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Frobenius
Frobenius is a surname. Notable people with the surname include: * Ferdinand Georg Frobenius (1849–1917), mathematician ** Frobenius algebra ** Frobenius endomorphism ** Frobenius inner product ** Frobenius norm ** Frobenius method ** Frobenius group ** Frobenius theorem (differential topology) * Georg Ludwig Frobenius (1566–1645), German publisher * Johann Froben, Johannes Frobenius (1460–1527), publisher and printer in Basel * Hieronymus Frobenius (1501–1563), publisher and printer in Basel, son of Johannes * Ambrosius Frobenius (1537–1602), publisher and printer in Basel, son of Hieronymus * Leo Frobenius (1873–1938), ethnographer * Nikolaj Frobenius (born 1965), Norwegian writer and screenwriter * August Sigmund Frobenius (died 1741), German chemist See also * Frobenius Orgelbyggeri, Danish organ building firm {{surname, Frobenius ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Endomorphism
In commutative algebra and field theory (mathematics), field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative Ring (mathematics), rings with prime number, prime characteristic (algebra), characteristic , an important class that includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general. Definition Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism ''F'' is defined by :F(r) = r^p for all ''r'' in ''R''. It respects the multiplication of ''R'': :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory , . Jean Dieudonné used this to characterize Frobenius algebras . Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory. Definition A finite-dimensional, unital, associative algebra ''A'' defined over a field ''k'' is said to be a Frobenius algebra if ''A'' is equipped with a nondegenerate bilinear form that satisfies the following e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Group
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius. Structure Suppose ''G'' is a Frobenius group consisting of permutations of a set ''X''. A subgroup ''H'' of ''G'' fixing a point of ''X'' is called a Frobenius complement. The identity element together with all elements not in any conjugate of ''H'' form a normal subgroup called the Frobenius kernel ''K''. (This is a theorem due to ; there is still no proof of this theorem that does not use character theory, although see .) The Frobenius group ''G'' is the semidirect product of ''K'' and ''H'': :G=K\rtimes H. Both the Frobenius kernel and the Frobenius complement have very restricted structures. proved that the Frobenius kernel ''K'' is a nilpotent group. If ''H'' has even order then ''K'' is abelian. The Frobenius complement ''H'' has the property th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leo Frobenius
Leo Viktor Frobenius (29 June 1873 – 9 August 1938) was a German self-taught ethnologist and archaeologist and a major figure in German ethnography. Life He was born in Berlin as the son of a Prussian officer and died in Biganzolo, Lago Maggiore, Piedmont, Italy. He undertook his first expedition to Africa in 1904 to the Kasai district in Congo, formulating the African Atlantis theory during his travels. During World War I, between 1916 and 1917, Leo Frobenius spent almost an entire year in Romania, travelling with the German Army for scientific purposes, as well as undertaking covert missions in Arabia and Ethiopia in 1914-1915. His team performed archaeological and ethnographic studies in the country, as well as documenting the day-to-day life of the ethnically diverse inmates of the Slobozia prisoner camp. Numerous photographic and drawing evidences of this period exist in the image archive of the Frobenius Institute. Until 1918 he travelled in the western and cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds. Biography Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin, from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Theorem (differential Topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern differential geometry, geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the Picard–Lindelöf theorem, existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in differential topology and Differentiable manifold, calculus on manifolds. Contact geometry studies 1-forms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Norm
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. Preliminaries Given a field \ K\ of either real or complex numbers (or any complete subset thereof), let \ K^\ be the -vector space of matrices with m rows and n columns and entries in the field \ K ~. A matrix norm is a norm on \ K^~. Norms are often expressed with double vertical bars (like so: \ \, A\, \ ). Thus, the matrix norm is a function \ \, \cdot\, : K^ \to \R^\ that must satisfy the following properties: For all scalars \ \alpha \in K\ and matrices \ A, B \in K^\ , * \, A\, \ge 0\ (''positive-valued'') * \, A\, = 0 \iff A=0_ (''definite'') * \left\, \alpha\ A \right\, = \left, \alpha \\ \left\, A\right\, \ (''absolutely homogeneous'') * \, A + B \, \le \, A \, + \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Method
