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# Topic: Vector bundle

 Vector - Wikipedia, the free encyclopedia Vector (spatial): In physics and engineering, vector most often refers specifically to an object that has a special relationship to the spatial coordinates/directions, that is, an element of a tangent bundle. A biological vector is a mechanism that transmits genes or organisms. Vector is the capital of the Empire in the videogame Final Fantasy VI. en.wikipedia.org /wiki/Vector   (427 words)

 PlanetMath: vector bundle A vector bundle is a fiber bundle having a vector space as a fiber and the general linear group of that vector space (or some subgroup) as structure group. Sepcifically, if we want a topological vector bundle, we must supply a topological space for the base space, a topological space for the whole space, and the projection map must be continuous; this specifies a topology on each fiber. In the algebraic category, that is, vector bundles over schemes, there is a very nice correspondence between vector bundles and locally free sheaves; when the dimension is one and the scheme is nice enough, there is a further correspondence with Cartier divisors. planetmath.org /encyclopedia/Section3.html   (624 words)

 Vector bundle   (Site not responding. Last check: 2007-11-05) In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vectorspaces, "glued together", form another topological space (or manifold or variety). A typical example is the tangent bundle of a differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point. Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π : E → X be a smooth map, and the local trivialization maps φ be diffeomorphisms. www.sciencedaily.com /encyclopedia/vector_bundle   (979 words)

 Vector bundle -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-05) The class of all vector bundles together with bundle morphisms forms a (A general concept that marks divisions or coordinations in a conceptual scheme) category. Two vector bundles on X, over the same field, have a Whitney sum, with fibre at any point the (A union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets) direct sum of fibres. Vector bundles are special (A bundle of fibers (especially nerve fibers)) fiber bundles, loosely speaking those where the fibers are vector spaces. www.absoluteastronomy.com /encyclopedia/v/ve/vector_bundle.htm   (1345 words)

 Fiber bundle - Wikipedia, the free encyclopedia Fiber bundles generalize vector bundles, where the main example is the tangent bundle of a manifold. A sphere bundle is a fiber bundle whose fiber is an n-sphere. Given a vector bundle E with a metric (such as the tangent bundle to a Riemannian manifold) one can construct the associated unit sphere bundle, for which the fiber over a point x is the set of all unit vectors in E en.wikipedia.org /wiki/Fiber_bundle   (1166 words)

 PlanetMath: tangent bundle forgetting the tangent vector and remembering the point, is a vector bundle. Cross-references: fibers, base, obvious, section, vector field, vector bundle, tangent vector, projection, differentiable, bijective, derivative, map, diffeomorphism, neighborhood, isomorphic, structure, tangent spaces, disjoint union, differentiable manifold This is version 2 of tangent bundle, born on 2003-10-06, modified 2003-10-06. planetmath.org /encyclopedia/TangentBundle.html   (137 words)

 Cotangent bundle - Wikipedia, the free encyclopedia Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out. The one-form assigns to a vector in the tangent bundle of the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold). The cylinder is the cotangent bundle of the circle. en.wikipedia.org /wiki/Cotangent_bundle   (480 words)

 Tensor field - Encyclopedia, History and Biography There is the idea of vector bundle, which is a natural idea of 'vector space depending on parameters' — the parameters being in a manifold. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion. The other vector bundles of tensors have comparable cocycles, which come from applying functorial properties of tensor constructions to the chain rule itself; this is why they also are intrinsic (read, 'natural') concepts. www.arikah.net /encyclopedia/Tensor_field   (1231 words)

 Fiber bundle In mathematics, in particular topology, a fiber bundle is a continuous surjective map, π from a topological space E to another topological space B, satisfying a further condition making it locally of a particularly simple form. For example in the case of a vector bundle, F is a vector space over the real numbers. An example of a principal bundle that occurs naturally in geometry is the bundle of all bases for the tangent space to a manifold, with G the general linear group ; restricting in Riemannian geometry to orthonormal bases, one would limit G to the orthogonal group. www.nebulasearch.com /encyclopedia/article/Fiber_bundle.html   (573 words)

