source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Wymark%2C%20Saskatchewan | Wymark is a hamlet in Swift Current Rural Municipality No. 137, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a population of 144 in the Canada 2006 Census. The hamlet is located on Highway 628 about 2 km north of Highway 363, and 15 km south of Swift Current.
Etymology
Wymark was named after William Wymark Jacobs, an English writer best known for his 1902 story The Monkey's Paw.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Wymark had a population of 148 living in 54 of its 57 total private dwellings, a change of from its 2016 population of 138. With a land area of , it had a population density of in 2021.
See also
List of communities in Saskatchewan
Hamlets of Saskatchewan
References
Designated places in Saskatchewan
Organized hamlets in Saskatchewan
Swift Current No. 137, Saskatchewan |
https://en.wikipedia.org/wiki/Vantage%2C%20Saskatchewan | Vantage is a hamlet in Sutton Rural Municipality No. 103, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a reported population of zero in the Canada 2006 Census.
Demographics
Heritage sites
Vantage Methodist or (Grace United)
The church was built in Vantage in 1917. Vantage Church shared a minister with Mossbank and Ettington. Rev. Bert Howard was the first ordained minister.
See also
List of communities in Saskatchewan
Hamlets of Saskatchewan
References
Former designated places in Saskatchewan
Sutton No. 103, Saskatchewan
Hamlets in Saskatchewan
Ghost towns in Saskatchewan
Division No. 3, Saskatchewan |
https://en.wikipedia.org/wiki/Link%20group | In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link complement.
Definition
The link group of an n-component link is essentially the set of (n + 1)-component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through immersions), knotting or unknotting itself, but is not allowed to move through other components. This is a weaker condition than isotopy: for example, the Whitehead link has linking number 0, and thus is link homotopic to the unlink, but it is not isotopic to the unlink.
The link group is not the fundamental group of the link complement, since the components of the link are allowed to move through themselves, though not each other, but thus is a quotient group of the link complement's fundamental group, since one can start with elements of the fundamental group, and then by knotting or unknotting components, some of these elements may become equivalent to each other.
Examples
The link group of the n-component unlink is the free group on n generators, , as the link group of a single link is the knot group of the unknot, which is the integers, and the link group of an unlinked union is the free product of the link groups of the components.
The link group of the Hopf link, the simplest non-trivial link – two circles, linked once – is the free abelian group on two generators, Note that the link group of two unlinked circles is the free nonabelian group on two generators, of which the free abelian group on two generators is a quotient. In this case the link group is the fundamental group of the link complement, as the link complement deformation retracts onto a torus.
The Whitehead link is link homotopic to the unlink – though it is not isotopic to the unlink – and thus has link group the free group on two generators.
Milnor invariants
Milnor defined invariants of a link (functions on the link group) in , using the character which have thus come to be called "Milnor's μ-bar invariants", or simply the "Milnor invariants". For each k, there is an k-ary function which defines invariants according to which k of the links one selects, in which order.
Milnor's invariants can be related to Massey products on the link complement (the complement of the link); this was suggested in , and made precise in and .
As with Massey products, the Milnor invariants of length k + 1 are defined if all Milnor invariants of length less than or equal to k vanish. The first (2-fold) Milnor invariant is simply the linking number (just as the 2-fold Massey product is the cup product, which is dual to intersection), while the 3-fold Milnor invariant measures whether 3 pairwise unlinked circles are Borromean rings, and if so, in some sense, how many times (that is to |
https://en.wikipedia.org/wiki/1950%E2%80%9351%20Detroit%20Red%20Wings%20season | The 1950–51 Detroit Red Wings season was the Red Wings' 25th season.
Offseason
Regular season
Final standings
Record vs. opponents
Schedule and results
Playoffs
Player statistics
Regular season
Scoring
Goaltending
Playoffs
Scoring
Goaltending
Note: GP = Games played; G = Goals; A = Assists; Pts = Points; +/- = Plus-minus PIM = Penalty minutes; PPG = Power-play goals; SHG = Short-handed goals; GWG = Game-winning goals;
MIN = Minutes played; W = Wins; L = Losses; T = Ties; GA = Goals against; GAA = Goals-against average; SO = Shutouts;
Awards and records
Transactions
See also
1950–51 NHL season
References
External links
Detroit Red Wings season, 1950-51
Detroit Red Wings season, 1950-51
Detroit Red Wings seasons
Detroit Red Wings
Detroit Red Wings |
https://en.wikipedia.org/wiki/1952%E2%80%9353%20Detroit%20Red%20Wings%20season | The 1952–53 Detroit Red Wings season was the Red Wings' 27th season.
Offseason
Regular season
Final standings
Record vs. opponents
Schedule and results
Playoffs
Player statistics
Regular season
Scoring
Goaltending
Playoffs
Scoring
Goaltending
Note: GP = Games played; G = Goals; A = Assists; Pts = Points; +/- = Plus-minus PIM = Penalty minutes; PPG = Power-play goals; SHG = Short-handed goals; GWG = Game-winning goals;
MIN = Minutes played; W = Wins; L = Losses; T = Ties; GA = Goals against; GAA = Goals-against average; SO = Shutouts;
Awards and records
All-Star teams
Transactions
The following is a list of all transactions that have occurred for the Detroit Red Wings during the 1952–53 NHL season. It lists which team each player has been traded to and for which player(s) or other consideration(s), if applicable.
See also
1952–53 NHL season
References
External links
Detroit
Detroit
Detroit Red Wings seasons
Detroit Red Wings
Detroit Red Wings |
https://en.wikipedia.org/wiki/1955%E2%80%9356%20Detroit%20Red%20Wings%20season | The 1955–56 Detroit Red Wings season was the Red Wings' 30th season.
Offseason
Regular season
Final standings
Record vs. opponents
Schedule and results
Playoffs
Player statistics
Regular season
Scoring
Goaltending
Playoffs
Scoring
Goaltending
Note: GP = Games played; G = Goals; A = Assists; Pts = Points; +/- = Plus-minus PIM = Penalty minutes; PPG = Power-play goals; SHG = Short-handed goals; GWG = Game-winning goals;
MIN = Minutes played; W = Wins; L = Losses; T = Ties; GA = Goals against; GAA = Goals-against average; SO = Shutouts;
Awards and records
Transactions
See also
1955–56 NHL season
References
External links
Detroit
Detroit
Detroit Red Wings seasons
Detroit Red Wings
Detroit Red Wings |
https://en.wikipedia.org/wiki/Fake%20projective%20plane | In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.
History
Severi asked if there was a complex surface homeomorphic to the projective plane but not biholomorphic to it. showed that there was no such surface, so the closest approximation to the projective plane one can have would be a surface with the same Betti numbers (b0,b1,b2,b3,b4) = (1,0,1,0,1) as the projective plane.
The first example was found by using p-adic uniformization introduced independently by Kurihara and Mustafin.
Mumford also observed that Yau's result together with Weil's theorem on the rigidity of discrete cocompact subgroups of PU(1,2) implies that there are only a finite number of fake projective planes. found two more examples, using similar methods, and found an example with an automorphism of order 7 that is birational to a cyclic cover of degree 7 of a Dolgachev surface. , found a systematic way of classifying all fake projective planes, by showing that there are twenty-eight classes, each of which contains at least an example of
fake projective plane up to isometry, and that there can at most be five more classes which were later shown not to exist. The problem of listing all fake projective planes is reduced to listing all subgroups of appropriate index of an explicitly given lattice associated to each class. By extending these calculations showed that the twenty-eight classes exhaust all possibilities for fake projective planes and that
there are altogether 50 examples determined up to isometry, or 100 fake projective planes up to biholomorphism.
A surface of general type with the same Betti numbers as a minimal surface not of general type must have the Betti numbers of either a
projective plane P2 or a quadric P1×P1. constructed some "fake quadrics": surfaces of general type with the same Betti numbers as quadrics. Beauville surfaces give further examples.
Higher-dimensional analogues of fake projective surfaces are called fake projective spaces.
The fundamental group
As a consequence of the work of Aubin and Yau on solution of Calabi Conjecture in the case of negative Ricci curvature, see , any fake projective plane is the quotient of a complex unit ball in 2 dimensions by a discrete subgroup, which is the fundamental group of the fake projective plane. This fundamental group must therefore be a torsion-free and cocompact discrete subgroup of PU(2,1) of Euler-Poincaré characteristic 3. and showed that this fundamental group must also be an arithmetic group. Mostow's strong rigidity results imply that the fundamental group determines the fake plane, in the strong sense that any compact surface with the same fundamental group must be isometric to it.
Two fake projective planes are defined to be in the same class if their fundamental groups are both |
https://en.wikipedia.org/wiki/Chinese%20people%20in%20Kyrgyzstan | Chinese people in Kyrgyzstan have been growing in numbers since the late 1980s. 2008 police statistics showed 60,000 Chinese nationals living in the country. However, the 2009 census showed just 1,813 people who declared themselves to be of Chinese ethnicity.
History
During the Mongol Empire, Han Chinese were moved to Central Asian areas like Besh Baliq, Almaliq, and Samarqand by the Mongols where they worked as artisans and farmers. The Daoist Chinese master Qiu Chuji travelled through Kyrgyzstan to meet Genghis Khan in Afghanistan.
As China and Kyrgyzstan are neighbouring countries, there is a long history of population movements between the lands that today make up their national territories. The Dungan people (Chinese-speaking Muslims from Northwest China) fled to Kyrgyzstan in 1877 after the failure of their uprising against the Qing Dynasty; they settled in Semirechie as well as the Ferghana Valley. In the early 20th century, Uyghurs, Dungans, and Han Chinese alike came to the Ferghana Valley as migrant workers in coal mines, cotton mills, and farms; some settled down permanently in Kyrgyzstan. The agricultural failures incurred during the 1950s Great Leap Forward spurred many people from Xinjiang to flee to the Soviet Union, including Kyrgyzstan, to escape hardships in China. However, as the Sino-Soviet split worsened, the border was closed and such migration made impossible.
Migration would begin again in the late 1980s, centred on Chüy Region, Bishkek and its surroundings; people from Xinjiang would come to rent land, and grow vegetables. Others came as cross-border traders, selling Chinese alcoholic beverages and buying up clothing—especially coats made from Karakul sheep pelts—for sale in Xinjiang. In the early 2000s, the majority of PRC nationals in Kyrgyzstan were of Uyghur ethnicity, but since then, an increasing number of Han Chinese have been arriving.
Kyrgyzstan and other post-Soviet states are popular destinations for people from Xinjiang because they offer the opportunity to learn Russian, which has become important in urban job markets such as Urumqi. Recent migrants state they chose Kyrgyzstan as their destination, rather than join the large numbers of Chinese people in Russia or in Kazakhstan because Kyrgyzstan is cheaper, and because they perceive public safety as being better in Kyrgyzstan than in Russia where there have been cases of attacks on migrant workers.
Business and employment
Chinese traders often employ local Dungans as assistants. Kyrgyz university students of all ethnicities also often seek out employment with Chinese traders, using their job as an opportunity to learn the Chinese language. On the outskirts of Bishkek, there is a large Chinese market, described as a "city within a city"; it has its own hospital, mosque, and apartment buildings.
Migrants from China also work in the construction sector, especially on housing projects for low-income people. President Kurmanbek Bakiyev once gave a speech pr |
https://en.wikipedia.org/wiki/Norman%20Shapiro | Norman Zalmon Shapiro was an American mathematician, who was the co-author of the Rice–Shapiro theorem.
Education
Shapiro obtained a BS in Mathematics at University of Illinois in 1952.
Shapiro spent the summer of 1954 at Bell Laboratories in Murray Hill, New Jersey where, in collaboration with Karel de Leeuw, Ed Moore, and Claude Shannon, he investigated the question of whether providing a Turing machine augmented with an oracle machine producing an infinite sequence of random events (like the tosses of a fair coin) would enable the machine to output a non-computable sequence. The well-known efficacy of Monte Carlo methods might have led one to think otherwise, but the result was negative. Stated precisely:
An infinite string, S, on a finite alphabet is computable if it can be output with probability one by a Turing machine augmented by an oracle machine giving an infinite sequence of equal-probability zeroes and ones.
Moreover, the result continues to hold if the output probability is any positive number, and the probability of an oracle machine inquiry yielding 1 is any computable real number.
Shapiro obtained his Ph.D from Princeton University in 1955 under the advisership of Alonzo Church. In 1955, as a Princeton PhD student, Shapiro coined the phrase "strong reducibility" for a computability theory currently called the many-one reduction. His thesis was titled Degrees of Computability and was published in 1958.
Career
Shapiro was a leading mathematician and computer scientist at the RAND Corporation think tank from 1959 until 1999. In the late 1960s and early 1970s Shapiro was the lead designer of one of the first computer-based mapping and cartography systems.
In the 1970s Shapiro co-designed the MH Message Handling System. MH was the first mail system to utilize Unix design principles by using shell commands to manipulate messages as individual files.
In 1972, Norman Z. Shapiro was a creative lead in his essays on e-mail etiquette, introducing concepts that were rarely considered until over 15 years later. His work may be the first substantial writing about netiquette. The primary essay was "Toward an Ethics and Etiquette for Electronic Mail".
In the 1970s through 1990s Shapiro developed many new and unique contributions to computer science, mathematics, and modeling. In the early 1980s, he was the software architect for large and complex game-structured simulation (the RAND Strategy Assessment System) at the RAND Corporation. That represented regional or global crisis and war with agents optionally substituting for human teams in making high-level decisions. These decisions then directed actions represented in a large global combat model. Different versions of the agents could be substituted (e.g., to reflect a change in government). Agents could run the simulation within itself to test potential strategies with "lookahead." The system was successfully implemented and was used in the late 1980s before the end of the Cold War |
https://en.wikipedia.org/wiki/%C4%B0smail%20Hakk%C4%B1%20Duru | İsmail Hakkı Duru is a Turkish theoretical physicist and emeritus professor of Mathematics at the Izmir Institute of Technology where he is former dean of the science faculty.
Publications
Complete list at Google Scholar
Complete list at SPIRES
References
External links
Turkish non-fiction writers
Turkish physicists
Living people
1946 births
Theoretical physicists |
https://en.wikipedia.org/wiki/Fyllinge | Fyllinge was a locality situated in Halmstad Municipality, Halland County, Sweden, with 2,927 inhabitants in 2010. Since 2015 the locality is now counted by Statistics Sweden as part of Halmstad.
References
Populated places in Halmstad Municipality |
https://en.wikipedia.org/wiki/Dolgachev%20surface | In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds, no two of which are diffeomorphic.
Properties
The blowup of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some .
The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature (so it is the unimodular lattice ). The geometric genus is 0 and the Kodaira dimension is 1.
found the first examples of homeomorphic but not diffeomorphic 4-manifolds and . More generally the surfaces and are always homeomorphic, but are not diffeomorphic unless .
showed that the Dolgachev surface has a handlebody decomposition without 1- and 3-handles.
References
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Jerome%20Cornfield | Jerome Cornfield (1912–1979) was an American statistician. He is best known for his work in biostatistics, but his early work was in economic statistics and he was also an early contributor to the theory of Bayesian inference. He played a role in the early development of input-output analysis and linear programming. Cornfield played a crucial role in establishing the causal link between smoking and incidence of lung cancer.
He introduced the Rare disease assumption and the "Cornfield condition" that allows one to assess whether an unmeasured (binary) confounder can explain away the observed relative risk due to some exposure like smoking.
He was born on October 30, 1912, in The Bronx, New York City. He graduated from New York University in 1933 and was briefly a graduate student at Columbia University. He also studied statistics and mathematics at the Graduate School of the US Department of Agriculture while employed by the Bureau of Labor Statistics, where he remained until 1947. He later worked at the National Cancer Institute, the Department of Biostatistics at Johns Hopkins School of Hygiene and Public Health, the National Heart Institute, the University of Pittsburgh, and George Washington University.
In 1951 he was elected as a Fellow of the American Statistical Association. He was the R. A. Fisher Lecturer in 1973 and President of the American Statistical Association in 1974.
Cornfield married Ruth Bittler in 1937. They had two daughters, Ann and Ellen.
He died on September 17, 1979, in Great Falls, Virginia.
References
Jerome Cornfield Papers: Historical Note. Special Collections Department, Iowa State University.
Jerome Cornfield: Chair 1958–1960. Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health.
Jerome Cornfield: Papers by and Comments On. Compiled by Milo Schield, Editor of www.StatLit.org
1912 births
1979 deaths
American statisticians
Fellows of the American Statistical Association
Presidents of the American Statistical Association
University of Pittsburgh faculty
20th-century American mathematicians
Mathematicians from New York (state)
Scientists from the Bronx |
https://en.wikipedia.org/wiki/Zolt%C3%A1n%20F%C3%BCredi | Zoltán Füredi (Budapest, Hungary, 21 May 1954) is a Hungarian mathematician, working in combinatorics, mainly in discrete geometry and extremal combinatorics. He was a student of Gyula O. H. Katona. He is a corresponding member of the Hungarian Academy of Sciences (2004). He is a research professor of the Rényi Mathematical Institute of the Hungarian Academy of Sciences, and a professor at the University of Illinois Urbana-Champaign (UIUC).
Füredi received his Candidate of Sciences degree in mathematics in 1981 from the Hungarian Academy of Sciences.
Some results
In infinitely many cases he determined the maximum number of edges in a graph with no C4.
With Paul Erdős he proved that for some c>1, there are cd points in d-dimensional space such that all triangles formed from those points are acute.
With Imre Bárány he proved that no polynomial time algorithm determines the volume of convex bodies in dimension d within a multiplicative error dd.
He proved that there are at most unit distances in a convex n-gon.
In a paper written with coauthors he solved the Hungarian lottery problem.
With Ilona Palásti he found the best known lower bounds on the orchard-planting problem of finding sets of points with many 3-point lines.
He proved an upper bound on the ratio between the fractional matching number and the matching number in a hypergraph.
References
External links
Füredi's UIUC home page
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Members of the Hungarian Academy of Sciences
Combinatorialists
University of Illinois Urbana-Champaign faculty
1954 births
Living people |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20Vancouver%20Canucks%20season | The 1986–87 Vancouver Canucks season was the team's 17th in the National Hockey League (NHL).
