|
Eilenberg
Eilenberg is a surname, and may refer to: * Samuel Eilenberg (1913–1998), Polish mathematician * Richard Eilenberg (1848–1927), German composer Named after Samuel * Eilenberg–MacLane space * Eilenberg–Moore algebra * Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ... * Eilenberg machine See also * Eilenburg * Eulenberg (other) {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a Jewish family. He spent much of his career as a professor at Columbia University. He earned his Ph.D. from University of Warsaw in 1936, with thesis ''On the Topological Applications of Maps onto a Circle''; his thesis advisors were Kazimierz Kuratowski and Karol Borsuk. He died in New York City in January 1998. Career Eilenberg's main body of work was in algebraic topology. He worked on the axiomatic treatment of homology theory with Norman Steenrod (and the Eilenberg–Steenrod axioms are named for the pair), and on homological algebra with Saunders Mac Lane. In the process, Eilenberg and Mac Lane created category theory. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book ''Homological Algebra''. Later ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. If n > 1 then ''G'' must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence, therefore any such space is often just called K(G,n). The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Steenrod Axioms
In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod. One can define a homology theory as a sequence of functors satisfying the Eilenberg–Steenrod axioms. The axiomatic approach, which was developed in 1945, allows one to prove results, such as the Mayer–Vietoris sequence, that are common to all homology theories satisfying the axioms.http://www.math.uiuc.edu/K-theory/0245/survey.pdf If one omits the dimension axiom (described below), then the remaining axioms define what is called an extraordinary homology theory. Extraordinary cohomology theories first arose in K-theory and cobordism. Formal definition The Eilenberg–Steenrod axioms apply to a sequence of functors H_n from the category of pairs (X,A) of topological spaces to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Eilenberg
Richard Eilenberg (13 January 1848 – 5 December 1927) was a German composer. Life Born in Merseburg, Eilenberg's musical career began with the study of piano and composition. At 18 years old, he composed his first work, a concert overture. As a volunteer he participated in the Franco-Prussian War from 1870 to 1871. In 1873, Eilenberg became the music director and conductor in Stettin. In 1889, he decided to move to Berlin as a freelance composer, where his second marriage with his wife Dorothee started. They lived on 73 Bremer Street. Eilenberg composed marches and dances for orchestra, harmony and military music, and a ballet ''The Rose of Shiras'', Op. 134. He also composed the operettas ''Comtess Cliquot'' (1909), ''King Midas,'' ''Marietta,'' and ''The Great Prince''. The most notable music that he composed were his marches, including ''The Coronation March'' (for Alexander III of Russia), and ''Janitscharen-Marsch'', Op. 295. Some of his music pieces, attributable to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–Moore Algebra
In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes and in functional programming languages, allowing languages with non-mutable states to do things such as simulate for-loops; see Monad (functional programming). Introduction and definition A monad is a certain type of endofunctor. For example, if F and G are a pair of adjoint functors, with F left adjoint to G, then the composition G \circ F is a monad. If F and G are inverse functors, the corresponding monad is the identity functor. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg Machine
The X-machine (''XM'') is a theoretical model of computation introduced by Samuel Eilenberg in 1974.S. Eilenberg (1974) ''Automata, Languages and Machines, Vol. A''. Academic Press, London. The ''X'' in "X-machine" represents the fundamental data type on which the machine operates; for example, a machine that operates on databases (objects of type ''database'') would be a ''database''-machine. The X-machine model is structurally the same as the finite-state machine, except that the symbols used to label the machine's transitions denote relations of type ''X''→''X''. Crossing a transition is equivalent to applying the relation that labels it (computing a set of changes to the data type ''X''), and traversing a path in the machine corresponds to applying all the associated relations, one after the other. Original theory Eilenberg's original X-machine was a completely general theoretical model of computation (subsuming the Turing machine, for example), which admitted determin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenburg
Eilenburg (; hsb, Jiłow) is a town in Germany. It lies in the district of Nordsachsen in Saxony, approximately 20 km northeast of the city of Leipzig. Geography Eilenburg lies at the banks of the river Mulde at the southwestern edge of the Düben Heath wildlife park. The town is subdivided into three urban districts: ''Berg'', ''Mitte'' and ''Ost'' and six rural districts named ''Behlitz'', ''Hainichen'', ''Kospa'', ''Pressen'', ''Wedelwitz'' and ''Zschettgau''. Neighbouring towns and cities are Leipzig (20 kilometres distant), Delitzsch (21), Bad Düben (16), Torgau (25) and Wurzen (12). History Eilenburg Castle was first mentioned on 29 July 961 in a document by Otto I. as ''civitas Ilburg''. The name has Slavic origin and means ''town with clay deposits''. A settlement of tradespeople probably developed from the 11th century in the vicinity of the castle. The town was incorporated in the Margravate of Meissen in 1386. In the 16th century Eilenburg was cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
