Narrow Results

In this paper cone convex and related functions have been studied. The concept of cone semistrictly convex functions on topological vector spaces is introduced as a generalization of semistrictly convex functions. Certain properties of these functions have been established and their interrelations with cone convex and cone subconvex functions have been explored. Assuming the functions to be cone subconvex, sufficient optimality conditions are proved for a vector valued minimization problem over topological vector spaces, involving Gâteaux derivatives. A Mond-Weir type dual is associated and weak and strong duality results are proved.

We investigate an optimal control problem for a hierarchical size-structured population model, which incorporates the inoculation delay into the globally feedbacked boundary condition. After the establishment of the continuity of the states with respect to the harvesting efforts, the existence of the optimal solutions is proved by a result in convex analysis, and the maximum principle is obtained via conjugate system and tangent-normal cones.

This is a leisurely introductory account addressed to non-experts and based on previous work by the authors, on how methods borrowed from physics can be used to “count” an infinite number of points. We begin with the classical case of counting integer points on the non-negative real axis and the classical Euler-Maclaurin formula. As an intermediate stage, we count integer points on product cones where the roles played by the coalgebra and the algebraic Birkhoff factorization can be appreciated in a relatively simple setting. We then consider the general case of (lattice) cones for which we introduce a connected cograded coalgebra of cones, with applications to renormalization of conical zeta values. When evaluated at zero arguments conical zeta functions indeed “count” integer points on cones.