Offseason
Regular season
Final standings
Schedule and results
Playoffs
Player statistics
Awards and records
Transactions
Draft picks
Vancouver's draft picks at the 1986 NHL Entry Draft held at the Montreal Forum in Montreal, Quebec.
Farm teams
See also
1986–87 NHL season
References
External links
Vancouver Canucks seasons
Vancouver C
Vancouver |
https://en.wikipedia.org/wiki/Fermat%E2%80%93Catalan%20conjecture | In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation
has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying
The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = am + bn) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples).
Known solutions
As of 2015, the following ten solutions to equation (1) which meet the criteria of equation (2) are known:
(for to satisfy Eq. 2)
The first of these (1m + 23 = 32) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since one can pick any m for m > 6), these solutions only give a single triplet of values (am, bn, ck).
Partial results
It is known by the Darmon–Granville theorem, which uses Faltings's theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (a, b, c) solving (1) exist. However, the full Fermat–Catalan conjecture is stronger as it allows for the exponents m, n and k to vary.
The abc conjecture implies the Fermat–Catalan conjecture.
For a list of results for impossible combinations of exponents, see Beal conjecture#Partial results. Beal's conjecture is true if and only if all Fermat–Catalan solutions have m = 2, n = 2, or k = 2.
See also
Sums of powers, a list of related conjectures and theorems
References
External links
Perfect Powers: Pillai's works and their developments. Waldschmidt, M.
Conjectures
Unsolved problems in number theory
Diophantine equations |
https://en.wikipedia.org/wiki/Polyadic%20algebra | Polyadic algebras (more recently called Halmos algebras) are algebraic structures introduced by Paul Halmos. They are related to first-order logic analogous to the relationship between Boolean algebras and propositional logic (see Lindenbaum–Tarski algebra).
There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras (when equality is part of the logic) and Lawvere's functorial semantics (a categorical approach).
References
Further reading
Paul Halmos, Algebraic Logic, Chelsea Publishing, New York (1962)
Algebraic logic |
https://en.wikipedia.org/wiki/Saleh%20Khalilazad | Mohammad-Saleh Khalil-Azad (, born 17 April 1990 in Shiraz) is an Iranian professional football goalkeeper who currently plays for Fajr Sepasi.
Club career
Club career statistics
External links
Mohammad Saleh Khalil-Azad at PersianLeague.com
1990 births
Living people
Iranian men's footballers
Bargh Shiraz F.C. players
Rah Ahan Tehran F.C. players
Fajr Sepasi Shiraz F.C. players
Footballers at the 2010 Asian Games
Men's association football goalkeepers
Asian Games competitors for Iran
Footballers from Shiraz |
https://en.wikipedia.org/wiki/Wilfried%20Brauer | Wilfried Brauer (8 August 1937 – 25 February 2014) was a German computer scientist and professor emeritus at Technical University of Munich.
Life and work
Brauer studied Mathematics, Physics, and Philosophy at the Free University of Berlin. He received a PhD in Mathematics 1966 from the University of Bonn for a dissertation on the theory of profinite groups.
Wilfried Brauer and his wife Ute were two of the 19 founding members of the German Informatics Society. From 1998 to 2001, he was chairman of the German Informatics Society. From 1994 to 1999, he was vice president of the International Federation of Information Processing.
He received several awards and honours:
Felix Hausdorff-Gedächtnispreis (1966)
IFIP Silver Core (1986)
honorary doctor of the University of Hamburg (1996)
Werner Heisenberg Medal (2000)
IFIP Isaac L. Auerbach Award (2002)
honorary doctor of the Freie Universität Berlin (2004)
One of ten inaugural fellows of the European Association for Theoretical Computer Science (2014, posthumous).
Publications
Below is a selection of books written by Brauer.
Über das Turingsche Modell einer Rechenmaschine und den Begriff des Algorithmus, Diploma Thesis, Freie Universität Berlin, 1962
Zur Theorie der pro-endlichen Gruppen. Doctorate Thesis, Universität Bonn, 1968 (Advisor: Wolfgang Krull)
with Klaus Indermark: Algorithmen, Rekursive Funktionen und Formale Sprachen (in German), 1968.
Automatentheorie (in German), Teubner 1984.
with Friedrich L. Bauer and H. Schwichtenberg: Logic and Algebra of Specifications, 1993
Editorship
Wilfried Brauer (ed.): Gesellschaft für Informatik e.V., 3. Jahrestagung, Hamburg, 8.-10. Oktober 1973. Lecture Notes in Computer Science Volume 1, Springer 1973
Texts in Theoretical Computer Science. An EATCS Series. Series Editors: Brauer, W., Hromkovič, J., Rozenberg, G., Salomaa, A., publisher's page.
References
External links
Wilfried Brauer at the mathematics genealogy project
German computer scientists
Theoretical computer scientists
Academic staff of the Technical University of Munich
1937 births
2014 deaths
Presidents of the German Informatics Society |
https://en.wikipedia.org/wiki/Subordinator | Subordinator may refer to
Subordination (linguistics), hierarchical organization in linguistics
Subordinator (mathematics), a stochastic process |
https://en.wikipedia.org/wiki/Roger%20Federer%20career%20statistics | This is a list of the main career statistics of Swiss former professional tennis player Roger Federer. All statistics are according to the ATP Tour website. Federer won 103 ATP singles titles including 20 major singles titles, 28 ATP Masters titles, and a shared record of six ATP Finals. Federer was also a gold medalist in men's doubles with Stan Wawrinka at the 2008 Beijing Olympics and a silver medalist in men's singles at the 2012 London Olympics.
Representing Switzerland, Federer assisted in winning the 2014 Davis Cup and a record three Hopman Cup titles (2001, 2018 and 2019). He is the first Swiss male player to win a major title, the only Swiss male player to hold the No. 1 ranking in singles, and the only Swiss player, male or female to win all four majors. At the international level, he helped Team Europe win three consecutive Laver Cup titles, the 2017, 2018 and 2019 editions.
Historic achievements
Federer has won 20 Grand Slam men's singles titles, third behind Djokovic (24) and Nadal (22). He was the first male player to win more than 14 Grand Slams. He has reached 31 Grand Slam singles finals, second-most behind Djokovic (10 consecutive, and another 8 consecutive—the two longest streaks in men's tennis history), 23 consecutive semifinal appearances, and 36 consecutive quarterfinal appearances. He is one of eight men to have won a career Grand Slam (winning all four majors at least once) and the second of four players to have won a career Grand Slam on three different surfaces, hard, grass, and clay courts, after Andre Agassi and before Nadal and Djokovic.
Federer is the only male player to win five consecutive US Open titles (2004–08) in the Open Era and in the process win 40 consecutive matches at the US Open. Federer achieved a record streak of 10 consecutive major finals (2005 Wimbledon to 2007 US Open) and never lost 2 consecutive finals during this streak.
Federer is the second male player to reach French Open and Wimbledon finals in the same year for four consecutive years (2006–2009), after Björn Borg (1978–81). Federer is the only male player to appear in seven consecutive Wimbledon finals (2003–2009), second behind Ivan Lendl's record of eight consecutive US Open finals (1982–1989). Federer is second male player to win 40 consecutive Wimbledon matches after Borg and in the process became the only male player to win 40 consecutive matches at two Grand Slams (Wimbledon and the US Open). Federer has won 11 hard court major titles (6 at the Australian Open and 5 at the US Open), second behind only Novak Djokovic (14).
Federer appeared in the French Open, Wimbledon and US Open finals in the same year for four consecutive years (2006–2009), surpassing the old record of Borg who achieve the same task three times in his (Borg) career (1978, 1980–81). Federer is the only male player to appear in Wimbledon and US Open finals in the same year for 6 consecutive years (2004–2008) and won both of them in the same year for 4 consecuti |
https://en.wikipedia.org/wiki/Fake%20projective%20space | In mathematics, a fake projective space is a complex algebraic variety that has the same Betti numbers as some projective space, but is not isomorphic to it.
There are exactly 50 fake projective planes. found four examples of fake projective 4-folds, and showed that no arithmetic examples exist in dimensions other than 2 and 4.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Deflator | In statistics, a deflator is a value that allows data to be measured over time in terms of some base period, usually through a price index, in order to distinguish between changes in the money value of a gross national product (GNP) that come from a change in prices, and changes from a change in physical output. It is the measure of the price level for some quantity. A deflator serves as a price index in which the effects of inflation are nulled. It is the difference between real and nominal GDP.
In the United States, the import and export price indexes produced by the International Price Program are used as deflators in national accounts. For example, the gross domestic product (GDP) equals consumption expenditures plus net investment plus government expenditures plus exports minus imports. Various price indexes are used to "deflate" each component of the GDP to make the GDP figures comparable over time. Import price indexes are used to deflate the import component (i.e., import volume is divided by the Import Price index) and the export price indexes are used to deflate the export component (i.e., export volume is divided by the Export Price index).
It is generally used as a statistical tool to convert dollars purchasing power into "inflation-adjusted" purchasing power, thus enabling the comparison of prices while accounting for inflation in various time periods.
See also
Bureau of Labor Statistics
GDP Deflator
Gross domestic product
Deflation
Inflation
Economic indicators
Producer price index (PPI)
Consumer price index (CPI)
References
External links
Deflator in glossary, U.S. Bureau of Labor Statistics Division of Information Services
Deflator, Investorwords
Can inflation be prevented?, MIT
Economic data
Time series
Inflation |
https://en.wikipedia.org/wiki/Trivial%20semigroup | In mathematics, a trivial semigroup (a semigroup with one element) is a semigroup for which the cardinality of the underlying set is one. The number of distinct nonisomorphic semigroups with one element is one. If S = { a } is a semigroup with one element, then the Cayley table of S is
{| class="wikitable"
|-
!
! a
|-
| a
| a
|}
The only element in S is the zero element 0 of S and is also the identity element 1 of S. However not all semigroup theorists consider the unique element in a semigroup with one element as the zero element of the semigroup. They define zero elements only in semigroups having at least two elements.
In spite of its extreme triviality, the semigroup with one element is important in many situations. It is the starting point for understanding the structure of semigroups. It serves as a counterexample in illuminating many situations. For example, the semigroup with one element is the only semigroup in which 0 = 1, that is, the zero element and the identity element are equal.
Further, if S is a semigroup with one element, the semigroup obtained by adjoining an identity element to S is isomorphic to the semigroup obtained by adjoining a zero element to S.
The semigroup with one element is also a group.
In the language of category theory, any semigroup with one element is a terminal object in the category of semigroups.
See also
Trivial group
Zero ring
Field with one element
Empty semigroup
Semigroup with two elements
Semigroup with three elements
Special classes of semigroups
References
Algebraic structures
Semigroup theory |
https://en.wikipedia.org/wiki/Nowhere%20commutative%20semigroup | In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab = ba then a = b. A semigroup S is nowhere commutative if and only if any two elements of S are inverses of each other.
Characterization of nowhere commutative semigroups
Nowhere commutative semigroups can be characterized in several different ways. If S is a semigroup then the following statements are equivalent:
S is nowhere commutative.
S is a rectangular band (in the sense in which the term is used by John Howie).
For all a and b in S, aba = a.
For all a, b and c in S, a2 = a and abc = ac.
Even though, by definition, the rectangular bands are concrete semigroups, they have the defect that their definition is formulated not in terms of the basic binary operation in the semigroup. The approach via the definition of nowhere commutative semigroups rectifies this defect.
To see that a nowhere commutative semigroup is a rectangular band, let S be a nowhere commutative semigroup. Using the defining properties of a nowhere commutative semigroup, one can see that for every a in S the intersection of the Green classes Ra and La contains the unique element a. Let S/L be the family of L-classes in S and S/R be the family of R-classes in S. The mapping
ψ : S → (S/R) × (S/L)
defined by
aψ = (Ra, La)
is a bijection. If the Cartesian product (S/R) × (S/L) is made into a semigroup by furnishing it with the rectangular band multiplication, the map ψ becomes an isomorphism. So S is isomorphic to a rectangular band.
Other claims of equivalences follow directly from the relevant definitions.
See also
Special classes of semigroups
References
Algebraic structures
Semigroup theory |
https://en.wikipedia.org/wiki/Packing%20in%20a%20hypergraph | In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k-uniform hypergraphs. One of them is a random greedy algorithm which was proposed by Joel Spencer. He used a branching process to formally prove the optimal achievable bound under some side conditions. The other algorithm is called the Rödl nibble and was proposed by Vojtěch Rödl et al. They showed that the achievable packing by the Rödl nibble is in some sense close to that of the random greedy algorithm.
History
The problem of finding the number of such subsets in a k-uniform hypergraph was originally motivated through a conjecture by Paul Erdős and Haim Hanani in 1963. Vojtěch Rödl proved their conjecture asymptotically under certain conditions in 1985. Pippenger and Joel Spencer generalized Rödl's results using a random greedy algorithm in 1989.
Definition and terminology
In the following definitions, the hypergraph is denoted by H=(V,E). H is called a k-uniform hypergraph if every edge in E consists of exactly k vertices.
is a hypergraph packing if it is a subset of edges in H such that there is no pair of distinct edges with a common vertex.
is a (,)-good hypergraph if there exists a such that for all and and both of the following conditions hold.
where the degree of a vertex is the number of edges that contain and the codegree of two distinct vertices and is the number of edges that contain both vertices.
Theorem
There exists an asymptotic packing P of size at least for a -uniform hypergraph under the following two conditions,
All vertices have the degree of in which tends to infinity.
For every pair of vertices shares only common edges.
where is the total number of vertices. This result was shown by Pippenger and was later proved by Joel Spencer. To address the asymptotic hypergraph packing problem, Joel Spencer proposed a random greedy algorithm. In this algorithm, a branching process is used as the basis and it was shown that it almost always achieves an asymptotically optimal packing under the above side conditions.
Asymptotic packing algorithms
There are two famous algorithms for asymptotic packing of k-uniform hypergraphs: the random greedy algorithm via branching process, and the Rödl nibble.
Random greedy algorithm via branching process
Every edge is independently and uniformly assigned a distinct real "birthtime" . The edges are taken one by one in the order of their birthtimes. The edge is accepted and included in if it does not overlap any previously accepted edges. Obviously, the subset is a packing and it can be shown that its size is almost surely. To show that, let stop the process of adding new edges at time . For an arbitrary , pick such that for any -good hypergraph where denotes the probability of vertex survival (a |
https://en.wikipedia.org/wiki/Quinn%20McNemar | Quinn Michael McNemar (February 20, 1900 – July 3, 1986) was an American psychologist and statistician. He is known for his work on IQ tests, for his book Psychological Statistics (1949) and for McNemar's test, the statistical test he introduced in 1947.
Life
McNemar was born in Greenland, West Virginia in 1900. He obtained his bachelor's degree in mathematics in 1925 from Juniata College, studied for his doctorate in psychology under Lewis Terman at Stanford University, and joined the faculty at Stanford in 1931. In 1942 he published The Revision of the Stanford–Binet Scale, the IQ test released in 1916 by Terman. By the time he retired from Stanford in 1965 he held professorships in psychology, statistics and education. He taught for another five years at the University of Texas before retiring to Palo Alto, where he died in 1986.
He was president of the Psychometric Society in 1951 and of the American Psychological Association in 1964.
References
1900 births
1986 deaths
American statisticians
Stanford University Department of Psychology faculty
University of Texas at Austin faculty
People from Grant County, West Virginia
Juniata College alumni
Stanford University alumni
Educators from West Virginia
Presidents of the American Psychological Association
20th-century American psychologists
Quantitative psychologists |
https://en.wikipedia.org/wiki/Asger%20Aaboe | Asger Hartvig Aaboe (26 April 1922 – 19 January 2007) was a historian of the exact sciences and of mathematics who is known for his contributions to the history of ancient Babylonian astronomy. In his studies of Babylonian astronomy, he went beyond analyses in terms of modern mathematics to seek to understand how the Babylonians conceived their computational schemes.
Aaboe studied mathematics and astronomy at the University of Copenhagen, and in 1957 obtained a PhD in the History of Science from Brown University, where he studied under Otto Neugebauer, writing a dissertation "On Babylonian Planetary Theories". In 1961 he joined the Department of the History of Science and Medicine at Yale University, serving as chair from 1968 to 1971, and continuing an active career there until retiring in 1992. At Yale, his doctoral students included Alice Slotsky and Noel Swerdlow.
He was elected to the Royal Danish Academy of Sciences and Letters in 1975, served as president of the Connecticut Academy of Arts and Sciences from 1970 to 1980, and was a member of many other scholarly societies.
Aaboe married Joan Armstrong on 14 July 1950. The marriage produced four children: Kirsten Aaboe, Erik Harris Aaboe, Anne Aaboe, Niels Peter Aaboe.
Selected publications
Episodes from the Early History of Mathematics, New York: Random House, 1964.
"Scientific Astronomy in Antiquity", Philosophical Transactions of the Royal Society of London, A.276, (1974: 21–42).
"Mesopotamian Mathematics, Astronomy, and Astrology", The Cambridge Ancient History (2nd. ed.), Vol. III, part 2, chap. 28b, Cambridge: Cambridge University Press, 1991,
Episodes from the Early History of Astronomy, New York: Springer, 2001, .
Notes
References
1922 births
2007 deaths
20th-century Danish mathematicians
21st-century Danish mathematicians
Historians of science
Danish historians of mathematics
Historians of astronomy
Danish expatriates in the United States
University of Copenhagen alumni
Brown University Graduate School alumni
Yale University faculty |
https://en.wikipedia.org/wiki/G%C3%A9za%20Fodor%20%28mathematician%29 | Géza Fodor (6 May 1927 in Szeged – 28 September 1977 in Szeged) was a Hungarian mathematician, working in set theory. He proved Fodor's lemma on stationary sets, one of the most important, and most used results in set theory. He was a professor at the Bolyai Institute of Mathematics at the Szeged University. He was vice-president, then president of the Szeged University. He was elected a corresponding member of the Hungarian Academy of Sciences.
