are a type of Bol loop. Gyrocommutative gyrogroups are equivalent to K-loops although defined differently. The terms Bruck loop and dyadic symset are also in use.
Mathematics of gyrovector spaces
Gyrogroups
Axioms
A gyrogroup (G, ) consists of an underlying set G and a binary operation satisfying the following axioms:
In G there is at least one element 0 called a left identity with 0 a = a for all a in G.
For each a in G there is an element a in G called a left inverse of a with (a) a = 0.
For any a, b, c in G there exists a unique element gyr[a,b]c in G such that the binary operation obeys the left gyroassociative law: a (b c) = (a b) gyr[a,b]c
The map gyr[a,b]: G → G given by c ↦ gyr[a,b]c is an automorphism of the magma (G, ) – that is, gyr[a,b] is a member of Aut(G, ) and the automorphism gyr[a,b] of G is called the gyroautomorphism of G generated by a, b in G. The operation gyr: G × G → Aut(G, ) is called the gyrator of G.
The gyroautomorphism gyr[a,b] has the left loop property gyr[a,b] = gyr[a b,b]
The first pair of axioms are like the group axioms. The last pair present the gyrator axioms and the middle axiom links the two pairs.
Since a gyrogroup has inverses and an identity it qualifies as a quasigroup and a loop.
Gyrogroups are a generalization of groups. Every group is an example of a gyrogroup with gyr[a,b] defined as the identity map for all a and b in G.
An example of a finite gyrogroup is given in .
Identities
Some identities which hold in any gyrogroup (G, ) are:
(gyration)
(left associativity)
(right associativity)
Furthermore, one may prove the Gyration inversion law, which is the motivation for the definition of gyrocommutativity below:
(gyration inversion law)
Some additional theorems satisfied by the Gyration group of any gyrogroup include:
(identity gyrations)
(gyroautomorphism inversion law)
(gyration even property)
(right loop property)
(left loop property)
More identities given on |
https://en.wikipedia.org/wiki/Patrik%20Kaminsk%C3%BD | Patrik Kaminský (born 27 October 1978 in Prešov) is a professional Slovak football defender who currently plays for the 3. liga club FC Lokomotíva Košice.
Career statistics
Last updated: 28 December 2009
External links
Player profile at mfkkosice.sk
Living people
1978 births
Slovak men's footballers
FC Steel Trans Ličartovce players
FC VSS Košice players
FK Bodva Moldava nad Bodvou players
FC Lokomotíva Košice players
Slovak First Football League players
Men's association football central defenders
Footballers from Prešov |
https://en.wikipedia.org/wiki/XYZ%20inequality | In combinatorial mathematics, the XYZ inequality, also called the Fishburn–Shepp inequality, is an inequality for the number of linear extensions of finite partial orders. The inequality was conjectured by Ivan Rival and Bill Sands in 1981. It was proved by Lawrence Shepp in
. An extension was given by Peter Fishburn in .
It states that if x, y, and z are incomparable elements of a finite poset, then
,
where P(A) is the probability that a linear order extending the partial order has the property A.
In other words, the probability that increases if one adds the condition that . In the language of conditional probability,
The proof uses the Ahlswede–Daykin inequality.
See also
FKG inequality
References
Inequalities
Theorems in combinatorics
Independence (probability theory) |
https://en.wikipedia.org/wiki/Niklas%20Anger | Niklas Anger (born 14 July 1977) was a Swedish professional ice hockey player. He was drafted 112th overall in the 1995 NHL Entry Draft by the Montreal Canadiens
Career statistics
Regular season and playoffs
International
References
External links
1977 births
Living people
AIK IF players
Almtuna IS players
Brynäs IF players
Djurgårdens IF Hockey players
EHC Basel players
HC Alleghe players
HC Ambrì-Piotta players
HC Gardena players
HC Sierre players
Linköping HC players
Montreal Canadiens draft picks
Ice hockey people from Stockholm
Swedish expatriate sportspeople in Switzerland
Swedish ice hockey forwards
Timrå IK players
Wings HC Arlanda players |
https://en.wikipedia.org/wiki/Teemu%20Aalto | Teemu Matias Aalto (born 30 March 1978) is a Finnish former professional ice hockey player. He played in the for Ilves in the Finnish Liiga.
Career statistics
Regular season and playoffs
References
External links
1978 births
Living people
Sportspeople from Kokkola
Finnish ice hockey defencemen
HPK players
Timrå IK players
SC Bern players
Tappara players
Linköping HC players
Lukko players
21st-century Finnish people |
https://en.wikipedia.org/wiki/Train%20track%20map | In the mathematical subject of geometric group theory, a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations. This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge e of the graph and for every positive integer n the path fn(e) is immersed, that is fn(e) is locally injective on e. Train-track maps are a key tool in analyzing the dynamics of automorphisms of finitely generated free groups and in the study of the Culler–Vogtmann Outer space.
History
Train track maps for free group automorphisms were introduced in a 1992 paper of Bestvina and Handel. The notion was motivated by Thurston's train tracks on surfaces, but the free group case is substantially different and more complicated. In their 1992 paper Bestvina and Handel proved that every irreducible automorphism of Fn has a train-track representative. In the same paper they introduced the notion of a relative train track and applied train track methods to solve the Scott conjecture which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n. In a subsequent paper Bestvina and Handel applied the train track techniques to obtain an effective proof of Thurston's classification of homeomorphisms of compact surfaces (with or without boundary) which says that every such homeomorphism is, up to isotopy, either reducible, of finite order or pseudo-anosov.
Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Train tracks are particularly useful since they allow to understand long-term growth (in terms of length) and cancellation behavior for large iterates of an automorphism of Fn applied to a particular conjugacy class in Fn. This information is especially helpful when studying the dynamics of the action of elements of Out(Fn) on the Culler–Vogtmann Outer space and its boundary and when studying Fn actions of on real trees. Examples of applications of train tracks include: a theorem of Brinkmann proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups; and others.
Train tracks were a key tool in the proof by Bestvina, Feighn and Handel that the group Out(Fn) satisfies the Tits alternative.
The machinery of train tracks for injective endomorphisms of free groups was later developed by Dicks and Ventura.
Formal definition
Combinatorial map
For a finite graph Γ (which is thought of here as a 1-dimensional cell |
https://en.wikipedia.org/wiki/Regular%20paperfolding%20sequence | In mathematics the regular paperfolding sequence, also known as the dragon curve sequence, is an infinite sequence of 0s and 1s. It is obtained from the repeating partial sequence
by filling in the question marks by another copy of the whole sequence. The first few terms of the resulting sequence are:
If a strip of paper is folded repeatedly in half in the same direction, times, it will get folds, whose direction (left or right) is given by the pattern of 0's and 1's in the first terms of the regular paperfolding sequence. Opening out each fold to create a right-angled corner (or, equivalently, making a sequence of left and right turns through a regular grid, following the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal:
Properties
The value of any given term in the regular paperfolding sequence, starting with , can be found recursively as follows. Divide by two, as many times as possible, to get a factorization of the form where is an odd number. Then
Thus, for instance, : dividing 12 by two, twice, leaves the odd number 3. As another example, because 13 is congruent to 1 mod 4.
The paperfolding word 1101100111001001..., which is created by concatenating the terms of the regular paperfolding sequence, is a fixed point of the morphism or string substitution rules
11 → 1101
01 → 1001
10 → 1100
00 → 1000
as follows:
11 → 1101 → 11011001 → 1101100111001001 → 11011001110010011101100011001001 ...
It can be seen from the morphism rules that the paperfolding word contains at most three consecutive 0s and at most three consecutive 1s.
The paperfolding sequence also satisfies the symmetry relation:
which shows that the paperfolding word can be constructed as the limit of another iterated process as follows:
1
1 1 0
110 1 100
1101100 1 1100100
110110011100100 1 110110001100100
In each iteration of this process, a 1 is placed at the end of the previous iteration's string, then this string is repeated in reverse order, replacing 0 by 1 and vice versa.
Generating function
The generating function of the paperfolding sequence is given by
From the construction of the paperfolding sequence, it can be seen that G satisfies the functional relation
Paperfolding constant
Substituting into the generating function gives a real number between and whose binary expansion is the paperfolding word
This number is known as the paperfolding constant and has the value
General paperfolding sequence
The regular paperfolding sequence corresponds to folding a strip of paper consistently in the same direction. If we allow the direction of the fold to vary at each step we obtain a more general class of sequences. Given a binary sequence (fi), we can define a general paperfolding sequence with folding instructions (fi).
For a binary word w, let w‡ denote the reverse of the complement of w. Define an operator Fa as
and then define a sequence of words depending on the (fi) |
https://en.wikipedia.org/wiki/Satoshi%20Hida | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Association football people from Mie Prefecture
People from Matsusaka, Mie
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Singapore Premier League players
Kawasaki Frontale players
Vegalta Sendai players
Albirex Niigata Singapore FC players
Zweigen Kanazawa players
Veertien Mie players
Japanese expatriate men's footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Junichi%20Misawa | is a former Japanese football player.
Club statistics
References
External links
1985 births
Living people
University of Tsukuba alumni
Association football people from Hokkaido
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Vegalta Sendai players
FC Ryukyu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Tam%C3%A1s%20Laps%C3%A1nszki | Tamás Lapsánszki (born 9 November 1974) is a Hungarian footballer who plays for Békéscsabai Előre FC as midfielder.
External links
Career statistics
1974 births
Living people
Hungarian men's footballers
Men's association football defenders
Békéscsaba 1912 Előre footballers
Footballers from Kecskemét |
https://en.wikipedia.org/wiki/Graph%20dynamical%20system | In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
The work on GDSs considers finite graphs and finite state spaces. As such, the research typically involves techniques from, e.g., graph theory, combinatorics, algebra, and dynamical systems rather than differential geometry. In principle, one could define and study GDSs over an infinite graph (e.g. cellular automata or probabilistic cellular automata over or interacting particle systems when some randomness is included), as well as GDSs with infinite state space (e.g. as in coupled map lattices); see, for example, Wu. In the following, everything is implicitly assumed to be finite unless stated otherwise.
Formal definition
A graph dynamical system is constructed from the following components:
A finite graph Y with vertex set v[Y] = {1,2, ... , n}. Depending on the context the graph can be directed or undirected.
A state xv for each vertex v of Y taken from a finite set K. The system state is the n-tuple x = (x1, x2, ... , xn), and x[v] is the tuple consisting of the states associated to the vertices in the 1-neighborhood of v in Y (in some fixed order).
A vertex function fv for each vertex v. The vertex function maps the state of vertex v at time t to the vertex state at time t + 1 based on the states associated to the 1-neighborhood of v in Y.
An update scheme specifying the mechanism by which the mapping of individual vertex states is carried out so as to induce a discrete dynamical system with map F: Kn → Kn.
The phase space associated to a dynamical system with map F: Kn → Kn is the finite directed graph with vertex set Kn and directed edges (x, F(x)). The structure of the phase space is governed by the properties of the graph Y, the vertex functions (fi)i, and the update scheme. The research in this area seeks to infer phase space properties based on the structure of the system constituents. The analysis has a local-to-global character.
Generalized cellular automata (GCA)
If, for example, the update scheme consists of applying the vertex functions synchronously one obtains the class of generalized cellular automata (CA). In this case, the global map F: Kn → Kn is given by
This class is referred to as generalized cellular automata since the classical or standard cellular automata are typically defined and studied over regular graphs or grids, and the vertex functions are typically assumed to be identical.
