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Invariants of the Rotation Group

  • Valery Romanovski
  • Douglas Shafer
Chapter

In Section 3.5 we stated the conjecture that the center variety of family (3.3), or equivalently of family (3.69), always contains the variety V(Isym) as a component. This variety V(Isym) always contains the set R that corresponds to the time-reversible systems within family (3.3) or (3.69), which, when they arise through the complexiӿcation of a real family (3.2), generalize systems that have a line of symmetry passing through the origin. In Section 3.5 we had left incomplete a full characterization of R. To derive it we are led to a development of some aspects of the theory of invariants of complex systems of differential equations. Using this theory, we will complete the characterization of R and show that V(Isym) is actually its Zariski closure, the smallest variety that contains it. In the ӿnal section we will also apply the theory of invariants to derive a sharp bound on the number of axes of symmetry of a real planar system of differential equations.

Keywords

Characteristic Vector Phase Portrait Rotation Group Polynomial System Conjugate Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Center for Applied Mathematics & Theorectical PhysicsUniversity of MariborSlovenia
  2. 2.Mathematics Dept.University of North CarolinaCharlotteUSA

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