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Calculus on Manifolds
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Although both the pushforward and pullback will always be defined, they may not be invertible. However, if one of them is invertible, so is the other. Any map f that induces an invertible pushforward and pullback is called an immersion of M into N. If, in addition, f itself is invertible, then it is an embedding of M in N. For any embedding, the image f(M) in N is called a submanifold of N.
A new key performance indicator oriented industrial process monitoring and operating performance assessment method based on improved Hessian locally linear embedding
Published in International Journal of Systems Science, 2022
Hongjun Zhang, Chi Zhang, Jie Dong, Kaixiang Peng
The theoretical framework of HLLE is developed on the quadratic form defined on the mapping . denotes the Hessian matrix of f. is also the averaged Frobenius norm over M. If the manifold M, as the Riemannian submanifold on space , is locally isometric to an connected open subset on the Euclidean space , then will have the -dimensional null space. Since is not required to be a convex set here, this framework is more available than ISOMAP in dealing with manifolds with different curvatures.
Estimates for the volume variation of compact submanifolds driven by a stochastic flow
Published in Dynamical Systems, 2022
Diego Sebastian Ledesma, Robert Andres Galeano Anaya, Fabiano Borges da Silva
Still in this context, but now with a different approach, which inspired this work, Kinateder and McDonald (cf. [20]) then present an Itô formula for , where is a smooth map that interprets the behaviour of some geometric parameter, and is obtained by the action of a stochastic flow on an initial domain . Here is the set of domains with smooth boundary in with compact closure, which is a Fréchet manifold (i.e. a manifold locally modelled on a Fréchet space). In this article, we extend this Itô formula to the volume () of a random compact submanifold over a differentiable manifold with a Riemannian metric h, and as a main result, we get estimates for the variation of this volume in terms of Ricci curvature.
Cohomological equations for linear involutions
Published in Dynamical Systems, 2021
Erwan Lanneau, Stefano Marmi, Alexandra Skripchenko
The notion of the Roth-type IET (interval exchange transformations) was slightly modified in [14] and adapted to the question of the conjugacy of generalized (affine) interval exchange transformations and obstructions for the linearization. The set of considered IETs is called restricted Roth type and has a full Lebesgue measure in the parameter space. For any IET from this class with s>0 singularities. For a given we consider -deformation of that are tangent to at the singularities. It was shown that those that are conjugated to by a -diffeomorphism close to the identity form a -submanifold of codimension .