Let @math be a contact form on a manifold @math , and @math a closed Legendrian submanifold. I prove that @math intersects some characteristic for @math at least twice if all characteristics are closed and of the same period, and @math embeds nicely into the product of @math and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most @math . This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.
Let @math be a bounded star-shaped domain with smooth boundary and @math a closed Legendrian submanifold of nonpositive curvature. The last condition means that @math that admits a Riemannian metric of nonpositive sectional curvature. In the recent preprint @cite_20 K. Cieliebak and K. Mohnke proved that @math possesses a Reeb chord of length bounded above by the (toroidal) Lagrangian capacity of @math , see [Corollary 1.12] CM . Using [Corollary 1.3] CM , they deduced that @math admits a Reeb chord of length bounded above by @math , if @math , and @math , i.e., @math is the unit sphere.
@cite_20
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A (tandem) duplication of length @math is an insertion of an exact copy of a substring of length @math next to its original position. This and related types of impairments are of relevance in modeling communication in the presence of synchronization errors, as well as in several information storage applications. We demonstrate that Levenshteins construction of binary codes correcting insertions of zeros is, with minor modifications, applicable also to channels with arbitrary alphabets and with duplication errors of arbitrary (but fixed) length @math . Furthermore, we derive bounds on the cardinality of optimal @math -ary codes correcting up to @math duplications of length @math and establish the following corollaries in the asymptotic regime of growing block length: 1) the presented family of codes is optimal for every @math , in the sense of the asymptotic scaling of code redundancy; 2) the upper bound, when specialized to @math , @math , improves upon Levenshteins bound for every @math ; and 3) the bounds coincide for @math , thus yielding the exact asymptotic behavior of the size of optimal single-duplication-correcting codes.