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Borel
Borel may refer to: People * Borel (author), 18th-century French playwright * Pierre Borel, 17th-century French chemist * Borel (1906–1967), pseudonym of the French actor Jacques Henri Cottance * Émile Borel (1871 – 1956), a French mathematician known for his founding work in the areas of measure theory and probability * Armand Borel (1923 – 2003), a Swiss mathematician * Mary Grace Borel (1915 – 1998), American socialite Places * Borel (crater), a lunar crater, named after Émile Borel Mathematics * Borel algebra, operating on Borel sets, named after Émile Borel, also: ** Borel measure, the measure on a Borel algebra * Borel distribution, a discrete probability distribution, also named after Émile Borel * Borel subgroup, in the theory of algebraic groups, named after Armand Borel Other uses * Borel (surname), a surname * Etablissements Borel, an aircraft manufacturing company founded by Gabriel Borel See also * Borrel *Borrell Borrell () is a common surname in mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Algebra
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups. Biography He studied at the ETH Zürich, where he came under the influence of the topologist Heinz Hopf and Lie-group theorist Eduard Stiefel. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of Jean Leray and Henri Cartan. With Hirzebruch, he significantly developed the theory of characteristic classes in the early 1950s. He collaborated with Jacques Tits in fundamental work on algebraic groups, and with Harish-Chandra on their arithmetic subgroups. In an algebraic group ''G'' a ''Borel subgroup'' ''H'' is one mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Formal definition Let X be a locally compact Hausdorff space, and let \mathfrak(X) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. A Borel measure is any measure \mu defined on the σ-algebra of Borel sets. A few authors require in addition that \mu is locally finite, meaning that \mu(C) 0 and μ(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the Hausdorff dimension dimHaus(''X'') ≥ ''s''. A partial converse is provided by the Frostman lemma: Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s'' > 0. Then the following are equivalent: *''H''''s''(''A'') > 0, where ''H''''s'' den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup of invertible upper triangular matrices is a Borel subgroup. For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups. Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive) algebraic groups, in Jacques Tits' theory of groups with a (B,N) pair. Here the group ''B'' is a Borel subgroup and ''N'' is the normalizer of a maximal torus contained in ''B''. The notion was introduced by Armand Borel, who played a leading role in the development of the theory of algebraic groups. Parabolic subgroups Subgroups between a Borel subgroup ''B'' and the ambient group ''G'' are called parabolic subgroups. Parabolic subgroups ''P'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel (author)
Borel (18th-century, Rouen – ? ) was an 18th-century French playwright from Normandy. Already made known by some poetic essays and an epigram in Latin entitled ''Nicetas'', crowned by the Académie des Palinods in 1749, Borel Borel addressed the dramatic genre with a play, ''le Méfiant'' ("The distrustful"), comedy in five acts and in verse, premiered in Paris at the Comédie Italienne, 20 December 1785, where it was well received. Despite this happy first attempt in literary career, the silence of the biographies on Borel from that time suggests he stopped there. Works *1786: ''Le Méfiant, comédie en 5 actes et en vers'', Paris, Cailleau. Sources * Théodore-Éloi Lebreton''Biographie rouennaise'' Rouen, Le Brument, 1865, (p. 46-7). External links Borel o [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Borel
Pierre Borel ( la, Petrus Borellius; c. 1620 Р1671) was a French savant: a chemist (and reputed alchemist), physician, and botanist. Borel was born in Castres. He became a doctor of medicine at the University of Montpellier in 1640. In 1654 he became physician to the King of France, Louis XIV. In 1663 he married Esther de Bonnafous. In 1674 he became a member of the Acad̩mie fran̤aise. He died in Paris. He concerned himself with an eclectic range of subjects: optics, ancient history, philology and bibliography. His biographers have tended to deplore his spreading of himself over so many areas. In '' The Case of Charles Dexter Ward'', H. P. Lovecraft (mis)represents Borellus (sic) as a potent necromancer Necromancy () is the practice of magic or black magic involving communication with the dead by summoning their spirits as apparitions or visions, or by resurrection for the purpose of divination; imparting the means to foretell future events .... In fact, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Brunius
