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Pseudo-differential operators
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
If V is a vector bundle over Rm, we can always choose a global trivialization of V; this is just a bundle isomorphism V≃1k. However, S(V) depends on the global trivialization chosen. If f→ϵS(1k), let () ||f→||s2:=||f1||s2+…||fk||s2
Structural stability and a characterization of Anosov families
Published in Dynamical Systems, 2019
Jeovanny de Jesus Muentes Acevedo
If E is a vector bundle over a compact Riemannian manifold M, set the Banach -vector space of continuous sections of E over M, endowed with the -topology. An automorphism is called hyperbolic if , where is the spectrum of L. For , let denote the fibre of E over p. For a diffeomorphism f on M, let be the bounded linear operator given by , where TM is the tangent bundle over M. Mather [10, 11] proved that f is an Anosov diffeomorphism if and only if is a hyperbolic automorphism. We can define the operator for non-stationary dynamical systems. In this case, that operator could be unbounded or non-hyperbolic (see Section 4). We will give some conditions on an Anosov family to obtain the hyperbolicity of defined for the family (see Theorem 4.5).
An open mapping theorem for nonlinear operator equations associated with elliptic complexes
Published in Applicable Analysis, 2023
We need to introduce appropriate functional spaces. Namely, let be a volume form on X and a denotes a Riemannian metric in the fibres of . As usual, we equip each bundle with a smooth bundle homomorphism defined by for all . Then we can consider the space with the unitary structure and the Lebesgue space with the norm . In this case the formal adjoint to operator is defined in the following way for the sections and : Let be a finite open cover of M by coordinate neighbourhoods over which is trivial and a corresponding partition of unity, As usual, denote by , , , the Sobolev space under the smooth vector bundle (see, for instance, [10]). It is a Banach space of sections with the norm In particular, it is a Hilbert space for , we denote it by .
Exergetic port-Hamiltonian systems: modelling basics
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Markus Lohmayer, Paul Kotyczka, Sigrid Leyendecker
For tensorial quantities, we use (abstract) index notation with Einstein’s convention: Indices of contravariant slots are written as superscript and indices of covariant slots are written as subscript. Repeated indices (up-down pairs) imply contraction. With a smooth manifold, denotes the tangent bundle and the cotangent bundle over . We write for a general vector bundle with total space and base space . When the latter is clear from the context, we just write . For vector bundles and , is the vector bundle over where . A section of a bundle is a function satisfying where is the bundle projection. The set of all sections of is denoted by . Given a contravariant -tensor field , the sharp map is the (curried) function defined by . Dually, the flat map corresponding to a covariant -tensor field is a bundle map from the tangent to the cotangent bundle. Its name derives from the fact that in index notation, it lowers the up-index of a tangent vector into the down-index of the covector .