 Vector article - Vector Latin mathematics element vector space field Vector (spatial) - What-Means.com   (Site not responding. Last check: 2007-11-05) Vector (spatial): In physics and engineering, vector most often refers specifically to an object that has a special relationship to the spatial coordinates/directions, i.e. Related concepts include: four-vector (the generalization to space and time in relativity), pseudovector, vector calculus, vector bundle, unit vector, null vector and normal vector. In biology, a vector is a mechanism that transmits genes or organisms. www.what-means.com /encyclopedia/Vector   (307 words)

 Vector bundle - InfoSearchPoint.com   (Site not responding. Last check: 2007-11-05) Vector bundles of rank 1 are called line bundles. Vector bundles are special fiber bundles, loosely speaking those where the fibers are vector spaces. Vector spaces over other topological fields may also be used, but that is comparatively rare. www.infosearchpoint.com /display/Vector_bundle   (914 words)

 Encyclopedia: Connection (fiber bundle)   (Site not responding. Last check: 2007-11-05) In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. That is an application to tangent bundles; there are more general connections, used in differential geometry and other fields of mathematics to formulate intrinsic differential equations. A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act as differential operators on vector bundle sections. www.nationmaster.com /encyclopedia/Connection-(fiber-bundle)   (515 words)

 Vector bundle moduli, small instanton transitions and nonperturbative superpotentials in heterotic M-theory. Vector bundle moduli, small instanton transitions and nonperturbative superpotentials in heterotic M-theory. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. repository.upenn.edu /dissertations/AAI3095862   (252 words)

 [No title] In other words a combinatorial vector bundle (B; M) is a matroid bundle (F(B0); M) where F(B0) is the poset of cells of a PL subdivision B0 of B. Note every regular cell complex can be given the structure of a PL space via a barycentric subdivision. B) is either a vector bundle with the 0-section deleted, a combinatorial sphere bundle, or a spherical (quasi)- fibration. The evaluation of the Euler class of the tangent bundle of a manifold on the fundamental class of that manifold is the Euler characteristic of the manifold (cf. hopf.math.purdue.edu /Anderson-DavisJ/MacPherson.txt   (10614 words)

 [No title] This means that a decomposition of a vector bundle in indecomposable components exists and is unique up to isomophism.\\ \indent We want to generalize a theorem of M. Nori on finite vector bundles. The vector bundle $E\cong L\otimes F_2$ trivializes on the $\G_m \times \G_a$-bundle $P_L \times_X P$, where $P$ is the principal $\G_a$-bundle from proposition 2, where $F_2$ and hence all the $F_r$ trivialize.\\ Let now L be torsion and $n\in \N$, $n\ge 2$, the minimal number with $L^{\otimes n}\cong{\mathcal O}_X$. We illustrate this for bundles of rank two.\\ Let now $E$ be a vector bundle of rank 2 on ${\P}^1$, $E={\mathcal O}(a)\oplus {\mathcal O}(b)$.\\ The case $(a,b)=(0,0)$ is trivial. www.uni-essen.de /~mat903/preprints/lekaus.tex   (1340 words)

 Tangent bundle Definition / Tangent bundle Research   (Site not responding. Last check: 2007-11-05) In mathematics, the tangent bundle of a manifoldIn mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. [click for more] is a vector bundleIn mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topological space (or manifold or variety). tangent bundle is the surface of an infinitely. www.elresearch.com /Tangent_bundle   (431 words)

 Which Equation Explains the Market Cycle ? best Associated Vector Bundle   (Site not responding. Last check: 2007-11-05) Vector bundle representations of groups in quantum physics Vector bundle representations of groups in quantum physics The theory of vector bundle representations of Lie groups G is developed. Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory The non-perturbative superpotential generated by a heterotic superstring wrapped once... Associated quantum vector bundles and symplectic structure on a quantum plane Associated quantum vector bundles and symplectic structure on a quantum plane We define a quantum generalization of the algebra of functions over an associated... ascot.pl /th/Fourier1/Associated-Vector-Bundle.htm   (628 words)

 8.6 Vector Bundles and regular schemes   (Site not responding. Last check: 2007-11-05) Thus L is a vector bundle of rank 1 or a line bundle. Note that any vector bundle is a vector space scheme and an exact sequence of vector bundles is also an exact sequence of vector space schemes. As a particular case we have the Jacobian criterion'' which says that a scheme is regular if the Zariski tangent vector space scheme is a vector bundle; note however that this is not in general necessary. www.imsc.ernet.in /~kapil/crypto/notes/node44.html   (443 words)