Example: Let Y be the circle graph on vertices {1,2,3,4} with edges {1,2}, {2,3}, {3,4} and {1,4}, denoted Circ4. Let K = {0,1} be the state space for each vertex and use the function nor3 : K3 → K defined by nor3(x,y,z) = (1 + x)(1 + y)(1 + z) with arithmetic modulo 2 for all vertex functions. Then for example t |
https://en.wikipedia.org/wiki/L%20%E2%80%93%20A%20Mathemagical%20Adventure | L – A Mathemagical Adventure is an educational adventure game that was created for the BBC/Acorn class of computers in 1984. It was written by members of the Association of Teachers of Mathematics and found its way into school computers, predominantly in the UK. The game is controlled by a contemporaneous two-word input system in the form of "<verb> <noun>", e.g. "get book", and includes several mathematical and logical puzzles. The target audience for L is mid-late primary (elementary) school children, and the puzzles are based on this level of critical thinking.
Introduction text
It is a very hot day. You are sitting on the grass outside a crumbling palace. Your sister is reading a book called "Fractions and the Four Rules - 5000 Carefully Graded Problems". You are bored and the heat is making you feel a little sleepy. Suddenly you see an old man dressed as an abbot. He glances at you nervously and disappears through a small door in the side of the palace.
Memorable characters/puzzles
The recommended usage of L was for a class of students (or smaller groups) to attempt solving the various puzzles in the game in short pieces over the course of a semester, year or similarly protracted time and is hence somewhat difficult to remember.
Drogo Robot Guard - These appear somewhat randomly at different times during the game and attempt to capture the player. It is possible to evade capture (a small puzzle in itself), and avoid being taken to the attic.
Runia - It is possible to solve the game by completing most of the puzzles, but a full solution includes rescuing Runia, a princess who is imprisoned within the game.
Three Blind Mice - One of the puzzles is a musical puzzle, where you have to play the correct tune on a piano in order to obtain a special item.
Use and reception
L has been extensively cited as a tool to help children learn spatial reasoning by scholars. A book on the BBC Micro noted that L integrated math concepts into the characters and story instead of just simply displaying sums. A UK teaching guide in 2010 said that L was an "old, excellent example of a well-thought-out, text-based mathematical game" that was "is worth buying a copy for yourself." One higher education teaching book noted that the game was an excellent example of "a mathematical game, not explicitly about skills practice" in a positive note. The game was featured in a 1993 fair to help make parents more maths literate with their children in St Neots.
References
External links
L - a walkthrough
L - video playthrough
L at the Classic Adventures Solution Archive
L at the Interactive Fiction Database
Mathematical education video games
Children's educational video games
Video games developed in the United Kingdom |
https://en.wikipedia.org/wiki/Rashad%20Jamal%20Salem | Rashad Jamal Salem (born January 18, 1979) is a Bahraini footballer currently playing for Al-Najma of Bahrain and the Bahrain national football team.
National team career statistics
Goals for Senior National Team
External links
1979 births
Living people
Bahraini men's footballers
Men's association football forwards
Bahrain men's international footballers |
https://en.wikipedia.org/wiki/Rudolf%20Kippenhahn | Rudolf Kippenhahn (24 May 1926 – 15 November 2020) was a German astrophysicist and science author.
Biography
Rudolf Kippenhahn was born in Pernink, Czechoslovakia. He originally studied mathematics and physics at the University of Erlangen-Nuremberg before changing to Astronomy. From 1975 to 1991, Kippenhahn was director of the Max Planck Institute For Astrophysics in Garching, Munich, Germany. After 1991, Kippenhahn was an active published author in Göttingen, trying to popularise astronomical science research, in the same vein as Stephen Hawking's writing, for which he won the Bruno H. Bürgel prize. His books covered such diverse topics as astronomy, cryptology and atomic physics. In 2005, Kippenhahn was honoured by the Royal Astronomical Society with the Eddington medal for his scientific research into the computation of the structure of star and of stellar evolution. A diagram displaying how the interior of a star evolves from the zero age main sequence to the later stages of its evolution are known as Kippenhahn Diagrams. Normally these diagrams display information such as convective borders, sites of nuclear energy generation and sites of shell burning.
Selected publications
Awards
1992 Bruno H. Bürgel Prize.
2005 Eddington Medal
2007 Karl Schwarzschild Medal
References
External links
sightandsound.com website showcasing some of Dr Rudolphs work.
Interview
Dr Rudolph delivering a lecture on Quasars at the University of Jena
1926 births
2020 deaths
People from Karlovy Vary District
Sudeten German people
Naturalized citizens of Germany
Officers Crosses of the Order of Merit of the Federal Republic of Germany
Max Planck Institute directors
University of Erlangen-Nuremberg alumni |
https://en.wikipedia.org/wiki/Jung%20Hoon | Jung Hoon (born August 31, 1985) is a South Korean football player.
Club career statistics
References
Jeonbuk Hyundai Motors website
External links
1985 births
Living people
South Korean men's footballers
Jeonbuk Hyundai Motors players
Gimcheon Sangmu FC players
Suwon FC players
K League 2 players
K League 1 players
Men's association football central defenders |
https://en.wikipedia.org/wiki/Karl%20Ferdinand%20Ignatius | Karl Emil Ferdinand Ignatius (27 September 1837 – 11 September 1909) was a Finnish historian, the head of the Main Office of Statistics and a Senator.
Biography
Ignatius was born in Pori, the son of vice-pastor Johan Ferdinand Ignatius and Sofia Fleming. He graduated secondary school in 1855 and enrolled in the Imperial Alexander University in Helsinki, gaining his undergraduate and graduate degrees in 1860, a Licenciate degree in 1862 and finally his Ph.D in 1864. Ignatius worked as a civil servant in the Finnish Main Office of Statistics from 1865 to 1868 and then as the head of the office from 1868 to 1885. He was the head of the chamber committee of the Senate of Finland during 1885–1900 and 1905–1908. Ignatius worked also as a Docent of Nordic History and Statistics from 1865 to 1870 and as the Curator of the Western Finland's Student Nation (combination of Turkulainen and Satakuntalainen Osakunta during 1846–1904) from 1868 to 1870.
Ignatius took part in the Diet of Finland as a member of the burghers estate in 1877–1885 and 1904–1905. He was also a member of the Helsinki City Council during 1875–1878 and 1903–1905. Ignatius was among of the founders of both the Finnish Historical and Geographical Societies, was a committee member of the Society for Culture and Education (Kansanvalistusseura) during 1873–1887 and was the Chairman of the Finnish Antiquarian Society in 1875–1885. He died in Helsinki, aged 71.
Family
Ignatius married Amanda Kristina Bergman (1841–1921) in 1863. The couple had five daughters and five sons, the most famous of them being the President of the Court of Appeals Kaarlo Yrjö Benedictus Ignatius (1869–1942), Lieutenant General Hannes Ignatius (1871–1941) and Provincial Governor Gustaf Ignatius (1873–1949).
Works
Bidrag till södra Österbottens äldre historia (1861)
Finlands historia under Karl X Gustafs regering (1865)
Renseignements sur la population de Finlande (1869)
Statistisk handbok för Finland (1872)
Storfurstendömet Finland. Statistiska anteckningar (1876)
Le Grand-duchiè de Finlande, notice statistique (1878)
Suomen maantiede kansalaisille 1. (1880–1890)
Oikea tie (1902)
References
External links
Biography at The National Biography of Finland
University of Helsinki Register 1640–1917
Wikisource: Kuka kukin oli 1961
See also
Karl Ferdinand Ignatius in 375 humanists – 11 May 2015. Faculty of Arts, University of Helsinki.
1837 births
1909 deaths
People from Pori
People from Turku and Pori Province (Grand Duchy of Finland)
Finnish Party politicians
Young Finnish Party politicians
Finnish senators
Members of the Diet of Finland
20th-century Finnish historians
Finnish statisticians
19th-century Finnish historians |
https://en.wikipedia.org/wiki/Sch%C3%B6nhardt%20polyhedron | In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who described it in 1928. The same polyhedra have also been studied in connection with Cauchy's rigidity theorem as an example where polyhedra with two different shapes have faces of the same shapes.
Construction
One way of constructing the Schönhardt polyhedron starts with a triangular prism, with two parallel equilateral triangles as its faces. One of the triangles is rotated
around the centerline of the prism, breaking the square faces of the prism into pairs of triangles. If each of these pairs is chosen to be non-convex, the Schönhardt polyhedron is the result.
Properties
The Schönhardt polyhedron has six vertices, twelve edges, and eight triangular faces. The six vertices of the Schönhardt polyhedron can be used to form fifteen unordered pairs of vertices. Twelve of these fifteen pairs form edges of the polyhedron: there are six edges in the two equilateral triangle faces, and six edges connecting the two triangles. The remaining three edges form diagonals of the polyhedron, but lie entirely outside the polyhedron.
The convex hull of the Schönhardt polyhedron is another polyhedron with the same six vertices, and a different set of twelve edges and eight triangular faces; like the Schönhardt polyhedron, it is combinatorially equivalent to a regular octahedron. The symmetric difference of the Schönhardt polyhedron consists of three tetrahedra, each lying between one of the concave dihedral edges of the Schönhardt polyhedron and one of the exterior diagonals. Thus, the Schönhardt polyhedron can be formed by removing these three tetrahedra from a convex (but irregular) octahedron.
Impossibility of triangulation
It is impossible to partition the Schönhardt polyhedron into tetrahedra whose vertices are vertices of the polyhedron. More strongly, there is no tetrahedron that lies entirely inside the Schönhardt polyhedron and has vertices of the polyhedron as its four vertices. This follows from the following two properties of the Schönhardt polyhedron:
Every triangle formed by its edges is one of its faces. Therefore, because it is not a tetrahedron itself, every tetrahedron formed by four of its vertices must have an edge that it does not share with the Schönhardt polyhedron.
Every diagonal that connects two of its vertices but is not an edge of the Schönhardt polyhedron lies outside the polyhedron. Therefore, every tetrahedron that uses such a diagonal as one of its edges must also lie in part outside the Schönhardt polyhedron.
Jumping polyhedron
In connection with the theory of flexible polyhedra, instances of the Schönhardt polyhedron form a "jumping polyhedron": a polyhedron that has two different rigid states, both having the same face shapes and the same orientation (convex or concave) of each edge. A model whose surface is made of a |
https://en.wikipedia.org/wiki/Reduced%20chi-squared%20statistic | In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation (MSWD) in isotopic dating and variance of unit weight in the context of weighted least squares.
Its square root is called regression standard error, standard error of the regression, or standard error of the equation
(see )
Definition
It is defined as chi-square per degree of freedom:
where the chi-squared is a weighted sum of squared deviations:
with inputs: variance , observations O, and calculated data C.
The degree of freedom, , equals the number of observations n minus the number of fitted parameters m.
In weighted least squares, the definition is often written in matrix notation as
where r is the vector of residuals, and W is the weight matrix, the inverse of the input (diagonal) covariance matrix of observations. If W is non-diagonal, then generalized least squares applies.
In ordinary least squares, the definition simplifies to:
where the numerator is the residual sum of squares (RSS).
When the fit is just an ordinary mean, then equals the sample standard deviation.
Discussion
As a general rule, when the variance of the measurement error is known a priori, a indicates a poor model fit. A indicates that the fit has not fully captured the data (or that the error variance has been underestimated). In principle, a value of around indicates that the extent of the match between observations and estimates is in accord with the error variance. A indicates that the model is "over-fitting" the data: either the model is improperly fitting noise, or the error variance has been overestimated.
When the variance of the measurement error is only partially known, the reduced chi-squared may serve as a correction estimated a posteriori.
Applications
Geochronology
In geochronology, the MSWD is a measure of goodness of fit that takes into account the relative importance of both the internal and external reproducibility, with most common usage in isotopic dating.
In general when:
MSWD = 1 if the age data fit a univariate normal distribution in t (for the arithmetic mean age) or log(t) (for the geometric mean age) space, or if the compositional data fit a bivariate normal distribution in [log(U/He),log(Th/He)]-space (for the central age).
MSWD < 1 if the observed scatter is less than that predicted by the analytical uncertainties. In this case, the data are said to be "underdispersed", indicating that the analytical uncertainties were overestimated.
