Abstract
Consider the algebra D (P, G) of the functions D = {çφ1(x), φ2(x),...} in the domain \(\tilde G = I \times G,G \in {R^n}\) , of existence and uniqueness of autonomous system⋆
solution. Define the algebra of functions D as a set of functions over a field P which contains every function together with its product on any number α ∈ P and contains every two functions together with their sum and product. This algebra is commutative, associative, and hass a unit. The functions D can be real or complexvalued, analytical, smooth (i.e. infinitely differentiable) or simply belong to the class C k, k ≺ +∞. The open set G, together with the differentiable structure D(G), is called a differentiable manifold. In the case when the algebra D(G) is produced by all possible analytical (real or complex) functions, D(G) is called an analytical (real or complex) manifold. To emphasize the field over which the manifold is considered, let us write D (K, G) for the complex case and D (R, G) for the real case. These notations can be represented in the unique form D(P, G). Further, if the opposite statement is not mentioned, the analytical manifold will be considered.
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© 1995 Springer Science+Business Media Dordrecht
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Mitropolsky, Y.A., Lopatin, A.K. (1995). Vector Fields, Algebras and Groups Generated by a System of Ordinary Differential Equations and their Properties. In: Nonlinear Mechanics, Groups and Symmetry. Mathematics and Its Applications, vol 319. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8535-4_2
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DOI: https://doi.org/10.1007/978-94-015-8535-4_2
Publisher Name: Springer, Dordrecht
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