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List Of Things Named After Felix Klein
{{Short description, none These are things named after Felix Klein (1849 – 1925), a German mathematician. Mathematics *Klein bottle **Solid Klein bottle * Klein configuration * Klein cubic threefold * Klein four-group * Klein geometry * Klein graphs *Klein's inequality * Klein model * Klein polyhedron * Klein surface *Klein quadric * Klein quartic * Kleinian group * Kleinian integer * Kleinian model * Kleinian ring *Kleinian singularity * Klein's icosahedral cubic surface * Klein's j-invariant * Beltrami–Klein model * Cayley–Klein metric *Clifford–Klein form * Schottky–Klein prime form Other * Klein's Encyclopedia of Mathematical Sciences * The Felix Klein Protocols * Felix Klein medal, named after the first president of the ICMI (1908–1920), honours a lifetime achievement in mathematics education research. * ThKlein projectof the IMU IMU may refer to: Science and technology * Inertial measurement unit, a device that measures acceleration and rotation, used for examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Integer
In mathematical cryptography, a Kleinian integer is a complex number of the form m+n\frac, with ''m'' and ''n'' rational integers. They are named after Felix Klein. The Kleinian integers form a ring called the Kleinian ring, which is the ring of integers in the imaginary quadratic field \mathbb(\sqrt). This ring is a unique factorization domain. See also *Eisenstein integer In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form :z = a + b\omega , where and are integers and :\omega = \f ... * Gaussian integer References * .Review. * Quadratic irrational numbers Ring theory {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Commission On Mathematical Instruction
The International Commission on Mathematical Instruction (ICMI) is a commission of the International Mathematical Union and is an internationally acting organization focussing on mathematics education. ICMI was founded in 1908 at the International Congress of Mathematicians (ICM) in Rome and aims to improve teaching standards around the world, through programs, workshops and initiatives and publications. It aims to work a great deal with developing countries, to increase teaching standards and education which can improve life quality and aid the country. HistoryICMIwas founded at the ICM, and mathematician Felix Klein was elected first president of the organisation. Henri Fehr and Charles Laisant created the international research journal '' L'Enseignement Mathématique'' in 1899, and from early on this journal became the official organ of ICMI. A bulletin is published twice a year by ICMI, and from December 1995 this bulletin has been available at the organisation's official websi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein Protocols
"The Felix Klein Protocols" is a collection of handwritten records of the Göttingen seminar lectures of Felix Klein and his school. They span over 8000 pages in 29 volumes, and are regarded as one of the richest records of mathematical activity in modern times. The previously unpublished Klein Protocols were made available digitally in 2006. A searchable index of the protocols can be found a''Felix Klein Protokolle'' Years covered From 1872 to 1896 Klein conducted his seminars alone, mainly in pure mathematics. The years 1897–1913 show collaborations with mathematicians such as David Hilbert, Karl Schwarzschild, Ludwig Prandtl, Carl Runge and Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t .... References Mathematics manuscripts {{mathpublicati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein's Encyclopedia Of Mathematical Sciences
Felix Klein's ''Encyclopedia of Mathematical Sciences'' is a German mathematical encyclopedia published in six volumes from 1898 to 1933. Klein and Wilhelm Franz Meyer were organizers of the encyclopedia. Its full title in English is ''Encyclopedia of Mathematical Sciences Including Their Applications'', which is ''Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'' (EMW). It is 20,000 pages in length (6 volumes, ''i.e. Bände'', published in 23 separate books, 1-1, 1-2, 2-1-1, 2-1-2, 2-2, 2-3-1, 2-3-2, 3-1-1, 3-1-2, 3-2-1, 3-2-2a, 3-2-2b, 3-3, 4-1, 4-2, 4-3, 4-4, 5-1, 5-2, 5-3, 6-1, 6-2-1, 6-2-2) and was published by B.G. Teubner Verlag, publisher of ''Mathematische Annalen''. Today, Göttinger Digitalisierungszentrum provides online access to all volumes, while archive.org hosts some particular parts. Overview Walther von Dyck acted as chairman of the commission to publish the encyclopedia. In 1904 he contributed a preparatory report on the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky–Klein Prime Form