MSWD > 1 if the observed scatter exceeds that predicted by the analytical uncertainties. In this case, the data are said to be "overdispersed". This situation is the rule rather than the exception in (U-Th)/He geochronology, indicating an incomplete understanding of the isotope system. Several reasons have been proposed to explain the overdispersion of (U-Th)/He data, including unevenly distributed U-Th distributions and radiation damage.
Often |
https://en.wikipedia.org/wiki/Ryuichi%20Kamiyama | is a Japanese football player.
Club statistics
Updated to 23 February 2018.
1Includes Promotion Playoffs to J1.
References
External links
Profile at Avispa Fukuoka
1984 births
Living people
Sportspeople from Sakai, Osaka
Association football people from Osaka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Avispa Fukuoka players
ReinMeer Aomori players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Satoshi%20Nagano | is a Japanese football player.
Club statistics
References
External links
1982 births
Living people
Fukuoka University alumni
Association football people from Fukuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Avispa Fukuoka players
Tokyo Verdy players
Giravanz Kitakyushu players
Satoshi Nagano
Satoshi Nagano
Japanese expatriate men's footballers
Japanese expatriate sportspeople in Thailand
Expatriate men's footballers in Thailand
Men's association football defenders |
https://en.wikipedia.org/wiki/Tamari%20lattice | In mathematics, a Tamari lattice, introduced by , is a partially ordered set in which the elements consist of different ways of grouping a sequence of objects into pairs using parentheses; for instance, for a sequence of four objects abcd, the five possible groupings are ((ab)c)d, (ab)(cd), (a(bc))d, a((bc)d), and a(b(cd)). Each grouping describes a different order in which the objects may be combined by a binary operation; in the Tamari lattice, one grouping is ordered before another if the second grouping may be obtained from the first by only rightward applications of the associative law (xy)z = x(yz). For instance, applying this law with x = a, y = bc, and z = d gives the expansion (a(bc))d = a((bc)d), so in the ordering of the Tamari lattice (a(bc))d ≤ a((bc)d).
In this partial order, any two groupings g1 and g2 have a greatest common predecessor, the meet g1 ∧ g2, and a least common successor, the join g1 ∨ g2. Thus, the Tamari lattice has the structure of a lattice. The Hasse diagram of this lattice is isomorphic to the graph of vertices and edges of an associahedron. The number of elements in a Tamari lattice for a sequence of n + 1 objects is the nth Catalan number Cn.
The Tamari lattice can also be described in several other equivalent ways:
It is the poset of sequences of n integers a1, ..., an, ordered coordinatewise, such that i ≤ ai ≤ n and if i ≤ j ≤ ai then aj ≤ ai .
It is the poset of binary trees with n leaves, ordered by tree rotation operations.
It is the poset of ordered forests, in which one forest is earlier than another in the partial order if, for every j, the jth node in a preorder traversal of the first forest has at least as many descendants as the jth node in a preorder traversal of the second forest .
It is the poset of triangulations of a convex n-gon, ordered by flip operations that substitute one diagonal of the polygon for another.
Notation
The Tamari lattice of the Cn groupings of n+1 objects is called Tn, but the corresponding associahedron is called Kn+1.
In The Art of Computer Programming T4 is called the Tamari lattice of order 4 and its Hasse diagram K5 the associahedron of order 4.
References
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Lattice theory |
https://en.wikipedia.org/wiki/Hisashi%20Jogo | is a Japanese footballer who plays and captains for Avispa Fukuoka in the J1 League.
Club career statistics
Updated to end of 2022 season.
1Includes J1 & J2 Playoffs.
References
External links
Profile at Avispa Fukuoka
1986 births
Living people
Association football people from Fukuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Avispa Fukuoka players
Men's association football forwards |
https://en.wikipedia.org/wiki/Tomoki%20Ikemoto | is a Japanese footballer who plays for Giravanz Kitakyushu.
Club career statistics
Updated to 1 January 2020.
References
External links
Profile at Giravanz Kitakyushu
1985 births
Living people
Association football people from Fukuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Giravanz Kitakyushu players
FC Gifu players
Kashiwa Reysol players
Yokohama FC players
Matsumoto Yamaga FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yuki%20Ozawa | is a Japanese football player currently playing for Kamatamare Sanuki.
He previously played in the Netherlands.
Career statistics
Updated to 23 February 2016.
References
External links
1983 births
Living people
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Mito HollyHock players
Shonan Bellmare players
SC Sagamihara players
Kamatamare Sanuki players
Men's association football defenders |
https://en.wikipedia.org/wiki/Geneva%20score | The Geneva score is a clinical prediction rule used in determining the pre-test probability of pulmonary embolism (PE) based on a patient's risk factors and clinical findings. It has been shown to be as accurate as the Wells Score, and is less reliant on the experience of the doctor applying the rule. The Geneva score has been revised and simplified from its original version. The simplified Geneva score is the newest version for the general population, and predicted to have the same diagnostic utility as the original Geneva score. A version of the revised score was modified to be applicable to pregnant patients.
Original Geneva Score
The original Geneva score was developed in 2001 in Geneva, Switzerland.
It's calculated using 7 risk factors and clinical variables:
The score obtained relates to the probability of the patient having had a pulmonary embolism (the lower the score, the lower the probability):
<5 points indicates a low probability of PE (10%)
5 - 8 points indicates a moderate probability of PE ( 38%)
>8 points indicates a high probability of PE. (81%)
Revised Geneva Score
More recently, the revised Geneva score has been introduced. This simplifies the scoring process, and has also been shown to be as effective as the Wells score.
The revised score uses 8 parameters, but does not include figures which require an arterial blood gas sample to be performed:
The score obtained relates to probability of PE:
0 - 3 points indicates low probability (8%)
4 - 10 points indicates intermediate probability (29%)
11 points or more indicates high probability (74%)
The probabilities derived from the scoring systems can be used to determine the need for, and nature of, further investigations such as D-dimer, ventilation/perfusion scanning and CT pulmonary angiography to confirm or refute the diagnosis of PE.
Simplified Geneva Score
A newer revision referred to as the simplified revised Geneva score has been prospectively studied and reported in the Archives of Internal Medicine on October 27 of 2008. The simplified scoring system replaced the weighted scores for each parameter with a 1-point score for each parameter present to reduce the likelihood of error when the score is used in a clinical setting. The report noted that the simplified Geneva score does not lead to a decrease in diagnostic utility in evaluating patients for a PE when compared to previous Geneva scores.
The simplified Geneva score:
Patients with a score of 2 or less are considered unlikely to have a current PE. Authors suggest that the likelihood of patients having a PE with a simplified Geneva score less than 2 and a normal D-Dimer is 3 percent.
Pregnancy Adapted Geneva (PAG)
In 2021, the items of the Revised Geneva Score were re-evaluated on pregnant women. Some items were removed, and the threshold values for the remaining items were modified to better discriminate patients even with the altered physiologic baseline of pregnancy (e.g. higher cut-off value fo |
https://en.wikipedia.org/wiki/Handedness%20and%20mathematical%20ability | Researchers have suggested a link between handedness and ability with mathematics. This link has been proposed by Geschwind, Galaburda, Annett, and Kilshaw. The suggested link is that a brain without extreme bias towards locating language in the left hemisphere would have an advantage in mathematical ability.
Body of research
Douglas study
A 1967 study by Douglas found no evidence to correlate mathematical ability with left-handedness or ambidexterity. The study compared the people who came in the top 15% of a mathematics examination with those of moderate mathematical ability, and found that the two groups' handedness preferences were similar. However, it did find that those who came lowest in the test had mixed hand preferences. A study in 1979 by Peterson found a trend towards low rates of left-handedness in science students.
Jones and Bell study
A 1980 study by Jones and Bell also obtained negative results. This study compared the handedness of a group of engineering students with strong mathematics skills against the handedness of a group of psychology students (of varying mathematics skills). In both cases, the distribution of handedness resembled that of the general population.
Annett and Kilshaw study
Annett and Kilshaw themselves support their hypothesis with several examples, including a handedness questionnaire given to undergraduates. Annett observes that studies that depend from voluntary returns of a handedness questionnaire are going to be biased towards left-handedness, and notes that this was a weakness of the study. However, the results were that there were significantly more left-handers amongst male mathematics undergraduates than male non-mathematics undergraduates (21% versus 11%) and significantly more non-right-handers (44% versus 24%), and that there was a similar but smaller left-handedness difference for female undergraduates (11% versus 8%). Annett reports the results of this study as being consistent with the hypothesis, for explaining the cause of handedness, of an absent genetic right-shift factor.
Other examples used by Annett include a study that observed the hand use of teachers of mathematics and other sciences from various universities and polytechnics, as they underwent a personal interview, compared to a control group comprising non-mathematics teachers. Again, a statistically significant difference was found for males, and again Annett states this to be consistent with the right-shift model. Further examples are a 1986 study by Benbow and a 1990 study by Temple of staff at the University of Oxford, which, Annett states, show not that there is a predominance of left-handers in talented groups, whether that talent be with mathematics or otherwise, but rather that there is a shortfall in such groups of people who are strongly right-handed.
Benbow study
A study by C. P. Benbow did not work to prove the mathematical abilities of study participants who are left-hand dominant but to prove the weak |
https://en.wikipedia.org/wiki/Hussain%20Salman | Hussain Salman Makki (; born 10 December 1982) is a Bahraini footballer.
He scored the only goal in a friendly against Inter Milan in January 2007 which ended 5–1.
National team career statistics
Goals for senior national team
External links
1982 births
Living people
Bahraini men's footballers
Bahrain men's international footballers
Al-Arabi SC (UAE) players
Al Wasl F.C. players
UAE Pro League players
Expatriate men's footballers in the United Arab Emirates
Bahraini expatriate sportspeople in the United Arab Emirates
Footballers at the 2002 Asian Games
UAE First Division League players
Men's association football midfielders
Asian Games competitors for Bahrain |
https://en.wikipedia.org/wiki/Saleh%20Farhan | Saleh Ahmed Farhan (born 1 January 1981) is a Bahraini footballer currently playing for Al-Riffa of Bahrain and the Bahrain national football team. He is of Syrian descent.
Career statistics
International
References
External links
1981 births
Living people
Bahraini men's footballers
Qatar SC players
Bahraini people of Syrian descent
Qatar Stars League players
Al-Hala SC players
Men's association football midfielders
Bahrain men's international footballers
2004 AFC Asian Cup players |
https://en.wikipedia.org/wiki/Iatromathematicians | Iatromathematicians (from Greek ἰατρική "medicine" and μαθηματικά "mathematics") were a school of physicians in 17th-century Italy who tried to apply the laws of mathematics and mechanics in order to understand the functioning of the human body. They were also keen students of anatomy. These iatromathematicians made an effort to prove that applying a purely mechanical conception to the study of the human body is futile. The mechanical conceptions that they had referred to was Leonardo da Vinci’s studies of the human body, and the writings of Aristotle about the motion of animals related to geometric analysis. Iatromathematicians considered the bodies functioning to be measured by quantifiable numbers, weights, and measures.
Iatromathematics
The field of iatromathematics is allied to science; however, it lacks the applicability of the proper scientific method and is therefore considered a form of pseudoscience. It applies the study of astrology to medicine.
Iatromathematicians viewed the human body through astrological reasoning as well as mechanics. They associate various stars, or zodiac signs with the functioning of the human body. The twelve astrological signs contribute to each part of the body from head to toe. Moreover, planets and existing cosmos in space are correlated with certain parts of the body. Through examining a natal chart, iatromathematicians attempt to predict biological setbacks in an individual.
Iatromathematicians examine the active and energetic temperament of the human body. Moreover, they explore the causes of various health problems and attempt to find ways to treat certain detrimental diseases. In iatromathematics, there is a particular assumption that there is an impact of various energetic fields caused on the star bodies. The star body of an individual is often referred to by astrologers as an energetic matrix and is believed to be spawned by heavenly bodies such as the sun, moon, planets, and several other astrological signs.
