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Göpel Tetrad
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution ''x'' ↦ −''x''. The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a (possibly nonalgebraic) torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces. Other surfaces closely related to Kummer surfaces include Weddle surfaces, wave surfaces, and tetrahedroids. Geometry of the Kummer surface Singular quartic surfaces and the double plane model Let K\subset\mathbb^3 be a quartic surface with an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Surface
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety of a smooth hyperelliptic curve of genus 2; i.e. a quotient of the Jacobian by the Kummer involution ''x'' ↦ −''x''. The Kummer involution has 16 fixed points: the 16 2-torsion point of the Jacobian, and they are the 16 singular points of the quartic surface. Resolving the 16 double points of the quotient of a (possibly nonalgebraic) torus by the Kummer involution gives a K3 surface with 16 disjoint rational curves; these K3 surfaces are also sometimes called Kummer surfaces. Other surfaces closely related to Kummer surfaces include Weddle surfaces, wave surfaces, and tetrahedroids. Geometry of the Kummer surface Singular quartic surfaces and the double plane model Let K\subset\mathbb^3 be a quartic surface with an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blowing Up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion. Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The weak factorization theorem says that every birational map can be factored as a composition of particularly simple blowups. The Cremona group, the group of birational automorphisms of the plane, is generated by blowups. Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces. For instance, most procedures for resolution of singularities proceed by bl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adolph Göpel 1812-1847
Adolf (also spelt Adolph or Adolphe, Adolfo and when Latinised Adolphus) is a given name used in German-speaking countries, Scandinavia, the Netherlands and Flanders, France, Italy, Spain, Portugal, Latin America and to a lesser extent in various Central European and East European countries with non-Germanic languages, such as Lithuanian Adolfas and Latvian Ādolfs. Adolphus can also appear as a surname, as in John Adolphus, the English historian. The female forms Adolphine and Adolpha are far more rare than the male names. The name is a compound derived from the Old High German ''Athalwolf'' (or ''Hadulf''), a composition of ''athal'', or ''adal'', meaning "noble" (or '' had(u)''-, meaning "battle, combat"), and ''wolf''. The name is cognate to the Anglo-Saxon name '' Æthelwulf'' (also Eadulf or Eadwulf). The name can also be derived from the ancient Germanic elements "Wald" meaning "power", "brightness" and wolf (Waldwulf). Due to negative associations with Adolf Hitler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rosenhain Tetrad
Rosenhain is surname of: * Johann Georg Rosenhain Johann Georg Rosenhain (10 June 1816 in Königsberg – 14 March 1887 Berlin) was a German mathematician who introduced theta characteristic In mathematics, a theta characteristic of a non-singular algebraic curve ''C'' is a divisor class Θ suc ... (1816 - 1887), German mathematician * Jakob ''(Jacob, Jacques)'' Rosenhain (1813, Mannheim – 1894, Baden-Baden), a Jewish German pianist and composer * Walter Rosenhain (1875, Berlin – 1934), an Australian metallurgist See also * {{surname, Rosenhain German-language surnames Jewish surnames Yiddish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Tetrahedra
Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" ideas based on faith in a system of thought * ''The Fundamentals'', a set of books important to Christian fundamentalism * Any of a number of fundamental theorems identified in mathematics, such as: ** Fundamental theorem of algebra, awe theorem regarding the factorization of polynomials ** Fundamental theorem of arithmetic, a theorem regarding prime factorization * Fundamental analysis, the process of reviewing and analyzing a company's financial statements to make better economic decisions Music * Fun-Da-Mental, a rap group * ''Fundamental'' (Bonnie Raitt album), 1998 * ''Fundamental'' (Pet Shop Boys album) * ''Fundamental'' (Puya album), 1999 * ''Fundamental'' (Mental As Anything album) * ''The Fundamentals'' (album) Other uses * " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Quadric
Fundamental may refer to: * Foundation of reality * Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental" * Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" ideas based on faith in a system of thought * ''The Fundamentals'', a set of books important to Christian fundamentalism * Any of a number of fundamental theorems identified in mathematics, such as: ** Fundamental theorem of algebra, awe theorem regarding the factorization of polynomials ** Fundamental theorem of arithmetic, a theorem regarding prime factorization * Fundamental analysis, the process of reviewing and analyzing a company's financial statements to make better economic decisions Music * Fun-Da-Mental, a rap group * ''Fundamental'' (Bonnie Raitt album), 1998 * ''Fundamental'' (Pet Shop Boys album) * ''Fundamental'' (Puya album), 1999 * ''Fundamental'' (Mental As Anything album) * ''The Fundamentals'' (album) Other uses * " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Configuration