1927 births
1977 deaths
Members of the Hungarian Academy of Sciences
Set theorists
20th-century Hungarian mathematicians
People from Szeged |
https://en.wikipedia.org/wiki/Cancellative%20semigroup | In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form a·b = a·c, where · is a binary operation, one can cancel the element a and deduce the equality b = c. In this case the element being cancelled out is appearing as the left factors of a·b and a·c and hence it is a case of the left cancellation property. The right cancellation property can be defined analogously. Prototypical examples of cancellative semigroups are the positive integers under addition or multiplication. Cancellative semigroups are considered to be very close to being groups because cancellability is one of the necessary conditions for a semigroup to be embeddable in a group. Moreover, every finite cancellative semigroup is a group. One of the main problems associated with the study of cancellative semigroups is to determine the necessary and sufficient conditions for embedding a cancellative semigroup in a group.
The origins of the study of cancellative semigroups can be traced to the first substantial paper on semigroups, .
Formal definitions
Let S be a semigroup. An element a in S is left cancellative (or, is left cancellable, or, has the left cancellation property) if implies for all b and c in S. If every element in S is left cancellative, then S is called a left cancellative semigroup.
Let S be a semigroup. An element a in S is right cancellative (or, is right cancellable, or, has the right cancellation property) if implies for all b and c in S. If every element in S is right cancellative, then S is called a right cancellative semigroup.
Let S be a semigroup. If every element in S is both left cancellative and right cancellative, then S is called a cancellative semigroup.
Alternative definitions
It is possible to restate the characteristic property of a cancellative element in terms of a property held by the corresponding left multiplication and right multiplication maps defined by and : an element a in S is left cancellative if and only if La is injective, an element a is right cancellative if and only if Ra is injective.
Examples
Every group is a cancellative semigroup.
The set of positive integers under addition is a cancellative semigroup.
The set of nonnegative integers under addition is a cancellative monoid.
The set of positive integers under multiplication is a cancellative monoid.
A left zero semigroup is right cancellative but not left cancellative, unless it is trivial.
A right zero semigroup is left cancellative but not right cancellative, unless it is trivial.
A null semigroup with more than one element is neither left cancellative nor right cancellative. In such a semigroup there is no element that is either left cancellative or right cancellative.
Let S be the semigroup of real square matrices of order n under matrix multiplication. Let a be any element in S. If a is nonsi |
https://en.wikipedia.org/wiki/Igor%20Dolgachev | Igor V. Dolgachev (born 7 April 1944) is a Russian–American mathematician specializing in algebraic geometry. He has been a professor at the University of Michigan since 1978. He introduced Dolgachev surfaces in 1981.
Dolgachev completed his Ph.D. at Moscow State University in 1970, with thesis On the purity of the degeneration locus of families of curves written under the supervision of Igor Shafarevich.
References
External links
Dolgachev's website at University of Michigan
1944 births
Living people
Moscow State University alumni
Soviet mathematicians
Russian mathematicians
20th-century American mathematicians
21st-century American mathematicians
University of Michigan faculty
Algebraic geometers |
https://en.wikipedia.org/wiki/Gauss%27s%20inequality | In probability theory, Gauss's inequality (or the Gauss inequality) gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode.
Let X be a unimodal random variable with mode m, and let τ 2 be the expected value of (X − m)2. (τ 2 can also be expressed as (μ − m)2 + σ 2, where μ and σ are the mean and standard deviation of X.) Then for any positive value of k,
The theorem was first proved by Carl Friedrich Gauss in 1823.
Extensions to higher-order moments
Winkler in 1866 extended Gauss' inequality to rth moments where r > 0 and the distribution is unimodal with a mode of zero. This is sometimes called Camp–Meidell's inequality.
Gauss' bound has been subsequently sharpened and extended to apply to departures from the mean rather than the mode due to the Vysochanskiï–Petunin inequality. The latter has been extended by Dharmadhikari and Joag-Dev
where s is a constant satisfying both s > r + 1 and s(s − r − 1) = rr and r > 0.
It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions.
See also
Vysochanskiï–Petunin inequality, a similar result for the distance from the mean rather than the mode
Chebyshev's inequality, concerns distance from the mean without requiring unimodality
Concentration inequality – a summary of tail-bounds on random variables.
References
Probabilistic inequalities |
https://en.wikipedia.org/wiki/Moffat%20distribution | The Moffat distribution, named after the physicist Anthony Moffat, is a continuous probability distribution based upon the Lorentzian distribution. Its particular importance in astrophysics is due to its ability to accurately reconstruct point spread functions, whose wings cannot be accurately portrayed by either a Gaussian or Lorentzian function.
Characterisation
Probability density function
The Moffat distribution can be described in two ways. Firstly as the distribution of a bivariate random variable (X,Y) centred at zero, and secondly as the distribution of the corresponding radii
In terms of the random vector (X,Y), the distribution has the probability density function (pdf)
where and are seeing dependent parameters. In this form, the distribution is a reparameterisation of a bivariate Student distribution with zero correlation.
In terms of the random variable R, the distribution has density
Relation to other distributions
Pearson distribution
Student's t-distribution for
Normal distribution for , since for the exponential function
References
A Theoretical Investigation of Focal Stellar Images in the Photographic Emulsion (1969) – A. F. J. Moffat
Continuous distributions
Equations of astronomy |
https://en.wikipedia.org/wiki/K.%20R.%20Parthasarathy%20%28graph%20theorist%29 | K. R. Parthasarathy is a professor emeritus of graph theory from the Department of Mathematics, Indian Institute of Technology Madras, Chennai. He received his Ph.D. (1966) in graph theory from the Indian Institute of Technology Kharagpur. Parthasarathy is known for his work (with his student G. Ravindra) proving the special case of the strong perfect graph conjecture for claw-free graphs. Parthasarathy guided and refereed Ph.D. students in graph theory, among them S. A. Choudum. Parthasarathy wrote a book on graph theory, Basic Graph Theory (Tata McGraw-Hill, 1994).
References
Tamil scholars
Graph theorists
IIT Kharagpur alumni
Academic staff of IIT Madras
Living people
Indian combinatorialists
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Light%27s%20associativity%20test | In mathematics, Light's associativity test is a procedure invented by F. W. Light for testing whether a binary operation defined in a finite set by a Cayley multiplication table is associative. The naive procedure for verification of the associativity of a binary operation specified by a Cayley table, which compares the two products that can be formed from each triple of elements, is cumbersome. Light's associativity test simplifies the task in some instances (although it does not improve the worst-case runtime of the naive algorithm, namely for sets of size ).
Description of the procedure
Let a binary operation ' · ' be defined in a finite set A by a Cayley table. Choosing some element a in A, two new binary operations are defined in A as follows:
x y = x · ( a · y )
x y = ( x · a ) · y
The Cayley tables of these operations are constructed and compared. If the tables coincide then x · ( a · y ) = ( x · a ) · y for all x and y. This is repeated for every element of the set A.
The example below illustrates a further simplification in the procedure for the construction and comparison of the Cayley tables of the operations ' ' and ' '.
It is not even necessary to construct the Cayley tables of ' ' and ' ' for all elements of A. It is enough to compare Cayley tables of ' ' and ' ' corresponding to the elements in a proper generating subset of A.
When the operation ' . ' is commutative, then x y = y x. As a result, only part of each Cayley table must be computed, because x x = x x always holds, and x y = x y implies y x = y x.
When there is an identity element e, it does not need to be included in the Cayley tables because x y = x y always holds if at least one of x and y are equal to e.
Example
Consider the binary operation ' · ' in the set A = { a, b, c, d, e } defined by the following Cayley table (Table 1):
The set { c, e } is a generating set for the set A under the binary operation defined by the above table, for, a = e · e, b = c · c, d = c · e. Thus it is enough to verify that the binary operations ' ' and ' ' corresponding to c coincide and also that the binary operations ' ' and ' ' corresponding to e coincide.
To verify that the binary operations ' ' and ' ' corresponding to c coincide, choose the row in Table 1 corresponding to the element c :
This row is copied as the header row of a new table (Table 3):
Under the header a copy the corresponding column in Table 1, under the header b copy the corresponding column in Table 1, etc., and construct Table 4.
The column headers of Table 4 are now deleted to get Table 5:
The Cayley table of the binary operation ' ' corresponding to the element c is given by Table 6.
Next choose the c column of Table 1:
Copy this column to the index column to get Table 8:
Against the index entry a in Table 8 copy the corresponding row in Table 1, against the index entry b copy the corresponding row in Table 1, etc., and construct Table 9.
The index ent |
https://en.wikipedia.org/wiki/Representability | Representability in mathematics can refer to
the existence of a representable functor in category theory
Birch's theorem about the representability of zero by odd degree forms
Brauer's theorem on the representability of zero by forms over certain fields in sufficiently many variables
Brown's representability theorem in homotopy theory
See also
Representation theory |
https://en.wikipedia.org/wiki/Michel%20Deza | Michel Marie Deza (27 April 1939 – 23 November 2016) was a Soviet and French mathematician, specializing in combinatorics, discrete geometry and graph theory. He was the retired director of research at the French National Centre for Scientific Research (CNRS), the vice president of the European Academy of Sciences, a research professor at the Japan Advanced Institute of Science and Technology, and one of the three founding editors-in-chief of the European Journal of Combinatorics.
Deza graduated from Moscow University in 1961, after which he worked at the Soviet Academy of Sciences until emigrating to France in 1972. In France, he worked at CNRS from 1973 until his 2005 retirement.
He has written eight books and about 280 academic papers with 75 different co-authors, including four papers with Paul Erdős, giving him an Erdős number of 1.
The papers from a conference on combinatorics, geometry and computer science, held in Luminy, France in May 2007, have been collected as a special issue of the European Journal of Combinatorics in honor of Deza's 70th birthday.
Selected papers
. This paper solved a conjecture of Paul Erdős and László Lovász (in , p. 406) that a sufficiently large family of k-subsets of any n-element universe, in which the intersection of every pair of k-subsets has exactly t elements, has a common t-element set shared by all the members of the family. Manoussakis writes that Deza is sorry not to have kept and framed the US$100 check from Erdős for the prize for solving the problem, and that this result inspired Deza to pursue a lifestyle of mathematics and travel similar to that of Erdős.
. This paper considers functions ƒ from subsets of some n-element universe to integers, with the property that, when A is a small set, the sum of the function values of the supersets of A is zero. The strength of the function is the maximum value t such that all sets A of t or fewer elements have this property. If a family of sets F has the property that it contains all the sets that have nonzero values for some function ƒ of strength at most t, F is t-dependent; the t-dependent families form the dependent sets of a matroid, which Deza and his co-authors investigate.
. This paper in polyhedral combinatorics describes some of the facets of a polytope that encodes cuts in a complete graph. As the maximum cut problem is NP-complete, but could be solved by linear programming given a complete description of this polytope's facets, such a complete description is unlikely.
. This paper with his son Antoine Deza, a fellow of the Fields Institute who holds a Canada Research Chair in Combinatorial Optimization at McMaster University, combines Michel Deza's interests in polyhedral combinatorics and metric spaces; it describes the metric polytope, whose points represent symmetric distance matrices satisfying the triangle inequality. For metric spaces with seven points, for instance, this polytope has 21 dimensions (the 21 pairwise distances between the p |
https://en.wikipedia.org/wiki/Sophie%20Germain%27s%20theorem | In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation of Fermat's Last Theorem for odd prime .
Formal statement
Specifically, Sophie Germain proved that at least one of the numbers , , must be divisible by if an auxiliary prime can be found such that two conditions are satisfied:
No two nonzero powers differ by one modulo ; and
is itself not a power modulo .
Conversely, the first case of Fermat's Last Theorem (the case in which does not divide ) must hold for every prime for which even one auxiliary prime can be found.
History
Germain identified such an auxiliary prime for every prime less than 100. The theorem and its application to primes less than 100 were attributed to Germain by Adrien-Marie Legendre in 1823.
Notes
References
Laubenbacher R, Pengelley D (2007) "Voici ce que j'ai trouvé": Sophie Germain's grand plan to prove Fermat's Last Theorem
Theorems in number theory
Fermat's Last Theorem |
https://en.wikipedia.org/wiki/J%C3%A1nos%20Koml%C3%B3s%20%28mathematician%29 | János Komlós (born 23 May 1942, in Budapest) is a Hungarian-American mathematician, working in probability theory and discrete mathematics. He has been a professor of mathematics at Rutgers University since 1988. He graduated from the Eötvös Loránd University, then became a fellow at the Mathematical Institute of the Hungarian Academy of Sciences. Between 1984–1988 he worked at the University of California, San Diego.
Notable results
Komlós' theorem: He proved that every L1-bounded sequence of real functions contains a subsequence such that the arithmetic means of all its subsequences converge pointwise almost everywhere. In probabilistic terminology, the theorem is as follows. Let ξ1,ξ2,... be a sequence of random variables such that E[ξ1],E[ξ2],... is bounded. Then there exist a subsequence ξ'1, ξ'2,... and a random variable β such that for each further subsequence η1,η2,... of ξ'0, ξ'1,... we have (η1+...+ηn)/n → β a.s.
With Miklós Ajtai and Endre Szemerédi he proved the ct2/log t upper bound for the Ramsey number R(3,t). The corresponding lower bound was established by Jeong Han Kim only in 1995, and this result earned him a Fulkerson Prize.
The same team of authors developed the optimal Ajtai–Komlós–Szemerédi sorting network.
Komlós and Szemerédi proved that if G is a random graph on n vertices with
edges, where c is a fixed real number, then the probability that G has a Hamiltonian circuit converges to
With Gábor Sárközy and Endre Szemerédi he proved the so-called blow-up lemma which claims that the regular pairs in Szemerédi's regularity lemma are similar to complete bipartite graphs when considering the embedding of graphs with bounded degrees.
Komlós worked on Heilbronn's problem; he, János Pintz and Szemerédi disproved Heilbronn's conjecture.
Komlós also wrote highly cited papers on sums of random variables, space-efficient representations of sparse sets, random matrices, the Szemerédi regularity lemma, and derandomization.
Degrees, awards
Komlós received his Ph.D. in 1967 from Eötvös Loránd University under the supervision of Alfréd Rényi. In 1975, he received the Alfréd Rényi Prize, a prize established for researchers of the Alfréd Rényi Institute of Mathematics. In 1998, he was elected as an external member to the Hungarian Academy of Sciences.
See also
Komlós–Major–Tusnády approximation
References
1942 births
Living people
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Members of the Hungarian Academy of Sciences
Theoretical computer scientists
Eötvös Loránd University alumni
Rutgers University faculty |
https://en.wikipedia.org/wiki/Dummy%20variable | The term dummy variable can refer to either of the following:
Bound variable, in mathematics and computer science, a placeholder variable
Dummy variable (statistics), an indicator variable |
https://en.wikipedia.org/wiki/Bruce%20Lee%20Rothschild | Bruce Lee Rothschild (born August 26, 1941) is an American mathematician and educator, specializing in combinatorial mathematics. He is a professor emeritus of mathematics at the University of California, Los Angeles.
Early life and education
Rothschild was born in 1941 in Los Angeles, California.
He earned a Ph.D. from Yale University in 1967 under the supervision of Øystein Ore.
Career
Rothschild, together with Ronald Graham, formulated one of the most monumental results in Ramsey theory, the Graham–Rothschild theorem. He has collaborated with American mathematicians Joel Spencer and Ronald Graham on key texts related to Ramsey theory. Rothschild wrote several papers with Paul Erdős, giving him an Erdős number of 1.
Awards and honors
In 1971, Rothschild shared the Pólya Prize (SIAM) with four other mathematicians for his work on Ramsey theory. In 2012, he became a fellow of the American Mathematical Society.
References
Combinatorialists
20th-century American mathematicians
21st-century American mathematicians
University of California, Los Angeles faculty
Yale University alumni
Living people
1941 births
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Wally%20Barnard | Walter Eric Barnard (9 October 1898 – 1982) was an English professional footballer who played in the Football League for Gillingham as a right back.
Career statistics
References
1898 births
English men's footballers
Footballers from Tottenham
Gillingham F.C. players
Tottenham Hotspur F.C. players
Brentford F.C. players
1982 deaths
Men's association football fullbacks
Clapton F.C. players
Isthmian League players |
https://en.wikipedia.org/wiki/Sebastian%20Idoff | Sebastian Idoff (born December 2, 1990) is a Swedish professional ice hockey goaltender, currently playing for Lørenskog of the Norwegian GET-ligaen.
Career statistics
Regular season and playoffs
External links
1990 births
Living people
Asplöven HC players
Borås HC players
Diables Rouges de Briançon players
Frölunda HC players
Lørenskog IK players
Manglerud Star Ishockey players
Örebro HK players
Södertälje SK players
Swedish ice hockey goaltenders
IF Troja/Ljungby players
Swedish expatriate ice hockey players in Norway
Swedish expatriate sportspeople in France
Ice hockey people from Malmö |
https://en.wikipedia.org/wiki/Rees%20factor%20semigroup | In mathematics, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after David Rees, is a certain semigroup constructed using a semigroup and an ideal of the semigroup.
Let S be a semigroup and I be an ideal of S. Using S and I one can construct a new semigroup by collapsing I into a single element while the elements of S outside of I retain their identity. The new semigroup obtained in this way is called the Rees factor semigroup of S modulo I and is denoted by S/I.
The concept of Rees factor semigroup was introduced by David Rees in 1940.
Formal definition
A subset of a semigroup is called an ideal of if both and are subsets of (where , and similarly for ). Let be an ideal of a semigroup . The relation in defined by
x ρ y ⇔ either x = y or both x and y are in I
is an equivalence relation in . The equivalence classes under are the singleton sets with not in and the set . Since is an ideal of , the relation is a congruence on . The quotient semigroup is, by definition, the Rees factor semigroup of modulo
. For notational convenience the semigroup is also denoted as . The Rees factor
semigroup has underlying set , where is a new element and the product (here denoted by
) is defined by
The congruence on as defined above is called the Rees congruence on modulo .