Iatromathematicians study these conceptions and try to regulate the path of the star body of individuals so that it will give a positive, rather than a negative result. By doing so, they believe that it will contribute to a healthier lifestyle. Its doctrine is based on cosmobiology in which several emotional and physiological dilemmas in the body are associated with the positioning of celestial bodies in outer space.
Iatromathematics is closely correlated with biomechanics because the field of biomechanics investigates macrobiotic bodies to a macroscopic degree through the appliance of several engineering principles. The perspective of iatromathematicians differed from that of iatrophysicists and iatrochemists in terms of the way human bodies function. Iatrophysicists predicted the deviations from the biological norm of the body through the appliance of physics, while iatrochemists measured the detrimental problems of the body by chemical means.
Ibn Ezra
Several individuals contributed |
https://en.wikipedia.org/wiki/List%20of%20West%20Bromwich%20Albion%20F.C.%20records%20and%20statistics | West Bromwich Albion Football Club are an English professional association football club based in West Bromwich, West Midlands. The club was founded in 1878 as West Bromwich Strollers, by workers from George Salter's Spring Works and turned professional in 1885. A founder member of the Football League in 1888, the team has spent the majority of its history in the top division of English football.
This list encompasses records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Albion players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at The Hawthorns, the club's home ground since 1900, are also included. Records generally refer only to first team, competitive matches in national or European competitions. Reserve matches, youth matches, friendlies, testimonials, war-time matches and regional competitions are not considered, except where indicated.
The club's record appearance maker and goalscorer is Tony Brown, who scored 279 goals in 720 appearances between 1963 and 1981.
Player records
Appearances
Youngest first-team league player: Charlie Wilson, 16 years 73 days, vs Oldham Athletic, First Division, 1 October 1921
Youngest Premier League player: Isaiah Brown, 16 years 117 days vs Wigan Athletic, 4 May 2013
Oldest first-team player: George Baddeley, 39 years 345 days, vs Sheffield Wednesday, First Division, 18 April 1914
Oldest Premier League player: Dean Kiely, 38 years 226 days vs Blackburn Rovers, 24 May 2009
Most Premier League appearances: 269, Chris Brunt
Most substitute appearances: 93, Hal Robson-Kanu, 2016–21
Most appearances
Total appearances including substitute appearances are listed below, with the number of substitute appearances shown in parentheses.
Goalscorers
Most goals in a season: 40, W. G. Richardson, 1935–36
Most league goals in a season: 39, W. G. Richardson, 1935–36
Most league goals in one match: 6, Jimmy Cookson, vs Blackpool, Second Division, 17 September 1927
First league goal: Joe Wilson, vs Stoke, The Football League, 8 September 1888
First FA Cup hat-trick: Jem Bayliss, vs Old Westminsters, sixth round, 13 February 1886
First league hat-trick: Tom Pearson, vs Bolton Wanderers, The Football League, 4 November 1889
Most hat-tricks: 14, W. G. Richardson, 1931–1938
Most Premier League goals: 30, Peter Odemwingie
Oldest Premier League goalscorer: Gareth McAuley, 37 years 87 days, 25 February 2017 vs Bournemouth
Top goalscorers
Tony Brown is the all-time top goalscorer for West Bromwich Albion.
Appearances, including substitute appearances, are marked in parentheses.
International caps
This section refers only to caps won while a West Bromwich Albion player.
First representative honour: Bob Roberts for North vs South, played in London on 26 January 1884
First i |
https://en.wikipedia.org/wiki/Hermann%20K%C3%BCnneth | Hermann Lorenz Künneth (July 6, 1892 Neustadt an der Haardt – May 7, 1975 Erlangen) was a German mathematician and renowned algebraic topologist, best known for his contribution to what is now known as the Künneth theorem.
In the winter semester 1910/11, Künneth joined the students' fraternity “Studentengesangverein Erlangen“, now “Akademisch-Musikalische Verbindung Fridericiana Erlangen“ (“Students' Choral Society Erlangen“, now “Akademic Musical Fraternity Fridericiana Erlangen“). He carried out a variety of posts during his studies as well as after having left university in 1914.
He participated in the First World War and was captured by British forces.
His 1922 doctoral thesis at the University of Erlangen was titled Über die Bettischen Zahlen einer Produktmannigfaltigkeit (“On the Betti numbers of a product manifold”), it was supervised by Heinrich Franz Friedrich Tietze.
From 1923, he was a teacher at secondary schools in Kronach and Erlangen. After retiring, he became professor at the University of Erlangen.
In 1964, he was decorated with the Bundesverdienstkreuz, the highest German decoration.
References
Literature
Haupt, Otto: “Hermann Künneth zum Gedenken”, Jahresbericht der Deutschen Mathematiker-Vereinigung, 78 (1976) pp. 61–66 (online)
Haas, Karl Eduard: “Die Akademisch-Musikalische Verbindung Fridericiana im Sondershäuser Verband“, Erlangen 1982, p. 295
1892 births
1975 deaths
20th-century German mathematicians
Topologists
German military personnel of World War I
People from Neustadt an der Weinstraße
People from the Palatinate (region)
Recipients of the Cross of the Order of Merit of the Federal Republic of Germany |
https://en.wikipedia.org/wiki/Island%20Gazette | Island Gazette was a weekly newspaper covering local news, state news, obituaries, real estate statistics, and classifieds based in Carolina Beach, North Carolina. The newspaper was owned by Seaside Press Co. Inc..
References
Weekly newspapers published in North Carolina
New Hanover County, North Carolina
1978 establishments in North Carolina |
https://en.wikipedia.org/wiki/Indagationes%20Mathematicae | Indagationes Mathematicae (from Latin: inquiry, search, investigation of the mathematics) is a Dutch mathematics journal.
The journal originates from the Proceedings of the Royal Netherlands Academy of Arts and Sciences (or Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen), founded in 1895. From 1939, mathematics articles in this journal were published separately, under the alternative title Indagationes Mathematicae.
In 1951 the proceedings officially split into three journals, keeping the same name but distinguished from each other by being in separate series. They were Series A (Mathematical Sciences), Series B (Physical Sciences), and Series C (Biological and Medical Sciences). At that time, Series A became published by the North-Holland Publishing Company; the volumes from this time are now listed by the publisher as Indagationes Mathematicae (Proceedings).
In 1971, North-Holland merged with Elsevier. Beginning in 1990 the journal dropped the "(Proceedings)" from its title, leaving the journal's name in its current form as Indagationes Mathematicae.
In 2010, sponsorship of the journal was transferred from the Royal Netherlands Academy to the Royal Dutch Mathematical Society, while it continued to be published by Elsevier.
The typesetting from this journal, including its mathematical formulae, was chosen by Donald Knuth as one of three examples of typesetting quality when he designed the TeX digital typesetting software from 1978.
It is indexed in Scopus and Zentralblatt MATH with a year 2020 impact factor of 0.956.
References
Mathematics journals
English-language journals
Academic journals established in 1951 |
https://en.wikipedia.org/wiki/Unitary%20element | In mathematics, an element x of a *-algebra is unitary if it satisfies
In functional analysis, a linear operator A from a Hilbert space into itself is called unitary if it is invertible and its inverse is equal to its own adjoint A and that the domain of A is the same as that of A. See unitary operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is unitary if and only if the matrix describing A with respect to this basis is a unitary matrix.
See also
References
Abstract algebra
Linear algebra |
https://en.wikipedia.org/wiki/Russo%E2%80%93Dye%20theorem | In mathematics, the Russo–Dye theorem is a result in the field of functional analysis. It states that in a unital C*-algebra, the closure of the convex hull of the unitary elements is the closed unit ball.
The theorem was published by B. Russo and H. A. Dye in 1966.
Other formulations and generalizations
Results similar to the Russo–Dye theorem hold in more general contexts. For example, in a unital *-Banach algebra, the closed unit ball is contained in the closed convex hull of the unitary elements.
A more precise result is true for the C*-algebra of all bounded linear operators on a Hilbert space: If T is such an operator and ||T|| < 1 − 2/n for some integer n > 2, then T is the mean of n unitary operators.
Applications
This example is due to Russo & Dye, Corollary 1: If U(A) denotes the unitary elements of a C*-algebra A, then the norm of a linear mapping f from A to a normed linear space B is
In other words, the norm of an operator can be calculated using only the unitary elements of the algebra.
Further reading
An especially simple proof of the theorem is given in:
Notes
C*-algebras
Theorems in functional analysis
Unitary operators |
https://en.wikipedia.org/wiki/Delta%20set | In mathematics, a Δ-set S, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.As an example, suppose we want to triangulate the 1-dimensional circle . To do so with a simplicial complex, we need at least three vertices, and edges connecting them. But delta-sets allow for a simpler triangulation: thinking of as the interval [0,1] with the two endpoints identified, we can define a triangulation with a single vertex 0, and a single edge looping between 0 and 0.
Definition and related data
Formally, a Δ-set is a sequence of sets together with maps
with for that satisfy
whenever .
This definition generalizes the notion of a simplicial complex, where the are the sets of n-simplices, and the are the face maps. It is not as general as a simplicial set, since it lacks "degeneracies."
Given Δ-sets S and T, a map of Δ-sets is a collection of set-maps
such that
whenever both sides of the equation are defined.
With this notion, we can define the category of Δ-sets, whose objects are Δ-sets and whose morphisms are maps of Δ-sets.
Each Δ-set has a corresponding geometric realization, defined as
where we declare that
Here, denotes the standard n-simplex, and
is the inclusion of the i-th face. The geometric realization is a topological space with the quotient topology.
The geometric realization of a Δ-set S has a natural filtration
where
is a "restricted" geometric realization.
Related functors
The geometric realization of a Δ-set described above defines a covariant functor from the category of Δ-sets to the category of topological spaces. Geometric realization takes a Δ-set to a topological space, and carries maps of Δ-sets to induced continuous maps between geometric realizations.
If S is a Δ-set, there is an associated free abelian chain complex, denoted , whose n-th group is the free abelian group
generated by the set , and whose n-th differential is defined by
This defines a covariant functor from the category of Δ-sets to the category of chain complexes of abelian groups. A Δ-set is carried to the chain complex just described, and a map of Δ-sets is carried to a map of chain complexes, which is defined by extending the map of Δ-sets in the standard way using the universal property of free abelian groups.
Given any topological space X, one can construct a Δ-set as follows. A singular n-simplex in X is a continuous map
Define
to be the collection of all singular n-simplicies in X, and define
by
where again is the -th face map. One can check that this is in fact a Δ-set. This defines a covariant functor from the category of topological spaces to the category of Δ-sets. A topological space is carried to the Δ-set just described, |
https://en.wikipedia.org/wiki/Maximum%20spacing%20estimation | In statistics, maximum spacing estimation (MSE or MSP), or maximum product of spacing estimation (MPS), is a method for estimating the parameters of a univariate statistical model. The method requires maximization of the geometric mean of spacings in the data, which are the differences between the values of the cumulative distribution function at neighbouring data points.
The concept underlying the method is based on the probability integral transform, in that a set of independent random samples derived from any random variable should on average be uniformly distributed with respect to the cumulative distribution function of the random variable. The MPS method chooses the parameter values that make the observed data as uniform as possible, according to a specific quantitative measure of uniformity.
One of the most common methods for estimating the parameters of a distribution from data, the method of maximum likelihood (MLE), can break down in various cases, such as involving certain mixtures of continuous distributions. In these cases the method of maximum spacing estimation may be successful.
Apart from its use in pure mathematics and statistics, the trial applications of the method have been reported using data from fields such as hydrology, econometrics, magnetic resonance imaging, and others.
History and usage
The MSE method was derived independently by Russel Cheng and Nik Amin at the University of Wales Institute of Science and Technology, and Bo Ranneby at the Swedish University of Agricultural Sciences. The authors explained that due to the probability integral transform at the true parameter, the “spacing” between each observation should be uniformly distributed. This would imply that the difference between the values of the cumulative distribution function at consecutive observations should be equal. This is the case that maximizes the geometric mean of such spacings, so solving for the parameters that maximize the geometric mean would achieve the “best” fit as defined this way. justified the method by demonstrating that it is an estimator of the Kullback–Leibler divergence, similar to maximum likelihood estimation, but with more robust properties for some classes of problems.
