72 research outputs found

    An n-order (F,a,p,d)- Convex Function and Duality Problem

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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 C3C^3 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 C1C^1 (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
    • …
    corecore