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The permutation action of finite symplectic groups of odd characteristic on their standard modules
Motivated by the incidence problems between points and flats of a symplectic
polar space, we study a large class of submodules of the space of functions on
the standard module of a finite symplectic group of odd characteristic. Our
structure results on this class of submodules allow us to determine the
-ranks of the incidence matrices between points and flats of the symplectic
polar space. In particular, we give an explicit formula for the -rank of the
generalized quadrangle , where is an odd prime power.
Combined with the earlier results of Sastry and Sin on the 2-rank of , it completes the determination of the -ranks of .Comment: 22 page
Evolutionary outcomes for pairs of planets undergoing orbital migration and circularization: second order resonances and observed period ratios in Kepler's planetary systems
In order to study the origin of the architectures of low mass planetary
systems, we perform numerical surveys of the evolution of pairs of coplanar
planets in the mass range (1-4)\ \rmn{M}_{\oplus}. These evolve for up to
2\times10^7 \rmn{yr} under a range of orbital migration torques and
circularization rates assumed to arise through interaction with a
protoplanetary disc. Near the inner disc boundary, significant variations of
viscosity, interaction with density waves or with the stellar magnetic field
could occur and halt migration, but allow ircularization to continue. This was
modelled by modifying the migration and circularization rates. Runs terminated
without an extended period of circularization in the absence of migration
torques gave rise to either a collision, or a system close to a resonance.
These were mostly first order with a few terminating in second order
resonances. Both planetary eccentricities were small and all resonant
angles liberated. This type of survey produced only a limited range of period
ratios and cannot reproduce Kepler observations. When circularization alone
operates in the final stages, divergent migration occurs causing period ratios
to increase. Depending on its strength the whole period ratio range between
and can be obtained. A few systems close to second order commensurabilities
also occur. In contrast to when arising through convergent migration, resonant
trapping does not occur and resonant angles circulate. Thus the behaviour of
the resonant angles may indicate the form of migration that led to near
resonance.Comment: 15 pages, 12 figures, 2014, MNRAS, 449, 304
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