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Heinz Prüfer
Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian groups, graph theory, algebraic numbers, knot theory and Sturm–Liouville theory. In 1915 he began his University studies in Mathematics, Physics and Chemistry in Berlin. After that he started his Doctorate degree with Issai Schur as his advisor at Friedrich Wilhelm University, Berlin. In 1921 he obtained his Doctorate degree. His thesis was named ''Unendliche Abelsche Gruppen von Elementen endlicher Ordnung'' (Infinite abelian groups of elements of finite order). This thesis set the road for his contributions on abelian groups. In 1922 he worked with mathematician Paul Koebe in the University of Jena, and in 1923 he obtained tenure and was at this University until 1927. In that year he moved to Münster University where he worked until the end of his life. His final work was about projective geometry, but it was posth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Münster University
Münster (; nds, Mönster) is an independent city (''Kreisfreie Stadt'') in North Rhine-Westphalia, Germany. It is in the northern part of the state and is considered to be the cultural centre of the Westphalia region. It is also a state district capital. Münster was the location of the Anabaptist rebellion during the Protestant Reformation and the site of the signing of the Treaty of Westphalia ending the Thirty Years' War in 1648. Today it is known as the bicycle capital of Germany. Münster gained the status of a ''Großstadt'' (major city) with more than 100,000 inhabitants in 1915. , there are 300,000 people living in the city, with about 61,500 students, only some of whom are recorded in the official population statistics as having their primary residence in Münster. Münster is a part of the international Euregio region with more than 1,000,000 inhabitants ( Enschede, Hengelo, Gronau, Osnabrück). History Early history In 793, Charlemagne sent out Ludger as a mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphs And Combinatorics
''Graphs and Combinatorics'' (ISSN 0911-0119, abbreviated ''Graphs Combin.'') is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. Its editor-in-chief is Katsuhiro Ota of Keio University. The journal was first published in 1985. Its founding editor in chief was Hoon Heng Teh of Singapore, the president of the Southeast Asian Mathematics Society, and its managing editor was Jin Akiyama. Originally, it was subtitled "An Asian Journal". In most years since 1999, it has been ranked as a second-quartile journal in discrete mathematics and theoretical computer science by SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journal and the prestige of the journals where the citations come from. Rationale Citati ..... References {{reflist Publications established in 1985 Combinatorics journal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. Originally it only covered topology, set theory, and foundations of mathematics: it was the first specialized journal in the field of mathematics..... It is published by the Mathematics Institute of the Polish Academy of Sciences. History The journal was conceived by Zygmunt Janiszewski as a means to foster mathematical research in Poland.According to and to the introduction to the 100th volume of the journal (1978, pp=1–2). These two sources cite an article written by Janiszewski himself in 1918 and titled "''On the needs of Mathematics in Poland''". Janiszewski required that, in order to achieve its goal, the journal should not force Polish mathematicians to submit articles written exclusively in Polish, and should be devote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jahresbericht Der Deutschen Mathematiker-Vereinigung
The German Mathematical Society (german: Deutsche Mathematiker-Vereinigung, DMV) is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). It was founded in 1890 in Bremen with the set theorist Georg Cantor as first president. Founding members included Georg Cantor, Felix Klein, Walther von Dyck, David Hilbert, Hermann Minkowski, Carl Runge, Rudolf Sturm, Hermann Schubert, and Heinrich Weber. The current president of the DMV is Ilka Agricola (2021–2022). Activities In honour of its founding president, Georg Cantor, the society awards the Cantor Medal. The DMV publishes two scientific journals, the ''Jahresbericht der DMV'' and ''Documenta Mathematica''. It also publishes a quarterly magazine for its membership the ''Mitteilungen der DMV''. The annual meeting of the DMV is called the ''Jahrestagung''; the DMV traditionally meets every four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Theorems
In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov. Statement Let ''A'' be an abelian group. If ''A'' is finitely generated then by the fundamental theorem of finitely generated abelian groups, ''A'' is decomposable into a direct sum of cyclic subgroups, which leads to the classification of finitely generated abelian groups up to isomorphism. The structure of general infinite abelian groups can be considerably more complicated and the conclusion needs not to hold, but Prüfer proved that it remains true for periodic groups in two special cases. The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups. The second Prüfer theorem states that a countable abelian ''p''-group whose non-trivial elements have finite ''p''-height is isomorphic to a direct sum of cyclic groups. Examples show that the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Group
In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots. The Prüfer ''p''-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century. Constructions of Z(''p''∞) The Prüfer ''p''-group may be identified with the subgroup of the circle group, U(1), consisting of all ''p''''n''-th roots of unity as ''n'' ranges over all non-negative integers: :\mathbf(p^\infty)=\ = \.\; The group operation here is the multiplication of complex numbers. There is a presentation :\mathbf(p^\infty) = \langle\, g_1, g_2, g_3, \ldots \mid g_1^p = 1, g_2^p = g_1, g_3^p = g_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Manifold
In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by and named after Heinz Prüfer. Construction The Prüfer manifold can be constructed as follows . Take an uncountable number of copies ''X''''a'' of the plane, one for each real number ''a'', and take a copy ''H'' of the upper half plane (of pairs (''x'', ''y'') with ''y'' > 0). Then glue the ''open upper half'' of each plane ''X''''a'' to the upper half plane ''H'' by identifying (''x'',''y'')∈''X''''a'' for ''y'' > 0 with the point in ''H''. The resulting quotient space Q is the Prüfer manifold. The images in Q of the points (0,0) of the spaces ''X''''a'' under identification form an uncountable discrete subset. See also *Long line (topology) In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Rank
In mathematics, especially in the area of algebra known as group theory, the Prüfer rank of a pro-p group measures the size of a group in terms of the ranks of its elementary abelian sections.. The rank is well behaved and helps to define analytic pro-p-groups. The term is named after Heinz Prüfer. Definition The Prüfer rank of pro-p-group G is ::\sup\ where d(H) is the rank of the abelian group :H/\Phi(H), where \Phi(H) is the Frattini subgroup of H. As the Frattini subgroup of H can be thought of as the group of non-generating elements of H, it can be seen that d(H) will be equal to the ''size of any minimal generating set'' of H. Properties Those profinite groups with finite Prüfer rank are more amenable to analysis. Specifically in the case of finitely generated pro-p groups, having finite Prüfer rank is equivalent to having an open normal subgroup that is powerful. In turn these are precisely the class of pro-p groups that are p-adic analytic - that is g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout Domain
In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain (PID) is a Bézout domain, but a Bézout domain need not be a Noetherian ring, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain (UFD), but still is a GCD domain. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property. Bézout domains are named after the French mathematician Étienne Bézout. Examples * All PIDs are Bézout domains. * Examples of Bézout domains that are not PIDs include the ring of entire functions (functions holomorphic on the whole complex plane) and the ring of all algebraic integers. In case of entire functions, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer. Examples The ring of entire functions on the open complex plane C form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring \mathbb /math> of integer polynomials is not . While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Prüfer domain. Just as a Dedekind domain is locally a discrete valuation ring, a Prüfer domain is locally a valuation ring, so that Prüfer domains act as non-noetherian analogues of Dedekind domains. Indeed, a domain that is the direct limit of subrings that are Prüfer domains is a Prüfer domain . Many Prü ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