__NOTOC__ Jacques B. Brunius (born Jacques Henri Cottance, 16 September 1906 – 24 April 1967) was a French actor, director and writer, who was born in Paris and died in Exeter, UK. He was cremated in Sidmouth, with a tribute by Mesens. Assistant director to Luis Buñuel on ''L'Âge d'or'', he appeared in more than 30 movies, using several alternate names: Jacques Borel, J.B. Brunius, Jacques-Bernard Brunius, Jacques Brunius, Brunius, J.B.Brunius. He acted in many of the early, more political, movies of his friend Jean Renoir. During World War II he broadcast from England to France over '' Radio Londres''. He married French-English actress Cecile Chevreau in 1951. Their son Richard was born in 1956. Member of the surrealist group in France and then in England, with his friend E.L.T. Mesens, Conroy Maddox, Ithell Colquhoun, Simon Watson Taylor and Roland Penrose. Brunius attacked Toni del Renzio, who was married to Colquhoun and who was attempting to reanimate an inactive Eng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ... and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was born in Saint-Affrique, Aveyron, the son of a Protestant pastor. He studied at the Collège Sainte-Barbe and Lycée Louis-le-Grand before applying to both the École normale supérieure (Paris), École normale supérieure and the École Polytechnique. He qualified in the first position for both and chose to attend the former institution in 1889. That year he also won the concours général, an annual national mathematics competition. After graduating in 1892, he placed first in the agrégati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mary Grace Borel
Mary Grace Borel Shumate Marshal Sweet Paxton (October 31, 1915May 18, 1998) was an American socialite and film actress. She was the granddaughter of Antoine Borel, a San Francisco banker and consul general of Switzerland, and her family was prominent on the San Francisco social scene. Her debut was attended by 200 guests in 1934. Her 1935 marriage to the physician son of the San Francisco Police Commissioner was said to be "the highlight of the 1935 social season". Two years later she sued for divorce and, in 1938, remarried to film actor Alan Marshal, with whom she had one son. She sued for divorce in 1947; she later remarried two more times. Using the stage name Mary Marshall, she acted in two films and two television series. Upon her death in 1998, she was buried in the same crypt as her second husband, Alan Marshal. Early life and education Mary Grace Borel was born in San Francisco, California, on October 31, 1915. She was the eldest daughter of Antoine A. Borel and his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel (crater)
Borel is a tiny lunar impact crater located in the southeast part of Mare Serenitatis. It was named after French mathematician Émile Borel. To the northeast is the crater Le Monnier and to the southeast is the crater Abetti. Borel was previously identified as Le Monnier C. This is a roughly circular, cup-shaped formation with inner floors that slope down to the midpoint of the crater. The interior has a higher albedo than the surrounding dark lunar mare The lunar maria (; singular: mare ) are large, dark, basaltic plains on Earth's Moon, formed by ancient asteroid impacts on the far side on the Moon that triggered volcanic activity on the opposite (near) side. They were dubbed , Latin for 'seas' .... References * * * * * * * * * * * Impact craters on the Moon Mare Serenitatis {{Craters on the Moon: A–B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Distribution
The Borel distribution is a discrete probability distribution, arising in contexts including branching processes and queueing theory. It is named after the French mathematician Émile Borel. If the number of offspring that an organism has is Poisson-distributed, and if the average number of offspring of each organism is no bigger than 1, then the descendants of each individual will ultimately become extinct. The number of descendants that an individual ultimately has in that situation is a random variable distributed according to a Borel distribution. Definition A discrete random variable ''X'' is said to have a Borel distribution with parameter ''μ'' ∈ ,1if the probability mass function of ''X'' is given by :P_\mu(n)= \Pr(X=n)= \frac for ''n'' = 1, 2, 3 .... Derivation and branching process interpretation If a Galton–Watson branching process has common offspring distribution Poisson with mean ''μ'', then the total number of i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel (surname)
Borel is a surname. Notable people with the surname include: * Adrien Borel (1886–1966), psychiatrist and psychanalyst *Aldo Borel (1912–1979), Italian football player * André Borel d'Hauterive (1812–1896), French historian *Armand Borel (1923–2003), Swiss mathematician * Calvin Borel, American jockey *Cleopatra Borel (born 1979), Trinidadian athlete * Daniel Borel (born 1950), Swiss engineer *Émile Borel (1871–1956), French mathematician and politician *Éric Borel (1978–1995), French serial killer *Ernesto Borel (1889–1951), Italian footballer *Eugène Borel (1835–1892), Swiss politician * Felice Borel (1914–1993), Italian footballer * Frédéric Borel (born 1959), French architect * Gabriel Borel, French aircraft designer * George Frederik Willem Borel, Dutch military figure * Henri Borel, Dutch writer, son of George Frederik Willem Borel * Jacques Borel, French novelist *Jean-Louis Borel (1809–1884), French general * Marguerite Borel (1883–1969), French f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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