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Complex Torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', where ''n'' is the complex dimension of ''M''. All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to C''n'' considered as real vector space; then the quotient group V/\Lambda is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For ''n'' = 1 this is the classical period lattice construction of elliptic curves. For ''n'' > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when ''n'' > 1, and are re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Torsion Points
Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornamental pattern of crossing strips of pastry Companies * Lattice Engines, a technology company specializing in business applications for marketing and sales * Lattice Group, a former British gas transmission business * Lattice Semiconductor, a US-based integrated circuit manufacturer Science, technology, and mathematics Mathematics * Lattice (group), a repeating arrangement of points ** Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure ** Lattice (module), a module over a ring which is embedded in a vector space over a field ** Lattice graph, a graph that can be drawn within a repeating arrangement of points ** Lattice-based cryptography, encryption systems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equations Defining Abelian Varieties
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension ''d'' ≥ 2, however, it is no longer as straightforward to discuss such equations. There is a large classical literature on this question, which in a reformulation is, for complex algebraic geometry, a question of describing relations between theta functions. The modern geometric treatment now refers to some basic papers of David Mumford, from 1966 to 1967, which reformulated that theory in terms from abstract algebraic geometry valid over general fields. Complete intersections The only 'easy' cases are those for ''d'' = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In ''P''3, an elliptic curve can be obtained as the intersection of two quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''- simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Of Abelian Varieties
Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space \mathcal_ over characteristic 0 constructed as a quotient of the upper-half plane by the action of SL_2(\mathbb), there is an analogous construction for abelian varieties \mathcal_g using the Siegel upper half-space and the symplectic group \operatorname_(\mathbb). Constructions over characteristic 0 Principally polarized Abelian varieties Recall that the Siegel upper-half plane is given byH_g = \ \subseteq \operatorname_g(\mathbb)which is an open subset in the g\times g symmetric matrices (since \operatorname(\Omega) > 0 is an open subset of \mathbb, and \operatorname is continuous). Notice if g=1 this gives 1\times 1 matrices with positive imaginary part, hence this set is a generalization of the upper half plane. Then any point \Omega \in H_g gives a complex torus X_\Omega = \mathbb^g/(\Omega\mathbb^g + \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Coset Space
A double is a look-alike or doppelgänger; one person or being that resembles another. Double, The Double or Dubble may also refer to: Film and television * Double (filmmaking), someone who substitutes for the credited actor of a character * ''The Double'' (1934 film), a German crime comedy film * ''The Double'' (1971 film), an Italian film * ''The Double'' (2011 film), a spy thriller film * ''The Double'' (2013 film), a film based on the Dostoevsky novella * '' Kamen Rider Double'', a 2009–10 Japanese television series ** Kamen Rider Double (character), the protagonist in a Japanese television series of the same name Food and drink * Doppio, a double shot of espresso * Dubbel, a strong Belgian Trappist beer or, more generally, a strong brown ale * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Dou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Bilinear Relations
In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data: * A lattice Λ in a complex vector space Cg. * An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations: # the real linear extension αR:Cg × Cg→R of α satisfies αR(''iv'', ''iw'')=αR(''v'', ''w'') for all (''v'', ''w'') in Cg × Cg; # the associated hermitian form ''H''(''v'', ''w'')=αR(''iv'', ''w'') + ''i''αR(''v'', ''w'') is positive-definite. (The hermitian form written here is linear in the first variable.) Riemann forms are important because of the following: * The alternatization of the Chern class of any factor of automorphy In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. Facto ... is a Riemann form. * Conversely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonsingular
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry And Analytic Geometry
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Main statement Let ''X'' be a projective complex algebraic variety. Because ''X'' is a complex variety, its set of complex points ''X''(C) can be given the structure of a compact complex analytic space. This analytic space is denoted ''X''an. Similarly, if \mathcal is a sheaf on ''X'', then there is a corresponding sheaf \mathcal^\text on ''X''an. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating ''X'' and ''X''an says that for any t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