There are certain distributions, especially those with three or more parameters, whose likelihoods may become infinite along certain paths in the parameter space. Using maximum likelihood to estimate these parameters often breaks down, with one parameter tending to the specific value that causes the likelihood to be infinite, rendering the other parameters inconsistent. The method of maximum spacings, however, being dependent on the difference between points on the cumulative distribution function and not individual likelihood points, does not have this issue, and will return valid results over a much wider array of distributions.
The distributions that tend to have likelihood issues are often those used to model physical phenomena. seek to analyze flood |
https://en.wikipedia.org/wiki/Arthur%E2%80%93Selberg%20trace%20formula | In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of on the discrete part of in terms of geometric data, where is a reductive algebraic group defined over a global field and is the ring of adeles of F.
There are several different versions of the trace formula. The first version was the unrefined trace formula, whose terms depend on truncation operators and have the disadvantage that they are not invariant. Arthur later found the invariant trace formula and the stable trace formula which are more suitable for applications. The simple trace formula is less general but easier to prove. The local trace formula is an analogue over local fields.
Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups.
Notation
F is a global field, such as the field of rational numbers.
A is the ring of adeles of F.
G is a reductive algebraic group defined over F.
The compact case
In the case when is compact the representation splits as a direct sum of irreducible representations, and the trace formula is similar to the Frobenius formula for the character of the representation induced from the trivial representation of a subgroup of finite index.
In the compact case, which is essentially due to Selberg, the groups G(F) and G(A) can be replaced by any
discrete subgroup of a locally compact group with compact. The group acts on the space of functions on
by the right regular representation , and this extends to an action of the group ring of , considered as the ring of functions on . The character of this representation is given by a generalization of the Frobenius formula as follows.
The action of a function on a function on is given by
In other words, is an integral operator on (the space of functions on ) with kernel
Therefore, the trace of is given by
The kernel K can be written as
where is the set of conjugacy classes in , and
where is an element of the conjugacy class , and is its centralizer in .
On the other hand, the trace is also given by
where m(π) is the multiplicity of the irreducible unitary representation of in
Examples
If and are both finite, the trace formula is equivalent to the Frobenius formula for the character of an induced representation.
If is the group of real numbers and the subgroup of integers, then the trace formula becomes the Poisson summation formula.
Difficulties in the non-compact case
In most cases of the Arthur–Selberg trace formula, the quotient is not compact, which causes the following (closely related) problems:
The representation on contains not only discrete components, but also continuous components.
The kernel is no longer integrable over the diagonal, and the operators are no longe |
https://en.wikipedia.org/wiki/Elementary%20Calculus%3A%20An%20Infinitesimal%20Approach | Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover.
Textbook
Keisler's textbook is based on Robinson's construction of the hyperreal numbers. Keisler also published a companion book, Foundations of Infinitesimal Calculus, for instructors, which covers the foundational material in more depth.
Keisler defines all basic notions of the calculus such as continuity (mathematics), derivative, and integral using infinitesimals. The usual definitions in terms of ε–δ techniques are provided at the end of Chapter 5 to enable a transition to a standard sequence.
In his textbook, Keisler used the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct hyperreal numbers infinitely close to each other. Similarly, an infinite-resolution telescope is used to represent infinite numbers.
When one examines a curve, say the graph of ƒ, under a magnifying glass, its curvature decreases proportionally to the magnification power of the lens. Similarly, an infinite-magnification microscope will transform an infinitesimal arc of a graph of ƒ, into a straight line, up to an infinitesimal error (only visible by applying a higher-magnification "microscope"). The derivative of ƒ is then the (standard part of the) slope of that line (see figure).
Thus the microscope is used as a device in explaining the derivative.
Reception
The book was first reviewed by Errett Bishop, noted for his work in constructive mathematics. Bishop's review was harshly critical; see Criticism of nonstandard analysis. Shortly after, Martin Davis and Hausner published a detailed favorable review, as did Andreas Blass and Keith Stroyan. Keisler's student K. Sullivan, as part of her PhD thesis, performed a controlled experiment involving 5 schools, which found Elementary Calculus to have advantages over the standard method of teaching calculus. Despite the benefits described by Sullivan, the vast majority of mathematicians have not adopted infinitesimal methods in their teaching. Recently, Katz & Katz give a positive account of a calculus course based on Keisler's book. O'Donovan also described his experience teaching calculus using infinitesimals. His initial point of view was positive, but later he found pedagogical difficulties with the approach to nonstandard calculus taken by this text and others.
G. R. Blackley remarked in a letter to Prindle, Weber & Schmidt, concerning Elementary Calculus: An Approach Using Infinitesimals, "Such problems as might arise with the book will be political. It is revolutionary. Revolutions are seldom welcomed by the established party, although revolutionaries often are."
Hrbacek writes that the definitions of continuity, derivative, and integra |
https://en.wikipedia.org/wiki/Clairaut%27s%20relation%20%28differential%20geometry%29 | In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ is a parametrization of a great circle then
where ρ(P) is the distance from a point P on the great circle to the z-axis, and ψ(P) is the angle between the great circle and the meridian through the point P.
The relation remains valid for a geodesic on an arbitrary surface of revolution.
A statement of the general version of Clairaut's relation is:
Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle moves along a geodesic under no forces other than those that keep it on the surface.
References
M. do Carmo, Differential Geometry of Curves and Surfaces, page 257.
Differential geometry
Differential geometry of surfaces
Geodesy |
https://en.wikipedia.org/wiki/Miroslav%20Viazanko | Miroslav Viazanko (born 27 October 1981) is retired Slovak football winger.
Career statistics
External links
MFK Košice profile
1981 births
Living people
Footballers from Prešov
Men's association football wingers
Slovak men's footballers
1. FC Tatran Prešov players
FC VSS Košice players
FK Železiarne Podbrezová players
Slovak First Football League players
2. Liga (Slovakia) players |
https://en.wikipedia.org/wiki/Milton%20Keynes%20urban%20area | The Milton Keynes urban area or Milton Keynes Built-up Area is a designation established by the United Kingdom's Office for National Statistics. Milton Keynes has no statutory boundary: the 1967 designated area only determined the area assigned to the Milton Keynes Development Corporation for development. The wider urban area outside that designation includes Newport Pagnell and Woburn Sands as well as Aspley Guise (Bedfordshire) and part of Stoke Hammond civil parish.
At the 2021 census, the population of Milton Keynes's Built-up Area was 256,385, an increase of 11.5% over the figure of 229,941 recorded at the 2011 census. That in turn was an increase of almost 25% on the population recorded in the 2001 census of 184,506.
Built-up area sub-divisions
These are the 'built-up area subdivisions' of the Milton Keynes urban area (built-up area) as defined by the ONS.
2021
, information about the definitions and populations of the built-up area subdivisions of the 2021 census has yet to be released.
2011
2001
In defining the sub-areas of the Milton Keynes urban area for the 2001 census, the ONS used the pre-designation urban and rural districts, subdividing the larger rural district by the chronological phases of urbanisation within them). (These designations were largely dropped for the 2011 Census.) These were:
Bletchley Urban District
Wolverton Urban District
Newport Pagnell Urban District
Newport Pagnell Rural District
The Central and North Milton Keynes enumeration districts covered that part of the former Newport Pagnell Rural District that is west of the River Ouzel. The "Walnut Tree" and "Browns Wood" sub-areas together cover the remainder of the Rural District from the Ouzel to the M1. The areas covered by these names are far larger than the civil parishes of the same name, except for 'North Milton Keynes', which does not exist otherwise. (The precise boundary mapping is no longer available: the information in the notes below was derived at the time.)
The corresponding 2001 Urban Sub-areas were:
At that time, Woburn Sands was not contiguous with the rest of the urban area and thus is not included. Its population was 4,963.
Notes
References
External links
A Vision of Britain - Milton Keynes.
Milton Keynes
Urban areas of England |
https://en.wikipedia.org/wiki/Enid%20Charles | Dorothy Enid Charles (29 December 1894 – 26 March 1972) was a British socialist, feminist and statistician who was a pioneer in the fields of demography and population statistics.
She was born in Denbigh, Wales. She obtained a bachelor's degree in mathematics, economics and statistics at Newnham College, Cambridge, where she became friends with Dorothy (Dodo) Crowther, sister of the writer James Crowther. She gained a Ph.D. in physiology from the University of Cape Town, South Africa.
Charles met the conscientious objector Lancelot Hogben while at Cambridge; they married in 1918. Out of a dozen or so socialist and feminist couples in Britain in the early 20th century, Charles was the only wife to keep her name. The couple, who had two sons and two daughters, separated in 1953 and divorced in 1957.
Charles worked on fertility rates and marriage rates for the Dominion Bureau of Statistics in Canada. In 1934, Charles projected drastic decline in population of the United Kingdom if fertility rates continued to fall. These results led her to speak out against the then commonly accepted principle of eugenics. She subsequently worked as a Regional Adviser in Epidemiology and Health Statistics and as a Population Statistics Consultant for the World Health Organization in Singapore and New Delhi.
She died in Torquay, England, in 1972, aged 77.
Bibliography
See also
Leslie Hogben, granddaughter of Enid
References
1894 births
1972 deaths
British statisticians
Women statisticians
British demographers
Alumni of Newnham College, Cambridge
University of Cape Town alumni
British feminists
People from Denbigh
British socialist feminists |
https://en.wikipedia.org/wiki/Lorry%20Gloeckner | Lorry Gloeckner (born January 25, 1956) is a Canadian former ice hockey player who played 13 games in the National Hockey League for the Detroit Red Wings during the 1978–79 season.
Career statistics
Regular season and playoffs
External links
1956 births
Living people
Boston Bruins draft picks
Calgary Cowboys draft picks
Canadian ice hockey defencemen
Detroit Red Wings players
Ice hockey people from Saskatchewan
Johnstown Red Wings players
Kansas City Red Wings players
Sportspeople from Kindersley
Victoria Cougars (WHL) players |
https://en.wikipedia.org/wiki/Definite%20quadratic%20form | In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every non-zero vector of . According to that sign, the quadratic form is called positive-definite or negative-definite.
A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively. In other words, it may take on zero values for some non-zero vectors of .
An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form.
More generally, these definitions apply to any vector space over an ordered field.
Associated symmetric bilinear form
Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form and its associated symmetric bilinear form are related by the following equations:
The latter formula arises from expanding
Examples
As an example, let , and consider the quadratic form
where and and are constants. If and the quadratic form is positive-definite, so Q evaluates to a positive number whenever If one of the constants is positive and the other is 0, then is positive semidefinite and always evaluates to either 0 or a positive number. If and or vice versa, then is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If and the quadratic form is negative-definite and always evaluates to a negative number whenever And if one of the constants is negative and the other is 0, then is negative semidefinite and always evaluates to either 0 or a negative number.
In general a quadratic form in two variables will also involve a cross-product term in ·:
This quadratic form is positive-definite if and negative-definite if and and indefinite if It is positive or negative semidefinite if with the sign of the semidefiniteness coinciding with the sign of
This bivariate quadratic form appears in the context of conic sections centered on the origin. If the general quadratic form above is equated to 0, the resulting equation is that of an ellipse if the quadratic form is positive or negative-definite, a hyperbola if it is indefinite, and a parabola if
The square of the Euclidean norm in -dimensional space, the most commonly used measure of distance, is
In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the axis and the axis.
