We give an alternative proof of the stable manifold theorem as an application of the (right and left) inverse mapping theorem on a space of sequences. We investigate on the diffeomorphism class of the global stable manifold, a problem which in the general Banach setting gives rise to subtle questions about the possibility of extending germs of diffeomorphisms. AMS Subject Classification: 37D10,...

We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.

We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem fields spaces is proved and applied to linear SDDEs. In Part II this article, we use our results prove stable manifold theorem for non-linear

*Correspondence: [email protected]; [email protected]; [email protected] 1Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China 2School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland Abstract In this paper, we investigate the existence of local center stable manifolds of Langevin differential equations wi...

Among the many problems attendant to the discovery of exotic generalized manifolds [2, 3] is the \normal bundle" problem, that is, the classi cation of neighborhoods of generalized manifolds tamely embedded in generalized manifolds (with the disjoint disks property). In this paper we study the normal structure of tame embeddings of a closed generalized manifold X into topological manifolds V n+...

We introduce a new technique for proving the classical Stable Manifold theorem for hyperbolic fixed points. This method is much more geometrical than the standard approaches which rely on abstract fixed point theorems. It is based on the convergence of a canonical sequence of “finite time local stable manifolds” which are related to the dynamics of a finite number of iterations.