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Logarithmic Transformation
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected Fiber (mathematics), fibers to an algebraic curve such that almost all fibers are smooth scheme, smooth curves of genus (mathematics), 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 theorem, 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 rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of implies that the homeomorphism type of the manifold only depends on this intersection form, and on a \Z/2\Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and ... [...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 Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), 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 graph, directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected graph, undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' ... [...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 integer 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 d ... [...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 ... [...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, Gérard Ligozat, and Dimitrios Poulakis. 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 eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate's Algorithm
In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve ''E'' over \mathbb, or more generally an algebraic number field, and a prime or prime ideal ''p''. It returns the exponent ''f''''p'' of ''p'' in the conductor of ''E'', the type of reduction at ''p'', the local index : c_p= (\mathbb_p):E^0(\mathbb_p) where E^0(\mathbb_p) is the group of \mathbb_p-points whose reduction mod ''p'' is a non-singular point. Also, the algorithm determines whether or not the given integral model is minimal at ''p'', and, if not, returns an integral model with integral coefficients for which the valuation at ''p'' of the discriminant is minimal. Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surfaces: in turn this determines the exponent ''f''''p'' of the conductor ''E''. Tate's algorithm can be greatly simplified if the characteristic of the residue class field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shioda Modular Surface
In mathematics, a Shioda modular surface is one of the elliptic surface 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 fi ...s studied by . References * * * Complex surfaces Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dolgachev Surface
In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds, no two of which are diffeomorphic. Properties The blowup X_0 of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface X_q is given by applying logarithmic transformations of orders 2 and ''q'' to two smooth fibers for some q\ge 3. The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature (1,9) (so it is the unimodular lattice I_). The geometric genus p_g is 0 and the Kodaira dimension is 1. found the first examples of simply-connected homeomorphic but not diffeomorphic 4-manifolds X_0 and X_3. More generally the surfaces X_q and X_r are always homeomorphic, but are not diffeomorphic unless q=r. showed that the Dolgachev surfac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Surface
In mathematics, a Kodaira surface is a compact 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 section, then quotient out the fibers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