Matrix form
A quadratic form can be written in terms of matrices as
where is any ×1 Cartesian vector in which at least one element is not 0; is an symmetric matrix; and superscript denotes a matrix transpose. If is diagonal this is equivalent to a non-matrix form containing solely ter |
https://en.wikipedia.org/wiki/Reeve%20tetrahedra | In geometry, the Reeve tetrahedra are a family of polyhedra in three-dimensional space with vertices at , , and where is a positive integer. They are named after John Reeve, who in 1957 used them to show that higher-dimensional generalizations of Pick's theorem do not exist.
Counterexample to generalizations of Pick's theorem
All vertices of a Reeve tetrahedron are lattice points (points whose coordinates are all integers). No other lattice points lie on the surface or in the interior of the tetrahedron. The volume of the Reeve tetrahedron with vertex is . In 1957 Reeve used this tetrahedron to show that there exist tetrahedra with four lattice points as vertices, and containing no other lattice points, but with arbitrarily large volume.
In two dimensions, the area of every polyhedron with lattice vertices is determined as a formula of the number of lattice points at its vertices, on its boundary, and in its interior, according to Pick's theorem. The Reeve tetrahedra imply that there can be no corresponding formula for the volume in three or more dimensions. Any such formula would be unable to distinguish the Reeve tetrahedra with different choices of from each other, but their volumes are all different.
Despite this negative result, it is possible (as Reeve showed) to devise a more complicated formula for lattice polyhedron volume that combines the number of lattice points in the polyhedron, the number of points of a finer lattice in the polyhedron, and the Euler characteristic of the polyhedron.
Ehrhart polynomial
The Ehrhart polynomial of any lattice polyhedron counts the number of lattice points that it contains when scaled up by an integer factor.
The Ehrhart polynomial of
the Reeve tetrahedron of height is
Thus, for , the coefficient of in the Ehrhart polynomial of is negative. This example shows that Ehrhart polynomials can sometimes have negative coefficients.
References
Digital geometry
Lattice points |
https://en.wikipedia.org/wiki/Supertoroid | In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surfaces (technically, a topological torus) whose shape is defined by mathematical formulas similar to those that define the superellipsoids. The plural of "supertorus" is either supertori or supertoruses.
The family was described and named by Alan Barr in 1994.
Barr's supertoroids have been fairly popular in computer graphics as a convenient model for many objects, such as smooth frames for rectangular things. One quarter of a supertoroid can provide a smooth and seamless 90-degree joint between two superquadric cylinders. However, they are not algebraic surfaces (except in special cases).
Formulas
Alan Barr's supertoroids are defined by parametric equations similar to the trigonometric equations of the torus, except that the sine and cosine terms are raised to arbitrary powers. Namely, the generic point P(u, v) of the surface is given by
where , , and the parameters u and v range from 0 to 360 degrees (0 to 2π radians).
In these formulas, the parameter s > 0 controls the "squareness" of the vertical sections, t > 0 controls the squareness of the horizontal sections, and a, b ≥ 1 are the major radii in the X and Y directions. With s=t=1 and a=b=R one obtains the ordinary torus with major radius R and minor radius 1, with the center at the origin and rotational symmetry about the Z axis.
In general, the supertorus defined as above spans the intervals in X, in Y, and in Z. The whole shape is symmetric about the planes X=0, Y=0, and Z=0. The hole runs in the Z direction and spans the intervals in X and in Y.
A curve of constant u on this surface is a horizontal Lamé curve with exponent 2/t, scaled in X and Y and displaced in Z. A curve of constant v, projected on the plane X=0 or Y=0, is a Lamé curve with exponent 2/s, scaled and horizontally shifted. If v is 0, the curve is planar and spans the interval in X, and in Z; and similarly if v is 90, 180, or 270 degrees. The curve is also planar if a = b.
In general, if a≠b and v is not a multiple of 90 degrees, the curve of constant v will not be planar; and, conversely, a vertical plane section of the supertorus will not be a Lamé curve.
The basic supertoroid shape defined above is often modified by non-uniform scaling to yield supertoroids of specific width, length, and vertical thickness.
Plotting code
The following GNU Octave code generates plots of a supertorus:
function supertoroid(epsilon,a)
n=50;
d=.1;
etamax=pi;
etamin=-pi;
wmax=pi;
wmin=-pi;
deta=(etamax-etamin)/n;
dw=(wmax-wmin)/n;
k=0;
l=0;
for i=1:n+1
eta(i)=etamin+(i-1)*deta;
for j=1:n+1
w(j)=wmin+(j-1)*dw;
x(i,j)=a(1)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(cos(w(j)))*abs(cos(w(j)))^epsilon(2);
y(i,j)=a(2)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(sin(w(j)))*abs(sin(w(j)))^epsilon(2);
z(i,j)=a(3)*sign(sin( |
https://en.wikipedia.org/wiki/Takuo%20%C5%8Ckubo | is a Japanese football player currently playing for Shimizu S-Pulse.
Club statistics
Updated to 24 July 2022.
References
External links
Profile at FC Tokyo
1989 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Yokohama FC players
JEF United Chiba players
V-Varen Nagasaki players
FC Tokyo players
Sagan Tosu players
Shimizu S-Pulse players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Yuta%20Nakano%20%28footballer%29 | is a Japanese football player. He plays for FC Kariya.
Club statistics
References
External links
1989 births
Living people
Association football people from Aichi Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Yokohama FC players
Fagiano Okayama players
Verspah Oita players
FC Kariya players
Men's association football forwards
Japanese expatriate men's footballers
Expatriate men's footballers in Singapore
Japanese expatriate sportspeople in Singapore |
https://en.wikipedia.org/wiki/Tsuyoshi%20Hakkaku | is a former Japanese footballer who last played for Giravanz Kitakyushu.
Club statistics
Updated to 2 February 2018.
References
External links
Profile at Giravanz Kitakyushu
1985 births
Living people
Komazawa University alumni
Association football people from Chiba Prefecture
Japanese men's footballers
J2 League players
Yokohama FC players
Giravanz Kitakyushu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Masaki%20Yoshida | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Hosei University alumni
Association football people from Niigata Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Yokohama FC players
Tokyo Verdy players
Matsumoto Yamaga FC players
FC Ryukyu players
Men's association football defenders
Sportspeople from Niigata (city) |
https://en.wikipedia.org/wiki/Myroslav%20Reshko | Myroslav Pavlovych Reshko (; born 18 August 1970, in Mukachevo) is a Ukrainian football player who played for BVSC Budapest.
External links
Statistics at FFU website
1970 births
Living people
People from Mukachevo
Ukrainian men's footballers
Ukrainian expatriate men's footballers
Expatriate men's footballers in Hungary
Men's association football defenders
Kiskőrösi FC footballers
Budapesti VSC footballers
Footballers from Zakarpattia Oblast
Soviet men's footballers |
https://en.wikipedia.org/wiki/Karen%20Vogtmann | Karen Vogtmann (born July 13, 1949 in Pittsburg, California) is an American mathematician working primarily in the area of geometric group theory. She is known for having introduced, in a 1986 paper with Marc Culler, an object now known as the Culler–Vogtmann Outer space. The Outer space is a free group analog of the Teichmüller space of a Riemann surface and is particularly useful in the study of the group of outer automorphisms of the free group on n generators, Out(Fn). Vogtmann is a professor of mathematics at Cornell University and the University of Warwick.
Biographical data
Vogtmann was inspired to pursue mathematics by a National Science Foundation summer program for high school students at the University of California, Berkeley.
She received a B.A. from the University of California, Berkeley in 1971. Vogtmann then obtained a PhD in mathematics, also from the University of California, Berkeley in 1977. Her PhD advisor was John Wagoner and her doctoral thesis was on algebraic K-theory.
She then held positions at University of Michigan, Brandeis University and Columbia University. Vogtmann has been a faculty member at Cornell University since 1984, and she became a full professor at Cornell in 1994. In September 2013, she also joined the University of Warwick. She is married to the mathematician John Smillie. The couple moved in 2013 to England and settled in Kenilworth. She is currently a professor of mathematics at Warwick, and a Goldwin Smith Professor of Mathematics Emeritus at Cornell.
Vogtmann has been the vice-president of the American Mathematical Society (2003–2006). She has been elected to serve as a member of the board of trustees of the American Mathematical Society for the period February 2008 – January 2018.
Vogtmann is a former editorial board member (2006–2016) of the journal Algebraic and Geometric Topology and a former associate editor of Bulletin of the American Mathematical Society.
She is currently an associate editor of the Journal of the American Mathematical Society, an editorial board member Geometry & Topology Monographs book series, and a consulting editor for the Proceedings of the Edinburgh Mathematical Society.
She is also a member of the ArXiv advisory board.
Since 1986 Vogtmann has been a co-organizer of the annual conference called the Cornell Topology Festival that usually takes places at Cornell University each May.
Awards, honors and other recognition
Vogtmann gave an invited lecture at the International Congress of Mathematicians in Madrid, Spain, in August 2006.
She gave the 2007 annual AWM Noether Lecture titled "Automorphisms of Free Groups, Outer Space and Beyond" at the annual meeting of American Mathematical Society in New Orleans in January 2007. Vogtmann was selected to deliver the Noether Lecture for "her fundamental contributions to
geometric group theory; in particular, to the study of the automorphism group of a free group".
On June 21–25, 2010 a 'VOGTMANNFEST' Geometric Grou |
https://en.wikipedia.org/wiki/William%20Edward%20Hodgson%20Berwick | William Edward Hodgson Berwick (11 March 1888 in Dudley Hill, Bradford – 13 May 1944 in Bangor, Gwynedd) was a British mathematician, specializing in algebra, who worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals.
Academic career
Berwick was educated at a small private school before entering Bradford Grammar School. He completed his schooling in 1906, securing a Brown Scholarship to assist him in his university studies; he was also awarded an Entrance Scholarship by Clare College, Cambridge, where he went to study for the Mathematical Tripos. He took Part I of the degree in 1909, placing joint fourth in the class, and Part II in 1910.
During his undergraduate years, under the tutelage of G B Matthews, Berwick became interested in number theory. He submitted an essay entitled An illustration of the theory of relative corpora for the Smith's Prize in 1911; the essay was placed second in the prize competition. He then co-wrote, with Matthews, a paper On the reduction of arithmetical binary cubics which have a negative determinant: it was published after Berwick had left Cambridge to take up an assistant lectureship at the University of Bristol, and was the only paper Berwick co-authored in his career.
Berwick taught at Bristol until 1913 when he took up another lectureship at the University College of Bangor. With the outbreak of the First World War in 1914 Berwick began war work on the Technical Staff of the Anti-Aircraft Experimental Section of the Munitions Inventions Department at Portsmouth. For the 1919–20 academic year Berwick was appointed acting head of the Bangor mathematics department; he then took up a lectureship at the University of Leeds, earning promotion to a Readership in Mathematical Analysis there in 1921. He was also elected to a fellowship at Clare College, Cambridge, in 1921.
In 1926, with thirteen research papers to his name, Berwick returned to Bangor to serve as Chairman of Mathematics. He had in 1925 become a member of the Council of the London Mathematical Society; in 1929 he was appointed Vice-President. He retired the post in 1941, at which point he was created Emeritus Professor.
Research and publications
Berwick was an algebraist, and worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals, and studied rings in algebraic integers. In 1927 he published Integral Bases, an ambitious account that used heavy numerical computations in place of practical proofs.
He published sixteen papers, ten of them — including a 1915 paper giving sufficient conditions for a quintic expression to be solved by radicals — in Proceedings of the London Mathematical Society. Much of his work gained recognition only in the 1960s, when it was republished.
Personal life
Berwick was described as a tall man with a distinctive voice and forthright personal style. He was a keen chess player, pa |
https://en.wikipedia.org/wiki/Radoslav%20%C5%A0koln%C3%ADk | Radoslav Školník (born on 14 November 1979 in Košice) is a former Slovak football defender who last played for Slovan Velky Folkmar.
