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Eckmann–Hilton Argument
In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two unital magma structures on a Set (mathematics), set where one is a homomorphism for the other. Given this, the structures can be shown to coincide, and the resulting magma (algebra) , magma demonstrated to be commutative monoid. This can then be used to prove the commutativity of the higher homotopy groups. The principle is named after Beno Eckmann and Peter Hilton, who used it in a 1962 paper. The Eckmann–Hilton result Let X be a set equipped with two Binary operation, binary operations, which we will write \circ and \otimes, and suppose: # \circ and \otimes are both unital algebra, unital, meaning that there are elements 1_\circ and 1_\otimes of X such that 1_\circ \circ a= a =a \circ 1_\circ and 1_\otimes \otimes a= a =a \otimes 1_\otimes, for all a\in X. # (a \otimes b) \circ (c \otimes d) = (a \circ c) \otimes (b \circ d) for all a,b,c,d \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interchange Law
Interchange may refer to: Transport * Interchange (road), a collection of ramps, exits, and entrances between two or more highways * Interchange (freight rail), the transfer of freight cars between railroad companies * Interchange station, a railway station where two or more routes meet and allow passengers to change trains * Cross-platform interchange, the transfer between trains across a station platform * Transport interchange or transport hub to include jointly operated interchange flights by two or more airlines Other uses * Interchange (de Kooning), ''Interchange'' (de Kooning), a 1955 painting by Willem de Kooning * Interchange (album), ''Interchange'' (album), a 1994 album by guitarist Pat Martino * Interchange (Australian rules football), a team position in Australian rules football * Interchange circuit, a circuit that facilitates the exchange of data and signaling information * Interchange fee, a fee paid between banks in the payment card industry * Interchange (film ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurewicz
Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on the textile industry. His father, Mieczysław Hurewicz, was an industrialist born in Wilno, which until 1939 was mainly populated by Poles and Jews. His mother was Katarzyna Finkelsztain who hailed from Biała Cerkiew, a town that belonged to the Kingdom of Poland until the Second Partition of Poland (1793) when it was taken by Russia. Hurewicz attended school in a German-controlled Poland but with World War I beginning before he had begun secondary school, major changes occurred in Poland. In August 1915 the Russian forces that had held Poland for many years withdrew. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly, a strong school of mathematics grew ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology). In brief, singular homology is constructed by taking maps of the standard ''n''-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation – mapping each ''n''-dimensional simplex to its (''n''−1)-dimensional boundary – induces the singular chain complex. The singular homology is then the homology of the chain complex. The resulting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seifert–Van Kampen Theorem
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in terms of the fundamental groups of two open, path-connected subspaces that cover X. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones. Van Kampen's theorem for fundamental groups Let ''X'' be a topological space which is the union of two open and path connected subspaces ''U''1, ''U''2. Suppose ''U''1 ∩ ''U''2 is path connected and nonempty, and let ''x''0 be a point in ''U''1 ∩ ''U''2 that will be used as the base of all fundamental groups. The inclusion maps of ''U''1 and ''U''2 into ''X'' induce group homomorphisms j_1:\pi_1(U_1,x_0)\to \pi_1(X,x_0) and j_2:\pi_1(U_2,x_0)\to \pi_1(X,x_0). Then ''X'' is path connected and j_1 and j_2 form a commutative pushout dia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witold Hurewicz
Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on the textile industry. His father, Mieczysław Hurewicz, was an industrialist born in Wilno, which until 1939 was mainly populated by Poles and Jews. His mother was Katarzyna Finkelsztain who hailed from Biała Cerkiew, a town that belonged to the Kingdom of Poland until the Second Partition of Poland (1793) when it was taken by Russia. Hurewicz attended school in a German-controlled Poland but with World War I beginning before he had begun secondary school, major changes occurred in Poland. In August 1915 the Russian forces that had held Poland for many years withdrew. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly, a strong school of mathematics gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth (née Kirchner) and Wilhelm Hopf. His father was born Jewish and converted to Protestantism a year after Heinz was born; his mother was from a Protestant family. Hopf attended Karl Mittelhaus higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age. In 1913 he entered the Silesian Friedrich Wilhelm University where he attended lectures by Ernst Steinitz, Adolf Kneser, Max Dehn, Erhard Schmidt, and Rudolf Sturm. When World War I broke out in 1914, Hopf eagerly enlisted. He was wounded twice and received the iron cross (first class) in 1918. After the war Hopf continued his mathematical education in Heidelberg (winter 1919/20 and summer 1920) and Berl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, making important contributions to set theory and topology. In topology, the Alexandroff compactification and the Alexandrov topology are named after him. Biography Alexandrov attended Moscow State University where he was a student of Dmitri Egorov and Nikolai Luzin. Together with Pavel Urysohn, he visited the University of Göttingen in 1923 and 1924. After getting his Ph.D. in 1927, he continued to work at Moscow State University and also joined the Steklov Institute of Mathematics. He was made a member of the Russian Academy of Sciences in 1953. Personal life Luzin challenged Alexandrov to determine if the continuum hypothesis is true. This still unsolved problem was too much for Alexandrov and he had a creative crisis at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or ''holes'', of a topological space. To define the ''n''-th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''-th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent ( homeomorphic), but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduard Čech
Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topology) and the notion of Čech cohomology. He was the first to publish a proof of Tychonoff's theorem in 1937. Biography He was born in Stračov, then in Bohemia, Austria-Hungary, now in the Czech Republic. His father was Čeněk Čech, a policeman, and his mother was Anna Kleplová. After attending the gymnasium in Hradec Králové, Čech was admitted to the Philosophy Faculty of Charles University of Prague in 1912. In 1915 he was drafted into the Austro-Hungarian Army and participated in World War I, after which he completed his undergraduate degree in 1918. He received his doctoral degree in 1920 at Charles University; his thesis, titled ''On Curves and Plane Elements of the Third Order'', was written under the direction of Karel Petr. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelianisation
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Commutators For elements g and h of a group ''G'', the commutator of g and h is ,h= g^h^gh. The commutator ,h/math> is equal to the identity element ''e'' if and only if gh = hg , that is, if and only if g and h commute. In general, gh = hg ,h/math>. However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