In geometry, the Klein configuration, studied by , is a geometric configuration related to Kummer surface In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety o ...s that consists of 60 points and 60 planes, with each point lying on 15 planes and each plane passing through 15 points. The configurations uses 15 pairs of lines, 12 . 13 . 14 . 15 . 16 . 23 . 24 . 25 . 26 . 34 . 35 . 36 . 45 . 46 . 56 and their reverses. The 60 points are three concurrent lines forming an odd permutation, shown below. The sixty planes are 3 coplanar lines forming even permutations, obtained by reversing the last two digits in the points. For any point or plane there are 15 members in the other set containing those 3 lines. udson, 1905 Coordinates of points and planes A possible set of coordinates for points ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apolar Complex
In chemistry, polarity is a separation of electric charge leading to a molecule or its chemical groups having an electric dipole moment, with a negatively charged end and a positively charged end. Polar molecules must contain one or more polar bonds due to a difference in electronegativity between the bonded atoms. Molecules containing polar bonds have no molecular polarity if the bond dipoles cancel each other out by symmetry. Polar molecules interact through dipole–dipole intermolecular forces and hydrogen bonds. Polarity underlies a number of physical properties including surface tension, solubility, and melting and boiling points. Polarity of bonds Not all atoms attract electrons with the same force. The amount of "pull" an atom exerts on its electrons is called its electronegativity. Atoms with high electronegativitiessuch as fluorine, oxygen, and nitrogenexert a greater pull on electrons than atoms with lower electronegativities such as alkali metals and alkaline ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Line
The Polar Line ( no, Polarbanen, german: Polarbahn) is an incomplete and abandoned railway line in Norway, from Fauske to Narvik and, if finished, ultimately would have run to Kirkenes. The railway was constructed by the ''Wehrmacht'' in occupation of Norway by Nazi Germany, occupied Norway during the Second World War as part of ''Festung Norwegen''. At Fauske, the line connected with the Nordland Line, and construction stretched as far north as Drag, Norway, Drag, Tysfjord. After the war, the plans were abandoned by Norwegian authorities, although from the 1970s, they were revitalized as part of the proposed Northern Norway Line. Some tunnels and bridges remain and part of the route has been used to build European route E06, European Road E6. Route Organizationally, the construction of the Polar Line started at Finneid. It ran through the Bratthaugen Tunnel before reaching Fauske, where Fauske Station was planned. At the time of End of World War II in Europe, German capitul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear in each argument separately: * and * and The dot product on \R^n is an example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When is the field of complex numbers , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. Coordinate representation Let be an -dimensional vector space with basis . The matrix ''A'', defined by is called the ''matrix of the bilinear form'' on the basis . If the matrix represents a vector with respect to this basis, and analogously, represents another vector , then: B(\mathbf, \mathbf) = \mathbf^\textsf A\mathbf = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Configuration
In geometry, the Kummer configuration, named for Ernst Kummer, is a geometric configuration of 16 points and 16 planes such that each point lies on 6 of the planes and each plane contains 6 of the points. Further, every pair of points is incident with exactly two planes, and every two planes intersect in exactly two points. The configuration is therefore a biplane, specifically, a 2−(16,6,2) design. The 16 nodes and 16 tropes of a Kummer surface form a Kummer configuration. There are three different non-isomorphic ways to select 16 different 6-sets from 16 elements satisfying the above properties, that is, forming a biplane. The most symmetric of the three is the Kummer configuration, also called "the best biplane" on 16 points. Construction Following the method of Jordan (1869), but see also Assmus and Sardi (1981), arrange the 16 points (say the numbers 1 to 16) in a 4x4 grid. For each element in turn, take the 3 other points in the same row and the 3 other points in the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
