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Elliptic Fibration
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change. The surface and the base curve are assumed to be non-singular ( complex manifolds or regular schemes, depending on the context). The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in F-theory. Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds and smooth 4-manifo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Surface
In mathematics, a Kodaira surface is a compact space, compact algebraic surface, complex surface of Kodaira dimension 0 and odd first Betti number. The concept is named after Kunihiko Kodaira. These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces. Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order ''k'' = 1,2,3,4,6, then the plurigenera ''P''''n'' are 1 if ''n'' is divisible by ''k'' and 0 otherwise. Hodge diamond: Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Fiber A1 A
is a city located in the western portion of Tokyo Metropolis, Japan. , the city had an estimated population of 195,207 in 93,654 households, and a population density of 9500 persons per km². The total area of the city was . Geography Kodaira is located in the Musashino Terrace near the geographic centre of Tokyo Metropolis. Surrounding municipalities Tokyo Metropolis *Nishitokyo *Tachikawa *Higashimurayama * Higashiiyamato *Higashikurume * Kokubunji *Koganei Climate Kodaira has a Humid subtropical climate (Köppen ''Cfa'') characterized by warm summers and cool winters with light to no snowfall. The average annual temperature in Kodaira is 14.0 °C. The average annual rainfall is 1647 mm with September as the wettest month. The temperatures are highest on average in August, at around 25.5 °C, and lowest in January, at around 2.6 °C. Demographics Per Japanese census data, the population of Kodaira increased rapidly in the 1950s and 1960s. History The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine A1 Diagram
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certain kind of constrained linear combination * Affine connection, a connection on the tangent bundle of a differentiable manifold * Affine Coordinate System, a coordinate system that can be viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. * Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group * Affine gap penalty, the most widely used scoring function used for sequence alignment, especially in bioinformatics * Affine geometry, a geometry characterized by parallel lines * Affine group, the group of all invertible affine transformations from any affine space over a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Fiber 0 B
is a city located in the western portion of Tokyo Metropolis, Japan. , the city had an estimated population of 195,207 in 93,654 households, and a population density of 9500 persons per km². The total area of the city was . Geography Kodaira is located in the Musashino Terrace near the geographic centre of Tokyo Metropolis. Surrounding municipalities Tokyo Metropolis *Nishitokyo *Tachikawa *Higashimurayama * Higashiiyamato *Higashikurume * Kokubunji *Koganei Climate Kodaira has a Humid subtropical climate (Köppen ''Cfa'') characterized by warm summers and cool winters with light to no snowfall. The average annual temperature in Kodaira is 14.0 °C. The average annual rainfall is 1647 mm with September as the wettest month. The temperatures are highest on average in August, at around 25.5 °C, and lowest in January, at around 2.6 °C. Demographics Per Japanese census data, the population of Kodaira increased rapidly in the 1950s and 1960s. History The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Fiber 0 A
is a city located in the western portion of Tokyo Metropolis, Japan. , the city had an estimated population of 195,207 in 93,654 households, and a population density of 9500 persons per km². The total area of the city was . Geography Kodaira is located in the Musashino Terrace near the geographic centre of Tokyo Metropolis. Surrounding municipalities Tokyo Metropolis *Nishitokyo *Tachikawa *Higashimurayama * Higashiiyamato *Higashikurume * Kokubunji *Koganei Climate Kodaira has a Humid subtropical climate (Köppen ''Cfa'') characterized by warm summers and cool winters with light to no snowfall. The average annual temperature in Kodaira is 14.0 °C. The average annual rainfall is 1647 mm with September as the wettest month. The temperatures are highest on average in August, at around 25.5 °C, and lowest in January, at around 2.6 °C. Demographics Per Japanese census data, the population of Kodaira increased rapidly in the 1950s and 1960s. History The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine A0 Diagram
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine combination, a certain kind of constrained linear combination * Affine connection, a connection on the tangent bundle of a differentiable manifold * Affine Coordinate System, a coordinate system that can be viewed as a Cartesian coordinate system where the axes have been placed so that they are not necessarily orthogonal to each other. See tensor. * Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group * Affine gap penalty, the most widely used scoring function used for sequence alignment, especially in bioinformatics * Affine geometry, a geometry characterized by parallel lines * Affine group, the group of all invertible affine transformations from any affine space over a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynkin Diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' Dynkin diagrams will be explicitly so named. Classification of semisimple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Cartan Matrix
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan. Lie algebras A (symmetrizable) generalized Cartan matrix is a square matrix A = (a_) with integral entries such that # For diagonal entries, a_ = 2 . # For non-diagonal entries, a_ \leq 0 . # a_ = 0 if and only if a_ = 0 # A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix. For example, the Cartan matrix for ''G''2 can be decomposed as such: : \begin 2 & -3 \\ -1 & 2 \end = \begin 3&0\\ 0&1 \end\begin \frac & -1 \\ -1 & 2 \end. The third condition is not independent but is really a consequence of the first and fourth conditions. We can always choose a ''D'' with positive diagonal entries. In that case, if ''S'' in the above ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m \times n matrices, and is denoted by the symbol O or 0 followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are : 0_ = \begin 0 \end ,\ 0_ = \begin 0 & 0 \\ 0 & 0 \end ,\ 0_ = \begin 0 & 0 & 0 \\ 0 & 0 & 0 \end .\ Properties The set of m \times n matrices with entries in a ring K forms a ring K_. The zero matrix 0_ \, in K_ \, is the matrix with all entries equal to 0_K \, , where 0_K is the additive identity in K. : 0_ = \begin 0_K & 0_K & \cdots & 0_K \\ 0_K & 0_K & \cdots & 0_K \\ \vdots & \vdots & \ddots & \vdots \\ 0_K & 0_K & \cdots & 0_K \end_ The zero matrix is the additive identity in K_ \, . That is, for all A \in K_ \, it satisfies the equation :0_+A = A + 0_ = A. There is exactly one zero matrix of any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
André Néron
André Néron (November 30, 1922, La Clayette, France – April 6, 1985, Paris, France) was a French mathematician at the Université de Poitiers who worked on elliptic curves and abelian varieties. He discovered the Néron minimal model of an elliptic curve or abelian variety, the Néron differential, the Néron–Severi group, the Néron–Ogg–Shafarevich criterion, the local height and Néron–Tate height of rational points on an abelian variety over a discrete valuation ring or Dedekind domain, and classified the possible fibers of an elliptic fibration. Life and career He was a student of Albert Châtelet, and his PhD students were Jean-Louis Colliot-Thélène and Gérard Ligozat. He gave invited talks at the International Congress of Mathematicians in 1954 and 1966 . In 1983 the Académie des sciences awarded him the Émile Picard Medal The Émile Picard Medal (or Médaille Émile Picard) is a medal named for Émile Picard awarded every 6 years to an outstanding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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