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List Of Things Named After Henri Poincaré
In physics and mathematics, a number of ideas are named after Henri Poincaré: * Euler–Poincaré characteristic * Hilbert–Poincaré series * Poincaré–Bendixson theorem * Poincaré–Birkhoff theorem * Poincaré–Birkhoff–Witt theorem, usually known as the PBW theorem * Poincaré algebra ** K-Poincaré algebra **Super-Poincaré algebra * Poincaré–Bjerknes circulation theorem * Poincaré complex * Poincaré conjecture, one of the Millennium Prize Problems ** Generalized Poincaré conjecture * Poincaré disk model, a model of hyperbolic geometry * Poincaré duality **Twisted Poincaré duality * Poincaré–Einstein synchronization * Poincaré expansion * Poincaré group, the group of isometries of Minkowski spacetime, named in honour of Henri Poincaré ** K-Poincaré group * Poincaré half-plane model, a model of two-dimensional hyperbolic geometry * Poincaré homology sphere * Poincaré–Hopf theorem * Poincaré inequality ** Poincaré–Wirtinger inequality * Poinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Poincaré Duality
In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system. Twisted Poincaré duality for de Rham cohomology Another version of the theorem with real coefficients features de Rham cohomology with values in the orientation bundle. This is the flat real line bundle denoted o(M), that is trivialized by coordinate charts of the manifold M, with transition maps the sign of the Jacobian determinant of the charts transition maps. As a flat line bundle, it has a de Rham cohomology, denoted by :H^* (M; \R^w) or H^* (M; o(M)). For ''M'' a compact manifold, the top degree cohomology is equipped with a so-called trace morphism :\theta\colon H^d (M; o(M)) \to \R, that is to be interpreted as integration on ''M'', ''i.e.'', evaluating against the fundamental class. Poincaré duality for different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Abelian Variety
In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''K''. Definition To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a ''k''-variety ''T'' is defined to be a line bundle ''L'' on ''A''×''T'' such that # for all t \in T, the restriction of ''L'' to ''A''× is a degree 0 line bundle, # the restriction of ''L'' to ×''T'' is a trivial line bundle (here 0 is the identity of ''A''). Then there is a variety ''A''v and a line bundle P \to A \times A^\vee,, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by ''A''v in the sense of the above definition. Moreover, this family is universal, that is, to any family ''L'' parametrized by ''T'' is associated a unique morphism ''f'': ''T'' → ''A''v so that ''L'' is isomorphic to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré–Lindstedt Method
In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method removes secular terms—terms growing without bound—arising in the straightforward application of perturbation theory to weakly nonlinear problems with finite oscillatory solutions. The method is named after Henri Poincaré, and Anders Lindstedt. Example: the Duffing equation The undamped, unforced Duffing equation is given by :\ddot + x + \varepsilon\, x^3 = 0\, for ''t'' > 0, with 0 < ''ε'' ≪ 1.J. David Logan. ''Applied Mathematics'', Second Edition, John Wiley & Sons, 1997. . Consider initial conditions : A perturbation-series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Duality
In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by , at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality. Formulations Let ''M'' be an orientable compact manifold of dimension ''n'', with boundary \partial(M), and let z\in H_n(M,\partial(M); \Z) be the fundamental class of the manifold ''M''. Then cap product with ''z'' (or its dual class in cohomology) induces a pairing of the (co)homology groups of ''M'' and the relative (co)homology of the pair (M,\partial(M)). Furthermore, this gives rise to isomorphisms of H^k(M,\partial(M); \Z) with H_(M; \Z), and of H_k(M,\partial(M); \Z) with H^(M; \Z) for all k. Here \partial(M) can in fact be empty, so Poincaré duality appears as a special case of Lefs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Lemma
In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another differential form ''β''. Thus, an ''exact'' form is in the ''image'' of ''d'', and a ''closed'' form is in the ''kernel'' of ''d''. For an exact form ''α'', for some differential form ''β'' of degree one less than that of ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since the exterior derivative of a closed form is zero, ''β'' is not unique, but can be modified by the addition of any closed form of degree one less than that of ''α''. Because , every exact form is necessarily closed. The question of whether ''every'' closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this kind on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré–Lelong Equation
In mathematics, the Poincaré–Lelong equation, studied by , is the partial differential equation :i\partial\overline\partial u=\rho on a Kähler manifold, where ρ is a positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ... (1,1)-form. References * * Complex manifolds Partial differential equations {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Inequality
In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is Friedrichs' inequality. Statement of the inequality The classical Poincaré inequality Let ''p'', so that 1 ≤ ''p'' < ∞ and Ω a subset bounded at least in one direction. Then there exists a constant ''C'', depending only on Ω and ''p'', so that, for every function ''u'' of the ''W''01,''p''(Ω) of zero-trace (a.k.a. zero on the boundary) functions, : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré–Hopf Theorem
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case of the hairy ball theorem, which simply states that there is no smooth vector field on an even-dimensional n-sphere having no sources or sinks. Formal statement Let M be a differentiable manifold, of dimension n, and v a vector field on M. Suppose that x is an isolated zero of v, and fix some local coordinates near x. Pick a closed ball D centered at x, so that x is the only zero of v in D. Then the index of v at x, \operatorname_x(v), can be defined as the degree of the map u : \partial D \to \mathbb S^ from the boundary of D to the (n-1)-sphere given by u(z)=v(z)/\, v(z)\, . Theorem. Let M be a compact differentiable manifold. Let v be a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Homology Sphere
Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Lucien Poincaré (1862–1920), physicist, brother of Raymond and cousin of Henri * Raymond Poincaré (1860–1934), French Prime Minister or President ''inter alia'' from 1913 to 1920, cousin of Henri See also *List of things named after Henri Poincaré In physics and mathematics, a number of ideas are named after Henri Poincaré: * Euler–Poincaré characteristic * Hilbert–Poincaré series * Poincaré–Bendixson theorem * Poincaré–Birkhoff theorem * Poincaré–Birkhoff–Witt theorem, usu .... * * {{DEFAULTSORT:Poincare French-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Half-plane Model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the ''y'' coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami who used it, along with the Klein model and the Poincaré disk model, to show that hyperbolic geometry was equiconsistent with Euclidean geometry. This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane. The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model. This model can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number ''x'' by a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-Poincaré Group
In physics and mathematics, the κ-Poincaré group, named after Henri Poincaré, is a quantum group, obtained by deformation of the Poincaré group into a Hopf algebra. It is generated by the elements a^\mu and _\nu with the usual constraint: : \eta^ _\rho _\sigma = \eta^ ~, where \eta^ is the Minkowskian metric: : \eta^ = \left(\begin -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end \right) ~. The commutation rules reads: * _j ,a_0= i \lambda a_j ~, \; _j,a_k0 * ^\mu , _\sigma = i \lambda \left\ In the (1 + 1)-dimensional case the commutation rules between a^\mu and _\nu are particularly simple. The Lorentz generator in this case is: : _\nu = \left( \begin \cosh \tau & \sinh \tau \\ \sinh \tau & \cosh \tau \end \right) and the commutation rules reads: * a_0 , \left( \begin \cosh \tau \\ \sinh \tau \end \right) = i \lambda ~ \sinh \tau \left( \begin \sinh \tau \\ \cosh \tau \end \right) * a_1 , \left( \begin \cosh \tau \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
