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Parabolic Geometry (other)
Parabolic geometry may refer to: * Parabolic geometry, former name for Euclidean geometry, a comprehensive and deductive mathematical system * Parabolic geometry (differential geometry): The homogeneous space defined by a semisimple Lie group modulo a parabolic subgroup, or the curved analog of such a space * Cartan parabolic geometry, geometry induced by Cartan connection#Parabolic Cartan connections, parabolic Cartan inclusions {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a metric tensor, Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be Smoothness, smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz continuity, Lipschitz Riemannian metrics or Measurable function, measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential operator, then * the Green's function G is the solution of the equation \operatorname G = \delta, where \delta is Dirac's delta function; * the solution of the initial-value problem \operatorname y = f is the convolution (G \ast f). Through the superposition principle, given a linear ordinary differential equation (ODE), \operatorname y = f, one can first solve \operatorname G = \delta_s, for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Geometry (differential Geometry)
In differential geometry and the study of Lie groups, a parabolic geometry is a homogeneous space ''G''/''P'' which is the quotient of a semisimple Lie group ''G'' by a parabolic subgroup ''P''. More generally, the curved analogs of a parabolic geometry in this sense is also called a parabolic geometry: any geometry that is modeled on such a space by means of a Cartan connection. Examples The projective space ''P''''n'' is an example. It is the homogeneous space PGL(''n''+1)/''H'' where ''H'' is the isotropy group of a line. In this geometrical space, the notion of a straight line is meaningful, but there is no preferred ("affine") parameter along the lines. The curved analog of projective space is a manifold in which the notion of a geodesic makes sense, but for which there are no preferred parametrizations on those geodesics. A projective connection is the relevant Cartan connection that gives a means for describing a projective geometry by gluing copies of the projective s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Parabolic Geometry
Cartan may refer to: * Élie Cartan (1869–1951), French mathematician who worked with Lie groups * Henri Cartan (1904-2008), French mathematician who worked in algebraic topology, son of Élie Cartan *Anna Cartan (1878-1923), French mathematician and teacher, sister of Élie Cartan * Cartan (crater), a lunar crater named for Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ... * Badea Cârțan (1849-1911), Austro-Hungarian Romanian activist {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