Example
Consider the semigroup S = { a, b, c, d, e } with the binary operation defined by the following Cayley table:
Let I = { a, d } which is a subset of S. Since
SI = { aa, ba, ca, da, ea, ad, bd, cd, dd, ed } = { a, d } ⊆ I
IS = { aa, da, ab, db, ac, dc, ad, dd, ae, de } = { a, d } ⊆ I
the set I is an ideal of S. The Rees factor semigroup of S modulo I is the set S/I = { b, c, e, I } with the binary operation defined by the following Cayley table:
Ideal extension
A semigroup S is called an ideal extension of a semigroup A by a semigroup B if A is an ideal of S and the Rees factor semigroup S/A is isomorphic to B.
Some of the cases that have been studied extensively include: ideal extensions of completely simple semigroups, of a group by a completely 0-simple semigroup, of a commutative semigroup with cancellation by a group with added zero. In general, the problem of describing all ideal extensions of a semigroup is still open.
References
Semigroup theory |
https://en.wikipedia.org/wiki/Weak%20inverse | In mathematics, the term weak inverse is used with several meanings.
Theory of semigroups
In the theory of semigroups, a weak inverse of an element x in a semigroup is an element y such that . If every element has a weak inverse, the semigroup is called an E-inversive or E-dense semigroup. An E-inversive semigroup may equivalently be defined by requiring that for every element , there exists such that and are idempotents.
An element x of S for which there is an element y of S such that is called regular. A regular semigroup is a semigroup in which every element is regular. This is a stronger notion than weak inverse. Every regular semigroup is E-inversive, but not vice versa.
If every element x in S has a unique inverse y in S in the sense that and then S is called an inverse semigroup.
Category theory
In category theory, a weak inverse of an object A in a monoidal category C with monoidal product ⊗ and unit object I is an object B such that both and are isomorphic to the unit object I of C. A monoidal category in which every morphism is invertible and every object has a weak inverse is called a 2-group.
See also
Generalized inverse
Von Neumann regular ring
References
Monoidal categories
Semigroup theory |
https://en.wikipedia.org/wiki/Heiko%20Harborth | Heiko Harborth (born 11 February 1938, in Celle, Germany) is Professor of Mathematics at Braunschweig University of Technology, 1975–present, and author of more than 188 mathematical publications. His work is mostly in the areas of number theory, combinatorics and discrete geometry, including graph theory.
Career
Harborth has been an instructor or professor at Braunschweig University of Technology since studying there and receiving his PhD in 1965 under Hans-Joachim Kanold. Harborth is a member of the New York Academy of Sciences, Braunschweigische Wissenschaftliche Gesellschaft, the Institute of Combinatorics and its Applications, and many other mathematical societies. Harborth currently sits on the editorial boards of Fibonacci Quarterly, Geombinatorics, Integers: Electronic Journal of Combinatorial Number Theory. He served as an editor of Mathematische Semesterberichte from 1988 to 2001. Harborth was a joint recipient (with Stephen Milne) of the 2007 Euler Medal.
Mathematical work
Harborth's research ranges across the subject areas of combinatorics, graph theory, discrete geometry, and number theory. In 1974, Harborth solved the unit coin graph problem, determining the maximum number of edges possible in a unit coin graph on n vertices. In 1986, Harborth presented the graph that would bear his name, the Harborth graph. It is the smallest known example of a 4-regular matchstick graph. It has 104 edges and 52 vertices.
In connection with the happy ending problem, Harborth showed that, for every finite set of ten or more points in general position in the plane, some five of them form a convex pentagon that does not contain any of the other points.
Harborth's conjecture posits that every planar graph admits a straight-line embedding in the plane where every edge has integer length. This open question () is a stronger version of Fáry's theorem. It is known to be true for cubic graphs.
In number theory, the Stolarsky–Harborth constant is named for Harborth, along with Kenneth Stolarsky.
Private life
Harborth married Karin Reisener in 1961, and they had two children. He was widowed in 1980. In 1985 he married Bärbel Peter and with her has three stepchildren.
Notes
1938 births
Living people
Combinatorialists
20th-century German mathematicians
21st-century German mathematicians
People from Celle
Academic staff of the Technical University of Braunschweig |
https://en.wikipedia.org/wiki/Chris%20Rogers%20%28mathematician%29 | Leonard Christopher Gordon Rogers (born 29 April 1954) is a mathematician working in probability theory and quantitative finance. He is Emeritus Professor of Statistical Science in the Statistical Laboratory, University of Cambridge.
Rogers' specialist fields include stochastic analysis and applications to quantitative finance. With David Williams he has written two influential textbooks on diffusion processes.
He was awarded the Mayhew Prize of Cambridge University in 1976, and the Rollo Davidson Prize in 1984. He was elected an Honorary Fellow of the Institute of Actuaries in 2003.
Rogers was an undergraduate at St John's College, Cambridge, where he graduated in 1975 and completed his PhD in 1980. He has held positions at several UK universities, including Warwick University (1980–1983), University College of Swansea (1983–1985), Cambridge University (1985–1991), Queen Mary and Westfield College (1991–1994), and the University of Bath (1994–2002). He was elected to the Cambridge Professorship of Statistical Science in 2002.
Selected publications
References
20th-century English mathematicians
21st-century English mathematicians
Living people
Alumni of St John's College, Cambridge
Cambridge mathematicians
Probability theorists
1954 births
Academics of the University of Warwick
Academics of Swansea University
Academics of Queen Mary University of London
Academics of the University of Bath
Professors of the University of Cambridge |
https://en.wikipedia.org/wiki/Kostant%20polynomial | In mathematics, the Kostant polynomials, named after Bertram Kostant, provide an explicit basis of the ring of polynomials over the ring of polynomials invariant under the finite reflection group of a root system.
Background
If the reflection group W corresponds to the Weyl group of a compact semisimple group K with maximal torus T, then the Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifold K/T, also isomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup. Armand Borel showed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by the invariant homogeneous polynomials of positive degree. This ring had already been considered by Claude Chevalley in establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with André Weil, Jean-Louis Koszul and Henri Cartan; the existence of such a basis was used by Chevalley to prove that the ring of invariants was itself a polynomial ring. A detailed account of Kostant polynomials was given by and independently as a tool to understand the Schubert calculus of the flag manifold. The Kostant polynomials are related to the Schubert polynomials defined combinatorially by for the classical flag manifold, when G = SL(n,C). Their structure is governed by difference operators associated to the corresponding root system.
defined an analogous basis when the polynomial ring is replaced by the ring of exponentials of the weight lattice. If K is simply connected, this ring can be identified with the representation ring R(T) and the W-invariant subring with R(K). Steinberg's basis was again motivated by a problem on the topology of homogeneous spaces; the basis arises in describing the T-equivariant K-theory of K/T.
Definition
Let Φ be a root system in a finite-dimensional real inner product space V with Weyl group W. Let Φ+ be a set of positive roots and Δ the corresponding set of simple roots. If α is a root, then sα denotes the corresponding reflection operator. Roots are regarded as linear polynomials on V using the inner product α(v) = (α,v). The choice of Δ gives rise to a Bruhat order on the Weyl group
determined by the ways of writing elements minimally as products of simple root reflection. The minimal length for an element s is denoted
. Pick an element v in V such that α(v) > 0 for every positive root.
If αi is a simple root with reflection operator si
then the corresponding divided difference operator is defined by
If and s has reduced expression
then
is independent of the reduced expression. Moreover
if and 0 otherwise.
If w0 is the longest element of W, the element of greatest length or equivalently the element sending Φ+ to −Φ+, then
More generally
for some constants as,t.
Set
and
Then Ps is a homogeneous polynomial of degree .
These polynomials are the Kostant polynomials.
Properties
Theorem. The Kostant p |
https://en.wikipedia.org/wiki/Surface%20of%20class%20VII | In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with
no rational curves with self-intersection −1) are called surfaces of class VII0. Every class VII surface is birational to a unique minimal class VII surface, and can be obtained from this minimal surface by blowing up points a finite number of times.
The name "class VII" comes from
, which divided minimal surfaces into 7 classes numbered I0 to VII0.
However Kodaira's class VII0 did not have the condition that the Kodaira dimension is −∞, but instead had the condition that the geometric genus is 0. As a result, his class VII0 also included some other surfaces, such as secondary Kodaira surfaces, that are no longer considered to be class VII as they do not have Kodaira dimension −∞. The minimal surfaces of class VII are the class numbered "7" on the list of surfaces in .
Invariants
The irregularity q is 1, and h1,0 = 0. All plurigenera are 0.
Hodge diamond:
Examples
Hopf surfaces are quotients of C2−(0,0) by a discrete group G acting freely, and have vanishing second Betti numbers. The simplest example is to take G to be the integers, acting as multiplication by powers of 2; the corresponding Hopf surface is diffeomorphic to S1×S3.
Inoue surfaces are certain class VII surfaces whose universal cover is C×H where H is the upper half plane (so they are quotients of this by a group of automorphisms). They have vanishing second Betti numbers.
Inoue–Hirzebruch surfaces, Enoki surfaces, and Kato surfaces give examples of type VII surfaces with b2 > 0.
Classification and global spherical shells
The minimal class VII surfaces with second Betti number b2=0 have been classified by , and are either Hopf surfaces or Inoue surfaces. Those with b2=1 were classified by under an additional assumption that the surface has a curve, that was later proved by .
A global spherical shell is a smooth 3-sphere in the surface with connected complement, with a neighbourhood biholomorphic to a neighbourhood of a sphere in C2. The global spherical shell conjecture claims that all class VII0 surfaces with positive second Betti number have a global spherical shell. The manifolds with a global spherical shell are all Kato surfaces which are reasonably well understood, so a proof of this conjecture would lead to a classification of the type VII surfaces.
A class VII surface with positive second Betti number b2 has at most b2 rational curves, and has exactly this number if it has a global spherical shell. Conversely
showed that if a minimal class VII surface with positive second Betti number b2 has exactly b2 rational curves then it has a global spherical shell.
For type VII surfaces with vanishing second Betti number, the primary Hopf surfaces have a global spherical shell, but secondary Hopf surfaces and Inoue surfaces do not because their fundamental groups are not infinite cycl |
https://en.wikipedia.org/wiki/Crossbar%20theorem | In geometry, the crossbar theorem states that if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC.
This result is one of the deeper results in axiomatic plane geometry. It is often used in proofs to justify the statement that a line through a vertex of a triangle lying inside the triangle meets the side of the triangle opposite that vertex. This property was often used by Euclid in his proofs without explicit justification.
Some modern treatments (not Euclid's) of the proof of the theorem that the base angles of an isosceles triangle are congruent start like this: Let ABC be a triangle with side AB congruent to side AC. Draw the angle bisector of angle A and let D be the point at which it meets side BC. And so on. The justification for the existence of point D is the often unstated crossbar theorem. For this particular result, other proofs exist which do not require the use of the crossbar theorem.
See also
Foundations of geometry
Jordan curve theorem
Notes
References
Euclidean plane geometry
Theorems in plane geometry
Foundations of geometry |
https://en.wikipedia.org/wiki/Andr%C3%A1s%20P%C3%A1l | András Pál (born 19 August 1985) is a Hungarian soccer player.
Career statistics
References
HLSZ
MLSZ
1985 births
Footballers from Budapest
Living people
Hungarian men's footballers
Hungary men's youth international footballers
Men's association football forwards
Újpest FC players
Vasas SC players
BFC Siófok players
MTK Budapest FC players
Soroksár SC players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players |
https://en.wikipedia.org/wiki/Tam%C3%A1s%20Kecsk%C3%A9s | Tamás Kecskés (born 15 January 1986) is a Hungarian former football player.
Career statistics
Club
External links
HLSZ
1986 births
People from Szentes
Sportspeople from Csongrád-Csanád County
Living people
Hungarian men's footballers
Men's association football midfielders
MTK Budapest FC players
BFC Siófok players
Pécsi MFC players
Paksi FC players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Spherical%20code | In geometry and coding theory, a spherical code with parameters (n,N,t) is a set of N points on the unit hypersphere in n dimensions for which the dot product of unit vectors from the origin to any two points is less than or equal to t. The kissing number problem may be stated as the problem of finding the maximal N for a given n for which a spherical code with parameters (n,N,1/2) exists. The Tammes problem may be stated as the problem of finding a spherical code with minimal t for given n and N.
External links
A library of putatively optimal spherical codes
Coding theory |
https://en.wikipedia.org/wiki/Calera%20de%20Tango | Calera de Tango is a Chilean commune in the Maipo Province, Santiago Metropolitan Region.
Demographics
According to the 2002 census of the National Statistics Institute, Calera de Tango spans an area of and has 18,235 inhabitants (9,243 men and 8,992 women). Of these, 9,932 (54.5%) lived in urban areas and 8,303 (45.5%) in rural areas. The population grew by 54% (6,392 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Calera de Tango is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2012-2016 alcalde is Erasmo Valenzuela Santibáñez (IND). The communal council has the following members:
Juan Irarrazaval Rossel (UDI)
Marco Jofre Muñoz (PS)
Carolina Saavedra Rojas (IND)
Sandra Meza Zumelzu (PS)
Marcelo Riquelme Yagi (PDC)
Lilian Farias Nallar (RN)
Within the electoral divisions of Chile, Calera de Tango is represented in the Chamber of Deputies by Ramón Farías (PPD) and José Antonio Kast (UDI) as part of the 30th electoral district, (together with San Bernardo, Buin and Paine). The commune is represented in the Senate by Guido Girardi Lavín (PPD) and Jovino Novoa Vásquez (UDI) as part of the 7th senatorial constituency (Santiago-West).
References
External links
Municipality of Calera de Tango
Communes of Chile
Populated places in Maipo Province |
https://en.wikipedia.org/wiki/Hirosi%20Ooguri | is a theoretical physicist working on quantum field theory, quantum gravity, superstring theory, and their interfaces with mathematics. He is Fred Kavli Professor of Theoretical Physics and Mathematics and the Founding Director of the Walter Burke Institute for Theoretical Physics at California Institute of Technology. He is also the director of the Kavli Institute for the Physics and Mathematics at the University of Tokyo and is the chair of the board of trustees of the Aspen Center for Physics in Colorado.
Ooguri aims at discovering mathematical structures in these theories and exploiting them to invent new theoretical tools to solve fundamental questions in physics. In particular, he developed the topological string theory to compute Feynman diagrams in superstring theory and used it to study mysterious quantum mechanical properties of black holes. He also made fundamental contributions to conformal field theories in two dimensions, D-branes in Calabi-Yau manifolds, the AdS/CFT correspondence, and properties of supersymmetric gauge theories and their relations to superstring theory.
Career
Finishing his graduate study in two years, Ooguri became a tenured faculty member at the University of Tokyo in 1986. He was a member the Institute for Advanced Study in Princeton and was appointed an assistant professor at the University of Chicago before receiving his Ph.D. in 1989. He was an associate professor at Kyoto University in 1990–1994 and returned to the United States as a professor of physics at the University of California at Berkeley in 1994. He moved California Institute of Technology (Caltech) in 2000, where he is the inaugural holder of the Fred Kavli Chair.
At Caltech, Ooguri served as the deputy chair of the Division of Physics, Mathematics and Astronomy, equivalent of a vice dean of physical sciences. He led the establishment of the Walter Burke Institute for Theoretical Physics and was appointed its founding director in 2014.
Ooguri also helped establish the Kavli Institute for the Physics and Mathematics of the Universe in Japan in 2007. After serving as its principal investigator for 11 years, he became the director in 2018.
Ooguri has been a member of the Aspen Center for Physics since 2003. After serving as the scientific secretary a trustee, and the president, he was elected the chair of the board of trustees of the center in 2021.
Ooguri is a Fellow of the American Academy of Arts and Sciences and an investigator of the Simons Foundation. He has received the Eisenbud Prize from the American Mathematical Society, the Humboldt Research Award and the Hamburg Prize in Germany, and the Nishina Memorial Prize in Japan.
Ooguri's popular science books have sold over a quarter million copies in Japan, and one of them was awarded the Kodansha Prize for Science Books. He also supervised a science movie, which was selected for the Best Educational Production Award from the International Planetarium Society.
Awards
Leonard Eisenbud Pr |
https://en.wikipedia.org/wiki/Robust%20measures%20of%20scale | In statistics, robust measures of scale are methods that quantify the statistical dispersion in a sample of numerical data while resisting outliers. The most common such robust statistics are the interquartile range (IQR) and the median absolute deviation (MAD). These are contrasted with conventional or non-robust measures of scale, such as sample standard deviation, which are greatly influenced by outliers.
These robust statistics are particularly used as estimators of a scale parameter, and have the advantages of both robustness and superior efficiency on contaminated data, at the cost of inferior efficiency on clean data from distributions such as the normal distribution. To illustrate robustness, the standard deviation can be made arbitrarily large by increasing exactly one observation (it has a breakdown point of 0, as it can be contaminated by a single point), a defect that is not shared by robust statistics.
IQR and MAD
One of the most common robust measures of scale is the interquartile range (IQR), the difference between the 75th percentile and the 25th percentile of a sample; this is the 25% trimmed range, an example of an L-estimator. Other trimmed ranges, such as the interdecile range (10% trimmed range) can also be used.
For a Gaussian distribution, IQR is related to as .
Another familiar robust measure of scale is the median absolute deviation (MAD), the median of the absolute values of the differences between the data values and the overall median of the data set; for a Gaussian distribution, MAD is related to as (the derivation can be found here).
Estimation
Robust measures of scale can be used as estimators of properties of the population, either for parameter estimation or as estimators of their own expected value.
For example, robust estimators of scale are used to estimate the population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation. For example, dividing the IQR by 2 erf−1(1/2) (approximately 1.349), makes it an unbiased, consistent estimator for the population standard deviation if the data follow a normal distribution.
In other situations, it makes more sense to think of a robust measure of scale as an estimator of its own expected value, interpreted as an alternative to the population standard deviation as a measure of scale. For example, the MAD of a sample from a standard Cauchy distribution is an estimator of the population MAD, which in this case is 1, whereas the population variance does not exist.
Efficiency
These robust estimators typically have inferior statistical efficiency compared to conventional estimators for data drawn from a distribution without outliers (such as a normal distribution), but have superior efficiency for data drawn from a mixture distribution or from a heavy-tailed distribution, for which non-robust measures such as the standard deviation should not be used.