 Vector bundle on projective spaces   (Site not responding. Last check: 2007-11-05) In the theorey of vector bundles on projective spaces, it is an important problem to find indecomposable vector bundles of small rank. One of the most powerful tools for constucting or classifying vector bundles is the technique of monad. The basic idea of monad construction is to represent a given bundle in terms of simpler bundles, such as bundles of differentials or direct sums of line bundles, and morphisms between these simpler bundles. www.math.colostate.edu /~abo/Research/bundle.html   (197 words)

 talks in UC Berkeley   (Site not responding. Last check: 2007-11-05) This conjecture is closely related to the question of whether there are vector bundles of small rank on the projective space, which are not direct sum of line bundles. For a given vector bundle, the theorem of Quillen and Suslin guarantees us the existence of sections that generate the vector bundle on the complement of a hyperplane. Kumar gave necessary and sufficient conditions for a vector bundle on a hyperplane of projective space to be obtained from a vector bundle on the projective space in this way. www.math.colostate.edu /~abo/Research/Talks/berkeley.html   (348 words)

 line_bundle   (Site not responding. Last check: 2007-11-05) A line bundle is a special case of vector bundle with dim(F)=1. Ex: a wave function in quantum mechanics is a section of a line bundle. For a magnetic monopole, remove of a point from M gives twisting of the bundle. www.cs.cmu.edu /People/jcl/classnotes/math/geometry/line_bundle/line_bundle.html   (50 words)

 Vector Power Inverters   (Site not responding. Last check: 2007-11-05) Vector (spatial) : In physics and engineering, vector most often refersspecifically to an object that has a special relationship to the spatial coordinates/directions, i.e. If the vector space is finite-dimensional, its vectors arecommonly denoted by matrices with dimensions n ×1(column vector) or 1× n (row vector). Vector (computing) is the method that malicious code(viruses, etc) uses to propagate itself. www.witchware.com /File/40724-Vector.Power.Inverters.Html   (412 words)

 [No title]   (Site not responding. Last check: 2007-11-05) To see how the structure arises, first note that the vector bundle in R^n is R^2n, since there is a canonical isomorphism between all the tangent spaces. Also, a vector field on M is a differentiable mapping > v:M->TM which satisfies P(v(p)) = p, where P is the projection from TM onto > M. The tangent bundle on a smooth manifold is just one of many examples of vector bundles. Essentially a vector bundle over a topolgical space X is an assignement of a vector space V_x to each point x of X such that the V_x "vary continously". www.math.niu.edu /~rusin/known-math/99/vec_bundle   (654 words)

 Ample vector bundle -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-05) An ample line bundle is one which becomes very ample after it is raised to some tensor power, i.e. The relationship with projective space is that the for a very ample will correspond to the hyperplane sections (intersection with some (Click link for more info and facts about hyperplane) hyperplane) of the embedded. There is a more general theory of ample (Click link for more info and facts about vector bundle) vector bundles. www.absoluteastronomy.com /encyclopedia/A/Am/Ample_vector_bundle.htm   (250 words)

 Relation Between The Dimensions Of The Ring Generated By A Vector Bundle Of Degree Zero On An Elliptic Curve And A ...   (Site not responding. Last check: 2007-11-05) Relation Between The Dimensions Of The Ring Generated By A Vector Bundle Of Degree Zero On An Elliptic Curve And A Torsor Trivializing This Bundle Abstract: this article we consider the family of vector bundles of degree zero on an elliptic curve. Dimension relation for vector bundles of degree zero... citeseer.ist.psu.edu /434714.html   (289 words)

 Meningar.com om cotangent. bundle, space, manifold mm. All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the cotangent bundle In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotan.. The cotangent bundle as phase space Symplectic form The cotangent bundle has a canonical symplectic 2-form In mathematics, in particular in abstract formulations of classical mechanics and analytical mechanics, a symplectic manifold is a smooth manifold.. "...The one-form assigns to a vector in the tangent bundle of the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the.. www.meningar.com /cotangent.html   (1366 words)

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