In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form z^2 u'' + p(z)z u'+ q(z) u = 0 with u' \equiv \frac and u'' \equiv \frac. in the vicinity of the regular singular point z=0. One can divide by z^2 to obtain a differential equation of the form u'' + \fracu' + \fracu = 0 which will not be solvable with regular power series methods if either or is not analytic at . The Frobenius method enables one to create a power series solution to such a differential equation, provided that ''p''(''z'') and ''q''(''z'') are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite). History Frobenius' contribution was not so much in all the possible ''forms'' of the series solutions involved (see below). These forms had all been established earlier, by Lazarus Fuchs. The ''indicial polynomial'' (see bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
August Sigmund Frobenius
August Sigmund Frobenius (earliest date mentioned 1727, died 1741), FRS, also known as Sigismond Augustus Frobenius, Joannes Sigismundus Augustus Frobenius, and Johann Sigismund August Froben, was a German-born chemist in the 18th century who is known for the first detailed description of the properties of diethyl ether and the naming of this substance (''Spiritus Vini Æthereus''). Not much is known about his life. He worked in London, Paris, Germany, and Italy. In the laboratory or Ambrose Godfrey in London he produced ether, following a method of Isaac Newton. His first article about ether was published 1729 in the Philosophical Transactions of the Royal Society under the title ''An Account of a Spiritus Vini Æthereus, Together with Several Experiments Tried'' and sparked a new scientific interest in this substance. This article contains an extensive description of the properties of ether but did not include experimental procedures. These missing details were, however, deposit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Inner Product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though they are vectors, and satisfies the axioms for an inner product. The two matrices must have the same dimension—same number of rows and columns—but are not restricted to be square matrices. It is named after Ferdinand Georg Frobenius. Definition Given two complex-number-valued ''n''×''m'' matrices A and B, written explicitly as : \mathbf = \,, \quad \mathbf =, the Frobenius inner product is defined as :\langle \mathbf, \mathbf \rangle_\mathrm =\sum_\overline B_ \, = \mathrm\left(\overline \mathbf\right) \equiv \mathrm\left(\mathbf^ \mathbf\right), where the overline denotes the complex conjugate, and \dagger denotes the Hermitian conjugate. Explicitly, this sum is :\begin \langle \mathbf, \mathbf \rangle_\mathrm = & \overl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolaj Frobenius
Nikolaj Frobenius (born 29 September 1965) is a Norwegian novelist and screen writer. Frobenius was born in Oslo, but grew up at Rykkinn. He studied film writing and research at LCP, London. He has written several books and screenplays, including the screenplay for the classic Nordic film thriller ''Insomnia'', which was adapted into a major Hollywood production in 2002. His international breakthrough as a novelist came with the novel ''Latours katalog'' (1996). His books have been translated into twenty-five languages, including English, French, Spanish, Italian, German, Russian and Danish. His novels have received critical acclaim both in Norway and internationally, and he has won several literary prizes for his writing. Nikolaj Frobenius is a former editor of the periodical ''Vinduet'' and worked as a commissioning editor for Norsk filmfond from 2005 to 2008. Frobenius has written several successful screenplays, including ''Dragonfly'' (2001). In 2011 he adapted his own novel, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Froben
Johann Froben, in Latin: Johannes Frobenius (and combinations), (c. 1460 – 27 October 1527) was a famous printer, publisher and learned Renaissance humanist in Basel. He was a close friend of Erasmus and cooperated closely with Hans Holbein the Younger. He made Basel one of the world's leading centres of the book trade. He passed his printing business on to his son, Hieronymus, and grandson, Ambrosius Frobenius. Early life and printing partnership Johann Froben was born in Hammelburg, Franconia and appears the first time at the workshop of the printer of Anton Koberger of Nuremberg in 1486. He moved to Basel in the 1480s. He graduated from the University in Basel, where he made the acquaintance of the famous printer Johann Amerbach (c. 1440–1513). Froben established himself as a printer in that city about 1491, when he published the first manageable bible in the octavo format. He soon attained a European reputation for accuracy and taste. In 1500, he married the daught ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
