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Lagrangian Grassmannian
In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous space :, where is the unitary group and the orthogonal group. Following Vladimir Arnold it is denoted by Λ(''n''). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V. A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space ''V'' of dimension 2''n''. It may be identified with the homogeneous space of complex dimension ''n''(''n'' + 1) :, where is the compact symplectic group. Topology The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: \Omega(\mathrm/\mathrm U) \simeq \mathrm U/\mathrm O, and \Omega(\mathrm U/ \ ... [...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] |
Homology Group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Riemann Equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be holomorphic (complex differentiable). This system of equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851. The Cauchy–Riemann equations on a pair of real-valued functions of two real variables and are the two equations: Typically ''u'' and ''v'' are taken to be the real and imaginary parts respectively of a complex-valued function of a single complex variable , . Suppose that and are real-differentiable at a point in an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Flow
''Spectral'' is a 2016 3D military science fiction, supernatural horror fantasy and action-adventure thriller war film directed by Nic Mathieu. Written by himself, Ian Fried, and George Nolfi from a story by Fried and Mathieu. The film stars James Badge Dale, Max Martini, Emily Mortimer, Clayne Crawford and Bruce Greenwood. The film was released worldwide on December 9, 2016 on Netflix. On February 1, 2017, Netflix released a prequel graphic novel of the film called ''Spectral: Ghosts of War'' which was made available digitally through the website ComiXology. Plot DARPA researcher Dr. Mark Clyne flies from Virginia to Moldova, the current deployment location of the US military in the ongoing Moldovan War, to be consulted on one of his creations, a line of hyperspectral imaging goggles that have been issued to troops there. After arriving at a US military airbase on the outskirts of Chișinău, he meets with US Army General Orland and CIA officer Fran Madison. They show him fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Manifold
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for ' complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reeb Vector Field
In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ... including: * in a contact manifold, given a contact 1-form \alpha, the Reeb vector field satisfies R \in \mathrm\ d\alpha, \ \alpha (R) = 1 ,http://www2.im.uj.edu.pl/katedry/K.G/AutumnSchool/Monday.pdf * in particular, in the context of Sasakian manifold#The Reeb vector field. References * * Contact geometry {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Vector Field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics. Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions ''f'' and ''g'' on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of ''f'' and ''g''. Definition Suppose that is a symplectic ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Bundle
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The term "symplectic" is a calque of "complex" introduced by Hermann Weyl in 1939. In mathematics it may refer to: * Symplectic Clifford algebra, see Weyl algebra * Symplectic geometry * Symplectic group * Symplectic integrator * Symplectic manifold * Symplectic matrix * Symplectic representation * Symplectic vector space It can also refer to: * Symplectic bone, a bone found in fish skulls * Symplectite, in reference to a mineral intergrowth texture See also * Metaplectic group * Symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation. Formal definition A diffeomorphism between two symplectic manifolds f: (M,\omega) \rightarrow (N,\omega') is called a symplectomorphism if :f^*\omega'=\omega, where f^* is the pullback of f. The symplectic diffeomorphisms from M to M are a (pseudo-)group, called the symplectomorphism group (see below). The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field X \in \Gamma^(TM) is called symplectic if :\mathcal_X\omega=0. Also, X is symplectic iff the flow \phi_t: M\rightarrow M of X is a symplectomorphism for every t. These vector fields build a Lie subalgebra of \Gamma^(TM). Here, \Gamma^(TM) is the set of smooth vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p'', namely a normal vector to ''X'' at ''p''. The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator. Gauss first wrote a draft on the topic in 1825 and published in 1827. There is also a Gauss map for a link, which computes linking number. Generalizations The Gauss map can be defined for hypersurfaces in R''n'' as a map from a hypersurface to the unit sphere ''S''''n'' &min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