For example, for data drawn |
https://en.wikipedia.org/wiki/Rafael%20Nadal%20career%20statistics | This is a list of the main career statistics of professional tennis player Rafael Nadal. All statistics are according to the ATP Tour website. To date, Nadal has won 92 ATP singles titles, including 22 Grand Slam men's singles titles and 36 ATP Tour Masters 1000 titles. He is one of two men to achieve the Career Golden Slam in men's singles, with titles at all four majors and the Olympic singles gold. He is the first man in history to win Grand Slam singles titles on three different surfaces in a calendar year (Surface Slam) and is the youngest (24) in the Open Era to achieve the Career Grand Slam. Following his triumph at the 2022 Australian Open, he became the fourth man in history to complete the double Career Grand Slam in singles, after Roy Emerson, Rod Laver, and Novak Djokovic. He is the first man to win multiple majors and rank world No. 1 in three different decades. Representing Spain, Nadal has won two Olympic gold medals including a singles gold at the 2008 Beijing Olympics and a doubles gold at the 2016 Rio Olympics. In the process, he became the first male player in history to complete the Career Grand Slam and win Olympic gold medals in both singles and doubles. Nadal is the only Spanish player, male or female, to win all four majors twice, to rank world No. 1 for more than 200 weeks, and to win more than 20 majors. He has led Spain to five Davis Cup titles in 2004, 2008, 2009, 2011, and 2019. At the international level, he has won the 2017 and 2019 editions of the Laver Cup with Team Europe.
Historic achievements
Nadal has been the most successful player in history on clay courts. He has a 63–8 record in clay court tournament finals and has lost only three times in best-of-five-set matches on clay. He has won 14 French Open titles (unbeaten in finals), 12 Barcelona Open titles (unbeaten in finals), 11 Monte-Carlo Masters titles, and 10 Rome Masters titles, and has won at least one of the three clay-court Masters 1000 tournaments every year between 2005 and 2014. His 9th French Open crown in 2014 made him the first man in the Open Era to win a single tournament 9 times, breaking a 32-year record held by Guillermo Vilas, who won the Buenos Aires title 8 times. He subsequently won his 9th title at three more tournaments; 2016 Monte Carlo, 2016 Barcelona, and the 2019 Italian Open. In 2018, he became the sole record-holder for most titles at the ATP 500 (Barcelona), Masters 1000 (Monte Carlo), and Grand Slam (French Open) levels.
He also holds the longest single-surface win streak in the Open Era, having won 81 consecutive matches on clay between April 2005 and May 2007. Nadal has never been taken to five sets in 14 French Open finals, and has never lost consecutive matches on clay since the start of his professional career. Many tennis critics and top players regard him as the greatest clay-court player of all time. Nadal's 14 French Open titles are a record for one player (male or female) at a single major, surpassing the old recor |
https://en.wikipedia.org/wiki/Tibor%20Szele | Tibor Szele (Debrecen, 21 June 1918 – Szeged, 5 April 1955) Hungarian mathematician, working in combinatorics and abstract algebra. After graduating at the Debrecen University, he became a researcher at the Szeged University in 1946, then he went back at the Debrecen University in 1948 where he became full professor in 1952. He worked especially in the theory of Abelian groups and ring theory. He generalized Hajós's theorem. He founded the Hungarian school of algebra. Tibor Szele received the Kossuth Prize in 1952.
References
A panorama of Hungarian Mathematics in the Twentieth Century, p. 601.
External links
Grave of Tibor Szele
Algebraists
Combinatorialists
Probability theorists
1918 births
1955 deaths
University of Debrecen alumni
Academic staff of the University of Debrecen
20th-century Hungarian mathematicians |
https://en.wikipedia.org/wiki/List%20of%20Women%27s%20National%20Basketball%20Association%20career%20scoring%20leaders | The following is a list of the players who have scored the most points during their WNBA careers.
Scoring leaders
All statistics are up to date as of August 17, 2022.
Progressive list of scoring leaders
This is a progressive list of scoring leaders showing how the record increased through the years.
Statistics accurate as of April 9, 2022.
Notes
References
External links
WNBA Career Leaders and Records for Points at Basketball Reference
https://stats.wnba.com/alltime-leaders/
Lists of Women's National Basketball Association players
Women's National Basketball Association statistics
Women's National Basketball Association lists |
https://en.wikipedia.org/wiki/List%20of%20Women%27s%20National%20Basketball%20Association%20career%20rebounding%20leaders | The following is a list of the players who have collected the most rebounds during their WNBA careers.
All statistics are up to date as of the close of the 2022 WNBA season.
Progressive list of rebounding leaders
This is a progressive list of rebounding leaders showing how the record increased through the years.
Statistics accurate as of August 18, 2022.
Notes
References
External links
Updated list
Lists of Women's National Basketball Association players
Women's National Basketball Association statistics |
https://en.wikipedia.org/wiki/Extremal%20orders%20of%20an%20arithmetic%20function | In mathematics, specifically in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and
we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and
we say that M is a maximal order for f. Here, and denote the limit inferior and limit superior, respectively.
The subject was first studied systematically by Ramanujan starting in 1915.
Examples
For the sum-of-divisors function σ(n) we have the trivial result because always σ(n) ≥ n and for primes σ(p) = p + 1. We also have proved by Gronwall in 1913. Therefore n is a minimal order and is a maximal order for σ(n).
For the Euler totient φ(n) we have the trivial result because always φ(n) ≤ n and for primes φ(p) = p − 1. We also have proven by Landau in 1903.
For the number of divisors function d(n) we have the trivial lower bound 2 ≤ d(n), in which equality occurs when n is prime, so 2 is a minimal order. For ln d(n) we have a maximal order , proved by Wigert in 1907.
For the number of distinct prime factors ω(n) we have a trivial lower bound 1 ≤ ω(n), in which equality occurs when n is a prime power. A maximal order for ω(n) is .
For the number of prime factors counted with multiplicity Ω(n) we have a trivial lower bound 1 ≤ Ω(n), in which equality occurs when n is prime. A maximal order for Ω(n) is
It is conjectured that the Mertens function, or summatory function of the Möbius function, satisfies though to date this limit superior has only been shown to be larger than a small constant. This statement is compared with the disproof of Mertens conjecture given by Odlyzko and te Riele in their several decades old breakthrough paper Disproof of the Mertens Conjecture. In contrast, we note that while extensive computational evidence suggests that the above conjecture is true, i.e., along some increasing sequence of tending to infinity the average order of grows unbounded, that the Riemann hypothesis is equivalent to the limit being true for all (sufficiently small) .
See also
Average order of an arithmetic function
Normal order of an arithmetic function
Notes
Further reading
A survey of extremal orders, with an extensive bibliography.
Arithmetic functions |
https://en.wikipedia.org/wiki/Marcell%20Moln%C3%A1r | Marcell Molnár (born 26 August 1990) is a Hungarian football player who currently plays for the Austrian club TSU Jeging (Coach Erwin Dankl).
Career statistics
External links
1990 births
Living people
People from Sátoraljaújhely
Hungarian men's footballers
Men's association football forwards
MTK Budapest FC players
BFC Siófok players
BKV Előre SC footballers
Nyíregyháza Spartacus FC players
Mezőkövesdi SE footballers
FC Veszprém footballers
Nemzeti Bajnokság I players
Footballers from Borsod-Abaúj-Zemplén County |
https://en.wikipedia.org/wiki/Geometric%20design | Geometrical design (GD) is a branch of computational geometry. It deals with the construction and representation of free-form curves, surfaces, or volumes and is closely related to geometric modeling. Core problems are curve and surface modelling and representation. GD studies especially the construction and manipulation of curves and surfaces given by a set of points using polynomial, rational, piecewise polynomial, or piecewise rational methods. The most important instruments here are parametric curves and parametric surfaces, such as Bézier curves, spline curves and surfaces. An important non-parametric approach is the level-set method.
Application areas include shipbuilding, aircraft, and automotive industries, as well as architectural design. The modern ubiquity and power of computers means that even perfume bottles and shampoo dispensers are designed using techniques unheard of by shipbuilders of 1960s.
Geometric models can be built for objects of any dimension in any geometric space. Both 2D and 3D geometric models are extensively used in computer graphics. 2D models are important in computer typography and technical drawing. 3D models are central to computer-aided design and manufacturing, and many applied technical fields such as geology and medical image processing.
Geometric models are usually distinguished from procedural and object-oriented models, which define the shape implicitly by an algorithm. They are also contrasted with digital images and volumetric models; and with mathematical models such as the zero set of an arbitrary polynomial. However, the distinction is often blurred: for instance, geometric shapes can be represented by objects; a digital image can be interpreted as a collection of colored squares; and geometric shapes such as circles are defined by implicit mathematical equations. Also, the modeling of fractal objects often requires a combination of geometric and procedural techniques.
Geometric problems originating in architecture can lead to interesting research and results in geometry processing, computer-aided geometric design, and discrete differential geometry.<ref>H. Pottmann, S. Brell-Cokcan and J. Wallner: Discrete surfaces for architectural design
</ref>
In architecture, geometric design''' is associated with the pioneering explorations of Chuck Hoberman into transformational geometry as a design idiom, and applications of this design idiom within the domain of architectural geometry.
See also
Architectural geometry
Computational topology
CAD/CAM/CAE
Digital geometry
Geometric design of roads
List of interactive geometry software
Parametric curves
Parametric surfaces
Solid modeling
Space partitioning
Wikiversity:Topic:Computational geometry
References
External links
Evolute Research and Consulting
Computer Aided Geometric Design
Geometric algorithms
Computational science
Computer-aided design
Applied geometry |
https://en.wikipedia.org/wiki/Independence%20of%20premise | In proof theory and constructive mathematics, the principle of independence of premise states that if φ and ∃x θ are sentences in a formal theory and is provable, then is provable. Here x cannot be a free variable of φ, while θ can be a predicate depending on it.
The main application of the principle is in the study of intuitionistic logic, where the principle is not generally valid. The principle is valid in classical logic.
Discussion
As is common, the domain of discourse is assumed to be inhabited. That is, part of the theory is at least some term. For the discussion we distinguish one such term as a. In the theory of the natural numbers, this role may be played by the number 7. Below, φ and ψ denote propositions not depending on x, while θ is a predicate that can depend on in.
The following is easily established:
Firstly, if φ is established to be true, then if one assumes to be provable, there is an x satisfying φ → θ.
Secondly, if φ is established to be false, then, by the explosion, any proposition of the form φ → ψ holds. Then, any x formally satisfies φ → θ (and indeed any predicate of this form.)
In the first scenario, some x bound in the premise is reused in the conclusion, and it is generally not the apriori a that validates it. In the second scenario, the value a in particular validates the conclusion of the principle. So in both of these two cases, some x validates the conclusion.
Thirdly, now in contrast to the two points above, consider the case in which it is not known how to prove or reject φ. A core case is when φ is the formula , in which case the antecedent becomes trivial: "If θ is satisfiable then θ is satisfiable." For illustration purposes, let it be granted that θ is a decidable predicate in arithmetic, meaning for any given number b the proposition θ(b) can easily be inspected for its truth value. More specifically, θ shall express that x is the index of a formal proof of some mathematical conjecture whose provability is not known. Certainly here, one way to establish would be to provide a particular index x for which it can be shown (then aided by the assumption that some value z satisfies θ) that it genuinely satisfies θ. However, explicating a such x is not possible (not yet and possibly never), as such x exactly encodes the proof of a conjecture not yet proven or rejected.
In intuitionistic logic
The arithmetical example above provides what is called a weak counterexample. The existence claim cannot be provable by intuitionistic means: Being able to inspect an x validating φ → θ would resolve the conjecture.
For example, consider the following classical argument: Either the Goldbach conjecture has a proof or it does not. If it does not have a proof, then to assume is has a proof is absurd and anything follows - in particular, it follows that it has a proof. Hence, there is some natural number index x such that if one assumes the Goldbach conjecture has a proof, that x is an index of such a proof.
|
https://en.wikipedia.org/wiki/Molecular%20models%20of%20DNA | Molecular models of DNA structures are representations of the molecular geometry and topology of deoxyribonucleic acid (DNA) molecules using one of several means, with the aim of simplifying and presenting the essential, physical and chemical, properties of DNA molecular structures either in vivo or in vitro. These representations include closely packed spheres (CPK models) made of plastic, metal wires for skeletal models, graphic computations and animations by computers, artistic rendering. Computer molecular models also allow animations and molecular dynamics simulations that are very important for understanding how DNA functions in vivo.
The more advanced, computer-based molecular models of DNA involve molecular dynamics simulations and quantum mechanics computations of vibro-rotations, delocalized molecular orbitals (MOs), electric dipole moments, hydrogen-bonding, and so on. DNA molecular dynamics modeling involves simulating deoxyribonucleic acid (DNA) molecular geometry and topology changes with time as a result of both intra- and inter- molecular interactions of DNA. Whereas molecular models of DNA molecules such as closely packed spheres (CPK models) made of plastic or metal wires for skeletal models are useful representations of static DNA structures, their usefulness is very limited for representing complex DNA dynamics. Computer molecular modeling allows both animations and molecular dynamics simulations that are very important to understand how DNA functions in vivo.
History
From the very early stages of structural studies of DNA by X-ray diffraction and biochemical means, molecular models such as the Watson-Crick nucleic acid double helix model were successfully employed to solve the 'puzzle' of DNA structure, and also find how the latter relates to its key functions in living cells. The first high quality X-ray diffraction patterns
of A-DNA were reported by Rosalind Franklin and Raymond Gosling in 1953. Rosalind Franklin made the critical observation that DNA exists in two distinct forms, A and B, and produced the sharpest pictures of both through X-ray diffraction technique. The first calculations of the Fourier transform of an atomic helix were reported one year earlier by Cochran, Crick and Vand, and were followed in 1953 by the computation of the Fourier transform of a coiled-coil by Crick.
Structural information is generated from X-ray diffraction studies of oriented DNA fibers with the help of molecular models of DNA that are combined with crystallographic and mathematical analysis of the X-ray patterns.
The first reports of a double helix molecular model of B-DNA structure were made by James Watson and Francis Crick in 1953. That same year, Maurice F. Wilkins,
A. Stokes and H.R. Wilson, reported the first X-ray patterns
of in vivo B-DNA in partially oriented salmon sperm heads.
The development of the first correct double helix molecular model of DNA by Crick and Watson may not have been possible without the biochem |
https://en.wikipedia.org/wiki/Fermat%20quotient | In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as
or
.
This article is about the former; for the latter see p-derivation. The quotient is named after Pierre de Fermat.
If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime.
Properties
From the definition, it is obvious that
In 1850, Gotthold Eisenstein proved that if a and b are both coprime to p, then:
Eisenstein likened the first two of these congruences to properties of logarithms. These properties imply
In 1895, Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:
From this, it follows that:
Lerch's formula
M. Lerch proved in 1905 that
Here is the Wilson quotient.
Special values
Eisenstein discovered that the Fermat quotient with base 2 could be expressed in terms of the sum of the reciprocals modulo p of the numbers lying in the first half of the range {1, ..., p − 1}:
Later writers showed that the number of terms required in such a representation could be reduced from 1/2 to 1/4, 1/5, or even 1/6:
Eisenstein's series also has an increasingly complex connection to the Fermat quotients with other bases, the first few examples being:
Generalized Wieferich primes
If qp(a) ≡ 0 (mod p) then ap−1 ≡ 1 (mod p2). Primes for which this is true for a = 2 are called Wieferich primes. In general they are called Wieferich primes base a. Known solutions of qp(a) ≡ 0 (mod p) for small values of a are:
{| class="wikitable"
|-----
! a
! p (checked up to 5 × 1013)
! OEIS sequence
|-----
| 1 || 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes)
|
|-----
| 2 || 1093, 3511
|
|-----
| 3 || 11, 1006003
|
|-----
| 4 || 1093, 3511
|
|-----
| 5 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
|
|-----
| 6 || 66161, 534851, 3152573
|
|-----
| 7 || 5, 491531
|
|-----
| 8 || 3, 1093, 3511
|
|-----
| 9 || 2, 11, 1006003
|
|-----
| 10 || 3, 487, 56598313
|
|-----
| 11 || 71
|
|-----
| 12 || 2693, 123653
|
|-----
| 13 || 2, 863, 1747591
|
|-----
| 14 || 29, 353, 7596952219
|
|-----
| 15 || 29131, 119327070011
|
|-----
| 16 || 1093, 3511
|
|-----
| 17 || 2, 3, 46021, 48947, 478225523351
|
|-----
| 18 || 5, 7, 37, 331, 33923, 1284043
|
|-----
| 19 || 3, 7, 13, 43, 137, 63061489
|
|-----
| 20 || 281, 46457, 9377747, 122959073
|
|-----
| 21 || 2
|
|-----
| 22 || 13, 673, 1595813, 492366587, 9809862296159
|
|-----
| 23 || 13, 2481757, 13703077, 15546404183, 2549536629329
|
|-----
| 24 || 5, 25633
|
|-----
| 25 || 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
|
|-----
| 26 || 3, 5, 71, 486999673, 6695256707
|
|-----
| 27 || 11, 1006003
|
|-----
| 28 || 3, 19, 23
|
|-----
| 29 || 2
|
|-----
| 30 || 7, 160541, 94727075783
|
|}
For more information, see and.
The s |
https://en.wikipedia.org/wiki/Pete%20Sampras%20career%20statistics | The career of American former tennis player Pete Sampras started when he turned professional in 1988 and lasted until his official retirement in August 2003. During his career Sampras played in 265 official tournaments and won 64 singles titles, including 14 titles at Grand Slam events. He competed in 16 ties for the United States Davis Cup team between 1991 and 2002 and was a member of the Davis Cup winning team in 1992 and 1995. Sampras reached the No. 1 ranking on April 12, 1993, and in total held that position for 286 weeks, third behind Novak Djokovic at 373 weeks and Roger Federer at 310 weeks. He finished the year as the No. 1 ranked player six consecutive times. His career win–loss record is 762–222 (77.4%).
Sampras is 7–0 in Wimbledon finals and is the only male player to win 3 or more consecutive Wimbledon titles twice in his career (1993–1995, 1997–2000) . He is the first player to win 14 Grand Slam Men's singles titles, since surpassed by Federer, Rafael Nadal Parera and Djokovic. His win–loss record in Grand Slam finals is unbeaten at 78% (14 wins in 18 finals) for players who have appeared in at least 10 Grand Slam finals. He is the only American male player to win more than 10 Grand Slams.
