The chains studied in this paper generalize Chern–Moser chains for CR structures. They form a distinguished family of one dimensional submanifolds in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be equivalently encoded as Cartan geometries (of different types). The aim of this paper is to study the relation between
these two Cartan geometries for Lagrangean contact structures and partially integrable almost CR structures.
We develop a general method for extending Cartan geometries which generalizes the Cartan geometry interpretation of Fefferman’s
construction of a conformal structure associated to a CR structure. For the two structures in question, we show that the Cartan geometry associated to the family of chains can be obtained in that way if and only if the original parabolic contact structure is torsion free. In particular, the procedure works exactly on the subclass of (integrable) CR structures.
This tight relation between the two Cartan geometries leads to an explicit description of the Cartan curvature associated to the
family of chains. On the one hand, this shows that the homogeneous models for the two parabolic contact structures give rise to examples of non–flat path geometries with large automorphism groups. On the other hand, we show that one may (almost) reconstruct
the underlying torsion free parabolic contact structure from the Cartan curvature associated to the chains. In particular, this leads to a very conceptual proof of the fact that chain preserving contact diffeomorphisms are either isomorphisms or antiisomorphisms of parabolic contact structures.
We consider the zeros on the boundary @ of a Neumann eigen-
function '¸j of a real analytic plane domain . We prove that the
number of its boundary zeros is O(¸j) where ¡¢'¸j = ¸2j
'¸j . We
also prove that the number of boundary critical points of either a
Neumann or Dirichlet eigenfunction is O(¸j ). It follows that the
number of nodal lines of '¸j (components of the nodal set) which
touch the boundary is of order ¸j . This upper bound is of the
same order of magnitude as the length of the total nodal line, but
is the square root of the Courant bound on the number of nodal
components in the interior. More generally, the results are proved
for piecewise analytic domains.
The limiting behavior of the normalized KÄahler-Ricci °ow for manifolds with positive ¯rst Chern class is examined under certain
stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges
uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue
of the ¹@y ¹@ operator on smooth vector ¯elds is bounded away from0 along the °ow, then the metrics converge exponentially fast in
C1 to a KÄahler-Einstein metric.
Gal O L, Sanz F, Speissegger P, et al. Non-interlaced solutions of 2-dimensional systems of linear ordinary differential equations[J]. Proceedings of the American Mathematical Society, 2013, 141(7): 2429-2438.
Dmitry Batenkov · Niv Sarig · Yosef Yomdin. Decoupling of Fourier Reconstruction System for Shifts of Several Signals. 2013.
We prove that the rotation in time T of a trajectory of a K-Lipschitz vector ¯eld in Rn around a given point (stationary or non-stationary) is bounded by A + BKT with A;B absolute constants. In particular, trajectories of a Lipschitz vector ¯eld in ¯nite time cannot have an in¯nite rotation around a given point (while trajectories of a C1 vector ¯eld may have an in¯- nite rotation around a straight line in ¯nite time). The bound above extends to the mutual rotation of two trajectories (for the time intervals T and T0, respectively) of a K-Lipschitz vector ¯eld in R3: this rotation is bounded from above by the quantity CK min(T; T0) + DK2TT0.
We prove the factoriality of a nodal hypersurface in P4 of degree d that has at most 2(d ¡ 1)2=3 singular points, and we prove the factoriality of a double cover of P3 branched over a nodal surface of degree 2r having less than (2r ¡ 1)r singular points.