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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 * 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 Frobenius is a Danish firm of organ builders. History Theodor Frobenius was born into a family of organ builders on 7 October 1885 in Weikersheim, Bavaria. From the age of 13, he trained as an ... [...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 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 equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which 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 denominator, of the explicit formula of the binomial coefficients :\frac, i ... [...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. 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 Hermitian conjugate. Explicitly this sum is :\begin \langle \mathbf, \mathbf \rangle_\mathrm = & \overline_ B_ + \overline_ B_ + \cdots + \overline_ B_ \\ & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, 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^. This article will always write such norms 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\, +\, B\, (''sub-additive'' or satisfying the ''triangle inequality'') The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative: *\le ... [...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 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 are 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). Explanation The method of Frobenius is to seek a power series solution of the form u(z)=z^r \sum_^\infty A_k z^k, \qquad (A_0 \neq 0) Differentiating: u'(z)=\sum_^\infty (k+r)A_kz^ u''(z)=\sum_^\infty (k+r-1)(k+r)A_kz^ Substit ... [...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 t ... [...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 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 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 calculus on manifolds. Introduction In its most elementary form, the theorem addresses the problem of finding a maximal set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georg Ludwig Frobenius
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Georg may refer to: * ''Georg'' (film), 1997 *Georg (musical), Estonian musical * Georg (given name) * Georg (surname) * , a Kriegsmarine coastal tanker See also * George (other) George may refer to: People * George (given name) * George (surname) * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Washington, First President of the United States * George W. Bush, 43rd Preside ... [...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. Biography Froben was born in Hammelburg, Franconia. After completing his university career at Basel, where he made the acquaintance of the famous printer Johann Amerbach (c. 1440 — 1513), Froben established a printing house in that city about 1491, and this soon attained a European reputation for accuracy and taste. In 1500, he married the daughter of the bookseller Wolfgang Lachner, who entered into a partnership with him. It was part of Froben's plan to print editions of the Greece, Greek Fathers. Between 1496 and 1512 he was in a printing alliance together wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hieronymus Froben
Hieronymus Froben (1501–1563) was a famous pioneering printer in Basel and the eldest son of Johann Froben. He was educated at the University of Basel and traveled widely in Europe. He, his father and his brother-in-law Nicolaus Episcopius were noted for their working friendship with Erasmus and for making Basel an important center of Renaissance printing. Their editions include the first Latin edition of Georgius Agricola's '' De Re Metallica'' in 1556, and some of them incorporate artwork by Hans Holbein the Younger. Through his own sons, Ambrosius Ambrosius or Ambrosios (a Latin adjective derived from the Ancient Greek word ἀμβρόσιος, ''ambrosios'' "divine, immortal") may refer to: Given name: *Ambrosius Alexandrinus, a Latinization of the name of Ambrose of Alexandria (before 21 ... and Aurelius, the family continued their printing concern through the end of the next century. References {{DEFAULTSORT:Froben, Hieronymus 1501 births 1563 deaths People ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ambrosius Frobenius
Ambrosius Froben, or in Latin Frobenius (1537–1602), was a Basel printer, and publisher of an almost complete Hebrew Talmud, 1578–1580.The way Jews lived: five hundred years of printed words and images p51 Constance Harris - 2009 "Froben's grandson, Ambrosius, published an important Talmud, 1578–1580, under the supervision of the Jewish editor" He was son of Hieronymus Froben (1501–65), and grandson of 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 t ... (1460–1527) the Swiss scholar and printer. References {{DEFAULTSORT:Frobenius, Ambrosius Swiss book publishers (people) 1537 births 1602 deaths ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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