In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ..., the Schottky–Klein prime form ''E''(''x'',''y'') of a compact Riemann surface ''X'' depends on two elements ''x'' and ''y'' of ''X'', and vanishes if and only if ''x'' = ''y''. The prime form ''E'' is not quite a holomorphic function on ''X'' × ''X'', but is a section of a holomorphic line bundle over this space. Prime forms were introduced by Friedrich Schottky and Felix Klein. Prime forms can be used to construct meromorphic functions on ''X'' with given poles and zeros. If Σ''n''''i''''a''''i'' is a divisor linearly equivalent to 0, then Π''E''(''x'',''a''''i'')''n''''i'' is a meromorphic function with given poles and zeros. See also * Fay's trisecant identity References * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford–Klein Form
In mathematics, a Clifford–Klein form is a double coset space :Γ\''G''/''H'', where ''G'' is a reductive Lie group, ''H'' a closed subgroup of ''G'', and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space ''G''/''H''. A suitable discrete subgroup Γ may or may not exist, for a given ''G'' and ''H''. If Γ exists, there is the question of whether Γ\''G''/''H'' can be taken to be a compact space, called a compact Clifford–Klein form. When ''H'' is itself compact, classical results show that a compact Clifford–Klein form exists. Otherwise it may not, and there are a number of negative results. History According to Moritz Epple, the Clifford-Klein forms began when W. K. Clifford used quaternions to ''twist'' their space. "Every twist possessed a space-filling family of invariant lines", the Clifford parallels. They formed "a particular structure embedded in elliptic 3-space", the Clifford surface, which demonstrated that "the same loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Klein Metric
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance"Cayley (1859), p 82, §§209 to 229 where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics. Foundations The algebra of throws by Karl von Staudt (1847) is an approach to geometry that is independent of metric. The idea was to use the relation of projective harmonic conjugates and cross-ratios as fundamental to the measure on a line. Another important insight was the Laguerr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beltrami–Klein Model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or ''n''-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere. The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "Cayley" in Cayley–Klein model refers to the English geometer Arthur Cayley. The Beltrami–Klein model is analogous to the gnomonic projection of spherical geometry, in that geodesics (great circles in spherical geometry) are mapped to straight lines. This model is not conformal, meaning that angles and circles are distorted, whereas the Poincaré disk model preserves these. In this model, lines and segments are straight Euclidean segments, whereas in the Poincaré disk model, lines are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J-invariant
In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that :j\left(e^\right) = 0, \quad j(i) = 1728 = 12^3. Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The -invariant can be defined as a function on the upper half-plane :j(\tau) = 1728 \frac = 1728 \frac = 1728 \frac with the third definition implying j(\tau) can be expressed as a cube, also since 1728 = 12^3. The given functions are the modular discriminant \Delta(\tau) = g_2(\tau)^3 - 27g_3(\tau)^2 = (2\pi)^\,\eta^(\tau) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch Surface
In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its blowup at the 10 Eckardt points. Definition The Clebsch surface is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4) of P4 satisfying the equations :x_0 + x_1 + x_2 + x_3 + x_4 = 0, :x_0^3 + x_1^3 + x_2^3 + x_3^3 + x_4^3 = 0. Eliminating ''x''0 shows that it is also isomorphic to the surface :x_1^3 + x_2^3 + x_3^3 + x_4^3 = (x_1 + x_2 + x_3 + x_4)^3 in P3. Properties The symmetry group of the Clebsch surface is the symmetric group ''S''5 of order 120, acting by permutations of the coordinates (in ''P''4). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group. The 27 exceptional lines are: * The 15 images (under ''S''5) of the line of points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Singularity
In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by Patrick du Val and Felix Klein. The Du Val singularities also appear as quotients of \Complex^2 by a finite subgroup of SL2(\Complex); equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups. The rings of invariant polynomials of these finite group actions were computed by Klein, and are essentially the coordinate rings of the singularities; this is a classic result in invariant theory. Classification The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
