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Invariants of the Rotation Group

In: The Center and Cyclicity Problems

Author

Listed:
  • Valery Romanovski

    (University of Maribor, Center for Applied Mathematics & Theorectical Physics)

  • Douglas Shafer

    (University of North Carolina, Mathematics Dept.)

Abstract

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.

Suggested Citation

  • Valery Romanovski & Douglas Shafer, 2009. "Invariants of the Rotation Group," Springer Books, in: The Center and Cyclicity Problems, chapter 0, pages 1-35, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4727-8_5
    DOI: 10.1007/978-0-8176-4727-8_5
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