Grand Slam finals
Singles: 18 (14 titles, 4 runner-ups)
Other significant finals
Year-end championships finals
Singles: 6 (5 titles, 1 runner-up)
Grand Slam Cup
Singles: 3 (2 titles, 1 runner-up)
ATP Super 9 / ATP Masters Series finals
Singles: 19 (11 titles, 8 runner-ups)
Doubles: 1 (1 title)
Career finals
Singles: 88 (64 titles, 24 runner-ups)
Wins (64)
Runner-ups (24)
Doubles: 4 (2 titles, 2 runner-ups)
Team competition: 4 (2 titles, 2 runner-ups)
Wins (2)
Runners-up (2)
Singles performance timeline
1This event was held in Stockholm through 1994, Essen in 1995, and Stuttgart from 1996 through 2001.
Record against other players
Sampras' record against players who held a top 10 ranking, with those who reached No. 1 in bold
Andre Agassi (20–14)
Todd Martin (18–4)
Jim Courier (16–4)
Patrick Rafter (12–4)
Petr Korda (12–5)
Goran Ivanišević (12–6)
Boris Becker (12–7)
Michael Chang (12–8)
Yevgeny Kafelnikov (11–2)
Cédric Pioline (9–0)
Jonas Björkman (9–1)
Greg Rusedski (9–1)
Thomas Enqvist (9–2)
Thomas Muster (9–2)
Stefan Edberg (8–6)
Karol Kučera (7–1)
Mark Philippoussis (7–3)
Magnus Larsson (7–4)
Wayne Ferreira (7–6)
Tim Henman (6–1)
Andriy Medvedev (6–2)
Magnus Gustafsson (5–0)
Aaron Krickstein (5–1)
Tommy Haas (5–3)
Ivan Lendl (5–3)
Guy Forget (5–4)
Brad Gilbert (5–4)
Magnus Norman (4–1)
Marc Rosset (4–1)
Àlex Corretja (4–2)
Lleyton Hewitt (4–5)
Michael Stich (4–5)
Richard Krajicek (4–6)
Sébastien Grosjean (3–0)
Nicolás Lapentti (3–0)
John McEnroe (3–0)
Nicolas Kiefer (3–1)
Carlos Moyá (3–1)
Marat Safin (3–4)
Guillermo Cañas (2–0)
Jimmy Connors (2–0)
Andrés Gómez (2–0)
Anders Järryd (2–0)
Alberto Mancini (2–0)
Marcelo Ríos (2–0)
Thomas Johansson (2–1)
Gustavo Kuerten (2–1)
Emilio Sánchez (2–1)
Mats Wilander (2–1)
Henri Le |
https://en.wikipedia.org/wiki/Justine%20Henin%20career%20statistics | This is a list of the main career statistics of professional Belgian tennis player Justine Henin.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup and Olympic Games are included in win–loss records.
Singles
Doubles
Grand Slam tournament finals
Singles: 12 (7 titles, 5 runner-ups)
Other significant finals
Olympic finals
Singles: 1 (1 gold medal)
WTA Championships finals
Singles: 2 (2 titles)
WTA Premier Mandatory & 5 finals
Singles: 14 (10 titles, 4 runner-ups)
Doubles: 1 (1 title)
WTA career finals
Singles: 61 (43 titles, 18 runner-ups)
Doubles: 3 (2 titles, 1 runner–up)
Team competition finals: 3 finals (1 title, 2 runner-ups)
ITF Circuit finals
Singles: 7 (7 titles)
Doubles: 3 (2 titles, 1 runner–up)
WTA Tour career earnings
Henin earned more than 20 million dollars during her career.
Career Grand Slam statistics
Grand Slam tournament seedings
The tournaments won by Henin are in boldface, and advanced into finals by Henin are in italics.
Singles
Best Grand Slam results details
Grand Slam winners are in boldface, and runner–ups are in italics.
Singles
Head-to-head record against other players
Henin's win–loss record against certain players who have been ranked World No. 10 or higher is as follows. Active players are in boldface:
Top 10 wins
Longest winning streaks
32-match win streak (2007–08)
Comparison of the 2003–present year-end number ones
Comparisons between the WTA singles year-end number one ranked players from 2003, the year Henin was first ranked year-end number one.
Ranking points
Adjusted to reflect the raising of the points scale (2007 and 2009), reduction of tournaments counted (2009) and removal of quality points (2006).
Not an exact figure but a good approximation of how they would have scored under the current system.
Original total in brackets.
Serena Williams 2013 – 13,260
Justine Henin 2007 – 12310 (6155)
Justine Henin 2003 – 11514 (6628)
Justine Henin 2006 – 11423 (3998)
Victoria Azarenka 2012 10595
Lindsay Davenport 2005 – 9334 (4910)
Jelena Jankovic 2008 – 9120 (4710)
Serena Williams 2009 – 9075
Lindsay Davenport 2004 – 8317 (4760)
Caroline Wozniacki 2010 – 8035
Caroline Wozniacki 2011 – 7485
Tournaments won
1. Serena Williams 2013 – 11 (2 GS + YEC)
2. Justine Henin 2007 – 10 (2 GS + YEC)
3. Justine Henin 2003 – 8 (2 GS)
4. Lindsay Davenport 2004 – 7
5. Justine Henin 2006 – 6 (1 GS + YEC) Victoria Azarenka 2012 – 6 (1 GS) Lindsay Davenport 2005 – 6 Caroline Wozniacki 2010 – 6 Caroline Wozniacki 2011 – 6
10. Jelena Jankovic 2008 – 4
11. Serena Williams 2009 – 3 (2 GS + YEC)
Tournaments played
1. Jelena Jankovic 2008 – 22 Caroline Wozniacki 2010 – 22 Caroline Wozniacki 2011 – 22
4. Justine Henin 2003 – 18
5. Lindsay Davenport 2004 – 17
6. Lindsay Davenport 2005 – 16 Serena Williams 2009 – 16
8. Justine Henin 2007 – 15
9. Serena Williams 2013 – 14
10. Justine Henin 2006 – 13
Winning percentage
Serena Williams 2013 – 95. |
https://en.wikipedia.org/wiki/Kim%20Clijsters%20career%20statistics | This is a list of the main career statistics of tennis player Kim Clijsters.
Performance timelines
Only results in WTA Tour (incl. Grand Slams) main-draw, Olympic Games and Fed Cup are included in win–loss records.
Singles
Doubles
Grand Slam tournament finals
Singles: 8 finals (4 titles, 4 runner-ups)
Doubles: 3 finals (2 titles, 1 runner-up)
Mixed doubles: 1 final (1 runner-up)
Other significant finals
Year-end championships
Singles: 3 finals (3 titles)
Doubles: 1 final (1 runner-up)
Tier I / Premier-Mandatory & Premier-5
Singles: 10 finals (7 titles, 3 runner-ups)
Doubles: 3 finals (1 title, 2 runner-ups)
WTA career finals
Singles: 60 (41 titles, 19 runner-ups)
Doubles: 20 (11 titles, 9 runner-ups)
ITF Circuit finals
Singles (3–1)
Doubles (3–0)
Team competition finals: 1 final (1 title)
Junior Grand Slam finals
Singles: 1 (1 runner-up)
Doubles: 3 (2 titles, 1 runner-up)
WTA Tour career earnings
*As of October 10, 2021
Record against other top players
Head-to-head vs. top-ten ranked players
Clijsters' record against players who have been ranked in the top 10.
Wins over reigning World No. 1's
Top-10 wins
Clijsters has a record against players who were, at the time the match was played, ranked in the top 10.
Longest winning streak
21-match win streak (2005)
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Andre%20Agassi%20career%20statistics | This is a list of the main career statistics of former tennis player Andre Agassi.
Finals
Grand Slam finals
Singles: 15 (8 titles, 7 runner-ups)
By winning the 1999 French Open, Agassi completed a men's singles Career Grand Slam. He is the 5th of 8 male players in history (after Budge, Perry, Laver, Emerson and before Rafael Nadal, Roger Federer, and Novak Djokovic) to achieve this.
Year-end championships finals
Singles: 4 (1 title, 3 runner-ups)
Grand Slam Cup
Singles: 1 (1 runner-up)
ATP Masters Series finals (since 1990)
Singles: 22 (17 titles, 5 runner-ups)
Agassi won 17 Masters Series singles titles, which is currently the fourth highest of all time, behind Novak Djokovic, Rafael Nadal and Roger Federer. It is also the overall sixth highest total of 'tier one' titles (including those which preceded Masters 1000 events, such as the Super Nine) behind Novak Djokovic (36), Rafael Nadal (36) and Roger Federer (28).
Olympic finals
Singles: 1 (1 gold medal)
Career finals
Singles: 90 (60 titles, 30 runner-ups)
Doubles: 4 (1 title, 3 runner-ups)
Team competition: 3 (2 titles, 1 runner-up)
Career ITF and exhibition finals
Singles
Wins (11)
Losses (3)
Singles performance timeline
Note: Tournaments were designated as the 'Masters Series' only after the ATP took over the running of the men's tour in 1990.
1This event was held in Stockholm through 1994, Essen in 1995, and Stuttgart from 1996 through 2001.
ATP Tour career earnings
* As of September 18, 2006.
Career Grand Slam tournament seedings
The tournaments won by Agassi are in boldface.
Record against top players
Agassi's win–loss record against top opponents is as follows:
Michael Chang 15–7
Pete Sampras 14–20
Todd Martin 13–5
Wayne Ferreira 11–0
Jan-Michael Gambill 11–2
Boris Becker 10–4
Patrick Rafter 10–5
Greg Rusedski 9–2
Yevgeny Kafelnikov 8–4
Jeff Tarango 7–0
Petr Korda 7–1
Sergi Bruguera 7–2
Gustavo Kuerten 7–4
Michael Stich 6–0
Todd Woodbridge 6–0
Nicolas Kiefer 6–0
Sargis Sargsian 6–0
Xavier Malisse 6–0
Thomas Johansson 6–1
Jan Siemerink 6–1
Mark Philippoussis 6–2
MaliVai Washington 6–2
Stefan Edberg 6–3
David Wheaton 6–3
Tommy Haas 6–4
Taylor Dent 5–0
Younes El Aynaoui 5–0
Jonas Björkman 5–0
Andy Roddick 5–1
Jason Stoltenberg 5–1
Nicolas Escudé 5–1
Jiří Novák 5–1
Guillermo Coria 5–2
Mats Wilander 5–2
Aaron Krickstein 5–3
Àlex Corretja 5–3
Thomas Muster 5–4
Thomas Enqvist 5–5
Jim Courier 5–7
Goran Prpić 4–0
Davide Sanguinetti 4–0
Robby Ginepri 4–0
Mikael Pernfors 4–1
Alberto Berasategui 4–1
James Blake 4–1
Emilio Sánchez 4–1
Rainer Schüttler 4–1
Albert Costa 4–1
Jim Grabb 4–1
Gastón Gaudio 4–1
Vincent Spadea 4–2
Marc Rosset 4–2
Arnaud Clément 4–2
Mark Woodforde 4–2
Francisco Clavet 4–2
Richard Krajicek 4–3
Sébastien Grosjean 4–3
Goran Ivanišević 4–3
Lleyton Hewitt 4–4
Brad Gilbert 4–4
Paul Annacone 3–0
Cédric Pioline 3–0
Paul-Henri Mathieu 3–0
Kenneth Carlsen 3–0
Todd Witsken 3–0
Mariano Zabal |
https://en.wikipedia.org/wiki/Nikolay%20Konstantinov | Nikolay Nikolayevich Konstantinov (; 2 January 1932 – 3 July 2021) was a leading Soviet and Russian mathematical educator and organizer of numerous mathematics competitions for high school students. He is best known as the creator of the system of math schools and math classes and as the creator and chief organizer of the Tournament of the Towns. For his work he was awarded the Paul Erdős award in 1992.
Biography
Konstantinov was born and grew up in Moscow, Soviet Union. He graduated from the Physics Department of the Moscow State University in 1954, and later received a Ph.D. in physics.
In the 1950s, he started a math circle in Moscow University and since the 1960s in a number of Moscow high schools. He continued working with schools developing special classes with mathematics concentration and individual approach to learning. His students went on to win mathematics competitions on all levels and dozens of them became well-known mathematicians.
In 1978, Konstantinov started the Lomonosov tournament, a multi-subject science competition. This tournament is continued every year since then. In 1980, he started the international Tournament of the Towns which is now organized in over 150 towns in 25 countries.
Until his late 80s, Konstantinov continued working in Moscow High School 179 and was an editor of Kvant magazine, a popular Russian science publication.
In 1990, Konstantinov was one of the founders of Independent University of Moscow, one of the leading institutions of higher learning in mathematics in Russia.
Konstantinov died from the effects of COVID-19 on 3 July 2021, during the COVID-19 pandemic in Russia. He was 89 years old.
Books
Scholarly publications
References
20th-century Russian mathematicians
1932 births
2021 deaths
Moscow State University alumni
Academic staff of the Independent University of Moscow
Mathematicians from Moscow
Soviet mathematicians
Mathematics educators
Soviet educators
Deaths from the COVID-19 pandemic in Russia |
https://en.wikipedia.org/wiki/Slater%20integrals | In mathematics and mathematical physics, Slater integrals are certain integrals of products of three spherical harmonics. They occur naturally when applying an orthonormal basis of functions on the unit sphere that transform in a particular way under rotations in three dimensions. Such integrals are particularly useful when computing properties of atoms which have natural spherical symmetry. These integrals are defined below along with some of their mathematical properties.
Formulation
In connection with the quantum theory of atomic structure, John C. Slater defined the integral of three spherical harmonics as a coefficient . These coefficients are essentially the product of two Wigner 3jm symbols.
These integrals are useful and necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator and Exchange operator are needed. For an explicit formula, one can use Gaunt's formula for associated Legendre polynomials.
Note that the product of two spherical harmonics can be written in terms of these coefficients. By expanding such a product over a spherical harmonic basis with the same order
one may then multiply by and integrate, using the conjugate property and being careful with phases and normalisations:
Hence
These coefficient obey a number of identities. They include
References
Atomic physics
Quantum chemistry
Rotational symmetry |
https://en.wikipedia.org/wiki/Beauville%20surface | In mathematics, a Beauville surface is one of the surfaces of general type introduced by . They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.
Construction
Let C1 and C2 be smooth curves with genera g1 and g2.
Let G be a finite group acting on C1 and C2 such that
G has order (g1 − 1)(g2 − 1)
No nontrivial element of G has a fixed point on both C1 and C2
C1/G and C2/G are both rational.
Then the quotient (C1 × C2)/G is a Beauville surface.
One example is to take C1 and C2 both copies of the genus 6 quintic
X5 + Y5 + Z5 =0, and G to be an elementary abelian group of order 25, with suitable actions on the two curves.
Invariants
Hodge diamond:
References
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Burniat%20surface | In mathematics, a Burniat surface is one of the surfaces of general type introduced by .
Invariants
The geometric genus and irregularity are both equal to 0. The Chern number is either 2, 3, 4, 5, or 6.
References
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Campedelli%20surface | In mathematics, a Campedelli surface is one of the surfaces of general type introduced by Campedelli.
Surfaces with the same Hodge numbers are called numerical Campedelli surfaces.
Construction
Invariants
Hodge diamond:
References
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Castelnuovo%20surface | In mathematics, a Castelnuovo surface is a surface of general type such that the canonical bundle is very ample and
such that c12 = 3pg − 7. Guido Castelnuovo proved that if the canonical bundle is very ample for a surface of general type then c12 ≥ 3pg − 7.
Construction
Invariants
References
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Catanese%20surface | In mathematics, a Catanese surface is one of the surfaces of general type introduced by .
Construction
The construction starts with a quintic V with 20 double points. Let W be the surface obtained by blowing up the 20 double points. Suppose that W has a double cover X branched over the 20 exceptional −2-curves. Let Y be obtained from X by blowing down the 20 −1-curves in X. If there is a group of order 5 acting freely on all these surfaces, then the quotient Z of Y by this group of order 5 is a Catanese surface. Catanese found a 4-dimensional family of curves constructed like this.
Invariants
The Catanese surface is a numerical Campedelli surface and hence has Hodge diamond
and canonical degree . The fundamental group of the Catanese surface is , as can be seen from its quotient construction.
References
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Sch%C3%BCtzenberger%20group | In abstract algebra, in semigroup theory, a Schützenberger group is a certain group associated with a Green of a semigroup. The Schützenberger groups associated with different are different. However, the groups associated with two different contained in the same of a semigroup are isomorphic. Moreover, if the itself were a group, the Schützenberger group of the would be isomorphic to the . In fact, there are two Schützenberger groups associated with a given and each is antiisomorphic to the other.
The Schützenberger group was discovered by Marcel-Paul Schützenberger in 1957 and the terminology was coined by A. H. Clifford.
The Schützenberger group
Let S be a semigroup and let S1 be the semigroup obtained by adjoining an identity element 1 to S (if S already has an identity element, then S1 = S). Green's in S is defined as follows: If a and b are in S then
a H b ⇔ there are u, v, x, y in S1 such that ua = ax = b and vb = by = a.
For a in S, the set of all b''' s in S such that a H b is the Green of S containing a, denoted by Ha.
Let H be an of the semigroup S. Let T(H) be the set of all elements t in S1 such that Ht is a subset of H itself. Each t in T(H) defines a transformation, denoted by γt, of H by mapping h in H to ht in H. The set of all these transformations of H, denoted by Γ(H), is a group under composition of mappings (taking functions as right operators). The group Γ(H) is the Schützenberger group associated with the H.
Examples
If H is a maximal subgroup of a monoid M, then H is an , and it is naturally isomorphic to its own Schützenberger group.
In general, one has that the cardinality of H and its Schützenberger group coincide for any H''.
Applications
It is known that a monoid with finitely many left and right ideals is finitely presented (or just finitely generated) if and only if all of its Schützenberger groups are finitely presented (respectively, finitely generated). Similarly such a monoid is residually finite if and only if all of its Schützenberger groups are residually finite.
