72 research outputs found
An n-order (F,a,p,d)- Convex Function and Duality Problem
A class of n-order (F,a,p, d)-convex function and their generalization on functions is introduced. Using the assumption on the functions involved,weak, strong ,and converse duality theorems are established for the n-order dual proble
Weakly convex sets and modulus of nonconvexity
We consider a definition of a weakly convex set which is a generalization of
the notion of a weakly convex set in the sense of Vial and a proximally smooth
set in the sense of Clarke, from the case of the Hilbert space to a class of
Banach spaces with the modulus of convexity of the second order. Using the new
definition of the weakly convex set with the given modulus of nonconvexity we
prove a new retraction theorem and we obtain new results about continuity of
the intersection of two continuous set-valued mappings (one of which has
nonconvex images) and new affirmative solutions of the splitting problem for
selections. We also investigate relationship between the new definition and the
definition of a proximally smooth set and a smooth set
Explicit bounds for rational points near planar curves and metric Diophantine approximation
The primary goal of this paper is to complete the theory of metric
Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007),
p.367-426] for non-degenerate planar curves. With this goal in mind, here
for the first time we obtain fully explicit bounds for the number of rational
points near planar curves. Further, introducing a perturbational approach we
bring the smoothness condition imposed on the curves down to (lowest
possible). This way we broaden the notion of non-degeneracy in a natural
direction and introduce a new topologically complete class of planar curves to
the theory of Diophantine approximation. In summary, our findings improve and
complete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and
extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of
Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of
non-degeneracy.Comment: 24 page
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