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Free Group
In mathematics, the free group ''F''''S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1''t'', but ''s'' ≠''t''−1 for ''s'',''t'',''u'' ∈ ''S''). The members of ''S'' are called generators of ''F''''S'', and the number of generators is the rank of the free group. An arbitrary group ''G'' is called free if it is isomorphic to ''F''''S'' for some subset ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in exactly one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''−1''t''). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History Free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F2 Cayley Graph
F, or f, is the sixth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''. History The origin of 'F' is the Semitic letter ''waw'' that represented a sound like or . Graphically it originally probably depicted either a hook or a club. It may have been based on a comparable Egyptian hieroglyph such as that which represented the word ''mace'' (transliterated as ḥ(dj)): T3 The Phoenician form of the letter was adopted into Greek as a vowel, ''upsilon'' (which resembled its descendant ' Y' but was also the ancestor of the Roman letters ' U', ' V', and ' W'); and, with another form, as a consonant, ''digamma'', which indicated the pronunciation , as in Phoenician. Latin 'F,' despite being pronounced differently, is ultimately descended from digamma and closely resembles it in form. After sound changes eliminated ... [...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] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Tarski Paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox". The theorem is called a paradox because it contradicts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Lie Algebra
In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity. Definition The definition of the free Lie algebra generated by a set ''X'' is as follows: : Let ''X'' be a set and i\colon X \to L a morphism of sets (function) from ''X'' into a Lie algebra ''L''. The Lie algebra ''L'' is called free on ''X'' if i is the universal morphism; that is, if for any Lie algebra ''A'' with a morphism of sets f\colon X \to A, there is a unique Lie algebra morphism g\colon L\to A such that f = g \circ i. Given a set ''X'', one can show that there exists a unique free Lie algebra L(X) generated by ''X''. In the language of category theory, the functor sending a set ''X'' to the Lie algebra generated by ''X'' is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Central Series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algrebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras. A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable. Definition A central series is a sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Magnus
Hans Heinrich Wilhelm Magnus known as Wilhelm Magnus (February 5, 1907 in Berlin, Germany – October 15, 1990 in New Rochelle, New York) was a German-American mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations. Biography In 1931, Magnus received his PhD from the University of Frankfurt, in Germany. His thesis, written under the direction of Max Dehn, was entitled ''Über unendlich diskontinuierliche Gruppen von einer definierenden Relation (der Freiheitssatz)''. Magnus was a faculty member in Frankfurt from 1933 until 1938. He refused to join the Nazi Party and, as a consequence, was not allowed to hold an academic post during World War II. In 1947 he became a professor at the University of Göttingen. In 1948 he emigrated to the United States to collaborate on the Bateman Manuscript Project as a co-editor, while a visiting professor at the California Institute o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology. A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still ''combinatorial'' in 1942, it had become ''algebraic'' by 1944. This corresponds also to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurt Reidemeister
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany. Life He was a brother of Marie Neurath. Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Göttingen. In 1920, he got the staatsexamen (master's degree) in mathematics, philosophy, physics, chemistry, and geology. He received his doctorate in 1921 with a thesis in algebraic number theory at the University of Hamburg under the supervision of Erich Hecke. He became interested in differential geometry; he edited Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Education and career Blaschke was the son of mathematician Josef Blaschke, who taught ...'s 2nd volume about that issue, and both made an acclaimed contribution to the Jena DMV conference in Sep 1921. In October 1922 (or 1923) he was appointed assistan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Schreier
Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. Life His parents were the architect Theodor Schreier (1873-1943) and his wife Anna (b. Turnau) (1878-1942). From 1920 Otto Schreier studied at the University of Vienna and took classes with Wilhelm Wirtinger, Philipp Furtwängler, Hans Hahn, Kurt Reidemeister, Leopold Vietoris, and Josef Lense. In 1923 he obtained his doctorate, under the supervision of Philipp Furtwängler, entitled ''On the expansion of groups (Über die Erweiterung von Gruppen)''. In 1926 he completed his habilitation with Emil Artin at the University of Hamburg ''(Die Untergruppen der freien Gruppe. Abhandlungen des Mathematischen Seminars der Universität Hamburg, Band 5, 1927, Seiten 172–179)'', where he had also given lectures before. In 1928 he became a professor at the University of Rost ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
