Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint]. Beijing: CAS. Computing Center, 1991 Home/Search Document Not in Database Summary Related Articles Check |
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\Gamma \Psi\Xi \Delta \Phi \Upsilon\Omega \Piff \Sigma \Theta - Preliminaries We Study (Correct)
....problem is to construct and analyse the algorithmic approximations f s a e t a fi fi t=s = e s a in a proper way. For this purpose we propose a unified framework based on the apparatus of formal power series, Lie algebra of vector fields and the corresponding Lie group of diffeomorphisms [1,2]. x2 Near 0 and Near 1 Formal Power Series Among the formal power series P 1 0 s k a k , a k : R N R N , we pick out two special classes. The first class consists of those with a 0 = 0, called near 0 formal vector fields; the second class consists of those with a 0 = 1N , called near 1 ....
Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint]. Beijing: CAS. Computing Center, 1991
\Delta \Omega \Phi\Xi \Gamma\Upsilon \Phi\Xi\Pi \Theta \Psi.. - Ma Phi Xi (Correct)
....s of I s , E s . Using ordinary versions, the methods C s = E s=2 ffi I s=2 , E s (0; b; c) ffi I s (a; 0; 0) and E s (a; 0; 0) ffi I s (0; b; c) which are slightly different from e C s , e P s , e Q s , are noncontact For futher development of contact algorithms, see [11, 13, 19]. We give in Fig 5, 6 comparisons of 3 D contact vs non contact computations. The contact systems are those describing the geodesic lines on ellipsoidal and torus surface respectively. A geodesic line will be closed or dense on the surface according as a frequency ratio is rational or irrational. ....
Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint]. Beijing: CAS. Computing Center, 1991
\Gamma fi \Upsilon \Sigma \Xi\Phifl \Theta \Piff\Omega \Delta.. - If Fi Psi (Correct)
....system, so this formal theory has important implications for the construction, analysis, assessment and understanding of numerical algorithms. x1 Preliminaries We study vector fields and their flows in R N together with their approximations, specifically from the formal power series approach [1]. We use coordinate description and matrix notation. The point vector x 2 R N and vector function a : R N Gamma R N are denoted by column matrix x = x 1 ; Delta Delta Delta ; xN ) 0 , a(x) a 1 (x) Delta Delta Delta ; aN (x) 0 . Prime 0 denotes matrix transpose. Jacobian ....
....problem seems to be difficult and is related to the KAM type theories. x3 Dual Properties between Formal Fields and Flows The near 1 formal map and the formal vector field completely characterize each other. We give a series of dual properties between them, f s a s ; e s b s , [1]. 1. f s ffi e t = e t ffi f s , 8s; t ( a s ; b t ] 0; 8s; t; a i ; b j ] 0; 8 i; j. 2. f s is a one parameter group ( a s is independent of s, i.e. a s = a 0 . Note that the one parameter group property is a strong structural property, all coefficients f k are ....
Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint]. Beijing: CAS. Computing Center, 1991
\Gamma \Omega \Sigma fl\Psi \Upsilon ff\Phi\Xifi \Pi.. - Problems And Motivations (Correct)
....laws. The crucial fact is that any symplectic algorithm possesses its own formal integrals of motion depending on the step size parameter and approximating the original integrals of motion (with the same order of accuracy as that of the method itself) and being invariant under the algorithm [4, 5]. It can also be proved that the majority of invariant tori of integrable systems are preserved under symplectic algorithms; resulting in a new version of the celebrated K.A.M. theorems and the theoretical infinite time tracking capability of the symplectic algorithms [6] although practically ....
Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint]. Beijing: CAS. Computing Center, 1991
Symplectic Algorithms For Hamiltonian Systems - Feng, Wang (Correct)
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Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint]. Beijing: CAS. Computing Center, 1991
\Gamma \Delta \Phiff\Omega fi\Psi\Xi \Theta \Upsilon \Sigma \Pifl - Ma Pi Fl (Correct)
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Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint]. Beijing: CAS. Computing Center, 1991
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