We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in...

⎯In this paper, we construct a pair of Mond-Weir type second order symmetric dual problems, in which the objective function contains support function and is, therefore, nondifferentiable. For this pair of problems, we validate weak, strong and converse duality theorems under pseudobonvexity – pseudoboncavity assumption on the kernel function that appears in the problems. A second order self dua...

Koszul property was generalized to homogeneous algebras of degree N > 2 in [5], and related to N -complexes in [7]. We show that if the N -homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincaré duality between Hochschild homology and cohomology of A, as for N = 2. Mathem...