Read e-book online Applications of Lie Groups to Differential Equations PDF

By Peter J. Olver

ISBN-10: 0387940073

ISBN-13: 9780387940076

ISBN-10: 3540940073

ISBN-13: 9783540940074

Symmetry tools have lengthy been well-known to be of serious significance for the examine of the differential equations. This booklet presents a pretty good creation to these purposes of Lie teams to differential equations that have proved to be beneficial in perform. The computational tools are provided in order that graduate scholars and researchers can effortlessly discover ways to use them. Following an exposition of the purposes, the ebook develops the underlying idea. the various subject matters are provided in a unique means, with an emphasis on specific examples and computations. additional examples, in addition to new theoretical advancements, look within the routines on the finish of every bankruptcy.

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7 (Real analytic Carathéodory section). Let κ : E → M be a real analytic vector bundle and let T ⊆ R be an interval. A Carathéodory section of class Cω of E is a map Σ : T × M → E with the following properties: (i) Σ(t, x) ∈ Ex for each (t, x) ∈ T × M; 36 3 Time-Varying Vector Fields and Control Systems (ii) for each t ∈ T, the map Σt : M → E defined by Σt (x) is of class Cπ ; (iii) for each x ∈ M, the map Σ x : T → E defined by Σ x (t) = Σ(t, x) is Lebesgue measurable. We shall call T the time-domain for the section.

In the following list, we let F → {R, C} and let U and V be F-locally convex spaces defined by families Q and P, respectively, of seminorms. 1. 37]. Locally convex spaces are often assumed to be Hausdorff, and we shall suppose this to be true for our statements below. 2. 37]. 3. We say that a subset B is von Neumann bounded if, for any neighbourhood N of 0, there exists λ → R>0 such that B ⊆ λN. 37(b)]. 4. A locally convex topology is normable if it can be defined by a single seminorm which is a norm.

Institute of Electrical and Electronics Engineers. Trans Autom Control 27(4):904–914 50. Nijmeijer H, van der Schaft AJ (1990) Nonlinear dynamical control systems. Springer, Berlin 51. Pasillas-Lépine W, Respondek W (2002) Contact systems and corank one involutive subdistributions. Acta Appl Math 69(2):105–128 52. Pietsch A (1969) Nuclear locally convex spaces. No. 66 in Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin 53. Polderman JW, Willems JC (1998) Introduction to mathematical systems theory.

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Applications of Lie Groups to Differential Equations by Peter J. Olver

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