References
Semigroup theory |
https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Rado%20theorem | In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named after Paul Erdős and Richard Rado. It is sometimes also attributed to Đuro Kurepa who proved it under the additional assumption of the generalised continuum hypothesis, and hence the result is sometimes also referred to as the Erdős–Rado–Kurepa theorem.
Statement of the theorem
If r ≥ 0 is finite and κ is an infinite cardinal, then
where exp0(κ) = κ and inductively expr+1(κ)=2expr(κ). This is sharp in the sense that expr(κ)+ cannot be replaced by expr(κ) on the left hand side.
The above partition symbol describes the following statement. If f is a coloring of the r+1-element subsets of a set of cardinality expr(κ)+, in κ many colors, then there is a homogeneous set of cardinality κ+ (a set, all whose r+1-element subsets get the same f-value).
Notes
References
</ref>
Set theory
Theorems in combinatorics
Rado theorem |
https://en.wikipedia.org/wiki/Ron%20Larson | Roland "Ron" Edwin Larson (born October 31, 1941) is a professor of mathematics at Penn State Erie, The Behrend College, Pennsylvania. He is best known for being the author of a series of widely used mathematics textbooks ranging from middle school through the second year of college.
Personal life
Ron Larson was born in Fort Lewis near Tacoma, Washington, the second of four children of Mederith John Larson and Harriet Eleanor Larson. Mederith Larson was an officer in the 321st Engineer Battalion of the United States Army. Mederith Larson served in active duty during World War II, where he was awarded a Bronze Star Medal and a Purple Heart, and the Korean War, where he was awarded an Oak Leaf Cluster and a Silver Star. During the years that Ron was growing up, his father was stationed in several military bases, including Chitose, Hokkaido, Japan and Schofield Barracks, Hawaii. While in Chitose, Ron attended a small DoDDS school, where he was one of only three students in the sixth grade. When Mederith Larson retired from the Army in 1957, he moved with his family to Vancouver, Washington, where he lived until he died (at the age of 89) in 2005. Harriet Larson died (at the age of 95) in the fall of 2009.
Larson spent his first two years of high school at Leilehua High School in Wahiawa, Hawaii. In 1957, when his family moved to Vancouver, Washington, Larson enrolled in Battle Ground High School, where he graduated in 1959. On October 29, 1960, at the age of 18, he married Deanna Sue Gilbert, also of Vancouver, Washington. Deanna Gilbert was the second child of Herbert and Dorothy Gilbert. Ron and Deanna Larson have two children, Timothy Roland Larson and Jill Deanna Larson Im, five living grandchildren, and two great-grandchildren. Their first grandchild, Timothy Roland Larson II, died at birth on summer solstice, June 21, 1983.
Larson is the third generation of Norwegian and Swedish immigrants who left Scandinavia to homestead in Minnesota in the late 1800s. The surnames and immigration dates of his great-grandparents are Bangen (1866, Norway), Berg (1867, Norway), Larson (1868, Norway), and Watterburg (1879, Sweden).
Larson has contributed several thousand dollars to Republican politicians, including Rand Paul, Marco Rubio, Mitt Romney, and Scott Brown.
Education
From 1959 until 1962, Ron and Deanna Larson started and operated a small business, called Larson's Custom Quilting. In 1962, they sold the business and Ron began attending Clark College in Vancouver, Washington. In 1964, he obtained his associate degree from Clark. Upon graduation from Clark College, Larson was awarded a scholarship from the Alcoa Foundation, which he used to attend Lewis & Clark College in Portland, Oregon. He graduated, with honors, from Lewis & Clark in 1966. During the four years from 1962 through 1966, Ron worked full-time, first at a restaurant and then at a grocery store, in Vancouver and Deanna worked full-time as the secretary to the president of R |
https://en.wikipedia.org/wiki/Harrison%20Randolph | Harrison Randolph (December 8, 1871 – 1954) was the 13th President and professor of mathematics at the College of Charleston from 1897 to 1945.
Randolph was born in New Orleans, Louisiana to John Feild Randolph and Virginia Dashiell Randolph, née Bayard. He was a lineal descendant of Edward Randolph of the Bremo Plantation, who was his great-great-great grandfather and Benjamin Harrison V, a paternal ancestor who signed the Declaration of Independence. He attended the University of Virginia, graduating in 1892 with a Master of Arts degree, and continued graduate study there from 1892 through 1895 while also serving as an instructor in mathematics. During this time he also served as the organist in the University of Virginia Chapel and directed the Virginia Glee Club, leading the latter organization on tours through the Southeast. Randolph had been elected of the President of the University of Arkansas in 1892, but declined the position. In 1895, he was elected chair of Mathematics at the University of Arkansas, remaining there until 1897.
College of Charleston
In 1897, Randolph was elected president and Chair of Mathematics at the College of Charleston. When he arrived, the college principally enrolled students from the city of Charleston, South Carolina. Under his presidency, the student body population changed as he led the building of residence halls and created scholarships to attract students from throughout South Carolina. He also oversaw the admission of women to the college in 1917. Under his leadership, the college grew from 68 students in 1905 to more than 400 in 1935.
Randolph was a member of Phi Beta Kappa and Alpha Tau Omega.
In August 2008, Charleston Magazine named Randolph the 72nd most influential individual in Charleston's history, citing his work to modernize the College of Charleston.
References
University of Virginia alumni
University of Virginia faculty
University of Arkansas faculty
Presidents of the College of Charleston
1954 deaths
1871 births |
https://en.wikipedia.org/wiki/James%20R.%20Norris | James Ritchie Norris (born 29 August 1960) is a mathematician working in probability theory and stochastic analysis. He is the Professor of Stochastic Analysis in the Statistical Laboratory, University of Cambridge.
He has made contributions to areas of mathematics connected to probability theory and mathematical analysis, including Malliavin calculus, heat kernel estimates, and mathematical models for coagulation and fragmentation. He was awarded the Rollo Davidson Prize in 1997.
Norris was an undergraduate at Hertford College, Oxford where he graduated in 1981. He completed his D.Phil in 1985 at Wolfson College, Oxford under the supervision of David Edwards. He was a research assistant from 1984 to 1985 at the University College of Swansea before moving in 1985 to a lectureship at Cambridge University and a Fellowship of Churchill College, Cambridge. He was appointed Professor of Stochastic Analysis in 2005. He is the director of the Statistical Laboratory, a trustee of the Rollo Davidson Trust and co-Director of the Cambridge Centre for Analysis.
Selected publication
References
20th-century English mathematicians
Alumni of Hertford College, Oxford
Alumni of Wolfson College, Oxford
21st-century English mathematicians
Fellows of Churchill College, Cambridge
1960 births
Living people
Cambridge mathematicians
Probability theorists |
https://en.wikipedia.org/wiki/Newey%E2%80%93West%20estimator |
A Newey–West estimator is used in statistics and econometrics to provide an estimate of the covariance matrix of the parameters of a regression-type model where the standard assumptions of regression analysis do not apply. It was devised by Whitney K. Newey and Kenneth D. West in 1987, although there are a number of later variants. The estimator is used to try to overcome autocorrelation (also called serial correlation), and heteroskedasticity in the error terms in the models, often for regressions applied to time series data. The abbreviation "HAC," sometimes used for the estimator, stands for "heteroskedasticity and autocorrelation consistent." There are a number of HAC estimators described in, and HAC estimator does not refer uniquely to Newey-West. One version of Newey-West Bartlett requires the user to specify the bandwidth and usage of the Bartlett Kernel from Kernel density estimation
Regression models estimated with time series data often exhibit autocorrelation; that is, the error terms are correlated over time. The heteroscedastic consistent estimator of the error covariance is constructed from a term , where is the design matrix for the regression problem and is the covariance matrix of the residuals. The least squares estimator is a consistent estimator of . This implies that the least squares residuals are "point-wise" consistent estimators of their population counterparts . The general approach, then, will be to use and to devise an estimator of . This means that as the time between error terms increases, the correlation between the error terms decreases. The estimator thus can be used to improve the ordinary least squares (OLS) regression when the residuals are heteroskedastic and/or autocorrelated.
where T is the sample size, is the residual and is the row of the design matrix, and is the Bartlett Kernel and can be thought of as a weight that decreases with increasing separation between samples. Disturbances that are farther apart from each other are given lower weight, while those with equal subscripts are given a weight of 1. This ensures that second term converges (in some appropriate sense) to a finite matrix. This weighting scheme also ensures that the resulting covariance matrix is positive semi-definite. L=0 reduces the Newey-West estimator to Huber–White standard error. L specifies the "maximum lag considered for the control of autocorrelation. A common choice for L" is .
Software implementations
In Julia, the CovarianceMatrices.jl package supports several types of heteroskedasticity and autocorrelation consistent covariance matrix estimation including Newey–West, White, and Arellano.
In R, the packages sandwich and plm include a function for the Newey–West estimator.
In Stata, the command newey produces Newey–West standard errors for coefficients estimated by OLS regression.
In MATLAB, the command hac in the Econometrics toolbox produces the Newey–West estimator (among others).
In Python, the |
https://en.wikipedia.org/wiki/M.%20G.%20Nadkarni | Mahendra G. Nadkarni is a professor emeritus at University of Mumbai. Nadkarni obtained his Ph.D. in mathematics from Brown University, the US in 1964 for his work on Ergodic theory. His research interests include Ergodic Theory, Harmonic Analysis, and Probability Theory.
Nadkarni has taught at Washington University in St. Louis, University of Minnesota, Indian Statistical Institute (ISI), Calcutta University (1968–1981), University of Mumbai (1981–1998), Indian Institute of Technology Indore (2010–2012), and Centre for Excellence in Basic Sciences (2012-present). He teaches Measure Theory, Probability Theory and Stochastic Calculus to undergraduates at CEBS.
He was Head of the Department of Mathematics, at the University of Mumbai. He is a fellow of the Indian National Science Academy as well as the Indian Academy of Sciences. Nadkarni is an author of books on Ergodic theory.
Selected publications
with Sadanand G. Telang:
References
University of Mumbai
UM-DAE CEBS
Scientists from Mumbai
Konkani people
Brown University alumni
Living people
Year of birth missing (living people)
People from Uttara Kannada
Academic staff of the University of Calcutta
Washington University in St. Louis mathematicians
University of Minnesota faculty
Washington University in St. Louis faculty
Academic staff of the Indian Statistical Institute
Academic staff of the University of Mumbai |
https://en.wikipedia.org/wiki/Coverage%20probability | In statistics, the coverage probability, or coverage for short, is the probability that a confidence interval or confidence region will include the true value (parameter) of interest. It can be defined as the proportion of instances where the interval surrounds the true value as assessed by long-run frequency.
Concept
The fixed degree of certainty pre-specified by the analyst, referred to as the confidence level or confidence coefficient of the constructed interval, is effectively the nominal coverage probability of the procedure for constructing confidence intervals. Hence, referring to a "nominal confidence level" or "nominal confidence coefficient" (e.g., as a synonym for nominal coverage probability) generally has to be considered tautological and misleading, as the notion of confidence level itself inherently implies nominality already. The nominal coverage probability is often set at 0.95. By contrast, the (true) coverage probability is the actual probability that the interval contains the parameter.
If all assumptions used in deriving a confidence interval are met, the nominal coverage probability will equal the coverage probability (termed "true" or "actual" coverage probability for emphasis). If any assumptions are not met, the actual coverage probability could either be less than or greater than the nominal coverage probability. When the actual coverage probability is greater than the nominal coverage probability, the interval is termed a conservative (confidence) interval; if it is less than the nominal coverage probability, the interval is termed anti-conservative, or permissive. For example, suppose the interest is in the mean number of months that people with a particular type of cancer remain in remission following successful treatment with chemotherapy. The confidence interval aims to contain the unknown mean remission duration with a given probability. In this example, the coverage probability would be the real probability that the interval actually contains the true mean remission duration.
A discrepancy between the coverage probability and the nominal coverage probability frequently occurs when approximating a discrete distribution with a continuous one. The construction of binomial confidence intervals is a classic example where coverage probabilities rarely equal nominal levels. For the binomial case, several techniques for constructing intervals have been created. The Wilson score interval is one well-known construction based on the normal distribution. Other constructions include the Wald, exact, Agresti-Coull, and likelihood intervals. While the Wilson score interval may not be the most conservative estimate, it produces average coverage probabilities that are equal to nominal levels while still producing a comparatively narrow confidence interval.
The "probability" in coverage probability is interpreted with respect to a set of hypothetical repetitions of the entire data collection and analysis procedure. In these hy |
https://en.wikipedia.org/wiki/Irregularity%20of%20a%20surface | In mathematics, the irregularity of a complex surface X is the Hodge number , usually denoted by q. The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic.
The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P3, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes.
For a complex analytic manifold X of general dimension, the Hodge number is called the irregularity of , and is denoted by q.
Complex surfaces
For non-singular complex projective (or Kähler) surfaces, the following numbers are all equal:
The irregularity;
The dimension of the Albanese variety;
The dimension of the Picard variety;
The Hodge number ;
The Hodge number ;
The difference of the geometric genus and the arithmetic genus.
For surfaces in positive characteristic, or for non-Kähler complex surfaces, the numbers above need not all be equal.
Henri Poincaré proved that for complex projective surfaces the dimension of the Picard variety is equal to the Hodge number h0,1, and the same is true for all compact Kähler surfaces. The irregularity of smooth compact Kähler surfaces is invariant under bimeromorphic transformations.
For general compact complex surfaces the two Hodge numbers h1,0 and h0,1 need not be equal, but h0,1 is either h1,0 or h1,0+1, and is equal to h1,0 for compact Kähler surfaces.
Positive characteristic
Over fields of positive characteristic, the relation between q (defined as the dimension of the Picard or Albanese variety), and the Hodge numbers h0,1 and h1,0 is more complicated, and any two of them can be different.
There is a canonical map from a surface F to its Albanese variety A which induces a homomorphism from the cotangent space of the Albanese variety (of dimension q) to H1,0(F). Jun-Ichi Igusa found that this is injective, so that , but shortly after found a surface in characteristic 2 with and Picard variety of dimension 1, so that q can be strictly less than both Hodge numbers. In positive characteristic neither Hodge number is always bounded by the other. Serre showed that it is possible for h1,0 to vanish while
h0,1 is positive, while Mumford showed that for Enriques surfaces in characteristic 2 it is possible for h0,1 to vanish while h1,0 is positive.
Alexander Grothendieck gave a complete description of the relation of q to in all characteristics. The dimension of the tangent space to the Picard scheme (at any point) is equal to . In characteristic 0 a result of Pierre Cartier showed that all groups schemes of finite type are non-singular, |
https://en.wikipedia.org/wiki/Lindsay%20Davenport%20career%20statistics | This is a list of the main career statistics of American former professional tennis player, Lindsay Davenport.
Major finals
Grand Slam tournament finals
Singles: 7 finals (3 titles, 4 runners-up)
Doubles: 13 finals (3 titles, 10 runners-up)
Summer Olympics
Singles: 1 Gold Medal Match (1-0)
WTA Tour Championships
Singles: 4 finals (1 title, 3 runner-ups)
(i) = Indoor
Doubles: 3 finals (3 titles)
WTA Tier I
Singles: 21 finals (11 titles, 10 runners-up)
Doubles: 14 finals (9 titles, 5 runners-up)
Career WTA Tour finals
Singles: 94 (55 titles, 38 runner-ups)
Doubles
Wins (38)
Runners-up (23)
Singles performance timeline
Grand Slam doubles performance timeline
WTA tour career earnings
Record against top 10 players
Davenport's record against players who have been ranked in the top 10:
Top 10 wins
Longest winning streak
22-match win streak (2004)
References
Tennis career statistics |
https://en.wikipedia.org/wiki/Clementine%20Maersk | Clementine Maersk is a container ship of the Maersk Line. The ship was built in 2002 in the shipyard of Odense Steel and has a capacity of 6,600 TEUs according to company statistics and calculations.
Design
Clementine Maersk was built in 2002 in the ship-yard of Odense Steel in Denmark and sails under the Danish flag. Clementine Maersk has a deadweight of 109,696 metric tons and a gross tonnage of 91,921 gross tons. The ship has a net tonnage of 53,625 net tons and a cargo capacity of 6,600 container (TEU). The length of the vessel is 347.00 meters, while the moulded beam is 43.00 meters. When the ship is fully loaded with cargo she reaches the maximum draft of 14,50 meters.
References
Container Ships
Container ships
Ships built in Odense
Merchant ships of Denmark
2002 ships
Ships of the Maersk Line |
https://en.wikipedia.org/wiki/Central%20Bureau%20of%20Statistics | Central Bureau of Statistics may refer to:
Central Bureau of Statistics (Aruba)
Israel Central Bureau of Statistics
Central Bureau of Statistics (Namibia)
Central Bureau of Statistics (Nepal)
Central Bureau of Statistics (North Korea)
Palestinian Central Bureau of Statistics
Central Bureau of Statistics (Sudan)
Central Bureau of Statistics (Syria)
Statistics Netherlands, formerly known as the Central Bureau of Statistics
See also
List of national and international statistical services
Central Statistical Office |
https://en.wikipedia.org/wiki/Dirichlet%20eigenvalue | In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function u ≠ 0 and eigenvalue λ
Here Δ is the Laplacian, which is given in xy-coordinates by
The boundary value problem () is the Dirichlet problem for the Helmholtz equation, and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues for the corresponding Neumann problem. The Laplace operator Δ appearing in () is often known as the Dirichlet Laplacian when it is considered as accepting only functions u satisfying the Dirichlet boundary condition. More generally, in spectral geometry one considers () on a manifold with boundary Ω. Then Δ is taken to be the Laplace–Beltrami operator, also with Dirichlet boundary conditions.
It can be shown, using the spectral theorem for compact self-adjoint operators that the eigenspaces are finite-dimensional and that the Dirichlet eigenvalues λ are real, positive, and have no limit point. Thus they can be arranged in increasing order:
where each eigenvalue is counted according to its geometric multiplicity. The eigenspaces are orthogonal in the space of square-integrable functions, and consist of smooth functions. In fact, the Dirichlet Laplacian has a continuous extension to an operator from the Sobolev space into . This operator is invertible, and its inverse is compact and self-adjoint so that the usual spectral theorem can be applied to obtain the eigenspaces of Δ and the reciprocals 1/λ of its eigenvalues.
