the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits...
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particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general...
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neighbourhood. Linear approximation Stable manifold theorem Arrowsmith, D. K.; Place, C. M. (1992). "The Linearization Theorem". Dynamical Systems: Differential...
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direction defines the stable manifold, the stretching direction defining the unstable manifold, and the neutral direction is the center manifold. While geometrically...
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Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical...
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topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the...
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manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold,...
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equilibrium point (x*, y*) is called the community matrix. By the stable manifold theorem, if one or both eigenvalues of A {\displaystyle \mathbf {A} } have...
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In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...
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Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving...
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separability of the classes, and measures of geometry, topology, and density of manifolds. For non-binary classification problems, instance hardness is a bottom-up...
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non-embedded) manifold with a given stable trivialisation of the tangent bundle. A related notion is the concept of a π-manifold. A smooth manifold M {\displaystyle...
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Nonabelian Hodge correspondence (redirect from Nonabelian Hodge theorem)
Kähler manifold. The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem which defines a correspondence between stable vector...
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Richard S. Hamilton (section Nash–Moser theorem)
heat flow, using a convergence theorem for the flow to show that any smooth map from a closed manifold to a closed manifold of nonpositive curvature can...
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opposite in sign. Hence by the stable manifold theorem, the equilibrium is a saddle point and there exists a unique stable arm, or “saddle path”, that converges...
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a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data...
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mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable...
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the annulus theorem in dimensions n ≥ 5 {\displaystyle n\geq 5} . It was also employed in further investigations of topological manifolds with Laurent...
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correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence...
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The most common is the stable manifold or its kin, the unstable manifold. Ushiki's theorem was published in 1980. The theorem appeared in print again...
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version of the transversality theorem. Let f : X → Y {\displaystyle f\colon X\rightarrow Y} be a smooth map between smooth manifolds, and let Z {\displaystyle...
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} The stable equivalence is an equivalence relation on semialgebraic subsets generated by stable projections and rational equivalence. Theorem (Mnëv's...
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eigenvalues of the community matrix have negative real part, then by the stable manifold theorem the system converges to a limit point. Since the determinant is...
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Curve-shortening flow (redirect from Gage–Hamilton–Grayson theorem)
1093/imanum/drw020, MR 3649420. Epstein, C. L.; Weinstein, M. I. (1987), "A stable manifold theorem for the curve shortening equation", Communications on Pure and...
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Thom space (redirect from Thom isomorphism theorem)
intimately related to the transversality properties of smooth manifolds—see Thom transversality theorem. By reversing this construction, John Milnor and Sergei...
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formula for the area of a stable minimal hypersurface of a three-dimensional Riemannian manifold. The Gauss–Bonnet theorem then highly constrains the...
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Differential geometry (redirect from Analysis of manifolds)
Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global...
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Structural stability (redirect from Structurally stable)
equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms. Structurally stable systems were introduced by Aleksandr...
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studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness...
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Limit cycle (redirect from Stable limit cycle)
equations. Attractor Hyperbolic set Periodic point Self-oscillation Stable manifold Thomas, Jeffrey P.; Dowell, Earl H.; Hall, Kenneth C. (2002), "Nonlinear...
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