Career statistics
Last updated: 28 December 2009
External links
Player profile at official club website
Living people
1979 births
Slovak men's footballers
FC VSS Košice players
Slovak First Football League players
Footballers from Košice
Men's association football central defenders |
https://en.wikipedia.org/wiki/Dennis%20Bailey%20%28rugby%20league%29 | Dennis Bailey is a former professional rugby league footballer who played in the 1990s. He played at club level for Dewsbury, as a , i.e. number 2 or 5.
External links
Dennis Bailey Statistics at rugbyleagueproject.org
Tetley's Yorkshire County Cup Final 2003
ĎŔƑ Dewsbury's Last Game, Ever
ĎŔƑ Memory Lane
Angry Leigh hit out at ref
RUGBY LEAGUE: Henare completes the Lynx
Cougars slump in Rams rampage
Video of try 1:39 -to- 1:59
Dewsbury Rams players
Living people
Year of birth missing (living people)
Place of birth missing (living people)
Rugby league wingers |
https://en.wikipedia.org/wiki/%C3%9Dusup%20Orazm%C3%A4mmedow | Ýusup Orazmämmedov is a professional Turkmen football player. He currently plays for FC Merw from Mary.
International Career Statistics
Goals for Senior National Team
External links
Turkmenistan men's footballers
Turkmenistan men's international footballers
Living people
1986 births
Men's association football forwards |
https://en.wikipedia.org/wiki/Rustam%20Saparow | Rustam Saparov (born April 10, 1978) is a retired Turkmenistani footballer.
Career
During 2007, Saparov played 14 times for Uzbek League club FC Nasaf.
Career statistics
International
Scores and results list Uzbekistan's goal tally first, score column indicates score after each Saparow goal.
References
External links
Living people
1978 births
Turkmenistan men's footballers
2004 AFC Asian Cup players
Turkmenistan men's international footballers
Men's association football midfielders
FC Nasaf players
FC Aşgabat players |
https://en.wikipedia.org/wiki/Arif%20Mirzo%C3%BDew | Arif Mirzoýew (born January 13, 1980) is a professional Turkmen football player, currently playing for FC Altyn Asyr.
International career statistics
Goals for Senior National Team
References
Living people
1980 births
Turkmenistan men's footballers
Turkmenistan expatriate men's footballers
Turkmenistan men's international footballers
2004 AFC Asian Cup players
Footballers from Ashgabat
Footballers at the 2002 Asian Games
Men's association football forwards
Asian Games competitors for Turkmenistan
Expatriate men's footballers in Azerbaijan
Expatriate men's footballers in Uzbekistan
Turkmenistan expatriate sportspeople in Azerbaijan
Turkmenistan expatriate sportspeople in Uzbekistan
Neftçi PFK players
FC Aşgabat players
navbahor Namangan players
Qarabağ FK players
Nisa Aşgabat players
FK Dinamo Samarqand players |
https://en.wikipedia.org/wiki/Klein%20surface | In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex numbers analytically, the former are used to study algebraic curves over the real numbers analytically. Klein surfaces were introduced by Felix Klein in 1882.
A Klein surface is a surface (i.e., a differentiable manifold of real dimension 2) on which the notion of angle between two tangent vectors at a given point is well-defined, and so is the angle between two intersecting curves on the surface. These angles are in the range [0,π]; since the surface carries no notion of orientation, it is not possible to distinguish between the angles α and −α. (By contrast, on Riemann surfaces are oriented and angles in the range of (-π,π] can be meaningfully defined.) The length of curves, the area of submanifolds and the notion of geodesic are not defined on Klein surfaces.
Two Klein surfaces X and Y are considered equivalent if there are conformal (i.e. angle-preserving but not necessarily orientation-preserving) differentiable maps f:X→Y and g:Y→X that map boundary to boundary and satisfy fg = idY and gf = idX.
Examples
Every Riemann surface (analytic manifold of complex dimension 1, without boundary) is a Klein surface. Examples include open subsets of the complex plane (non-compact), the Riemann sphere (compact), and tori (compact). Note that there are many different inequivalent Riemann surfaces with the same underlying torus as manifold.
A closed disk in the complex plane is a Klein surface (compact, with boundary). All closed disks are equivalent as Klein surfaces. A closed annulus in the complex plane is a Klein surface (compact, with boundary). Not all annuli are equivalent as Klein surfaces: there is a one-parameter family of inequivalent Klein surfaces arising in this way from annuli. By removing a number of open disks from the Riemann sphere, we obtain another class of Klein surfaces (compact, with boundary). The real projective plane can be turned into a Klein surface (compact, without boundary), in essentially only one way. The Klein bottle can be turned into a Klein surface (compact, without boundary); there is a one-parameter family of inequivalent Klein surfaces structures defined on the Klein bottle. Similarly, there is a one-parameter family of inequivalent Klein surface structures (compact, with boundary) defined on the Möbius strip.
Every compact topological 2-manifold (possibly with boundary) can be turned into a Klein surface, often in many different inequivalent ways.
Properties
The boundary of a compact Klein surface consists of finitely many connected components, each of which being homeomorphic to a circle. These components are called the ovals of the Klein surface.
Suppose Σ is a (not necessarily connected) Riemann surface and τ:Σ→Σ is an anti-holomorphic (orientation-re |
https://en.wikipedia.org/wiki/Experimental%20event%20rate | In epidemiology and biostatistics, the experimental event rate (EER) is a measure of how often a particular statistical event (such as response to a drug, adverse event or death) occurs within the experimental group (non-control group) of an experiment.
This value is very useful in determining the therapeutic benefit or risk to patients in experimental groups, in comparison to patients in placebo or traditionally treated control groups.
Three statistical terms rely on EER for their calculation: absolute risk reduction, relative risk reduction and number needed to treat.
Control event rate
The control event rate (CER) is identical to the experimental event rate except that is measured within the scientific control group of an experiment.
Worked example
In a trial of hypothetical drug "X" where we are measuring event "Z", we have two groups. Our control group (25 people) is given a placebo, and the experimental group (25 people) is given drug "X".
Event "Z" in control group : 4 in 25 people
Control event rate : 4/25
Event "Z" in experimental group : 12 in 25 people
Experimental event rate : 12/25
Another worked example is as follows:
See also
Absolute risk reduction
Relative risk reduction
Number needed to treat
References
Biostatistics
Epidemiology
Medical statistics
Statistical ratios |
https://en.wikipedia.org/wiki/Unimodular%20polynomial%20matrix | In mathematics, a unimodular polynomial matrix is a square polynomial matrix whose inverse exists and is itself a polynomial matrix. Equivalently, a polynomial matrix A is unimodular if its determinant det(A) is a nonzero constant.
References
.
External links
Polynomial matrix glossary at Polyx (A matlab toolbox)
Matrices
Polynomials |
https://en.wikipedia.org/wiki/R%C3%B3bert%20Cicman | Róbert Cicman (born 3 September 1984) is a professional Slovak football centre back who plays for FK Čaňa..
Career statistics
Honours
MFK Košice
Slovak Cup (1): 2008–09
External links
MFK Košice profile
Living people
1984 births
Footballers from Košice
Men's association football central defenders
Slovak men's footballers
FC VSS Košice players
FC Steel Trans Ličartovce players
SK Slavia Prague players
FC Nitra players
Sandecja Nowy Sącz players
FK Haniska players
FC Lokomotíva Košice players
FK Slavoj Trebišov players
Slovak First Football League players
Czech First League players
Slovak expatriate men's footballers
Expatriate men's footballers in the Czech Republic
Expatriate men's footballers in Poland
Slovak expatriate sportspeople in the Czech Republic
2. Liga (Slovakia) players |
https://en.wikipedia.org/wiki/Bayesian%20vector%20autoregression | In statistics and econometrics, Bayesian vector autoregression (BVAR) uses Bayesian methods to estimate a vector autoregression (VAR) model. BVAR differs with standard VAR models in that the model parameters are treated as random variables, with prior probabilities, rather than fixed values.
Vector autoregressions are flexible statistical models that typically include many free parameters. Given the limited length of standard macroeconomic datasets relative to the vast number of parameters available, Bayesian methods have become an increasingly popular way of dealing with the problem of over-parameterization. As the ratio of variables to observations increases, the role of prior probabilities becomes increasingly important.
The general idea is to use informative priors to shrink the unrestricted model towards a parsimonious naïve benchmark, thereby reducing parameter uncertainty and improving forecast accuracy.
A typical example is the shrinkage prior, proposed by Robert Litterman (1979) and subsequently developed by other researchers at University of Minnesota, (i.e. Sims C, 1989), which is known in the BVAR literature as the "Minnesota prior". The informativeness of the prior can be set by treating it as an additional parameter based on a hierarchical interpretation of the model.
In particular, the Minnesota prior assumes that each variable follows a random walk process, possibly with drift, and therefore consists of a normal prior on a set of parameters with fixed and known covariance matrix, which will be estimated with one of three techniques: Univariate AR, Diagonal VAR, or Full VAR.
This type model can be estimated with Eviews, Stata, Python or R Statistical Packages.
Recent research has shown that Bayesian vector autoregression is an appropriate tool for modelling large data sets.
See also
Bayesian econometrics
Dynamic Stochastic General Equilibrium
Macroeconomic Modeling
References
Further reading
Bayesian statistics
Multivariate time series |
https://en.wikipedia.org/wiki/Bayesian%20econometrics | Bayesian econometrics is a branch of econometrics which applies Bayesian principles to economic modelling. Bayesianism is based on a degree-of-belief interpretation of probability, as opposed to a relative-frequency interpretation.
The Bayesian principle relies on Bayes' theorem which states that the probability of B conditional on A is the ratio of joint probability of A and B divided by probability of B. Bayesian econometricians assume that coefficients in the model have prior distributions.
This approach was first propagated by Arnold Zellner.
Basics
Subjective probabilities have to satisfy the standard axioms of probability theory if one wishes to avoid losing a bet regardless of the outcome. Before the data is observed, the parameter is regarded as an unknown quantity and thus random variable, which is assigned a prior distribution with . Bayesian analysis concentrates on the inference of the posterior distribution , i.e. the distribution of the random variable conditional on the observation of the discrete data . The posterior density function can be computed based on Bayes' Theorem:
where , yielding a normalized probability function. For continuous data , this corresponds to:
where and which is the centerpiece of Bayesian statistics and econometrics. It has the following components:
: the posterior density function of ;
: the likelihood function, i.e. the density function for the observed data when the parameter value is ;
: the prior distribution of ;
: the probability density function of .
The posterior function is given by , i.e., the posterior function is proportional to the product of the likelihood function and the prior distribution, and can be understood as a method of updating information, with the difference between and being the information gain concerning after observing new data. The choice of the prior distribution is used to impose restrictions on , e.g. , with the beta distribution as a common choice due to (i) being defined between 0 and 1, (ii) being able to produce a variety of shapes, and (iii) yielding a posterior distribution of the standard form if combined with the likelihood function . Based on the properties of the beta distribution, an ever-larger sample size implies that the mean of the posterior distribution approximates the maximum likelihood estimator
The assumed form of the likelihood function is part of the prior information and has to be justified. Different distributional assumptions can be compared using posterior odds ratios if a priori grounds fail to provide a clear choice. Commonly assumed forms include the beta distribution, the gamma distribution, and the uniform distribution, among others. If the model contains multiple parameters, the parameter can be redefined as a vector. Applying probability theory to that vector of parameters yields the marginal and conditional distributions of individual parameters or parameter groups. If data generation is sequential, Bayesian principl |
https://en.wikipedia.org/wiki/List%20of%20settlements%20in%20Devon%20by%20population | This list is of towns and cities in Devon in order of their population, according to the 2011 census data from the Office for National Statistics. It comprises the Key Statistics for local authorities, civil parishes and wards that attempt to show their populations. The largest settlement in Devon is the city and unitary authority of Plymouth with a population of 256,720, whereas the smallest settlement was the town and civil parish of Beer with a population of 1,317. The city of Exeter, which is home to Exeter Cathedral, is the county town and headquarters of Devon County Council. The ceremonial county of Devon includes unitary authority areas of Plymouth and Torbay, but the non-metropolitan county of Devon excludes such unitary authority areas. It is governed by Devon County Council, whereas Plymouth and Torbay can govern themselves on matters such as transport and education.
