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...

the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits...

direction defines the stable manifold, the stretching direction defining the unstable manifold, and the neutral direction is the center manifold. While geometrically...

separability of the classes, and measures of geometry, topology, and density of manifolds. For non-binary classification problems, instance hardness is a bottom-up...

neighbourhood. Linear approximation Stable manifold theorem Arrowsmith, D. K.; Place, C. M. (1992). "The Linearization Theorem". Dynamical Systems: Differential...

manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold,...

self-dual. Every weakly stable Yang–Mills field over a compact orientable homogenous Riemannian 4 {\displaystyle 4} -manifold with gauge group SU ⁡ (...

Kähler manifold. The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem which defines a correspondence between stable vector...

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...

Rokhlin's theorem that the signature of a compact smooth spin 4-manifold is divisible by 16. Stable homotopy groups of spheres are used to describe the group...

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...

Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving...

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...

correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence...

construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the...

4-manifolds, complex differential geometry and symplectic geometry. The following theorems have been mentioned:[by whom?] The diagonalizability theorem...

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...

certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It is called hyperbolic in analogy with the linear theory...

mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable...

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...

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...

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...

equations. Attractor Hyperbolic set Periodic point Self-oscillation Stable manifold Thomas, Jeffrey P.; Dowell, Earl H.; Hall, Kenneth C. (2002), "Nonlinear...

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...

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...

theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which...

Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global...

formula for the area of a stable minimal hypersurface of a three-dimensional Riemannian manifold. The Gauss–Bonnet theorem then highly constrains the...

satisfying a transversality condition on the stable and unstable manifolds. Morse–Smale systems are structurally stable and form one of the simplest and best...

Gauss-Bonnet theorem, Schoen and Yau were able to rule out the existence of several types of stable minimal surfaces in three-dimensional manifolds of positive...