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Euler Class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this. Throughout this article E is an oriented, real vector bundle of rank r over a base space X. Formal definition The Euler class e(E) is an element of the integral cohomology group :H^r(X; \mathbf), constructed as follows. An orientation of E amounts to a continuous choice of generator of the cohomology :H^r(\mathbf^, \mathbf^ \setminus \; \mathbf)\cong \tilde^(S^;\mathbf)\cong \mathbf of each fiber \mathbf^ relative to the complement \mathbf^ \setminus \ of zero. From the Thom isomorphism, this induces an orientation class :u \in H^r(E, E \setminus E_0; \mathbf) in the cohomology of E relative to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Instability
In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior. In structural engineering, a structural beam or column can become unstable when excessive compressive load is applied. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability. Atmospheric instability is a major component of all weather systems on Earth. Instability in control systems In the theory of dynamical systems, a state variable in a system is said to be unstable if it evolves without bounds. A system itself is said to be unstable if at least one of its state variables is unstable. In continuous time control theory, a system is unstable if any of the roots of its charac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Classifying Space For SO(n)
In mathematics, the classifying space \operatorname(n) for thspecial orthogonal group'' \operatorname(n) is the base space of the universal \operatorname(n) principal bundle \operatorname(n)\rightarrow\operatorname(n). This means that \operatorname(n) principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into \operatorname(n). The isomorphism is given by pullback. Definition There is a canonical inclusion of real oriented Grassmannians given by \widetilde\operatorname_n(\mathbb^k)\hookrightarrow\widetilde\operatorname_n(\mathbb^), V\mapsto V\times\. Its colimit is:Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151 : \operatorname(n) :=\widetilde\operatorname_n(\mathbb^\infty) :=\lim_\widetilde\operatorname_n(\mathbb^k). Since real oriented Grassmannians can be expressed as a homogeneous space by: : \widetilde\operatorname_n(\mathbb^k) =\operatorname(n+k)/(\operatorname(n)\times\operatorname( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Classifying Space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of ''G''. It has the property that any ''G'' principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG \to BG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group ''G'', ''BG'' is a path-connected topological space ''X'' such that the fundamental group of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Pontryagin Class
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundle E over M, its k-th Pontryagin class p_k(E) is defined as :p_k(E) = p_k(E, \Z) = (-1)^k c_(E\otimes \Complex) \in H^(M, \Z), where: *c_(E\otimes \Complex) denotes the 2k-th Chern class of the complexification E\otimes \Complex = E\oplus iE of E, *H^(M, \Z) is the 4k-cohomology group of M with integer coefficients. The rational Pontryagin class p_k(E, \Q) is defined to be the image of p_k(E) in H^(M, \Q), the 4k-cohomology group of M with rational coefficients. Properties The total Pontryagin class :p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\Z), is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e., :2p(E\oplus F)=2p(E)\smile p(F) for two vector bundles E and F over M. In terms of the individual Pon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Characteristic Number
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent to which the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Definition Let ''G'' be a topological group, and for a topological space X, write b_G(X) for the set of isomorphism classes of principal ''G''-bundles over X. This b_G is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map f\colon X\to Y to the pullback operation f^*\co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fundamental Class
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold. Definition Closed, orientable When ''M'' is a connected orientable closed manifold of dimension ''n'', the top homology group is infinite cyclic: H_n(M;\mathbf) \cong \mathbf, and an orientation is a choice of generator, a choice of isomorphism \mathbf \to H_n(M;\mathbf). The generator is called the fundamental class. If ''M'' is disconnected (but still orientable), a fundamental class is the direct sum of the fundamental classes for each connected component (corresponding to an orientation for each component). In relation with de Rham cohomology it represents ''integration over M''; namely for ''M'' a smooth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Self-intersection
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov–Witten theory and the extension of intersection theory from schemes to stacks. Topological intersection form For a connected oriented manifold M of dimension 2n the intersection form is defined on the n-th cohomology group (what is usually called the 'middle dimension') by the evaluation of the cup product on the fundamental class /math> in H_(M,\partial M). Stated precisely, there is a bilinear form :\lambda_M \colon H^n(M,\partial M) \times H^n(M,\partial M)\to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Normal Bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian manifold, and S \subset M a Riemannian submanifold. Define, for a given p \in S, a vector n \in \mathrm_p M to be '' normal'' to S whenever g(n,v)=0 for all v\in \mathrm_p S (so that n is orthogonal to \mathrm_p S). The set \mathrm_p S of all such n is then called the ''normal space'' to S at p. Just as the total space of the tangent bundle to a manifold is constructed from all tangent spaces to the manifold, the total space of the normal bundle \mathrm S to S is defined as :\mathrmS := \coprod_ \mathrm_p S. The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle. General definition More abstractly, given an immersion i: N \to M (for instance an em ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Poincaré Dual
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, ... * {{DEFAULTSORT:Poincare French-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |