3,909 research outputs found
Transitive decomposition of symmetry groups for the -body problem
Periodic and quasi-periodic orbits of the -body problem are critical
points of the action functional constrained to the Sobolev space of symmetric
loops. Variational methods yield collisionless orbits provided the group of
symmetries fulfills certain conditions (such as the \emph{rotating circle
property}). Here we generalize such conditions to more general group types and
show how to constructively classify all groups satisfying such hypothesis, by a
decomposition into irreducible transitive components. As examples we show
approximate trajectories of some of the resulting symmetric minimizers
On the weak order of Coxeter groups
This paper provides some evidence for conjectural relations between
extensions of (right) weak order on Coxeter groups, closure operators on root
systems, and Bruhat order. The conjecture focused upon here refines an earlier
question as to whether the set of initial sections of reflection orders,
ordered by inclusion, forms a complete lattice. Meet and join in weak order are
described in terms of a suitable closure operator. Galois connections are
defined from the power set of W to itself, under which maximal subgroups of
certain groupoids correspond to certain complete meet subsemilattices of weak
order. An analogue of weak order for standard parabolic subsets of any rank of
the root system is defined, reducing to the usual weak order in rank zero, and
having some analogous properties in rank one (and conjecturally in general).Comment: 37 pages, submitte
Families of L-functions and their Symmetry
In [90] the first-named author gave a working definition of a family of
automorphic L-functions. Since then there have been a number of works [33],
[107], [67] [47], [66] and especially [98] by the second and third-named
authors which make it possible to give a conjectural answer for the symmetry
type of a family and in particular the universality class predicted in [64] for
the distribution of the zeros near s=1/2. In this note we carry this out after
introducing some basic invariants associated to a family
Searching for integrable Hamiltonian systems with Platonic symmetries
In this paper we try to find examples of integrable natural Hamiltonian
systems on the sphere with the symmetries of each Platonic polyhedra.
Although some of these systems are known, their expression is extremely
complicated; we try here to find the simplest possible expressions for this
kind of dynamical systems. Even in the simplest cases it is not easy to prove
their integrability by direct computation of the first integrals, therefore, we
make use of numerical methods to provide evidences of integrability; namely, by
analyzing their Poincar\'e sections (surface sections). In this way we find
three systems with platonic symmetries, one for each class of equivalent
Platonic polyhedra: tetrahedral, exahedral-octahedral,
dodecahedral-icosahedral, showing evidences of integrability. The proof of
integrability and the construction of the first integrals are left for further
works. As an outline of the possible developments if the integrability of these
systems will be proved, we show how to build from them new integrable systems
in dimension three and, from these, superintegrable systems in dimension four
corresponding to superintegrable interactions among four points on a line, in
analogy with the systems with dihedral symmetry treated in a previous article.
A common feature of these possibly integrable systems is, besides to the rich
symmetry group on the configuration manifold, the partition of the latter into
dynamically separated regions showing a simple structure of the potential in
their interior. This observation allows to conjecture integrability for a class
of Hamiltonian systems in the Euclidean spaces.Comment: 22 pages; 4 figure
Deformation of geometry and bifurcation of vortex rings
We construct a smooth family of Hamiltonian systems, together with a family
of group symmetries and momentum maps, for the dynamics of point vortices on
surfaces parametrized by the curvature of the surface. Equivariant bifurcations
in this family are characterized, whence the stability of the Thomson heptagon
is deduced without recourse to the Birkhoff normal form, which has hitherto
been a necessary tool.Comment: 26 page
Oka's conjecture on irreducible plane sextics
We partially prove and partially disprove Oka's conjecture on the fundamental
group/Alexander polynomial of an irreducible plane sextic. Among other results,
we enumerate all irreducible sextics with simple singularities admitting
dihedral coverings and find examples of Alexander equivalent Zariski pairs of
irreducible sextics.Comment: Final version accepted for publicatio
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