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Search: MSC category 58D19 ( Group actions and symmetry properties )

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1. CJM 2008 (vol 60 pp. 1149)

Petersen, Kathleen L.; Sinclair, Christopher D.
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of $\R^{N-1}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

Categories:11C08, 28A75, 15A52, 54H10, 58D19

2. CJM 2003 (vol 55 pp. 247)

Cushman, Richard; Śniatycki, Jędrzej
Differential Structure of Orbit Spaces: Erratum
This note signals an error in the above paper by giving a counter-example.

Categories:37J15, 58A40, 58D19, 70H33

3. CJM 2003 (vol 55 pp. 266)

Kogan, Irina A.
Two Algorithms for a Moving Frame Construction
The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains challenging, despite its later significant development and generalization. This paper presents two new variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.

Categories:53A55, 58D19, 68U10

4. CJM 2001 (vol 53 pp. 715)

Cushman, Richard; Śniatycki, Jędrzej
Differential Structure of Orbit Spaces
We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applied to analyze the differential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds. Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with singular reduced spaces. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed.

Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds
Categories:37J15, 58A40, 58D19, 70H33

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