Traditionally a town is any settlement which has received a charter of incorporation, more commonly known as a town charter, approved by the monarch. However, since 1974, any civil parish has the right to declare itself as a town. Prior to 1888, city status was given to settlements home to a cathedral of the Church of England such as Exeter. After 1888 it was no longer a necessary condition, leading to Plymouth gaining city-status in 1928. Historical towns such as Plympton, Stonehouse and Devonport, which were merged into the city of Plymouth, have not been included, as well as Topsham, which became a part of Exeter's urban district, and St Marychurch, which was annexed by Torquay. However, the unitary authority area of Torbay recognises the three towns of Torquay, Paignton and Brixham. The ward for Ottery St Mary is also included, as it is titled Ottery St Mary Town.
Changes to population structures have, however, led to explosions in non-traditional settlements that do not fall into traditional, bureaucratic definitions of 'towns'. Several villages, which are not included in this list, have grown steadily and are more populous than many towns. For example, the ward of Fremington, with a population of 4,310. would be ranked 34 whilst its neighbour Braunton civil parish, with a population of 8,128, would be ranked 21.
See also
List of urban areas in the United Kingdom
List of English districts by population
Subdivisions of Scotland § Council areas
List of Welsh principal areas
List of districts in Northern Ireland (pre-2015)
Travel to work area
References
Towns and cities
Devon |
https://en.wikipedia.org/wiki/Kuznetsov%20trace%20formula | In analytic number theory, the Kuznetsov trace formula is an extension of the Petersson trace formula.
The Kuznetsov or relative trace formula connects Kloosterman sums at a deep level with the spectral theory of automorphic forms. Originally this could have been stated as follows. Let
be a sufficiently "well behaved" function. Then one calls identities of the following type Kuznetsov trace formula:
The integral transform part is some integral transform of g and the spectral part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of g. The Kuznetsov trace formula was found by Kuznetsov while studying the growth of weight zero automorphic functions. Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases where Pierre Deligne's proof of the Weil conjectures was not applicable.
It was later translated by Jacquet to a representation theoretic framework. Let be a reductive group over a number field F and be a subgroup. While the usual trace formula studies the harmonic analysis on G, the relative trace formula is a tool for studying the harmonic analysis on the symmetric space . For an overview and numerous applications Cogdell, J.W. and I. Piatetski-Shapiro, The arithmetic and spectral analysis of Poincaré series, volume 13 of Perspectives in mathematics. Academic Press Inc., Boston, MA, (1990).
References
Automorphic forms
Spectral theory
Theorems in analytic number theory |
https://en.wikipedia.org/wiki/Petersson%20trace%20formula | In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula.
In its simplest form the Petersson trace formula is as follows. Let be an orthonormal basis of , the space of cusp forms of weight on . Then for any positive integers we have
where is the Kronecker delta function, is the Kloosterman sum and is the Bessel function of the first kind.
References
Henryk Iwaniec: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics 17, American Mathematics Society, Providence, RI, 1991.
Theorems in analytic number theory
Automorphic forms |
https://en.wikipedia.org/wiki/Paul%20F.%20Velleman | Paul F. Velleman (born April 16, 1949) is an American academic who is a professor of statistics at Cornell University.
Education
Velleman earned a Bachelor of Arts degree in mathematics and social science from Dartmouth College, followed by Master of Arts and PhD from Princeton University. Velleman's thesis was written on the topic of non-linear data smoothing.
Career
He is the author and designer of the multimedia statistics CD-ROM, ActivStats, for which he was awarded the EDUCOM Medal for innovative uses of computers in teaching statistics, and the ICTCM Award for Innovation in Using Technology in College Mathematics. He developed the statistics program, Data Desk and the Internet site Data and Story Library (DASL), which provides datasets for teaching statistics. In 1987, he was elected as a Fellow of the American Statistical Association.
Books
He is co-author (with Richard De Veaux and David Bock) of Intro Stats, Stats: Modeling the World, and Stats: Data and Models, Business Statistics and Business Statistics: A First Course (with Norean Sharpe and Richard De Veaux) and the co-author (with David Hoaglin) of ABCs of Exploratory Data Analysis. Velleman taught statistics at Cornell University from 1975 to 2018. He is a Fellow of the American Statistical Association and of the American Association for the Advancement of Science.
See also
Data Desk
J. David Velleman
References
External links
People involved with Data Desk
1949 births
American statisticians
Princeton University alumni
Cornell University faculty
Living people
Fellows of the American Statistical Association |
https://en.wikipedia.org/wiki/Mart%C3%ADn%20Boasso | Martín Mariano Boasso Danielle (born 11 April 1975 in El Trebol, Santa Fe) is an Argentine naturalized Mexican footballer.
External links
Argentine Primera statistics
1975 births
Living people
Argentine men's footballers
People from San Martín Department, Santa Fe
Naturalized citizens of Mexico
Argentine emigrants to Mexico
Rosario Central footballers
Gimnasia y Esgrima de Jujuy footballers
Unión de Santa Fe footballers
C.D. Irapuato footballers
C.F. Pachuca players
Tecos F.C. footballers
Men's association football forwards
Expatriate men's footballers in Mexico
Expatriate men's footballers in El Salvador
C.D. FAS footballers
Santos Laguna footballers
Footballers from Santa Fe Province |
https://en.wikipedia.org/wiki/Ahmed%20Lari | Ahmed Lari is a member of the Kuwaiti National Assembly, representing the first district. Born in 1955, Lari studied Statistics and worked in the Municipal Council before being elected to the National Assembly in 2006. While political parties are technically illegal in Kuwait, Lari is a member of the National Islamic Alliance, a Shia party.
Expulsion from Popular Action Bloc
On February 19, 2008, the Popular Action Bloc expelled Lari and fellow Shiite MP Adnan Zahid Abdulsamad for taking part in a ceremony eulogizing Hezbollah's slain top commander, Imad Mughniyeh. The ceremony's description of the fugitive Lebanese militant — killed in a February 12 car bombing in Syria — as a hero sparked public outrage in a country that holds him responsible for hijacking a Kuwait Airways flight and killing two of its Kuwaiti passengers 20 years prior. The two lawmakers were only expelled from their bloc, and remained in the legislature. After the expulsions of the two, the seven member bloc was down to five members.
Opposition to Grilling of Prime Minister Nasser
In November 2008, Waleed Al-Tabtabaie, Mohammed Al-Mutair, and Mohammed Hayef Al-Mutairi filed a request to grill Prime Minister Nasser Mohammed Al-Ahmed Al-Sabah for allowing prominent Iranian Shiite cleric Mohammad Baqir al-Fali to enter Kuwait despite a legal ban. Lari was quoted as saying that the majority of MPs, like him, were against the grilling.
References
Members of the National Assembly (Kuwait)
Living people
Kuwaiti Shia Muslims
Kuwaiti people of Iranian descent
Popular Action Bloc politicians
National Islamic Alliance politicians
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Birkhoff%27s%20representation%20theorem | This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.
The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.
Understanding the theorem
Many lattices can be defined in such a way that the elements of the lattice are represented by sets, the join operation of the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. For instance, the Boolean lattice defined from the family of all subsets of a finite set has this property. More generally any finite topological space has a lattice of sets as its family of open sets. Because set unions and intersections obey the distributive law, any lattice defined in this way is a distributive lattice. Birkhoff's theorem states that in fact all finite distributive lattices can be obtained this way, and later generalizations of Birkhoff's theorem state a similar thing for infinite distributive lattices.
Examples
Consider the divisors of some composite number, such as (in the figure) 120, partially ordered by divisibility. Any two divisors of 120, such as 12 and 20, have a unique greatest common factor 12 ∧ 20 = 4, the largest number that divides both of them, and a unique least common multiple 12 ∨ 20 = 60; both of these numbers are also divisors of 120. These two operations ∨ and ∧ satisfy the distributive law, in either of two equivalent forms: (x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z) and (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z), for all x, y, and z. Therefore, the divisors form a finite distributive lattice.
One may associate each divisor with the set of prime powers that divide it: thus, 12 is associated with the set {2,3,4}, while 20 is associated with the set {2,4,5}. Then 12 ∧ 20 = 4 is associated with the set {2,3,4} ∩ {2,4,5} = {2,4}, while 12 ∨ 20 = 60 is associated with the set {2,3,4} ∪ {2,4,5} = {2,3,4,5}, so the join and meet operations of the lattice correspond to union and in |
https://en.wikipedia.org/wiki/Jean-Pierre%20Sydler | Jean-Pierre Sydler (1921-1988) was a Swiss mathematician and a librarian, well known for his work in geometry, most notably on Hilbert's third problem.
Biography
Sydler was born in 1921 in Neuchâtel, Switzerland. He graduated from ETH Zürich in 1943 and received a doctorate in 1947. In 1950 he became a librarian at the ETH while continuing to publish mathematical papers in his spare time. In 1960 he received a prize of the Danish Academy of Sciences for his work on scissors congruence. In 1963 he became a director of the ETH library and pioneered the use of automatisation. He continued serving as a director until the retirement in 1986. He died in 1988 in Zürich.
References
Greg N. Frederickson, Dissections: Plane and Fancy, Cambridge University Press, 2003.
External links
1921 births
1988 deaths
People from Neuchâtel
ETH Zurich alumni
20th-century Swiss mathematicians
Swiss librarians |
https://en.wikipedia.org/wiki/Sketch%20%28mathematics%29 | In the mathematical theory of categories, a sketch is a category D, together with a set of cones intended to be limits and a set of cocones intended to be colimits. A model of the sketch in a category C is a functor
that takes each specified cone to a limit cone in C and each specified cocone to a colimit cocone in C. Morphisms of models are natural transformations. Sketches are a general way of specifying structures on the objects of a category, forming a category-theoretic analog to the logical concept of a theory and its models. They allow multisorted models and models in any category.
Sketches were invented in 1968 by Charles Ehresmann, using a different but equivalent definition. There are still other definitions in the research literature.
References
.
.
.
.
.
.
External links
Sketches: Outline with references (updated 2009).
Category theory |
https://en.wikipedia.org/wiki/Local%20trace%20formula | In mathematics, the local trace formula is a local analogue of the Arthur–Selberg trace formula that describes the character of the representation of G(F) on the discrete part of L2(G(F)), for G a reductive algebraic group over a local field F.
References
Automorphic forms
Theorems in number theory |
https://en.wikipedia.org/wiki/Steinitz%20exchange%20lemma | The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Steinitz. The result is often called the Steinitz–Mac Lane exchange lemma, also recognizing the generalization
by Saunders Mac Lane
of Steinitz's lemma to matroids.
Statement
Let and be finite subsets of a vector space . If is a set of linearly independent vectors, and spans , then:
1. ;
2. There is a set with such that spans .
Proof
Suppose and . We wish to show that , and that after rearranging the if necessary, the set spans . We proceed by induction on .
For the base case, suppose is zero.
In this case, the claim holds because there are no vectors , and the set spans by hypothesis.
For the inductive step, assume the proposition is true for . By the inductive hypothesis we may reorder the so that spans . Since , there exist coefficients such that
.
At least one of must be non-zero, since otherwise this equality would contradict the linear independence of ; it follows that . By reordering the if necessary, we may assume that is nonzero. Therefore, we have
.
In other words, is in the span of . Since this span contains each of the vectors , by the inductive hypothesis it contains .
Applications
The Steinitz exchange lemma is a basic result in computational mathematics, especially in linear algebra and in combinatorial algorithms.
References
Julio R. Bastida, Field extensions and Galois Theory, Addison–Wesley Publishing Company (1984).
External links
Mizar system proof: http://mizar.org/version/current/html/vectsp_9.html#T19
Lemmas in linear algebra
Matroid theory |
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