One of the primary tools in the study of the Dirichlet eigenvalues is the max-min principle: the first eigenvalue λ1 minimizes the Dirichlet energy. To wit,
the infimum is taken over all u of compact support that do not vanish identically in Ω. By a density argument, this infimum agrees with that taken over nonzero . Moreover, using results from the calculus of variations analogous to the Lax–Milgram theorem, one can show that a minimizer exists in . More generally, one has
where the supremum is taken over all (k−1)-tuples and the infimum over all u orthogonal to the .
Applications
The Dirichlet Laplacian may arise from various problems of mathematical physics;
it may refer to modes of at idealized drum, small waves at the surface of an idealized pool,
as well as to a mode of an idealized optical fiber in the paraxial approximation.
The last application is most practical in connection to the double-clad fibers;
in such fibers, it is important, that most of modes of the fill the domain uniformly,
or the most of rays cross the core. The poorest shape seems to be |
https://en.wikipedia.org/wiki/Martina%20Navratilova%20career%20statistics | This is a list of the main career statistics of former Czechoslovak-born American tennis player Martina Navratilova.
Significant finals
Grand Slam finals
Singles: 32 (18–14)
By winning the 1983 US Open title, Navratilova completed the Career Grand Slam. She became only the seventh female player in history to achieve this.
Doubles: 37 (31–6)
By winning the 1980 Australian Open title, Navratilova completed the women's doubles Career Grand Slam. She became the ninth female player in history to achieve this.
Mixed doubles: 16 (10–6)
By winning the 2003 Australian Open title, Navratilova completed the mixed doubles Career Grand Slam. She became only the third female player in history to achieve this. Having also completed Career Grand Slams in singles and doubles, Navratilova completed the "Career Boxed Set", only the second player in the Open Era after Margaret Court to do so.
Year-End Championships finals
Singles: 14 (8–6)
Doubles: 11 (11–0)
Singles performance timelines
Grand Slam tournaments
Note: Australian Open was held twice in 1977, in January and December, and was not held in 1986.
* World Rank before the 1975 inception of WTA rankings.
See also
Performance timelines for all female tennis players who reached at least one Grand Slam final
Other tournaments
Career singles statistics
Navratilova did not play an official WTA tour singles match from 1995 through 2001.
Doubles performance timeline
Grand Slam doubles
Grand Slam mixed doubles
Note: The Australian Open was held twice in 1977, in January and December, and was not held in 1986.
WTA singles finals
Titles: (167)
Runner-ups: (72)
Doubles finals
Doubles: 223 (177 wins, 46 losses)
Grand Slam Mixed doubles: 16
Non-Grand Slam mixed doubles: 5
Grand Slam seedings
The tournaments won by Martina are in boldface, and advances into finals by her are in italics.
Women's singles
Women's doubles
WTA Tour career earnings
Record against top 10 players
Navratilova's record against players who have been ranked in the top 10:
See also
Evert–Navratilova rivalry
Graf–Navratilova rivalry
References
Statistics
Tennis career statistics |
https://en.wikipedia.org/wiki/Charles%20Hermite | Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré.
He was the first to prove that e, the base of natural logarithms, is a transcendental number. His methods were used later by Ferdinand von Lindemann to prove that π is transcendental.
Life
Hermite was born in Dieuze, Moselle, on 24 December 1822, with a deformity in his right foot that would impair his gait throughout his life. He was the sixth of seven children of Ferdinand Hermite and his wife, Madeleine née Lallemand. Ferdinand worked in the drapery business of Madeleine's family while also pursuing a career as an artist. The drapery business relocated to Nancy in 1828, and so did the family.
Hermite obtained his secondary education at Collège de Nancy and then, in Paris, at Collège Henri IV and at the Lycée Louis-le-Grand. He read some of Joseph-Louis Lagrange's writings on the solution of numerical equations and Carl Friedrich Gauss's publications on number theory.
Hermite wanted to take his higher education at École Polytechnique, a military academy renowned for excellence in mathematics, science, and engineering. Tutored by mathematician Eugène Charles Catalan, Hermite devoted a year to preparing for the notoriously difficult entrance examination. In 1842 he was admitted to the school. However, after one year the school would not allow Hermite to continue his studies there because of his deformed foot. He struggled to regain his admission to the school, but the administration imposed strict conditions. Hermite did not accept this, and he quit the École Polytechnique without graduating.
In 1842, Nouvelles Annales de Mathématiques published Hermite's first original contribution to mathematics, a simple proof of Niels Abel's proposition concerning the impossibility of an algebraic solution to equations of the fifth degree.
A correspondence with Carl Jacobi, begun in 1843 and continued the next year, resulted in the insertion, in the complete edition of Jacobi's works, of two articles by Hermite, one concerning the extension to Abelian functions of one of the theorems of Abel on elliptic functions, and the other concerning the transformation of elliptic functions.
After spending five years working privately towards his degree, in which he befriended eminent mathematicians Joseph Bertrand, Carl Gustav Jacob Jacobi, and Joseph Liouville, he took and passed the examinations for the baccalauréat, which he was awarded in 1847. He married Joseph Bertrand's sister, Louise Bertrand, in 1848.
In 1848, Hermite returned to the École Polytechnique as répétiteur and examinateur d'admission. In 1856 he contracted smallpox. Through the influence o |
https://en.wikipedia.org/wiki/CECM | CECM may refer to:
Centre for Experimental and Constructive Mathematics at the Simon Fraser University,
Montreal Catholic School Commission (Commission des écoles catholiques de Montréal),
,
Certified in Ethics and Compliance Management at the John Cook School of Business (St. Louis University),
Computer Engineering Course Management website of University of Tehran |
https://en.wikipedia.org/wiki/Additive%20Markov%20chain | In probability theory, an additive Markov chain is a Markov chain with an additive conditional probability function. Here the process is a discrete-time Markov chain of order m and the transition probability to a state at the next time is a sum of functions, each depending on the next state and one of the m previous states.
Definition
An additive Markov chain of order m is a sequence of random variables X1, X2, X3, ..., possessing the following property: the probability that a random variable Xn has a certain value xn under the condition that the values of all previous variables are fixed depends on the values of m previous variables only (Markov chain of order m), and the influence of previous variables on a generated one is additive,
Binary case
A binary additive Markov chain is where the state space of the chain consists on two values only, Xn ∈ { x1, x2 }. For example, Xn ∈ { 0, 1 }. The conditional probability function of a binary additive Markov chain can be represented as
Here is the probability to find Xn = 1 in the sequence and
F(r) is referred to as the memory function. The value of and the function F(r) contain all the information about correlation properties of the Markov chain.
Relation between the memory function and the correlation function
In the binary case, the correlation function between the variables and of the chain depends on the distance only. It is defined as follows:
where the symbol denotes averaging over all n. By definition,
There is a relation between the memory function and the correlation function of the binary additive Markov chain:
See also
Examples of Markov chains
Notes
References
A.A. Markov. (1906) "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga". Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2-ya seriya, tom 15, 135–156
A.A. Markov. (1971) "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wiley and Sons
Ramakrishnan, S. (1981) "Finitely Additive Markov Chains", Transactions of the American Mathematical Society, 265 (1), 247–272
Markov processes |
https://en.wikipedia.org/wiki/Lajos%20P%C3%B3sa%20%28mathematician%29 | Lajos Pósa (born 9 December 1947 in Budapest) is a Hungarian mathematician working in the topic of combinatorics, and one of the most prominent mathematics educators of Hungary, best known for his mathematics camps for gifted students. He is a winner of the Széchenyi Prize.
Paul Erdős's favorite "child", he discovered theorems at the age of 13. Since 2002, he has worked at the Rényi Institute of the Hungarian Academy of Sciences; earlier he was at the Eötvös Loránd University, at the Departments of Mathematical Analysis, Computer Science.
Biography
He was born in Budapest, Hungary on 9 December 1947. His father was a chemist, his mother a mathematics teacher. He was a child prodigy. While still in elementary school, the educator Rózsa Péter, friend of his mother introduced him to Paul Erdős, who invited him
for lunch in a restaurant, and bombarded him with mathematical questions. Pósa finished the problems sooner than his soup, which impressed Erdős, who himself had been a child prodigy, and who supported young talents with much care and competence. That is how Pósa’s first paper was born, co-authored with Erdős (hence his Erdős number is 1).
He went to the first special mathematics class of the country at Fazekas Mihály Secondary School from 1962 to 1966, where his classmates included Miklós Laczkovich, László Lovász, , Zsolt Baranyai, István Berkes, Katalin Vesztergombi, Péter Major. He won the first prize on the International Mathematical Olympiad in 1966 (Bulgaria) and second prize in 1965 (Germany).
He started his Mathematics studies at ELTE University in 1966, and graduated in 1971. From 1971 to 1982 he worked at the Department of Mathematical Analysis at ELTE University, and he obtained a doctorate in 1983 with his dissertation about Hamiltonian circuits of random graphs. From 1984 to 2002 he worked at the Department of Computer Science at ELTE University, and since 2002 he has been a member of the Rényi Mathematical Institute.
Despite his significant results in mathematical research, he stopped research and devoted himself fully to Mathematics Education. Erdős, who preferred him among all his protégés, expressed his regret that Pósa had stopped research with the typical Erdős style phrase "Pósa is dead."
Mathematics education
He started teaching mathematics very early. He tutored his secondary school classmates, and during his first year at university he started teaching extracurricular courses at his former secondary school. His students at that time included: László Babai, György Elekes, Péter Komjáth, Imre Z. Ruzsa.
At the beginning of the 1970s he got involved with the school reform movement called complex teaching of mathematics led by Tamás Varga. Pósa worked on the reform of secondary mathematics teaching, while he taught at Radnóti Miklós Secondary School from 1976 to 1980. From 1982 to 1989 he was a member of the Research Group on Mathematics Education led by János Surányi. From 1982 to 1991 he had two experimental
classe |
https://en.wikipedia.org/wiki/Identifiability | In statistics, identifiability is a property which a model must satisfy for precise inference to be possible. A model is identifiable if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining an infinite number of observations from it. Mathematically, this is equivalent to saying that different values of the parameters must generate different probability distributions of the observable variables. Usually the model is identifiable only under certain technical restrictions, in which case the set of these requirements is called the identification conditions.
A model that fails to be identifiable is said to be non-identifiable or unidentifiable: two or more parametrizations are observationally equivalent. In some cases, even though a model is non-identifiable, it is still possible to learn the true values of a certain subset of the model parameters. In this case we say that the model is partially identifiable. In other cases it may be possible to learn the location of the true parameter up to a certain finite region of the parameter space, in which case the model is set identifiable.
Aside from strictly theoretical exploration of the model properties, identifiability can be referred to in a wider scope when a model is tested with experimental data sets, using identifiability analysis.
Definition
Let be a statistical model with parameter space . We say that is identifiable if the mapping is one-to-one:
This definition means that distinct values of θ should correspond to distinct probability distributions: if θ1≠θ2, then also Pθ1≠Pθ2. If the distributions are defined in terms of the probability density functions (pdfs), then two pdfs should be considered distinct only if they differ on a set of non-zero measure (for example two functions ƒ1(x) = 10 ≤ x < 1 and ƒ2(x) = 10 ≤ x ≤ 1 differ only at a single point x = 1 — a set of measure zero — and thus cannot be considered as distinct pdfs).
Identifiability of the model in the sense of invertibility of the map is equivalent to being able to learn the model's true parameter if the model can be observed indefinitely long. Indeed, if {Xt} ⊆ S is the sequence of observations from the model, then by the strong law of large numbers,
for every measurable set A ⊆ S (here 1{...} is the indicator function). Thus, with an infinite number of observations we will be able to find the true probability distribution P0 in the model, and since the identifiability condition above requires that the map be invertible, we will also be able to find the true value of the parameter which generated given distribution P0.
Examples
Example 1
Let be the normal location-scale family:
Then
This expression is equal to zero for almost all x only when all its coefficients are equal to zero, which is only possible when |σ1| = |σ2| and μ1 = μ2. Since in the scale parameter σ is restricted to be greater than zero, we conclude that the model is identifiable: ƒθ1 = ƒθ2 |
https://en.wikipedia.org/wiki/Barnard%27s%20test | In statistics, Barnard’s test is an exact test used in the analysis of contingency tables with one margin fixed. Barnard’s tests are really a class of hypothesis tests, also known as unconditional exact tests for two independent binomials. These tests examine the association of two categorical variables and are often a more powerful alternative than Fisher's exact test for contingency tables. While first published in 1945 by G.A. Barnard, the test did not gain popularity due to the computational difficulty of calculating the value and Fisher’s specious disapproval. Nowadays, even for sample sizes n ~ 1 million, computers can often implement Barnard’s test in a few seconds or less.
Purpose and scope
Barnard’s test is used to test the independence of rows and columns in a contingency table. The test assumes each response is independent. Under independence, there are three types of study designs that yield a table, and Barnard's test applies to the second type.
To distinguish the different types of designs, suppose a researcher is interested in testing whether a treatment quickly heals an infection.
One possible study design would be to sample 100 infected subjects, and for each subject see if they got the novel treatment or the old, standard, medicine, and see if the infection is still present after a set time. This type of design is common in cross-sectional studies, or ‘field observations’ such as epidemiology.
Another possible study design would be to give 50 infected subjects the treatment, 50 infected subjects the placebo, and see if the infection is still present after a set time. This type of design is common in clinical trials.
The final possible study design would be to give 50 infected subjects the treatment, 50 infected subjects the placebo, and stop the experiment once a pre-determined number of subjects has healed from the infection. This type of design is rare, but has the same structure as the lady tasting tea study that led R.A. Fisher to create Fisher's exact test.
Although the results of each design of experiment can be laid out in nearly identical-appearing tables, their statistics are different, and hence the criteria for a "significant" result are different for each:
The probability of a table under the first study design is given by the multinomial distribution; where the total number of samples taken is the only statistical constraint. This is a form of uncontrolled experiment, or "field observation", where experimenter simply "takes the data as it comes".
The second study design is given by the product of two independent binomial distributions; the totals in one of the margins (either the row totals or the column totals) are constrained by the experimental design, but the totals in other margin are free. This is by far the most common form of experimental design, where the experimenter constrains part of the experiment, say by assigning half of the subjects to be provided with a new medicine and the other half |
https://en.wikipedia.org/wiki/Pitman%E2%80%93Yor%20process | In probability theory, a Pitman–Yor process denoted PY(d, θ, G0), is a stochastic process whose sample path is a probability distribution. A random sample from this process is an infinite discrete probability distribution, consisting of an infinite set of atoms drawn from G0, with weights drawn from a two-parameter Poisson-Dirichlet distribution. The process is named after Jim Pitman and Marc Yor.
The parameters governing the Pitman–Yor process are: 0 ≤ d < 1 a discount parameter, a strength parameter θ > −d and a base distribution G0 over a probability space X. When d = 0, it becomes the Dirichlet process. The discount parameter gives the Pitman–Yor process more flexibility over tail behavior than the Dirichlet process, which has exponential tails. This makes Pitman–Yor process useful for modeling data with power-law tails (e.g., word frequencies in natural language).
The exchangeable random partition induced by the Pitman–Yor process is an example of a Poisson–Kingman partition, and of a Gibbs type random partition.
Naming conventions
The name "Pitman–Yor process" was coined by Ishwaran and James after Pitman and Yor's review on the subject. However the process was originally studied in Perman et al.
It is also sometimes referred to as the two-parameter Poisson–Dirichlet process, after the two-parameter generalization of the Poisson–Dirichlet distribution which describes the joint distribution of the sizes of the atoms in the random measure, sorted by strictly decreasing order.
See also
Chinese restaurant process
Dirichlet distribution
Latent Dirichlet allocation
References
Stochastic processes
Nonparametric Bayesian statistics
Cluster analysis algorithms |
https://en.wikipedia.org/wiki/Imre%20B%C3%A1r%C3%A1ny | Imre Bárány (Mátyásföld, Budapest, 7 December 1947) is a Hungarian mathematician, working in combinatorics and discrete geometry. He works at the Rényi Mathematical Institute of the Hungarian Academy of Sciences, and has a part-time appointment at University College London.
Notable results
He gave a surprisingly simple alternative proof of László Lovász's theorem on Kneser graphs.
He gave a new proof of the Borsuk–Ulam theorem.
Bárány gave a colored version of Carathéodory's theorem.
He solved an old problem of James Joseph Sylvester on the probability of random point sets in convex position.
With Van H. Vu proved a central limit theorem on random points in convex bodies.
With Zoltán Füredi he gave an algorithm for mental poker.
With Füredi he proved that no deterministic polynomial time algorithm determines the volume of convex bodies in dimension d within a multiplicative error dd.
With Füredi and János Pach he proved the following six circle conjecture of László Fejes Tóth: if in a planar circle packing each circle is tangent to at least 6 other circles, then either it is the hexagonal system of circles with identical radii, or there are circles with arbitrarily small radius.
Career
Bárány received the Mathematical Prize (now Paul Erdős Prize) of the Hungarian Academy of Sciences in 1985. He was an invited speaker at the Combinatorics session of the International Congress of Mathematicians, in Beijing, 2002. He was an Erdős Lecturer at Hebrew University of Jerusalem in 2004. He was elected a corresponding (2010), full (2016) member of the Hungarian Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society. Since 2021, he is a member of the Academia Europaea
He is an editor-in-chief for the journal Combinatorica, and an Editorial Board member for Mathematika and the Online Journal of Analytic Combinatorics".
He is area editor of the journal Mathematics of Operations Research''.
References
External links
Mathematicians from Budapest
Academics of University College London
Geometers
1947 births
Living people
Members of the Hungarian Academy of Sciences
Fellows of the American Mathematical Society
Combinatorialists |
https://en.wikipedia.org/wiki/Jordan%20frame | Jordan frame may refer to:
Jordan and Einstein frames, arising in the theory of relativity
Jordan frame (Jordan algebra), complete sets of pairwise orthogonal minimal idempotents in a Jordan algebra
A specific type of spinal board used in Australia |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.