# Algebraic Groups and Discontinuous Subgroups by A. Borel, G. Mostow

By A. Borel, G. Mostow

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Example text

First of all, here we mean the conservation problem for symplecticity property when one replaces an exactly symplectic map (produced by any Hamiltonian system) with approximating map. At this step, the role of symbolic computation is very important. Indeed, the approaches suggested in [5,6] can be realized up to necessary order of approximation (for approximating invariants and symmetries) and the algorithm for explicit solutions in a closed form as described in [7]. N. 2 Symbolic Computation Algorithm for the Matrix Formalism The above approach is based on the so-called matrix formalism for Lie algebraic tools [6,8] up to necessary order of approximation.

T. S. The ﬁber cardinality is often not immediately available, as we may be unable to take inequations into account. However, we can use all information contained in the equations using reduction, if all equations are contained in ST . Thus we require (SQ )=

In addition, these relations reduce the computational costs, indeed, in the case of dim X = 4, for 40 elements of Q12 we obtain 24 restrictions, and for 80 elements for Q13 we obtain 60 restrictions of type (17). N. Andrianov Kinematic and Dynamic Nonlinear Invariants Any numerical computational process leads to distortion of qualitative properties (for example, some dynamic and kinematical invariants [4]). These quantities can be evaluated using, for example, Casimirs operators. According to this theory for Lie groups generated by dynamical systems we can construct invariants using special forms and use these data for computational process controlling [8].