Cocycle

	Cocycle In mathematics a cocycle is a closed cochain (algebraic topology), cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous system (mathematics), autonomous dynamical systems, cocycles are used to describe particular kinds of map, as in Oseledets theorem. Definition Algebraic Topology Let ''X'' be a CW complex and C^n(X) be the singular Chain complex, cochains with coboundary map d^n: C^(X) \to C^n(X). Then elements of \textd are cocycles. Elements of \text d are coboundaries. If \varphi is a cocycle, then d \circ \varphi = \varphi \circ \partial =0 , which means cocycles vanish on boundaries. See also * Čech cohomology * Cocycle condition References Algebraic topology Cohomology theories Dynamical systems {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Group Cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''- simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Oseledets Theorem In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 1966. A conceptually different proof of the multiplicative ergodic theorem was found by M. S. Raghunathan. The theorem has been extended to semisimple Lie groups by V. A. Kaimanovich and further generalized in the works of David Ruelle, Grigory Margulis, Anders Karlsson, and François Ledrappier. Cocycles The multiplicative ergodic theorem is stated in terms of matrix cocycles of a dynamical system. The theorem states conditions for the existence of the defining limits and describes the Lyapunov exponents. It does not address the rate of convergence. A cocycle of an autonomous dynamical system ''X'' is a map ''C'' : ''X×T'' → R''n×n'' satisfying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Čech Cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topological space, and let \mathcal be an open cover of ''X''. Let N(\mathcal) denote the nerve of a covering, nerve of the covering. The idea of Čech cohomology is that, for an open cover \mathcal consisting of sufficiently small open sets, the resulting simplicial complex N(\mathcal) should be a good combinatorial model for the space ''X''. For such a cover, the Čech cohomology of ''X'' is defined to be the simplicial homology, simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of ''X'', ordered by Open cover#Refinement, refinement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cocycle Condition In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous dynamical systems, cocycles are used to describe particular kinds of map, as in Oseledets theorem. Definition Algebraic Topology Let ''X'' be a CW complex and C^n(X) be the singular cochains with coboundary map d^n: C^(X) \to C^n(X). Then elements of \textd are cocycles. Elements of \text d are coboundaries. If \varphi is a cocycle, then d \circ \varphi = \varphi \circ \partial =0 , which means cocycles vanish on boundaries. See also * Čech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard Čech. Moti ... * Cocycle condition Refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cochain (algebraic Topology) In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Algebraic Topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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]
	Closed Manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only Connected space, connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP''n'' is a closed ''n''-dimensional manifold. The complex projective space CP''n'' is a closed 2''n''-dimensional manifold. A Real line, line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If M is a closed connected n-manifold, the n-th homology group H_(M;\mathbb) is \mathbb or 0 depending on whether M is Orientability, orientable or not. Moreover, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Autonomous System (mathematics) In mathematics, an autonomous system or autonomous differential equation is a simultaneous equations, system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the Physical law, laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]