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1. CJM 2008 (vol 60 pp. 1149)
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)
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)
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)
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 |