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Maslov Index
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 homogeneous space, 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 symplectic group, compact symplectic group. As a homogeneous space To see that the Lagrangian Grassmannian Λ(''n'') can be identified with , note that \mathbb^n is a 2''n''-dimensional real vector space, with the imaginary part of its usual inner product making ... [...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] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Flow
''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Mark Clyne, with Max Martini, Emily Mortimer, Clayne Crawford, and Bruce Greenwood in supporting roles. The film is set in a civil war-ridden Moldova as invisible entities slaughter any living being caught in their path. 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 Mark Clyne is sent to a US military airbase on the outskirts of Chișinău, to consult his created line of hyperspectral imaging goggles issued to US Army Special Forces led by Army General James Orland, who is covertly supporting the Moldovan government in an ongoing civil war against ... [...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 (differential geometry), 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 'integrable system, 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 (differential topology), 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 clas ... [...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 including: * in a contact manifold, given a contact 1-form \alpha, the Reeb vector field satisfies R \in \mathrm\ d\alpha, \ \alpha (R) = 1 , * in particular, in the context of Sasakian manifold. If a contact manifold arises as a constant-energy hypersurface inside a symplectic manifold, then the Reeb vector field is the restriction to the submanifold of the Hamiltonian vector field associated to the energy function. (The restriction yields a vector field on the contact hypersurface because the Hamiltonian vector field preserves energy levels.) The dynamics of the Reeb field can be used to study the structure of the contact manifold or even the underlying manifold using techniques of Floer homology such as symplectic field theory and, in three dimensions, embedded contact homology. Different contact forms whose kernels give t ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Vector Field
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian matrix, a matrix with certain special properties commonly used in linear algebra * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of ... [...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 category * Symplectic Clifford algebra, see Weyl algebra * Symplectic geometry * Symplectic group, and corresponding symplectic Lie algebra * Symplectic integrator * Symplectic manifold * Symplectic matrix * Symplectic representation * Symplectic vector space, a vector space with a symplectic bilinear form 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 ... [...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 if 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 vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Map
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface ''X'' in Euclidean space R3, the Gauss map is a map ''N'': ''X'' → ''S''2 (where ''S''2 is the unit sphere) such that for each ''p'' in ''X'', the function value ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p''. The Gauss map is named after Carl F. Gauss. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the tangent space at a point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
